children's mathematical understandings of tessellations
TRANSCRIPT
The Dissertation Committee for Robert Scott Eberle Certifies that this is the
approved version of the following dissertation:
Children’s Mathematical Understandings of Tessellations:
A Cognitive and Aesthetic Synthesis
Committee:
Guadalupe Carmona-Dominguez, Supervisor
Leema Berland
Susan Empson
Nathalie Sinclair
Michael Starbird
Walter Stroup
Children’s Mathematical Understandings of Tessellations:
A Cognitive and Aesthetic Synthesis
by
Robert Scott Eberle, B.A.; M.A.; M.Ed.
Dissertation
Presented to the Faculty of the Graduate School of
The University of Texas at Austin
in Partial Fulfillment
of the Requirements
for the Degree of
Doctor of Philosophy
The University of Texas at Austin
December, 2011
Dedication
To Lucia, for her enduring support and her inspiring love of teaching
And to Ana, Christina, and Sara, for their love of geometry
v
Acknowledgements
First of all, I want to thank my family, and especially Lucia, for their support,
encouragement, and patience during the pursuit of this doctorate. I thank my father for
introducing me to the beauty of mathematics via the writings of Martin Gardner.
Secondly I want to thank my supervisor for her incredible support and detailed feedback
during all phases of my dissertation. Her input was beyond all expectations.
My thanks also go to everyone else who contributed to the success of this
research: the members of my committee for their great feedback—many of the ideas in
this dissertation were inspired by my discussions with them; my colleagues here at UT
Austin who gave me encouragement and help; and friends all over the world who sent my
wife and me a steady stream of encouraging messages to persevere to the end. My special
thanks go to all those in SIM and many churches who encouraged me to follow this road.
Without their support, this doctorate would not have been possible.
Most of all, my thanks goes to God for the strength, ability, and desire he gave me
to complete this work. To him be all the glory.
vi
Children’s Mathematical Understandings of Tessellations:
A Cognitive and Aesthetic Synthesis
Robert Scott Eberle, Ph.D.
The University of Texas at Austin, 2011
Supervisor: Guadalupe Carmona-Dominguez
Tessellations have a rich mathematical structure and are especially appropriate as
a context for teaching geometry in the middle grades. Few studies have researched how
children conceptualize and learn tessellations in spite of their international use in
educational contexts. This exploratory study looks at how fourth grade students
conceptualize tessellations before instruction. The analysis is done from a Piagetian,
cognitive viewpoint and from an aesthetic viewpoint. It is argued that the aesthetic
viewpoint is crucial and foundational to children‘s mathematical understanding, just as it
is for mathematicians. A series of clinical interviews was conducted with six fourth grade
children. The results identified common themes of children‘s understanding, strategies,
reasoning, and aesthetic criteria for tessellations. Children‘s ontology varied between
object and process conceptions of tessellations. Children struggled especially with the
infinite space of mathematical tessellations. Children‘s aesthetics, including symmetry,
influenced their choices in creating tessellations and are shown to have played a cognitive
role in children‘s mathematical exploration of tessellation structures. Mathematics
influences students‘ aesthetic appreciation of tessellations and, more importantly,
aesthetics drives the study of the mathematical structure of tessellations. Children‘s
aesthetic criteria were the same as mathematicians‘, but with much different emphases.
vii
Other results are discussed, including the mathematical content elicited by the tasks, the
influence of the tools used to create tessellations, the children‘s epistemology of their
tessellations, and the role symmetry played in giving children confidence.
Recommendations for future research and possible implications for curriculum and
instruction are noted.
viii
Table of Contents
List of Tables ........................................................................................................ xii
List of Figures ...................................................................................................... xiii
Chapter 1: Introduction ............................................................................................1
Problem ...........................................................................................................1
Approach .........................................................................................................2
Research Question ..........................................................................................2
Chapter 2: Literature Review ...................................................................................4
Theoretical Framework ...................................................................................4
Cognitive Perspective ............................................................................4
Piaget.............................................................................................5
Vygotsky .....................................................................................10
Process and Object Conceptions .................................................14
Aesthetic Perspective ...........................................................................19
Nature of Mathematics ................................................................20
Mathematical Aesthetics .............................................................27
Mathematical Aesthetics for Education ......................................46
Connection to the Cognitive Perspective ....................................55
Summary of the Aesthetic Perspective .......................................57
Research Related to This Study ....................................................................59
Mathematical Theory of Tessellations .................................................59
Definitions...................................................................................60
Symmetry and Transformations ..................................................62
Polygonal Tilings ........................................................................67
Regular Polygons ........................................................................69
Periodicity and Symmetry...........................................................72
Lack of Symmetry.......................................................................75
Non-Regular Polygons ................................................................80
ix
Polyforms ....................................................................................82
Color ...........................................................................................84
Vocabulary Summary .................................................................85
Children‘s Understandings of Tessellations ........................................87
Geometric Properties in Tessellations
(van Hiele-Geldof and Fuys et al.) .....................................87
Rectangular Arrays of Squares (Outhred & Mitchelmore) .........89
Combining Tessellations With Art
(Upitis, Phillips, & Higginson) ..........................................91
Unitizing (Wheatley & Reynolds) ..............................................93
Covering Shapes With Tiles (Owens & Outhred) ......................93
The Logic of Mixing Tiles (Vitale & Zinder) .............................94
Other Geometry-Related Research ......................................................95
Children‘s Understanding of Infinite Space ...............................95
Children‘s Understanding of Symmetry .....................................97
Summary .....................................................................................................101
Chapter 3: Methodology ......................................................................................104
Clinical Interviews ......................................................................................104
Theory of Clinical Interviews ............................................................105
Methodology of Clinical Interviews ..................................................105
Validity and Reliability ......................................................................108
Method ........................................................................................................114
Pilot Study ..........................................................................................114
Population ..........................................................................................118
Pre-test ...............................................................................................121
Interviews ...........................................................................................122
Criteria ......................................................................................122
Overview of the Interviews .......................................................124
Structure of the Interviews ........................................................126
Computer software ....................................................................132
Analysis..............................................................................................133
x
Aesthetic Analysis ....................................................................136
Symmetry Analysis ...................................................................137
Chapter 4: Results ................................................................................................139
Cognitive Understanding ............................................................................139
Piagetian stage ...................................................................................154
Tiling Criteria.....................................................................................156
How Shapes Fit (Gaps and Overlaps) .......................................157
Infinite Space ............................................................................159
Ontology of Tilings (Process versus Object) .....................................164
Epistemology of Tilings .....................................................................168
Children‘s Strategies ..........................................................................173
Children‘s Reasoning .........................................................................177
Other General Results ........................................................................180
Aesthetic Influences ....................................................................................181
Children‘s Evaluation Criteria ...........................................................187
Mathematicians‘ Evaluation Criteria .................................................197
Children‘s Creation Criteria ...............................................................199
Interaction of the Two Perspectives ............................................................205
Tasks and Tools .................................................................................206
Role of Symmetry ..............................................................................210
The Cognitive Role of Other Aesthetic Criteria ................................214
Differences Between Schools or Across Time ...........................................217
Differences Between Schools ............................................................217
Differences Across Time ...................................................................217
Chapter 5: Conclusions ........................................................................................222
Discussion of the Results ............................................................................222
Children‘s Cognitive Understanding of Tessellations .......................222
Children‘s Aesthetic Understanding of Tessellations ........................228
Interaction of the Two Perspectives ............................................................234
Implications for Future Research ................................................................236
xi
Implications for Curriculum and Instruction ..............................................238
What Students Learn With Tessellations ...........................................239
Mathematical Concepts Elicited by the Tiling Tasks ...............239
The Teaching of Tessellations ...........................................................244
Importance of the Study of Tessellations ...........................................246
Appendix A: Pre-test............................................................................................248
Appendix B: Protocols .........................................................................................251
Appendix C: Software..........................................................................................267
Appendix D: Analysis Codes ...............................................................................270
Appendix E: Results by Task ...............................................................................285
Appendix F: Analysis of Change .........................................................................303
Appendix G: Analyzing Greater Symmetry ........................................................307
References ............................................................................................................309
Vita .....................................................................................................................318
xii
List of Tables
Table 1: Important tessellation terms ....................................................................86
Table 2: Summary of Clement‘s viability criteria ...............................................112
Table 3: Student participants in this study ...........................................................120
Table 4: The task sets ..........................................................................................126
Table 5: Interview structure .................................................................................132
Table 6: Aesthetic themes ....................................................................................188
Table 7: Some unanimous preferences at both schools during Comparisons
in Pairs ............................................................................................196
Table 8: Escher drawings used in the sixth interview .........................................265
Table 9: Codes used in the analysis of the data ...................................................270
Table 10: Groups formed by student sorting .......................................................301
Table 11: Analysis of code changes across first five interviews .........................303
xiii
List of Figures
Figure 1: Bhaskara II‘s visual proof of the Pythagorean Theorem.
Color and labels have been added to show a2, b
2, and c
2..................22
Figure 2: Analysis of a near-Penrose pattern from the Alhambra
(Lu & Steinhardt, 2007) ....................................................................23
Figure 3: A small portion of the Mandelbrot set near ..................40
Figure 4: Some symmetries of a tessellation .........................................................63
Figure 5: Rhombus tiling showing image of rhombus after dilating by factors
of 3 and 5 ..........................................................................................65
Figure 6: An example of a tiling with dilation symmetry .....................................65
Figure 7: A monohedral tiling that is neither isohedral nor edge-to-edge ............67
Figure 8: The ―brick wall‖ tiling of rectangles is not edge-to-edge. .....................68
Figure 9: Six edges of a brick ................................................................................68
Figure 10: Another non-edge-to-edge tiling..........................................................68
Figure 11: Portions of the three regular tessellations ............................................69
Figure 12: Two tessellations with squares and triangles .......................................70
Figure 13: The 3.3.4.3.4 semi-regular tessellation ................................................71
Figure 14: The six other semi-regular tessellations ..............................................71
Figure 15: Two symmetric semi-regular tessellations ..........................................74
Figure 16: A random tessellation of polygons all different from each other ........76
Figure 17: A non-periodic tiling of squares and triangles occurring naturally in
crystal growth (Krumeich, Conrad, Nissen, & Harbrecht, 1998) .....77
Figure 18: The 36/3
2.4.3.4 tiling with three dodecagons highlighted ....................78
xiv
Figure 19: Non-periodic, asymmetric, tiling created by rotating three
dodecagons 30° .................................................................................78
Figure 20: A 4-fold symmetric tiling without translation symmetry ....................79
Figure 21: Isosceles triangle tiling with 8-fold symmetry transformed into a
spiral tiling with rotation symmetry but no reflection symmetry .....80
Figure 22: Tiling of a non-convex quadrilateral ...................................................81
Figure 23: Four of the several possible isohedral tessellations with a scalene
triangle. The two on the right are not edge-to-edge. .........................81
Figure 24: Order-3 and order-4 polyominoes and their common letter names .....83
Figure 25: Polyiamonds up to order 5 ...................................................................84
Figure 26: A triabolo that can be made in two different ways ..............................84
Figure 27: A tiling whose symmetry depends on whether color is taken
into consideration ..............................................................................85
Figure 28: Rhombus tiling with ―saw‖ (red) and ―ladder‖ (orange) highlighted ..88
Figure 29: Regular triangle tiling with oppositely oriented triangles
colored differently .............................................................................94
Figure 30: The six finite tiling problems .............................................................130
Figure 31: Screenshot of ―Polygon Tiler‖ software ............................................133
Figure 32: Some of Rachel‘s tilings during the first interview ...........................140
Figure 33: A student‘s idea of how four equilateral triangles fit around a
common vertex................................................................................142
Figure 34: Kelsey‘s parallelogram tiling drawing...............................................143
Figure 35: The beginning of Michelle‘s regular pentagon tiling ........................144
Figure 36: Moses points out the inevitable gap left by regular pentagons. .........145
Figure 37: The 36/3
2.4.3.4 tiling made by Kobe ..................................................148
xv
Figure 38: A step in Kelsey‘s attempt to squeeze five triangles around a
square corner ...................................................................................148
Figure 39: Kelsey‘s responses to the first pre-test problem and the first
problem of Set B .............................................................................149
Figure 40: An attempt by Michelle to tile a right scalene triangle by using
rectangular units ..............................................................................150
Figure 41: Kelsey‘s L-tromino tiling ..................................................................151
Figure 42: Beginning of an L-tromino tiling by Michelle drawn on
blank paper ......................................................................................152
Figure 43: Kelsey‘s random triangle and hexagon tiling ....................................154
Figure 44: An I-tromino drawn by Michelle in a diagonal orientation ...............155
Figure 45: S-tetromino tiling by Kobe ................................................................156
Figure 46: Tilings by Rachel (left), Marie (middle), and Moses (right) where
equilateral triangles have been stretched to fit gaps large enough
for two triangles ..............................................................................158
Figure 47: Kelsey‘s first two triangle tessellations .............................................161
Figure 48: Rhombus tiling border by Moses .......................................................164
Figure 49: Failed attempt envisioned by Kobe for fitting
T-tetrominoes together ....................................................................172
Figure 50: Semi-random rhombus tiling by Kelsey ............................................173
Figure 51: Two stages of Marie‘s growing tower in the
software environment......................................................................175
Figure 52: Two regular polygon tilings by Michelle ..........................................178
Figure 53: Tiling of a parallelogram on Michelle‘s pre-test ...............................180
Figure 54: Two of the tilings from the Comparisons in Pairs task .....................183
xvi
Figure 55: Both schools thought the mathematician would prefer this tiling. ....183
Figure 56: The two least favorite Alhambra tilings at both schools ...................184
Figure 57: Escher‘s ―Regular Division of the Plane #58,‖ the second most
popular Escher tiling at the private school ......................................185
Figure 58: Rachel‘s favorite from her created triangle tilings ............................191
Figure 59: 4.82 tiling chosen by both schools for the kitchen .............................191
Figure 60: Tiling J in the Comparisons in Pairs task ..........................................192
Figure 61: L-tromino tiling perceived to have pleasing complexity ...................193
Figure 62: The favorite Alhambra tiling at both schools ....................................195
Figure 63: Escher‘s ―Regular Division of the Plane #66,‖ a favorite tiling at
both schools ....................................................................................195
Figure 64: Three I-tromino tilings of increasing complexity by Kobe ...............202
Figure 65: Kelsey‘s rhombus tessellation drawing .............................................202
Figure 66: Kelsey‘s tilings of the acute isosceles, right scalene, and obtuse
scalene triangles ..............................................................................203
Figure 67: Kelsey‘s attempt to draw a tiling with an obtuse scalene triangle .....207
Figure 68: Kelsey‘s tiling of an obtuse scalene triangle in the
software environment......................................................................207
Figure 69: 2-isogonal tiling creation by Rachel in the software environment ....208
Figure 70: How Michelle extracted a symmetric tiling from a
semi-random one .............................................................................212
Figure 71: Comparison of mathematicians‘ and children‘s aesthetic themes .....220
Figure 72: Two tilings with unusual symmetry in the aesthetic
evaluation tasks ...............................................................................231
Figure 73: Mutual influence of aesthetics and mathematics ...............................233
xvii
Figure 74: First page of the pre-test, shown half size .........................................248
Figure 75: Second page of the pre-test, shown half size .....................................249
Figure 76: Third page of the pre-test, shown half size ........................................250
Figure 77: Pre-test Item 1 ....................................................................................256
Figure 78: Set B, Problems 1 and 2 .....................................................................257
Figure 79: Set B, Problems 3 and 4 .....................................................................257
Figure 80: Set B, Problems 5 and 6 .....................................................................258
Figure 81: Students‘ tilings for the first aesthetic task ........................................262
Figure 82: Most of the tilings for three of the aesthetic tasks .............................263
Figure 83: The seven Alhambra tilings ...............................................................264
Figure 84: Mo and I discussing square and hexagon tilings with Polydrons ......288
Figure 85: Kelsey‘s drawing of a pentagon tiling with Polydrons ......................289
Figure 86: One of Michelle‘s attempts at making a tiling with squares
and triangles ....................................................................................290
Figure 87: Michelle‘s attempt to solve problem B1 ............................................293
Figure 88: Marie‘s attempt to solve Problem B6 ................................................295
Figure 89: Michelle‘s attempt to solve Problem B6 ...........................................295
Figure 90: Rachel‘s attempt to solve Problem B6...............................................296
Figure 91: Moses‘ ―mail with wings‖ .................................................................297
Figure 92: Tessellation with orbifold notation *632 ...........................................307
1
Chapter 1: Introduction
The study of tessellations is a rich topic that connects many geometric concepts.
Tessellations are appropriate for study in some form for all ages, from kindergarten
through college. Between third grade and eighth grade, tessellations connect with much
or even most of the geometry curriculum, as well as with other areas of mathematics.
Many states, provinces, and countries that rank high on international tests of mathematics
require students to study tessellations in these grades. A few teaching experiments have
demonstrated that tessellations provide a powerful learning environment in geometry for
this age group.
PROBLEM
Despite their importance, there has been very little study done on children‘s
understanding of tessellations. There have been almost no studies to understand
children‘s initial conceptualizations of tessellations. This study was designed to explore
fourth grade children‘s understandings before they received any classroom instruction on
tessellations.
Tessellations (or tilings) are rich in mathematical structure. The simplest
tessellation, a square grid, is a basic concept in mathematics, essential for elementary
studies of area and graphing. Studies (Battista, Clements, Arnoff, Battista, & Borrow,
1998; Outhred & Mitchelmore, 1992, 2000) have shown that children do not achieve a
thorough understanding of finite square arrays before third or fourth grade. Before age 8,
most children do not conceptualize square arrays as structured in rows and columns. No
one has yet thoroughly investigated how children conceptualize the structure of other
types of tessellations.
2
APPROACH
I study children‘s understandings of tessellations from two complementary
perspectives: the cognitive viewpoint frequently used for such studies; and the aesthetic
viewpoint, using a framework suggested by Nathalie Sinclair (2006). The cognitive
viewpoint seeks to understand how children perceive tessellations and how they
understand and find the mathematical patterns they are making. The aesthetic viewpoint
looks at the motivating factors behind children‘s creation and evaluation of tessellations.
Both the aesthetic and cognitive viewpoints seek to understand what children are
focusing on as they tile, how children choose the patterns they make, and how children
understand the mathematical structure of their tilings.
The aesthetic viewpoint is usually neglected in studies of this sort. However,
recent studies have begun to show the importance of mathematical aesthetics in
education. Mathematics is itself an aesthetic study and, contrary to popular opinion,
mathematicians use aesthetics in deep ways in their research. Children are capable of
using their aesthetic abilities to pursue mathematics just as mathematicians do, though of
course in different ways. Indeed, it is not even possible to do more than rote mathematics
without using aesthetics, whether children are aware of it or not. By investigating how
children understand tessellations aesthetically as well as cognitively, we can obtain a
more complete understanding of children‘s conceptualizations of tessellations and deduce
how to develop children‘s mathematical aesthetics in classroom settings.
RESEARCH QUESTION
My research question can be stated this way: As understood from both cognitive
and aesthetic viewpoints, how do fourth grade children from various schools
3
conceptualize geometric tessellations in the context of creating and evaluating
tessellations? In particular:
1. What are children‘s cognitive understandings of tessellations? What strategies and
types of reasoning do they use to create tessellations? What mathematical
concepts are elicited?
2. What aesthetic considerations guide students‘ thinking about tessellations? How
do they evaluate tessellations made by themselves and by others? What
mathematical value is there in their aesthetics of tessellations?
3. How do the cognitive and aesthetic viewpoints inform each other for a more
complete understanding of children‘s conceptualizations of tessellations?
4
Chapter 2: Literature Review
In this study I draw from a number of different areas. This literature review is
divided into two parts. The first part (Theoretical Framework) informs the methodology
of this study and the two viewpoints from which the data are analyzed. First I introduce
the background needed for understanding the cognitive viewpoint, and then I present the
theory of the aesthetic viewpoint in mathematics.
The second part of this literature review (Research Related to This Study) focuses
on studies directly related to my research question on tessellations. I begin with a brief
presentation of the mathematical theory of tessellations, followed by a summary of the
research related to children‘s understanding of tessellations.
THEORETICAL FRAMEWORK
This study seeks to understand how children conceptualize tessellations. The
approach is from two viewpoints which are complementary: a cognitive perspective and
an aesthetic perspective (page 19).
Cognitive Perspective
This research is supported by the psychological constructivist perspective where
the individual is understood to construct his or her own knowledge. This perspective is
rooted in the work of Jean Piaget, and I spend some time in this section discussing the
part of his work that informs my study, especially his explanation of how children‘s
understanding of geometry develops. From this foundation, I continue by discussing the
contributions of Lev Vygotsky as well as process and object theories based on Piaget‘s
work. While there are differences between these theoretical viewpoints, they are
compatible. Each focuses on different aspects of cognition from a constructivist
5
perspective. As some researchers (e.g. Bergsten, 2008; Cobb, 2007) have pointed out,
mathematics education researchers must often turn to several theories in order to cover all
the perspectives needed for addressing their research questions. Each theory brings its
own contribution by studying human cognition from a slightly different viewpoint: Piaget
focused on genetic epistemology; Vygotsky sought to understand cognition in its social
context; and process and object theorists explain students‘ ontological perceptions. All of
these viewpoints can be understood as part of a larger constructivist framework which
attempts to understand how students construct their understanding in their local context.
It is within this larger framework that I analyze how fourth grade children initially
understand tessellations before they receive any formal instruction in this subject.
Piaget
Jean Piaget was a Swiss biologist who spent most of his career in the area of
developmental psychology. Much of Piaget‘s research focused on genetic epistemology,
the study of the origins of knowledge. According to Piaget, knowledge is organized in
schemata, mental representations of the world that have been abstracted from repeated
actions. When a subject encounters a novel situation that seems to fit an existing schema,
that new knowledge is assimilated into the schema. If perception of the situation
contradicts existing schemata, or reveals previously unnoticed contradictions between
schemata, cognitive dissonance occurs and the subject must accommodate this new
knowledge by changing or abandoning certain schemata. Assimilation is easier than
accommodation, so if data can be interpreted to fit existing schemata, they will be, even if
unwarranted assumptions or changes have to be made in order for the information to fit.
We tend to hear what we expect, even if this distorts what was actually said (Duckworth,
1996). The important point to Piaget‘s theories is that we never start with a blank slate.
6
All knowledge is filtered through existing schemata. Knowledge is constructed by the
subject as schemata are created, organized, modified, and abandoned.
An important aspect of Piaget‘s theory is that all conceptual knowledge is rooted
in sensorimotor experiences in the first few years of life. Piaget referred to this as the
sensorimotor stage of development. A baby begins by making movements at random, and
then discovers that certain types of movements have consistent consequences. By
noticing these patterns, the child develops schemata that abstract understanding of the
world. From there, the child continues to develop increasingly sophisticated concepts of
space, quantity, logic, and physics as various events lead the child to assimilate new data
into existing schemata or to accommodate data into new schemata. These concepts are
developed jointly and are not always distinguished for young children. Consider, for
example, Piaget‘s well-known experiment showing that preoperational children (before
age 6 or 7) do not understand that quantity is conserved; they believe that when tokens
are spread out they become more numerous. This is explained by the fact that for young
children, space, quantity, and logic are not fully distinguished. The child believes that
collections that take more space are necessarily more numerous, which is a correct
conclusion in some other situations. It is only when children learn that operations (such
as ―spreading out‖) can be reversed that they can understand how number is conserved
and is therefore a property of the objects independent of space.
Note that a proper adaptation to the world requires a balance of assimilation and
accommodation. If children were only to assimilate, they would never create new
concepts. All sensorimotor data would be forced into pre-existing schemata and learning
would never take place. If children were only to accommodate, then a new schema would
be created for every new experience. Children would never generalize.
7
Students‘ schemata can be studied through clinical interviews, a technique
developed by Piaget, which I describe in Chapter 3. Piaget noted (1971) that many people
felt confident inventing various psychological theories by simply depending on
reflection. Piaget moved away from such speculation by depending on experimentation to
analyze how children think and perceive the world. His clinical interview method (or
méthode critique, ―critical method,‖ as he preferred calling it in the original French) was
used to put children in situations that revealed how they thought about the world.
Stages. Piaget is best known for his four developmental stages of children‘s
understanding: sensorimotor, preoperational, concrete operational, and formal
operational. For my purposes, I focus just on the stages of geometric understanding as
Piaget described them in Psychogenèse et histoire des sciences (Piaget & Garcia, 1983).
Though Piaget does not discuss tessellations or symmetry in this work, his analysis of
children‘s understanding of geometric ideas is applicable to those concepts.
According to Piaget, children pass through three developmental stages of
understanding geometry. The first stage is the intrafigural stage, which fits into Piaget‘s
preoperational stage of development. At the intrafigural stage a child considers geometric
objects holistically without appreciating the relationships between them. At this stage, the
child only focuses on what is inherent in a figure and not the relationship between two
geometric objects. For example, young children cannot copy a vertical chimney on a
sloping roof. Instead of drawing the chimney vertically, they draw it perpendicular to the
roof. A right angle can be drawn as a single figure, but young children cannot coordinate
the chimney and the roof in order to form the proper acute angle between them. Young
children can see that the locus of points equidistant from themselves is a circle because
this involves a single figure. However, children at the intrafigural stage cannot conceive
8
the locus of points equidistant from two people because this requires understanding the
relationship between two different measurements.
The second geometric stage is the interfigural stage, which corresponds to
Piaget‘s concrete operational stage. In this stage, children are able to coordinate multiple
figures as well as distances. They begin to understand rigid transformations such as
translations, reflections, and rotations and can visualize how figures are changed under
such transformations. As research by Richard Kidder (1976) shows, children at this stage
can only understand these transformations at the simplest, most basic level and do not
understand principles (such as distance preservation) associated with these
transformations. Children at this stage have interiorized objects and are able to
manipulate a geometric figure mentally in simple ways to see its relationship with
another figure.
Piaget believed there are several factors that allow children to move into this
stage. The first is the homogenization and filling of space. At the intrafigural level,
children do not understand space to be homogenous. If a barrier is placed between two
objects, young children will say the distance between the objects is now shorter. At the
interfigural stage, children realize that distance in space is constant and does not depend
on the objects it contains. Another factor that allows children to begin to reason at the
interfigural stage is the ability to coordinate distance and direction. Children at this stage
realize that it takes two coordinated measurements to locate a point on a sheet of paper.
Both of these factors—homogenization of space and coordination of distance and
direction—are necessary conditions for the most basic understanding of tessellations and
the transformations that assure that tessellation patterns can continue indefinitely in all
directions. Without the ability to relate different figures, it is not possible for children to
9
understand how a single tile is transformed to cover the entire plane, nor is it possible to
understand basic ideas of symmetry beyond a simple appreciation of reflective vertical
symmetry, which is present even in infants (Muir, Humphrey, & Humphrey, 1994). It
therefore would seem that children younger than 6 or 7 cannot be expected to understand
tessellations to any significant depth. And in fact researchers (e.g. Vitale & Zinder, 1991)
have found that young children seem unable to appreciate that tessellations are patterns
that can be understood to continue indefinitely.
Piaget‘s third geometric stage is transfigural and is reached when a person moves
into Piaget‘s formal operational stage. At this stage, people can not only coordinate
individual figures, but also two simultaneous systems of figures, even if such
coordination requires a calculation. For example, if a snail moves along a board a certain
distance and then stops, after which the board itself moves a certain distance carrying the
snail, children at the interfigural stage can visualize and deduce the total movement of the
snail. But if the snail and the board are moving at the same time, a person must be able to
reason at the transfigural stage in order to visualize and deduce the resulting motion, even
though the calculation is precisely the same. Children at the interfigural stage cannot
conceptualize such complex motions to be a single transformation because it requires
coordinating two different systems of motion. I conjecture that children at the interfigural
stage can also not conceptualize a glide reflection as a single transformation, making such
transformations beyond the reach of children at that stage.
Piaget did not believe that children should wait until the transfigural stage to
begin studying geometry, though this was the predominant curriculum worldwide until
recent decades. It was partly because Piaget believed there were appropriate geometric
10
concepts that children can and should learn at each stage that geometry is now taught at
every primary school level in most countries, including the United States.
I chose to study children in fourth grade (age 9) because they should all be firmly
into the interfigural stage. This is the earliest stage at which children can meaningfully
study tessellations. Some second grade students might still be in the intrafigural stage
where they cannot understand that tessellations continue indefinitely in all directions, do
not tend to perceive patterns in more than one direction, and cannot relate an individual
shape to the overall pattern. The National Council of Teachers of Mathematics (NCTM,
2006) recommends studying tessellations in fourth grade because children at this age can
begin to study transformations. At this age, children can coordinate shapes with global
patterns, appreciate symmetry, and create patterns in a variety of relatively sophisticated
ways.
For my study, I used the clinical interview method developed by Piaget for his
studies. Piaget recommends that researchers have a theoretical expectation for what they
will find in a clinical interview, though of course this expectation may not always be met.
According to Piaget, children do not move from the interfigural to the transfigural stage
until about 11 or 12 years of age. Therefore I expected the fourth grade children I worked
with to be still at the interfigural stage, and in fact they showed evidence they were. They
had little difficulty handling very simple translations, reflections, and rotations, but
showed no evidence of noticing glide reflections.
Vygotsky
Lev Vygotsky was a Soviet contemporary of Piaget who unfortunately died after
only 10 years of contributions to psychological research. Vygotsky both conducted his
own research and summarized the research of others as he investigated the psychological
11
origins of thought and speech. Vygotsky‘s distinctive contribution was his insistence that
these psychological phenomena had social and cultural origins. Knowledge is a cultural
product which we develop in the context of social interactions with others. Vygotsky
believed that knowledge started at a social level, and then was later internalized, perhaps
after an extended period of learning (Vygotsky, 1978).
Both Piaget and Vygotsky were interested in analyzing the origins of thought.
Like Piaget, whom he admired, Vygotsky distanced himself from quantitative
experiments of the sort that measured how many correct answers a child could get on a
psychological test, and instead performed qualitative experiments that looked at what
children were doing and how they were doing it. Also like Piaget, Vygotsky did not
believe that people could learn by direct instruction alone. Forming simple association
bonds, as behaviorists suggested, was insufficient for the complex learning needed to
develop higher order thinking skills. On the other hand, Vygotsky also thought that
formal knowledge was distinctly different from the spontaneous knowledge we develop
from everyday interactions. Such ―scientific‖ knowledge could only be learned in formal
classroom settings (Vygotsky, 1978).
Zone of proximal development. Though Vygotsky held that formal knowledge
could only be learned in school, the classroom setting Vygotsky had in mind was not of
the lecture and rote learning type. Higher order mental processes could only come about
through social influences. Vygotsky realized that having children drill knowledge they
had already mastered was useless. Equally useless was trying to teach them knowledge
they were not ready to understand. Between these two extremes, there lies a zone of
proximal development, where the abilities children are ready to begin to learn are found.
For Vygotsky, two children capable of solving tasks for eight-year-olds are not
12
necessarily at the same level. One might be ready to learn new tasks all the way up to the
twelve-year-old level, while the other might only be ready to begin learning tasks
appropriate for nine-year-olds. This difference between the children‘s actual level and
their potential level was what Vygotsky called the zone of proximal development.
Vygotsky criticized Piaget for focusing only on the actual developmental level of the
child and neglecting the potential level.
Classroom instruction therefore should involve the student working with teachers
or peers who help him accomplish a new task slightly beyond what he is currently
capable of handling. As the child learns the new material, help is slowly withdrawn and
the child internalizes this new knowledge, which is actually an understanding he shares
with others.
The zone of proximal development was the main inspiration behind the modern
practice of scaffolding, where a teacher provides various support and guidance as a
student learns a new topic and then gradually withdraws the support as the student
succeeds in constructing the new concepts for himself (Bigge & Shermis, 1999).
The zone of proximal development does not have an impact on the analysis of my
data because I did not conduct a teaching experiment. However, this concept is important
for understanding certain aspects of the role of mathematical aesthetics.
Tools. Vygotsky took interest in the fact that human action is mediated by tools
and signs. Signs are inwardly oriented and tools are outwardly oriented. Both tools and
signs allow humans to work and reason in ways that are not predetermined by our genetic
development.
Tools could be physical or psychological. Physical tools are concrete devices that
enable us to accomplish a task, such as hammers, pencils, or pattern blocks. Later
13
researchers extended this research to conclude that tools have a profound influence on
mathematical thinking and the way students develop mathematical understanding (Cobb
& Bowers, 1999). It is clear that students may not understand a concept with one tool, but
may succeed with another. (See Ward, 2003, for an example with tessellations.) The
effect of the tool on the student‘s mathematical thinking has focused in recent years on
the case of computer technology (Hollebrands, 2003), but the tool shapes thinking in
important ways no matter what the environment. In this research study, I look at the
impact of using different tools to create tessellations.
Tools are not only physical; they can also be psychological, such as sign and
language systems and mnemonic devices. Vygotsky was primarily interested in
psychological tools and believed that speech was the most important tool at our disposal.
Most of Vygotsky‘s work centered on the relationship between thought and speech.
Thought and speech. Vygotsky took issue with Piaget‘s portrayal of private
speech1 as a mere indication that young children were not able to understand others‘
viewpoints. Piaget had portrayed private speech as an intermediate form bridging the
autistic thinking2 of babies and the phenomenon of social speech. Instead, Vygotsky
showed that private speech was a tool that young children used to reason with. Social
speech actually precedes private speech, and therefore private speech cannot be an
intermediate form leading to social speech. Young children take the social phenomenon
of speech and use it as a reasoning tool to enable themselves to solve problems they are
not capable of solving with non-verbal reasoning. One experiment showed that young
1 The older term ―egocentric speech‖ was used in translations of Vygotsky‘s work. ―Private‖ or
―egocentric‖ speech refers to the phenomenon of young children talking out loud to themselves. 2 ―Autistic thinking‖ in Piaget‘s and Vygotsky‘s writings does not refer to the pathological condition
known as ―autism‖ today, but rather the extreme egocentrism of babies, who do not yet comprehend the
distinction between self and other. Autistic thinking is disconnected from the reality of the external world.
14
children who were prevented from using private speech became incapable of solving
problems that they otherwise would have been able to solve. Vygotsky determined that,
rather than fading away as children lose their egocentrism, private speech is internalized
and becomes inner speech, the tool we use as adults to reason logically with (Vygotsky,
1934/1986). Piaget later came to agree with this view. Piaget agreed with Vygotsky that
logic had deeper roots than language (Piaget, 1971), though Piaget still believed an
important phenomenon of egocentrism had been overlooked by Vygotsky. Piaget also
agreed that learning and reasoning had a social nature. ―All logical thought is socialized
because it implies the possibility of communication between individuals‖ (Piaget, Vakar,
& Hanfmann, 1962, p. 10). In the section on Mathematical Aesthetics, I will connect
these ideas of thought and speech to the role of mathematical aesthetics in cognition.
Process and Object Conceptions
Piaget described how actions are encapsulated as thematized objects through one
of three possible mental operations. Empirical abstraction operates on physical objects to
abstract their properties. Pseudo-empirical abstraction is reasoning on operations on an
object (such as counting or ordering) to form concepts. Reflective abstraction operates on
these mental constructs to create more abstract representations (Piaget, 1977).
Educational researchers have used this idea of reflective abstraction to investigate the
construction of mathematical concepts. Piaget identified two types of thinking—
figurative, which is thinking about a static concept, and operative, which deals with
transformations and actions (Piaget & Inhelder, 1971). Many different terms (with
slightly different meanings) have been used by subsequent researchers to describe
figurative and operative thinking with respect to mathematics. The most popular terms
have been object and process.
15
Anna Sfard (1991) noticed that mathematicians tend to have visual, concrete
images of abstract mathematical ideas—they ―see‖ these ideas and manipulate their
conceptualizations of them. She compared this to Piaget‘s figurative thinking. She
referred to these abstract conceptualizations as structural objects, as opposed to
operational processes, such as algorithms, which she compared to Piaget‘s operational
thinking. Sfard described a series of three transitions that transform processes into
objects, the last of which she called reification.
Sfard extended these psychological ideas by analyzing the ontological differences
between processes and objects. It is this ontological difference that is of particular interest
for my research. Sfard noted that reification—the transition from an operational, dynamic
process to a static object—may happen rather suddenly and represents an ontological
shift in the student‘s thinking about the concept. In order to work with mathematical
concepts at a higher, more abstract level in connection with other concepts, it is usually
necessary to conceptualize them first as objects. Sfard also notes that objects may
sometimes be ―debased‖ when an inferior conceptualization is reified in place of the
preferred mathematical one, for example when students adopt a prototype in place of the
full concept. Objects and processes are not mutually exclusive conceptualizations; in fact,
they are ―different sides of the same coin.‖ Those who have an object conception can still
operate with the process conception of a mathematical idea.
Sfard notes that one important exception to the sequence of moving from
processes to objects is in the field of geometry, which is a highly visual subject by its
very nature. Many geometric ideas, such as shapes and symmetry, are grasped first as
objects and then are analyzed until they are also understood as processes. According to
Sfard, symmetry is first understood as a visual property of a shape, which is a static
16
concept—an object. Later it is understood as a transformation, which is a dynamic
operation—a process. Sfard does not discuss the case of geometry beyond this.
Ed Dubinsky. Piaget‘s theories focused on the development of knowledge in
children, though many of his ideas are applicable to all people. Ed Dubinsky and other
researchers have developed a practically-oriented extension of Piaget‘s theories which
helps to explain how mathematical thinking develops in older students. This theory is
known by the acronym APOS, which stands for the four main elements of the theory:
action, process, object, and schema (Dubinsky & McDonald, 2001). Researchers use this
framework to describe how students use reflective abstraction on procedural knowledge
to construct mathematical objects.
According to APOS theory, mathematical understanding usually starts as an
external algorithm, called an action. The steps for solving a mathematical problem are
not yet understood, but are in the form of a list of instructions to follow. If the steps are
memorized at all, it is only at the level of rote memory. When the action becomes
internalized to the point that the student can think about the sequence of steps, including
how to reverse them or compose them with other actions, the concept is called a process.
However, the concept is still in the form of an algorithm. At some point, the student may
understand the process as a totality, knowing what transformations can be carried out on
it as a whole. The student no longer thinks about the steps at all, unless needed to carry
out a calculation. The concept has then become an object, which enables flexible
thinking. A collection of actions, processes, and objects that are connected in a common
framework is a schema, as in Piaget‘s theory. The sequence from action to process to
object is not a simple linear process. When a student is internalizing a process, he may
17
return to the action level in order to further develop the process. However, in general
actions and processes must precede objects and schemata.
Dubinsky agrees that the case of geometry is different. Like Sfard, he considers
geometry to be object-based. Instead of processes that are encapsulated to form objects,
processes are used to study the geometric objects they act on. Geometry concepts are
developed largely through empirical abstraction. Actions are used mainly to study a
geometric object, rather than to form the basis of a geometric concept. Geometric
concepts also develop through reflective abstraction as students develop linguistic
categories to describe the concepts.
David Tall and Eddie Gray support APOS theory, but question whether it is
capable of analyzing all student mathematical thinking. In particular Tall (1999)
questions the primacy of action for all types of mathematical thinking. He has shown, for
example, that it is possible to teach certain calculus concepts in such a way that the object
conceptualization develops in parallel with procedural understanding. Like Sfard and
Dubinsky, Tall sees that for the domain of geometry certain types of object thinking can
precede process thinking. He also sees formal, axiomatic mathematics as requiring yet
another kind of thinking not well analyzed by APOS theory. According to Tall, analysis
of process and object thinking applies principally to symbolic mathematics such as
algebra and calculus.
To supplement APOS theory, Gray and Tall (1991; 1994) propose a slightly
different understanding of the relationship between processes and objects. They noticed
that symbols are ambiguously related to either processes or objects. This flexibility is
ideally utilized by students when they understand both conceptualizations. Tall and Gray
introduced the idea of procept to describe this flexible thinking. They first defined an
18
elementary procept as ―the amalgam of three components: a process that produces a
mathematical object, and a symbol that represents either the process or the object‖ (1994,
p. 121). They then noted that symbols are flexible and several symbols may point to the
same object, from which they drew their definition of procept as ―a collection of
elementary procepts that have the same object‖ (1994, p. 121). These procepts are the
result of pseudo-empirical and reflective abstraction so that the student focuses on
conceptual rather than perceptual understanding. Children who focus on the appearance
of a prototypical square with horizontal and vertical sides are not likely to recognize one
presented at a non-standard orientation. Gray and Tall noted that procepts are not fixed.
Students can revise their encapsulated procedures as needed. Like APOS theory, they saw
procept theory as useful mainly for symbolic mathematics. They understood geometry as
reflective abstraction on perceived, geometric objects, rather than on processes, as is the
case for symbolic mathematics.
Hartwig Meissner explored the procept idea for geometry and found that it could
be profitably applied to most concepts in this domain as well. He showed that geometric
representations such as nets can behave as symbols in procept theory (2001). A geometric
concept, such as triangle, is both a percept and a concept—it is an object (2006). This
object conceptualization undergoes change as properties are associated with the triangle
and these properties are actively put to use in problem solving. The triangle now requires
proceptual thinking—understanding as both process and object. Meissner shows how
other geometric concepts, such as basic theorems, must be understood as both object and
process in order to be used to solve problems.
What all of these theories have in common is that many mathematical ideas must
be understood as both object and process in order to be used flexibly in problem solving.
19
Geometry is more object-oriented than other domains of mathematics and therefore
students can understand many geometric ideas first as objects before analyzing them as
processes.
In the case of tessellations, we are dealing with a geometric construct that cannot
be fully represented in finite space and therefore cannot be fully grasped by empirical
abstraction alone. Some form of proceptual thinking will therefore be required for a full
understanding of tessellations. Tessellations need to be understood both as mathematical
process and as mathematical object. In this dissertation I analyze the students‘ ontological
conceptions of tessellations and the relationship of those conceptions to students‘
understanding of how tessellations extend into infinite space.
Aesthetic Perspective
La beauté apparaît souvent aux festins où l’on n’avait invité que l’utilité ou la vérité.
(―Beauty often shows up at banquets where only usefulness or truth was invited,‖ Le
Lionnais, 1948, p. 437)
In addition to the traditional cognitive perspective of understanding children‘s
mathematical thinking, this research considers the aesthetic perspective of understanding
what children find appealing in tessellation tasks. This perspective is much newer than
the cognitive perspective for modern mathematics education (though it has roots in
antiquity) and deserves more careful explanation. I first explore the issue of aesthetics in
mathematics generally and then in children‘s mathematics education specifically. I focus
primarily on carefully developing the reasons why aesthetics is foundational to a
complete understanding of mathematics and children‘s mathematical thinking.
In order to explore mathematical aesthetics, I first lay some groundwork by
discussing certain philosophical issues concerning the nature of mathematics.
20
Nature of Mathematics
In order to properly understand the role of aesthetics in mathematics, we must
first be clear on the nature of mathematics. I believe the foundational role of aesthetics
can be seen by understanding the nature of mathematics. However, this understanding is
not simple, all the more so because the understanding of what exactly mathematics is has
changed over the course of history and there may not be an agreed, universal view of the
nature of mathematics.
Historical development. In antiquity, mathematics began as a purely practical
matter. Arithmetic developed over commercial needs and geometry developed to solve
problems in surveying and architecture. The Greeks were the first to take these practical
matters and begin to explore the reasons for the formulas that had been discovered.
Possibly because of logical concerns, such as the paradoxes of Zeno, the Greeks refused
to depend directly on visual evidence and instead began to derive theorems by deductive
methods. They found reasoning by deductive methods to be more beautiful than mere
induction because it led to order and consistency (Kline, 1964).
Pythagoras (6th century BCE), one of the earliest Greek mathematicians, founded
a mystic society that considered mathematics to be an expression of divine reality.
Number and form were not considered to be abstract concepts, but were rather the only
true reality. Mathematical beauty was the highest possible aesthetic, as seen in the
harmonies of number and form. ―The Pythagoreans were overwhelmed by the aesthetic
appeal of the theorems they discovered and were perennially preoccupied with the
interconnectedness of the mathematical and the aesthetic‖ (Sinclair & Pimm, 2006, p. 4).
The Greeks pursued mathematics for mathematics‘ sake in just the same way as
people pursue art for art‘s sake—for purely aesthetic reasons (Sinclair & Pimm, 2006).
Aristotle confirmed the supreme place of mathematical aesthetics in his Metaphysics, ―Of
21
what is fair, however, the most important species are order and symmetry, and that which
is definite, which the mathematical sciences make manifest in a most eminent degree‖
(Aristotle, trans. 1896, p. 358). For the next few centuries, the Greeks considered
mathematics more of an art than a science (Le Lionnais, 1948).
Greek mathematics reached its highest point with Euclid‘s Elements (c. 300
BCE). This work, the foundation of Western mathematical thought for two millennia, laid
out geometry and number theory as a purely deductive system based on five common
sense axioms and five geometric postulates, which were considered visually obvious.
Though the geometric theorems were accompanied by diagrams, the reasoning was
supposedly entirely based on deductive logic, not on any visual inferences from the
drawings.3
Other cultures developed mathematics differently. Indians, for example,
prioritized arithmetic, and later algebra, inventing a symbol for zero and our current
Hindu-Arabic numeral system. Geometry was the basis and inspiration for their
mathematics, but geometry was never developed to the same extent as arithmetic, nor did
Indians take much interest in deductive proofs in geometry. When Indians did begin to
include geometric proofs, these proofs tended to be diagrams that were meant to be
understood visually in a single step (Arnheim, 1969), such as Bhaskara II‘s well-known
12th
century proof of the Pythagorean theorem (Figure 1 below), which was actually
discovered by the Chinese many centuries previously (Schattschneider, 2006). The only
text that accompanied Bhaskara II‘s visual proofs was the single word, ―Behold!‖ For
3 There were no obvious visual inferences from the diagrams. As we will see, when mathematicians began
to develop more rigorous methods in the 19th
century, it was discovered that there were many unstated
assumptions in Euclid‘s Elements and that the proofs sometimes did, in fact, depend upon inferences from
the drawings.
22
whatever reason, Indian mathematicians understood the need for proofs in arithmetic and
algebra, but were satisfied with visual reasoning in geometry.
Figure 1: Bhaskara II‘s visual proof of the Pythagorean Theorem. Color and labels have
been added to show a2, b
2, and c
2.
During the European Middle Ages, Arabic mathematics began to soar. Muslim
mathematicians took the best from Greek and Indian mathematics, adopting the Hindu
numeral system and inventing algebra to solve ever more important problems. The
development of Islamic geometry was slow and complex. The most important event was
the translation of Euclid‘s Elements, which ―caused a cultural revolution in the Islamic
world in the ninth century similar to the one it created in the West after the Latin
translation by [Adelard] of Bath in 1142‖ (Powell, 1998). Islamic writers were hesitant to
call mathematics beautiful (though some did), preferring to appreciate it for being
constant and certain (Sinclair & Pimm, 2006). Muslims did not hesitate, however, to
apply geometry to works of beauty. Their work with geometric mosaics is well known.
Mathematics was a tool for art, rather than an art itself. Muslims also believed that
geometry provided a door to the spiritual world allowing us to glimpse perfection (Abas,
2001).
23
Islamic artisans and mathematicians worked together to develop increasingly
interesting mosaic designs (Özdural, 2000). The Alhambra in Spain (14th
century)
contains a rich variety of tessellations, including examples of most of the 17
mathematically possible periodic patterns (Grünbaum, 2006). I used postcards of some of
these tilings in the final clinical interview in this study. By 1500, Islamic artisans were
even creating patterns which are now known to be based on nearly perfect Penrose
tilings, an especially interesting and mathematically complex aperiodic tessellation not
even known to exist in the West until the 1970s (Lu & Steinhardt, 2007). (See Figure 2
below.)
Figure 2: Analysis of a near-Penrose pattern from the Alhambra (Lu & Steinhardt, 2007)
In the Western tradition after the Greeks, mathematics became more closely
linked to science. Nevertheless, until the 18th
century, Western philosophers continued to
claim that mathematics was aesthetic because mathematics was the height of reason, a
pure expression of eternal truth (Chernyak & Kazhdan, 1996), though this claim of
beauty was rarely made explicit by mathematicians. In practice, mathematics was often
seen as a mere tool for analyzing the real world. Mathematicians and philosophers in the
24
late 18th
century, such as Euler and Kant, defined mathematics simply as the study of
quantity (J. Gray, 2008).
In the 19th century Western mathematics began to face a crisis as mathematicians
began trying to bring greater rigor to their subject. Attempts to provide a more rigorous
basis for calculus proved very difficult. In geometry, the need for greater rigor seemed
especially urgent. Euclid‘s Elements was discovered to have unstated assumptions and
had depended at times on visual reasoning, rather than strict deduction. Consequences of
these flaws were easily seen as mathematicians created deliberately fallacious proofs (e.g.
all triangles are isosceles) within Euclid‘s system. Most importantly, dissatisfaction with
Euclid‘s fifth postulate, the parallel postulate, eventually led to the discovery of non-
Euclidean geometries—complete and consistent axiomatic systems which seemed to
describe geometries that did not exist in the real world.
The development of projective geometry and other geometries during the 19th
century drove mathematicians to the realization that mathematical definitions are
arbitrary. In certain geometries, the terms ―line‖ and ―point‖ could be interchanged, along
with ―concurrent‖ and ―collinear,‖ resulting in entirely new theorems, equally as valid as
the originals. This and other developments led to the realization that such words should
be understood uniquely in terms of their axioms without any connection to their
traditional representations. Definitions, axioms, theorems, and proofs became nothing
more than a logical, abstract manipulation of otherwise meaningless symbols.
Similar changes in other branches of mathematics during the late 19th
century led
to a decisive break between pure mathematics on the one hand and applied mathematics
and the natural sciences on the other. Jeremy Gray (2008) credits the sense of uncertainty
created by 19th
century questions of error and rigor with the desire to separate pure
25
mathematics from the natural sciences in order to avoid ―contamination.‖ Universities set
up separate departments for mathematics and physics towards the end of the 19th
century.
The modernist transformation of mathematics, which began at the end of the
19th
century, saw a definitive shift in the ontology and epistemology of mathematics (J.
Gray, 2008). The ontology of mathematics shifted from an objective description of the
real world to an aesthetic creation of the human mind, which could not be both ―true‖ and
provable. Its epistemology was intensely debated in the context of at least three important
movements: formalism, logicism, and intuitionism.
Formalism evolved from the previously mentioned view that pure mathematics is
a completely abstract study with no necessary connection with the real world. Axioms
and definitions are entirely arbitrary; they need no connection with any real
representation. Logicists attempted to set mathematics on a symbolic logical foundation.
One of the logicists‘ hopes was that mathematics would be reduced to pure symbolic
logic and that their system would lead to an algorithmic method by which all
mathematical theorems could be proved—a calculus ratiocinator. Mathematics would
then have a perfect foundation that would bring absolute certainty to every theorem.
Intuitionists pointed out some of the limits of logicism and formalism. They felt that,
even if axioms were in some sense arbitrary, they could still be rooted in some common
sense notions. For example, Henri Poincaré, who was the leading and most read figure in
public debates in the early 20th
century, noted that all attempts to define the number zero
inevitably contained within the definition the common sense notion of ―zero.‖ The
number ―zero‖ was sometimes defined as the number of elements in the null set; but the
null set was conceived as the set containing no (zero) elements. If one needed to have a
previous understanding of this and other primitive notions in order to create rigorous
26
definitions, why not simply depend on these common sense notions to begin with (J.
Gray, 2008)? Poincaré believed that only aesthetics could drive the identification of the
most fruitful axiomatic systems. Poincaré believed that the human mind, guided by an
interior aesthetic, was the only possible mechanism for discovering rich and deep truths
in mathematics (Poincaré, 1908/2000).
In 1931 Gödel showed the limits of logicism when he proved that it was not
possible to set up an axiomatic system that could prove all true arithmetic propositions.
There must by necessity be theorems which cannot be proved from within the system.
Logic alone is insufficient for establishing mathematics.
Today, mathematicians continue to hold a variety of philosophical positions
about the nature of mathematics. However, it is generally agreed that axioms are
basically arbitrary and not necessarily connected with the real world (though mathematics
proves to be remarkably useful for describing it). As such, pure mathematics is an
aesthetic subject because it is pure invention. Pure mathematicians study mathematics for
mathematics‘ sake, much as artists study art for art‘s sake. It is also recognized, as a
result of Gödel‘s theorem, that no calculus ratiocinator can be devised to find and verify
all theorems in mathematics. Without appeal to human abilities other than logic,
mathematics is deadlocked and can neither get started nor progress further.
Definition of mathematics. Mathematics is an aesthetic study. But we are still
left with the question of what exactly mathematics is. It is surprisingly difficult to pin
down a precise definition of mathematics. Many dictionaries and encyclopedias define
mathematics in terms of its primary domains: arithmetic, geometry, analysis, and so forth.
But this does not explain what unifies these domains. The Encyclopedia Britannica
defines mathematics as ―the science of structure, order, and relation‖ (Mathematics,
27
2006), which is a more satisfying definition. G. H. Hardy saw mathematics as the study
of patterns, a definition many others have agreed with. ―A mathematician, like a painter
or poet, is a maker of patterns. If his patterns are more permanent than theirs, it is because
they are made with ideas‖ (Hardy, 1940, p. 84). Warwick Sawyer agreed, defining
mathematics as the ―classification and study of all possible patterns‖ believing that
―where there is pattern there is significance‖ (1955/1982, pp. 12 and 36, author‘s italics).
Many mathematicians today agree with these definitions. It is this idea that pattern and
structure are the objects of study in mathematics that will lead us to understand more
fully the fundamental role of aesthetics in mathematics.
Mathematical Aesthetics
―It‘s a thing that nonmathematicians don‘t realize.
Mathematics is actually an aesthetic subject almost entirely‖
(John Horton Conway, quoted in Spencer, 2001, p. 165).
Before turning to the field of education, I define mathematical aesthetics more
precisely. I look at the foundational role given to aesthetics by the modernist
transformation of mathematics, pause to look at the theory of aesthetics in general and its
connection to mathematics, propose precise criteria for identifying mathematical
aesthetics, consider the problem of subjectivity in mathematical aesthetics, and conclude
with a summary of the importance of aesthetics to the field of mathematics.
If mathematics is the study of patterns, it is natural that mathematics should be
aesthetic because patterns are naturally appealing. This could even suggest that
mathematics begins with beauty as we are drawn to study its patterns. Indeed, some
mathematicians admit that they are driven by the initial beauty of certain patterns to
investigate the mathematics of these patterns.
28
As I discussed above, the modernist transformation of mathematics at the end of
the 19th
century brought about a radically different understanding of mathematics.
Whereas mathematics had been previously understood to be a rigorous study rooted in
―obvious‖ axioms, common sense definitions, and logical deductions, by the end of the
19th
century it was understood that axioms and definitions are basically arbitrary. If the
foundations are arbitrary and not based on either empirical evidence or self-evidence,
how are they selected? Being ―obvious‖ is no longer considered a valid criterion. Instead
the criteria for choosing axioms include aesthetic criteria such as simplicity,
connectedness, and elegance, as well as certain logical criteria such as consistency,
mutual independence, and relative completeness.
Not only were the foundations of mathematics partially based in aesthetics, but
the methods were also seen to depend on aesthetics. Gödel proved that a calculus
ratiocinator could never exist to logically crank out all mathematical proofs. Aesthetics is
needed to determine the most fruitful paths for discovering important theorems and
proofs. Poincaré took special interest in writing about this creative process of finding
patterns and relationships, which requires an aesthetic sense to find generative
combinations of logic. Aesthetics is important for filtering what enters the conscious
(Poincaré, 1908/2000). As the Greeks found aesthetics in the ontology of mathematics,
Poincaré found aesthetics in the epistemology of mathematics.
I conclude from the above discussion that mathematics today depends
fundamentally on aesthetics in at least two respects. First, axiomatic systems are now
considered arbitrary rather than ―obvious.‖ In theory, any set of consistent axioms could
establish a mathematical theory; however in practice mathematicians use aesthetics to
derive systems that are useful, elegant, and simple. Second, once an axiomatic system is
29
chosen, no purely logical, mechanical procedure can exist that will deduce the truth of all
possible theorems.4 Aesthetics is needed to pursue a useful and important system of
theorems that can be proved from the axioms. Aesthetics is also needed in the search for
proofs of these theorems. In actual practice, mathematicians usually explore new areas of
mathematics first, and then establish an axiomatic foundation to support it last.5 But in
any case, the entire mathematical enterprise is based on and guided by aesthetics.
Aesthetics. At this point I should say something about aesthetics in general. So I
pause here briefly to discuss the theory of aesthetics in general and show how it connects
to the question of aesthetics for mathematics.
For the Greeks, beauty was an objective quality, a permanent property of the thing
itself. Aristotle considered order and symmetry to be the most important aesthetic criteria
(Aristotle, trans. 1896).
For the modern Western tradition, many of our ideas about aesthetics can be
traced back to Kant‘s Critique of Judgement (Kant, 1790/2008). According to Kant, there
are four kinds of judgment: the agreeable, the good, the sublime, and the beautiful. The
agreeable is what appeals to my senses and is purely subjective. The good, on the other
hand, is objective. It refers to that which is morally or ethically right. Both the agreeable
and the good address our interests—our interests in pleasure or moral right. The sublime
4 Indeed, from a purely formalist point of view, no theorem in mathematics is actually ―true‖ because
everything is based on arbitrary assumptions without real world connections. Theorems can only be
―proved‖ with respect to a set of axioms. This is a distinction mathematicians recognize in theory but
usually ignore in practice, except in certain cases where the provability of a theorem depends on the set of
axioms chosen. For example, the Banach-Tarski paradox gives the counter-intuitive result that a solid ball
can be cut into a finite number of rigid pieces which can be rearranged to make two solid balls the same
size as the original. This theorem is only ―true‖ if one accepts the ―truth‖ of the Axiom of Choice, an
―obvious‖ axiom which mathematicians found controversial in the early 20th
century because of similar
counter-intuitive results, but which is usually accepted today. 5 In fact, Keith Devlin (2008) notes that even when axiomatic systems change, the body of mathematics
supposedly built on those axioms generally is not affected and remains the same.
30
and the beautiful do not address interests. The sublime is a subjective concept describing
that which is awe-inspiring, perhaps to a point bordering on fear. The sublime appeals to
us because it shows reason to be superior to our imagination. For this reason mathematics
may sometimes be sublime when we study, for example, infinity. We delight in the fact
that our reason can understand a concept that we cannot fully imagine.
Unlike the agreeable or the sublime, Kant considered the beautiful to be an
objective category. Like the sublime, the beautiful describes an object which evokes
delight in a disinterested way. By ―disinterested‖ he means that it pleases us not because
of any further benefit we might obtain, but because of the sheer pleasure of
contemplating the object.
For Kant, beauty is its own end, not the means to satisfying some other desire.
Neither the sublime nor the beautiful have any ulterior purpose, and yet beauty has a
certain ―purposiveness‖ or ―finality‖ (Zweckmässigkeit) to it, as if it had an inner
purpose. Geometric figures have a fruitfulness in their use that may be related to their
beauty. Unlike the agreeable, which is judged agreeable only because it gives us pleasure
first, the beautiful is first of all beautiful, and as a result of this judgment of beauty we
take pleasure in it. I know that something agreeable to me may not be agreeable to others.
But there is a sense that what we find beautiful is really objectively beautiful. Even
though we know that all aesthetics is inherently subjective and that taste may need to be
acquired to appreciate certain beauty, we feel that others ought to be able to perceive the
same beauty that we perceive. Kant also noted that beauty requires variety. The overly
familiar does not appeal to us.
Kant anticipated some future developments, seeing for example that mathematics,
like poetry, was ―pure invention‖ (1938/1993), nearly a century before the modernist
31
transformation. Many of Kant‘s ideas have been rejected in the last century, but his ideas
still run deep in Western philosophical thought. Some of his ideas, such as the objectivity
of beauty, are commonly accepted by popular opinion (though rejected by many
philosophers). Other ideas have been accepted by one or several branches of modern
philosophy.
One philosopher who extended and contradicted some of Kant‘s ideas was John
Dewey, who in his book Art as Experience (1934) argued that art was an intensive form
of common experience. Each experience is unified by a pervasive quality. This quality
does not arise by analysis of the parts of the experience, but from the overall pattern or
structure of the experience. This quality that rounds out the experience is the aesthetic.
The aesthetic emerges therefore from the complete experience, not from any of its
elements. Dewey rejected Kant‘s distinctions of the beautiful, sublime, and so forth,
because such labels were too restrictive. Just as Poincaré showed that logic alone could
not create mathematics, Dewey argued that reason alone, without aesthetics, was
insufficient for attaining truth in any domain.
This idea of the fundamental role of aesthetic knowing has been taken up by many
other philosophers. Robert Root-Bernstein (2002) argued strongly for the recognition of
―aesthetic cognition,‖ the ability that all scientists depend on for the source of their
insights. He argues against the commonly held idea that only ideas that can be verbally
expressed can be actually thought. Aesthetic cognition precedes the verbal logic in which
it is later expressed. Just as we depend on a sense of judgment, and not mathematical
formulas, for physical activities such as throwing a ball, so scientists depend on an
aesthetic feeling to direct them. We use this aesthetic cognition to reason in ways that are
not linguistic. Linguistic and aesthetic cognition are quite different because linguistic
32
reasoning must be linear, whereas aesthetic cognition is not—it is a different way of
reasoning. Root-Bernstein quotes Kant, ―The intellect can intuit nothing. The sense can
think nothing. Only through their union can knowledge arise‖ (Root-Bernstein, 2002, p.
69).
Many of Root-Bernstein‘s ideas come from Poincaré, such as the insistence that
discoveries are made through our intuition (which Root-Bernstein equates to aesthetic
cognition), and then afterward confirmed by verbal logic. Logic‘s role is only to prove; it
is useless for identifying which paths will be fruitful. Poincaré wrote, ―Without
[intuition], the geometrician would be like a writer well up in grammar but destitute of
ideas‖ (Poincaré, 1914/2010).
Root-Bernstein accused philosophers of an obsession to understand the logic of
science and of neglecting the role of aesthetics, resulting in a failure to understand
creativity in science. Root-Bernstein believes that philosophers may also avoid the study
of aesthetics in connection to mathematics because of aesthetics‘ deep connection to the
emotions. It is sometimes supposed that emotions impede logical thinking, but in reality
emotions are necessary for all rational decisions. Experiments have shown that people
whose emotions are impaired cannot make decisions, even if their logical abilities are
intact (Root-Bernstein, 2002). Israel Scheffler (1991) has pointed out that scientists
depend on a wide range of cognitive emotions for their research, including love of truth,
concern for accuracy, abhorrence of error, admiration of theoretical achievement, respect
for good arguments, shame at failing, self-respect, joy of verification, and surprise at new
discoveries. Root-Bernstein explains the apparent paradox of the mutual dependence of
feelings and thinking by noting that thinking is ultimately grounded in our senses, and
33
therefore in aesthetics. ―The best science, like the best art, is that which appeals to the
widest range of emotion and intellect‖ (Root-Bernstein, 2002, p. 65).
Root-Bernstein‘s idea of aesthetic cognition is similar to ideas by other writers.
The psychologist Rudolf Arnheim (1969) argued that perception was not ontologically
different from cognition. Dewey (1934) argues that we first intuit an aesthetic quality
from a global situation before analyzing it into discrete logical parts. Note, however, that
after we analyze and understand the parts, we may come back to a greater and different
appreciation of the aesthetic whole. Our understanding enhances aesthetic appreciation.
Language is a tool which we adopt in order to reason logically about situations,
but language is not the totality of our reasoning ability. We all have an innate reasoning
ability which is not verbal. This primary capacity, variously termed aesthetic cognition,
intuition, visual reasoning, meta-logic, super-logic, etc., is where our reasoning begins.
This reasoning is non-verbal, and therefore does not depend on symbolic logic. For
example, as I look at a rectangle, I may notice a pleasing symmetry that leads me to ―see‖
that the diagonals must be congruent and mutually bisecting. This realization takes place
before I have had a chance to carry out a symbolic proof of this fact, or even verbalize it.
Such knowledge is not innate—it does not occur spontaneously to children—but it is
built up through geometric experience with rectangles and their properties. This non-
verbal knowledge is then translated through the tool of inner speech into a form that can
be communicated and checked for logical consistency. According to Root-Bernstein, the
mathematical expression that is communicated is not the actual content of our
understanding, but is merely a way of expressing it. Rather than being some superfluous,
illogical feeling, aesthetic cognition is seen to be a foundational step in any cognitive
task.
34
Nor is aesthetic cognition vague. Mark Johnson (2007) points out that writers will
often struggle to find just the right word to express their idea. The idea is not yet verbal—
they have not found the word for it—and yet it is quite precise, as evidenced by the effort
to find the perfect word to express it.
Aesthetics is therefore foundational not only for establishing the axiomatic basis
of mathematics and for guiding mathematicians towards generative mathematical ideas,
but aesthetics is also foundational for the reasoning process itself. Root-Bernstein writes,
―Only when we feel that we know and know what we feel do we truly understand‖ (Root-
Bernstein, 2002, p. 70). Mathematical aesthetics is a ―way of knowing‖ (Sinclair, 2006)
mathematics prior to verbal reasoning.
One consequence for the field of education is that both aesthetic and logical
cognition should be taught, as well as the connection between the two. Students must
develop their sense of mathematical aesthetics to guide them in problem solving. They
must also learn how to translate their non-verbal ideas into communicable, logical
descriptions, just as they must learn to interpret others‘ verbal descriptions in a way that
makes sense to them. As mathematician Michael Starbird would say, the task of
education is to teach students to say what they think and think what they say (personal
communication, September 24, 2010). Helping children form this deep connection
between the verbal and the non-verbal—the cognitive and the aesthetic—is fundamental
to a clear understanding of mathematics.
So we have now returned to the question of aesthetics in mathematics. Next we
need to know how to recognize mathematical aesthetics. What form does the aesthetic
take in mathematics?
35
Criteria for mathematical aesthetics. The idea that pure mathematics is an
aesthetic study is generally accepted by most mathematicians today (Burton, 1999). As
we saw above, since the modernist transformation aesthetics is at the very foundation of
mathematics. Aesthetics is essential for establishing axioms, for providing direction, and
even for the reasoning process itself. I mention these roles first because they are
frequently overlooked.
The role of mathematical aesthetics most commonly recognized today is in the
evaluation of results, particularly theorems and proofs. Morris Kline noted that
mathematicians are rarely satisfied with an ugly theorem or proof. Instead, they are
always seeking to refine theorems and proofs to be as elegant as possible. Kline believed
that this desire to refine proofs showed that mathematics had an artistic nature. ―Much
research for new proofs of theorems already correctly established is undertaken simply
because the existing proofs have no aesthetic appeal‖ (1964, p. 470). Wolfgang Krull
noted that some mathematicians have even refused to publish correct theorems for years
until they found an aesthetically satisfying proof. He stated in his inaugural address at
Erlangen, ―Mathematicians are not concerned merely with finding and proving theorems;
they also want to arrange and assemble the theorems so that they appear not only correct
but evident and compelling. Such a goal, I feel, is aesthetic rather than epistemological‖
(1930/1987, p. 49).6 Krull went on to assert that even a flawed work by Klein (whose
work was usually highly visual and aesthetic) ranked far above many similar flawless
(but less aesthetic) mathematical works by other mathematicians. The goal of
aesthetically pleasing proofs is to demonstrate a theorem in a convincing way which
leads the reader to see that a particular truth could not be any other way. Such aesthetics
6 Of course the goal is also epistemological, but it is primarily aesthetic.
36
may even lead mathematicians to prove the same theorem in a variety of ways. ―I believe
this state of affairs shows beyond doubt that aesthetic viewpoints play a large role in
mathematics‖ (Krull, 1930/1987, p. 51).
There have been various attempts to define modern mathematical aesthetics.
English-speaking mathematicians often refer to the work of G. H. Hardy (1940) in
defining what makes a mathematical result aesthetic. Hardy tied his understanding of
mathematics as the study of patterns to the aesthetic appeal of mathematics. Hardy
identified significance, unexpectedness, inevitability, and economy as factors that
contribute to a mathematical aesthetic. Doris Schattschneider (2006) proposes aesthetic
criteria of elegance, ingenuity, insight (the quality that makes you go ―Aha!‖—think of
Bhaskara‘s one-step visual proofs), connections, and ―paradigm of technique‖ (novel,
useful methods that evoke admiration, such as the pigeon-hole principle). Nathalie
Sinclair (2006) suggests including connectedness, visual appeal, apparent simplicity, and
surprise as further criteria for what mathematicians find beautiful.
French mathematicians often turn to the writer and mathematician François Le
Lionnais (1948), who identified two general types of mathematical aesthetics (which I
look at in more detail later). ―Classical‖ aesthetics is characterized by elegance,
simplicity, and a sense of control. ―Romantic‖ aesthetics is that which shocks and
overturns established ideas—the non-conforming and bizarre.
Root-Bernstein claims that elegance is a key criterion. ―The ability to concentrate
meaning and connections maximizes understanding and its emotional impact, whereas
simply following a logical path to a conclusion often yields neither insight, connections,
surprises, nor joy‖ (Root-Bernstein, 2002, p. 70). He also notes that Einstein identified
37
the aesthetic criteria for science as simplicity, connections, and significance, claiming
that these were needed for convincing, rather than mere conviction.
I propose summarizing all these criteria in five categories: significance, surprise,
simplicity, connectedness, and visual appeal.
1. Significance – Though significance alone is rarely sufficient to make a
mathematical theorem aesthetic, a sense of significance adds to the perceived
beauty. The Fundamental Theorem of Calculus is considered beautiful partly
because of its extreme importance in mathematics and partly because it connects
two branches of mathematics (differential and integral calculus) in a very elegant
way. A method meeting Schattschneider‘s criterion of ―paradigm of technique‖ is
significant because it has many uses. ―Depth‖ could refer to significance in the
sense of not being trivial, or to the many connections the result makes.
2. Surprise, unexpectedness, shock, new ideas, and insight – This criterion tends to
be mentioned by mathematicians more than any other (Wells, 1990). Sinclair
(2006) notes an example where students were casually told that angles inscribed
in a semicircle are always right angles. Students were so shocked they excitedly
interrupted the lesson to explore this unexpected fact. Le Lionnais mentions that
we are delighted by the appearance of order where we expected only chaos.
However, surprising disorder where we expect order can also create beauty, such
as in Gödel‘s Incompleteness Theorem or Penrose tilings. Israel Scheffler (1991)
identifies surprise as a crucial cognitive emotion for all of science because it leads
us to change our currently held theories. Only by being open to such challenges to
our beliefs can we grow in our understanding.
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3. Simplicity, elegance, economy, control, and ingenuity – Mathematicians find
beauty when a theorem or proof is delightfully shorter or more powerful than one
would expect. A brief, clever solution creates a feeling of joy. Euler‘s formula
( ) is often considered the most beautiful equation in mathematics
because of its extreme simplicity in relating the five most important mathematical
constants (Wells, 1990). Many people who discuss mathematical aesthetics cite
Euclid‘s proof of an infinite number of primes as a supreme example of beauty,
probably because of its ingenious nature, being far simpler than one would
anticipate. Hardy associated ―inevitability‖ with economy: Simple proofs leave no
room for doubt; they compel the reader towards an inevitable conclusion.
Simplicity is often associated with surprise. There are many trivial results in
mathematics which are not considered beautiful because they are not expected to
be complex. Wells notes that complexity is paradoxically also an aesthetic
criterion. For example, Penrose noted that his aperiodic tilings are more beautiful
than simple square tilings (Wells, 1990). In this case, I believe we are dealing
with depth. We find results beautiful when they have complex, significant order,
not random complexity. However, results that are overly complex fail to be
beautiful. Beauty arises from simple expressions of complex depth.
4. Connectedness – A sense of connectedness brings a feeling of unity and
inevitability to our appreciation of mathematics. Euler‘s formula defines a
surprising connection between two numbers, e and π, which otherwise have no
apparent relationship. Non-trivial results have many connections and are often
said to be ―deep.‖ Le Lionnais waxes eloquent over Descartes‘s connection of
geometry with algebra or Klein‘s connection of group theory with geometry. I
39
believe tessellations to be beautiful not only because of their visual appeal, but
also because they connect so many different ideas in geometry.
5. Visual appeal – Bhaskara‘s proof of the Pythagorean Theorem is often
considered beautiful because it relies entirely on visual reasoning. Le Lionnais
(1948) points out that the beauty of Klein‘s work is often tied to its visual nature.
Even non-visual results are considered more appealing if they can be explained by
graphs, diagrams, or other visual means. Presence of visual symmetry, unusual
shapes, a moderate amount of visual complexity, or patterns that are clear to the
eye may make the result more beautiful.
Beautiful ideas in mathematics almost always satisfy more than one criterion. The
Mandelbrot Set (Figure 3) is considered by many to be one of the most beautiful objects
in mathematics and seems to satisfy all five of the above criteria: It is significant for
helping to open the field of fractal geometry; its infinite complexity is very surprising
given its extremely simple generating function;7 it has surprising properties such as self-
similarity and topological connectedness; it connects several fields of mathematics,
including complex numbers, geometry, topology, analysis, and complexity theory; and it
has enormous visual appeal as evidenced by the popularity of images and videos in books
and on the Internet.
7 The recursive complex function that defines the Mandelbrot Set is
; . If zn does not
grow arbitrarily large as n goes to infinity, c is in the Mandelbrot Set. That‘s it. The result is surprisingly
complex and beautiful. Colors are usually added according to how long it takes zn to get large. This
increases both comprehension and visual appeal.
40
Figure 3: A small portion of the Mandelbrot set near
Trying to be more specific than a list of criteria in trying to define exactly what
makes a mathematical theorem or proof beautiful continues to be difficult, probably
because most mathematicians have not given it much thought (Burton, 2001).
Buckminster Fuller claimed that he never thought about beauty unless a solution was not
beautiful. In that case the lack of aesthetics told him that something was wrong (Fadiman,
1985). In a similar way, mathematicians show concern over ugly theorems, but rarely
give explicit thought to the philosophy of mathematical aesthetics. Questions of
mathematical beauty have not preoccupied modern mathematicians as they did the
ancient Greeks. It is perhaps for this reason that the general public seems unaware of the
role of aesthetics in mathematics.
It is important to note how aesthetics is tied to importance in mathematics.
Criteria such as significance and connectedness are not only criteria which are appealing
to mathematicians, but also important for identifying those elements of mathematics that
will move the field forward. The criterion of surprise is similar to the concept of
41
cognitive dissonance; it implies delight in novel situations that advance our ideas. The
criterion of visual appeal is perhaps related to that of inevitability (simplicity). When we
can perceive a mathematical truth with our eyes, we become more convinced of its
correctness because we more readily understand the result at an intuitive level. The visual
perception may even trigger an unexpected insight (surprise). Sinclair (2006) notes that
simple equations and round numbers (mathematical objects having simplicity) are more
generative. This intersection of aesthetic criteria with what is important and generative
for doing mathematics suggests a close relationship between the aesthetic and the
cognitive in mathematics.
Subjectivity in mathematical aesthetics. Another reason that a mathematical
aesthetic has been difficult to define is that aesthetics is partly contextual, varying
between cultures, across time, and even between individuals of the same time and culture.
If we claim that a mathematical aesthetic exists, then it is important to explain these
differences. There is a certain subjectivity in any aesthetic. But is mathematical aesthetics
so hopelessly subjective that it is futile to think of considering it in a classroom setting?
Or is there a certain objectivity about mathematical beauty that all students should learn
to appreciate?
The fact that aesthetics varies from culture to culture is well known in the arts. In
the West, for example, there is a tradition that values symmetry and balance. In Eastern
tradition, artists make deliberate attempts to break symmetry, considering perfect
symmetry to be too artificial and simplistic (Sinclair, 2009). And yet, people from both
cultures can appreciate the works of art from the other culture. In fact, appreciation of
both perspectives requires an understanding of symmetry. Both types of aesthetics yield
artistic creations which are considered beautiful by everyone, though perhaps for
42
different reasons. In a similar way, some mathematicians may value certain aspects of
aesthetics differently from other mathematicians. And yet, there seems to be a general
agreement that, for whatever reason, certain mathematical ideas are consistently
considered more beautiful than others. Certain aesthetic qualities, in mathematics as in
any domain, are universally recognized as aesthetic even though individuals‘ personal
taste may prefer one quality over another.
Perhaps more bothersome to the idea of the importance of mathematical aesthetics
is the fact that aesthetics seems to change over time. What may be beautiful to one
generation of mathematicians may seem trivial to another. Schattschneider (2006) notes
that the Pythagorean theorem was probably a wondrous thing when it was first
discovered, but of course today it is just a basic fact to be learned. Nevertheless, many
mathematical truths remain beautiful from one generation to the next, and even when
theorems begin to become too commonplace to be considered truly beautiful,
mathematicians can still often appreciate the beauty that previous generations of
mathematicians must have found in these now familiar theorems. Even though the
Pythagorean Theorem is now too common to be considered beautiful by most
mathematicians, certain surprising visual proofs of the theorem can still show us the
original beauty today (Schattschneider, 2006). Even today, geometry students often find
Bhaskara‘s proof of the Pythagorean Theorem beautiful when they first see it. (See
Figure 1 on page 22.) The writer and mathematician Lewis Carroll (1890) believed the
Pythagorean Theorem was still just as beautiful today as it was to the ancient Greeks.
The change in aesthetics that the mathematical community experiences across
generations is similar to that of individuals as their understanding of mathematics grows.
Kant found it natural that we grow tired of frequent exposures to the same beautiful
43
objects. Beauty requires a certain novelty. We are more likely to find new and surprising
theorems beautiful than old and familiar ones. Le Lionnais (1948) noted that ideas that
once seemed wonderful and mysterious to him became banal once their inner workings
were well understood. This will be important when we consider mathematical aesthetics
for education because it means students will be attracted to new and important ideas
rather than old, familiar ones.
I have explained how aesthetics may vary with time and culture and yet retain a
certain consistent objectivity. But we are still left with the argument that mathematical
aesthetics may be hopelessly subjective because it varies from person to person in a way
making it entirely inappropriate for the classroom. Just as aesthetics may differ from
person to person in other domains, different mathematicians may have somewhat
different criteria for mathematical beauty. Some aesthetic criteria, such as surprising
simplicity, are agreed upon by nearly all mathematicians. But differences do exist, and
mathematicians frequently disagree whether certain theorems are beautiful or not.
These differences are not arbitrary, however. They may, for example, depend on
the mathematician‘s field of interest (Wells, 1990). They may also indicate important
distinctions in what exactly a mathematician is looking for or considers significant. Le
Lionnais (1948) distinguished two types of mathematics, the classical and the romantic.
The former is the mathematics of equilibrium, harmony, and order in mathematics.
Classical mathematics tries to identify general rules that are true globally. Its beauty lies
in surprising order where we expected to find only chaos. What could be more beautiful
than the surprising unity expressed by Euler‘s simple identity ( ) which brings
together the five most important constants in mathematics without any superfluous
44
numbers?8 Romantic mathematics, on the other hand, is the mathematics of irregularities,
paradoxes, anomalies, and even ―monstrosities.‖ It is emotional and non-conformist.
Such beauty includes the bizarre and the unusual, such as one often finds with initial
developments like set theory9 or the upsetting discovery by Gödel that a logical
foundation for mathematics could not be both complete and consistent. Romantic
mathematics, such as the discovery of non-Euclidean geometries, upsets the established
order. Classical mathematics, such as Klein‘s Erlangen Program, restores order. Both are
beautiful. The fact that two such different aesthetics should exist is no more surprising
than the fact that such differences exist in all the arts. And just as different artists have
different styles, so might some mathematicians be more drawn to certain types of
mathematical aesthetics rather than others.
What is important to realize is that both types of mathematics—the classical and
the romantic—are needed, so both of these very different aesthetics are useful for
different purposes. The classical aesthetic seeks to find the general rules that bring order
and harmony to mathematics, such as various general theorems about continuous
functions. The romantic aesthetic leads mathematics forward by blazing trails into new
territory and by finding the limits of classical mathematics, such as Riemann and
Weierstrass‘s discovery that there exist continuous functions which are nowhere
differentiable, a fact that greatly troubled and repulsed 19th
century mathematicians.
8 On the one hand, Le Lionnais (1948) noted that Euler‘s formula had lost most of its original beauty
among mathematicians because it is now so well understood. The overly familiar no longer attracts us as it
once did. On the other hand, a more recent poll of mathematicians (Wells, 1990) found that mathematicians
still rank Euler‘s formula as the most beautiful result in mathematics, even when expressed in the less
aesthetic form . 9 According to Le Lionnais (1948), set theory was initially accused of being ―un insolent défi au bons
sens,‖ an insolent challenge to common sense.
45
Romantic ideas sometimes take time, but are often eventually accepted by the
expert community. Most mathematicians today find continuous, non-differentiable
functions fascinating and beautiful. We have no trouble today appreciating the aesthetic
behind them. There are many examples in mathematics of the romantic aesthetic, often
labeled according to their initially repellant nature: the ―monster‖ group; ―pathological‖
functions; ―irrational,‖ ―complex,‖ and ―transcendental‖ numbers; and so forth. Le
Lionnais (1948) notes that it is significant that mathematicians named these objects
according to their emotional aesthetic, rather than some term more clearly linked to their
definitions. All of these fascinating discoveries represent important advances in
mathematics.
This leads us to the conclusion that, just as in other aesthetic domains, in
mathematics there exist different types of aesthetics according to the goals that one is
pursuing. When aesthetic criteria change, they often do so in order to serve different
purposes. Different aesthetics reveal different truths and guide in different ways. In some
ways, conflicting aesthetics is like cognitive conflict—it may reveal important ideas that
have not yet been assimilated by the mathematics community. It is therefore wrong to
suppose that conflicting aesthetics means that aesthetics is completely subjective and not
useful. Instead, we conclude that different aesthetics have different purposes. Most
importantly, we see that aesthetics changes according to what we are capable of
appreciating. Aesthetics is tied to our cognitive comprehension.
Importance of mathematical aesthetics. I conclude that aesthetics is essential in
mathematics and varies according to purpose. We have already seen that aesthetics is
essential for laying the foundations of mathematics, as well as for guiding
mathematicians in order to find interesting theorems and promising paths towards their
46
proofs. Aesthetics is even foundational to the reasoning process itself. Aesthetics also
seems to be fundamental in mathematicians‘ evaluations of results. Such aesthetic
evaluation is important because it leads to mathematics that is more compelling and
therefore more likely to be free of error.
Some aesthetic criteria seem to be universal, while other aesthetic criteria may
vary from person to person. But different aesthetics serve different needs and all lead to
important results. Certain mathematical criteria are nearly universally valued by
mathematicians, though individual mathematicians may rank these criteria differently. I
maintain that this sense of aesthetics is needed not only by mathematicians, but by
anyone doing true mathematics, including children. Le Lionnais did not hesitate to speak
of his fascination with mathematical beauty as a young student. It is especially important
to note that the things he found beautiful as a student were precisely those things that he
was beginning to learn and had not yet mastered. When a mathematical idea becomes
well known and commonplace, it is no longer as beautiful. Mathematical aesthetics draws
us towards those ideas that are fresh and new and still a bit mysterious. Therefore I claim
that mathematical aesthetics are actually of importance in guiding and driving children to
those aspects of mathematics that are most important for them.
Mathematical Aesthetics for Education
I have made the argument that mathematical aesthetics attracts mathematicians
and children alike towards ideas that are important. Aesthetic criteria such as significance
and connectedness apply not only to what is beautiful but also to the identification of
important and generative results. Furthermore, when a person has sufficient knowledge to
grasp a concept, but not enough to fully understand its working, the concept may appear
beautiful—surprising and delightful. As such, aesthetics often serves as a natural
47
motivation to work with others in our zone of proximal development, enticing us to
contemplate facts and methods that we only half understand. These are the reasons I have
argued mathematical aesthetics is important and should have a place in education. In this
section I now concentrate on the reasons promoted by other researchers as I look at the
role of mathematical aesthetics in education.
In spite of its clear importance to mathematics, most mathematicians have not
given much thought to aesthetics, and educators have given even less thought to the place
of aesthetics in mathematics education. The National Council of Teachers of
Mathematics (NCTM, 2000) has called for a greater appreciation of aesthetics in
mathematics instruction, but few curricula have given aesthetics more than a cursory
glance. There is almost no place for aesthetics in most school mathematics curricula, nor
is aesthetics taken into consideration by teachers10 except in rare artistic activities.
Seymour Papert wrote, ―If mathematical aesthetics gets any attention in the schools, it is
as an epiphenomenon, an icing on the mathematical cake, rather than as the driving force
which makes mathematical thinking function‖ (1980, p. 192). Nevertheless, some
educators, going back to John Dewey, have seen the importance of aesthetics for
education in general and have argued for its inclusion in the curriculum, though their
arguments have largely gone unheard by the general public. Other researchers who have
written recently of the importance of mathematical aesthetics in education include
Reuben Hersh, Vera John-Steiner, André Mack, and Rena Upitis (Hersh & John-Steiner,
2011; Mack, 2006, 2007; Upitis, Phillips, & Higginson, 1997).
In recent years, Nathalie Sinclair has researched mathematical aesthetics for
education extensively. She has argued that children have a natural attraction to
10 At least, they do not consider aesthetics consciously. Sinclair (2008) has shown that aesthetics is often
implicit in good teaching.
48
mathematical beauty. Many of her arguments are laid out in Mathematics and Beauty:
Aesthetic Approaches to Teaching Children (2006). As Dewey argued for all academic
domains, Sinclair contends that Western society has introduced a false dichotomy
between mathematics, which is seen as the domain of logic, and art, which is seen as the
domain of beauty. (This false conception is perhaps at the root of many students‘ dislike
of mathematics.) All people have a sense of aesthetics, probably because of an innate
desire to impose order on their perception of the world. As such, it is only natural to
suppose that aesthetics, this innate ability to see order in our experience, would guide all
people who do mathematics, which we have seen to be the study of pattern and structure.
Seen in this way, aesthetics gives us the ability to do mathematics. Without aesthetics,
our ability to do mathematics is limited to that of a mindless machine programmed to
calculate without appreciating what is important. As mathematicians such as Poincaré
argued a century ago, true mathematics cannot be done in this way.
Most curricula include justification and reasoning as important mathematical
skills to be learned. In this area we seek to teach students to find logic that is compelling.
As Krull (1930/1987) noted, such compelling logic will naturally appeal to our aesthetic
sense. As we have seen, most mathematicians would agree that it is rarely sufficient to
find a proof that is correct. We need mathematics to make us say, ―Oh, of course that
must be right.‖ A good sense of aesthetics is needed to appreciate a good argument in
mathematics just as much as it is needed to craft a good argument in, say, literature or
philosophy. Studies have found that many American students do not learn to reason in
proof-oriented geometry courses, probably at least partly because they are taught to do
two-column proofs by rote rather than learning to appreciate a good argument (Mayberry,
1981; Usiskin, 1982).
49
Sinclair (2006) shows through several examples that it is not just mathematically
inclined people that are attracted to the beauty of mathematics, but all children. Children
have a natural desire to put things in order and to create patterns and this should be
encouraged. In order for students to make the proper connections, they must have an
expectation of regularity. Otherwise learning dissolves into a mess of unrelated facts. The
student should always desire to integrate everything into a pleasing whole.
Sinclair proposed three primary roles of mathematical aesthetics: the motivational
role, the generative role, and the evaluative role. These roles are true for everyone who
does mathematics, whether mathematician or schoolchild.
Motivational role. Aesthetics motivates us to do mathematics. Indeed, we have
seen that the aesthetic appeal of mathematical patterns is often what drives the initial
investigation of mathematics. We choose problems because they are aesthetically
appealing. One powerful aesthetic motivator, especially for children, is symmetry.
Sinclair (2006) gives the example of Casey, a student who was struggling with a problem.
Instead of asking the child to consider the type of heuristics questions that Pólya
recommends we use, Sinclair helped the child focus on the symmetry of the problem.
Once Casey grasped the aesthetics of the problem, she would not let go again until she
had a satisfactory solution. Likewise, I found that children in this study tried to introduce
symmetry into the tasks I gave them. Such aesthetic aspects of a mathematical task give
strong intrinsic motivation.
Aesthetics gives us a sense of enjoyment when we do mathematics. This element
of enjoyment is important. Mann (2006) has noticed that most future elementary school
teachers‘ most memorable childhood mathematics experience was unpleasant. This
removes any intrinsic motivation to do mathematics. Clearly children will progress much
50
further if they have a sense of enjoyment when doing mathematics. Children generally
like math in the early grades. Hersh and John-Steiner (2011) noted that attitudes polarized
after fourth grade with about 40% saying they hate math, perhaps because of the way
mathematics is taught. The harm we do children by training children to detest
mathematics is incalculable. It is all the more inexcusable when we realize that aesthetics
is not an artificial gimmick added to make math more attractive, but a genuine,
overlooked aspect of the foundations of mathematics.
Aesthetics dictates how mathematicians choose the problems they work on and
this choice is what motivates them to pursue a solution. Unfortunately, students usually
cannot choose the problems they work on. Problems are usually imposed by the teacher,
or else the curriculum may not be covered. Clearly this removes a large motivational
factor. In order to overcome this problem, educators often try to make exercises more
appealing, either by turning them into ―real world‖ problems, or else by adding fun
activities that are exterior to the problem. The clear message that children get is that
mathematics is unappealing and needs sugar coating. Instead of such artificial motivators,
it may be preferable to realize that children have a natural attraction to the exploration of
pure mathematics.
Generative role. One of the most important roles of mathematical aesthetics is
the generative role. This is the role that Poincaré explored. Directions of mathematical
thinking that are likely to be productive have aesthetic appeal. When mathematicians
look for a solution to a problem, their belief that the solution should be beautiful leads
them to explore directions with aesthetic appeal. This generative role is just as important
for children as for mathematicians. When children are given an open-ended task, they
tend to examine possibilities that they find appealing. I believe that, more often than not,
51
these turn out to be the same possibilities which lead to mathematically correct solutions.
This study supports this belief. I show, for example, that when children incorporated
aesthetic symmetry into their solutions, they were more likely to find a mathematically
valid solution and to be confident of their answer.
There is very little empirical research in the area of mathematical aesthetics for
children, though sometimes other research touches on this area. Lehrer et al. (1998)
describe a rich five-week second grade geometry activity based on a study of quilt
patterns. Children frequently found the quilts to be ―cool.‖ The authors note that at the
beginning of the unit, ―cool‖ was associated with colors. Later in the unit, talk of ―cool‖
quilts included aspects of symmetry, shape complexity, transformations, and the aesthetic
constraints of certain pattern choices. In other words, as children grew familiar with the
mathematical aspects of their unit on quilts, their aesthetics focused more on
mathematical structure. Again we see a connection between the aesthetic and the
cognitive in mathematics.
Sinclair notes that the generative role is usually harder to observe than the other
two roles because it takes time to manifest. Aesthetics does not typically lead to
immediate flashes of insight. The process is slower and may not be as easy to observe as
the motivational or evaluative role.
Evaluative role. We have noted that mathematicians are frequently unsatisfied
with perfectly correct proofs for aesthetic reasons alone. A sense of aesthetics pushes
mathematicians to seek what they consider to be ―cleaner,‖ more elegant results. A sense
of aesthetics can also help students to evaluate the quality of their solutions. Sinclair
(2006) notes an example of an open-ended sorting exercise where one child, Zoe,
defended her choice as ―better‖ by pointing out that the choice of another group of
52
students was too ―simple.‖ Zoe‘s choice revealed something deeper and less obvious
(symmetry again) about the shapes that were being sorted, compared to the simple side
counting that the other group was engaged in. Zoe‘s aesthetic sense led her to evaluate
her sorting as better and more mathematically interesting.
Unfortunately, many people believe that school mathematics is a finished product.
Students learn what they need to learn as efficiently as possible so that they can give what
is considered to be the single correct answer to each problem. Such mathematics has no
need to be evaluated. The answer is either right or wrong. Without a better understanding
of the place for open-ended inquiry in mathematics, and of the need to evaluate solutions
to such problems, there is little hope that aesthetics will find a place in the current
American curriculum.
In fact, much of this discussion of mathematical aesthetics is of little use for
classrooms where students cannot choose either the problems they will solve or the
methods they will use to solve them. If students are given the methods they must use, and
a system of external rewards and punishments to motivate them to do the problems, then
the aesthetic will play little or no role in their work. This type of mathematics instruction
bears as much resemblance to real mathematics as multiple choice grammar exercises
resemble creative writing. However, it is a reality that some classrooms are like this, and
for those classrooms mathematical aesthetics will be of limited use. In this study I am
assuming that students have at least some opportunity to do authentic mathematics.
Resistance to incorporating mathematical aesthetics in education. Sinclair
(2009) has pointed out that many see the consideration of aesthetics for mathematics
education as either frivolous or elitist. Those who see it as frivolous either wrongly
53
associate aesthetics with style and fashion or else believe aesthetics to be peripheral to
mathematics rather than the foundation on which mathematics is built.
The view that mathematical aesthetics for education is elitist comes from a
Platonic view that aesthetics is an inherent property of mathematical objects, independent
of culture (Sinclair, 2009). As such, it is believed that aesthetics can be properly
evaluated only by mathematics experts.11 Students do not always share the same aesthetic
appreciation for solutions as mathematicians; such evaluation is therefore believed to be
beyond children‘s abilities. Many see the purpose of mathematics education to be the
learning of skills needed for life and other coursework—the mere application of what
mathematicians have developed. Aesthetics, it is argued, is at best a concern for
mathematicians alone.
Children‘s mathematical aesthetics is indeed somewhat different from that of
experts, but appears to serve their needs better than expert aesthetics would because it
highlights the aspects of mathematical learning that are still new to the child and not
completely understood. For example, whereas mathematicians seek elegant solutions,
many students actually prefer messy solutions that encapsulate the history of their
struggle to find a solution. They seem to be more interested in telling the story of how
they solved the problem than in erasing those traces in order to present a ―clean‖ solution.
Sinclair (2006) recommends allowing such students to pursue the kind of mathematics
they find aesthetically pleasing, rather than imposing expert aesthetics that they may not
be ready for. The process of doing mathematics is an important aspect of mathematics
11 Certain mathematicians, such as Hardy and Poincaré, are unfortunately also partly to blame, as Sinclair
also points out. They tended to believe that mathematicians had special ability and that others could not
fully understand and do significant mathematics.
54
that students are still learning. It is therefore natural, instructive, and important that their
aesthetics should include telling the story of their mathematical discoveries.
Aesthetics is therefore just as important for children, at least when they are
involved in genuine mathematical inquiry. Many researchers (e.g. Hiebert et al., 1996)
have emphasized the importance of incorporating inquiry in mathematics education.
Brown, Collins, and Duguid (1989) have argued that learning should be situated in an
authentic context. But if students are to experience such an authentic cognitive
apprenticeship, their learning must include genuine inquiry guided, not by heuristics, but
by aesthetic criteria. An authentic cognitive apprenticeship in the domain of pure
mathematics should involve the exploration of abstract mathematical problems.
Researchers from diverse perspectives have emphasized that it is an error to attempt to
situate all learning in ―real-world‖ contexts (Cobb & Bowers, 1999). It is wrong to
believe that pure mathematics is somehow ―context-free,‖ in need of an artificial
situation. Pure mathematics is as valid a context as applied mathematics and both pure
and applied mathematics should be used appropriately in instruction. In the case of pure
mathematics, it must be recognized that aesthetics is a powerful and important driving
force. One of the goals of this research is to investigate the role of aesthetics in children‘s
mathematical thinking when they are engaged in open-ended tessellation tasks.
Sinclair (2008) notes that enculturation into mathematical aesthetics is in fact
already taking place implicitly in many classrooms. When teachers point out a ―secret
weapon,‖ a surprising fact, a discussion of mathematical style, or a humorous numerical
example, they are really exploiting their students‘ sense of mathematical aesthetics.
Children‘s aesthetics could be more finely developed if teachers gave more explicit
guidance in the aesthetics of mathematics. By becoming more aware of the ways
55
aesthetics guides thinking, we can become better able to give explicit guidance to
students on the aesthetic aspects of mathematics.
Connection to the Cognitive Perspective
In this discussion of mathematical aesthetics we have seen the tight relationship
between the aesthetic and the cognitive in several ways.
All cognition is rooted in the aesthetic. Thinking begins with aesthetic cognition
and is reasoned about and communicated through verbal logic.
Aesthetic appreciation is developed through increased cognitive understanding.
The criteria that mathematicians have identified for describing mathematical
aesthetics are seen to guide towards cognitive understanding.
Aesthetics guides mathematicians and children alike towards the most generative
cognitive ideas. The aesthetic guides our cognitive perception and the cognitive
informs our aesthetic perception.
Aesthetics adapts to our cognitive level. We tend to find most beautiful those
things that we do not yet fully understand. Mathematical aesthetics draws us into
our zone of proximal development as we work with others to understand these
new ideas.
We can analyze any open-ended mathematical task from either viewpoint. The
cognitive viewpoint looks at how students understand a concept from a logical point of
view—their strategies, reasoning, and conceptual understanding. The aesthetic viewpoint
looks at what students find appealing—their choices, preferences, and aesthetic criteria.
These two viewpoints are closely related and may at times be hard to distinguish,
especially when dealing with non-verbal thinking. Doris Schattschneider (1978a) notes
that in the case of tessellations, artistic (aesthetic) symmetry and mathematical
56
(cognitive) symmetry are the same. Does someone create a symmetric pattern because it
is understood how it will help make the tiling easier (a cognitive reason) or simply
because the symmetry is more pleasing (an aesthetic reason)? And to what extent does
such a distinction matter? This research is at the early exploratory stage of trying to
understand children‘s thinking about tessellations, so I am not concerned about overly
fine distinctions. However, I analyze both students‘ cognitive reasoning and their
aesthetic values for tessellations in order to gain as complete an understanding as possible
of children‘s thinking.
Research design. I note at this point what the close relationship of the cognitive
and the aesthetic means for the design of this research. It is natural that I, the researcher,
should have a mathematical aesthetic that is different from the aesthetic of the student
participants because of my different cognitive understanding. My aesthetic is drawn from
my understanding of the mathematics of tessellations, which is naturally much closer to
the canonical understanding of the mathematics research community than fourth graders‘
understanding would be. This aesthetic unavoidably influenced the design of this
exploratory study. Because I am interested in exploring children‘s understanding of
tessellations in relationship to the standard mathematical theory of tessellations and what
I believe children should study in schools, the tasks were designed to explore those
elements that are valued by the mathematical community. This does not mean that
children‘s aesthetic values were overlooked in the design of the tasks. All tasks were
designed so that children‘s aesthetic values could be analyzed. Some tasks were designed
to explore specifically the aesthetic values children had expressed during the pilot study,
as well as aesthetic values typically valued by mathematicians. The type of tessellation
that the task allowed necessarily imposed certain aesthetic constraints. However, this did
57
not prevent the children from expressing their aesthetic values. For example, none of the
tasks allowed children to choose the color of the tiles. Nevertheless, the analysis revealed
that color was one of the most commonly discussed aesthetic values.
Summary of the Aesthetic Perspective
Pure mathematics, which is best defined as the study of pattern and structure, is
entirely an aesthetic subject and, like the arts, it has been pursued for its own sake in
Western civilization from the Ancient Greeks to modern mathematicians. The aesthetic
appeal of patterns drives the study of mathematics. The foundations of mathematics are
partly determined by aesthetic criteria; the structure of mathematics is explored and built
up guided by aesthetic forces; and mathematical results are evaluated by aesthetic
criteria. This aesthetic epistemology became clear beginning in the late 19th
century with
the modernist transformation and the arguments of mathematicians, such as Poincaré,
who pointed out that aesthetics is part of all mathematical work. Rigor and logic alone are
insufficient for establishing and developing mathematics.
Many researchers believe that cognition takes place primarily at an aesthetic level.
Such understanding is translated into verbal expression for communication and logical
reasoning, but this verbal expression is not the understanding itself. Authentic
mathematics requires reasoning at both the verbal-symbolic and aesthetic levels.
Although a clear theory of mathematical aesthetics has not been developed in
modern times, mathematicians have described it using criteria of significance, surprise,
simplicity, connectedness, and visual appeal. The fact that these criteria also point to
what is important and generative in mathematics suggests a deep connection between the
aesthetic and the cognitive. It seems likely that an innate desire to impose order on our
perception of the world is at least partly responsible for the aesthetics that drives our
58
understanding of mathematical patterns and structure. Mathematical aesthetics drives
cognitive understanding and cognitive understanding develops our mathematical
aesthetics.
For mathematics education, Sinclair (2006) has proposed viewing aesthetics
through its three primary roles of motivation, generativity, and evaluation.
Mathematical aesthetics, like all beauty, is in some sense objective. There is some
agreement among mathematicians as to what is beautiful in mathematics and what some
of the criteria for this beauty are. However, mathematical aesthetics also has subjective
aspects and mathematicians may disagree on details. Mathematicians generally agree on
the criteria for a mathematical aesthetic, though they may value and apply these criteria
differently. Two important ways that mathematical aesthetics varies are in how aesthetics
adapts to our purpose and to our understanding. For example, Le Lionnais (1948) noted
that when our purpose is to confirm and build up existing mathematical structure, our
aesthetic tends in that direction (the ―classical‖); but when our purpose is to explore the
edges and push the limits of our knowledge, our aesthetic (the ―romantic‖) may value the
seemingly bizarre. Likewise, children may have different purposes from mathematicians
in their pursuit of mathematics, and their aesthetic adapts accordingly.
More importantly, aesthetics adapts to our understanding. For an idea to be
beautiful, we must understand it enough to appreciate it without understanding it too
deeply. Ideas that are common and well understood no longer seem as beautiful as they
once did, either to individuals or to the mathematical community. The fact that the most
beautiful ideas are those whose inner secrets we do not yet completely understand makes
mathematical aesthetics an ideal driving force, guiding us to study what is most
59
important. This is true for anyone doing mathematics, whether mathematician or
kindergartner.
Criticism that mathematical aesthetics is frivolous can be answered by pointing
out its foundational role in modern mathematics. Criticism that it is elitist can be
answered by noting how it serves children as well as mathematicians.
Because aesthetics is integral to mathematics at its very foundations, it should be
valued from a child‘s earliest years. Mathematics that relies on heuristics alone, with
artificial contexts for motivation, lacks the genuineness that an authentic apprenticeship
demands. Aesthetics is an essential foundation for any true mathematics education.
The cognitive and the aesthetic are deeply related in many ways. Both the
cognitive and the aesthetic viewpoints are needed for a complete understanding of
children‘s mathematical thinking.
RESEARCH RELATED TO THIS STUDY
In this second section of the literature review I summarize some of the research
that has been done directly relating to children‘s understanding of tessellations. I focus on
research conducted in the field of mathematics education, but first I start by explaining
certain aspects of the mathematical theory of tessellations, as well as introducing some
vocabulary and notation. After this, I summarize the scant research that has already been
done on children‘s understanding of tessellations. Finally, I look at the research that has
been done on children‘s understanding of related concepts, namely infinite space,
symmetry, and transformations.
Mathematical Theory of Tessellations
Tessellations have been explored as an art form since prehistoric times in all
cultures and are also found in nature (Grünbaum & Shephard, 1986). They have been
60
used extensively in art and architecture in every major civilization since antiquity. And
yet no mathematical study was made of them until Kepler‘s Harmonices Mundi
(1619/1997), which was ignored by mathematicians until the early 20th
century. After
Kepler‘s 17th
century work, no mathematical studies on tessellations were published until
the late 19th
century (Grünbaum & Shephard, 1986). Many problems in tessellations
remain unsolved. Their study has proven especially important in modern times for the
field of crystallography, but also for engineering, biology, geology, physics, computer
science, metallurgy, communication theory, and other scientific domains.
In order to be clear about the subject I am exploring, I briefly describe the theory
of tessellations as it is currently understood in the mathematical literature. I focus on
those elements that are important for mathematical interpretations of the tilings students
could make and evaluate in the tasks they were given in this study. It is important to be
clear about the terminology, especially because there is some variation in the literature. I
give a summary of vocabulary terms12 associated with tessellations at the end of this
section on page 85. The mathematical theory will also form the basis for the design of the
tasks.
Definitions
In the most general definition, a tessellation, or tiling, is a complete covering of a
space by a collection of non-overlapping, closed sets called tiles. By ―complete,‖ we
mean there are no gaps—the union of the tiles is the entire space. (I.e. the tiling is a
mathematical covering.) By ―non-overlapping‖ we mean that the interior of the each tile
is not shared with the interior of any other tile (the interiors are pairwise disjoint), though
12 This terminology is solely for the purposes of analysis and clear communication in this dissertation.
Simple everyday terms were used with the children and most of the concepts here were not shared with
them at all.
61
of course points on the boundaries of the tiles will belong to two or more tiles. (I.e. the
tiling is a mathematical packing.) In general tiling theory, the tiling can be embedded in
any geometric space and the tiles can be any shape at all, including unusual shapes that
are not connected or bounded. But for the purposes of this study, I limit my discussion to
tessellations created by a countable collection of closed topological disks13 in the
Euclidean plane. Each tile will usually be a filled polygon (a polygon together with its
interior), though in the case of artwork, such as Escher tilings, the tiles may have curved
edges.
The edges of the tessellation are made up of the pairwise intersections of the tiles.
A point where three or more tiles come together is called a vertex.
We will almost always require the shapes of the tessellation tiles T1, T2, … to be
chosen from a finite set P of possible shapes (prototiles) P1, P2, …, Pn called the
generating set. In other words, a generating set of n prototiles P = {P1, P2, …, Pn} will be
chosen and the tessellation will be constructed in such a way that each tile Ti is congruent
to one of the prototiles Pj from the generating set. If all the tiles in a tessellation are
congruent to a single shape, i.e. n = 1, then we say that the shape P1 can tessellate or tile
the plane, and the tessellation itself is said to be monohedral. Children understand
tessellations easily in reference to floor tilings, which are usually monohedral, or at most
dihedral (n = 2). They understand quickly that tilings should not have gaps or overlaps
between the tiles. There is no general method for deciding if a given generating set can
actually make a tessellation. In general, you cannot look at a prototile or set of prototiles
and know if they can tessellate without actually trying. Even if a shape can tile an
13 A topological disk is basically a closed and bounded set whose boundary is a simple closed curve. We
therefore exclude unbounded tiles, tiles with holes, tiles with several parts connected only at points, tiles
with empty interior, and other complex shapes. The edges of such simple tiles always lie on the boundary
of the tile.
62
arbitrarily large area, there is no general test to know if it can continue to tile the whole
plane. There are therefore many open problems in tiling theory (Schattschneider &
Senechal, 2004).
Symmetry and Transformations
Most interesting tessellations are symmetric. Symmetry of a tessellation is
described in terms of isometries. An isometry is a rigid transformation of the plane that
preserves shape and distance. If the isometry maps the tessellation back onto itself in
such a way that each tile is mapped onto a tile, then the isometry is called a symmetry of
the tessellation. In other words, after applying the isometry the tessellation looks exactly
as if it had not been transformed at all, though actually each tile may be in a new location.
The tiles have all ―switched places,‖ so to speak, while still preserving their relationships.
There are only four types of isometries in the Euclidean plane: translations,
rotations, reflections, and glide reflections.14 Every isometry is one of these four types.
All four of these can be seen in the example in Figure 4 below. (Of course, in all figures
the tiling must be imagined to continue indefinitely in all directions throughout the entire
plane with the same pattern. These figures are only finite portions of tessellations.) For
example, from the point O, the tessellation can be translated up one tile, or to the right
one tile. There are several possible reflection symmetries, all horizontal or vertical. The
14 Glide reflections can be decomposed into a ―glide‖ (translation) and a reflection across a line parallel to
the direction of the glide. Psychologically, glide reflections seem more complex than the other three kinds
of isometries, but mathematically, no isometry is more complex than another. A glide reflection is
considered a single transformation. Any isometry, not just a glide reflection, can be decomposed into two
other types of isometries and the composition of any two isometries is always one of these four isometries.
However, glide reflections are hard to visualize as a single movement and some textbooks omit them. They
are rarely included in elementary curricula. Glide reflections are psychologically more complex than
translations, reflections, and rotations. At the very least, they probably require Piaget‘s transfigural stage,
unless they are conceived as the separate applications of a translation followed by a reflection. It is not
expected that fourth graders be able to conceptualize glide reflections. However, patterns with glide
reflection symmetry are visually appealing and symmetric, and children may still notice a symmetry that
they cannot analyze or describe.
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tessellation could be reflected across line l or line m, for example. There are 180°
rotations about hexagon centers (such as point A) or the centers of any two touching
triangles (such as point B or C). A glide reflection can be created in Figure 4 by
combining a horizontal or vertical translation with a horizontal or vertical reflection
respectively. There are, of course, infinitely many symmetries in this tessellation.
Figure 4: Some symmetries of a tessellation
In each of these symmetries, one must imagine the entire plane being mapped to
itself in such a way that every hexagon moves to a hexagon and every triangle moves to a
triangle. When we say that the tiling is translated up, we mean that every tile is moved up
the same distance in the same direction. This idea is sometimes difficult for students to
grasp because they focus on the movement of a single tile rather than the entire plane.
Some mathematicians (e.g. Grünbaum & Shephard, 1986) have found it helpful to
imagine the tessellation copied onto an infinite transparency.15 The transparency is then
15 Laurie Edwards (2003) points out that a transformation is not really a motion and therefore it is wrong
for students to conceive them with the transparency model, especially because it leads to errors. She sees
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moved, rotated, flipped, or glide reflected to a new position that matches the original
tessellation.
We sometimes speak of one other type of symmetry that is not an isometry, but
which is nevertheless visually powerful and useful for creating and analyzing some
tessellations. Dilation symmetry (or scale symmetry) exists when a tessellation can be
mapped onto itself with a dilation (possibly in combination with an isometry). The
tessellation is shrunk or expanded in such a way that single tiles are not mapped onto
single tiles, but rather a single tile is mapped onto a finite collection of tiles (the tiling
expands), or vice versa (the tiling shrinks). This creates self-similarity such as one finds
in fractals or in certain complex tessellations like the Pinwheel Pattern (see Burger &
Starbird, 2005, p. 254 ff). Dilation symmetry also exists in simple tessellations such as
the rhombus tessellation in Figure 5 below. One of the rhombi has been highlighted, as
have its images after dilating (expanding) by a factor of 3 and a factor of 5. The dilated
images are mapped onto sets of 32 = 9 and 5
2 = 25 rhombi. Some form of dilation
symmetry was used by children in this study as a strategy for creating some of their
tessellations. A more complex example of dilation symmetry is in Figure 6 below.
the perceived motion as a remnant of our embodied understanding, in contradiction to a rigorous definition
of transformation as a function. However, mathematicians such as Grünbaum & Shephard (1986)
encourage such visualizations, as long as we keep in mind that this is only an aid and that we really only
care about the final position, and not the motion that got us there. The errors Edwards identifies have to do
with visualizing the transformation of only the object, rather than the entire plane as Grünbaum &
Shephard‘s model encourages us to do.
65
Figure 5: Rhombus tiling showing image of rhombus after dilating by factors of 3 and 5
Figure 6: An example of a tiling with dilation symmetry
Some tessellations are isogonal (vertex-transitive), isotoxal (edge-transitive), or
isohedral (tile-transitive). By this we mean that we can map any vertex, edge, or tile to
any other vertex, edge, or tile by some isometry which preserves the entire tessellation. In
less formal terms, all the vertices of an isogonal tiling are the ―same.‖ An ant standing on
any vertex of an isogonal tiling could not know which vertex it is standing on because
they all look exactly alike—the surrounding tessellation looks the same no matter which
vertex the ant stands on. (The same is true for any edge in an isotoxal tiling or any tile in
66
an isohedral tiling.) This is not the case for the non-isogonal tessellation in Figure 4 on
page 63 where there are two types of vertices: one where hexagons and triangles alternate
around the vertex (e.g. vertex B) and one where they do not (e.g. vertex D). An ant
standing on B would know by the pattern of surrounding edges and tiles that it is not
standing on D. There is no isometry that could map this tessellation onto itself in such a
way that vertex B would be mapped to vertex D. The tiling in Figure 4 is not isogonal;
however we can say that it is 2-isogonal, meaning it has precisely two types of vertices—
the kind at B and the kind at D. In general, tessellations with k types of vertices are said
to be k-isogonal. If a k-isogonal tessellation is made of regular polygons, it is called k-
uniform. In a similar way, we can speak of k-isohedral and k-isotoxal tilings. The tiling in
Figure 4 is 2-isogonal, 2-uniform, 3-isotoxal, and 2-isohedral.
A tiling that is isohedral is necessarily monohedral, but not vice versa. These
terms are easily confused. ―Monohedral‖ simply means that only one type of tile is used
to create the tessellation; all the tiles are congruent. ―Isohedral‖ means that you can map
any tile to any other tile using an isometry which preserves the tessellation. The regular
square tessellation is isohedral; any square can be mapped by a symmetry to any other
square. The tiling is Figure 4 is not monohedral (it is dihedral), so it cannot be isohedral.
The tiling in Figure 7 below is monohedral, but not isohedral. All the tiles are congruent,
but there does not exist an isometry of the tessellation that maps, say, one of the central
rectangles to one of the horizontally oriented tiles. Similarly, the tessellations in Figure
20 and Figure 21 on pages 79 and 80 are also monohedral but not isohedral.
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Figure 7: A monohedral tiling that is neither isohedral nor edge-to-edge
Polygonal Tilings
All of the tasks where students created tilings in this study involved tilings of
polygons where the sides are straight line segments. For the rest of this section, I focus on
tilings of polygons. We call the edges of polygons sides and the vertices of polygons
corners. This is in order to avoid confusion with the terms ―edge‖ and ―vertex‖ as applied
to tilings. Usually the polygon sides will be the same as the tiling edges and the polygon
corners will be the same as the tiling vertices, but this is not always the case. When they
are the same, we say that the polygonal tessellation is edge-to-edge. It is often assumed
that tilings are to be created in an edge-to-edge manner, but unless specifically stated, this
is not a requirement. Figure 7 above and Figure 8 below are examples of tilings that are
not edge-to-edge. Note that the ―brick wall‖ tessellation in Figure 8 could be considered
edge-to-edge if the ―bricks‖ were considered hexagons with two straight-angled corners.
(See Figure 9 below for the six edges of the orange brick.) I do not consider the bricks as
hexagons because such interpretations are very unlikely among children, but one
mathematician interviewed in this study did make such observations. Some tilings, such
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as Figure 10 below cannot be construed as edge-to-edge no matter how you consider the
sides of the polygons; the ―zigzag‖ edges between pairs of tiles are made of at least three
sides each.
Figure 8: The ―brick wall‖ tiling of rectangles is not edge-to-edge.
Figure 9: Six edges of a brick
Figure 10: Another non-edge-to-edge tiling
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Regular Polygons
Certain types of tilings have special interest. Mathematicians, including Kepler,
have frequently focused on tilings created from regular polygons. For monohedral tilings
(where all the tiles are congruent), the only regular polygons that can be used to tessellate
the plane are the equilateral triangle, the square, and the regular hexagon. When the tiles
are placed edge-to-edge, there are just three possibilities, called the three regular
tessellations, which are illustrated in Figure 11 below.16 (As always, the patterns must be
imagined to continue indefinitely in all directions.) The regular tessellations are isogonal,
isotoxal, and isohedral.
Figure 11: Portions of the three regular tessellations
If we create a tessellation with more than one type of regular polygon, then there
are infinitely many possibilities. For example, we could make the hexagon and triangle
tessellation in Figure 4 on page 63. Or we could make the tiling on the left in Figure 12
below with one row of triangles between each row of squares. Or we could make the 2-
isogonal tiling on the right in Figure 12 with two rows of triangles between each row of
16 Two tilings are considered the same if there exists a similarity mapping from one to the other. In other
words, we do not say that rotating, flipping, or shrinking a tiling creates a different tiling. In this sense,
there is only one possible way of creating an edge-to-edge tiling with an equilateral triangle, a square, or a
regular hexagon. This is the accepted view in mathematics (Grünbaum & Shephard, 1986). Children and
non-mathematicians, on the other hand, may or may not consider various orientations or different sizes of a
tiling as being the same tiling.
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squares. Or we could make a similar tiling with any numbers of rows of triangles and
squares in any pattern we wanted, or no pattern at all. Tessellations such as the one on the
right in Figure 12 are neither isogonal, isotoxal, nor isohedral. The tessellation on the left
is isogonal (vertex-transitive), but not isotoxal or isohedral. (It is not isohedral because
there are two types of tiles: squares and triangles. It is not isotoxal because there are three
types of edges: square-square, square-triangle, and triangle-triangle).
Figure 12: Two tessellations with squares and triangles
Of special interest among tessellations using more than one type of regular
polygon are the eight semi-regular (or Archimedean) tessellations. These are isogonal
tessellations made of two or more regular polygons, such as the tiling at left in Figure 12.
The only other semi-regular tessellation made only of triangles and squares is in Figure
13 below. This tessellation is visually much more complex. However, one child in my
pilot research attempted to create it as her own idea, and nearly succeeded. The other six
possible semi-regular tessellations are in Figure 14 below.
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Figure 13: The 3.3.4.3.4 semi-regular tessellation
Figure 14: The six other semi-regular tessellations
At this point it is helpful to introduce some notation. When tilings are isogonal,
we can define them by describing the pattern of polygons around each vertex. If the tiles
are regular polygons, we can simply list the number of sides of each polygon in order
around the vertex. By convention, we start with the smallest polygons. The notation for
the regular triangle tessellation (Figure 11 on page 69) would be 3.3.3.3.3.3 or 36 because
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there are six triangles around each vertex. The regular square tessellation is 44 and the
regular hexagon tessellation is 63. The semi-regular tessellation on the left in Figure 12
above would be 33.4
2. The tessellation in Figure 13 has the notation 3.3.4.3.4 (or
32.4.3.4). For 2-isogonal tessellations, we list the polygons around each type of vertex.
The tessellation on the right in Figure 12 would be described as 36/3
3.4
2 because some
vertices are surrounded by six triangles and others by three triangles and two squares.
Periodicity and Symmetry
Up to this point we have mainly considered tessellations that are symmetric and
periodic. A periodic tessellation consists of a finite region which repeats indefinitely in
the same orientation throughout the plane. It can be shown that all periodic tessellations
have two linearly independent (non-parallel) translation symmetries. For example, the
regular 44 unit square tessellation has horizontal and vertical translation symmetry. We
can translate the unit square tessellation up one unit, or to the right one unit, in order to
map it onto itself. All other translation symmetries of the unit square tessellation can be
expressed as a linear combination of these two base translations. Every square in the
tiling is the image of any other square under one of these translation symmetries. It can
also be shown that if an edge-to-edge tiling has translation symmetry in one direction, the
tiles can be used to make a tiling with translation symmetry in two directions (Grünbaum
& Shephard, 1986). However, it is also quite easy to make tessellations which have
translation symmetry in only one direction, such as the example in Figure 7 on page 67,
which only has horizontal translation symmetry.
Periodic tessellations can be further analyzed by the various types of symmetry
they possess other than translation symmetry. We can classify the symmetry of a
tessellation by the number and order of distinct, independent rotations, reflections, and
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glide reflections that map the tessellation onto itself. The set of all possible
transformations form a group with respect to composition of the transformations. There
are a finite number of possible symmetry groups. It can be proved that there are exactly
seven possible symmetry groups with translation symmetry in only one direction. These
are known as frieze groups. There are also 17 possible periodic groups (known as
crystallographic groups or wallpaper groups) of planar patterns possessing two
independent translation symmetries. All repeating tessellations have the same basic
symmetries as one of these 24 possible patterns.
In this study, I worked with children who had only a basic notion of reflection and
translation symmetry, and perhaps a visual sense of other symmetries. (See the summary
of research on Children’s Understanding of Symmetry on page 97.) Instead of using a
complete symmetry analysis, it was sufficient merely to note that certain patterns have (1)
(simple) reflection symmetry, (2) rotation symmetry of order n, or (3) n-fold symmetry.17
This analysis can be applied equally to tessellations or to finite patterns. Further analysis
was not really needed for this study.18
(1) Simple reflection symmetry means that there is one line of reflection
symmetry and no rotation symmetry.
(2) Rotation symmetry of order n means that we must rotate the pattern at least
360°/n about a point of rotation symmetry to map the pattern onto itself.19 The semi-
regular tessellation on the left in Figure 15 below has rotation symmetry of order 6 about
17 Note that the simple fact that a tile is symmetric does not mean that the tiling will inherit the same
symmetry. For example, the X-pentomino has 2-fold reflection symmetry, but can only form a tiling with
simple rotation symmetry (Schattschneider, 2010). 18 Appendix G describes the one case where deeper mathematical analysis was perhaps needed. However
the results from that analysis were inconclusive. 19 Many mathematicians (e.g. Grünbaum & Shephard, 1986) use the term ―n-fold rotational symmetry,‖
but I will reserve the term ―n-fold‖ to indicate presence of reflections in order to make a greater distinction.
Many other ways of describing symmetry are also in common use.
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the center of any hexagon. A rotation of 60° about such a center will map the tessellation
onto itself. It also has rotation symmetries of order 3 and order 2 about certain other
points on the triangles. I generally only note the highest order of rotation symmetry when
discussing a tiling.
Figure 15: Two symmetric semi-regular tessellations
(3) A pattern has n-fold symmetry if there are n axes of reflection symmetry that
intersect through a common point. There are six axes of reflection that intersect each
vertex in the regular triangle tessellation, so we say it has 6-fold symmetry. (Other points
happen to have other orders of reflection symmetry, but six is the maximum.) A pattern
with n-fold (reflection) symmetry (n > 1) necessarily has rotation symmetry of order n as
well. This is because the combination of two non-parallel reflections makes a rotation.
Therefore I invent the convention of saying such patterns have n-fold symmetry without
further mention of reflections or rotations. The semi-regular tiling on the right in Figure
15 above has both horizontal and vertical20 reflection symmetry as well as two oblique
axes of reflection symmetry about the centers of each tile, so it also has order-4 rotation
symmetry about those centers. I simply say these tilings have 4-fold symmetry. Note,
20 Reflection symmetry is said to be horizontal if the axis of reflection is horizontal. Vertical reflection
symmetry has a vertical line of reflection.
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however, that rotation symmetry does not imply reflection symmetry. For example, the
tiling on the left in Figure 15 above has order-6 rotation symmetry, but no reflection
symmetry.
I use these simplified analyses for finite patterns as well as tessellations. When a
pattern is finite, the symmetry must map the border of the pattern onto itself. Under this
restriction, the finite pattern on the right in Figure 15 above has horizontal reflection
symmetry, but no other kind of symmetry.
Lack of Symmetry
Note that a tessellation does not have to have any symmetry at all.21 A tessellation
can even be non-periodic (lacking translation symmetry). A random pattern of polygons
all different from each other, as in Figure 16 below would be one such example, as long
as it is imagined to continue randomly forever in all directions. However, tessellations
can still have great structure and be built from a small set of prototiles without having any
symmetry. Such tessellations gained special interest in the 1960s and 70s when aperiodic
prototiles were discovered—any tessellation made with such prototiles was necessarily
non-periodic. The most famous of such tilings are Penrose tilings, discovered by Roger
Penrose in 1974. These tilings are created from a generating set of only two prototiles
which force mathematically and aesthetically interesting non-periodic tilings (Gardner,
1977).
21 We are, of course, not considering the identity mapping as a symmetry, though technically, from a
mathematical point of view, it is. If we did consider the identity mapping, we could say that a tiling must
have a symmetry other than the identity mapping to be considered symmetric.
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Figure 16: A random tessellation of polygons all different from each other
It is quite easy to make simple non-periodic tessellations even with regular
polygons. One of the simplest examples would be to lay down dominoes at random, half
horizontal and half vertical, aligned without gaps or overlaps, making a tessellation
without any pattern. As long as reasonable care is made not to leave square-shaped holes,
the tiling will lack symmetry. Another type of random tiling is Voronoi (or Dirichlet)
tilings, which have been extensively studied by researchers in recent years. There are
many non-periodic tessellations with triangles and squares.22 In fact, children in this
study sometimes attempted to make such tessellations and these tessellations also
frequently occur in nature (see Figure 17 below). Making such tessellations by placing
the tiles at random, as the children did, is tricky because it is possible to get ―caught‖ in a
position where the tiles no longer fit. This sometimes occurs in nature as well—such
natural tessellations usually contain gaps in the pattern. But if the squares and triangles
22 In fact, there are uncountably many tessellations such that every vertex is surrounded by two squares and
three triangles. A complete description of all such tessellations has not yet been made (Grünbaum &
Shephard, 1986).
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are placed with reasonable care, convex patterns can always be extended to the whole
plane and the children in the pilot study did not leave holes in their patterns.23
Figure 17: A non-periodic tiling of squares and triangles occurring naturally in crystal
growth (Krumeich, Conrad, Nissen, & Harbrecht, 1998)
It is easy enough to show that non-periodic tessellations exist by simply creating
one. Start with the periodic 36/3
2.4.3.4 tessellation in Figure 18 below. (One of the
children in my study created this tessellation. See Figure 37 on page 148.) Next, rotate at
least three dodecagonal groupings, as in Figure 19. The flaw introduced into the pattern
destroys all symmetry, including translation symmetry (periodicity).
23 To be precise, it can be proved that any convex region of squares and triangles in which all vertices are
surrounded by two squares and three triangles can be extended to a tessellation of the entire plane with all
vertices surrounded by two squares and three triangles (Grünbaum & Shephard, 1986, p. 99).
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Figure 18: The 36/3
2.4.3.4 tiling with three dodecagons highlighted
Figure 19: Non-periodic, asymmetric, tiling created by rotating three dodecagons 30°
Several children in this study tried to create random tilings similar to those
occurring in crystals. However it is impossible to know if such tilings will eventually
contain a gap and this sometimes created uncertainty in the minds of the children. There
is no such uncertainty when there is translation symmetry in two independent directions.
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Note also that it is easy to create tessellations which have rotation or reflection
symmetry without translation symmetry. For example if we had rotated only two
dodecagons in Figure 19, the pattern would have lost translation symmetry, but would
still have retained rotation (and possibly reflection) symmetry. Another example is in
Figure 20 below which is a monohedral tiling made of L-trominoes (a type of non-convex
hexagon). It has 4-fold symmetry, with a unique center of symmetry from which the rest
of the pattern appears to radiate visually. Because it has only one center of symmetry, it
has no translation symmetry. One of the children in my pilot study attempted to make a
pattern similar to this one, but with 2-fold symmetry. I say that patterns with rotation and
reflection symmetry about a point without translation symmetry are radial patterns.
Figure 20: A 4-fold symmetric tiling without translation symmetry
A tessellation without translation symmetry can also be created from isosceles
triangles as on the left in Figure 21 below, which is a radial tessellation. If we translate
half the tessellation, we can even make a spiral tessellation which has rotation symmetry,
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but no reflection or translation symmetry, as on the right in Figure 21. (Certain lines have
been highlighted to emphasize the spiral nature of the tiling.)
Figure 21: Isosceles triangle tiling with 8-fold symmetry transformed into a spiral tiling
with rotation symmetry but no reflection symmetry
Non-Regular Polygons
It is interesting to note that, not only equilateral triangles and squares, but in fact
all triangles and quadrilaterals tessellate. That is, any triangle or quadrilateral, even non-
convex, can be used to create tessellations in various ways with various types of
symmetry, for example as in Figure 22 below.24 Triangles and quadrilaterals can actually
tessellate in a wide variety of ways. There are 14 ways to form an isohedral tessellation
with triangles and 56 ways to form an isohedral tessellation with quadrilaterals
(Grünbaum & Shephard, 1986). Figure 23 below shows four possible ways of tiling with
a scalene triangle.
24 It is also interesting to note that this basic fact appears to be not well known among mathematicians.
Rena Upitis (Upitis et al., 1997) recounts a trip to MIT where she casually mentioned her work with
children making tessellations with non-convex quadrilaterals. The mathematicians there did not believe
such tessellations were possible.
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Figure 22: Tiling of a non-convex quadrilateral
Figure 23: Four of the several possible isohedral tessellations with a scalene triangle. The
two on the right are not edge-to-edge.
Pentagon tessellations are much more complex and in fact the precise number of
possible pentagons that tessellate is an unsolved problem in mathematics. Currently there
are 14 known types of pentagons that can make monohedral tessellations. Four of these
were discovered by Marjorie Rice, a San Diego homemaker with a high school education,
who noticed the problem in her son‘s copy of Scientific American and decided to work on
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it. These 14 types of pentagons have many ways of tiling, some of which have been
discovered by high school classes working on the problem (Schattschneider, 1978b).
More recently, it has been discovered that the number of ways of creating monohedral
convex pentagon tilings is actually infinite (Sugimoto & Ogawa, 2000). It is of course
impossible to tessellate with regular pentagons because the angles about each vertex must
add up to 360.
Pentagons represent the only case of polygons which have not been completely
analyzed. There are precisely three types of hexagons which can tessellate, including of
course the regular hexagon. It can be proved that convex polygons with seven or more
sides cannot form a monohedral tessellation (Grünbaum & Shephard, 1986), though there
are non-convex polygons for any number of sides which can tessellate (polyforms, for
example, which I discuss next).
Polyforms
Another type of polygon whose tessellations have been extensively explored are
polyforms, which are polygons made by adjoining copies of some unit shape along their
sides. The most frequently studied polyform is the class of polyominoes—shapes made by
adjoining squares along their sides. A single square is called a monomino. Two squares
are called a domino because of their resemblance to the playing tile. (Other polyominoes
were named as a humorous back formation from ―domino,‖ as if the d- were the Greek
di- or ―two.‖) If we do not count congruent shapes as different, there are two ways to
assemble trominoes (three squares) and five ways to form four squares into a tetromino.
See Figure 24 below for the standard letter names given to these shapes. (The interior
lines are only meant to show the unit squares and have no role in making tessellations,
though some students insisted on adding these lines, perhaps to help them visualize the
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shape.) If we count reflected shapes as different, there are two versions of the L- and S-
tetrominoes, making the seven familiar tetrominoes used in the popular game of Tetris.
All of these shapes can tessellate the plane, as can the 12 pentominoes and the 35
hexominoes, though some shapes are more difficult to tile than others. Figure 10 and
Figure 20 show two ways to tile with the L-tromino. Searching for these tessellations
provides open-ended tasks at a range of difficulties, which I used as one of the sets of
tasks in this research.
Figure 24: Order-3 and order-4 polyominoes and their common letter names
Another common polyform is the class of polyiamonds, which are polyforms
made from equilateral triangles. Figure 25 below shows all the polyiamonds up to order
5. Any polyiamond up to order 6 can tessellate the plane (Grünbaum & Shephard, 1986).
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Figure 25: Polyiamonds up to order 5
A less well-known type of polyform is the polyaboloes, which are made of right
isosceles triangles. The only polyabolo used in this study is the trapezoid triabolo
pictured in Figure 26 below. Notice that this triabolo can be made in two different ways.
Figure 26: A triabolo that can be made in two different ways
Color
Note that it is possible to take color into consideration when analyzing
tessellations. Though normally color is not a consideration in geometry, it may become a
consideration for tessellations when analyzing symmetry and periodicity because we
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expect colored shapes to be mapped to shapes of the same color. The tiling in Figure 27
below would be considered a simple isohedral hexagon tiling with order-2 rotation
symmetry about four different points, as long as the colors are not taken into
consideration. If, however, we take color into consideration and require symmetry to map
tiles to tiles of the same color, the tiling no longer has rotation symmetry, though it is still
periodic. Most people, and especially children, are likely to take color into consideration
when evaluating a tessellation. Many tessellations in works of art, such as those at the
Alhambra or in Escher‘s work, rely on color to restrict the symmetry patterns in
interesting ways, leading the eye to notice symmetry that would not have been obvious
without color.
Figure 27: A tiling whose symmetry depends on whether color is taken into
consideration
Vocabulary Summary
Table 1 below has a summary of important technical terms used to discuss
tessellations.
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Table 1: Important tessellation terms
Term Meaning
Aperiodic Set of prototiles which can only form non-periodic tilings
Corner The term “vertex” has a special meaning for tilings, so “corner” is used to refer to what is usually called the vertex of a polygon.
Dihedral Tiling that uses two prototiles
Dilation symmetry Transformation mapping several tiles onto one tile or vice versa
Edge The intersection of two neighboring tiles; not always the same as a side
Edge-to-edge Polygonal tiling whose edges are the same as the polygon sides
Generating set The set of prototiles for a tiling
Isogonal Vertex-transitive: any vertex can be mapped to any other by a symmetry
Isohedral Tile-transitive: any tile can be mapped to any other by a symmetry
Isotoxal Edge-transitive: any edge can be mapped to any other by a symmetry
Isometry A rigid transformation: translation, reflection, rotation, or glide reflection
k-isogonal Tessellation with k different types of vertices that cannot be mapped to each other by any symmetry
k-uniform k-isogonal tiling made of regular polygons
Monohedral Tiling that uses only one prototile; all tiles are congruent
n-fold symmetry n lines of reflection symmetry through a point; this also implies rotation symmetry of order n (n > 1).
Non-periodic Tiling that lacks translation symmetry
Periodic Tiling with two independent translations; the pattern repeats in all directions
Prototile An initial shape used to tile with; all tiles are congruent to a prototile
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Table 1. cont.
Term Meaning
Radial Patterns with rotation and reflection symmetry about a certain point, but no translation symmetry
Regular An isohedral, edge-to-edge tiling made of regular polygons
Rotation symmetry of order n
A symmetry that is a rotation of 360°/n
Semi-regular Isogonal, edge-to-edge tiling of at least two types of regular polygons
Side The term “edge” has a special meaning for tilings, so “side” is used to refer to what is usually called the edge of a polygon.
Symmetry An isometry of the tiling that maps tiles onto tiles
Vertex The non-empty intersection (endpoint) of three or more edges
Children’s Understandings of Tessellations
Tessellations have been used successfully in many school classes, and are even
required in curricula of many countries and some states. Surprisingly, however, there has
been very little research into children‘s understanding of tessellations. The following
studies comprise the sum of all research that specifically informed the tiling aspect of this
study.
Geometric Properties in Tessellations (van Hiele-Geldof and Fuys et al.)
Perhaps the earliest educational study using tessellations was done by Dina van
Hiele-Geldof, who, with her husband Pierre van Hiele, developed the well known van
Hiele model of geometric reasoning. Dina‘s 1957 doctoral dissertation describes a
teaching experiment with 12-year-olds in a Dutch Montessori secondary school (Fuys et
al., 1984). She used a variety of methods to lead her students to explore many geometric
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ideas. She devoted the second school term to leading her students to construct and
analyze tessellations, including sidewalk paving; polygon tilings; possibilities with
triangles, quadrilaterals, and regular pentagons; parallelograms; and other shapes.
Various geometric properties were discovered in this way. For example, by noticing
parallel lines and congruent angles in certain tessellations, the students found patterns
they called ―saws‖ and ―ladders.‖ (See Figure 28 below.) These patterns led to the finding
of congruent angle theorems with parallel lines and how to reason with such theorems.
Her visual method of placing these theorems in the context of tessellations was much
more successful than the abstract, context-free way these theorems are usually presented
to students. Dina van Hiele reported a significant increase in students‘ ability to reason in
geometry through her teaching experiment. Her method was successfully repeated by the
Brooklyn College Project in the 1980s with similar results (Fuys, Geddes, & Tischler,
1988). These studies demonstrate some of the mathematical power of tessellations in
education.
Figure 28: Rhombus tiling with ―saw‖ (red) and ―ladder‖ (orange) highlighted
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Rectangular Arrays of Squares (Outhred & Mitchelmore)
Most of the studies of children‘s conceptualization of tilings have focused on
finite square arrays in order to understand how children understand the area formula for
rectangles. Outhred and Mitchelmore (1992) noted that children can easily arrange
wooden square tiles in rectangular arrays without necessarily understanding the row and
column structure of the array. This is because the physical tiles cannot overlap and
naturally organize themselves as children place them next to each other. Outhred and
Mitchelmore found that some children learn the area formula for a rectangle without
understanding—indeed without even the possibility of understanding—because they lack
a valid conceptualization of the row and column structure of a rectangular array. Battista,
Clements, Arnoff, Battista, and Borrow (1998) carried this research further, identifying
five levels of understanding that they believe children pass through in arriving at a robust
understanding of row and column structure. Outhred and Mitchelmore (2000) did further
studies, proposing a sequence of principles to be learned in order to understand the array
structure behind the area formula of a rectangle.
The goal of these studies is to lay the foundation for understanding the area
formula for a rectangle. Therefore the authors are only interested in finite examples of
square arrays, rather than infinite tessellations. Nevertheless, the authors report several
findings which are easily extended to all tessellations and therefore have a direct impact
on my study:
Tessellations require a mental model of the shape as well as a mental model of the
array. These models do not exist in young children. Most children do not develop
mental models of square arrays until third grade.25 Even in fourth grade, only
25 The new Common Core State Standards (2010) require students to study square arrays in second grade.
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about half of the students demonstrated a complete awareness of the array
structure in one of the tasks they were given. These mental models come about as
abstractions of one‘s actions in forming tessellations rather than any direct
abstraction of the tiling itself.
Use of concrete manipulatives ―prestructures‖ the tiling so that an accurate
conceptualization is not needed to create a successful tiling. It is easy for young
children to create successful tilings with physical square tiles, even if they have
no understanding of the row and column structure of the array. Other methods,
such as freehand drawing, must be used to analyze the child‘s thinking. (For
investigating student thinking, I found that it was even more helpful to have
children describe a tiling orally first, with one or two physical tiles for support,
before attempting to sketch the tessellation. I discuss this further in Chapter 3.)
Outhred and Mitchelmore also found that freehand drawing promoted
mathematically correct understanding of the structure of a tiling better than
concrete manipulatives.
Freehand drawing alone will not lead children to understand the structure of the
array unless they also realize when their drawings are insufficient. A predict-and-
check method may be best for guiding students to confront mathematically
erroneous schemata. It may be best to have children draw first and then check
with tiles.
Array structure is so well assimilated for adults that teachers may not realize that
array structure is not self-evident for children. This also appears to be the case for
some researchers before 1992, such as Vitale and Zinder (1991, see below), who
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assumed that all fourth graders understand the array structure of a parallelogram
tessellation.
One of the goals of my dissertation is to extend this study of children‘s
understandings of square arrays to children‘s understandings of other types of
tessellations. I show that some results are similar to those found by Outhred and
Mitchelmore for square arrays. I also show how and why Outhred and Mitchelmore‘s
reliance on drawing should be extended to oral discussion in order to get an even better
idea of children‘s initial conceptions of tessellations.
Combining Tessellations With Art (Upitis, Phillips, & Higginson)
Unlike some mathematical topics, tilings have obvious and immediate aesthetic
appeal to most people and many teachers see tessellations as an opportunity to do art
rather than math. One very common activity in schools is to modify the edges of a
tessellation so that the tiles take the shapes of animals or other objects, making an Escher-
type tessellation. These entertaining activities frequently have little or no mathematical
content, though they can.
An interesting study that combined both the mathematical and artistic aspects of
tessellations is recounted in Creative Mathematics: Exploring Children’s Understanding
(Upitis et al., 1997). The three authors—a researcher (Rena Upitis), a teacher (Eileen
Phillips), and a mathematician (Bill Higginson)—describe an aesthetic approach to
tessellations in the second chapter of their book. Rena begins the unit on tessellations
with the third graders five weeks before Eileen starts with the fourth graders, allowing the
third graders to teach certain discoveries to the fourth graders. Rena pursues the unit in a
very artistic manner, but with significant mathematical content, having the children create
patterns with a variety of materials, invent mathematical notations to describe the
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patterns, and make connections with the real world. The children eventually work out that
all triangles and quadrilaterals tessellate, but only some pentagons and hexagons. They
make Escher-type tessellations on dot paper, experiment with pattern blocks, discover
unknown tessellations, and end with a silk card project.
Eileen noticed a significant impact on students‘ ability in other mathematics units,
especially when using geoboards, which Eileen uses to teach certain fraction, geometry,
and multiplication concepts. Three months after the tessellation project, Eileen‘s students
had no trouble with these geoboard activities, though students in her 20 previous years of
teaching had always struggled. Furthermore, the children who studied tessellations
retained all the polygon vocabulary and showed impressive and surprising transfer of
their geometry knowledge in their accurate and complex geoboard results. ―Tessellations
were obviously real—really powerful, really useful, really adaptable‖ (p. 47). Bill notes
how the tessellation unit integrated all five of the bullet points from the NCTM Standards
for K-4 geometry (NCTM, 1989). He remarks that tessellations include a strong aesthetic
appeal. The process of ―fitting‖ is an aesthetic one and is basic to human instinct. This
explains the appeal of games such as Tetris, where the goal is to quickly create a gapless
tiling from tetrominoes. However, Bill notes that it is important for the teacher to have
significant knowledge of the subject. He applauds Rena‘s willingness to allow her
students to explore unknown paths, though she appropriately backed off from pentagons,
which, as we have noted, are still not fully explored by mathematicians.
This is one of the only teaching experiments on fourth grade tessellations in the
literature. It demonstrates the power of tessellations as a context for learning geometry at
this grade level.
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Unitizing (Wheatley & Reynolds)
Wheatley and Reynolds (1996) did a study comparing how children unitized in
arithmetic and tiling problems. The authors found that third grade children who unitized
well in arithmetic also tended to unitize in creating tilings, such as by putting two L-
trominoes together to form a rectangle. For example, the most successful student drew
rectangular units first and then subdivided them into the tiles that made up the unit. For
the student who had the most difficulty, even the tile was not yet perceived as a unit and
she had to draw the polyomino tile slowly one segment at a time. They concluded ―that
the construction and coordination of abstract units is central to mathematical activity in
both numerical and geometric settings‖ (p. 67).
The authors were surprised by the difficulty of the tasks. (Fifth and sixth graders
did much better.) The tiling exercises were done using blank paper or dot paper. The
polyomino tasks in my study are an extension of Wheatley and Reynolds‘s dot paper
exercises. The authors concluded, ―Students are likely to benefit greatly in their
mathematical development from opportunities to construct tilings of geometric shapes‖
(1996, p. 82).
Covering Shapes With Tiles (Owens & Outhred)
Owens and Outhred (1998) studied how children tiled finite areas. These finite,
monohedral tilings were intended to explore concepts necessary for understanding area
principles. Students had much more difficulty tiling with triangles than with rectangles
because triangular tiles had to be placed in two orientations. (Children in my study also
frequently commented on this dual orientation phenomenon. See, for example, Figure 29
below, where triangles oriented differently are also colored differently.) Unfamiliar
shapes to be covered, such as trapezoids, were more difficult than familiar shapes. All
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children had to draw in order to visualize the tessellations and be aware of their structure.
Children who worked on spatial activities improved on the tiling test. The authors
criticize common methods for teaching area concepts such as tiling with potato prints
because these highlight counting rather than area concepts. Children need to draw the
tilings by hand and understand the limitations of their drawing ability.
I based one of the tasks in my study on Owens and Outhred‘s finite tiling tasks in
order to explore students‘ understanding of fit.
Figure 29: Regular triangle tiling with oppositely oriented triangles colored differently
The Logic of Mixing Tiles (Vitale & Zinder)
Vitale and Zinder contributed a chapter on tessellations in Piaget and Garcia‘s
Toward a Logic of Meanings (1991). The subject of the book is on understanding
children‘s intensional logic. In Vitale and Zinder‘s chapter, the focus is on children‘s
intensional logic in deciding which combinations of polygons can form tessellations. This
particular chapter is flawed, in my opinion, because there does not seem to be any
rationale for the research question, either from a mathematical perspective or a child‘s
perspective. (There is no intensional logic in the combination of tiles.) Nevertheless,
there are some interesting points that come from this study. One is the fact that young
children do not understand that rectangular arrays of squares can be continued
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indefinitely. This is very likely due to the fact that young children do not yet
conceptualize tessellations in terms of rows.
Other Geometry-Related Research
The following studies do not explicitly address tiling, but do have results
important for my study concerning children‘s understanding of geometric concepts
important in tiling. In my study I pay special attention to students‘ conception of infinite
space with respect to tessellations and the role of symmetry in students‘ creation and
evaluation of tessellations.
A study by Clements, Wilson, and Sarama (2004) was potentially useful for
analyzing students‘ sense of fit, but the trajectory they proposed was developed for
younger children and did not seem to describe how the fourth grade students in my study
approached composition of units.
Children’s Understanding of Infinite Space
Plato held that space, the place where objects could be situated, was actually
infinite (Monaghan, 2001). Aristotle, however, taught that infinity itself did not actually
exist, but was merely a potentiality—straight lines could be extended as far and as often
as needed, but were not actually infinite. Some researchers have found that these two
ideas continue to be held in our minds, creating contradictory notions of how exactly
infinity is to be understood. Unfortunately, almost all research on infinity has been on
concepts such as limits, continuity, numbers, and the infinitely small. Almost no
researchers have looked at children‘s conceptions of infinite space.
Piaget and Inhelder were the first to look at children‘s conception of infinity, but
they did not cover it extensively. Aside from infinite numbers, they looked at ideas of
continuity, which according to Monaghan (2001) was mostly in the interest of
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establishing topological properties. They only worked with children up to 12 years of
age. Children at the concrete operational stage could imagine dividing a segment into a
large but finite number of pieces. Children at the formal operational stage could conceive
of dividing the segment an infinite number of times (Piaget & Inhelder, 1956).
Fischbein, Tirosh, and Hess (1979) extended Piaget and Inhelder‘s work to older
participants, whom they found to have contradictory ideas. They found that it was
important to distinguish the mathematical (Cantorian) idea of infinity and the self-
contradictory intuitive concepts of infinity. Our schemata are created through finite
contexts, so they are not well adapted to deal with mathematical infinity. They found that
children continued to grow in their understanding of infinity until about age 12, after
which their ideas stabilized, though these ideas were easily open to change. Whether
children gave the ―infinitist‖ answer or not depended on the question. Some questions
elicited 81% ―correct,‖ Cantorian answers and others only 10%, an indication that
students held contradictory notions. High achievers did not give better answers, but did
give wrong answers more consistently. Instruction had little or no effect on the answers,
especially for the non-standard questions.
Monaghan (2001) considered a variety of infinite concepts, including endless
processes, sequences, and operations. He cautioned that it is easy to lose sight of the fact
that infinity is a strange idea for children. Monaghan‘s studies found that students tended
to look at infinity more as a process than as an object. Like Fischbein et al., he found that
their concepts were contradictory, easily changed, and impervious to instruction. He
cautions that Cantorian infinity is not the only mathematically possible approach to
infinity and student answers should not be considered ―wrong‖ if they do not agree with
Cantorian set theory.
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One of the only studies so far of spatial infinity was done by Marchini (2003),
who looked at young children, aged 5–7, as they drew simple pattern sequences. Every
time they came to some sort of end, such as the edge of the paper, they were simply told
to ―continue.‖ Some students simply stopped anyway, whereas others found creative
ways to carry forward, such as making loops or attaching new sheets of paper. The
children made various comments. The data suggested that only a few had consistently
infinite ideas, though a larger number had what he called ―mixed‖ finite and infinite
conceptions. Marchini recognized that his methodology had limited power, but proposed
it as a first step towards understanding young children‘s idea of spatial infinity.
Children’s Understanding of Symmetry
Next I summarize the main findings on children‘s conceptualizations of
transformations and symmetry, which are key concepts for an understanding of
tessellations, as well as key factors in the aesthetics of tessellations. Symmetry connects
the cognitive with the aesthetic. Symmetry is an objective, mathematical concept which
explains much of the structure of a tessellation. As such it is understood from the
cognitive perspective. But symmetry is also universally recognized as an aesthetic
category which contributes to the perceived beauty of a tessellation. Arnheim (1969)
notes that perceptual symmetry can bring together the parts of a figure into a unified
whole, making perception easier. It is therefore important to understand to what extent
children understand symmetry, as well as the closely related topic of transformations,
which give mathematical definition to symmetry.
Most geometric concepts appear to be universal, and therefore innate in all
people. Dehaene, Izard, Pica, and Spelke (2006) confirmed this by testing the geometry
ability of the Mundurukú, an isolated Amazonian indigenous people, in order to explore
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what innate abilities we have without the schooling or other geometric context that
permeates modern society. Mundurukú participants demonstrated basic abilities in
topology, Euclidean geometry, and geometric figures. Both Mundurukú children and
adults scored at the same level as American children. (American adults scored
significantly higher.) The Mundurukú did less well with symmetry and metric properties,
though they still got most problems correct, except those involving chirality at oblique
angles. The Mundurukú did not do well at all with any problems on transformations,
including translations, dilations, reflections, and rotations. ―It is possible that geometrical
transformations are inherently more difficult mathematical concepts. Alternatively, such
transformations may be more difficult to detect in static images‖ (pp. 381–382).
Visualizing transformations may be a skill that requires experience.26
Understanding and working with symmetry, and to a greater extent
transformations, appears therefore to be at least partially a learned skill. Laurie Edwards
(2003) analyzes some of the inherent difficulty with transformations using the theory of
embodied cognition, a viewpoint that understands cognition to be situated in our
sensorimotor processes. Edwards discusses how children and adults struggle with
transformations, particularly rotations. Most people have difficulty conceiving rotations
where the center is not on the object. This difficulty arises because we do not think of the
entire space as being transformed, only the object. Edwards previously thought rotation
errors were just a ―misconception,‖ but now realizes that these errors occur because
students see transformations based on previous experiences. Even after explanation,
students talk about transformations as movements and struggle when centers are not on
26 It is unfortunate that the new Common Core State Standards (2010) postpone students‘ first work with
transformations until eighth grade, despite the heavy importance given to transformations in high school
geometry.
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the object. ―This tension between the mathematician‘s understanding of transformations
as mappings, implemented in the microworld, and the embodied, natural understanding of
motion that the learners brought to the experience is the source of their ‗misconceptions.‘
These ‗misconceptions‘ are in actuality, conceptions that are adaptive and functional
outside of the context of formal mathematics‖ (p. 9, sic for punctuation). Children at the
age studied in my research have a very physical understanding of transformations. It is
normal that they struggle to visualize rotations or rotation symmetry, especially because
the context of tessellations requires visualizing the transformation of the entire plane
rather than a single tile. It was not expected that children be able to verbalize any
recognition of rotation symmetry, and in fact none of them did, though rotation symmetry
clearly added to the aesthetic appeal of some of the tessellations for the children.
Researchers, such as Hoyles and Healy (1997), have found that reflections are
also a difficult transformation to visualize when the line of reflection is oblique. Hoyles
and Healy also found that 12-year-old students did not understand reflection as a
transformation of the entire plane, but rather just transformation of certain points. Some
transformations were difficult for the students because it was not clear how to connect
their visual perception with a formal procedure.
Piaget and Inhelder reported their studies on children‘s ability to understand
transformations in Mental Imagery in the Child (1971). They found that children were
successful with simple translation and reflection tasks from age 6 or 7, though much
younger children could gesture how shapes would be displaced in a translation. Simple
rotation tasks could be answered by age 7, though they sometimes described results with
oval imagery instead of circles. They concluded that children younger than age 7 or 8
have some understanding of motion, but cannot conceptualize transformations, which
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require an understanding of how an object changes due to the transformation. Young
children therefore reason in static configurations only.
Kidder (1976) found that children up to age 13 have difficulty with mental
representations of transformations. Kidder found this surprising because Piaget and
Inhelder believed children could begin to work with transformations at age 9. It should be
noted, however, that Piaget and Inhelder‘s tasks were much simpler than Kidder‘s
activities. The students in Kidder‘s study found translations to be the easiest
transformations. There was no significant difference in ability between rotations and
reflections for Kidder‘s activities. Compositions and inverses of transformations were
difficult for all students. Students were not consciously aware of the implications of
length conservation under any transformation. The numerous errors in students‘ answers
led Kidder to believe that his tasks required Piaget‘s formal operational (transfigural)
stage, which would have just been beginning for the oldest participants in Kidder‘s study.
In more recent studies, researchers have found that rotations are harder for
children to conceptualize than reflections, and that reflections are more difficult than
translations (Clements, 2003). Reflection symmetry is more easily noticed if the axis is
vertical or, to a lesser extent, horizontal (Genkins, 1975).
It was expected that children in my study would easily see vertical and horizontal
reflection symmetry as well as translation symmetry. Oblique reflection symmetry,
rotation symmetry, and glide reflection symmetry may be evident at the aesthetic level,
but it was not expected that students would be able to describe them, nor was it expected
they should be able to analyze any of the symmetry in terms of transformations except in
vague, localized terms.
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SUMMARY
This literature review has been necessarily lengthy because I am studying what is
essentially an unresearched area of mathematics education. There is no single theory that
can provide a sufficient theoretical framework for this study, especially because my
research question is analyzing the question of how children conceptualize tessellations
from two different theoretical viewpoints. These two viewpoints are complementary and
both are needed for a complete understanding of children‘s mathematical thinking, but
they are almost always treated separately in the literature. The second viewpoint
(aesthetic) is not well known, so it has required a more extensive presentation.
For the cognitive viewpoint, I have taken a broad Piagetian constructivist view.
Children construct their knowledge through assimilation and accommodation, passing
through various stages of understanding geometric concepts. Fourth grade children are
expected to be all at Piaget‘s interfigural stage, which is the earliest stage where
significant work with infinite tessellations can be handled. Children at this stage can
understand relations between figures, including basic symmetry and transformations.
They cannot however understand relations between systems of figures, which means they
probably will not understand glide reflections.
This Piagetian base is extended by other theories, all within the broader
constructivist framework. Vygotsky analyzes the social-cultural aspect of learning,
introducing the idea of zone of proximal development, remarking on the important
impact of tools on the child‘s thinking, and providing a theory for the relationship of
thought and speech. Theories by Sfard, Dubinsky, Tall and Gray, and Meissner enable us
to understand the relationship between a process conception and an object conception of
a mathematical idea.
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The aesthetic viewpoint has historically been overlooked in most research, but is
of importance for several reasons. Some researchers believe that cognition takes place
primarily at the level of aesthetic cognition. Aesthetics is also foundational to
establishing axiomatic systems and for finding generative paths of mathematical research.
Mathematicians recognize the important role of aesthetics in evaluating mathematical
results, using criteria of significance, surprise, simplicity, connectedness, and visual
appeal. The fact that these criteria also point to what is important in mathematics suggests
a deep connection between the aesthetic, the cognitive, and the mathematical. Children‘s
mathematical aesthetic is different because they have different purposes from
mathematicians in their pursuit of mathematics, and because children‘s mathematical
aesthetic adapts according to their goals and understanding. Nevertheless, children do
have a mathematical aesthetic, which educators should take into account, especially in the
context of mathematical inquiry. For mathematics education, Sinclair has proposed
viewing aesthetics through its three primary roles of motivation, generativity, and
evaluation.
I have summarized the mathematical theory of tessellations for clarity in analysis
and communication. A summary of technical terms used to discuss tessellations is given
on page 86.
I have also summarized the scant research that has been done directly related to
this study on children‘s understandings of tessellations. This body of research connects
the cognitive viewpoint with the mathematical content of this study. Several studies
demonstrate that tessellation lessons can have a powerful impact on the learning of
geometry by children aged 8–12. Studies on finite square arrays in the 1990s showed
some of the thinking that might be revealed by children in a more general study of
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tessellations, and in particular how the structure of the arrays is not understood by most
children until about age 8. Work by Wheatley and Reynolds showed that geometric
unitizing is an important strategy that is used by students at various levels. Owens and
Outhred investigated children‘s ability to tile a finite area.
In addition to these directly related studies, I also summarized the research on
children‘s understanding of infinite space, symmetry, and transformations. There seems,
in fact, to be no significant research on children‘s conception of infinite space, though
there are related studies on children‘s understanding of infinity in other contexts, showing
that students of all ages have limited ability to conceptualize infinity mathematically. As
Piagetian theory suggests, fourth grade children have only a simple grasp of basic
symmetry and transformations. Children understand translations best, followed by
vertical reflective symmetry, horizontal reflective symmetry, other reflective symmetry,
and then rotations. Glide reflections are probably too complex for fourth grade children to
understand at an analytical level.
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Chapter 3: Methodology
For this study of children‘s understanding of tessellations, I conducted a series of
clinical interviews of six fourth grade children. In this chapter I first describe the
methodology of clinical interviews—theory, practical details, and validity of the
methodology. I then give the details of the method used for this research.
CLINICAL INTERVIEWS
Clinical interviews have their origin in the work of Piaget, who used them
extensively to study children‘s psychological conceptions of their world. It is a method
which is suitable for use by both researchers and teachers, for whom it can give a much
deeper understanding of children‘s thinking than standardized tests.
In this study I am trying to understand children‘s understandings of tessellations
from cognitive and aesthetic viewpoints. What strategies, reasoning, and mathematical
structures do they use and how does aesthetics guide their thinking? Ginsburg (1997)
claims that a constructivist orientation requires the use of clinical interviews or similar
method in order to properly understand children‘s thinking. Indeed, for an exploratory
study such as this one, the clinical interview method is almost the only method that will
work. A more standardized format would limit the possibility of follow-up as new data
reveal children‘s new ways of thinking. The Piagetian constructivist theoretical
framework presented here therefore requires that I use clinical interviews, especially
because this research is exploratory in nature. Other researchers have done teaching
experiments with tessellations, but in this study I am trying to understand children‘s
natural thinking without specifically designing for any type of instruction.
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Theory of Clinical Interviews
A clinical interview is a social interactional event (diSessa, 2007). The goal is to
get children to expose their natural ways of thinking. This is not an easy task and requires
flexibility. In a clinical interview, a task is defined in advance and questions are prepared,
but as thinking is revealed, the researcher invents follow-up questions uniquely tailored
to the situation. These questions allow for deeper exploration of interesting answers.
The researcher is therefore not a passive questioner, but must be constantly
probing and creating hypotheses to explain how the child is thinking. These hypotheses
are tested by follow-up questions created during the dialogue between researcher and
student. The researcher must have background knowledge and an initial theory of what
might be expected in the interview in order to inform what will be looked for (Ginsburg,
1997). Piaget noticed that when new researchers start, ―they either suggest to the child all
they hope to find, or they suggest nothing at all, because they are not on the look-out for
anything, in which case, to be sure, they will never find anything‖ (1929/2007, p. 9). My
literature review informed the initial background of expectations for this study.
Methodology of Clinical Interviews
The interview is prepared in advance with a protocol or rough plan including the
tasks the interviewer will propose to the children. The interviewer must make clear that
the purpose is for the children to explain their thinking and engage in mutual inquiry, not
to display formal school knowledge (Russ, Lee, & Sherin, 2010). During the interview
the researcher can follow up on the children‘s responses with questions and tasks that are
not in the original protocol. Various researchers (e.g. diSessa, 2007; Ginsburg, 1997;
Hunting, 1997) have described techniques for clinical interviews in detail. The goal is to
set interviewees at ease in an informal atmosphere and gain their trust so they can feel
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free to think aloud. Interviewers do not normally teach during the interview. The purpose
of a clinical interview is to discern how the child is reasoning. Robert Hunting found that
35 to 50 minutes is ideal for interviewing 10- to 12-year-old children. The children in this
study are a bit younger (about 9 years old), so I chose to conduct half-hour interviews,
though some interviews lasted a bit longer.
Piaget believed in checking the robustness of children‘s beliefs by challenging
their assertions in order to see the children‘s reaction. This is to check the stability of
children‘s knowledge. Do children change their mind easily when challenged, or do they
maintain their beliefs? Ginsburg suggests that it could also be productive to help, rather
than challenge, the child in order to determine the child‘s fluid knowledge, what is in the
child‘s zone of proximal development. By providing limited help, we can explore not
only what a child already knows, but what the child is now capable of understanding. We
can distinguish concepts that the child is not ready to understand from concepts that are
ready to be grasped.
DiSessa (2007) notes that exposing natural ways of thinking is not easy because
thinking is complex. He tries to look at both robust, stable knowledge and fragile, fluid
knowledge. A schema is considered robust when it is used persistently even while the
child is learning ideas that should modify the schema. By interacting with children‘s
ideas and pushing them to think beyond what they already know, we can discover what
knowledge is stable and what knowledge is easily changed or abandoned. I believe the
exact nature of the interaction between interviewer and child depends on what the
interviewer is trying to investigate, but in any case it is a good idea to distinguish
between robust ideas held by the child and less stable ideas that change under
interrogation. I engaged the students in a variety of activities, as well as discussions of
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the aesthetics of tessellations. One of the tessellation activities was repeated to see if
increased familiarity with tessellations changed their answers. (I show in Chapter 4 that
no change was observed.) In many of the tasks, children were challenged to justify their
answers. This allowed me to check to see if their beliefs were firm, or only weakly-held
hunches.27
Though it could have been revealing to do so, I did not follow Ginsburg‘s
recommendation of helping children because of the length of the interviews. Each child
was interviewed six times. If help had been given during any interview except the last, I
could not have known what children were doing on their own in subsequent interviews
and what they were doing because they had learned from me in an earlier interview. I
believe that looking for consistency across the interviews was sufficient to identify
consistent, robust trends of thinking, especially because of the length of time each child
was interviewed.
After a clinical interview, the data are reviewed and models that explain the
child‘s thinking are created, evaluated, and revised according to the extrinsic criteria I
discuss below. The researcher begins with a broad analysis and gradually narrows the
focus in order to create a model that explains the data.
Clinical interviews have come under some criticism in recent years. In order to
assure the validity of this methodology, it is important to consider the intrinsic and
extrinsic criteria that researchers have established to respond to this criticism.
27 Exploring a child‘s fluid knowledge is in some ways a type of teaching experiment. Some teaching
experiments, in fact, are little more than a series of clinical interviews. This research was not a teaching
experiment because I did not do any explicit teaching. The term ―teaching experiment‖ is used by
researchers with a range of meaning (Cobb, 2000; Cobb & Steffe, 1983; Steffe & Thompson, 2000). The
fact that I looked at how children‘s thinking changed through experience with one of the tasks would lead
some to call what I did a teaching experiment, at least for that task. However, the emphasis of this study
was on discovering children‘s current knowledge and the analysis of possible change was only an isolated
observation.
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Validity and Reliability
Researchers have addressed concerns about the validity and reliability of clinical
interview research by describing the intrinsic and extrinsic criteria that should be used in
such research.
Intrinsic criteria describe how researchers maintain the validity of the clinical
interview process itself. Andrea diSessa (2007) defended clinical interviews against
criticism that the methodology lacked ecological validity, was coercive, and did not
reveal invariant knowledge that could transfer to other situations. DiSessa concedes that
clinical interviews are less developed than some other methodologies. He claims,
however, that the problems clinical interviews are sometimes associated with are often
the fault of the larger program which the clinical interview is a part of.
Some educational researchers have attacked the ecological validity of clinical
interviews, claiming that they are too far removed from the natural environment of actual
classrooms. DiSessa defends the ecological validity of clinical interviews by locating
them in the process of normal human inquiry and sense-making situations. Clinical
interviews are similar to the everyday phenomenon of mutual inquiry where people say,
―I wonder why….‖ Nevertheless, it is true that the clinical interview situation is not quite
the same as ordinary daily experiences, so the researcher must understand the relationship
between the child‘s answers in the context of the clinical interview and the child‘s
general beliefs in the world. DiSessa proposes as the Central Hypothesis that the clinical
interview is ―a form of [familiar] mutual inquiry that is developmentally derivative of
naturally occurring individual and mutual inquiry activities‖ (2007, p. 531). He proposes
as the Primary Constraint on clinical interviews that questions must be sensible in the
sense of being reasonably similar to questions someone might bring up in a normal sense-
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making inquiry. Therefore, questions must not be too difficult and the researcher must
constantly evaluate the participant‘s competence to respond to a given question.
Even if clinical interviews are not as ecologically valid as other forms of research,
they can still address questions that cannot be reasonably answered by other
methodologies. In particular, clinical interviews are ideally suited to studies that seek to
understand a concept that may not occur often in more natural settings. For certain
concepts, it is impractical to follow a student around, waiting for evidence of the
particular kind of thinking under investigation. In a classroom setting, even when the
child displays evidence of the concept of interest, it will likely be in too limited a fashion
to be helpful. The clinical interview provides a means for exploring the concept in depth
by temporarily creating a necessarily artificial situation where follow-up questions and
tasks can be proposed in order to obtain a deeper and more accurate understanding of the
participant‘s thinking.
DiSessa addresses the accusation that interviewers are unduly influencing the
participants and that coercion is producing unnatural responses in the clinical interview.
To address this concern, diSessa proposes four criteria for evaluating whether the intent
of the interview is respected: (1) Is there independent evidence of conviction on the part
of the participant? (2) Is it possible that the viewpoint was prompted? (3) Were the
responses systematic (e.g. always changing one‘s mind after a prompt) or were they
natural? (4) Is there converging evidence? If these criteria show that the participant was
in any way coerced, the interview has failed. During the analysis of my data, the
interviews were reviewed with these criteria in mind in order to be sure that intrinsic
criteria were being respected and that any answers that were possibly coerced were not
considered for the conclusions I reached. For example, when students were asked ―Are
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you sure?‖ and they answered with a quick, non-committal ―Yeah,‖ this was given less
weight than when students expressed long, unsolicited expressions of confidence. Quick
answers may only be an unthinking attempt to please the interviewer. Such answers were
therefore not considered sufficient evidence of student conviction unless supported by
other independent evidence.
Coercion is to be avoided, though diSessa freely admits that he may be
―seductive‖ during a clinical interview in the sense that he deliberately pulls participants
into areas that are most likely to reveal interesting thinking. This is not coercion so long
as there are no leading questions that would bias the participant‘s answer. The fact that
the interviewer retains a certain authority to direct the interview is also not considered
coercion as long as the participant retains the right to judge reasoning. Again, it is
important to make these points clear at the beginning of the interview. Children must feel
free in the clinical interview to express their beliefs without fear of being judged right or
wrong. This is not the usual environment in the classroom, so there may be a more or less
long time of transition at the beginning where this question of authority is negotiated and
the participant accepts that we are engaged in a form of mutual inquiry. This orientation
is maintained during the interview by withholding judgment on responses and
questioning the correctness of an answer only to the extent the participant is vague or is
aware that the answer is problematic. An answer may also be challenged to test
robustness, but the child should understand that this is not a judgment on the correctness
of the answer.
The third criticism to which diSessa responds is that knowledge is too fluid and
the results from the clinical interview do not transfer to the ―real‖ world. DiSessa
responds that researchers must therefore explore the relationship between robust and
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fragile knowledge in their clinical interviews. This can be done, as noted earlier, by
challenging answers in order to see what knowledge is stable. In the case of lengthy
interview time, as in my research, answers can also be checked for consistency across
time.
Extrinsic criteria are those that are used to evaluate the results of a clinical
interview or similar method. How well does the model of thinking proposed by the
researcher explain the data evaluated? John Clement (2000), basing his work on the
criteria offered by Kuhn and Darden for evaluating scientific theories, proposes an
extensive framework of criteria for evaluating the viability of a model derived from the
analysis of clinical interviews. Clement rejects the classical construct of validity,
claiming that it has had too many different interpretations. Instead he proposes the
construct of viability—the strength or believability of a model.
Clement outlines a framework for viability using criteria categories of
plausibility, empirical support, non-empirical support, and external viability. The
plausibility criteria require that the model explain the behavior in a believable way and be
internally consistent, e.g. not mixing conscious and unconscious thoughts. The criteria of
empirical support verify that there are sufficient, strongly supporting data and
insignificant confounding data. There are also non-empirical criteria requiring the model
to be clear and simple, without arbitrary contrived elements (no ―ad hocness‖), as well as
coherent with other known theories. External viability includes generalizability to other
populations (external generalizability) and other contexts (theoretical generalizability).
External viability criteria also include predictive validity, as well as the ability for the
model to be eventually extended to larger contexts and to be fruitful for identifying other
behaviors in line with the model; a model which is only successful in predicting behavior
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in one context would not be very useful. These external viability criteria are especially
important for exploratory studies such as this one where new models are being created,
though they can generally only be tested over time. Clement strongly urges researchers to
be explicit about all these criteria in research based on clinical interviews. The criteria are
summarized in Table 2 below.
Table 2: Summary of Clement‘s viability criteria
Category Criterion Explanation
Plausibility Explanatory adequacy The model is believable.
Internal coherence E.g., not mixing conscious and unconscious thought
Empirical Support Triangulation and number of supporting observations
There are strongly supporting observations.
Strength of connection to each observation
Reliable data directly support the model.
Lack of anomalies There are no significant confounding data.
Nonempirical Criteria Clarity The model is clear.
Simplicity The model is simple.
Lack of “ad hocness” The model is not contrived.
External coherence The model is coherent with other known theories.
External Viability Generalizability The model has external generalizability (to other populations) and internal generalizability (to other contexts).
Predictiveness The model has predictive validity.
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Table 2. cont.
Category Criterion Explanation
Extendibility The model can be extended to larger contexts.
Fruitfulness The model is fruitful for identifying other behaviors in line with the model. It is useful.
Clement also describes a spectrum of four possible approaches for clinical
interview studies, from generative, exploratory studies which investigate an initial model,
to convergent, independent coder studies. The former use larger units of analysis and a
process of abduction, criticism, and revision to interpret observations and propose models
that explain them. Clement proposes a cycle where generative studies lead to well-
defined observations that allow convergent studies which bring new observations to be
explained by more generative studies. There should always be a balance between
theoretical work, which develops models, and empirical work, which provides new data.
He notes that Ptolemy‘s geocentric system was able to predict the motion of the planets
with significant accuracy, but was nevertheless a wrong model. Likewise, a convergent
study with high replicability may still fail to identify a viable model. More in-depth
generative studies are needed alongside the convergent studies. My research is a
generative, exploratory study, developing an initial theory of how children conceptualize
tessellations.
Clement also considers criteria for reliability of the data. He defined viability as
the strength or believability of the model, and he defines reliability as the strength or
believability of the observations. This includes and goes beyond the classical definition
of reliability as consistency. The more convergent approaches have higher reliability than
the divergent, generative ones, but all kinds of studies are needed. Reliability can be
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improved at any level by standardizing procedures, estimating and reporting the relative
strength of observations, and increasing the quality of participants‘ self-reporting, all of
which were used to increase the reliability of this study.
All claims and their supporting data in this study were analyzed for viability and
reliability according to Clement‘s criteria.
Clement also discusses objectivity. Observations are theory laden. This is a
necessary thing, as Piaget pointed out. Researchers have certain expectations of how
children will respond to each situation. However, it is critical for researchers to be open
to new and unusual phenomena. This point was made clear to me during my pilot study
when I realized that I was frequently assuming students understood certain tilings in a
certain way, but further questioning showed that the children had different ideas. For this
reason, I lengthened the time given for each task in the main study, in order to explore
answers in depth and to be more certain that the children really were thinking what they
seemed to be thinking.
METHOD
The method used in this research involved a series of interviews with six children
as they solved various tiling tasks. Diversity was the key to the method design. Because
this was exploratory research, I designed a method where fourth grade students from
diverse backgrounds with diverse ways of reasoning used diverse tools to create tilings
with diverse shapes. In this way, the data were expected to identify a broader spectrum of
children‘s ways of thinking.
Pilot Study
In order to prepare for this study, I conducted a pilot study in which I tried out
several ideas for investigating children‘s conceptualizations of tessellations. The purpose
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of the pilot study was to see how various ideas for tessellation activities would work out
in practice and what kind of insights they might provide into children‘s thinking in order
to plan the dissertation study. I used a variety of tessellation tasks which were prepared in
an open-ended format that allowed tasks to be modified as needed during the interviews.
The 11 participants of the pilot study were in the fourth or fifth grade in two
elementary schools, one a charter school and the other a private school. The charter
school is a demonstration elementary school for a large state university serving students
from low income families. The private school is a small, multilingual elementary school
with accreditation from the French Ministry of Education. Many, but not all, of the
students at the private school come from high income families. The same two schools
were also used for the dissertation study. Three of the children in the pilot study were
French; the other eight were American.
For the pilot study, six of the eleven children were interviewed in pairs, the other
five individually, in whichever language they were most comfortable with. Students were
given a variety of tiling tasks similar to the tasks eventually used for the dissertation
study.
The pilot study showed me various practical ways that my clinical interview
techniques could be improved on for this study. More importantly, it showed that
interesting observations could be made which could throw light on how children
conceptualized tessellations.
From the cognitive viewpoint, the pilot study suggested a number of results,
which were further explored in the dissertation study. There were several strategies that
students used repeatedly in their creations during the pilot study, including making
simple or complex rows, unitizing, making random patterns, making complex patterns,
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and growing patterns. Students also used symmetry and other strategies. Children
justified their answers in a variety of ways, including using empirical evidence, using
previous experience, explicitly using another known tessellation, noting that gaps are
always filled, noting the resemblance with another known tessellation, using symmetry,
and using transformations. These types of strategies and reasoning were used as part of
the initial basis for analyzing cognitive thinking in the dissertation study.
It was also noticed in the pilot study that the children often focused on finite
patterns and did not always show a clear understanding that a tiling could continue
indefinitely. The children sometimes did not see how many polygons would fit around a
vertex. Several observations concerning symmetry and aesthetic influences were also
made. All of these observations formed the basis for the design of the tasks in the
dissertation study and the initial analysis of my dissertation data.
I concluded from my pilot study that an exploratory study of children‘s
conceptualizations of tessellations was feasible and that important conclusions were
possible from both cognitive and aesthetic viewpoints by using clinical interviews of
children as they discussed the possibilities of tiling. The study also resulted in a paper
presented at a PME-NA conference (Eberle, 2010) in which I showed that even though
children‘s mathematical aesthetic is different from that of mathematicians‘, their aesthetic
sense of symmetry was ideally suited for the kinds of explorations that they were engaged
in. Children are not capable of exploring sophisticated non-periodic tessellations to any
significant mathematical depth. And in fact children were dissatisfied with the tilings if
they did not possess symmetry. If symmetry was present, the children were more
confident of their tiling. This confidence was mathematically appropriate—more often
than not, the symmetric tilings were in fact mathematically correct, or at least on the right
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track, whereas the non-symmetric tilings were not. Aesthetics is therefore not only
important for motivational reasons, but it is important because it usually correctly leads to
generative and valid mathematical solutions. This important result from my pilot study
was further confirmed by my dissertation data, as I shall discuss in Chapter 4.
There were three main changes made from the pilot study as I designed the
dissertation study: (1) the dissertation study was considerably longer than the pilot study,
allowing for a greater variety of tasks at greater depth; (2) the dissertation study explicitly
included investigation of the aesthetic elements of the tasks; and (3) there were no
interviews in pairs.
(1) The dissertation study included six half-hour interviews instead of only one
for each student. With the 30-minute interview of the pilot study, I had to cut many of the
tasks short as soon as I thought the children‘s reasoning was sufficiently clear. However,
on later analysis, I realized that I often cut the task short too soon. When I did allow
students to continue, I was often surprised by the result. I clearly needed to allow the
students to continue the tasks further and to probe in greater depth as to how the students
were thinking. With some students in the pilot study, I was not even able to finish all the
tasks. Allowing more time enabled me to probe almost every task at sufficient depth,
letting students have enough time to express their thinking. I was also able to include a
greater variety of tasks in order to explore thinking in more ways.
(2) During the pilot study interviews, I was not yet considering the aesthetic
viewpoint. After analyzing the interviews it became clear that the aesthetics of the tasks
played a very important role in how the students approached the problems. This aesthetic
viewpoint was incorporated explicitly in the design of the dissertation study.
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(3) Interviewing in pairs seemed like a good idea for the pilot study. It
occasionally allowed students to share thinking in a way that made it easier for them to
verbalize thinking. However, as often as not, one student dominated the conversation
making it virtually impossible to know what the other student was thinking. At times,
students would collaborate on a tiling without a clear, common objective, creating a tiling
that was difficult to analyze because the students had different strategies and did not have
a common pattern in mind. I found it was much easier to discern a student‘s mental
model for a tessellation when they worked alone. For this reason, I followed the classic
procedure of interviewing students alone for this study, except for the final interview.
The final interview was done in a group format because it focused entirely on aesthetic
evaluation. There was no danger of confusing two children‘s thinking because no tilings
were created in the sixth interview. Using a group format in the sixth interview allowed
students to verbalize their aesthetic reasoning more easily.
Population
My pilot study, as well as the research discussed in the Literature Review, show
that fourth grade students are capable of beginning to analyze tessellations of varying
complexity, sometimes in rather sophisticated ways. This age group therefore forms a
baseline for early conceptualizations of tessellations at a meaningful level. Most children
in second grade still have not grasped the row and column structure of square arrays and
will therefore certainly not be ready to discern more complex structure in non-square
tessellations. It was therefore decided to analyze children‘s thinking in fourth grade in
order to understand thinking at an age where elementary but sufficiently sophisticated
tessellation ideas could be analyzed.
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For this dissertation study, I aimed to interview fewer students than the pilot
study, but to allow much more time for exploring their thought in depth. I nevertheless
wanted a sufficient number of students in order to gain a reasonable view of the diversity
of fourth grade thinking. It was important that the population be large enough that the
results could be considered generalizable. Individual case studies can provide powerful
insights, but may only be revealing thought that is peculiar to one particular student. I
therefore decided to interview six students in order to have a reasonable overview of how
students likely tend to think. More than six was not needed because this study was only
intended to be exploratory in nature. An exploratory study is intended to identify likely
models, which can then be further explored in a more convergent study, as explained on
page 113.
In order to survey as wide a spectrum of children‘s thinking as possible, the
participants were chosen in such a way as to create as much diversity as possible within
the small population of six children. Children were chosen from the same two schools as
the pilot study in order to sample students from populations of both low and high socio-
economical status using two different curricula, one American and one French. The
French curriculum tends to place a somewhat greater emphasis on geometry and
precision, while the American curriculum frequently emphasizes creativity. There are
therefore somewhat different emphases in the curricula from both cognitive and aesthetic
viewpoints. The goal in continuing to study two schools was not to compare them, but
simply to achieve an additional measure of diversity in the students and their
backgrounds. Most importantly, diversity in the selection of students was achieved
through a pre-test, which I describe below. No student at either school had yet studied
tessellations in their classes at the time of the interviews.
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In the French system, the nearest equivalent to fourth grade is the cours moyen 1
(CM1), though students in CM1 may be a few months younger than American fourth
grade students because of different cutoff birthdates for school admission. Most students
were nine years old at the beginning of the interviews in October, though one in the
French school was eight and one at the charter school was ten. The eight-year-old turned
nine before the final interview in January. The six students selected for the interviews are
described in Table 3 below. The names are not their real names.
Table 3: Student participants in this study
Student Grade Sex Age (1st interview)
Dominant language
Kelsey 4 Female 9 English
Rachel 4 Female 9 English
Moses 4 Male 10 English
Kobe CM1 Male 8 English
Marie CM1 Female 9 French
Michelle CM1 Female 9 English
The tasks for this study were longer and more detailed than the tasks in the pilot
study and some were new. Another girl aged eight years nine months from a public
school was selected to test the tasks before giving them to the six study participants.
Refinements were made to the protocol based on these trial interviews. Results from her
interviews are not reported in this study.
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Pre-test
In order to further diversify the population of students used for this study, students
were not selected at random. Instead, a pre-test was designed, which was administered to
the two classes. The pre-test had three questions assessing different aspects of tiling
possibilities. (See Appendix A.) The first question was a finite tiling question based on
one of the questions from the study done by Battista, Clements, Arnoff, Battista, and
Borrow (1998). A 5-by-7 square array was partially erased and students were asked how
many squares were in the original array. This was intended to evaluate how the students
understood the row and column structure of a square array. Such conceptions were well
studied in the literature and likely to be easier than understanding other tessellation
structures. The second question was to tile a 60° parallelogram, chosen partly to test
students‘ thinking with angles other than 90° and 45°. The third problem was to tile a
right-angled trapezoid with a 45° angle, chosen because it was not symmetric and it
probed understanding of 45° angles. The second problem was in the context of a blank
sheet of paper, whereas the third problem was done on dot paper, exploring tessellations
in the context of two different tools.
The test was created in two versions—in French for the private school (because
math was taught mainly in French in that school) and in English for the charter school.
The French pre-test was discussed with the CM1 teacher and modified according to his
suggestions. The English pre-test was discussed with the mathematics coordinator of the
charter school. The classroom teachers administered the 15-minute pre-test without the
investigator‘s presence.
Pre-tests from each school were analyzed in an attempt to sample a diversity of
thinking while still identifying common ways of thinking. Pre-tests from each school
were analyzed separately; the pre-tests from the two schools were not combined together.
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Each answer was coded for evidence of different types of thinking in comparison with
other answers. Coding was emergent as each test was analyzed. For example, the first
problem was coded according to whether the array was drawn; whether multiplication
was used; evidence of the lack of structural understanding (imperfect rows and columns,
rows but not columns, or neither rows nor columns); and whether students used two
methods with conflicting answers. After all the answers for a school were coded, the pre-
tests were sorted according to codes. I looked for ways to find three students who
represented frequently found solutions and yet who had maximally different ways of
thinking. For each school, I found several different ways of achieving such results,
resulting in the selection of eight children at each school. I then reviewed the sorted
analysis with the teacher of the private school or the coordinator of the charter school and
discussed which students would most likely do well in a clinical interview format
(speaking clearly, offering their own thinking, interacting well with a stranger, speaking
English or French as their first language, etc.). Together, we chose at least three students
at each school. We waited to see which students would return permission slips in order to
make the final selection.
Interviews
In this section, I describe the design of the interview tasks. I begin by noting the
extent of the constraints imposed by the investigator.
Criteria
In order to study students‘ aesthetic thinking, it is important to impose as few
constraints as possible. However, some sort of question or task must be posed using some
sort of tool, and therefore some constraints are inevitable. The goal of the clinical
interviews was to explore students‘ understandings of tessellations, and therefore the
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minimal constraints (no overlaps, no gaps, endless) were imposed to focus the interviews
on this specific mathematical task. Students were also given a specific tool for reasons
that I discuss in the Structure of the Interviews section below.
It was known from the pilot study that only the infinity constraint (tiling an
endless room without walls) was likely to be problematic. The infinity constraint could
have been omitted; the students could have been simply asked to make a ―tiling.‖
However, it was clear from both the pilot study and the pre-test results that many,
perhaps most, students would interpret this to mean ―Create a finite tiling that fits the
paper.‖ Because this is not the kind of mathematical thinking that the study was intended
to explore, it was important to specify this criterion up front. If children had begun with
finite tilings and then been told to switch to infinite tilings, the result might have been
confusing.28 Because this criterion might (and did) take some adjustment time, and
because it was fundamental to the study, it was decided to impose this criterion from the
beginning. The one task that involved tiling a finite figure was not given until the third
interview.
Students were briefly reminded of the criteria at the beginning of most interviews.
Once it was clear that students understood the three criteria, they were given freedom to
create the tilings any way they wanted, including ignoring the criteria. If students seemed
to be ignoring the criteria because they had forgotten, or if they explicitly asked about the
criteria, they were reminded, but then still allowed to create the tilings the way they
wanted. Reminders of the infinity criterion sometimes engendered interesting and
revealing conversations about this constraint.
28 One example of how the initial task can influence all the following tasks is with the special sixth
interview of this study. The sixth interview contained a variety of tasks, the first of which was to tile the
rooms of a house. None of the other tasks involved rooms of a house, and yet some students continued to
interpret all future tasks in the context of which tiling would look good in which room of a house.
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Overview of the Interviews
The interviews were organized as six half-hour sessions spaced at least a week
apart. Interviews were conducted in English or French according to whichever language
the child was most comfortable with. Marie was most comfortable in French, the others
in English. The first five interviews were organized as seven sets of tiling creation tasks,
labeled A through G, plus optional sets H and I, which were not used.
The framework for all the tasks was the problem of tiling a floor. In all but one
task, the floor was said to be infinite and without walls (or that it was so big we did not
need to worry about what would happen at the walls). The finite tiling task was not given
until the third interview. The three criteria were discussed at the beginning of the first
interview in order to check understanding.
I began with a dialog of how to tile a floor with squares in order to discuss the
task and its criteria. The initial interview also contained a discussion of the ground rules
for a clinical interview: why we are here; that there are no right or wrong answers; that
students should think aloud; and so on.
In addition to these largely cognitive tasks, there was a sixth interview which
consistently entirely of aesthetic evaluation tasks, labeled Set J. This final interview was
conducted by school with all three participants together so that students could interact
and share their criteria for evaluating the tessellations with each other. Aesthetic
evaluation tasks included tasks that evaluated tilings the students had made in previous
interviews as well as tilings made by other people.
I describe all tasks briefly below. Detailed protocols of the interviews can be
found in Appendix B.
It was decided to do the aesthetic evaluation after the creation tasks so that
exposure to professional tilings would not influence the children‘s tilings. If the aesthetic
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tasks had occurred earlier, children may have been under the impression that they were to
give artistic responses to the creation tasks, or they may have tried to imitate the
professional tilings. The goal was for the students to work in an environment for the
creation tasks where any aesthetic influences revealed during the analysis would be from
their own initiative.
The aesthetic evaluation tasks allowed me to identify the kinds of tilings the
children found more pleasing, and the reasons for these choices. By having the children
compare tilings I sought to identify the elements that students find beautiful or interesting
(presence of symmetry, type of symmetry, complexity, shapes, mathematical validity,
and so on, as well as non-mathematical criteria such as color and style). By having the
students work in groups and come to joint decisions concerning the choice of tilings, the
protocol focused on getting students to articulate what they found appealing (and
unappealing) about the tessellations and what aesthetic criteria they found most
important.
In addition to this formal aesthetic evaluation interview, Interviews 2–5 also
included evaluation questions about the tilings the children had just created. Did they like
the tiling they created? Which was their favorite tiling for this interview? Why? These
brief questions were generally towards the end of the interview. I did not want the
students to misunderstand the goal of the interviews to be an artistic one. Instead, I
wanted to probe their natural aesthetic reactions to the proposed tiling tasks. Almost no
aesthetic questions were asked during the first interview for the same reason. Analysis of
the interviews revealed no significant change in aesthetic criteria verbalized by the
students between the first five interviews and the final sixth interview. (I discuss this
further in Chapter 4.)
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Structure of the Interviews
The eight task sets and two optional sets are summarized in Table 4 below.
Table 4: The task sets
Set Task Tools
A Basic Tilings (Rhombi, regular hexagons and triangles, parallelograms)
Blank paper, pattern blocks, software
B Finite Tilings (Tiling completion tasks based on Owens & Outhred, 1998)
Prepared sheets
C Pentagons (Regular pentagons)
Polydrons
D Squares and Triangles (Dihedral tilings with squares and equilateral triangles)
Blank paper, pattern blocks, software
E Triangles (Acute isosceles, right scalene, and obtuse scalene triangles)
Blank paper, software
F Polyominoes (Trominoes and tetrominoes)
Dot paper, software
G Regular Polygons (Non-monohedral tilings of regular polygons)
Software
H Polyiamonds (Optional set as time allowed; not used)
Software
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Table 4. cont.
Set Task Tools
I Follow-up (Optional time for following up on any past interview difficulties—not needed)
As needed
J Aesthetic Evaluation Printed tilings, postcards, books
I began with the most basic tilings in Sets A and C in order to probe their basic
understanding of tessellations. Set B explored basic concepts of polygon fit. I then moved
into dihedral tessellations of squares and triangles, which I introduced early in order to
see if their answers to this task changed by the end of the series. (There was no
significant change, as I discuss in Chapter 4.) This was followed by increasingly
sophisticated tasks. Triangle tessellations are very basic, but sometimes difficult for
children to conceptualize (Owens & Outhred, 1998). Polyominoes introduce non-convex
tilings, some of which were seen to be challenging in the pilot study. I expected regular
polygon tilings to be the most complex because students had a large generating set to
draw from. The goal in ordering the tasks in this way was not to make a teaching
experiment, but to ensure that students were not overwhelmed by the more complex tasks
until they had a chance to get used to the tools used for the interviews.
The goal in using a variety of tasks with a variety of tools was to explore students‘
conceptions as thoroughly as possible. As discussed in the theoretical framework, the tool
has a significant impact on thinking and the subsequent task results. It was important to
propose tiling tasks with a variety of tools so that general results would not be due to the
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particular tool that was used. For most of the tiling creation tasks, the format was in three
phases: oral, drawing, and construction.
The oral phase involved asking the student if a shape could tile, and if so how.
Students were allowed to hold or look at one tile, but otherwise had no drawing or
construction tools. Owens and Outhred (1998) had determined that drawing forms
a closer link with the student‘s mental model than physical tiles. However, my
pilot study found that a more fruitful method for investigating initial student
thinking was to have them describe their ideas orally without access to a
construction tool. This was their opportunity to predict how the tiling would work
with the physical tiles. Drawing provided a context in which students could (and
did) make discoveries that changed their initial ideas. Sharing their ideas orally
was less likely to change their ideas, though this method (like any method) also
affected their thinking. Holding a single tile caused them to focus on describing
how neighboring tiles fit around the central tile. It was very rare during this phase
for students to describe the global pattern.
The drawing phase involved giving the students paper and marker (or pencil)
and allowing them to sketch how they thought the shape would tile (or very
rarely, why it would not tile). The oral phase had the constraint of not allowing
the students access to a tool to help their cognition, causing them to focus
thinking on a few neighboring tiles. The construction phase had the constraint of
tiles that would prestructure the tiling in various ways. The drawing phase
allowed students to express their ideas of how they thought the tiling would work
without either constraint.
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The construction phase involved allowing them to explore further than the first
two phases. The students tended to consider this tiling to be the ―real‖ tiling
where they could test their ideas and possibly explore new ideas. The ―real‖ tile
could be a pattern block, Polydron tile, or computer tile. Some tasks allowed
students to create physical tilings with more than one tool. Once students reached
this phase, if they asked to switch to a different tool, they were allowed to do so. I
usually ended the construction phase by asking them to describe how the tiling
would continue beyond what they had constructed.
A few of the tasks had unique formats. The pentagon task included a comparison
to square and hexagon tilings in order to encourage students to explore their reasoning
concerning the impossibility of tiling with a regular pentagon. The polyomino tasks used
dot paper for the drawing phase, which had aspects of a construction phase because of the
constraining nature of the dot grid. For polyominoes, students explored the two trominoes
first, followed by the five tetrominoes in any order they wanted. The regular polygon task
had only a computer phase because the task was too open-ended and complex for paper
and pencil. For the regular polygon task, students were able to choose freely from a
generating set of five regular polygons: triangle, square, pentagon, hexagon, and octagon.
Set B (Finite Tilings) was the only set that did not explore infinite tessellations.
Instead it examined questions of fit in a finite space. The problems were adapted from
Owens and Outhred‘s (1998) study of children‘s ability to cover shapes with tiles. The
six problems are reproduced in Figure 30 below. Children used the colored prototile at
top to tile the shape at bottom. The printed prototiles in the first four problems were the
same size, color, and shape as pattern blocks. These problems went through all three
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phases if needed. The last two had no physical tiles so the problem was limited to oral
and drawing phases. All problems were possible except for the fifth.
Figure 30: The six finite tiling problems
The aesthetic evaluation tasks of Set J were entirely different from the tiling
creation tasks. Rather than creating tilings, students were asked to evaluate previously
created tilings. These tasks took place during the sixth interview with all three students
from the same school together as a group. Tasks were presented in a variety of formats
that encouraged students to discuss with each other their reasons for choosing the tilings
they chose. There were six tasks:
1. The Student Tilings task involved choosing four tilings for four rooms of a house.
These tilings were chosen from a sample of printed tilings students had created in
the computer environment during the first five interviews. Each student chose a
tiling for each of the four rooms, resulting in three tiling proposals for each room.
Then as a group they selected one tiling for each room.
2. In the Other Tilings tasks I presented a sample of tilings ―made by adults,‖ which
were used in three tasks. These tilings had been chosen to represent a wide variety
of possible aesthetic and mathematical criteria.
a. The Student Sorting task asked students to agree as a group how to sort the
tilings and then to decide which set and tilings were their favorites.
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Students were encouraged to discuss their reasons together in order to
reach consensus.
b. For the Comparisons in Pairs task, the interviewer showed students a pair
(or larger set) of tilings and asked the group which was their favorite.
Students had to discuss and come to a group decision. The pairs had been
chosen according to some of the aesthetic criteria revealed in the
theoretical framework or pilot study. The goal in this and the other tasks
was not necessarily to test specific criteria, but to provide a space where as
wide a range of aesthetic values as possible could be expressed.
c. The Professional Preferences task asked which tiling an artist or a
mathematician would choose from among all the adult tilings.
3. The Professional Tilings (Alhambra) task involved showing students postcards of
Alhambra tilings, one at a time. Students worked as a group to put the tilings in
order from best to least favorite and then to justify their choices.
4. In the Professional Tilings (Escher) task I showed students a variety of tilings
from three books on artist M. C. Escher. Students worked as a group to choose the
best tilings and then award and justify first, second, and third prizes to their
favorites.
Table 5 below contains a summary of the structure of the interviews.29
29 Set B had been originally planned to be done after Set A. However, the responses for Set A were so rich
that I decided on the spot to lengthen the time for this set and moved Set B to the third interview. This also
helped assure that students were clear on the infinity criterion before introducing the one task where this
constraint was temporarily set aside. There was also an optional Set H of polyiamond tasks created in the
eventuality that a student would finish one of the tasks early. Sets H and I were not needed.
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Table 5: Interview structure
Interview Format Tasks (30 minutes per interview)
1 Individual Set A: Basic Tilings
2 Individual Set C: Pentagons
Set D: Squares and Triangles
3 Individual Set B: Finite Tilings
Set E: Triangles
4 Individual Set F: Polyominoes
5 Individual Set G: Regular Polygons
Set I: Follow-up
6 Group Set J: Aesthetic Evaluation
After the interviews were completed, I also informally interviewed two
mathematicians in order to obtain experts‘ viewpoints of the aesthetics of tessellations.
Those interviews used the same tessellations as the children‘s sixth interview.
Mathematicians were asked to give their opinion of which tilings they liked best, or
found beautiful, and why.
Computer software
Most of the tools used for the interviews are well known and have been used
extensively by myself and by countless teachers and researchers: pencil, marker, paper,
dot paper, pattern blocks, and Polydrons. The one exception is the computer software
used for certain tasks.
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I considered various possibilities for the software to use for this research. It
quickly became apparent that software was needed that was tailored to the unique
requirements for this research. Therefore I developed my own software environment
(Figure 31 below). Details of the software and the motivating factors behind the design
choices for the ―Polygon Tiler‖ software are in Appendix C.
Figure 31: Screenshot of ―Polygon Tiler‖ software
The purpose of the software was (a) to provide a different tool that may elicit
thinking different from the other tools; (b) to provide a second tool (in addition to the
pattern blocks) where inaccurate drawing does not play a role; and (c) to provide a
greater variety of possible tiles than the pattern block set could provide.
I tested the software informally with several children and also tested the final
version formally with the 8-year-old child with whom I tested all of my interview tasks.
Analysis
The data for this research consisted of the pre-test; video recordings of all
interviews; student drawings made during the interviews, including the finite tiling
worksheets from Set B; screenshots from each step of students‘ software creations; and
notes made during the two mathematicians‘ interviews.
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The interviews were transcribed using F4 transcription software (Lauterbach &
Kunath, 2008). The transcriptions were analyzed in two ways: (1) data were analyzed
using NVivo qualitative analysis software ("NVivo 8," 2009) and an inductive coding
method (Miles & Huberman, 1994); and (2) transcripts were printed and analyzed by
hand, looking for trends, general methods of tiling, and general results. Comparisons
were made across tasks for the same student and across students on the same task. I
constantly referred back to the actual video recordings while reading and interpreting the
transcripts. F4 and NVivo associated the transcripts and the videos via their timestamps,
so it was easy to flip back and forth between the two while coding.
(1) Coded analysis. Initial coding categories suggested by the literature review
and by the pilot study were analyzed using NVivo with the transcript and video data.
Other categories and codes emerged during the analysis. Categories of codes used during
the analysis included:
Aesthetic criteria used by students (and mathematicians);
Level of confidence in their answers;
Polygon used in the task;
Reasons why a tiling is impossible, or must be possible;
Type of reasoning used to justify confidence that the tiling will work;
Sorting criteria used in the sorting task of the sixth interview;
Strategy used to create a tiling;
Symmetry used to create a finite pattern;
Whether students understood how the tiles fit together;
Type of understanding of the tiling (To what extent their understanding was
apparently accurate from a mathematical point of view); and
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Whether students used active or descriptive wording during the first interview.
Other codes included awareness that a freehand drawing is representative and
may not be accurate, changing the center of symmetry, concern for accuracy, making an
explicit reference to symmetry (not necessarily using that word), and tracing a pattern
block to make the drawing. A complete list of codes is included in Appendix D.
Because this is an exploratory study, coding was not carried out at the level of
detailed behaviors (see Clement, 2000, pp. 558–559), but rather each interview was
broken into short episodes and coded according to broad identifications of the types of
thinking each student used. Each episode generally discussed a single idea. Episodes
ranged from about 3 seconds to more than 2 minutes in length, depending on the amount
of dialog in the exchange. Most episodes were less than 15 seconds long. Codes were
used to help find patterns in the responses, which were then analyzed through the process
recommended by Clement: interpretation, criticism, and revision by reading the passages
and reviewing the video recordings to see what interpretation made the most sense in
context.
Notes from the mathematicians‘ interviews were analyzed only according to
aesthetic criteria.
Because the interview process was carried out over an extended period of time,
analysis was also made to see if there was any change between the early interviews and
the later interviews by comparing both aesthetic and cognitive codes from early
interviews with codes from later interviews.
I used NVivo‘s query functions (especially matrix queries) to analyze patterns
relating different codes. In this way, relationships between the codes became apparent.
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(2) General analysis. In addition to coding, I also reviewed the printed transcripts
by hand. I analyzed the data and made notes summarizing what each student was doing
during each task. Unusual statements were noted. Attempts were made to understand the
thinking of each individual student for each task. Responses to each task were also
analyzed across all six students, looking for patterns in the types of answers, strategies,
and reasoning that tended to be used for each task.
I drew from suggested techniques by Clement for analyzing exploratory research
data in order to identify patterns and draw conclusions. Possible models and claims were
critiqued according to Clement‘s criteria of viability. Evidence in support of the claims
was analyzed for reliability.
Aesthetic Analysis
Aesthetics was analyzed according to the framework suggested by Nathalie
Sinclair (2006). Mathematical aesthetics has motivational, generative, and evaluative
roles. For the evaluative role, I coded students‘ aesthetic expressions whenever they
evaluated a completed tessellation, whether their own or someone else‘s. For the
generative role, I analyzed the aesthetic criteria students used in creating their
tessellations. This created two sets of aesthetic criteria: evaluation criteria and creation
criteria. The motivational role was often in the background, keeping the students engaged
on the tasks. (In fact, they were all quite enthusiastic throughout all six interviews.)
Students did not have many choices as to which task they would pursue except in the
fourth and fifth interviews where students were able to choose which prototiles to use.
Students were also able to choose which tool they would use in the later interviews.
Instances where aesthetic motivation played a role in students‘ choices were noted.
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Aesthetic criteria that emerged during the analysis proved to be particularly
diverse, especially for the evaluation criteria. All instances where students gave a reason
why they liked, preferred, or disliked a tiling were coded as an aesthetic criterion. There
were 327 instances in all of verbal aesthetic reasons from students and mathematicians. I
initially created codes for the aesthetic criteria which had been identified in the
theoretical framework or the pilot study. Whenever a participant expressed a positive or
negative aesthetic criterion that was different from the criteria that had been expressed
before, a new code was created. This resulted in the creation of 59 aesthetic criteria, too
many to analyze in a meaningful way, especially because many criteria were only used
once or twice. When all the aesthetic instances had been coded, the criteria were then
grouped into a more manageable number of themes by putting together negative and
positive criteria (such as good color and bad color) and other closely related criteria (such
as multi-colored and insufficient color). This resulted in 18 aesthetic themes which
included all criteria which had been frequently cited. There were 14 other aesthetic
criteria which could not be grouped, but these criteria were not frequently cited; none
were cited more than twice. Most of the aesthetic analysis was carried out on the 18
aesthetic themes.
Symmetry Analysis
Two types of symmetry analysis were considered—finite patterns and infinite
tessellations. Analysis of the finite patterns students made as they developed their
tessellations was straightforward. Patterns were coded according to the amount of n-fold
symmetry or order-n rotation symmetry they possessed. Translation and glide reflection
symmetry is not possible on finite patterns. See Periodicity and Symmetry on page 72 for
an explanation of types of symmetry analyzed in this study.
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Symmetry of infinite tessellations was only appropriate for patterns that clearly
represented infinite tessellations. This was not always evident for children‘s tilings, nor
was it clear how some students envisioned their infinite tessellations (as I discuss in
Chapter 4), so no attempt to analyze the theoretical infinite symmetry of their tilings was
made. We could ask if children preferred infinite tessellations with greater symmetry
during the aesthetic evaluation tasks of the sixth interview when such tessellations were
evaluated. For the most part, this would only apply to the Other Tilings tasks in the sixth
interview. Analysis of the symmetry of Alhambra and Escher tilings was considered too
complex, as evidenced by mathematicians‘ debates over the symmetries in the Alhambra
(Grünbaum, 2006; Jaworski, 2006; Rønning, 1995). Attempts were made to see if
students preferred certain types of symmetry during the Other Tilings tasks. Results were
inconclusive. See Appendix G for details of methodology and results.
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Chapter 4: Results
In this chapter I report on the results from the analysis of my data. I first focus on
the cognitive results, which come from the first five interviews. I describe the general
results from each of the tasks, and then report on certain patterns found in the data across
tasks. These results will help answer the first part of my research question about
children‘s cognitive understandings of tessellations and their strategies and reasoning
about tessellations.
Next I report in a similar fashion on the aesthetic results, most of which come
from the sixth interview, though there are also aesthetic data in the other interviews. This
will address the second part of my research question about how children evaluate
tessellations, how aesthetics guides their thinking, and the mathematical value of their
aesthetics.
Next I look at results which must be looked at from both viewpoints
simultaneously: results concerning the influence of tools, results concerning symmetry,
and the cognitive role of aesthetic criteria. This will address the third part of the research
question about how the aesthetic and cognitive viewpoints inform each other.
Finally, I look briefly at any differences revealed between the two schools. I also
look at differences between the early interviews and later interviews in order to evaluate
any change in thinking as well as differences between creation and evaluation task
results.
COGNITIVE UNDERSTANDING
As described in Chapter 3, there were seven sets of tasks given in the first five
interviews. Each set used a distinct set of tools and shapes, which influenced the kind of
thinking that was revealed. I first briefly summarize the results of each set of tasks. Fuller
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details of the results of each task are given in Appendix E.30 After the summary I discuss
results that cut across all the tasks.
Set A: Basic Tilings. Students were asked about the possibility of tiling with four
basic shapes. The rhombus, hexagon, and triangle tasks used pattern blocks. The
parallelogram task appeared as a paper-and-pencil task on the pre-test and was done in
the software environment during the interviews. Students described tilings through the
three phases described in Chapter 3 (page 128): an oral phase, a drawing phase, and a
construction phase. As an example of the construction phase, Figure 32 below shows
some of Rachel‘s tilings for this set.
Figure 32: Some of Rachel‘s tilings during the first interview
30 In this summary, I sometimes use terms like ―most,‖ ―some,‖ or ―a few students‖ when precise numbers
are not easily given. For example, if four of the six students believe something, and a fifth student hesitates
or gives more than one answer, I simply write that ―most students‖ believed this instead of ―four or five
students.‖ Fuller details are in Appendix E, though even there I sometimes avoid precise numbers if exact
counts are ambiguous. My goal is to indicate general trends, not precise quantitative analysis, which is not
warranted in this qualitative study. Nevertheless, all terms are used with care. ―Most‖ always indicates
more than half. Most of the counts come from the general analysis. (See page 137.) Descriptive statistics
that come from the coding are noted as they occur.
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All six students believed from the beginning that it would be possible to tile with
each of the four shapes, though Marie and Kobe were doubtful about the hexagon until
they tried sketching it. Descriptions tended to be local at first, describing how tiles would
fit around the initial tile. It was rare for students to describe the global structure, though
there were three instances where this happened. Moses and Michelle described how a
rhombus pattern would grow in all directions, and in the pre-test, Michelle had taken a
ruler and drawn a parallelogram grid.
Nearly all tilings in Set A were created edge-to-edge. (Figure 32 above shows one
of the two exceptions.) Aside from orientation, the hexagon and the equilateral triangle
can only tile one way in an edge-to-edge tessellation, but there are infinitely many ways
to create edge-to-edge tessellations with the rhombus and with the parallelogram, as well
as other ways that are not edge-to-edge. Students made a variety of tilings with the
rhombus and with the parallelogram, including tilings with tiles all in the same
orientation, alternate row tilings, non-edge-to-edge tilings, and random tilings. Some
students made several different rhombus and parallelogram tilings, sometimes
discovering new ways to tile through exploration. Two students thought that the standard
tiling was the only possible tiling. Some students considered different orientations to be
the same and others considered them different. Two students compared the process to
doing a puzzle. Students used a variety of strategies for every shape, including dilation
strategies, row strategies, and unitizing strategies. I discuss these strategies for creating
tilings in more depth on page 173.
Students tended to look for finite patterns in their rhombus and triangle
tessellations, identifying hexagons, stars, trapezoids, and rhombi patterns. Even though
there is only one way to make an edge-to-edge tiling with triangle pattern blocks,
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students used a multitude of different strategies to create these tilings. Each student
created several (identical) triangle tessellations, using a different method for constructing
each one. As I discuss on pages 159–168, at least two students did not appreciate that
these triangle tessellations were not actually different.
None of the students saw during their oral descriptions that six triangles could fit
around a vertex. Most students thought that four triangles would be enough, as it was for
the rhombus. During the pilot study, one student even drew how the four triangles would
fit in Figure 33 below. During the drawing phase of this study, Moses was surprised to
discover that it actually took six; he clearly experienced a moment of cognitive
dissonance, resulting in accommodation of the new fact: ―Oh my goodness. So it‘s kinda
like a hexagon.‖ Other students in this study also drew six, though half the students
occasionally drew five.
Figure 33: A student‘s idea of how four equilateral triangles fit around a common vertex
Almost no aesthetic questions were asked during the first set, but students
occasionally made aesthetic comments. Many students saw the standard parallelogram
tiling as three-dimensional, as if looking at a rectangular floor pattern from an angle.
Kelsey thought during the drawing phase that the rows of the parallelogram tiling would
be ―shorter and longer,‖ which would look cool (see Figure 34 below). It is not clear if
she was envisioning some kind of brick wall tiling, or if she thought parallelograms
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would naturally jut out beyond each other. She made a standard edge-to-edge tiling
during the construction phase.
Figure 34: Kelsey‘s parallelogram tiling drawing
At the end of the interview I asked if there were any shapes that were impossible
to tile with. Most students demonstrated that it was impossible to tile with a circle.
Rachel thought all non-polygons would fail to tile and recited polygon criteria she had
learned in third grade. Kobe repeatedly insisted that it was possible to tile with any shape.
Set B was done during the third interview, just before Set E.
Set C: Pentagons. I showed the students tilings made with square Polydrons and
with hexagon Polydrons in order to review these tilings from the first interview and to
introduce the new Polydron material. Then I asked them if it was possible to tile with
pentagon Polydrons. All students believed it was possible, though a few hesitated. Most
seemed to think it would look like the hexagon tiling and drew sketches to that effect,
with three or four pentagons around each vertex. Some even put five pentagons around a
vertex (see Figure 35 below).
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Figure 35: The beginning of Michelle‘s regular pentagon tiling
Students were not sure how to explain the unexpected result that it was, in fact,
impossible to tile with the regular pentagon. Three students simply pointed out the
inevitable gaps when asked why the pentagon cannot tile. (See Figure 36 below.) Kobe
was particularly bothered by the failure, having stated both in the previous interview and
at the beginning of this interview that all shapes can tile. He spent over five minutes
trying to find a solution, apparently trying to resolve his cognitive dissonance. Some
students correctly discerned that other types of pentagons could successfully tile. Kelsey
seemed confused by the tool itself and focused on the tiny tabs that make the Polydrons
snap together. She does not seem to have conceptualized the tiles as pentagons during
any of the phases.
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Figure 36: Moses points out the inevitable gap left by regular pentagons.
One consistent result was that the students perceived the hexagon as having much
more symmetry than the pentagon, possibly explaining the pentagon‘s inability to tile.
The hexagon clearly had six-fold symmetry, but the pentagon was usually perceived to
have only simple reflection symmetry. Some saw the regular pentagon as a square plus a
triangle, like a house. The pentagon was therefore not seen as isogonal and isotoxal. It
had a top and a bottom and the sides needed to play different roles in the structure.
Consequently, students did not see that it would have been sufficient to study how
pentagons fit around a single vertex. The following is Marie‘s description of the
difference between a hexagon and pentagon (with translation). The researcher is Scott.
Original Translation
Scott: Alors pourquoi les hexagones, ça
marche et les pentagones ça
marche pas ?
So why do the hexagons work
and the pentagons don‘t?
Marie: Je crois parce que l‘hexagone
c‘est tout pareil comme ça
[tracing with finger on table].
I think because the hexagon is all
the same like that [tracing with
finger on table].
Scott: Mm-hmm. Mm-hmm.
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Marie: Et celui-là [pentagon] c‘est un
peu différent.
And that one [pentagon] is a bit
different.
Scott: Comment c‘est différent ? How is it different?
Marie: Parce que c‘est comme une
forme de maison.
Because it‘s like a house shape.
Scott: C‘est comme une forme de
maison.
It‘s like a house shape.
Marie: Et pa-, et alors en fait c‘est tout
droit. C‘est comme un carré au
bout d‘un moment ça branche
[referring to “roof” of
pentagon].
And no-, and well in fact it‘s
straight. It‘s like a square and
after a while it branches
[referring to “roof” of
pentagon].
Scott: Uh-huh. Uh-huh.
Marie: Et [unintelligible] en tout cas, un
hexagone c‘est tout le temps,
comme, pareil, …
And [unintelligible] in any case,
a hexagon it‘s always, like, the
same …
Scott: Mm-hmm. Mm-hmm.
Marie: … c‘est, … … it‘s …
Scott: C‘est pareil partout ? It‘s the same everywhere?
Marie: … c‘est toutes les lignes c‘est pas
ça fait pas ça ou quelque chose
comme ça.
… it‘s all the lines it‘s not, it
doesn‘t do that or something like
that.
Set D: Squares and Triangles. Participants were asked if it was possible to make
a (dihedral) tiling with square and triangle pattern blocks. After using pattern blocks, they
were invited to continue exploring on the computer. This activity was repeated at the
beginning of Set G. Most students could not recall during Set G how they had created
their tiling in Set D, but they tended to give very similar answers both times, using
similar reasoning.
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Each student‘s approach was unique. Most thought it was possible from the
beginning, and they all sketched and created a variety of tilings. Several made the semi-
regular 33.4
2 tessellation by using a row strategy. Other common strategies included
putting squares around a central triangle, which often resulted in a radial pattern,
sometimes with 3-fold symmetry, or putting triangles around a core of squares, which
usually proved problematic because they wanted to fit only triangles around the corner of
each square, which cannot work. A few students succeeded in creating complex
mathematically valid tessellations, including one example of the 2-isogonal 36/3
2.4.3.4
tessellation. (See Figure 37 below.) Most tilings incorporated symmetry, though a few
appeared to be random or semi-random.31 Students tried to make the tilings edge-to-edge,
but in some cases which did not seem to be working, they shifted the tiles into non-edge-
to-edge positions in an effort to squeeze in certain tiles. This inevitably left gaps. See, for
example, Figure 38 below, where Kelsey has shifted the two bottom right triangles over a
bit to make more room. During Set D, two students reasoned that these gaps were
insignificant, but during Set G one student decided the tiling was impossible and another
abandoned the attempt and tried a different strategy. This may have indicated some
learning between the two interviews. However, other than this, there does not seem to be
any significant difference in tilings, reasoning, or strategy between the two interviews.
31 By ―random‖ I mean there was no single, clear repeating pattern. I do not mean to imply the students
deliberately incorporated randomness in any statistical sense. ―Semi-random‖ refers to a tiling where some
sections contained patterns, but there was no overall pattern to the tiling. See Figure 70 on page 212 for an
example.
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Figure 37: The 36/3
2.4.3.4 tiling made by Kobe
Figure 38: A step in Kelsey‘s attempt to squeeze five triangles around a square corner
Set B: Finite Tilings. This set was adapted from Owens and Outhred‘s (1998)
study of finite tilings. Students were asked if and how it was possible to tile the ―small
rooms‖ with the given tile. They were also asked how many tiles it would take. As with
the other tasks, the interview went through oral and drawing phases, as well as a
construction phase in the case of some of the pattern block tiles. The six tiling problems
are in Figure 30 on page 130. All the tilings were mathematically possible except for the
fifth problem, which most students saw at once to be impossible.
Students were generally more successful with the problems than I was expecting.
They sometimes misjudged the size of the prototile in both the oral and drawing phases,
particularly in the first few problems. Orientation only caused difficulty in a few cases.
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The most difficult problem proved to be the sixth, in which they needed to tile a right
trapezoid (a triabolo) with a right isosceles triangle. The difficulty seemed to come from
the fact that the prototile needed to be turned in two or three different orientations in
order to tile the trapezoid. Difficulty in judging the size of the prototile caused some
students to believe the tile would fit on the left with the hypotenuse at the bottom. (See
Figure 89 on page 295.)
In both the pre-test and the first finite tiling problem, Kelsey displayed a lack of
understanding of the row and column structure of a square array. (See Figure 39 below.)
In discussing her pre-test, she confirmed that some squares would necessarily ―be bigger
because some of them are shorter.‖
Figure 39: Kelsey‘s responses to the first pre-test problem and the first problem of Set B
As Owens and Outhred (1998) found, students tended to be more successful with
shapes that they were familiar with. It is impossible to compare these results with Owens
and Outhred‘s study because they interviewed both 2nd
and 4th
year students and did not
distinguish the two groups in their results.
Set E: Triangles. For this task, students were shown an acute isosceles triangle, a
right scalene triangle, and an obtuse scalene triangle in the software environment.
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Students initially believed all the triangles could tessellate, though a few changed their
minds if they were unsuccessful during the construction phase.
The acute isosceles triangle was tiled by most students in rows. Two students
attempted to create a radial pattern, one successfully.
The right scalene triangle prototile was oriented at a nonstandard angle. Only one
student realized at first that a rectangular unit could be made from two triangles. Two
other students realized this after exploring with the computer tiles, though one was
unsuccessful at tiling with the rectangular units, perhaps because of the non-standard
orientation. (See Figure 40 below.) Other students tried a variety of strategies, none of
which could make a mathematically valid tiling, though they believed their patterns
would eventually work. It is very likely that students would have seen the rectangular
unit tiling much more easily if the initial prototile had been oriented with horizontal and
vertical sides.
Figure 40: An attempt by Michelle to tile a right scalene triangle by using rectangular
units
Most students pursued strategies for the obtuse scalene triangle that were similar
to their strategies for the right scalene triangle. Upon exploration, many students tried
tilings made with parallelogram or kite units.
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Set F: Polyominoes. For the first part of this set, students were asked about tiling
each of the two trominoes. The drawing and construction phases were replaced by a
phase using dot paper. Three students insisted on dividing the trominoes into their unit
squares, even when I drew the initial tile. See for example Figure 41 below. This made it
very difficult to see the boundaries between the tiles. This may be due to the plastic tile
which served as a model; it had distinct square units. During the pilot study, no such
model was shown and none of the 11 students tried to divide the tromino into unit squares
on the dot paper. This phenomenon could also be an indication that the unit squares were
seen as an important part of the structure of the polyomino and students may have felt
that this needed to be seen clearly in order to have a clearer view of the local structure,
not realizing it would make the global structure nearly impossible to see. They may also
have understood the task as that of making a standard square tiling, in which case the
polyominoes were conceived not as tiles, but merely as units of tiles.
Figure 41: Kelsey‘s L-tromino tiling
Students were successful at creating a variety of tilings, including creative tilings
with the simple I-tromino. (See, for example, Figure 64 on page 202.)
Tilings with the L-trominoes were similar to those found by Wheatley and
Reynolds (1996). Most students made random patterns with the L-tromino. Two students
tiled with rectangular units made of two L-trominoes. Michelle believed there were other
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possibilities if she departed from the grid constraint. She showed, for example, how she
thought a radial pattern could be made with L-trominoes. See Figure 42 below.
Figure 42: Beginning of an L-tromino tiling by Michelle drawn on blank paper
In the second part of the interview, students were shown plastic models of all five
tetrominoes at once. They were asked which could tile and which couldn‘t. They were
then allowed to explore the tilings with any tile they chose using either dot paper or the
computer. Students generally thought most tetrominoes could tile, but in a few cases they
believed the tiling would be impossible. Students created a wide variety of tilings using
various strategies.
The T-tetromino was the most difficult. Three students doubted it was even
possible. Moses presented a demonstration showing why he thought the T-tetromino
could not tile. Three students made random patterns with the T-tetromino, but two got
stuck and decided it was not possible.
Sets F and G were the only sets where students had real choices over what
problems they would tackle first. Most students were motivated to work with the S-
tetromino, claiming it had a ―cool‖ shape. They seemed to be drawn by its rotational
symmetry. They described it as pointing in two directions at once.
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Set G: Regular Polygons. The last tiling creation task was open-ended and did
not have an oral or drawing phase because of its complexity. It was done entirely in the
computer environment. Students were first asked to revisit the squares and triangles task.
This served the purpose of seeing if any thinking had changed, as well as introducing the
next task. Students were slightly more successful with squares and triangles this time, but
for the most part they approached the problem the same way they had before, even if the
approach failed. The only difference was that students were more likely to recognize that
their approach failed during Set G than during Set D. Each student tended to use the same
sort of reasoning in both sets.
Next they were given five regular polygons (triangle, square, pentagon, hexagon,
and octagon) and invited to create tilings by combining any polygons they wished.
Students seemed attracted to trying to use the pentagon. None made reference to the
earlier pentagon task where they had discovered it was impossible to tile with the regular
pentagon. (It is also impossible to tile with the regular pentagon in combination with
other regular polygons, but none of the students realized this.) The numerous attempts to
include a pentagon in the tilings were probably due to the motivational appeal of
uniqueness. They were unfamiliar with any tilings using the pentagon and tried very hard
to find one.
Of the ten tiling attempts that did not use a pentagon, seven were successful.
Three students constructed the 4.82 semi-regular tessellation using squares and octagons.
Students made a wide variety of other constructions, including some complex
symmetrical tilings. All students except Marie were successful in creating at least one
tessellation. Kelsey was the only student to create a random pattern. See Figure 43 below.
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Figure 43: Kelsey‘s random triangle and hexagon tiling
Sets H and I were optional sets, which were not needed. Set H was an extra task
(polyiamonds) in case a student finished or stopped another task early. Set I was a follow-
up time for any unusual answers from the other tasks.
Set J: Aesthetic evaluation. The sixth interview had a completely different
format. Students were involved in evaluating previously created tilings. Data from this
interview were used (in combination with data from the first five interviews) to explore
students‘ aesthetic criteria for tessellations. See the Aesthetic Influences section
beginning on page 181 for a summary of the aesthetic results.
In addition to analyzing how students responded to each task, data were also
analyzed across tasks. The remainder of this section explores the findings from this cross-
task analysis.
Piagetian stage
One interesting incident occurred with Michelle‘s solution of the first problem of
Set B (tiling a 2×3 rectangle with squares). Her attempt to draw the tiling was
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unsuccessful, even though she had an idea how six or seven squares would fit in the
rectangle. The orientation of the square seems to have been the source of the difficulty.
She realized that the square could be turned, and conceptualized the square as rotated for
the rightmost column, but thought the square could be left in its original orientation for
the other two columns. This would seem to be the kind of thinking that occurs in Piaget‘s
intrafigural stage. Another indication that Michelle sometimes reasoned at the intrafigural
stage occurred during the polyomino task when she tried to draw an I-tromino in a
diagonal orientation during Set F. Though three of the sides of the tromino were drawn
correctly, one side and the lines for the unit squares in the tromino were oriented
according to the dot paper grid. See Figure 44 below.
Figure 44: An I-tromino drawn by Michelle in a diagonal orientation
Aside from these two instances, evidence suggests that all students, including
Michelle, were at the interfigural stage throughout all interviews, as expected. Children
had little difficulty handling very simple translations, reflections, and rotations of tiles.
They were able to describe these transformations orally before performing them, as
would be expected at the interfigural stage. They were able to coordinate tiles locally.
Evidence suggests that none had reached the transfigural stage. They created simple
global patterns requiring translations, but could not handle global symmetry requiring
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multiple mental transformations. For example, there was no evidence of noticing glide
reflections, which requires envisioning a translation and a reflection. Kobe did create the
tessellation in Figure 45 below, which could have glide reflection symmetry (depending
on how it is continued). However he gave no indication of noticing the glide reflection
symmetry that the infinite tessellation would have had, nor did he use glide reflections to
construct it. There was no clear evidence that children noticed rotation symmetry in any
tessellation. This may be because such patterns almost always had reflection symmetry as
well, which is easier to see, and children at the interfigural stage should not be able to
perceive two different types of symmetry simultaneously.
Figure 45: S-tetromino tiling by Kobe
Tiling Criteria
As explained in Chapter 3, three constraints were imposed on the tasks: tilings
should be gapless, without overlaps, and endless in all directions. I analyzed the data to
understand the students‘ reactions to these constraints.
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How Shapes Fit (Gaps and Overlaps)
Avoiding gaps and overlaps was natural for the students, even on the pre-test,
where no constraints were given. With some tools, such as pattern blocks and dot paper,
overlapping is basically impossible. Pattern blocks sometimes leave slight gaps, but they
cannot overlap. In the computer environment, which allowed both gaps and small
overlaps, if it was necessary to break a constraint because tiles did not fit perfectly,
students preferred overlapping. It is possible that, because the software rejected large
overlapping, the students depended on the program to protect them from illegal overlaps.
In that case, they may have accepted slight overlaps simply because the computer did.
Another explanation for the preference of overlaps is that gaps are more visible to the eye
than overlaps. On at least one occasion, a student did not notice overlapping of computer
tiles. They always noticed gaps, no matter how small.
Gaps. Two students noted that in real tilings there are small spaces between the
tiles. However all of the students easily accepted the gapless constraint for these
interviews. One student once left deliberate gaps in order to highlight units of tiles, but
even then he clarified that the gaps were only for explanatory purposes. If possible,
students would fiddle with the tiling in an attempt to squeeze out all perceived gaps. The
fit must be precise, or else something must be done about the remaining gap, possibly
moving other pieces around. Students sometimes rotated tiles around a vertex or put them
in a different order around a vertex to try to close up a gap. (This shows that they did not
understand the additivity of angles around a vertex.) If the gap was small, it sometimes
became acceptable to leave it in hopes that the gap was not real, but simply an
imperfection of some sort. It was not always clear if these imperfections were the fault of
imprecise drawing or if tilings might legitimately have tiny gaps. Michelle noted for two
of her tilings that ―there might be about one or two cracks about that big,‖ holding her
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fingers very close. She believed these potential gaps were an artifact of the lack of
symmetry in her tilings. ―It wouldn‘t be an exact pattern, but um, it should work.‖
Slight gaps raised doubts about the validity of the tiling. When drawing, students
would distort shapes in order to avoid leaving gaps. For example, Rachel stretched the
topmost triangle, Marie stretched the rightmost triangle, and Moses stretched the bottom
triangles in their tilings in Figure 46 below to much more than 60° in order to avoid
leaving a gap. In other words, when adding a tile to a drawing, students were often
guided by the need to fit rather than the shape of the tile.
Figure 46: Tilings by Rachel (left), Marie (middle), and Moses (right) where equilateral
triangles have been stretched to fit gaps large enough for two triangles
Overlaps are closely related to gaps. Both are questions of fit. Students generally
tried to avoid overlaps. Marie confirmed the importance of the overlapping constraint by
recounting a story of overlapping house tiles that made her trip. Overlapping was
sometimes cited as the reason circles could not tile a floor. However, the desire to avoid
overlapping seems slightly less strong than the desire to avoid gaps. Marie made a regular
polygon tiling with some slight overlapping. Rather than reject the tiling, she questioned
the overlapping constraint. She thought that with extra work the overlapping could
probably be fixed. Likewise Michelle thought some overlapping might be permissible as
long as it was not much. At one point Rachel asked my opinion whether two computer
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tiles were overlapping. I told her yes, but she decided to leave them anyway until it
became impossible to continue without fixing the problem.
This occasional ambiguity towards overlapping was also found by Owens and
Outhred (1998). In their study they noted that the one task in their study that could not be
tiled without overlapping proved to be the hardest. Students frequently answered it could
be tiled. Upon interview the students indicated that they knew the tiles would overlap, but
had answered that the tiling was possible anyway.
Infinite Space
Students quickly expressed comfort with the idea of a room that goes on forever
and ever without walls. Students repeated this constraint in their own words. For
example, Rachel said, ―Like, um, where it doesn‘t stop or where, it, can keep going and, a
long distance.‖ During the tasks, Moses would occasionally envision infinite tilings as
covering the world: ―Yeah, if it went on forever and ever and ever, around the globe, then
it would stop and connect [on the other side of the world], like a puzzle.‖
No constraints were given on the pre-test, but only one of the six children
interpreted the pre-test tiling problems to mean tiling only the rectangular sheet of paper.
The other students treated the problem in an apparently infinite space. In some cases, this
may be because they simply did not consider the problem of what happens when the tiles
meet the border. Rachel, for example, drew a rectangular space to tile in, but then
answered the problem as if the border did not exist.
During the interviews, the students showed significant signs of struggling with the
infinity constraint, even though they accepted it. This struggle was evidenced in several
ways: forgetting the constraint; finite patterns from the same tessellation representing
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different tessellations; expressing completion with a finite pattern; and, most importantly,
discomfort over tiling in an empty plane.
Forgetting the constraint. Students were briefly reminded at the beginning of
most interviews, and whenever they forgot or asked, that the rooms went on forever and
that we therefore do not worry about the walls—if the walls exist at all they are too far
away to care about. Nevertheless, students frequently forgot about this constraint,
especially during the earlier interviews. They spoke about walls and worried what would
happen when the tiling reached the walls.
Finite patterns from the same tessellation representing different tessellations.
At least two students, and possibly more, did not understand that different finite patterns
can represent the same infinite tiling. This was particularly evident for the equilateral
triangle tilings. Because all students made these tilings edge-to-edge, there was only one
possible tiling (barring orientation differences). Students found several different
strategies for making this tiling, sometimes in rows, sometimes by unitizing. If they
unitized, they might use hexagons, trapezoids, or more complex shapes. Kelsey and
Michelle believed that each strategy produced a unique tiling. This was true even when
both of Michelle‘s patterns were in front of her at the same time. They did not envision
the tiling extended to an infinite pattern, but only focused on the finite units they had
created. Kelsey, for example, insisted that one of her tilings would contain more
hexagons than another, though neither had been created by hexagonal units alone. (See
Figure 47 below.) Michelle, when asked if her favorite triangle tiling was more
interesting because of the way it was made or because of how it would look when it was
finished, stated that it was more interesting because of how it would look when it was
finished. She did not realize that the two tilings would look the same when extended
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indefinitely. Kobe and Rachel were not sure if their different strategies were creating
different tilings or not. Marie and Moses showed a firmer understanding of infinite space.
They explicitly acknowledged that their different finite patterns, if continued indefinitely,
would create the same infinite tiling.
Figure 47: Kelsey‘s first two triangle tessellations
I return to this phenomenon in the section on Ontology of Tilings (Process versus
Object) on page 164.
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Expressing completion with a finite pattern. Students sometimes announced
they were finished when they achieved a finite design, often when it was symmetric and
convex, sometimes without a clear idea of how the pattern would continue. For example,
Rachel talked about a hexagon tiling looking like a flower, ―when you‘re done with it.‖
After laying down a hexagon and six surrounding hexagons like a flower, she paused.
When I asked if she was done, she replied, ―Yes.‖ This would seem to suggest that
Rachel was focusing on how the finite pattern would look, rather than the infinite
tessellation. On the other hand, Rachel later pointed out for triangles that the finite
patterns were only temporary. They are destroyed as soon as you continue, which is what
makes all the tilings really the same. This suggests that Rachel may have been hesitating
between two ways to view tessellations.
During the pilot study, two students created a tower pattern with alternating rows
of squares and triangles. If extended indefinitely, it would become the semi-regular 33.4
2
tiling. However when challenged to extend their tiling further, both students put squares
in the triangle row and triangles in the square row, preserving the tower as a finite
pattern, as well as preserving the vertical reflection symmetry of the tiling. These
students seem to have been focusing on the symmetrical finite structure rather than
conceptualizing an infinite tiling.
Discomfort over tiling in an empty plane. During the pilot study, students
sometimes started a tiling in the corner of the paper, which caused them to create finite
patterns that incorporated the corner as part of the tiling. In order to avoid this problem, I
asked students in this study to begin all of their tilings in the center of the sheet of paper.
The computer software would automatically shrink and center any tiling that approached
the edges of the screen too closely. Marie complained about this. She insisted that she
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understood that there were no walls, but maintained that she needed some walls to help
her think about the tiling. She therefore used corners and walls in many of her tilings in
order to help her structure the tiling. When she did so, she would sometimes apologize,
saying she knew the wall was not really there, but that it was needed to make the task
easier. However, it was not clear that she had a conception of how the tiling continued
beyond her temporary walls.
Instead of working in an empty plane, students sometimes tried to create a tiling
with an implied or created frame. On the pre-test, the sheet of paper was interpreted by
Kobe as a frame within which the tiling had to fit. Marie saw that the computer screen
had an apparent border and tried to fit the tiles along this border.
Moses was more concerned about the borders of his rhombus pattern than the
interior structure, even with reminders not to worry about the walls. He spent a great deal
of time working out how the pattern could be closed off at the edges. He even doubted
that his alternating rows tiling would work because of the difficulty of finding a way to
enclose the pattern. At one point his border became very problematic as he attempted to
define a ―pit‖ in which he could attempt to complete his rhombus tiling. (See Figure 48
below.) In answer to the question whether his rhombus tiling would go on forever, he
talked about finishing it off with side borders.
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Figure 48: Rhombus tiling border by Moses
Some students enjoyed creating radial patterns—non-periodic tessellations with
rotation symmetry. The children in this study may have valued radial patterns
aesthetically. For example, in the Alhambra aesthetic evaluation task, students preferred
the radial patterns over the periodic patterns. (See, for example, Figure 62 on page 195.)
It may also be that students sometimes created radial patterns because they needed an
anchor point for their tessellations. The initial tile can serve as an anchor around which
the rest of the tessellation is constructed. Marie‘s corners may have served as anchor
points for her. Having an infinite, empty plane with no points of reference may be a
daunting idea for children at this age. Further research would be needed to know if this
were true.
Ontology of Tilings (Process versus Object)
There is evidence that participants sometimes conceptualized tiling as an object,
either mathematical or physical, and sometimes as a process or action. It is not likely that
in the context of this study we could distinguish between a mathematical and a physical
ontology, but we can look at evidence that points toward an object or process ontology. If
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the students truly conceptualized the tiling as infinite, we can safely say this is a
mathematical construct. Beyond this, I will not attempt to distinguish between
mathematical and physical tessellations for these children‘s conceptualizations.
One piece of evidence for a process ontology is in the episodes mentioned earlier
during discussion of participants‘ conceptualization of infinite space. Kelsey and
Michelle each created two mathematically identical tilings, and yet they believed the
tilings were different because each was created by different processes. For example, the
tilings in Figure 47 on page 161 were created by different procedures. Though Kelsey
was making the same regular 36 tessellation in each case, she perceived the two
tessellations to be different because of the different actions used to make them. Part of the
discussion of the difference between the two was as follows:
Scott: Ok. Umm, is there another way to tile with triangles?
Kelsey: You could make it like these put these here, just like um the,
[makes a hexagon from six pattern blocks] like this, then make it
do like a pair [adds six triangles to make a star] and then put this
right here, and this [fills in spaces between some star points to
form a more convex figure] and you keep going like that just like
in these. [See the bottom photo in Figure 47 on page 161.]
Scott: Ok. And that will make a different pattern?
Kelsey: Mm-hmm. Yes.
Scott: Ok.
Kelsey: ‗Cause this one has the shapes in the middle like that [points to
shapes near middle].
Scott: I‘m sorry, say it again. How is this different from the other pattern?
Kelsey: Um this one ha- gets these shapes [indicates two hexagonal
patterns on sides] like this and makes ‗em like that and then there‘s
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like one that makes a big triangle [shows a large triangle made of
six triangles at bottom] and one that …
Kelsey‘s perception of each tessellation is tied to how she actually created the
tessellation (in rows or with units) and not to what they currently look like. Even if we set
aside the issue of what the tessellations will look like when extended infinitely, she is
interpreting each tessellation in terms of the process used to create them rather than the
current (finite) result.
On the other hand, Marie and Moses expressed their belief that there was only one
triangle tessellation, no matter what method they used to achieve it. Moses was especially
clear about this. Like Kelsey and Michelle, Moses created triangle tessellations in two
different ways, but said that every method created the same tessellation. ―You could look
at it as a d-, um, in a different, like perspective and like uh if it was diamonds but it
would still be the same thing.‖ This points to an object conceptualization for Marie and
Moses. They distinguished the object tessellation from the process used to create it. Kobe
and Rachel were not sure if each creation was a different tessellation or not. Kobe
hesitated for over three minutes trying to decide whether the different creations were the
same or not.
An object conceptualization does not prevent someone from using a process
conceptualization. In geometrical contexts, an object conception may precede a process
conception, though it is difficult to see how one can completely understand an infinite
tessellation without also having a process conception. It is possible that students would
have access to both conceptions. Proceptual thinking requires flexible use of both.
Further evidence that Kelsey and Michelle used primarily a process conception
and Marie and Moses depended more on an object conception comes from an analysis of
the wording they used to describe their tessellations. Students sometimes used an active
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wording, describing the tiling in terms of their actions, e.g. ―Put this one right here like
that and then do it again.‖ Students also sometimes used descriptive wording, depicting
the tiling as an object they were passively describing, e.g. ―One line‘s gonna be facing
one way, the other, the next line‘s gonna be facing the other way.‖ Instances of such
wording in the first interview were coded by episode. There were 122 instances coded in
all. A ratio was calculated for each student of active wording to descriptive wording. A
high ratio indicated predominantly active wording and possibly a predominant process
conception. A low ratio indicated predominantly descriptive wording and may indicate a
preference for an object conception. Kelsey and Michelle, who interpreted their tilings in
terms of the actions used to create them, had ratios of 4.25 and 2.40 respectively. Marie
and Moses, who recognized the independence of their actions from the final result, had
ratios of 1.25 and 0.41 respectively. (The ratios for Kobe and Rachel, who were not sure,
were in between these two extremes.)
It is natural that participants should use active language to describe their
tessellations because they were in the process of creating them. Even I frequently used
active wording in my questions. However, the fact that Marie, and especially Moses, had
such low ratios, indicates that in spite of the active environment, they had a strong
tendency to view the tessellation as an object. The ratio measures point to the same
students as the observations about distinct tilings. Students with low ratios were precisely
those who were able to recognize the tessellation as an object distinct from their actions
and students with high ratios were not. This correspondence between two independent
sets of observations suggests that certain students predominantly conceptualized the
tessellations as an action and others as an object.
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It may be that all students used both object and process conceptions to some
extent, even if some privileged the process conception. For example, Marie, who seemed
to show an object conception for triangles, created the hexagon tiling in several different
ways and believed each way was different (though in this case the differences may be due
to orientation and focus on finite patterns rather than processes). Another example is
children‘s responses to the pentagon task. It is possible that children‘s reaction to this
task represents various process and object conceptions about pentagon tessellations.
Students believed that it was possible to tile with the pentagon, so when it did not work
they began trying different arrangements of pentagons. They seemed to believe the tiling
was not working because their actions were not yet correct—a process conception.
Eventually students would decide that the pentagon tessellation was impossible. They
usually pointed out a structural problem, the inevitable presence of a gap, to explain the
failure—an object conception.
Epistemology of Tilings
There is evidence that children sometimes had an idea of what the tiling looked
like, and perhaps even some of its structure, before they used any tool to construct it. The
extent to which they had a clear idea of the tiling in advance varied considerably. Usually
the descriptions during the oral phase were vague and local. Global structure was rarely
discussed. Oral descriptions were sometimes hard to understand, even on questioning.
The students would frequently express some doubt about the tiling during the oral phase,
which also contributed to the vagueness. If the tiling was more familiar, the description
was less vague. For example, most students were familiar with the 44 square tiling.
However, none of them could describe in any detail at all what the triangle tiling would
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look like, even though they were certain it existed. A typical partial description of a tiling
is the following oral description of a rhombus tiling by Michelle.
Scott: How would it work?
Michelle: Well if you, you could put one that goes (like) right there [on one
side of the original tile].
Scott: Mm-hmm.
Michelle: And there [a second side].
Scott: Mm-hmm.
Michelle: And there [a third side]. And it just sort of goes around and around
and around. [Spreading hands around table] Out out out.
Notice that this is primarily local in its description. There is no clear description
of the global structure, only how the tiles will fit around the initial tile and that it will
somehow spread ―out out out,‖ presumably by fitting around each tile in the same way.
When students say they could put a tile on each side of the initial tile, they typically do
not describe the orientation of the tiles, even when several orientations are possible. On
the other hand, Moses gave a global description of the rhombus tiling, noting that other
rhombi would line up from the first rhombus in four directions, all having the same
orientation.
Scott: Tell me first [tapping tile] and then we‘ll, we‘ll try it after you
describe what you‘re gonna do.
Moses: Wait. One, [pointing to sides of rhombus] like that like (one) in
there [pointing to spaces at opposite sides of rhombus], and they‘re
all gonna be facing the same way but they‘re gonna be, like it‘s
gonna be kinda like a diagonal line [indicating row coming from
two opposite sides of rhombus].
Scott: Ok.
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Moses: And then this [same gestures showing row from other two sides],
it‘s gonna be just a straight line of them.
Scott: Mm-hmm.
Moses: All facing the same way.
Drawings of tilings were usually more detailed than the initial oral descriptions.
Michelle admitted she really did not know what the hexagon tessellation would look like,
but ―I know that you can, you can connect them.‖ She then traced the hexagon pattern
block with pencil on paper until she discovered what the tiling actually looked like. She
perhaps recollected that she had seen the hexagon tiling before, but had not retained any
knowledge of its structure.
Students sometimes explicitly said they saw new ideas as they created their
tilings, confirming that the final tessellation was a combination of initial ideas and new
discoveries as they tiled. Moses began with his description of simple rhombus rows, but
then suddenly saw that he could make an aesthetically more interesting tiling by
alternating the orientations of the rows. On at least five occasions, Marie said she had just
had an idea while creating her tiling, changing her initial idea for the tiling.
This change from their initial ideas also occurred during the drawing phase.
Moses discovered as he was drawing his first triangle tessellation that six triangles make
a hexagon. At that point he reconceptualized his tiling and planned a radial hexagon
pattern of triangles.
Outhred and Mitchelmore (2000) noted that physical tiles prestructure square
arrays in such a way that students do not have to understand the structure of the tiling in
order to create them. This prestructuring frequently occurred for the tessellations created
by the children in this study as well. When Kobe began to construct his first triangle
tiling with pattern blocks, he placed the first six tiles next to each other and created a
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hexagon, at which point he said, ―I don‘t really know how I did that.‖ He later used this
hexagon pattern to help structure his tiling.
Physical tilings were often different from both the drawing and oral phases,
indicating an evolution of ideas, even when students claimed to be using their initial idea.
In one case, Kelsey said that her physical rhombus tiling was the same as her drawing,
but her drawing and oral description seem to have been of the standard tiling with all tiles
in the same orientation, whereas her physical tiling was semi-random. This is not because
Kelsey thought all tilings were the same. She continued to make other tilings that she
claimed were different.
Students sometimes talked about seeing tilings in their head. When asked how he
knew the S-tetromino could tile before trying it, Kobe traced with his finger where the
second tile would go and replied, ―I just pictured how it would work in my head.‖
Michelle explained how she could fix a flaw in her newly created S-tetromino tiling and
said, ―So mainly in my head I‘m looking at a quilt pattern.‖ Moses said he was sure his
triangle pattern would continue forever because ―I can picture, I can picture a lot of
things in my mind.‖
Of course, the structure they imagine in their head may not fit the structure that
appears in drawn explanations. Children often knew not to depend on imprecise
drawings, especially the children from the school with the French curriculum. Michelle
struggled with her explanation that trominoes could be drawn in patterns that did not fit
the dot paper lattice. When she drew Figure 42 on page 152, she saw that she had not
placed the tiles in the outer layer quite the way she wanted:
Scott: Ok. And you‘re sure that would work?
Michelle: Mm. It says so.
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Scott: It says so?
Michelle: From the drawing, no, but in my head yes, so it‘s like
[unintelligible].
Scott: Ah-ha. So in the drawing it doesn‘t look like it would work, but in
your head you‘re sure this will work?
Michelle: Yeah.
She also knew to distrust answers she could see only in her head, ―‗Cause in my
head most of the time I can make anything work.‖
In fact, students frequently had a mental model that could not work physically.
Kobe for example orally described the pattern in Figure 49 below, but when he tried it, he
saw that the tetromino on the left would not fit in the two-space gap left by the two
tetrominoes on the right.
Figure 49: Failed attempt envisioned by Kobe for fitting T-tetrominoes together
Sometimes students had an idea in their heads that they lost while trying to sketch
it. It is possible that their drawing conflicted with the structure they imagined in such a
way that it interfered with their initial idea, causing them to forget it. Or it may be that,
rather than trying to accommodate the new situation into their initial schema, they
discarded the schema altogether. ―Losing‖ an idea is more likely when the idea is vague
to begin with.
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Children’s Strategies
Students used a variety of strategies to create their tessellations. Strategies were
coded and counted. The most popular strategies used, in order of preference, were as
follows:
Simple rows. The most common strategy was to arrange tiles in simple rows with
straight borders between the rows. This was an effective strategy for simple
shapes such as parallelograms and rectangles. This strategy was also used with
squares and triangles to create the 33.4
2 tessellation, or some variation thereof.
Random (or semi-random). Students frequently used the ―Fit one at a time‖
strategy to create random patterns without symmetry. In many cases, this became
a semi-random strategy as students created sections with patterns, but not one
global pattern. The semi-random strategy was particularly popular for the
rhombus. See for example Figure 50 below. Even if they did not deliberately
create local patterns, they usually looked for local patterns and units after the fact.
Figure 50: Semi-random rhombus tiling by Kelsey
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Fit one at a time. When students were not sure what to do, they simply began
placing one tile at a time, as with a puzzle, to see what patterns would emerge.
Sometimes no pattern would emerge, in which case they continued to place tiles
with a random strategy. But often there would be a clear pattern, though not
organized by units or rows. Fitting one at a time was the most common strategy
for hexagons, whose structure was not always evident to the students, but which
was nevertheless not random. This strategy was also frequently used for rhombi,
equilateral triangles, and parallelograms, as well as the Regular Polygons task.
Unitizing. Students often grouped tiles into units that were easier to tile with.
Students did this most frequently when triangles were involved. They also
unitized with the L-tromino, as in Reynolds and Wheatley‘s (1996) study. All
students used the unitizing strategy at least twice, except Marie.
Growing. Students sometimes used some form of growing strategy to make a
small pattern larger and larger. For example, Marie created the tower pattern on
the left in Figure 51 below. When challenged to continue it, she expanded it both
vertically and horizontally, as in the photo on the right. In some cases, students
used dilation symmetry to grow their tiling. For example, Moses created large
triangles with the triangle pattern blocks which he kept adding on to in order to
make still larger triangles. This strategy also describes radial tilings, such as the
tilings in Figure 52 on page 178.
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Figure 51: Two stages of Marie‘s growing tower in the software environment
Alternating rows. For aesthetic reasons, students often preferred creating rows
with alternating orientations. Moses and Kelsey used this strategy frequently, but
every student used it at least once. Edge-to-edge row tilings of equilateral or
isosceles triangles were perceived to be in alternating rows because the triangle
orientations reversed with each row. This alternation may have made triangle
tilings slightly more difficult for the students; they commented on it several times.
Moses thought alternating triangle patterns looked like rows of teeth.
Complex. Sometimes students used a complex strategy that used neither rows nor
unitizing. For example, sophisticated tilings with 6-fold symmetry such as the one
in Figure 37 on page 148 were organized in overlapping units. Such tilings took
time, but students were able to analyze and defend them in a mathematically
correct way. Kobe described how the translation symmetry would continue the
pattern.
Other strategies were used less often. These included making complex rows
without straight borders, creating brick wall type tilings, making a two-dimensional grid,
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and creating spiral tilings. Another uncommon strategy was random units, which put tiles
together in units, but then placed the units together randomly instead of in rows.
Students, tools, and prototiles were coded and compared to strategies used.
Strategies depended more on the prototiles than on the student or the tool. For example,
the ―semi-random‖ strategy was one of the main strategies for the rhombus tilings, the
dihedral square and triangle tilings, and the pentagon tilings, but was hardly ever used for
the other tasks. The ―alternating rows‖ strategy was the main strategy used for the acute
isosceles triangle, and was frequently used for the parallelogram and the equilateral
triangle. But it was hardly ever used for other shapes.
There were few tendencies among the students. The popular strategies were used
by all six children with few exceptions. Rachel tended to avoid any random strategy.
Marie did not use the ―unitizing‖ strategy. Otherwise, everyone tended to use the same
strategies, though not always the same strategy for the same task.
The tool seems to have had little impact on the strategy used. The semi-random
strategy was used mostly with pattern blocks, but was also used with other tools. The
spiral strategy was only used with drawing on blank paper. Otherwise, there were no
clear tendencies associating strategies with students or tools, only with the prototile being
used. This is natural because certain strategies really only make sense with certain
shapes. The L-tromino cannot be used to make simple rows with straight borders unless it
is first put in rectangular units. No simple units can be made with the hexagon, and there
is only one way to fit hexagons together, so the ―fit one at a time‖ strategy makes sense.
However, most shapes can be tiled in more than one way and it is interesting to note that
certain strategies seem to be frequent modes of thinking among all students.
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Children’s Reasoning
Students frequently did not give a clear reason to justify why their tessellations
were valid or to explain how they knew how to construct the tessellation, even when
questioned. When they did give a reason, this was coded and counted. Reasons were
usually one of three kinds: justification by experience, empirical justification, and
justification by known unit.
Justification by experience. Students frequently said they knew a certain shape
would tile because they had seen it before. Kelsey knew that the hexagon could
tile because she had seen walls of hexagon tessellations in the film A Bee Movie.
Other students also mentioned honeycombs either before or after creating the
hexagon tessellation. (This visual, real world connection is actually an aesthetic
quality influencing their thinking.) During the Regular Polygons task, Michelle
made the squares and triangles tessellation on the left in Figure 52 below. Later in
the interview she made the tessellation on the right. When asked to justify how
she knew the second tessellation would continue to work, she suddenly realized
that there was a correspondence between the two tessellations. She had me call up
a copy of the previous tessellation so that she could compare and show me how
they had the same structure.
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Figure 52: Two regular polygon tilings by Michelle
Empirical justification. Students were frequently certain that a tiling was
possible simply because it was working so far. This inductive reasoning was
sufficient to assure them that the tiling would always continue to tile. Students
who used the ―fit one at a time‖ or ―random‖ strategy seemed to be satisfied that
the tiling was valid as long as the tiles kept fitting. Moses justified one of his
tilings by saying, ―I‘m just thinking, I‘m, I‘m not guessing but I‘m just, based on
what has happened already I think it should work.‖ Rachel justified one of her
polyomino tilings by saying, ―Because like, once (you‘ve) like make one, it just,
you have a space for another, and then a little space for another.‖
Justification by known unit. If copies of the prototile can be put together to form
a familiar unit that is known to tile, then the prototile can also tile. For example in
Set A (Basic Tilings), students often knew or discovered that equilateral triangles
can be put together to make a rhombus or hexagon. Students knew from the two
previous tasks that the rhombus and the hexagon could tile, therefore the triangle
could also tile. In Set E (Triangles), students who succeeded in making rectangle
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or parallelogram units usually showed how those units could be used to make a
simple periodic tessellation. Sometimes they were not successful with the new
unit, but they were still confident a tiling with the unit was possible.
Other justifications were used more rarely. Occasionally students would say a
shape was similar to another shape that was known to tile. Rachel said she knew the
rhombus could tile, ―‗Cause it‘s just like a square really, except it‘s kinda more,
smooshed together.‖32 Kobe sometimes used a transformational argument, showing how
his rows or units would repeat periodically.
Several students used an unusual subunit justification. Instead of putting tiles
together to make a larger unit, they broke the tile down into smaller subunits. Kobe
noticed that the parallelogram could be decomposed into four triangles. Rachel saw that
the hexagon could be broken down into six triangles. Both argued that the larger shape
must tile because the smaller shape can tile. From a mathematical viewpoint, this is not
quite correct (not all polyiamonds tile), but such reasoning can help to give an intuitive
understanding of how the shape might tile by imposing it on an imaginary triangular grid,
the same way polyominoes are imposed on a square grid. Moses also used this strategy
by noting that a hexagon could be broken down into two trapezoids, which he apparently
was more familiar with.
Symmetry may have been an unstated reason why students believed some of their
tessellations would work. When a pattern is symmetric, only part of the pattern needs to
be validated in order to be sure that the entire pattern will make a tessellation. Students
who made symmetric tessellations were usually more confident of their answers, as I
discuss later.
32 Steven Greenstein, in his doctoral dissertation (2010), shows how this intuitive qualitative
transformational knowledge is common in children and could be developed in educational settings.
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Other General Results
It is worth noting a few other cross-task results.
Dimensions. It was rare for students to see two dimensions simultaneously. They
usually thought locally or in linear rows. Many patterns were created either in rows,
which allow students to focus on one dimension at a time, or in units, which are local
finite patterns that are then often organized in rows. There were exceptions however.
Michelle showed how the parallelogram could tile on her pre-test by taking a ruler and
drawing a parallelogram grid, as in Figure 53 below. Moses demonstrated two-
dimensional thinking in his oral descriptions of both rhombus and L-tromino tessellations
by saying that the tiles or units would go off in rows in two different directions.
Figure 53: Tiling of a parallelogram on Michelle‘s pre-test
Orientation. Students had differing opinions on whether rotating a tessellation
created a different tessellation. Marie and Michelle both thought that each orientation
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should be counted as a new tessellation. Kelsey and Kobe insisted that orientation did not
matter.
Confidence. Levels of confidence were coded and compared to other codes.
Students displayed a great deal of confidence in their answers, even when those answers
changed. Strong confidence usually correlated with a mathematically accurate
understanding of fit and mathematically valid tilings. Confidence correlated even more
strongly with symmetry, as I discuss later.
AESTHETIC INFLUENCES
―Whoa! Ooh! Wow!‖
(Collective reaction at private school on seeing the first two Alhambra tilings)
In this section we look at the aesthetic values that influenced students‘ strategies
for creating tilings and for judging tilings. Children frequently expressed their aesthetic
criteria explicitly. I identify most of these explicit judgments with the evaluative role of
mathematical aesthetics. In addition to these explicit comments, children seemed to have
implicit preferences for the ways they created their tilings, particularly with respect to
visual symmetry. I identify these preferences with the generative role of mathematical
aesthetics. Occasionally children made choices of what problem they would work on or
what tool they would use. These choices come from the motivational role of
mathematical aesthetics.
In this section I first highlight a few results of the six aesthetic evaluation tasks
and then discuss cross-task observations from all interviews. The cross-task observations
are organized according to whether they are evaluative (Children’s Evaluation Criteria,
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page 187) or generative (Children’s Creation Criteria, page 199). Instances of the
motivational role are noted when they occur.
The aesthetic evaluation tasks were done during the last interview so that they
would not influence students‘ thinking in the other tasks. Further details of the results can
be found in Appendix E.
Student Tilings. Students were given cards with tilings they had previously
created in the other tasks and asked to choose tilings for four rooms of a house. The
charter school chose a finite pattern with slightly overlapping tiles for the bathroom
because it looked fun and resembled a sun. The private school almost picked this pattern
too, but Kobe insisted it was not valid. Both schools chose the 4.82 tiling for the kitchen.
Both schools chose one random pattern and both schools chose one pattern with six-fold
symmetry among their four choices.
Student Sorting. For this and the next two tasks, students were shown cards with
tessellations that others had created. In this task, students were asked to sort the cards into
groups and choose favorite groups and tilings. There was no coordination of criteria in
sorting the tilings to make groups. The private school sorted tilings into six groups with
criteria such as ―blue,‖ ―mazy,‖ and ―crazy.‖ The charter school made seven groups by
using criteria such as ―colorful,‖ ―same colors,‖ and ―maze.‖ Most of the preferred
groups of sorted tilings at both schools were associated with color.
Comparisons in Pairs. I asked students to choose their favorite from among two
(or more) tilings I showed them. Students may have had a slight preference for symmetric
patterns (see Appendix G), though both schools chose the random tiling on the left in
Figure 54 below over the tiling on the right. The explicit reason for choosing the random
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tiling was for its colors. The private school said the rejected tiling had bad patterns
(perhaps because it has no reflection or rotation symmetry).
Figure 54: Two of the tilings from the Comparisons in Pairs task
Professional Preferences. Students were asked which tilings an artist and a
mathematician would choose. Both schools thought the artist would like the colorful,
random pattern on the left in Figure 54 above. Both schools thought the mathematician
would pick the tiling in Figure 55 below because of its illusion of three dimensional
cubes. Other suggestions were also made for both professionals.
Figure 55: Both schools thought the mathematician would prefer this tiling.
Alhambra. Students were asked to put seven postcards of Alhambra tilings in
order from best to worst. Both schools preferred the tiling in Figure 62 on page 195. The
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least favorite tilings were the two in Figure 56 below because they were too simple and
boring. (The mathematicians, on the other hand, chose the tiling on the left in Figure 56
as one of their favorites. It may be that the symmetry was not as evident for the children.)
Figure 56: The two least favorite Alhambra tilings at both schools
Escher. Students were shown many Escher tilings from three books and asked to
award three prizes for best tilings. Both groups chose the tiling in Figure 63 on page 195
among their top two choices. The private school chose the tiling in Figure 57 below for
―2nd
prize‖ because of its perceived Asian style.
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All M.C. Escher works © 2011 The M.C. Escher Company - the Netherlands.
All rights reserved. Used by permission. www.mcescher.com
Figure 57: Escher‘s ―Regular Division of the Plane #58,‖ the second most popular Escher
tiling at the private school
All of the aesthetic evaluation tasks were organized as group activities where
students were told to discuss and make unanimous decisions in selecting the best tilings.
In some ways the revealed aesthetic values were diverse and subjective; in other ways
they were shared among all students. When it came to choosing individual tilings, there
was frequently disagreement between the students. Unanimity was often achieved by
compromise rather than coming to agreement over which tilings were best. In this way,
the students‘ aesthetic values were seen to be subjective. However, analysis of the values
expressed in their discussions revealed many common themes that were shared by all
students. As students discussed the aesthetic criteria, they would usually agree that a
tiling possessed certain criteria. This was objective. However, the decision to value one
criterion over the other was subjective. Everyone might agree that a particular tiling had
vibrant colors and exciting star patterns whereas another tiling had an interesting maze-
like quality and looked futuristic. Where subjectivity entered was in deciding which
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aesthetic criterion was most valuable. One student might prefer the vibrant colors of the
first tiling while another student might choose the maze-like quality of the second. This
was a matter of personal taste. Subjectivity might enter also in preferring certain colors
over others, or in the real world connections they made. A tiling might remind one child
of something at home, which would be outside of the other children‘s experience.
However children usually agreed on the presence of various aesthetic criteria in the
tilings, even if they disagreed over which criteria were most important.
Children sometimes mentioned more than one aesthetic quality for a tiling and I
imagine that the children always saw several aesthetic qualities, just as most people do.
Usually, however, children would express only one aesthetic value, which attracted their
attention above the others, unless the tiling was extremely appealing, in which case they
might express several qualities in its favor.
There were 228 instances of aesthetic expression from the students during all
interviews, and 99 instances from the mathematicians in their interviews. As I described
in Chapter 3, there were 59 aesthetic criteria that emerged from the coding of these 327
instances of aesthetic expression. These verbal aesthetic criteria were almost33 always in
the context of evaluating a tiling. The criteria were grouped into 18 aesthetic themes.
Each theme represents a type of aesthetic value which might be expressed positively or
negatively. The themes represent all codes that could be grouped with other codes, or
codes that could not be grouped but that occurred more than twice. There were 14 codes
that could not be grouped and occurred only once or twice. Most of the following
analysis is done on the 18 aesthetic themes, which represent 94% all aesthetic expressions
in the interviews.
33 The only exception is when Mo explained how he was making his tiling symmetric because he liked the
patterns that symmetry created.
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Analysis of the expressions revealed both diversity and unanimity. Students relied
on a wide range of aesthetic criteria in both their group discussions and in their individual
interviews. However, there were no clear preferences for any one aesthetic theme during
the group interviews. Both schools relied on all of the top 16 aesthetic themes during
their group interviews. There were very few differences between individuals revealed in
the individual interviews. (These are discussed below.)
Children’s Evaluation Criteria
The evaluative role of aesthetics is to judge the mathematical results either of
one‘s own work, or the work of others. This was studied by analyzing children‘s
expressions of aesthetic judgment.
Children could not always express why they liked certain tilings. Some tilings
simply looked ―cool‖ to them and they were not sure why. But when they could express
their reasons, the aesthetic criteria they used were rich and diverse. Table 6 below shows
the 18 aesthetic themes that emerged from the coding analysis of all interviews, in order
of frequency among students. This includes all aesthetic expressions. Most of these
expressions were evaluative, though a few concerned the creation process. The
percentages are calculated relative to each group.
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Table 6: Aesthetic themes
Theme Relative Frequency Description
Students Mathematicians
Real World Connection 21% 5%
Students were excited when the tiling, or a part of the tiling, suggested a real world object or context.
Color 18% 2% Color could be good or bad. Diversity of color was good.
Complexity 14% 16%
Simple tilings were often “boring.” Overly complex tilings were confusing. The best tilings were intriguing.
Uniqueness 6% 4%
Tilings that were different from what students had seen before were attractive. Students often called them “creative.”
Energy 6% 0% Students liked tilings that seemed “crazy,” “energetic,” or suggested exploding patterns.
Interconnectedness 5% 1% Students liked tilings that recalled mazes, puzzles, paths, or knots.
Dimensionality 4% 13%
Tilings which suggested surfaces other than the plane of the tiling were said to be “3D,” which was appealing to all. Mathematicians disliked tilings with only one interesting dimension.
Units 4% 1%
Students were impressed when they found special shapes made by several tiles. If simple units could not be formed, this might be bad.
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Table 6. cont.
Theme Relative Frequency Description
Students Mathematicians
Tiles 4% 4% The shape of the prototiles played a role in several aesthetic judgments, e.g. the animals in Escher tilings.
Culture 4% 2% Tilings were attractive if they suggested a foreign style such as “Egyptian,” “African,” or “Asian.”
Alternation 2% 1% Patterns that had alternating orientations were more interesting.
Symmetry 2% 5% Symmetry was a positive factor, but was rarely mentioned explicitly by students.
Surprise 1% 14% Mathematicians particularly enjoyed tilings that surprised them or were thought provoking.
Validity 1% 8% Mathematical validity (e.g. the three criteria) was an important aesthetic criterion to mathematicians.
Connectedness 1% 5%
Tilings that connected two different ideas, or that made a connection to ideas exterior to the tiling, were appealing. Mathematicians liked multiple possibilities in interpretation.
Gender 1% 0%
Students (of both genders) reacted against two “girlish” tilings: one having a pink theme and one having flowers.
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Table 6. cont.
Theme Relative Frequency Description
Students Mathematicians
Folds 1% 3% People were intrigued by tilings that seemed to be folded or foldable in surprising ways.
Randomness 0% 6% Mathematicians found randomness sometimes appealing, sometimes not.
Other (not grouped by theme)
5% 9% A wide diversity of qualities were mentioned only once or twice each, such as fun, futuristic, or relaxing.
By far the three most important aesthetic themes were Real World Connection,
Color, and Complexity. Together, these accounted for over half the children‘s aesthetic
expressions.
Real World Connection could be a real world object or symbol that the tiling, or
a part of the tiling, reminded people of. For example, Rachel chose her obtuse scalene
triangle tiling (see Figure 58 below) as her favorite because it reminded her of a vampire
coffin. The Real World Connection theme also refers to occasions when a tiling was
valued because it seemed appropriate for, or was frequently used for, a certain real world
tiling task. For example, the semi-regular 4.82 tiling in Figure 59 below seemed
especially appropriate by both groups of students for use as kitchen flooring, apparently
because they had seen this tiling on kitchen floors before. Some tilings seemed good for
stained glass windows. If the real world connection was repulsive or boring, this aesthetic
could be negative. For example, one tiling reminded Kobe of a ―sewer gate.‖ Real world
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connections were used by most students during the creation tasks, as well as the sixth
interview. If a real world object was perceived in part of their created tiling, this would
attract their attention. Marie especially enjoyed comparing her created tilings to windows,
bookcases, ladders, stairways, farms, houses, robots, judges, cartoon characters, cats, and
flowers.34
Figure 58: Rachel‘s favorite from her created triangle tilings
Figure 59: 4.82 tiling chosen by both schools for the kitchen
Color elicited a variety of comments, which were of two types. (1) Specific
colors could be either good or bad. (2) Having many colors was good; insufficient
34 Many of these comments were not counted in the coding if they served neither to guide Marie in her
tiling creation, nor to evaluate her tilings. If we decided to count them, the Real World Connection
frequency would have a slightly higher percentage.
192
coloring was bad. In the following example, students begin to discuss reasons from three
different aesthetic themes, including Color, Uniqueness, and Interconnectedness. The
tilings they discuss are Tiling J in Figure 60 below and Tiling P (a 4.82 tiling with green
squares and octagons in two shades of blue).
Scott: Why do you like J better than P?
Moses: It‘s more colorful and, it‘s more, …
Rachel: It‘s more unique. (I like that.)
Moses: … not co-, yeah it‘s more unique …
Moses: … and it looks like a pathway around.
Scott: How is it unique?
Rachel: If it was just, if it was just maybe uh like a purple background it
w-, it wouldn‘t be as interesting, but it‘s all these different colors
for (that background).
Figure 60: Tiling J in the Comparisons in Pairs task
Complexity. People seemed to think that tilings ranged from too simple to too
complex. The best tilings were somewhere in between. Overly simple tilings were
considered boring. The children sometimes said that anybody could do such tilings.
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Simplicity could also be a good quality, when appropriate. For example, students found
the 4.82 tiling in Figure 59 above to be ―kinda basic.‖ But they also felt that kitchens
should have simpler tilings, so this tiling was appropriate for that purpose. More often,
everyone seemed to prefer tilings that were moderately complex. Such tilings either had a
variety of shapes, or else the shapes were used in a complex way, such as the L-tromino
tiling in Figure 61 below, which had dilation symmetry that the students at the charter
school got excited about.
Figure 61: L-tromino tiling perceived to have pleasing complexity
Differences. Different students had different aesthetic preferences. During the
individual interviews, Kelsey only made four aesthetic comments (three from the
Alternation theme and one from the Uniqueness theme). Rachel made 14 aesthetic
comments from 10 different themes. The other students fell between these two extremes.
No single aesthetic theme was used by all students during the individual interviews. Most
students showed no clear preference for any particular theme, though Kelsey‘s comments
were mostly about alternation and half of Marie‘s comments concerned real world
connections.
194
There were no clear preferences when students perceived more than one aesthetic
quality in a tiling. Choices had to be made between tilings with different aesthetic
qualities. There was no hierarchy of themes to guide these choices. Sometimes students
preferred a real world connection over having many colors. Sometimes they preferred a
tiling with good color over one with a real world connection. The children disagreed
among themselves much of the time as to which tilings were aesthetically the best, often
because of these conflicting aesthetic possibilities. But in spite of their disagreements,
they tended to use the same themes of Color, Complexity, and Real World Connection
most of the time to defend their choices.
Students were not always in disagreement. They frequently agreed when a tiling
was ―boring‖ or when it was colorful. Both schools independently chose the tiling in
Figure 62 below as their favorite of the seven Alhambra tilings. Both groups cited criteria
from the themes of Energy and Color to defend this choice. Both groups also
independently chose the Escher tiling in Figure 63 below as one of their three prize-
winners out of the 54 tilings they considered. Both groups cited criteria from themes of
Color, as well as the fantasy/mythological element, to explain this choice. Both schools
were impressed with the way differently colored animals faced in opposite directions in
this tiling.
195
Figure 62: The favorite Alhambra tiling at both schools
All M.C. Escher works © 2011 The M.C. Escher Company - the Netherlands.
All rights reserved. Used by permission. www.mcescher.com
Figure 63: Escher‘s ―Regular Division of the Plane #66,‖ a favorite tiling at both schools
Some other common preferences are illustrated in Table 7 below, along with the
aesthetic themes cited to explain the preference. In each case, the tiling on the left was
unanimously preferred at both schools over the tiling or tilings on the right during direct
comparisons, though not for the same reasons. Note that in the first pair the theme of Real
World Connection was used to select the first tiling, not the second; it reminded the
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children of waves and a grandfather‘s bathroom tiling. Another unanimous preference is
illustrated in Figure 54 on page 183.
Table 7: Some unanimous preferences at both schools during Comparisons in Pairs
Preferred Not preferred Aesthetic themes
Charter school: Color, Complexity
Private school: Energy, Real World Connection
Both schools: Color
Private school: Interconnectedness, Energy
Charter school: “It’s fragments …”
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Mathematicians’ Evaluation Criteria
I only interviewed two mathematicians informally, but their responses can still
throw some light on interpreting children‘s mathematical aesthetic. These interviews
were informal because the goal of this research was not to evaluate mathematicians‘
aesthetics. Other studies referenced in my theoretical framework have already
investigated this question. The goal of interviewing two mathematicians was simply to
validate the value of the children‘s evaluation aesthetics. I think it is important to place
children‘s criteria in the context of the aesthetic criteria valued by the mathematics
community. This helps answer my research question ―What mathematical value is there
in children‘s aesthetics of tessellations?‖
For the most part, mathematicians used the same aesthetic themes as the students,
but tended to use the themes very differently. As Table 6 above shows, the only
significant aesthetic theme not used by mathematicians was the Energy theme. Students
frequently were impressed by the energy certain tilings seemed to possess. Students used
terms such as ―crazy,‖ ―lightning,‖ ―weird,‖ ―shocking,‖ ―energetic,‖ and ―exploding,‖ as
well as several invented words like ―Pfoom!‖ to describe tilings that made them excited.
The only significant aesthetic theme not used by the children was Randomness.
According to the mathematicians, randomness was usually appealing, but sometimes not.
This might apply to complete randomness or to semi-randomness where there are
elements of order and chaos combined. Either could be appealing or unappealing to the
mathematicians. Children never explicitly mentioned randomness in their evaluations,
though they did sometimes talk about the appeal of patterns.
Aside from Energy and Randomness, children and mathematicians shared the
same aesthetic themes. The difference was in emphasis. Whereas children turned mainly
to real world connections, color, and complexity for their aesthetic criteria,
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mathematicians turned mostly to complexity, surprise, and dimensionality. The children‘s
themes of real world connections and color were not frequently mentioned by
mathematicians, and the mathematicians‘ theme of surprise was not often brought up by
children. Mathematicians were especially appreciative of tilings that were ―thought
provoking.‖
The mathematicians used the Symmetry, Validity, Folds, and Connectedness
themes much more often than the children. The children used the themes of
Interconnectedness, Units, and Alternation much more often than mathematicians.
Both children and mathematicians shared the importance of complexity and
dimensionality. The only difference in the ways they approached these two themes is that
one of the mathematicians frequently brought up the lack of two interesting dimensions
in the tilings, a quality never considered by the children or by the other mathematician.
Both children and mathematicians shared a fascination for tilings that had the illusion of
having three dimensions.
Recall from the theoretical framework that mathematicians in general usually
identify criteria of significance, simplicity, surprise, connectedness, and visual appeal in
defining a mathematical aesthetic. Mathematicians‘ preference in this study for the
themes of Complexity and Surprise would align with the mathematical aesthetic criteria
of simplicity35 and surprise discussed in the theoretical framework. Connectedness was
noted by mathematicians five times. It was difficult to notice criteria of significance
because the tilings were not mathematical theorems that would contribute significantly to
the field. However, the theme of Validity may have stood for the place of significance in
these evaluations. Mathematicians mentioned the Validity theme eight times—the fourth
35 The Complexity theme contains aesthetic criteria of being simple as well as criteria of being
complicated.
199
most common theme for them. All of the criteria could probably be included in the
domain of ―visual appeal,‖ because tessellations are aesthetic primarily because of their
visual qualities. Symmetry is an especially important quality for visual appeal and was
noted by mathematicians much more often than children. The one theme that
mathematicians used with great frequency that does not seem to line up with the five
criteria is Dimensionality. However, Dimensionality was frequently brought up because
tessellations are supposed to be two-dimensional, and one-dimensional patterns or
apparent three-dimensional patterns are breaking this expectation of what a mathematical
tessellation is defined to be. This criterion may therefore be related to significance and
surprise.
Note that it is not so clear how to relate the children‘s preferred criteria of real
world connections and color to the five mathematical aesthetic criteria, except in the way
all criteria are related to visual appeal. Units and Alternation are mathematically
important themes, but they are mathematically simple. They are therefore probably more
important to children, for whom they are still fresh, than to mathematicians, for whom
these ideas are well known and therefore less beautiful. It is not surprising that children
used themes of Units and Alternation far more often than mathematicians.
Children’s Creation Criteria
In addition to the verbal remarks made by students in the evaluation of tilings,
there were indications that students were also guided by aesthetics in the creation of their
tilings. This is the generative role of mathematical aesthetics. They rarely talked about
these driving forces, though this generative aesthetic sometimes became explicit when I
asked them why they preferred certain tilings they had just made.
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By far the most important apparent aesthetic in the creation of children‘s tilings
was visual symmetry. Because symmetry is both an aesthetic criterion and a
mathematical (cognitive) property, I discuss it later. (See Role of Symmetry on page 210.)
Aside from symmetry, there is evidence that children also used criteria of complexity,
uniqueness, alternation, and units in creating their tilings. All four of these qualities were
also themes expressed verbally during the evaluation tasks.
Students had no control over the color of the tiles in their creation tasks. However,
in the Regular Polygons task, the more prototiles they used, the greater the variety of
color. Students mentioned this as a reason for preferring certain regular polygon creations
over others.
Michelle seemed to prefer radial patterns when she could make them. This may
have been a strategic choice or an aesthetic choice. Other students seemed to prefer
making periodic patterns, but they sometimes made radial patterns. Students‘ strategic
choice to structure tilings in rows and columns may also have been aesthetic.
In fact, any of the children‘s strategic choices can be seen as aesthetic as well as
strategic. As I discussed in the theoretical framework, our aesthetic cognition is what
drives us to make our mathematical choices. This aesthetic is developed by our
mathematical experience. However, it is certain that complexity, uniqueness, alternation,
and units were considered aesthetic by the students because these qualities were also
expressed in their tessellation evaluations. I will discuss these four qualities in more
depth.
In addition to these criteria which apply to infinite tessellations, students also
seemed to favor convexity when creating finite patterns.
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Complexity. Students tried to make tilings that were neither too simple nor too
complicated. Moses began with the standard tiling for rhombi with all the tiles in the
same orientation. When he saw that it was possible to complicate this pattern by
alternating the orientation of the rows, he immediately changed his structure, even
thought this created difficulties for his perceived border problem. The O-tetromino is
simply a square and all the students knew it could be tiled as a regular 44 tiling. Kelsey
decided to complicate her tiling by staggering the squares in a brick wall type tiling,
which she explicitly said she preferred. A clear example where a student chose moderate
complexity is in Kobe‘s three I-tromino tilings in Figure 64 below. The first one was the
one at the bottom left—the standard edge-to-edge tiling. The second one was the one at
the top left with a moderately complex 3×4 unit laid out in alternating orientations. The
third one was the one on the right, which has a 7×12 unit and is so complex that he could
only create one unit. Kobe envisioned repeating this 7×12 unit without alternating the
orientation. Kobe explicitly preferred the second tiling saying the first was ―too simple,‖
and the third was ―too hard to make.‖ There were few instances where students preferred
creating random tilings. If they knew how to make a non-random tiling they usually did.
This may be partly because randomness is complex. One example of creating a random
tiling despite an awareness of a periodic tessellation would be Kelsey‘s attempts to make
a rhombus tiling. She seems to describe the standard tessellation and attempts to draw it,
though her drawing is not clear. (See Figure 65 below.) This partial understanding leads
her to attempt the standard tiling twice, but each time she creates a semi-random pattern
instead, apparently because of focus on local fit rather than global pattern.
202
Figure 64: Three I-tromino tilings of increasing complexity by Kobe
Figure 65: Kelsey‘s rhombus tessellation drawing
Uniqueness. Students deliberately tried to create tilings that were different from
what they had seen or created before. Especially during the open-ended Regular Polygons
task, students shied away from familiar tilings and tried to create something new. For
example, early in the Regular Polygons task, Moses put two octagons and a square
together. This could have been the beginning of the semi-regular 4.82 tiling, and in fact
Moses said, ―Yeah I see a pattern that I can make with that.‖ But rather than pursuing this
203
pattern, he tried to create something with greater diversity of tiles. For the next 18
minutes, Moses struggled unsuccessfully to create a more complex pattern, before finally
creating the 4.82 tiling just as time was up. The reason for avoiding this tiling may have
been its simplicity or its familiarity. Children from both schools confirmed during the
sixth interview that this was a familiar tiling. Kelsey‘s creation of a brick wall tiling with
the O-tetromino was probably also partially motivated by familiarity with the regular 44
tiling. Marie made two parallelogram tilings, which were identical except for orientation.
She preferred the one with tiles oriented vertically because it was more unfamiliar and
harder to realize in the computer environment.
Alternation. On several occasions, students took pleasure in creating alternating
patterns, perhaps as a way to make the tilings unique or more complex. Kelsey preferred
her alternating row tiling of acute isosceles triangles over the tilings she created using
rectangle or parallelogram units for the other two triangles in Set E. (See Figure 66
below.) She thought it was ―cool‖ how the triangles pointed up and down, ―like one
upwards and then one down and one up and one down.‖ Moses explicitly preferred
alternating the orientation in his rows of rhombi.
Figure 66: Kelsey‘s tilings of the acute isosceles, right scalene, and obtuse scalene
triangles
204
Units. When students noticed certain finite patterns appear in their construction,
they would deliberately create more of those patterns, sometimes constructing the entire
tiling with the unit they discovered, as Kelsey did in the two tilings on the right in Figure
66 above. When students saw hexagons in their evolving triangle tessellations, they
would frequently point them out and begin creating more hexagons to extend the pattern.
In the case of the triangle tessellation, unitizing was not needed to create the tiling; the
units appear to have been used mainly for aesthetic reasons. Even if their tiling was
complete, students sometimes looked for unit patterns in their completed tilings,
especially if the tilings were random. Marie placed the eight triangles in the second Finite
Tiling problem without unitizing. But when she was finished, she reanalyzed the tiling as
four diamonds. See also Rachel‘s analysis of her semi-random tiling in Figure 70 on page
212. Unitizing is related to the aesthetics of fit, which all students evidenced. After initial
hesitation at being chosen to do interviews that had something to do with mathematics,
Marie decided that she thoroughly enjoyed making tilings. They reminded her of puzzles,
which she liked. Composition and decomposition of shapes into other shapes is not
simply a cognitive skill children need to learn; it is also an aesthetic process that they
enjoy. This includes putting tiles together to make larger pleasing units as well as
breaking them down into smaller subunits, as a few students did. This generative
aesthetic of units (as well as fit) would be an example of the motivational role of
aesthetics as well.
Convexity. Tilings are infinite and are therefore trivially convex.36 But children
must first create finite patterns as they explore how to create their tessellations. Students
seemed to prefer convex patterns when creating finite patterns. Units used to create
36 A set is convex if the line segment joining any two points of the set always lies entirely within the set. A
tessellation occupies the entire plane, which is trivially convex.
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tessellations were always convex. When students sought a stopping point to explain their
tessellations, they would often complete a convex pattern, if possible, though there is no
particular mathematical reason to do so. Such patterns are easier to relate to the infinite
tessellation. I would frequently push them to extend their tessellations beyond these
convex patterns in order to test their vision of how the tessellation continues.
Real world connections and color have less empirical support from my data as
creation criteria, but are nevertheless likely factors in children‘s creation criteria because
they were such important factors in children‘s evaluation criteria. The real world
connection of honeycombs was noted by four of the children for the hexagon tiling and
may have had aesthetic value. Children could not color, but the aesthetic appeal of having
many colors motivated some children to include as many polygons as possible in the
Regular Polygons task, which increased the complexity of the patterns.
Non-tessellation aesthetic events. On a few occasions, children began aesthetic
tasks that were not tessellations. Kobe, towards the end of one interview, decided to make
a man with the polygons at his disposal. He knew this was not a tessellation. Michelle put
all five tetromino pieces together to see if she could make a convex shape with them.
This was a finite tiling and a mathematical task, as well as an aesthetic task, though it was
not part of the task of creating monohedral infinite tessellations with tetrominoes.
INTERACTION OF THE TWO PERSPECTIVES
Some analyses must be carried out from the cognitive and aesthetic viewpoints
simultaneously. (1) I look at how the tool influences the motivating aesthetic factors of
the task and also how it both helps and constrains the cognitive possibilities. (2) I look at
symmetry results, which can be understood both from a visual, aesthetic viewpoint and
from a mathematical, cognitive viewpoint. Through tools and symmetry, we see how the
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aesthetic and cognitive viewpoints combine for a more complete understanding of
children‘s understanding of tessellations. (3) I also look at how the previously discussed
aesthetic creation criteria relate to the cognitive perspective.
Tasks and Tools
The tool and other aspects of the task inevitably both limit and help the aesthetic
and cognitive possibilities for tiling in various ways. By imposing the constraint of
making a mathematical tessellation, the child‘s aesthetic was partially constrained by the
mathematical aesthetic of the researcher. I write ―partially‖ because students sometimes
disregarded these constraints. The children‘s cognitive abilities were influenced by the
tool, which sometimes limited them from actions they may have wanted to explore, and
sometimes helped the child to see possibilities they may not have seen otherwise.
Some tools made certain aspects of tiling easier and other aspects more difficult.
At least one student wanted to select and copy units on the computer, but this was not a
possibility in the software environment that we used. This limitation imposed a constraint
on the complexity of possible tilings. Another example of a limitation was color. No tool
allowed students to choose the color of their tilings, and therefore questions of color
rarely entered into consideration when students created their tilings. The only exception
was the Regular Polygons task, where each prototile was a different color. Some students
tried to use as many prototiles as possible in order to make the tiling colorful.
Students frequently gave different responses according to which tool they were
using. For example, when asked if she could tile with the obtuse scalene triangle, Kelsey
at first responded yes. She vaguely described how the tiles could be matched up with
congruent sides touching. When asked to draw her vision, she began the drawing in
Figure 67 below, which at first she thought would work. But then as she continued to
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draw and erase, she decided the tiling would not work. Finally, when she attempted the
tiling in the computer environment, she began by placing two triangles together in such a
way that they formed a parallelogram. She immediately saw she could continue such a
pattern and created the standard parallelogram unit tiling in Figure 68 below. This is not a
case of tiles pre-structuring the tessellation because there are many ways of putting these
triangles together. However the way the software environment worked did help Kelsey to
see structural possibilities that she probably would never have seen in the pencil and
paper environment.
Figure 67: Kelsey‘s attempt to draw a tiling with an obtuse scalene triangle
Figure 68: Kelsey‘s tiling of an obtuse scalene triangle in the software environment
When Michelle was asked if she could tile with the L-tromino, she described the
creation of rectangular units. However, when she drew the tiling on dot paper, she made a
random tessellation. When asked if there was another way to tile, she demonstrated the
beginning of a five-fold symmetric radial pattern using blank paper instead of dot paper.
(See Figure 42 on page 152.) Each tool seemed to solicit a different type of tiling. The
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variety may also have been due to an attempt to create unique answers, but the five-fold
symmetric radial pattern was only possible on the blank paper and she deliberately chose
the blank paper because she knew she could not realize her vision on the dot paper.
The nature of the tool influenced the type of discoveries children were likely to
make. In a pattern block environment, children always matched up sides. In the computer
environment, the tiles behaved as if the corners were magnetized, and therefore students
sometimes attached them at the corners rather than the sides. This behavior in the
software environment led Rachel in Set G to create a row of hexagons touching only at
the corners, from which she constructed the 2-isogonal 32.6
2/3.6.3.6 tiling in Figure 69
below by filling the gaps with triangles. The environment influences more than just the
result; it also influences thinking. Rachel stated during this task that the tiling must be
edge-to-edge, and she explained why, even though it was the fifth interview and she had
seen and created non-edge-to-edge tilings in the previous interviews. It is likely that the
magnetic behavior of the tiles in the computer program was influencing her current
thinking.
Figure 69: 2-isogonal tiling creation by Rachel in the software environment
In the software environment, rotating a tile was easy, but not nearly as easy as it
was for pattern blocks. In the software environment, students had to make a deliberate
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decision to rotate a tile. Pattern blocks spun so easily that students were constantly
turning them whether they needed to or not. This almost surely had an impact on the
results. For example, most of the students did not recognize the right angle in the Set E
right scalene triangle task because the prototile was at an unusual orientation on the
computer screen. As a result, most students did not realize they could create rectangular
units until they began playing with the computer tiles and accidentally created a
rectangle. It is quite likely that this discovery would have happened more quickly if the
tiles had been made of wood or plastic. The original non-standard orientation would not
have been kept because the students would have been constantly turning the prototile to
study it. No student failed to recognize the right angles in the polyomino tasks because
these were always introduced as plastic tiles. As in the software environment, the
prototile was static during the oral and drawing phases of the finite tiling tasks and had to
be rotated mentally. The constraint of doing this rotation in their heads during the oral
and drawing phases created difficulties for the students that they were able to overcome
when they could handle a pattern block during the final construction phase.
During the oral descriptions, students were limited by their inability to use the
environment to help their thinking. As a result, patterns described orally were simple and
local, with little or no concern for symmetry. When students could draw or construct,
they developed more complex tilings, sometimes in an attempt to be unique and creative.
Outhred and Mitchelmore had noticed that ―drawings may be seen as reflecting, or at
least as being guided by, students‘ mental images of the array structure‖ (2000, p. 146).
However, the act of drawing may sometimes bring students quite far from their initial
ideas. Students sometimes even said they saw new ideas as they drew their tessellations.
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Oral descriptions, though limited, may be much closer to students‘ initial mental models
of tessellations than drawings.
There were many other ways the tool shaped student thinking. The squares in the
plastic polyomino models seemed to suggest to some students that the task was one of
tiling units of squares rather than a single shape. The complexity of the Polydrons
prevented one student from understanding that the mathematical question was one of
tiling pentagons and that the tabs therefore had no mathematical importance. Dot paper
constrained the positioning of the tiles in such a way that small gaps and overlaps were
impossible; tiles might leave a large hole, but not a small gap. The software had an
automatic zoom which reinforced the endless, borderless nature of the plane. Every tool
shaped thinking in some way.
Context. The task shaped the children‘s thinking. The context of the task also
influenced the children‘s aesthetic choices. This was particularly evident during the sixth
interview where the context of tiling certain rooms of the house influenced which tiling
they chose. At both schools, the choice for kitchen floor was swift and unanimous for the
semi-regular 4.82 tiling. Both groups felt that this type of tiling was appropriate in the
context of kitchens. During another task for the sixth interview without the house context,
I asked students which of two tilings they preferred. Michelle answered, ―Um. It depends
for what. What.‖ In some ways aesthetics, like cognition, is situated.
Role of Symmetry
There is evidence that students used symmetry in their creations, usually in the
visual, aesthetic sense, which understands symmetry as a mathematical object, but also in
the transformational sense, which understands symmetry as a mathematical process.
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A coding analysis was made of the symmetry of all finite patterns made with
more than six tiles. Analysis was limited to finite patterns because it was not always clear
exactly what infinite tessellations students intended by their patterns. Analysis was
limited to patterns of more than six tiles because small patterns frequently have
unintended, accidental symmetry. It was found that 77% of all students‘ patterns with
more than six tiles were symmetric.
The word ―symmetry‖ was used by one student on the pre-test, but was never
used in the interviews. However, other terms such as ―pattern‖ were sometimes used to
refer to visual symmetry, and words like ―copy‖ were used to describe transformational
symmetry. Explicit descriptions of transformational symmetry were fairly rare. Moses
was one of the few to describe how he achieved reflection symmetry, noting that he was
not sure if the dihedral squares and triangles tessellation could work if he did not impose
reflection symmetry.
Scott: Now how do you know when to put a triangle and when to put a
square?
Moses: Um. I can see, where I should do it and then I, and then, so, then I
copy that on the other side …
Scott: Oh.
Moses: … to get a pattern.
Scott: [echoing] Copy it on the other side to make a pattern.
Moses: So like, I do it on one side then, and I copy it on the other.
Scott: Ok. Now why do you do that?
Moses: Because I, I like more like patterns, I‘m (good) ...
Scott: Because you like patterns?
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Moses: Yeah I like patterns better than, regular sh-, [unintelligible
mumbling]
Scott: Would it work if you didn‘t copy the pattern? If you didn‘t copy
the same thing on both sides would it still work?
Moses: I don‘t think, don‘t, know.
Scott: You don‘t think so?
Moses: I don‘t know.
Students rarely created random tilings if they could find a way to make a
symmetric tiling. If they did make a random tiling, they often tried to identify patterns
contained in the tiling. Michelle first made the semi-random rhombus tessellation in the
photo on the left in Figure 70 below. She then explained some of the patterns she
perceived in it and separated the tiling into four parts, three of which had their own
patterns (middle photo). She then identified the top left pattern as her favorite and
showed how it could be extended to make a symmetric tiling, in the photo on the right.
This final symmetric tiling was described as ―more easy‖ and ―you could do this forever
and ever.‖
Figure 70: How Michelle extracted a symmetric tiling from a semi-random one
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Once students obtained a symmetric pattern with a significant number of tiles,
they usually kept the pattern symmetric as it grew, which demonstrates intent to use
symmetry. It was rare for a student to lose any form of symmetry once it was achieved in
a sizable pattern.
All students used a variety of symmetry in their patterns. Analysis of symmetry
codes revealed no clear preference for any one type of symmetry. All students created
patterns using reflection symmetry, rotation symmetry and combinations of reflection and
rotation symmetry. All students created patterns at some point with 6-fold symmetry,
with order-2 rotation symmetry, and with vertical reflection symmetry. Half the students
also used horizontal reflection symmetry. There was only one instance of a student using
oblique reflection symmetry.
The type of symmetry appearing in children‘s patterns was often constrained by
the tile. The parallelogram naturally led to patterns with order-2 rotation symmetry.
Equilateral triangles, with or without squares, led to 6-fold symmetric patterns more often
than other shapes. Vertical reflection symmetry was usually achieved with isosceles or
equilateral triangles.
An attempt was made to analyze whether students preferred tilings with greater
symmetry over tilings with less symmetry in the aesthetic evaluation tasks, but the results
were inconclusive. See Appendix G for details.
Symmetry and confidence. Students expressed confidence about most of their
tilings. But students were even more confident of tilings that incorporated symmetry. For
example, one of the reasons Michelle preferred a six-fold symmetric pattern of squares
and triangles over a random tiling of squares and triangles was because she was more
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confident the six-fold symmetric pattern would work. This result agrees with the similar
finding in my pilot study (Eberle, 2010).
Codes for symmetry and confidence were compared in the data for this study. Of
the 88 times students created tilings with some form of symmetry, they displayed strong
confidence in their tilings 85 times (97% of the time). Of the 11 times students expressed
some form of doubt about their created tilings, the tilings lacked symmetry 8 times (73%
of the time).
Note that confidence did not correlate so highly with mathematical accuracy. For
example, oral and drawing phases were coded according to the accuracy of the student‘s
understanding of how polygons fit together. (Saying that only four triangles can fit
around a common vertex is not mathematically accurate, as students discovered during
the construction phase.) Of the 110 accurate descriptions of fit, students showed strong
confidence of their answer only 75 times (68% of the time). Of the 65 times students
expressed doubt over their answer, their answers were mathematically inaccurate only 30
times (40% of the time). Similar percentages can be found for other measures of
accuracy. Symmetry was a much greater predictor of confidence than mathematical
correctness.
The Cognitive Role of Other Aesthetic Criteria
As discussed in Children’s Creation Criteria (page 199), children‘s aesthetic
creation criteria played an important role in their creation of tessellations. These criteria
were clearly aesthetic. Most of them also seemed to play a cognitive role in the children‘s
work. We have seen how symmetry is simultaneously an aesthetic and cognitive factor in
tessellations. Units are also both aesthetic and cognitive. Units played a clear aesthetic
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role, as demonstrated by their use as an aesthetic evaluation theme. Units also played a
cognitive role, as demonstrated by their use as a strategy for creating tessellations.
The aesthetic theme of Uniqueness had the cognitive role of seeking new
mathematical structures. If students had not sought unique structures, they would have
only repeated familiar cognitive structures rather than explored ideas they had not yet
assimilated. By pursuing unique patterns, students created a wider variety of tessellation
structures than they would have otherwise.
The aesthetic factor of complexity is related to that of uniqueness. Complexity led
students to create tessellations that pushed their cognitive limits without pushing so far
they could no longer grasp the mathematical structure of their creations. Kobe‘s I-
tromino tilings in Figure 64 on page 202 are an example of this. Kobe had already
demonstrated an understanding of the structure of square arrays and parallelogram arrays
in earlier tasks. The simplest I-tromino tiling, which was a simple rectangle tiling,
presented no new mathematical structure for his cognition. His third I-tromino tiling was
so complex that it was probably difficult for him to visualize as an infinite tiling. The
second tiling, which he found the most pleasing, presented a rich variety of mathematical
structures which were within his cognitive grasp: unitizing, alternation, row and column
structure, and so on. His ability to visualize this tiling was clear from his extended
drawing.
Alternation contributed to the complexity of the patterns. Rhombus patterns with
alternate orientations of the rows were more mathematically complex than those in which
the tiles were all in the same orientation. Such patterns were therefore somewhat more
cognitively demanding. In the case of triangles, the students had no choice concerning
alternation; simple rows of triangles can be formed only by alternating the orientations of
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the triangles. The children found this alternating orientation striking. Kelsey, Marie, and
Rachel commented on this alternating feature of isosceles or equilateral triangle
tessellations as a necessary part of their structure.
The aesthetic factor of surprise, of course, sometimes served the important
cognitive role of introducing cognitive dissonance, leading to new insights concerning the
structure of the tessellations. Moses, for example, was clearly surprised when he
discovered that six triangles fit around a vertex, forming a hexagon unit. He went on from
this discovery to use hexagonal structures in his triangle tessellation.
Validity has clear mathematical importance and is therefore mainly a cognitive
factor. Children were attempting to make mathematically valid tessellations that
respected the three criteria laid down at the beginning of the interview. And yet validity
was also an aesthetic factor, as evidenced by its occurrence as an aesthetic theme during
the evaluation tasks.
Connectedness was another aesthetic theme that served a cognitive role during
tessellation creation. For example, Michelle used the connection between her regular
polygon tiling and her triangle and square tiling to demonstrate the validity of the former.
(See Figure 52 on page 178.) The aesthetic quality of this connection was probably
evident in the excitement Michelle showed when she made the discovery of this
connection.
Even seemingly non-mathematical aesthetic criteria such as real world
connections and color may have occasionally played a cognitive role in the children‘s
work. The real world connection with honeycombs served as a confirmation of the
mathematical validity of children‘s hexagon tiling. Color may have been helpful to the
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children during the Regular Polygons task in order to see symmetry and structure more
easily.
DIFFERENCES BETWEEN SCHOOLS OR ACROSS TIME
I analyzed the coded data to discern any special differences between the two
schools or between earlier and later interviews.
Differences Between Schools
I was able to discover very few differences between the types of responses at the
two schools in this study. Only two students traced around the prototile to create precise
drawings, and both were from the private school. They did this 15 times. Students at the
private school showed more awareness of the fact that freehand drawings are only a
model of the tessellation and may not be accurate or to scale. These school differences
may be due to the greater emphasis on precision in geometry in the French curriculum, an
emphasis made even in the early grades. However, both schools expressed equal concern
for accuracy in their final results.
No other differences were found.
Differences Across Time
The format of this study is a clinical interview. It was not intended to be a
teaching experiment. It is therefore important to check that no major changes in thinking
were observed during the interviews. It was expected that some changes would occur as
the students got used to the format of the tasks, and some learning was inevitable, but
there were no expectations that large changes in cognitive or aesthetic understanding of
tessellations would happen. Consistency across the interviews can also serve as evidence
for robustness of thinking.
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Another concern was that the final interview was in an entirely different format
from the first five. Students may express aesthetic values differently simply because of
the new format, but these differences were not expected to be great. Students could have
expressed different aesthetics because the sixth interview focused on evaluation, whereas
the other five interviews focused on creation. However, I found that the aesthetics
expressed during the final interview appear to be fairly similar to the aesthetics expressed
during the other interviews, as described below.
Cognitive perspective. Responses to the dihedral squares and triangles tasks
were compared between the second and fifth interviews. Very little change in thinking
was noted. In fact, other than a greater readiness to abandon tilings that were not
working, each child‘s thinking was remarkably similar during both interviews.
Codes related to the cognitive perspective were analyzed for change across the
first five interviews. There was very little change other than certain tendencies that were
expected as the children got comfortable with tiling. Some changes were inevitable
simply because different tasks elicited different types of thinking. See Appendix F for
details.
The only clearly discernible changes were in codes that concerned accuracy. The
two children who traced tiles during the drawing phase did so only in the first interview.
(Marie used a tile to help her draw in the fourth interview without actually tracing around
it.) Children voiced questions about accuracy and how imprecision in drawing might not
reflect mathematical reality 28 times in the first interview, twice in the second interview,
7 times in the third interview, and not at all in the fourth or fifth interviews. The last two
interviews were almost entirely on dot paper and in the software environment, so there
were not many opportunities to discuss accuracy in drawings. This change in tools,
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together with increasing comfort with the clinical interview process, may account for
most of the change in expressions about accuracy.
There was no significant change in codes that measured various aspects of
understanding, confidence, or use of symmetry. The examples of children‘s finite,
process-oriented understanding of tilings all came from Set A‘s rhombus and triangle
tasks. (See pages 159 to 168.) Otherwise, there was no significant change in codes related
to types of reasoning or strategy.
Aesthetic perspective. Because the sixth interview was devoted entirely to
aesthetic evaluation, most of the aesthetic data came from that interview. But aesthetic
questions were also asked during Interviews 2 to 5, and students sometimes made
unprompted aesthetic comments throughout all interviews. It is important to ask if there
is evidence of aesthetic change between the cognitively oriented interviews, which
focused on creation of tessellations, and the final sixth interview, which focused on
evaluation of tessellations. If there is a difference, then the sixth interview may be
reflecting aesthetic values learned during the first five interviews, or aesthetic values
related to evaluation that do not apply to creation tasks.
The graph in Figure 71 below shows the number of codes in each of the 18
aesthetic themes from three sets of data: (1) mathematicians, (2) children during the first
five interviews, and (3) children during the sixth interview. With few exceptions, the
differences were as expected. Color was much more important during the sixth interview.
This was expected because the children had almost no control over color during the
creation tasks. The Interconnectedness theme was expressed much more often during the
sixth interview. This is expected because several of the tilings in the sixth interview did
have a complex path-like appearance. Such tilings are complicated and not likely to be
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created by children during their creation tasks, though some expressed this aesthetic for
tilings made with the S-tetromino. Units, Symmetry, and Alternation were mentioned
more frequently during the creation tasks, probably because these were important
strategies used in the creation of children‘s tessellations. It is nevertheless noteworthy
that children did not explicitly mention symmetry during the sixth interview, a theme
which the mathematicians believed to be very important. The only result in Figure 71 that
seems unexpected is the more frequent mention of the Culture theme during the creation
tasks.
Figure 71: Comparison of mathematicians‘ and children‘s aesthetic themes
I calculated the correlations between the three sets of data of the number of
occurrences of each code. The correlation between the creation tasks and the sixth
interview was .44. This seems to me a very high correlation considering the very different
nature of the interviews and the expected differences discussed above. It is in fact much
higher than the correlation between codes for mathematicians and the sixth interview
(.04) even though both groups evaluated the same tilings. In other words, children‘s
expression of aesthetic values seems to have been moderately consistent across the
interviews, whereas mathematicians‘ aesthetic values were quite different from the
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children‘s, especially during evaluation tasks. In fact the correlation of .04 between
mathematicians and children for the evaluation tasks suggests that mathematicians and
children approached the evaluation tasks in entirely different ways, even though they
used largely the same aesthetic themes. It is noteworthy that mathematicians‘ evaluation
aesthetic has a moderate correlation with the aesthetics expressed in children‘s creation
tasks (.22). This may suggest that children in the creation tasks were more focused on
mathematical criteria that mathematicians notice even when simply evaluating others‘
tessellations. Correlations with mathematicians‘ data must be used with caution because
there were only two interviews with mathematicians and they were of an informal nature.
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Chapter 5: Conclusions
There are few geometric topics as rich or as appropriate as tessellations for the
teaching of the geometry curriculum in late elementary school. This exploratory study
supports this claim and lays a foundation for further study of how children think and
learn about mathematical tessellations.
DISCUSSION OF THE RESULTS
Children‘s thinking is complex. However, we can describe children‘s
mathematical understandings of tessellations by identifying certain tendencies in their
early understanding, or certain phenomena that are true for many children and which we
should therefore be aware of. This description becomes more complete if we come at it
from more than one direction. In this study, I have analyzed children‘s understanding of
tessellations through a cognitive lens and through an aesthetic lens. The latter viewpoint
has been generally neglected in the literature, but in this study we see how the two
viewpoints together create a clearer picture of children‘s mathematical thinking.
Children’s Cognitive Understanding of Tessellations
The first part of the research question for this study is as follows:
What are children’s cognitive understandings of tessellations? What strategies
and types of reasoning do they use to create tessellations? What mathematical
concepts are elicited?
In Chapter 4, I identified some of the main components of children‘s initial
cognitive understanding of tessellations. Students used a variety of strategies to create
tessellations. These strategies were somewhat constrained by the shape of the prototile.
They also used a variety of reasoning to justify their tessellations. I look at some of the
mathematical concepts elicited on page 239.
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Among the more important conclusions are those concerning children‘s ontology
and epistemology of tessellations, and their understanding of the infinite space in which
tessellations are embedded. Ontologically, children see tilings as both an object and a
process. Epistemologically, children come to know tilings partly through their initial
ideas, and partly by subsequent constructions. Students understand the consequences of
tessellations‘ infinite extension in different ways.
Students sometimes see a tiling as a mathematical or concrete object and sometimes as
a process or action.
Some children did not understand that different finite tilings can represent the
same infinite tessellation. This seems to be partly true because students were focusing on
the process by which they made the tessellations. Students who described tessellations
primarily in terms of their own actions tended to see their tilings through the lens of the
actions that had created them. It was not obvious to these students, even after reflection,
that different actions could create the same object. For them, the tiling was understood as
a process or action, rather than as a mathematical object. Other students, who used more
descriptive terminology to explain their tessellations, tended to see their tilings as objects
independent of the actions they used to create them. Both types of students used both
active and descriptive wording in describing their tessellations, but when it came to
understanding their results, two students seemed to understand them primarily in terms of
actions and two others primarily in terms of final objects.
It is important for students to reach the point where they can understand a
mathematical concept in terms of either object or process, as needed. The evidence from
this study suggests that children who preferred the process understanding could not yet
conceptualize their creation as an infinite mathematical object.
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The situation is similar to that of other geometric constructions, which can be
conceptualized either as a series of actions or as the resultant constructed object. Both
ways of understanding a construction are necessary. A complete understanding of
geometric constructions or tessellations requires understanding the physical construction
process as well as the abstracted mathematical object that the actions represent.
The study of tessellations makes an ideal environment for children to construct
the idea that mathematical results may be independent of the processes that create them.
Tilings are a combination of initial ideas and further discovery.
Children‘s epistemology of tessellations is not simple. There rarely appears to be
any simple mental model in the students‘ minds prior to the construction of a tessellation.
Students have ideas concerning the tiling, and these ideas may have various levels of
clarity. The child may even believe that she knows the structure and appearance of the
tessellation. But the tessellation is not really known until it is realized in sufficient detail
via some construction tool. By the time this realization occurs, the initial idea the child
had has probably been modified. Children mentioned seeing new ideas as they made their
constructions. Drawings were inevitably more detailed than oral descriptions. Physical
constructions were more detailed still, and frequently different from the oral description.
The final construction is not independent of the initial idea, however. The child‘s tiling is
neither a pre-existing idea, nor a construction ex nihilo. It is a combination of certain
initial ideas about the structure and appearance of the tessellation and the final realization
constructed with tools. Sometimes children may initially have only a very hazy idea of
what the tessellation will be like, and then construct it until the pattern is clear. Such
constructions are perhaps better understood as the creation of a new understanding. Other
times children may have a very clear idea and the final construction is faithful to their
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idea. Even in this latter case, however, new structure becomes evident as the child sees
the actual appearance of the tessellation. There is no clear dividing line between these
two extreme cases of creation of a new idea and construction of a pre-existing idea. In
most cases, children had partial ideas that were then fleshed out through the construction
process.
In a certain sense, there is no point in time at which we can say, ―This is the
student‘s tiling.‖ Children‘s tilings are as much a process as they are a final construction
because the actual mathematical tilings are infinite. The tiling is never fully constructed;
the displayed finite construction can really only serve as an aid to describe what the
infinite mathematical object looks like. Even after construction of the pattern that
represents the tiling, the student may still not have a clear idea of the structure of the
infinite tiling. Consider for example Michelle‘s regular polygon tiling on the right in
Figure 52 on page 178. After she explained that the tiling was the same as the one on the
left, but with some triangles replaced by hexagons, it was still not clear to me which
triangles were to be replaced by hexagons, nor did it appear to be completely clear to her.
She assumed it would become obvious if she continued the construction.
Even if the student‘s ideas are clear, the finite pattern may or may not clearly
suggest to other people the structure of the infinite mathematical object. The pattern
displayed on the table or computer may be accompanied by verbal explanations of how
the student intends to extend the pattern to fill the plane. (For adult mathematicians, the
patterns may be accompanied by descriptions of the mathematical rules that will extend
the pattern.) Simple periodic tilings with small units will probably be clear to all with
little or no explanation. Radial tilings such as Michelle‘s, or tilings with large complex
units, may be interpreted differently by different people. As such, children‘s tilings are in
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some sense a social construct. Even more than other geometric objects, they cannot be
precisely realized in the real world. A tiling is an infinite mathematical concept that can
be communicated to others through visual and verbal description. The tiling that the child
displays is in one sense just a finite pattern and in another sense a representation of the
child‘s vision of how the mathematical tessellation can be constructed.
The child‘s initial understanding of the appearance and structure of the
tessellation may be vague or clear. If the understanding is clear and yet the constructed
tiling does not match pre-conceived ideas, then the child will experience cognitive
dissonance. Moses was the only one to show surprise that six triangles fit around a
common vertex. Other students did not notice this contradiction to their oral descriptions
of four triangles around a common vertex, even with prompting. It is likely that most of
the students had not thought through clearly exactly how triangles fit around a vertex.
Rachel was certain that equilateral triangles placed on the sides of a central triangle
would point left and right. When asked to demonstrate, she was surprised to find that they
formed a trapezoid in precisely the way she said they would not. These moments of
cognitive dissonance were rare during the interviews, partly because this was not a
teaching experiment and partly because these moments can only occur when the child has
a definite idea of what she expects to find.
In order to build up their intuition of what tessellations will look like before they
actually construct them, students need to practice, to play, to explore. By building up
their mental models, tilings can become more deliberate and initial ideas can be closer to
the constructed result. But construction will always be needed. Tiling is a verb as well as
a mathematical object. Even research articles on tessellations written by mathematicians
inevitably contain drawings to aid visualization. Hilbert succeeded in setting geometry
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free from the real world, but subjects such as tessellations are still very much grounded in
visual space.
Tessellations for children will probably never be fully interiorized as, say, mental
arithmetic might be. Tessellations will always need to maintain a tight contact with the
physical realizations from which students take their understanding. Children‘s
understanding of tessellations is intimately related to the action of tiling.
Students have limited understanding of infinite space.
Studies of children‘s understanding of infinity have been few and studies of
children‘s understanding of infinite space have been essentially non-existent. Though it
was not intended to be, this appears to be the first study of children‘s understanding of
infinite space at the concrete operational stage.
Previous studies of infinity (Fischbein et al., 1979; Monaghan, 2001) have shown
that students hold contradictory ideas of infinity because their ideas are based on
schemata constructed through experiences in a finite world. Students‘ ideas about infinity
change easily and are based more on a process understanding than on an object
understanding. These studies have focused on ideas such as numerical infinity and
continuity, but not spatial infinity.
I have shown that the observation that some children do not distinguish between
finite and infinite patterns is partly explained by process-oriented thinking. This
observation also shows that these students do not grasp that their finite patterns are
intended to represent infinite mathematical constructions. This failure to understand the
consequences of infinity limits these students‘ ability to the study of mere patterns, rather
than periodic tessellations. We already knew before this study that younger children at
the pre-operational stage cannot appreciate the infinite extension of tessellations. It would
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appear that at the concrete operational stage, students can appreciate that tessellations
continue indefinitely, but for some this understanding is not complete.
Other indications that children are struggling with the idea of infinite space are:
forgetting the infinity constraint; expressing completion with a finite pattern; and
showing discomfort over tiling in an empty plane without walls or reference points. Not
every piece of evidence applied to every student, but every student struggled with the
infinite space of the tessellations in some way.
Infinity is a fundamental concept and, like other concepts in mathematics, there is
no reason to postpone students‘ exposure to important mathematical concepts until they
reach calculus (Stroup, 2005). The infinite plane is an important aspect of Euclidean
geometry and tessellations are an ideal environment for allowing students to begin
exploring this idea for the first time.
Children’s Aesthetic Understanding of Tessellations
The second part of the research question for this study is:
What aesthetic considerations guide students’ thinking about tessellations?
How do they evaluate tessellations made by themselves and by others?
What mathematical value is there in their aesthetics of tessellations?
The aesthetic viewpoint has been largely neglected in the literature on children‘s
mathematical understanding. This study adds to the research that indicates that aesthetics
is not some optional epiphenomenon, but is an integral part of how mathematics is done,
including children‘s mathematics. Only the most rote, elementary mathematics can be
done without aesthetics, whether we are consciously aware of it or not.
I identified 18 aesthetic themes which categorized most of the verbal expressions
of aesthetics in this study. Nearly all of these themes were used by both children and
mathematicians. However, the children and the mathematicians used these themes in
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different ways and placed emphasis on them very differently. Mathematicians tended to
prefer themes that aligned with aesthetic qualities identified by previous researchers as
defining a mathematical aesthetic. Children tended to prefer other themes, principally
color and real world connections. Mathematicians‘ aesthetics are therefore not something
entirely foreign to children‘s experience, but they do reflect different awareness of what
is mathematically important. I conclude from this that mathematicians‘ aesthetics are not
something that children cannot appreciate, but are something children can learn as they
gain experience in the mathematical domain.
Children used aesthetics not only to evaluate tessellations, but also to create them.
As children created tessellations, they were guided in their creations by several aesthetic
factors, especially that of visual symmetry.
There is an implicit use of symmetry in most of the students’ construction and
evaluation of tilings.
In the study of tessellations, symmetry can be either a visual, aesthetic object or a
mathematical, physical process (Schattschneider, 1978a). Children tended to use
symmetry to guide their tiling creations. This was seen by the fact that once a pattern
attained significant symmetry, children took care to keep it symmetric as they expanded
it. In all, 77% of their finite patterns were symmetric. Children rarely created non-
symmetric patterns if they knew how to create symmetric ones.
When students create symmetric patterns, they are more confident that their
tilings are valid. This confidence is mathematically justifiable because the validity of
symmetric patterns is much easier to verify.
The mathematical role of symmetry in tessellations suggests a probable
mechanism for the role of aesthetic cognition in mathematical exploration. The aesthetic
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value of symmetry probably derives from the innate appeal of patterns. Mathematics
itself being the study of patterns, it seems natural that symmetry should be an essential
part of mathematics. Children, however, are not aware of this deep connection. At first,
they simply enjoy the pleasing quality the symmetry brings to their designs. However, as
they work with their designs they probably notice, at least subconsciously, that patterns
with symmetry are easier to analyze, hence the increase in their confidence that the
tessellations are valid. This increases their motivation to apply symmetry to future
tessellations.
Children also have a strong aesthetic preference for color. Certain colors are
appealing and a variety of colors makes the pattern more interesting. Color, however, has
little mathematical role in any of the tasks from this study. Mathematicians realized this
in their interviews. Mathematicians mentioned color as having aesthetic importance, but
rarely used it to evaluate the tessellations. If children work with tessellations beyond what
this introductory study could examine, they will probably discover that color does not
help them much with their mathematical task and will eventually turn to other aesthetic
qualities to guide them. Lehrer et al. (1998) found that children who studied the
mathematics of quilt patterns initially were attracted to the colors of quilts, but as they
grew in mathematical sophistication they increasingly expressed aesthetic appreciation
for symmetry, shape complexity, transformations, and the aesthetic constraints of certain
pattern choices. Mathematical aesthetics is not some vague force, but is a way of thinking
that is increasingly perfected as it is honed by our mathematical experiences. Aesthetics
guides our thinking, and our thinking about our mathematical experiences, in turn, shapes
our aesthetics.
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Conclusions about the types of symmetry children valued are difficult to confirm,
but I did notice that no tiling shocked them in the sense of Le Lionnais‘s romantic
aesthetic. This was not unexpected. Ideas about tiling cannot be disturbing if children are
just beginning to develop and formalize their ideas about tiling. I observed no cases of
cognitive dissonance as children observed some of the unusual tilings in the sixth
interview, such as the tilings in Figure 72 below. A few noticed the dilation symmetry of
Ls in the tiling on the left; others did not. If they did notice the unusual pattern, they
simply delighted in it. They never showed dismay or confusion over any tiling to be
evaluated.
Figure 72: Two tilings with unusual symmetry in the aesthetic evaluation tasks
Symmetry and transformations are foundational to the study of modern geometry.
As this study shows, symmetry is also a fundamental aesthetic quality that guides
children as well as mathematicians in the mathematical task of creating tessellations.
Symmetry is important mathematically both as an aesthetic object and as a
transformational process.
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Aesthetics had an influence on the way students tiled.
I have shown that aesthetic symmetry had a definite positive impact on children‘s
tessellation creations. I found other aesthetic qualities that drove children‘s mathematical
thinking about tessellations as well, including complexity, uniqueness, alternation, and
units. These qualities were clearly shown to be aesthetic in the children‘s thinking during
the evaluation tasks. They were also used by the children as they created their tilings.
Another aesthetic quality used by children for their finite patterns was the mathematical
concept of convexity. Real world connections and color may also have been aesthetic
factors in children‘s creation of tessellations. All of these aesthetic criteria, as well as
aesthetic criteria of surprise, validity, and connectedness, were shown to have cognitive
roles in addition to their aesthetic roles. (See The Cognitive Role of Other Aesthetic
Criteria on page 214.)
These qualities therefore highlight the deep connection between cognitive and
aesthetic understanding in the children‘s thinking. If the children could not grasp the
mathematics behind a particular concept used in a tiling, they could not see that concept
as beautiful. Aesthetic appreciation is linked to cognitive understanding. Students found
the tiling on the left in Figure 72 above beautiful if they noticed the dilation pattern of Ls.
Other students appreciated this tiling for its maze-like quality, but they only got really
excited if they saw the ever-increasing L patterns. One mathematician did not
aesthetically appreciate the tiling on the right in Figure 72 until he understood how it had
been made from a simpler radial pattern. His aesthetic appreciation was tied to his
cognitive understanding of the tessellation.
Mathematical understanding therefore has an impact on aesthetic appreciation.
Even more importantly, aesthetics guides and motivates students‘ mathematical
understanding. Aesthetics and understanding impact each other as in Figure 73 below.
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Mathematics is the study of patterns and the natural appeal of patterns motivates students
to investigate the mathematics behind these patterns. Aesthetics also guides students to
seek the most generative paths of mathematical inquiry. For example, we saw above that
the aesthetic quality of symmetry guided students in their search for mathematical
validity. Symmetry also served as a powerful motivating force to study the mathematical
structure of tessellations.
Figure 73: Mutual influence of aesthetics and mathematics
Aesthetic factors other than symmetry also drove students‘ investigation of
mathematics. The aesthetic quality of uniqueness, for example, drove students to seek
new patterns and new ways of understanding tessellations. Instead of simply making the
same tessellations they had seen before, students usually preferred trying to make a
tessellation that they were not yet familiar with. They enjoyed exploring the space of
possibilities and deliberately used strategies and shapes that they had not yet tried. The
aesthetic quality of units drove students to explore mathematical concepts of fit,
composition, and unitizing. The aesthetic quality of complexity drove students to explore
patterns that were richer than the simplest patterns and yet still within their cognitive
grasp. The aesthetic quality of alternation helped add an element of complexity. These
aesthetic qualities served as a force to drive students to explore their zone of proximal
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development—the space of ideas they are ready to understand with others, but have not
yet learned. American classroom instruction is often organized around teacher- or
curriculum-imposed exercises. This study would suggest that students should be allowed
more often to let their own aesthetics motivate them towards the concepts they need to
learn.
Even seemingly non-mathematical aesthetic qualities probably play an important
role in driving children‘s study of mathematics. Real world connections, for example,
may inspire the study of patterns known from experience. Children were immediately
comfortable with the regular hexagon tiling, even if they were constructing it for the first
time, because they had frequently seen it as a honeycomb pattern.
Aesthetics and mathematics have a mutual impact on each other, but aesthetics
comes before the mathematics. This is seen in the simple fact that students had an
immediate aesthetic attraction to various aspects of tiling, such as symmetry and
dimensionality, without yet understanding the mathematics to which these aesthetic
qualities will lead. Current mathematical curricula assume that aesthetic appreciation is
an optional application to mathematical study. But aesthetics is actually at the very
foundation of mathematics. Therefore, aesthetics should be understood as the initial
driving force towards mathematical understanding. Children are drawn towards
numerical and geometric patterns and this aesthetic force leads them to study and
understand mathematics. The consideration of aesthetics cannot be postponed to the role
of educational afterthought. It is the driving force behind the study of mathematics and its
role in learning should be considered explicitly in education.
INTERACTION OF THE TWO PERSPECTIVES
The third part of the research question is:
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How do the cognitive and aesthetic viewpoints inform each other for a more
complete understanding of children’s conceptualizations of tessellations?
The two perspectives together are needed for a more complete understanding of
children‘s mathematical thinking. In this study I found the two perspectives together
particularly useful for exploring the role symmetry played in children‘s understanding of
tessellations and how their aesthetic values influenced their mathematical results. (See
above.) It was also useful for understanding how tools influenced thinking. Also, an
essential part of this study was considering the cognitive role of the children‘s aesthetic
criteria. (See above.)
The tilings students created depended on the task given to the students and the tools
used to accomplish the task.
The task given to the students and the tools students had to accomplish the task
determine the space of possibilities for the result. Students were never given options of
color (independent of the shape), so color almost never entered as a mathematical aspect
of children‘s tilings. The software environment did not have the capacity to select and
copy units of tiles, so children were somewhat limited in the complexity they could
achieve. The magnetized behavior of the polygon corners in the software environment
influenced some students to create certain types of tessellations that they might not have
created with another tool. The relative difficulty of rotating a computer tile compared to
spinning a plastic tile meant that few students recognized the right angle in the right
scalene triangle task, but all recognized the right angles in the polyomino tasks. The task
of describing a tiling orally without construction tools requires students to visualize the
structure mentally, so their oral descriptions tended to be simple and local. Dot paper
helps students avoid overlaps and small gaps. The automatic zoom in the software
highlighted the endless, borderless nature of the tiling plane.
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Outhred and Mitchelmore (2000) point out that drawing tools get closer to
children‘s mental models of tessellations than physical construction tools do. My study
shows that oral descriptions get even closer to children‘s initial ideas about the structure
and appearance of tessellations. Drawings produce a combination of initial ideas and
fresh discoveries.
It seems reasonable that the heavy influence of tools should lead us to conclude
that a variety of tools is needed for exploring tessellations. Instruction that is limited to a
single tool will inevitably limit thinking along the lines constrained by the tool. One
reason the teaching experiment of Upitis, Phillips, and Higginson (1997) was so powerful
is that the teachers used a wide variety of tools and tasks to explore the mathematical
concept of tiling.
IMPLICATIONS FOR FUTURE RESEARCH
Tessellations have been neglected in the research literature, in spite of their
importance. My research is exploratory in nature. As such, it is intended to be the first
step in Clement‘s (2000) cycle of research extending from generative, exploratory studies
to convergent, confirmatory studies. All aspects of this research need to be followed up.
Tessellations are a broad field and I have touched on several different aspects of this
topic. Different types of tessellations use different types of reasoning and geometric
properties. Studies are needed which treat each kind of tessellation task in greater depth.
Children‘s understanding in each area can be pushed further to test which beliefs are
robust and which are easily changed.
There are many important mathematical concepts elicited by tessellations, as I
discuss on page 239. Studies are needed which explore the role of tessellations for each
of these concepts. How do students understand each of these mathematical concepts in
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the context of tessellations, and how can they best extend their understanding? What are
children‘s understandings of isotropic space with respect to tessellations? How does the
orientation of the figure affect children‘s ability to perceive acute, right, and obtuse
angles and see possibilities of fit? What are children‘s understandings of symmetry with
respect to infinite tessellations and how do children learn about symmetry in the context
of tessellations?
There is essentially no research on children‘s understanding of spatial infinity.
This study shows that this important topic can be studied in the context of tessellations.
Previous studies suggested that students‘ understanding of infinity was impervious to
instruction (Fischbein et al., 1979; Monaghan, 2001). Is this true in the case of children‘s
understanding of spatial infinity? If not, how can children learn to appreciate the
consequences of tiling in an infinite plane? How can students be taught to distinguish
when their pattern is sufficiently described to make a well-defined tessellation? Upitis et
al. (1997) encouraged children to invent notations to describe their patterns. Can this be
extended to help them understand that their finite patterns must be extended throughout
infinite space in a way that others will understand? How can such activities connect the
children‘s understandings of transformations and symmetry?
More research needs to be done on children‘s ontological conceptions. I have
presented evidence that some children fail to understand that two different finite creations
make the same infinite tessellation partly because their understanding of infinite space is
incomplete and partly because they are using a process-oriented ontology. This needs to
be confirmed and understood at greater depth.
This study also shows that tessellations can be used for the study of children‘s
aesthetics and the impact their aesthetics have on their mathematical choices. How do the
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aesthetic themes identified here compare with the aesthetics of other geometric tasks?
How do children‘s aesthetics evolve as they deepen their understanding of tessellations?
Most importantly, this study serves only to identify children‘s initial thinking.
Therefore, teaching experiments are needed which build on the results presented here.
How can students best be taught to understand the mathematical structure of
tessellations? How can we best teach angle properties and angle perceptions at various
orientations? How can teaching help students to attain a deeper understanding of
symmetry and the transformations that create tessellation structure? At present, symmetry
at this grade level is usually limited to identifying simple reflections and rotations. This is
clearly insufficient if students need to understand, say, the isogonal nature of the regular
pentagon. Students need to be able to identify richer forms of symmetry. Students also
need to be able to extend their reasoning beyond the simple empirical reasoning they
often used in this study. In what ways can this be done?
IMPLICATIONS FOR CURRICULUM AND INSTRUCTION
Many recent studies in geometry education attempt to identify levels or learning
progressions or trajectories that map out the steps students go through as they learn the
subject matter. Such an approach seems inappropriate for the results of this study for two
reasons. First, this is not a teaching experiment, so learning progressions or trajectories
cannot be reliably identified. Second, I do not feel it would be right to try to assign levels
to children‘s understandings of tessellations because their thinking in this area has so
many dimensions. Take for example Michelle. At times she came up with very
sophisticated results, such as drawing a two-dimensional parallelogram grid with a ruler,
or creating two different tessellations with six-fold symmetry and then correctly
analyzing the isometry between their structures. And yet she gave several answers that
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suggest she is still sometimes reasoning at Piaget‘s intrafigural stage. Or consider Kelsey.
She was one of only two students who found successful tilings for all three triangles in
Set E, using mathematically valid unitizing and row strategies, and yet she did not
demonstrate understanding of the row and column structure of simple square arrays.
Children‘s thinking is complex, but I have nevertheless identified several aspects
of their thinking which are true for all six students. I have also noted a few of the
immediate implications from these conclusions for curriculum and instruction. I now
return to my contention that there are few topics in geometry that can combine so many
powerful and important geometric ideas that this age group of children needs to be
studying.
What Students Learn With Tessellations
Students learn a wide variety of geometric concepts when studying tessellations.
Most importantly, they learn these concepts in the context of a genuine mathematical
task. Instead of studying concepts in isolation through exercises that either have no
connection to students‘ needs or are couched in highly artificial ―real world‖ contexts,
problems such as the tessellation tasks in this study provide an environment where
students can explore and discover the need for the geometric properties that they are
learning in a motivating context that naturally highlights children‘s mathematical
aesthetics.
Mathematical Concepts Elicited by the Tiling Tasks
A number of mathematical concepts were elicited during the tessellation creation
tasks in this study. The following is a brief summary of those concepts.
Fit. Students needed to be able to fit polygons together both as tiles and as units
of tiles in order to create interesting tilings. This requires coordination of sides and angles
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in a way that was not always obvious to the students. Students sometimes struggled with
fitting tetrominoes together in a way that did not leave holes. Misjudging the sizes of
sides and angles made certain tasks, such as the sixth finite tiling problem, difficult.
Angles. Students needed to work with angles in order to understand fit.
Sometimes students referred to angles as ―corners,‖ though their understanding of this
term was not explored. The word ―angle‖ was not used often. When it was, it was usually
used to mean ―orientation‖ or ―triangle.‖ An understanding of angles was needed to
analyze the pentagon task and to see the correct number of polygons that could go around
a common vertex. Additivity of angles, along with independence of order, is needed to
see that the order of regular polygons can be permuted—a fact that would have been very
helpful for the square and triangle task had the students realized it.37 Basic angle facts
were needed, such as four right angles or six equilateral triangles can fit around a vertex
precisely, facts that were not well known by the students. The abilities to estimate angles,
draw approximately congruent angles, and spot right angles at non-orthogonal
orientations were needed for several tiling tasks. Because most students did not recognize
a right angle at a non-standard orientation, they failed to see a simple solution to the right
scalene triangle task until they explored with the physical tiles. They also frequently
failed to judge if a gap angle was acute, right, or obtuse during the Regular Polygons
task. However, students did seem to realize that the prototiles had different sized angles,
so if one angle did not fit, they could choose a polygon with a smaller or larger angle.
37 Contrast the way additivity of angles can be explored in tessellations with the less meaningful way
complementary and supplementary angles are often taught by simply memorizing that they must add up to
90° or 180° without much real understanding of what the importance of these sums might be. These
concepts are usually introduced later than fourth grade. Do students understand the additivity of angles
when they study these concepts? They apparently do not at the fourth grade level.
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Sides. In addition to angles, students must pay attention to the length of polygon
sides when deciding which tiles can fit where. In the case of polyominoes, which are
usually not edge-to-edge, children must also implicitly work with the sums of the edges
in order to align two or more polyominoes against a given side. The fact that some
students expected the parallelograms not to match up in the standard tessellation shows
that they were not thinking through the consequences of the parallelogram‘s congruent
sides.
Congruence. Concepts of congruence are implicit in the concept of the prototiles,
which are congruent with the tiles in the tessellation. Students seemed to notice that the
sides of the regular polygons were all congruent and therefore they correctly focused on
the problem of getting the various angles to fit together in the Regular Polygons task.
Students realized in the Triangles tasks of Set E that it was important to match congruent
sides together. This sometimes proved to be difficult during the drawing phase and
students sometimes erased and redrew as they struggled to coordinate sides and
orientation.
Transformations. Transformations and symmetry are foundational tools for the
study of tessellations. But it is not easy to envision that the entire plane is transformed by
a mathematical transformation. Tessellations require such understanding and offer an
environment where such understanding can be made concrete.
Students sometimes misused reflections in their tilings. By flipping some tiles in
the computer environment and not remembering that some were flipped, they sometimes
made it impossible to create a successful tiling. In order to make certain tilings, students
need to be aware that a tile may have two forms related by reflection. Shapes with
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reflection symmetry have only one form, and therefore flipping them does not change
their shape. Michelle wanted to flip such tiles anyway in one of the tasks.
Symmetry. Students explicitly created tilings with reflection symmetry, though
they almost never used the word symmetry. Moses talked about copying what he did on
one side to the other side. Students seem to have also used visual rotation symmetry,
though they did not express this. Reflections and rotations are needed to create the
symmetry that students desired and that made tilings easier to create. Students explicitly
used translation symmetry to defend the infiniteness of some of their tilings. They could
show that a tiling continued forever by showing how a unit could be translated and still fit
into the pattern. It is difficult to justify that a tiling will be endless without translation
symmetry.
Dilations are another type of transformation that can be studied in the context of
tessellations. Dilations were noticed by the students in this study for some of the tilings.
Some students created rhombus tessellations by using a dilation strategy. During the
aesthetic evaluation tasks, some students noticed that one of the tilings had a
sophisticated dilation symmetry.
Orientation. Students needed to be aware of polygons and angles in non-typical
orientations. Failure to mentally rotate a tile to a more standard orientation created a
failure to see tiling possibilities, as was the case with the right scalene triangle and in
some of the Set B finite tiling tasks.
Infinity. Students were more successful at arriving at a mathematical
understanding of their tessellations if they could work comfortably in an infinite, empty
plane. Tasks of fitting needed to be carried out without any final finite shape as a goal.
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Students needed to understand and visualize how their tiling would look extended
indefinitely in all directions.
Dimensions. Students need to understand that planes have two dimensions and
that tiling translations in the plane need to go in two independent directions. Periodic
tilings can be created by extending first in one dimension (as a row) and then in the other
dimension (stacking rows). Tilings with 6-fold symmetry generally have some sort of
rows in three different directions, a fact students did not notice.
Systematic counting. One or two students had difficulty counting the sides of
polygons. Students need counting strategies for counting items arranged in a circuit, as
well as counting shapes contained in a unit that has not been visually delineated.
Inductive reasoning. In order to evaluate the validity of a tiling, students need to
use mathematical reasoning, eventually in the form of deductive reasoning. They need to
understand the limits of the inductive reasoning they frequently used. It is often pointed
out that teaching the limits of inductive reasoning is difficult because presenting good
examples is not easy. Tiling provides an accessible activity for exploring these limits.
Deductive reasoning. Students usually understood the role of counterexamples in
deductive reasoning. A failed tiling was not a valid counterexample to the hypothesis that
a shape could tile. A successful tiling was a valid counterexample to the hypothesis that a
shape could not tile. The fact that the regular pentagon could not tile does not imply that
all pentagons cannot tile.
Composition. Composing and decomposing shapes is a basic skill for geometry.
Students need to learn to unitize and to subdivide shapes into other shapes, both of which
were necessary skills for the tessellation tasks.
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Visualization. Tiling requires and aids two-dimensional visualization skills,
including the recognition of two-dimensional patterns, symmetry, and transformations.
Tessellations with polyominoes require visualization on a square grid. Students
sometimes failed to visualize polyominoes correctly with respect to the grid, resulting in
mathematically invalid tilings.
Parallelograms. Students need to learn that all parallelograms tile like squares,
with rows and columns. They need eventually to understand the properties of
parallelograms shared by squares and rectangles (themselves special parallelograms).
These properties are learned naturally in the context of tiling and students in my study
noticed the similarities.
In addition to the mathematical concepts elicited by the tasks in this study, the
study of tessellations can also be used to explore classification of polygons (e.g., squares
and rectangles share the properties of all parallelograms), properties of parallel lines (e.g.,
van Hiele-Geldof, 1957), polygon vocabulary (e.g., Upitis et al., 1997), and of course
area concepts (e.g., Owens & Outhred, 1998).
The Teaching of Tessellations
This was not a teaching experiment, but we can see certain factors that will
probably need to play a role in geometry instruction that uses tessellations. Most
especially, we can see the importance of allowing children to have a rich set of
tessellation experiences. Owens and Outhred (1998) pointed out that students did poorly
with finite tilings of shapes they were not familiar with. In order for students to gain the
aesthetic and cognitive knowledge they need for geometric conjecture and insight, they
need ample opportunity to explore and gain familiarity with geometric figures. Moses
noted that some of his insights came from having played with pattern blocks in previous
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years; he was already aware of the possibilities of combining them together. Students
need multiple experiences to learn geometric properties, preferably set in contexts that
highlight the usefulness of those properties. Tessellations provide an environment for
such experiences, provided students can explore possibilities and study why some things
work and other do not. The successful teaching experiments of van Hiele-Geldof (1957)
and Upitis et al. (1997) allowed students to conduct such guided explorations. More
teaching experiments are needed to determine the best ways for students to learn from
tessellations. What elements help students to learn? What elements detract from learning?
For example, color was very important to the children. The color pink stirred great
passions among the children at the private school, usually negative. Will allowing
students to choose their own colors increase the aesthetic and motivational factors for
creating tessellations? Will it allow a deeper investigation of symmetry (see page 84)? Or
will it distract from the more important properties that need to be the focus? These are
issues that need further research.
Teaching should also take into account mathematical aesthetics. Children
generally share the same basic aesthetic categories as mathematicians, but they use these
aesthetics in different ways. In some ways this difference is appropriate. Children need
the aesthetic qualities of alternation and units to guide them in the creation of
tessellations. In other ways, their aesthetics needs to be molded by geometric experience
that helps them learn which aesthetic criteria are most useful for the task of tiling and
which criteria are less useful. Color and real world connections may not be
mathematically helpful; complexity, connectedness, and surprise will guide students to
richer geometric learning.
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Importance of the Study of Tessellations
Tessellations provide a space for the geometric experience and knowledge
students need. Tessellations cover much of the geometry curriculum around this grade
level. They also provide an environment where students can explore and be guided by
appropriate mathematical aesthetics, which will increase motivation and can help them
focus on what is most important for them to learn at this grade level in geometry.
This claim that tessellations provide an appropriate space for this age level is not
new, though the theoretical base that demonstrates how children initially understand
tessellations has never been thoroughly investigated before. Teaching experiments have
already shown that tessellations are a powerful topic for the learning of geometry. The
study of tessellations has, in fact, already been mandated between third and seventh grade
(often in several grades) in some states, provinces, and countries, including many that are
among the highest ranking in comparative tests of mathematics, such as Japan,
Singapore, Ontario, Quebec, and Massachusetts. The NCTM Focal Points (2006)
recommends tessellations for fourth grade as a context for the study of transformations.
Unfortunately, tessellations are not required in the Texas Essential Knowledge
and Skills, nor in the new Common Core State Standards for Mathematics (CCSS-M;
Common Core State Standards Initiative, 2010). The CCSS-M are intended only to be
standards from which states must devise curricula that specify how the standards are to be
implemented. States may or may not decide to include tessellations in their curricula in
order to cover the content of the standards. Even if tessellations are not included in the
curriculum, teachers can use tessellations to cover the mandated material. The question in
today‘s American climate of intense testing pressure is whether teachers will feel the
freedom to introduce a significant unit on tessellations if it is not included in the
curriculum. These are details that remain to be worked out at the policy level. However, I
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strongly recommend that teachers and curriculum writers consider the rich possibilities
inherent in the study of tessellations for fourth and fifth grade children.
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Appendix A: Pre-test
This appendix contains a reduced copy of the English pre-test.
Figure 74: First page of the pre-test, shown half size
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Appendix B: Protocols
This appendix contains the original protocols for the six interviews. After the first
interview, Set B was moved to the third interview, so it has been deleted from the original
protocol for Interview #1. A brief two-page outline was made for each of the six
interview protocols to remind the interviewer of the tasks during the actual interviews.
INTERVIEW #1 PROTOCOL
Preparation
(Materials: video camera, batteries, and cord; tripod)
Prepare the materials for today‘s tasks and the master key. Position the video
camera in such a way that it will need to be moved as little as possible and yet can
capture the work space and students‘ actions. The camera must capture pattern block
tilings first, and then the computer screen later. (Check the equipment in advance to make
sure the microphone can capture all dialog.) Check which students will be participating
and prepare software codes for them.
Introduction
(Purpose: To set the ground rules and introduce the main task)
(Materials: blank paper, marker, square pattern blocks; pre-test solution)
Information – Collect the assent form if it has not been previously collected.
Return the main part of the consent form if submitted. Record the child‘s first name, age,
pseudonym (a two letter code written on all papers for later identification), gender, and
school on the master key.
Introduction – The interviewer introduces himself and explains that he is
studying how children create patterns with shapes. Confirm that the child understands
why a video recording is being made. Start the camera and say today‘s date, interview
number, and student pseudonym.
Ground rules – Attempt to make a point of connection which invites the student
to enjoy the tasks by asking students how they feel about math, geometry, art, patterns,
floor tiles, and so on. Discuss that the interviewer will be giving some problems to do and
that there are no right or wrong answers to the problems. In fact, there will be many good
ways of doing most of the problems and we will explore together possible ways to do
them. There is no grade attached to the work and the child‘s teacher will not know about
their work. We are doing these tasks ―for the fun of it.‖ Again, the reason we are here is
because the interviewer is studying how children think while making patterns with
shapes, so it is important to think out loud and explain all work as much as possible.
Explain that you will be asking the student how they know their answer works.
Initial problem – The problem we will be exploring is that of tiling a floor.
Discuss floor tiling including the facts that there can be no gaps or overlaps, and the tiling
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must be able to go on and on. We do not care what happens when the tiling reaches the
walls. Let‘s imagine it goes on forever. Ask the child what it means for the pattern to go
on forever.
Show the student his or her answer to the first task of the pre-test (see the
preliminary task of Set B) and ask if it could go on forever. Next, ask the student to make
the tiling with square pattern blocks and discuss any discrepancies with their oral
description, as well as apparent gaps or overlaps in the drawing and whether the
inaccuracies in the freehand sketch are important or not (Owens & Outhred, 1998). If
necessary, discuss again the constraints against gaps and overlaps and the condition of an
infinite pattern. If there is a square tiling on the floor, point it out.
Set A: Basic Tilings
(Purpose: To see how students perceive and construct basic monohedral
tessellations and to introduce the computer software)
(Materials: blank paper; marker [preferably] or pencil [if the student keeps starting
over]; triangle, hexagon, and rhombus pattern blocks; pre-test solution; computer
with software and mouse)
Rhombi – Give the student a single blue rhombus pattern block. Ask if it would
be possible to tile the floor with such a diamond. How certain is the student?
Ask the student to explain their answer orally, using a tile for support. (If needed,
the child can use a second tile.) Based on the child‘s answer, ask the child to explain on
paper (starting in the center of the page) either why the tiling cannot work or what the
tiling will look like. (If the student wants to trace the shape, allow this.) If the child
creates a tessellation, discuss how it works—the patterns, how it continues, or any
unusual features. Discuss any discrepancies with the oral explanation. Finally, allow the
child to create the tiling with the pattern blocks, or else demonstrate with the pattern
blocks why the shape cannot tile. Again, discuss any apparent differences with earlier
descriptions. Also, if the tessellation is not standard, discuss any consequences of unusual
features of the tiling. At some point in comparing the sub-tasks there may be cognitive
dissonance. If the cognitive dissonance is resolved during the interview, this might lead
the child to change his or her initial ideas. Help clarify in what ways these ideas changed.
Ask the student how certain they are that the tiling will or will not work. If the child
believes at this point that it is impossible to tile with this shape, end the task here.
Are there other ways to tile with rhombi? If so, how? How are these other tilings
different? How are they the same? Which one does the student like the best? Why?
Label the student‘s work with their pseudonym code and the principal orientation
of the paper with respect to the student and file away.
Triangles and hexagons – Repeat the above procedure first with hexagons, then
with triangles.
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Parallelograms – Repeat the above procedure for parallelograms. In place of the
pattern blocks, display a 60°/120° 1×2 parallelogram on the computer screen using the
software with the student‘s name. Ask if it would be possible to tile the floor with such a
shape. How certain is the student?
Ask the student to explain their answer orally. Discuss the child‘s solution on the
pre-test. Discuss any discrepancies with the oral explanation. Conclude by showing the
student how the software works and allow the student to construct the tiling with the
software (or to show why the tiling cannot work). Again discuss any apparent differences
with earlier descriptions. Finally, show the student his or her answer from the pre-test.
Discuss any differences. If any of the final tessellations are not standard, discuss any
consequences of unusual features of the tiling. At some point in comparing the sub-tasks
there may be cognitive dissonance. If the cognitive dissonance is resolved during the
interview, this might lead the child to change his or her initial ideas. Help clarify in what
ways these ideas changed. Ask the student how certain they are that the tiling will or will
not work. If the child believes at this point that it is impossible to tile with this shape, end
the task here.
Are there other ways to tile with parallelograms? If so, how? How are these other
tilings different? How are they the same? Which one does the student like the best? Why?
Final task – If the student created a tessellation for each of the four shapes, finish
this set by asking if there is a shape that cannot tile the floor. If the student believes there
is such a shape, give the student paper to draw such a shape.
Set B: Tiling Completion
(This set was moved to the third interview.)
The interviewer will thank the student for her or his participation and say he is
looking forward to next week‘s interview.
Guidelines for Interviews
The clinical interview will be carried out according to the guidelines explained in
Entering the Child’s Mind (Ginsburg, 1997). The interviewer will do his best to assure
the interviews are conducted with a fun and supportive atmosphere. It is important to give
ample time and material for the child to explore each task in searching for an answer that
clearly communicates the child‘s ideas. When possible, allow the student to continue the
tiling for as long as they like. For example, encouraging the child to continue the tiling is
better than asking ―And what goes here?‖
All responses by the child are to be accepted as honest attempts to explain their
thinking and occasional affirmation (―Ok,‖ ―Cool,‖ ―Good,‖ ―Nice,‖ ―Interesting,‖
―Wow, none of the other students have made that pattern‖) will be given equally to all
task results without regard to the interviewer‘s own aesthetic preference. Attempts will be
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made to modify or skip any tasks which appear unduly frustrating to the child. Clinical
interviews require the interviewer to hypothesize during the interview, invent appropriate
questions on the spot to test these hypotheses, and occasionally even create new tasks
during the interview, though this will usually be done as a follow-up during the last two
interviews.
Coercion and seduction – The interviewer will make every effort not to bias the
child‘s thinking by asking leading questions unless he is deliberately attempting to test
the robustness of the child‘s thinking. Answers are not to be judged right or wrong; the
only goal is to understand the child‘s thinking. Each week, the interviewer will review the
videos and note any instances of coercion according to diSessa‘s (2007) four criteria: (1)
Is there independent evidence of conviction on the part of the subject? (2) Is it possible
that the viewpoint was prompted? (3) Were the responses systematic (e.g. always
changing one‘s mind after a prompt) or were they natural? (4) Is there converging
evidence? ―Seduction,‖ the deliberate attempt to draw a student into areas that are likely
to reveal interesting thinking, is not the same as coercion and is acceptable as long as
there are no leading questions that would bias answers.
INTERVIEW #2 PROTOCOL
Preparation
(Materials: blank paper; marker; pencil; computer and mouse; square, pentagon,
and hexagon Polydrons; square and triangle pattern blocks)
Review each student‘s answer for the hexagon tiling. Prepare generating sets in
the software for regular pentagons and for squares and triangles.
Assemble a sample square tiling and a sample hexagon tiling with Polydrons.
Check camera positions and sound settings.
Set C: Pentagons
(Purpose: To investigate a shape that cannot tessellate)
(Materials: blank paper; marker; pencil; software; square, pentagon, and hexagon
Polydrons)
Show the students a square tiling and a hexagon tiling with Polydrons. Remind
students of the three constraints for making a tessellation.
Oral: Give them a single pentagon Polydron and ask if it is possible to tile with
that shape. Are they certain? (If necessary, discuss the difference between pentagons and
hexagons. Compare their answer to the answer they gave for the hexagon tiling.)
Written: After students attempt to justify their answer orally, ask them to
describe their thinking with paper, beginning in the center of the page. Discuss
similarities and differences with their oral answer.
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With either task, if they believe the tiling is impossible, ask why hexagons can
tile, but not pentagons.
Polydrons: Ask students to attempt to make the tiling with Polydrons. Discuss
any differences in the result with their oral and written descriptions. (If students have
difficult manipulating the Polydrons, move to the computer environment.)
Set D: Squares and Triangles
(Purpose: To explore a rich, dihedral tessellation task which can reveal both
conceptual and aesthetic thinking)
(Materials: paper, marker, software, square and triangle pattern blocks)
Show student square and triangle pattern blocks. Ask if it is possible to make a
tiling with both together. Are they certain?
Oral and written: Ask students briefly to justify their answer orally and with
paper.
Pattern blocks: Give students all the triangles and squares and ask them to make
a tiling. Will their tiling continue forever? How? Are they certain? After they complete
one tiling, ask them if they can make a different tiling. Which of their tilings do they like
best? Why? What are the similarities and differences?
Software: After two tilings with pattern blocks, ask students to make further
tilings with the computer software as time allows. Will their tiling continue forever?
How? Are they certain? Compare the different tilings they have made. How are they alike
or different? Which do they like best and why?
INTERVIEW #3 PROTOCOL
Preparation
(Materials: pre-printed sheets [see figures on page 257]; marker; pencil; triangle
and square pattern blocks; computer software)
Prepare the three triangles in the triangle set in the computer software—acute
isosceles, right scalene, and obtuse scalene triangles.
Analyze the students‘ responses to the first task of the pre-test. Refer to the article
by Battista, Clements, Arnoff, Battista, and Borrow (1998). Prepare questions to ask the
students concerning their response to this task.
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Set B: Tiling Completion
(Purpose: To identify mathematical structures in the context of a finite tiling; to
investigate the ability to visualize basic decompositions; and to provide a context for
exploring the constraints against gaps and overlaps)
(Materials: pre-printed sheets [see figures on page 257]; marker; pencil; triangle
and square pattern blocks)
This set was originally planned for Interview #1. However, Set A took more time
than anticipated, so Set B was moved to Interview #3.
The problems for this set are loosely based on the problems in Owens and
Outhred (1998).
Preliminary problem – Discuss the student‘s response to the first problem in the
pre-test (see Figure 77 below), which was based on a problem by Battista, Clements,
Arnoff, Battista, and Borrow (1998). Students were asked how many squares were
originally in the rectangle. The details of the protocol for this problem will be determined
based on the individual response on the pre-test and prepared in advance of the interview.
(The researcher also anticipates drawing from the responses to the tasks in Set A.)
Questions will be prepared to explore other aspects of thinking described in Battista et al.,
as well as exploring how much help a student needs to visualize the complete row- and
column-structure.
Figure 77: Pre-test Item 1
Problem 1 – Students will be shown the first sheet (on the left in Figure 78
below). Instead of a big, infinite room, we are now tiling a little room. Ask whether the
orange square can be used to tile the rectangle.
If the answer is no, students will be asked to justify their thinking.
If the answer is yes, students will be asked how many orange squares it will take
to complete the tiling. After oral discussion, or if the student is stuck, the student
will be asked to sketch the squares in the rectangle.
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Problems 2–6 – Problem 1 will be repeated with the five other sheets (figures
below). For Problem 2, note that lines indicate where someone has started to show the
tiling. If needed, students will be offered the chance at the end of Problems 1–4 to verify
their answer with pattern blocks. Label each paper with the student‘s pseudonym.
Figure 78: Set B, Problems 1 and 2
Figure 79: Set B, Problems 3 and 4
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Figure 80: Set B, Problems 5 and 6
The interviewer will thank the student for her or his participation and say he is
looking forward to next week‘s interview.
Set E: Triangles
(Purpose: To explore children’s beliefs and conceptualizations of non-equilateral
triangles)
(Materials: Computer software with triangle set—acute isosceles, right scalene,
obtuse scalene)
The interviewer will present the triangle set in the computer environment and ask
students if it is possible to tile first with the acute isosceles, then the right scalene, and
finally the obtuse scalene triangles. As in past interviews, students will be asked first to
describe the possibility or impossibility orally, and how certain they are. Next they will
describe their answer by sketching on blank paper. Finally students will attempt to make
the tiling in the computer environment. After creating a tiling, students will be asked if it
is possible to use the same triangle to tile in a different way. Which way does the student
prefer, and why? This process will be repeated for each of the three triangles. Finally,
students will be asked to compare all the triangle tessellations they have made. If
necessary, access the snapshots created by the program. Which do they prefer, and why?
Is the preference a factor of the tile selected, or the way they tile? What makes a beautiful
or interesting tiling?
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INTERVIEW #4 PROTOCOL
Preparation
(Materials: Plastic tromino and tetromino tiles; dot paper; pencil (no marker);
software with tromino and tetromino sets; a pre-drawn example of a non-edge-to-
edge tiling with the I-tromino)
Prepare a dot paper example of a non-edge-to-edge tiling of the I-tromino (a
―brick wall‖ tiling). Prepare the tromino and tetromino tiling sets in the software. Also
prepare polyiamonds tilings sets in case there is extra time at the end. (See Set H.)
Set F: Polyominoes
(Purpose: To study how students tile with non-convex, orthogonal shapes, and how
students choose which subtask to work on.)
(Materials: Plastic tromino and tetromino tiles; dot paper; pencil (no marker);
software with tromino and tetromino sets; a pre-drawn example of a non-edge-to-
edge tiling with the I-tromino)
Trominoes: Show the student the I-tromino plastic piece and ask if it is possible
to make a tiling with that piece. Explore the child‘s reasoning orally first. Is the child
certain of the answer? Is there more than one way to tile with this piece? Then, give the
child dot paper to sketch the solutions with. Which tiling does the child like the best? If
the child found only the edge-to-edge tiling, show the child a non-edge-to-edge tiling and
probe the reaction. Is this tiling acceptable? Which tiling is best?
Next show the student the L-tromino plastic piece and ask if it is possible to tile
with that shape. If not, why not? If it is possible, is there more than one way to tile? Is the
child certain? Give the child dot paper to sketch the solutions. Which tiling does the child
like best?
Tetrominoes: Show the student all five tetromino pieces. Which ones can be used
to tile the floor? How certain is the student? Ask for brief oral justifications of any tile
that the child is certain about.
Allow the student to choose any tile the student wants to create a tiling using dot
paper. Is that the only tiling possible? Why did the student choose that tile first? (If the
student asks if a tile can be flipped over, allow the child to decide if that should be
allowed or not.) After a couple of tilings, ask the student which tile the student would like
to use next, and so on for the remainder of the half hour. After a few tilings, or when the
child is stuck or tired, move to the computer environment and continue creating tilings
there.
Compare all the polyomino tilings made by the student. Which does the student
like best? Why? What makes a good tiling?
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INTERVIEW #5 PROTOCOL
Preparation
(Materials: computer software)
Prepare the regular polygon set with 15° rotations.
Set G: Regular Polygons
(Purpose: To observe the aesthetic choices of an open-ended tessellation task.)
(Materials: computer software with regular polygon tiling set)
Remind students of the task—to create infinite, gapless tilings.
Triangles and squares – Students will be asked to create one more square and
triangle tessellation.
Regular polygons – Show the student the regular polygon tiling set. Students will
be asked to continue making tilings using any combination of regular polygons.
After a few tessellations, ask if it is necessary that the vertices matched. Should
the snapping magnets be turned off? What kind of tiling is best—with or without
snapping? Why?
After each tessellation, ask the student‘s opinion of the tiling. Is it beautiful? Why
or why not? What makes a good tiling? Why did the student choose the polygons used?
At the end of the interview, show the student all the tilings made during this
interview and ask which tiling is the favorite. Why? What makes a beautiful tiling?
INTERVIEW #6 PROTOCOL
Preparation
(Materials: Selected tilings from subjects, public domain, the Alhambra, and
Escher; four sheets marked with names of rooms; sticky notes)
Select student tilings from first five interviews using aesthetic criteria from
theoretical framework and interviews to create as wide a space of tilings as possible. Print
them on card stock and sort into three diverse sets. (Put similar tilings in different sets.)
Prepare four sheets of paper with names of four rooms in a house (kitchen,
bathroom, etc.).
Use the aesthetic criteria to create more tilings. Draw also from tilings in the
public domain. Print these out on card stock. Prepare a list of pairs (or small groups) of
tilings to be compared according to specific aesthetic criteria.
Prepare a selection of postcards from the Alhambra.
Mark with sticky notes selected pages from books on Escher.
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Prepare sticky notes with ―1st‖, ―2
nd‖, and ―3
rd‖.
Aesthetic Tasks
(Purpose: To observe the aesthetic choices of tilings.)
(Materials: Selected tilings from subjects, public domain, the Alhambra, and
Escher; four sheets marked with names of rooms; sticky notes)
Remind students of tessellation criteria: infinite, gapless, and non-overlapping.
There are four separate tasks in this interview.
Student tilings
Give each student one of the three packets of student tilings. Explain that these
examples are from their own answers to the tiling tasks in earlier interviews. Lay out the
four sheets of paper. Ask the students to select nice tilings for each of the four rooms.
The cards are meant to be the pattern one would show to the person who has to lay the
tiles.
Next, ask the students to work together as a group to choose the ―best‖ tiling for
each room. Encourage them to discuss and explain their reasons.
(See Figure 81 below for students‘ tilings. The prototiles on the left were removed
from each card before the task, leaving only the tiling.)
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Other tilings (Student Sorting, Comparisons in Pairs, Professional Preferences)
(1) Spread out the cards representing tilings created by the interviewer and others.
(See Figure 82 below for the tilings used in these tasks. P was a blue and green 4.82
tiling.) Ask the students to put them in groups according to any criteria they want. Ask
which group is the best and why. Which tiling in that group is the best tiling? Ask the
same question of the other groups.
(2) Go through the list of prepared pairs of tilings (A-N, C-H, M-I, B-I, D-O, C-F,
M-R, D-I-G-O-P, and C-Q-L-N). For each pair (or set), ask the group which tiling is best.
Encourage the group to come to a consensus on each decision and to explain their
reasoning to each other. Do not allow them to spend too much time on any one decision.
(3) Which tiling do they believe an artist would prefer? Why?
Which tiling do they believe a mathematician would prefer? Why?
Figure 82: Most of the tilings for three of the aesthetic tasks
Professional tilings: Alhambra
Show the students the postcards from the Alhambra one at a time. Ask them to
rank them in order from ―best‖ to ―worst.‖ Encourage them to explain why the one or two
best tilings are their favorite.
Which tiling do they believe an artist would prefer? Why?
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Which tiling do they believe a mathematician would prefer? Why?
(See Figure 83 below for the seven Alhambra tilings used in this task.)
Figure 83: The seven Alhambra tilings
Professional tilings: Escher
Begin with the collection of tiling types drawn by Escher. Ask which one they
favor.
Explain that Escher is an artist who created tilings with animal shapes. They are
to be the judges of his work and award certain tilings first, second, and third prize. Flip
through the selected Escher drawings. Whenever the students say they really like a
certain tiling, mark it with a sticky note and encourage them to explain their reasons.
Once all the tilings have been looked at, have them discuss which of the tilings marked
with their sticky notes should win first, second, and third prize. Have them explain their
reasons for each choice.
The tilings used in this task are in Table 8 below. Most were prepared in advance.
A few were noticed by the children as I turned the pages and they stopped to comment on
them.
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Table 8: Escher drawings used in the sixth interview
Title (description) Book38 Page
Poster of eight tessellations used by Escher in lectures 1 32
“Fundamental Forms of Regular Division of the Plane” (Second illustration used by Escher in lectures)
1 33
“Regular Division of the Plane Drawing #6” (camels) 1 120
“Regular Division of the Plane Drawing #7” (squirrels) 1 121
“Regular Division of the Plane Drawing #58” (see Figure 57 on page 185)
1 161
“Regular Division of the Plane Drawing #67” (horsemen) 1 169
“Regular Division of the Plane Drawing #66” (see Figure 63 on page 195)
1 168
“Regular Division of the Plane Drawing #128” (black and white birds)
1 221
“Regular Division of the Plane Drawing #127” (angular black and white birds)
1 220
“Regular Division of the Plane Drawing #132” (red and blue flowers)
1 225
“Regular Division of the Plane Drawing #131” (black and white flowers)
1 224
“Regular Division of the Plane Drawing #34B” (birds and fish) 2 51
“Regular Division of the Plane Drawing #106” (birds) 2 89
“Regular Division of the Plane Drawing #104” (black and white lizards)
2 88
38 The three books used are:
1: M. C. Escher: Visions of Symmetry (Schattschneider & Escher, 2004)
2: M. C. Escher’s Universe of Mind Play (Escher & Seibu, 1983)
3: The Magic of M.C. Escher (Escher & Locher, 2000)
All authors use Escher‘s numbering system, but give the sketches somewhat different names. I am
following the titles given by Escher & Locher.
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Table 8. cont.
Title (description) Book Page
“Regular Division of the Plane Drawing #118” (lizards) 2 98
“Regular Division of the Plane Drawing #119” (flying fish) 2 99
“Regular Division of the Plane Drawing #6” (extended) 3 Inside cover
“Regular Division of the Plane Drawing #21” (basis for “Cycle”)
3 53
“Cycle” 3 55
“Regular Division of the Plane Drawing #69” (detail) 3 62
“Regular Division of the Plane Drawing #22” (red birds, white fish)
3 63
“Regular Division of the Plane Drawing #20” (extended) 3 90
“Reptiles” 3 175
An assortment of 35 regular division of the plane drawings, numbers 7, 10, 12, 13, 14, 16, 20, 24, 25, 28, 34, 37, 39, 42, 46, 54, 55, 65, 67, 71, 72, 76, 89, 97, 99, 111, 112, 118, 120, 121, 122, 123, 124, 128, 129
3 74–75
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Appendix C: Software
In this appendix I explain some of the design decisions I made for the software
used in this study.
One problem with all existing software is the relative complexity of rotating a tile.
A child typically needs to rotate a tile several times while placing it. However, some
existing software requires several mouse clicks each time the tile is rotated. With some
software, the user must release the tile, click on a rotation button, click and drag the tile
to rotate it, reclick the rotation button, and finally resume moving the tile. If the rotation
is not right, these steps must be repeated several times. With more sophisticated graphics
software, instead of a separate button, the user can often just click on the shape itself to
enter rotation mode. But such software tends to have an interface which is overwhelming
for children, and the number of clicks is sometimes still the same.
I know from experience that children adapt to this complexity, but it slows the
process down and is not nearly as intuitive as the placing of physical tiles on a table. I
wanted the software to be as simple and intuitive as possible so that the child could focus
on the tessellation rather than the tool. Some tessellation software designed for children
take these needs into consideration by allowing the child to click on the corner of a tile
and to begin immediately rotating it without having to click again, as if the mouse were
grabbing the corner of the tile and turning it. Unfortunately, the tile still does not usually
line up in the correct orientation until after several attempts of alternately rotating and
sliding the tile into position.
Some software (e.g., Tessellation Creator, NCTM, 2011) ―helps‖ the child by
automatically aligning the polygon in an edge-to-edge manner against any nearby
previously placed tile. This can make for quick tiling, but has two drawbacks with respect
to this research: (1) the software may help the child too much, placing the tile in positions
that the child had not thought of, thereby helping the child to create a tiling the child had
not conceived; and (2) the child in some cases may not want to make an edge-to-edge
tiling.
Other difficulties that I faced included: (1) children‘s software typically comes
with a restricted set of prototiles; (2) I needed frequent automatic screen captures to
document the actions of the children; and (3) I needed a flexible environment, e.g. one
that could make either edge-to-edge or non-edge-to-edge tilings, as needed.
I decided therefore to design my own software, ―Polygon Tiler,‖ using Visual
Basic (Microsoft, 2003). The advantage to creating my own software was that I could
tailor it very precisely for this research. In order to make both rotations and edge-to-edge
placements intuitive, the tile behaves as if the vertices were magnetized. The child begins
by simply dragging a tile to its destination. The tile initially moves in a simple translation
mode. Once a vertex of the moving tile comes close to the vertex of a previously placed
tile, the moving vertex ―snaps‖ to the already placed vertex and the moving tile then
268
begins to rotate about the snapped point until it reaches the desired position. If a second
vertex comes near another vertex, the tile snaps solidly into place and stops moving
unless the mouse pointer moves too far from the center of the tile, in which case the
moving tile unsnaps and can be moved freely again. Once the mouse button is released,
the tile stays put. The entire process of moving and rotating the tile from its initial
position to its final position is accomplished with a single click-and-drag in an intuitive
manner. A special algorithm prevents last-second slipping of the tile when the mouse
button is released. If the child desires to place the tiles in a non-edge-to-edge manner, the
magnets can be turned off either temporarily or permanently. If needed, a tile can also be
rotated by clicking on its vertex instead of the interior of the tile, as with other children‘s
software. These two rotation methods give the child a choice of intuitive methods for
placing the tile.
Flipping of tiles is much more rare. Tiles with reflection symmetry never need to
be flipped at all and other tiles never more than once. Flipping in Polygon Tiler therefore
simply relies on the traditional method of clicking a ―Flip‖ button (which is hidden if not
needed).
Because I designed my own software, I could tailor the program to the precise
needs for this research, including the need to take snapshots of all steps in each
construction and file these data by pseudonym and date. I included many other elements
in the program, some of which I included just in case I saw a need for them during the
interviews when I no longer had time to program. Some of the elements included in the
software are:
A spare, full screen with no more than three unobtrusive buttons;
An optional flip button which can be hidden or displayed as needed;
A hidden toolbar for seldom-used options, just in case;
Keyboard shortcuts for these options, including a shortcut for temporarily
disabling ―magnetization‖ when it occasionally interferes with tile placement in a
complex tessellation;
Automatic zooming and centering of the tessellation if it grows too close to the
edge of the screen, using a smooth animation;
The ability to zoom in or out and center manually, if needed;
The ability to easily disable snapping entirely if a non edge-to-edge tiling is
desired;
Control of snapping sensitivity, zooming factors, and graphics animation in order
to adjust to the child‘s dexterity and the computer‘s graphics capability;
Automatic sizing of the tiles in the initial generating set according to screen size;
An algorithm to prevent slipping of the tile when the mouse button is released;
Automatic screenshots of each tile placement organized by pseudonym, date, and
tiling;
Management of names and pseudonyms in a password protected environment;
269
Rejection of seriously overlapping tiles using animation (which the students found
amusing);
Various rotation modes to constrain the allowable rotations (usually I allowed
only 15° rotations);
The ability to change the color of the tiles (this was not used);
Automatic saves and the ability to resume a previous tiling;
Four different graphic methods for quickly defining any polygon needed for this
research;
The ability to define generating sets ready for specific interviews, including the
kinds of flipping and rotation allowed for each set;
The ability to edit polygons and generating sets;
The ability to choose file locations for tile definitions and screenshots;
And a password lock to prevent children from altering sensitive settings without
permission.
After using the software for the interviews, I only noticed two significant changes
that needed to be made. (1) The setting for the anti-slipping function needs to be
adjustable. This parameter needed to be increased for most of the students and there was
no way to adjust it quickly on the fly. The only aspect of the software that seemed to
frustrate the students was making fine adjustments without the mouse slipping at the last
moment, which might cause the tile to overlap and fall. Nevertheless, they found the
falling tiles to be amusing and sometimes enjoyed making the tiles overlap and fall on
purpose. (2) The automatic zooming function should be a bit less sensitive, or else
adjustable on the fly. The idea for the automatic zooming was to reinforce the idea that
there was no boundary to the tiling space. If tiles were placed close to opposite screen
borders, the program zoomed out automatically. Most students did not mind the
automatic zooming function, but a few did. Fortunately the automatic zooming was not
that important and I had allowed for an easy way to disable the automatic zooming
completely, so this did not present a problem. Aside from these two difficulties, the
software worked well and the children seemed to enjoy working in this environment.
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Appendix D: Analysis Codes
Table 9 below contains the list of codes that emerged from the analysis of my
data. Not all codes proved useful in identifying patterns in the data. Codes in italics are
just headings for other codes; no episodes were coded with these headings.
Table 9: Codes used in the analysis of the data
Category
Codes and sub-codes Description
Aesthetics Type of aesthetics expressed, if any. Instances where students could not express a reason were not coded.
Mathematical Aesthetics promoted by mathematicians
Connectedness Appealing because it relates otherwise unrelated ideas
Significance Appealing because it is mathematically important
Simplicity Simple, elegant, economical, controlled, or ingenuous
Surprise Surprising, unexpected, shocking, new, or insightful
Validity A tiling is appealing only if it meets the 3 criteria for a tiling
Visual appeal Appeals to the visual sense
Negative appeal Category for aesthetic qualities that are unattractive
Bad color Color is unappealing.
Bad real-world object Reminds students of a repulsive or boring real-world object
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Table 9. cont.
Category
Codes and sub-codes Description
Bad tiles Poor choice of prototile
Boring Tiling is too simple.
Can’t get a box unit Kobe disliked one tiling because he could never close it off straight.
Femininity Some tilings are considered “girl stuff.”
Finite Cannot be extended indefinitely
Insufficient color Needs more color
Invalid Not a valid tiling
Lack of pattern Too random
Not unique Tiling is not sufficiently different from other tilings.
One dimension Has only one interesting dimension
Overly complicated Too complex
Radial is bad Less interesting than periodic tilings
Semi-random is bad Having some order, but not completely patterned
Single orientation All tiles are in one orientation.
Too simple Even a child could create such tilings.
Other All other positive aesthetic criteria—these were used to describe appealing tilings
3D Tiling has three-dimensionality, whether real or illusion
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Table 9. cont.
Category
Codes and sub-codes Description
Alternating Tiling is cool because of alternation in the lengths or orientations, or makes a zigzag pattern.
Animal used An Escher tiling uses an animal which students like or find awesome.
Appropriateness Tiling is appropriate (frequently used) for a certain real world tiling task, such as tiling a kitchen floor. It is appropriate for the given or an imagined context.
Color Colors of tiling are appealing.
Complexity Variety of shapes, or the way shapes within shapes are used (such as L-tromino dilation tiling).
Craziness Tiling is appealing because it is “crazy” or “weird.”
Cultural style Tiling is appealing because it appears to be in a certain foreign style, such as “Asian,” “African,” or “Fantasy World.”
Diagonal Tiles or units appear in a diagonal pattern.
Energy Tiling is appealing because it is energetic or suggests explosions.
Folding Tiling suggests it could be folded in surprising ways.
Fun Tiling is considered fun.
Futuristic Pattern suggests a “future look.”
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Table 9. cont.
Category
Codes and sub-codes Description
Imagined or familiar context Tiling is appealing because of what person imagined could be done with it, or because of imagined similarity with real-world example.
Induction Mathematicians like the induction suggested by Tiling N. It is a “rep-tile.”
Intricacy Tilings have a labyrinthine, knotted, puzzle, or interconnected quality.
Masculinity Tilings that are “boyish” are more appealing than those that are “girlish.”
Movement Tiling suggests movement.
Multi-colored Tiling is colorful.
Multiple possibilities Mathematicians liked tilings that could be interpreted in multiple ways.
Non-tiling element A photo was appealing because of some elements (such as grass or “earthy”) that had nothing to do with the tiling itself.
Optical illusion Tiling suggests an optical illusion.
Proportions Appealing proportions
Randomness Randomness is appealing.
Realistic In one case, a photo was preferred over a drawing because it was “realistic.”
Real-world suggestion Tiling is appealing because it reminds people of a real-world object, symbol, or style (such as “Easter”).
Relaxing Tiling has relaxing quality.
274
Table 9. cont.
Category
Codes and sub-codes Description
Semi-random Tiling has elements of order and randomness.
Shapes contained Tiling is appealing because it contains certain shape elements, such as flowers, arrows, zig-zags, or stars.
Shapey The private school described one tiling as “shapey,” having curves and lines and looking good even without color, but I’m not sure what they mean.
Size Rachel said she preferred the L-tromino to the L-tetromino because she likes “little things.”
Symmetry Tiling is better when it includes symmetry, sometimes described as having patterns or being organized or “tidy.”
Thought-provoking Mathematicians liked tilings that were thought provoking.
Tiles used Tiling is appealing because of the specific shapes used or the diversity of shapes used to compose it.
Touch of assymetry A tiling is appealing if it is symmetric, but contains a small element that destroys the symmetry.
Unique-creative Describes tilings that are different, not like the others. Tiling is “creative.”
Confidence Level of confidence in answer, especially in validity of tiling
No confidence Student claims to be guessing or very uncertain
275
Table 9. cont.
Category
Codes and sub-codes Description
Strong confidence Student expresses strong confidence in voice and answer, e.g. “Are you sure? Yeah,” or “I’m pretty sure,” or “I think it’s gonna work.”
Weak confidence Student expresses hesitation or some uncertainty, e.g. “I’m not 100% sure,” or “Pretty probably maybe.”
Confidence at symmetry Confidence was coded for each episode where symmetry was coded using nearby statements of confidence so that symmetry and confidence could be compared.
Doubt tiling will work (at symmetry) Suspects tiling will not work, e.g. “Kind of think it won’t work,” but has not yet decided.
No confidence (at symmetry) Same as “No confidence”
Strong confidence (at symmetry) Same as “Strong confidence”
Weak confidence (at symmetry) Same as “Weak confidence”
Polygon Which polygon is the student working with?
Acute isosceles triangle Acute isosceles triangle
Obtuse scalene triangle Obtuse scalene triangle
Parallelogram Parallelogram
Pattern blocks Pattern blocks
Equilateral triangle Green triangle
Hexagon Yellow hexagon
Rhombus Blue rhombus
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Table 9. cont.
Category
Codes and sub-codes Description
Square Used in the introduction; not recorded for all students
Squares and triangles Dihedral tilings of squares and triangles
Pentagon Polydron pentagon
Polyominoes Polyominoes
Domino Domino (used in Finite Tiling task)
I-tetromino I-tetromino
I-tromino I-tromino
L-tetromino L-tetromino
L-tromino L-tromino
O-tetromino O-tetromino
S-tetromino S-tetromino
T-tetromino T-tetromino
Regular polygons Tilings made in Interview #5, generally non-monohedral and frequently changing the generating set
Right isosceles triangle Right isosceles triangle (Finite Tiling task)
Right scalene triangle Right scalene triangle
Possibility Which tiles can tile? Are there tiles that cannot tile?
All polygons can tile States that any polygon can tile
All shapes can tile Expresses the view that all shapes can tile
277
Table 9. cont.
Category
Codes and sub-codes Description
Impossibility reason Why will the shape not work?
Curves or no flat sides make tiling impossible
Curves or no flat sides make tiling impossible
Empirical—can’t find a tiling that works
Empirical reason; cannot find a tiling that works
Equilateral not right kind of triangle for square
Squares can only tile with right isosceles triangles
Gaps make tiling impossible The shape cannot work because it will leave gaps.
Known impossible shape makes tiling impossible
Kobe thought that because squares and triangles together made pentagons, which were known not to tile, squares and triangles cannot tile.
Neither round nor sharp enough to fit
Hexagons are round enough to tile; squares are sharp enough. Pentagons are neither.
Not a square-type tile, so cannot tile
Only square-type tiles can tile.
Overlap makes tiling impossible
Circles cannot tile because they must overlap.
Some pentagons don’t work (others will)
Some pentagons don’t work. (Others will.)
Impossibility shape Which shape cannot tile?
Bizarre shapes can’t tile Bizarre shapes cannot tile.
Circles can’t tile Circles cannot tile.
Curved edges can’t tile Shapes with curved edges cannot tile.
278
Table 9. cont.
Category
Codes and sub-codes Description
Non-polygons can’t tile Non-polygons cannot tile.
Pentagons can’t tile Pentagons cannot tile.
Rhombi can’t tile When attempting to sketch, student concludes the rhombus will not tessellate.
Squares and triangles can’t tile Dihedral tilings with squares and triangles are impossible.
Trapezoids can’t tile The right-angled trapezoid of the pre-test cannot tile.
Reasoning Type of reasoning displayed
Infinity Type of reasoning about infinity of tiling
Functional infinity Understands consequences of limitless tiling, but prefers working with finite boundaries to create tiling
Limited infinity Displays lack of understanding of consequences of limitless tiling, such as failing to see two patterns make same infinite tiling
Mathematical infinity Displays sound reasoning about the consequences of limitless tiling
Tiling justification Reasoning used to justify the validity of the tiling or explanation of how they know where to put the tiles
Empirical justification Uses empirical or inductive evidence
Experience justification Uses previous experience with tiling or similar tiling
Gapless justification Notes that gaps are always filled
279
Table 9. cont.
Category
Codes and sub-codes Description
Known unit justification Explicitly uses another known tiling by making units of known tiling, e.g. 2 L-trominoes make a rectangle
Similar to known tiling justification
Notes resemblance to another known tiling
Subunit justification Says a tiling can be made with a shape because it can be decomposed into smaller shapes known to tile
Symmetric justification Demonstrates an (unstated) use of symmetry in the justification
Transformational justification Makes an implied or explicit reference to a transformation, usually translation in one direction only
Sorting criteria What criteria were used in sorting exercises during Interview #6?
Sort by certain attribute Sets created based on a certain aesthetic attribute such as “mazy,” “crazy,” or 3D
Sort by color Sets were created using color, either certain colors or the attribute of being colorful.
Sort by context Sets created according to where the tiling would be used, such as “patio” or “outdoors”
Sort by media Sets created according to media used, such as stones
Sort by multiple criteria Sets were created with more than one criterion, such as “gray and stony.”
280
Table 9. cont.
Category
Codes and sub-codes Description
Sort by real-world resemblance Sets created because they reminded students of a certain category of real-world objects, such as “buildings”
Sort by shapes Sorting uses tile shapes.
Sort by simplicity vs complexity Sorting considers whether a tiling is basic or something unusual not often seen.
Strategy Strategy used to make tiling
Complex rows strategy Student makes complex rows. (If possible, use one of the three subcodes.)
Alternating rows strategy Makes rows with alternating orientations
Brick wall strategy Staggered, non-edge-to-edge rows
Units in rows strategy Student makes complex rows made of units. This occurs together with “Units strategy” coding.
Complex strategy Makes a sophisticated pattern using neither rows nor units
Fit one at a time strategy Student places one tile at a time resulting in apparent pattern or standard tiling, but without expressing or implying any organizing strategy such as rows or units.
Grid strategy Student sees square, rectangle, or parallelogram grid, simultaneously in two directions.
Growing strategy Makes a growing pattern, such as a tower, dilation, or radial pattern
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Table 9. cont.
Category
Codes and sub-codes Description
Random strategy Makes apparently random tiling
Semi-random strategy Makes apparently random pattern, though there are strong elements of order in parts of the tiling
Random units strategy Creates units, but then combines units in apparently random way
Simple rows strategy Student makes simple rows; this includes alternating square rows and triangle rows.
Spiral strategy Student sees tiles as expanding in a spiral.
Units strategy Student makes units. If the units are placed in rows, this is also coded as “Units in rows strategy.”
Symmetry Type of symmetry created by student, as observed during drawing, physical tiles, or computer creation; only patterns with more than 6 tiles are considered (significant progress towards actual tiling, not just exploring fit). Codes were applied when largest such symmetry was created (or noted in transcript) for entire tiling (not local symmetries). If there was no symmetry, but the infinite tiling was clearly intended to be symmetric (e.g. Moses’ parallelograms), I left the tiling uncoded.
2-fold symmetry 2-fold symmetry
3-fold symmetry 3-fold symmetry
4-fold symmetry 4-fold symmetry
6-fold symmetry 6-fold symmetry
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Table 9. cont.
Category
Codes and sub-codes Description
8-fold symmetry 8-fold symmetry
9-fold symmetry 9-fold symmetry
No symmetry No symmetry (other than necessary translation symmetry) was observed throughout the tiling creation, nor did the student try to create symmetry beyond first few tiles. (Coded at end of tiling.)
No symmetry at end Some significant symmetry was observed during the tiling creation, but the final product was distinctly not symmetric.
Order-2 rotation symmetry No reflection symmetry (e.g. standard parallelogram tiling)
Reflection symmetry Reflection symmetry (coded by axis orientation)
Horizontal reflection symmetry
Simple horizontal symmetry (horizontal axis)
Oblique reflection symmetry There was one case of an oblique axis.
Vertical reflection symmetry Simple vertical symmetry (vertical axis)
Tool With what tool was the tiling created or envisioned?
Blank paper drawing Blank paper drawing
Computer program Computer program
Dot paper drawing Dot paper drawing
Oral description Oral description
Pattern blocks Pattern blocks
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Table 9.cont.
Category
Codes and sub-codes Description
Polydrons Polydrons
Pre-test discussion Pre-test discussion
Understanding How well does the student understand how the tiles fit together to form a tiling? Is the description a mathematically valid way of making a tessellation?
Fit What is the level of understanding for how the polygon fits together around each vertex?
Accurate fit Describes accurately how tiles fit together
Inaccurate fit Does not put the correct number of polygons around a vertex or otherwise displays lack of understanding of how polygons fit together
Tiling Is the tiling mathematically valid?
Accurate tiling Accurately describes a valid tessellation
Apparently accurate tiling Tiling with errors, which student explains as being drawing imprecision that will not appear in physical tiling, or finite tilings that would be correct except for the size of the tile
Embodied tiling Student lacks a mental model for the tiling, but can construct it on paper. (The student knows it works, but cannot describe it.)
Inaccurate tiling Does not give a mathematically valid description that will work with the physical tiles
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Table 9.cont.
Category
Codes and sub-codes Description
Incomplete tiling The student sees how the tiles fit together, but did not draw enough to show how it would repeat in all directions; student possibly lacks an overall repeating pattern for forming an indefinite tessellation or possibly just was not pushed to extend tiling.
Partially accurate tiling Displays partly valid tessellation, with some errors
Wording Analysis of active or passive wording in first interview
Active wording Student uses active wording to describe tiling, e.g. “Put this one right here like that and then do it again.”
Descriptive wording Student uses passive description, e.g. “One line’s gonna be facing one way, the other, the next line’s gonna be facing the other way.”
(Uncategorized)
Abstractness of drawing Demonstrates understanding that a freehand drawing may not be accurate, but is only a representation of geometric ideal
Change of center As student makes a growing, symmetric tiling starting with a central core, the center of the tiling shifts to a new point.
Concern for accuracy Expresses concern over inaccuracy of drawing
Reference to symmetry Student makes an explicit reference to symmetry (not necessarily using that word).
Traces physical tile Uses pattern block to trace tiles in drawing
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Appendix E: Results by Task
This appendix provides greater detail about the specific results from each task.
References to the ―standard tiling‖ mean the simplest, edge-to-edge tiling. See Appendix
B for protocol details.
SET A: BASIC TILINGS
Students were asked about tiling various pattern blocks. The initial phase for each
shape was an oral phase where students were given one pattern block and asked to
describe how the tiling would work. This was followed by a drawing phase (with marker
and paper) and a construction phase (with a set of pattern blocks of the given shape).
Moses was also allowed to use the computer for the construction phase of the triangle.
All students used the computer for the parallelogram, which had no pattern block.
Rhombus
All six students thought from the beginning that it would be possible to tile with
the rhombus. Some were initially concerned about walls where tiles would have to be cut.
With the exception of Moses, each student described the tiling by describing how other
rhombi would fit around the first tile. Moses described how there would be rows of
rhombi going off in all four directions from the sides of the initial tile. This was a rare
instance of explicit awareness of structure in two different directions. After describing the
initial fit, Michelle described how the tiling would continue to grow in all directions. The
other four did not mention how the tiling would work beyond the tiles next to the initial
tile, limiting their thinking to how the tiles fit together. Describing how tiles fit around
the initial tile tended to be the primary way of describing a tiling during the oral phase of
each task.
All students, including Moses, sketched their ideas of how the tiling would work
one tile at a time. Three students drew the standard tiling with all tiles in the same
orientation. Kelsey also drew all tiles in the same orientation, but without any clear
structure and with one tile drawn as a large triangle. The other two students drew random
tilings.
When students were given pattern blocks, three created random tilings, two
created the standard tiling (often by making large rhombi), and one created a tiling with
alternating rows. Kelsey attempted a second tiling which she seemed to describe as the
standard tiling, ―just turn them sideways like that ... on and on and on.‖ But with the
eighth tile, the tiling turned random. She afterward described patterns in the random tiling
which she used to create portions of the tiling. This occurred with several students and
seems to reflect a focus on local fit, rather than the global structure of the tiling, at least
when the child is engaged in the process of creating the tiling. Kelsey‘s second tiling had
smaller sections of tiles with the same orientation, which Kelsey saw as aesthetically
preferable.
286
Michelle made a random tiling at first, but then separated out part to make a
double row with opposite orientations, which she said would be both easier and better
aesthetically. (See Figure 70 on page 212.) The two students who made the standard
tiling maintained that the only way to tile was the standard tiling with all rhombi in the
same orientation.
Moses had described the standard tiling in his oral description and drawing, but
partway through his construction with pattern blocks he realized he could make a tiling
with ―zigzagging‖ rows, which he felt was a much better tiling. At one point, Moses
began to focus entirely on the border of his pattern, despite frequent reminders that there
were no walls in this infinite room. (See Figure 48 on page 164.) He felt sure the sides
would get closer and closer at the top and spent much time trying to build up the border
with an empty interior before finally abandoning his idea when prompted to consider
whether tiles would actually fit inside his long curving borders.
Hexagon
All students thought the hexagon would tile, though Marie and Kobe were
doubtful until they actually tried to draw it. All were fairly successful in sketching a small
section of the tiling, some by tracing the pattern block. Because there is only one possible
tiling, all students successfully made it, though they used slightly different strategies to
tile. Some emphasized symmetry more than others. Most took a one-tile-at-a-time
approach, focusing on how the tiles fit together.
Moses focused on vertical columns that were being formed. Four students
recognized the tiling as a honeycomb tiling and Moses recalled previous experience with
pattern blocks. Kobe built up his tiling as diamonds with two-fold symmetry. Marie
found she could make the pattern by either focusing on rows like a sidewalk, or using a
growing pattern like a flower. She felt the latter was more creative. Kelsey made a
―flower‖ of seven hexagons and said she was done until prompted to tile further. She
carefully explained the two ways that hexagons fit together using sides and angles.
Michelle recalled that she could make the hexagon out of triangles (perhaps
meaning the rhombi) and therefore the hexagon should tile too. This subunit reasoning
occurred a couple of times to different students: If a shape can be divided into subunits
that can tile, then the shape itself should be able to tile. Moses used this strategy by
remembering that two trapezoids (with which he apparently was more familiar) could be
used to make a hexagon.
Triangle
During the oral phase of this task students described how other tiles would fit
around the initial triangle. No one saw that six triangles could fit around a common
vertex. In fact, most children thought that once three triangles had been placed on the
sides of the initial tile, a single triangle would be needed to complete the fit at each vertex
of the initial tile, as with the rhombi. Moses expressed surprise when he drew the tiling
287
and discovered that six tiles would actually fit together, ―kinda like a hexagon.‖ The other
children never realized that their constructed tilings contradicted their initial description,
even with prompting. Two students saw that two triangles would make a rhombus and
therefore the triangle would also tile.
All students drew and constructed the regular 36 triangle tiling. When drawing,
three students occasionally put five triangles around a single vertex. (See Figure 46 on
page 158.) Kelsey organized her drawings by rows, as did Moses for his pattern blocks.
Moses used a growing strategy in his drawing and on the computer, adding layers of
triangles around the initial hexagon. Moses also saw he could make a large triangle and
use that as a unit to tile with—a dilation strategy, though he did not mention further
levels. Many students organized their tiling using a variety of patterns without any clear
single way to structure the tiling. Kobe noted, ―With the triangle you can get into any of
the patterns, diamonds, octagons, hexagons.‖ Rachel kept noticing the alternating
orientations of the triangles. Students usually preferred creating a convex pattern.
Two students believed that each tiling was different, though if extended all would
result in the same regular tiling. Two students even identified certain tilings as their
favorite, though mathematically they were all equivalent. Kelsey believed there were
more hexagons in one pattern than another. Two students saw that all the tilings were
really the same. Rachel thought they were the different, though she showed some
hesitation and at the end changed her mind. Kobe was not sure.
Parallelogram
Five of the students answered on the pre-test that it was possible to tile with the
parallelogram. Kobe understood the problem to be whether the parallelogram could tile to
exactly fill the sheet of paper, which it cannot without cutting. Other students expressed
hesitation orally concerning the problem of walls. Kobe changed his answer once it was
explained that the tiling was to be infinite.
Students tended to structure the tiling in either rows or columns. Michelle noticed
both and drew the tiling on her pre-test by using a ruler to make a grid. (See Figure 53 on
page 180.) Nearly all students created the standard tiling on the computer, which many
saw as three-dimensional, as though it were a rectangular floor tiling observed from an
angle.
Kelsey drew two columns in her pre-test, but during the interview she made two
rows, which started edge-to-edge but did not stay together. (See Figure 34 on page 143.)
She talked about how the rows ―could be shorter and longer,‖ which would look cool. On
the computer she made the standard tiling.
Kobe used a subunit reasoning. The parallelogram could be divided into four
triangles, therefore it must tile the plane. He focused entirely on the triangles until
prompted to show how the parallelograms would actually tile, in which case he gave the
standard tiling.
288
Nearly all students made the standard tiling, several in two orientations, which
some recognized as basically the same. Many claimed these were the only possible ways
to tile with the parallelogram. Kelsey and Rachel also offered the brick wall tiling and
Moses added a tiling with rows of alternating orientations.
Impossible Shapes
Students were asked if there is a shape which cannot be used to tile a floor.
Most students mentioned the circle as an example of a shape that could not tile.
Students said either circles would have to overlap or else they would leave gaps. Moses
said that the circle lacked flat sides. Rachel said that non-polygons could not tile and
recited criteria she had learned for defining a polygon.
Kobe had maintained earlier in the interview that all shapes could tile, mentioning
that the only exception would be if we moved to other dimensions. (Kobe frequently
mentioned things that showed he was being exposed to some very deep mathematical
ideas.) Marie believed the right-angled trapezoid on the pre-test could not tile the floor.
She did not see that she could rotate the trapezoid. (Rachel had also thought the pre-test
trapezoid could not tile because it was not symmetric, which is the only instance of a
student using this word. During the interview, however, she only mentioned circles and
non-polygons as impossible.)
Note: Set B was done after Set D.
SET C: PENTAGON
Students were shown Polydron tilings of squares and hexagons to remind them of
tilings from the first interview. (See Figure 84 below.) They were then asked if it was
possible to tile with the regular pentagon.
Figure 84: Mo and I discussing square and hexagon tilings with Polydrons
All students believed that the regular pentagon could tile, some with hesitation,
until they tried to do it with the Polydrons, after which they concluded it was impossible.
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Most seemed to think the tiling would look somewhat like the hexagon tiling and drew
something to that effect, with three or four pentagons around each vertex. Michelle drew
a tiling with five pentagons around a common vertex. (See Figure 35 on page 144.)
Students were not sure what the difference was between the pentagon and the
other regular polygons to account for the fact that the pentagon could not tile. In the end,
three students said the pentagon would not work simply because it always left gaps.
Moses saw from his drawing that other pentagons could tile, namely ―skinnier‖
pentagons. He also noticed the Polydrons could be bent into a three dimensional shape
(the beginning of a dodecahedron). Michelle thought the sides and angles of the pentagon
were different, accounting for the impossibility of tiling; the hexagon could tile because it
was rounder, and the square could tile because it was sharper.
Kelsey seemed confused by the Polydrons themselves. She focused on the hinges
even in her drawing (see Figure 85 below) and did not seem to understand how the
Polydrons worked until she actually tried to construct the tiling. Even then, she seemed
confused by the Polydrons.
Figure 85: Kelsey‘s drawing of a pentagon tiling with Polydrons
Kobe reiterated his belief from the previous interview that ―you can tile with
almost any shape.‖ He wrestled for over four minutes with the Polydrons before finally
deciding the tiling would not work, and even then kept trying. This seemed to make a big
impression on him and during the next task (squares and triangles), he focused on making
tilings with pentagons composed of one square and one triangle, in order to show that
some pentagon tilings were possible.
Students saw the hexagon as more symmetric than the pentagon. They noticed the
bilateral symmetry of the pentagon, but not its five-fold symmetry. The pentagon was
observed to be a ―house‖ or a square with a triangle on top. It was not obvious that the
base could be any side. The hexagon was more easily seen to be the same from any side.
The hexagon was also seen to be composed of six triangles.
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Michelle had some unusual difficulty counting the sides of pentagons and
hexagons and persisted in believing for a long time that the pentagon had at least as many
sides as the hexagon.
Students never saw that it was sufficient to study how pentagons fit around a
single vertex. Though most saw that three pentagons created a gap that could not be
filled, they continued to try to create a larger tiling in hopes that somehow something
would eventually work.
SET D: SQUARES AND TRIANGLES (ALSO SET G)
The squares and triangles task was visited twice—once during the second
interview and again during the fifth interview. During the second interview, all students
described tiling possibilities in oral and drawing phases and then attempted at least one
tiling using pattern blocks. All but Michelle also attempted a tiling using the computer.
For the fifth interview, students used only the computer.
Approaches to the squares and triangles task varied considerably. Every response
during the oral phase was unique. Most students believed during the oral phase that a
tiling would be possible, though Michelle expressed some hesitation. Rachel believed
that a tiling was not possible because the triangle needed to be a right isosceles triangle
(half square) in order to go with the square. However Rachel seemed to believe that some
sort of overlapping was being asked for in the task and when this misunderstanding was
cleared up during the physical construction phase, she changed her mind.
During the oral phase Kobe focused on the pentagon formed by a square and
triangle, seeing how they could make both a 33.4
2 pattern, which would work, and a
triangle surrounded by squares, which he believed would not work. Moses thought the
latter pattern could work by placing a single triangle at each corner of the central triangle,
making a large hexagon. (See the drawing on the right in Figure 46 on page 158. During
the construction phase, he saw he needed two triangles at each corner, which did not form
a hexagon but still led to a valid tiling.) Marie saw how two squares and three triangles
would fit together in a 33.4
2 pattern. Michelle described a hexagon of six triangles
surrounded by squares, which she decided during the drawing phase (where she actually
drew seven triangles) could not work. See Figure 86 below.
Figure 86: One of Michelle‘s attempts at making a tiling with squares and triangles
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A few students began with a central triangle, which usually did not lead to
problems. However, three students tried to create a tiling centered about a square (or
larger square made of square tiles), and this usually failed because they believed the
square should be entirely surrounded by triangles. It did not occur to them that another
square could be placed at each corner of the central square. In each case the students
struggled to fit five triangles around the corner of a square, shifting the triangles so that
they were no longer edge-to-edge with the square in order to minimize the size of the
inevitable gaps that formed. (See, for example, Figure 38 on page 148.) During the
second interview, students usually believed that such a tiling would eventually be
successful. Michelle, for example, thought the small gaps were the result of the
randomness of her tiling. During the fifth interview, Kelsey and Michelle again tried to
surround a square corner with triangles. This time, Kelsey decided that squares and
triangles could not tile and Michelle eventually found a different strategy. Kobe was the
only student to successfully create a tiling centered around squares, making a complex
tiling having four-fold and translation symmetry with rectangles made of six squares at
the center of each unit.
With both the pattern blocks and the computer, Moses made patterns which began
around a triangle and had three-fold symmetry. Each time, as he built it, the center shifted
and he ended with bilateral symmetry. He described how he kept the symmetry by
copying one side to the other because he liked patterns. Moses believed his second
pattern could be used as a large unit to tile the floor.
Kobe and Michelle found strategies based on ―houses‖—pentagons formed by a
square and a triangle. Kobe remembered from the previous task that the regular pentagon
was not possible and initially believed this tiling would not work either, preferring at first
to focus on a row strategy instead. Later when he successfully made a more complex
pattern, he pointed out all the pentagons that made it up and how this was different from
the failure of the pentagon task. Michelle used houses to create a radial pattern. Her
strategy eventually abandoned using houses alone, but she still saw houses throughout
her pattern.
The most common strategy, used at some point by four different students, was to
use row patterns—nearly always the 33.4
2 pattern with horizontal rows. Kobe also made a
tiling with double square rows. In the fifth interview Marie made a tiling based on a
tower with a double row of triangles at the top. (See Figure 51 on page 175.) She
eventually decided the double row of triangles were to be used along the walls of the
room. Unlike the pilot study where some students saw the tower as a finite unit, Marie
indicated that the rows were infinite. No one else used a tower strategy.
Three students made patterns with six-fold symmetry. Kobe worked out the 2-
isogonal 36/3
2.4.3.4 tiling and described its translation symmetry. (See Figure 37 on page
148.) The other two students created more complex radial tilings without translation
symmetry.
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Three students used different methods to create more or less random patterns.
Michelle‘s and Kelsey‘s patterns had small gaps. Michelle explained the gaps as a
necessary consequence of the randomness of her pattern. In the second interview, Kelsey
thought her random tiling would eventually work, ―As long as you can do it right.‖
Most students did not seem to recall for Interview #5 how they did the task for
Interview #2. And yet answers were similar for both interviews for most students. Kobe
came up with complex symmetric patterns both times. (He also made a row pattern the
first time.) Rachel made row patterns both times, after struggling to fit triangles around a
square corner the first time. Marie made the 33.4
2 tiling both times and both times said it
reminded her of a 3D farm. The second time, her row pattern evolved into a tower
pattern. Kelsey tried the same unsuccessful strategy of fitting triangles around a square
corner both times. She believed it worked the first time, but not the second. Michelle
struggled to put triangles around a square corner both times. The second time she
discovered a six-fold radial pattern after struggling with the square corner. Moses made
complex patterns both times that began with trilateral symmetry and changed to either
bilateral or six-fold symmetry.
SET B: FINITE TILINGS
Comparison of my students‘ initial answers with answers given by students in
Owens and Outhred‘s (1998) study are very comparable, except my students tended to
get the correct answer more often, partly because Owens and Outhred‘s study also
included second graders, but also perhaps because of the interview format. (Owens &
Outhred had noticed that some students gave the wrong answer on their written test, but
then showed correct understanding on a follow-up interview.) Both my students‘ and
Owens and Outhred‘s tended to answer that the tiling was not possible more often for
problems B3 and B6. (B5 was not in Owens & Outhred‘s test.)
Pre-test
The first pre-test item showed a 5×7 grid with most of the interior erased.
Students were asked how many squares were in the original rectangle and how they
knew. This item was adapted from a problem by Battista, Clements, Arnoff, Battista, and
Borrow (1998). This item was discussed with students as an introduction to finite tilings.
Kelsey appeared to be at Battista et al.‘s Level 1—complete lack of row and
column structuring. (See Figure 39 on page 149.) Though there was some awareness of
alignment of squares, the awareness was local rather than global and squares appeared in
different sizes. The other five students used the row and column structure to deduce that
there were 35 squares in the original rectangle. All but Kobe redrew the squares, the
students from the French school using a ruler. Half the students used multiplication to
find the final answer and half counted the squares.
For the remaining finite tiling exercises, students were shown a shape to be tiled
and a prototile and then asked if the tiling would work and if so, how many tiles it would
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take. There was an oral phase, a drawing phase, and for the first few problems a
construction phase to check their answer.
Many students had a tendency to underestimate the size of the prototiles, usually
resulting in an overestimate of the number of tiles that would fit, both in the oral phase
and in the drawing phase. Their judgment often got better in the later problems. Marie,
for example, drew half-size tiles for problems B1, B2, and B3, and full-size tiles for the
remaining problems.
When students had pattern blocks (problems B1 – B4), they were always
successful at creating the standard physical tiling during the construction phase. No tiles
were available for the last two problems.
Problem B1
Students were asked about tiling a 2×3 rectangle (without any interior markings to
guide them) with a tilted square the size of a pattern block.
Most students saw that the square could be rotated and used to make the standard
square array tiling of the rectangle. Initial estimates for the number of tiles required
ranged from 4 to 12. After drawing, two students correctly saw there would be 6 tiles;
other answers after drawing included 9, 18 and 19. Kelsey revised her answer from 19 to
5 when she saw the actual pattern block. Apparently seeing the physical block enabled
her to do a more accurate estimate than seeing only a drawing of the prototile.
Michelle eventually described an unusual mental model where the tiles were left
in their original tilted orientation for the left and middle columns and turned to the
standard orientation for the right column, resulting in a failure to sketch a tiling. (See
Figure 87 below.) She was successful with the physical pattern blocks, after which she
explained what her initial mental model had been.
Figure 87: Michelle‘s attempt to solve problem B1
Kelsey drew different-sized squares and did not impose a global row and column
structure on her drawing. (See Figure 39 on page 149.)
Problem B2
Students were asked about tiling a 2×2 rhombus with a unit triangle. The rhombus
had markings on the side to suggest the standard tiling.
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All students thought the large rhombus could be tiled with triangles. Initial
estimates for the number of tiles ranged from 6 to 12. Four students were successful in
using the guide marks to sketch the standard tiling of eight tiles.
Marie drew small triangles with various angles; she abandoned her attempt to
draw a tiling about halfway through. She also struggled a bit with the physical tiling. In
the end, she used symmetry to find the positions of the last few tiles. She then analyzed
the tiling as composed of diamonds (small rhombi).
Kelsey left gaps between the triangles in the center of her drawing, but explained
she had just messed up a little. She explained that the gaps would not appear in the
physical tiling.
Problem B3
Students were asked about tiling an isosceles trapezoid with a unit triangle. The
triangle was turned to an orientation that did not match the trapezoid.
All students thought the trapezoid could be tiled with triangles. Five students were
certain the tiling could be made with three triangles and were successful at sketching the
standard tiling. Kelsey made two attempts because she drew the triangles too small. Her
final drawing did not fill the trapezoid, but she still felt sure that three tiles would work.
Michelle initially doubted that a tiling was possible, but after measuring the prototiles
with her fingers, she decided it could be done with three triangles. Marie thought perhaps
7 tiles were needed and drew 8 slightly overlapping triangles of different sizes.
All students were successful with the physical tiling, at which time Marie realized
her triangles had been drawn too small.
Problem B4
Students were asked about tiling a large equilateral triangle (side of length 2) with
a unit triangle turned to an orientation that did not match the large triangle.
Initial estimates for the number of tiles ranged from 3 to 4. Most were successful
at sketching the standard four-triangle tiling, though some struggled a bit.
Marie believed the tiling could be made with four tiles, but when she attempted to
draw it, she could not find the correct positions for the lines and claimed to no longer see
her initial idea.
Kelsey made two attempts, again drawing the triangles too small. Her final
drawing did not fill the large triangle, which left her uncertain if the tiling could work.
All were successful at finding the standard tiling with the physical pattern blocks.
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Problem B5
Students were asked about tiling an L-tromino with a domino. No tiles were
available to check their answers.
Four students saw immediately that the tiling was not possible, because a second
domino could not fit in the square that would be left after placing the first domino, or
because two tiles would necessarily overlap.
Michelle initially said the tiling was possible, apparently believing it would be
acceptable to complete the tiling with a square. She then showed the tiling was
impossible by showing how the dimensions of the domino matched the dimensions of the
L-tromino and that only room for a half-tile would be left.
Moses was also certain at first that the tiling was possible, but after a few seconds
of thinking and using his finger to imagine rotating and moving the prototile, he changed
his mind. The tiling would only work with squares.
Problem B6
Students were asked about tiling a right-angled trapezoid (a triabolo) with a unit
right isosceles triangle. The trapezoid can be tiled with three triangles in two different
ways. The triangle was in the correct orientation for one of the tiles of one method, but
not in the orientation that would fit the acute angle of the trapezoid. Students found this
task to be the most difficult and gave a variety of answers.
Moses and Kobe saw and sketched two different valid tilings. Marie thought it
could be done with three tiles, but was not sure. Her sketch seems to show the middle tile
turned with the hypotenuse at the bottom:
Figure 88: Marie‘s attempt to solve Problem B6
Michelle was sure the tiling could be done with two tiles:
Figure 89: Michelle‘s attempt to solve Problem B6
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Rachel thought a tiling would be impossible. She tried to insert the triangle in the
left part at the wrong orientation, making the rest of the tiling impossible. (See Figure 90
below.) Kelsey gave a similar response.
Figure 90: Rachel‘s attempt to solve Problem B6
SET E: TRIANGLES
Students were asked about tiling three triangles shown on the computer. There
was an oral phase, a drawing phase, and a computer construction phase for each triangle.
Acute Isosceles Triangle
The acute isosceles triangle had angles of 40°, 70°, and 70°. All students believed
it was possible to tile with this triangle.
Four students envisioned, sketched, and constructed the standard tiling by using
rows of triangles. Michelle thought two rows looked like teeth because of the opposing
orientations. Rachel organized her rows as trapezoids (sets of three triangles at a time),
two of which could combine to make a hexagon. In addition to a row pattern, Marie also
attempted a random pattern, which she abandoned as impossible.
Kobe saw that the triangles could be put together at their apex to form a nonagon,
which he believed could then tile a floor. After sketching, Kobe began to have some
doubts. After an unsuccessful attempt to create the tiling on the computer, Kobe switched
to a row tiling. Michelle thought the triangles would make an octagon and sketched a
radial pattern (with inconsistent numbers of triangles in the second layer) based on this
idea. On the computer, Michelle constructed a successful radial pattern starting with a
nonagon.
Right Scalene Triangle
Students were asked about tiling a right triangle with legs of lengths 2 units and 3
units. The orientation of the triangle was such that no side was vertical or horizontal.
Because of the unusual orientation, students did not easily see that two triangles
could be combined to create a rectangle, but they still believed a tiling was possible. Only
Kobe realized he could make rectangles during the initial oral and drawing phases. Other
students tried a variety of strategies, including fan patterns and radial patterns. Most
students eventually discovered rectangle units as they explored tiling possibilities.
Michelle struggled with the task. She eventually discovered rectangle units, but
was not successful in arranging them in a clear pattern. (See Figure 40 on page 150.)
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Moses envisioned and sketched a fan pattern. He made a complex version of this
pattern on the computer with a central rectangle (―mail with wings‖) and believed this
unit could be used to tile a floor. (See Figure 91 below.) Kelsey also sketched a fan
pattern, but abandoned it and eventually tiled with rectangle units.
Figure 91: Moses‘ ―mail with wings‖
After discovering the rectangle unit tiling, Kobe thought he saw a second way to
tile and created a parallelogram unit, which he quickly rejected as the basis for a new
tiling.
Obtuse Scalene Triangle
Students were asked about tiling an obtuse triangle. Responses were often similar
to responses for the right triangle. All students initially thought a tiling was possible.
Kelsey and Kobe struggled with sketches, deciding that a tiling was not possible
after all. (See Kelsey‘s drawing in Figure 67 on page 207.) On the computer they both
discovered the tiling with parallelogram units.
Time ran out before Marie and Michelle could sketch or construct a tiling, but
they both seemed to describe a parallelogram unit tiling.
Moses tried to make an ―arrow‖ unit (non-convex kite), but was not successful in
expanding this unit to a tiling. He decided the obtuse triangle could not tile the floor.
Rachel sketched a tiling with kites. Under prompting she showed how each kite
could be divided into two obtuse scalene triangles, but it is not clear if she initially
thought her kites were units or tiles. On the computer she began a complex tiling with
both parallelogram and kite units. (See Figure 58 on page 191.) She placed one unit in a
non-edge-to-edge position at the end, which created problems, but she still thought her
tiling could eventually work.
SET F: POLYOMINOES
Trominoes
Students were shown a single plastic tromino (first an I-tromino and later an L-
tromino) and asked if it could be used to tile a floor. There was an initial oral phase and
then a drawing phase on dot paper.
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Kelsey, Michelle, and Rachel drew their trominoes subdivided into squares,
which meant that it was impossible to see where the outlines of the trominoes were in
their tiling drawings. (See, for example, Figure 41 on page 151.) I quietly attempted to
suggest the shape be drawn without squares simply by drawing the first shape for them
without squares, but they still subdivided my drawing. (I did not otherwise prompt them
not to use squares.) I used the video recordings to reconstruct their tilings. The plastic tile
was itself divided into squares, which appears to be the source of the idea to subdivide
the tiles. (In the pilot study no plastic tile was shown and no student thought to subdivide
the shapes.) Wheatley and Reynolds (1996) did not report this phenomenon in their
study.
I-tromino
Students had no difficulty tiling with the 3×1 rectangle, most making several
possible tilings. Most students made standard tilings. Three students drew at least one
complex tiling with units and one student made a ―brick wall‖ tiling. (See Figure 64 on
page 202 for Kobe‘s complex unit tilings.)
At the end students were shown a brick wall tiling with the I-tromino and asked if
such a non-edge-to-edge tiling was possible. All thought it was a valid tiling.
Other than changing the orientation of the standard tiling, three students believed
that no other ways of tiling with the I-tromino were possible. After seeing the brick wall
tiling, they thought the brick wall and standard tilings were the only two possible tilings.
L-tromino
All students believed a tiling was possible with the L-tromino.
Four students drew random patterns but their initial explanations were different.
Kelsey envisioned stacking the tiles like arrows, but sketched a random tiling. Kobe
made rectangle units at first, but then his tiling became random. Marie struggled to draw
the L-tromino at first. She succeeded by drawing a 2×2 square and erasing one corner.
She explained her tiling with reference to walls (which she knew were not really there),
showing how the tiles could line up against two walls, but then she drew a random
pattern. Michelle described a tiling with rectangle units, but did not use them in making
her random pattern. Students who made random patterns were always certain they would
work, sometimes expressing empirical reasoning.
Only Moses and Rachel envisioned and drew the standard tiling using rectangle
units. Moses explained how the units would line up in rows and columns. Rachel
believed this was the only possible way to tile with the L-tromino.
Michelle had previously tried unsuccessfully to make a tiling with I-trominoes at
a slant across the dot paper. When asked if her random L-tromino tiling was the only way
to tile, Michelle made a sketch on blank paper showing how she believed L-trominoes
could be arranged in a radial pattern if she did not have to respect the dot grid. (See
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Figure 42 on page 152.) She had some doubts because her drawing was not accurate.
When asked if she was sure it would work, she replied, ―From the drawing, no, but in my
head, yes.‖ She later expressed further doubt, saying, ―‗cause in my head most of the time
I can make anything work.‖
Tetrominoes
For this more open-ended task, students were shown plastic tiles for each of the
five tetrominoes and asked which ones could tile. They then explored one or two tilings
of their choosing on dot paper, and further tilings on the computer.
Students generally believed that most or all of the tetrominoes could be used to
tile a floor. Rachel and Marie were not sure concerning the T-tetromino, but Marie
eventually decided the T could tile. Michelle was not sure about the L-tetromino. Moses
did not believe either the L or the T could be used to tile; he drew small random tilings to
demonstrate why they would not tile. Rachel thought the S-tetromino would not tile.
Whenever a student attempted to make a tiling, they almost always eventually
concluded it could tile, even if they initially believed otherwise, expressed doubts during
the tiling, or failed to make a valid tiling. This was true even if their tiling was random.
Few students had trouble drawing the shapes. Moses stretched all of his S-
tetrominoes into hexominoes, or even longer. Some of his L-tetrominoes were
pentominoes. At one point, Rachel began drawing S-tetrominoes as two dominoes
touching only at a vertex. Students who subdivided their polyominoes, making it hard to
see the boundaries between the tiles, occasionally slipped an extra square into their tiling.
There are two symmetric ways to tile using the S-tetromino if all tiles keep the
same orientation. Four students constructed the more common row pattern which has
straight lines separating the rows. Kelsey began making the less common tiling, which
has no such lines, though she only placed five tiles and did not describe the pattern, so
she may have had in mind a random tiling. Kobe made an interesting pattern with a single
glide reflection axis. (See Figure 45 on page 156.) Two students created random patterns
with the S. Most students expressed a preference for the S-tetromino, believing its shape
to be more ―cool‖ than the others.
Three students made random patterns with the L-tetromino. Kobe created a tiling
of diagonal rows. Rachel was the only one to create rectangle units, which she used to
create the standard rectangle unit tiling.
Students found the O- and I-tetrominoes very easy to tile with. Rachel created
both the standard I tiling and a complex tiling with a mix of both orientations. Rachel
created the standard tiling with the O by making a large rectangle. Kelsey created a brick
wall tiling with the O, saying this was more ―cool‖ than the standard square array.
Rachel created the unique tiling with the T-tetromino that keeps all tiles in the
same orientation. She believed this was the only way to tile, ―unless you did it upside
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down of course.‖ Three students tried to make random tilings with the T-tetromino, but
only Marie believed she was successful.
SET G: REGULAR POLYGONS
This final open-ended task presented students with five regular polygons (triangle,
square, pentagon, hexagon, and octagon) on the computer and asked them to make tilings
any way they wanted.
The presence of the pentagon proved an irresistible temptation to all students. No
student remembered the second interview where they learned the pentagon cannot tile by
itself. Indeed, it cannot tile with other regular polygons either, but some students focused
the majority of their attention on unsuccessfully trying to create a tiling with pentagons.
Of the 28 attempted tilings, 18 included pentagons and failed. Another 3 attempts did not
use pentagons and also failed. The other 7 tilings were successful.
Kelsey and Rachel created tilings using hexagons and triangles. Kelsey‘s pattern
was random. (See Figure 43 on page 154.) Rachel created the 2-isogonal 32.6
2/3.6.3.6
tiling. (See Figure 69 on page 208.) Kobe made the regular 63 hexagon tiling.
Three students found the semi-regular 4.82 tiling using octagons and squares.
Michelle created a radial pattern using triangles, squares, and hexagons. At one
point she realized it was identical to her last tiling using triangles and squares (see
above), except groups of six triangles had been replaced with hexagons. (See Figure 52
on page 178.)
Marie was the only student not to find a valid tiling, though she did create a large
radial pattern centered on an octagon using successive layers of triangles, pentagons, and
then squares. The tiles did not quite fit exactly and she could not find a way to add a fifth
layer. She made many other tiling attempts and remained optimistic that there was a
solution somewhere.
Sets H and I were optional and were never used.
SIXTH INTERVIEW (SET J)
The sixth interview consisted of six aesthetic evaluation tasks. These tasks were
done in two groups of three students, one group from each school. See Appendix B for
illustrations of the tilings used.
Student Tilings
Students had to choose four tilings for four rooms of a house. The charter school
chose a pattern with slightly overlapping tiles for the bathroom because it looked fun and
resembled a sun. The private school students almost picked this pattern as well, but
rejected it at Kobe‘s insistence because of the overlapping. It is also flawed because it is
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not perfectly symmetric and cannot be continued beyond the finite pattern shown, but this
generally went unnoticed by the children.
Both schools chose the 4.82 tiling for the kitchen. The charter school believed this
was typical of kitchens and that kitchens should be basic. Both schools chose one random
pattern and one rich six-fold pattern.
Other Tilings
For the next three tasks, I showed students 18 tilings ―done by adults.‖
Student Sorting
First, students had to sort the tilings, and then choose their favorite group and
tiling. The preferred groups all had to do with colors, except for Kobe‘s ―Mazy‖ group
using the Interconnectedness aesthetic theme. There was no coordination of criteria in
making the sorted groups. The groups students formed, roughly in order of preference,
along with descriptions and favorite tiling in the group, are summarized in Table 10
below.
Table 10: Groups formed by student sorting
School Group Description Favorite in group
Private H J O P Blues H
Private E N Mazy N
Private K M R Crazy K
Private I L Q They go together I
Private B G Windows G
Private A C D F Gray and stony D
Charter K M R Colorful R
Charter J P Same colors J
Charter E N Maze N
Charter B Q They go together Q
Charter L O Basic shapes and matching colors O
Charter H I Basic shapes H
Charter A C D F G Outdoors A
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Comparisons in Pairs
Next, students had to express preferences in pairs (or larger groups) of tilings I
showed them. Both schools chose R (the random pattern) over M, explicitly for the
colors, though M was also colorful. M was said by the private school to have bad
patterns. (It had only translation symmetry). See Table 7 on page 196 for other
unanimous results from this task.
Professional Preferences
Lastly, students were asked which tilings a mathematician or an artist would
prefer. Both schools thought the artist would like R (the random pattern). They also chose
J, K, and D. Both schools thought the mathematician would pick K (because of its
illusion of 3D cubes). The private school also suggested N and Q for the mathematician.
Professional Tilings: Alhambra
Students were shown seven postcards of Alhambra tilings. Both schools preferred
tiling #7. The private school also chose tiling #5. Tilings #5 and #7 are radial patterns
which they compared to stars and many other things. They liked the colors of tiling #7.
Their least favorite tilings were #2 and #6. The private school also included #4 among
least favorite tilings. These were all considered boring and too simple.
Professional Tilings: Escher
Students were shown a variety of tilings by Escher and asked to award prizes to
the best three. From Escher‘s first set of basic tilings, both groups liked #3, though for
different reasons. For the second set of basic tilings, both groups picked the rhombi
because they looked three-dimensional, though a majority at the private school actually
preferred the parallelograms because they looked like home.
Escher‘s ―Regular Division of the Plane #66‖ was chosen by the private school
for first place and by the charter school for second place. (See Figure 63 on page 195.)
They cited the colors and fantasy or mythology element as their reasons. Both groups
were impressed with the way the different colored lions went in opposite directions and
fit together perfectly.
The charter school chose Escher‘s ―Regular Division of the Plane #12,‖ a tiling of
blue and white butterflies, for first prize, citing themes of Color, Energy, and Real World
Connection. The private school chose Escher‘s ―Regular Division of the Plane #58‖ for
second prize claiming it had an Asian style. (See Figure 57 on page 185.) The charter
school chose Escher‘s ―Regular Division of the Plane #132‖ (blue and red flowers in a
complex pentagonal pattern) for the third prize. The private school chose Escher‘s
―Regular Division of the Plane #97‖ (black and white dogs) for third prize.
303
Appendix F: Analysis of Change
All of the most important cognitive codes were analyzed for change across the
interviews. Change in aesthetic codes was analyzed by comparing the first five interviews
with the sixth interview. (See page 219 for the analysis of aesthetic change. There were
insufficient aesthetic codes to analyze aesthetic change across the first five interviews.)
This appendix contains the details of the analysis of the change in cognitive codes. All
important non-aesthetic codes were totaled for each of the five interviews. Codes that
were task related, such as Possibility, Polygon, and Tool, were not under the control of
the students and were not analyzed. Certain codes, such as Wording and Attribute Focus,
were only coded for certain tasks; these were not analyzed either. Table 11 below shows
the code totals for each interview.
Table 11: Analysis of code changes across first five interviews
Code Interview #
1 2 3 4 5
Uncategorized codes Abstractness of drawing 7 0 2 0 0 Change of center 0 1 0 0 0 Concern for accuracy 6 0 5 0 0 Reference to symmetry 0 1 0 0 0 Traces physical tile 15 0 0 0 0
Understanding Fit
Accurate fit 41 4 29 10 0 Inaccurate fit 15 11 17 1 1
Tiling Accurate tiling 44 1 27 23 0 Apparently accurate tiling 1 0 7 1 0 Embodied tiling 4 0 0 0 0 Inaccurate tiling 12 13 20 12 0 Incomplete tiling 3 0 2 4 0 Multiple tiling 0 0 0 0 0 Partially accurate tiling 3 1 1 1 0
Confidence No confidence 7 4 5 2 0 Strong confidence 54 11 18 27 2 Weak confidence 23 9 9 5 1 Confidence at symmetry 0 0 0 0 0 Doubt tiling will work (at symmetry) 0 0 2 0 2 No confidence (at symmetry) 0 1 1 0 0 Strong confidence (at symmetry) 37 17 19 21 13
304
Table 11. cont.
Weak confidence (at symmetry) 2 2 1 0 0 Symmetry
2-fold symmetry 7 5 1 3 2 3-fold symmetry 3 3 0 0 0 4-fold symmetry 1 1 0 0 3 6-fold symmetry 5 3 0 0 5 8-fold symmetry 0 0 1 0 3 9-fold symmetry 0 0 1 0 0 No symmetry 10 2 2 10 2 No symmetry at end 2 3 0 0 1 Order-2 rotation symmetry 10 1 8 4 0 Reflection symmetry 1 0 0 0 0 Horizontal reflection symmetry 3 0 2 0 0 Oblique reflection symmetry 0 0 1 0 0 Vertical reflection symmetry 6 6 8 4 2
Reasoning Infinity 0 0 0 0 0
Functional infinity 3 3 2 2 2 Limited infinity 8 0 0 0 0 Mathematical infinity 2 0 1 0 0
Tiling justification Empirical justification 2 1 0 6 1 Experience justification 9 0 0 3 1 Gapless justification 0 0 0 0 0 Known unit justification 3 0 5 1 0 Similar to known tiling justification 3 0 0 0 0 Subunit justification 1 0 0 0 1 Symmetric justification 0 0 0 0 0 Transformational justification 0 1 0 0 1
Strategy Complex rows strategy 0 0 0 1 1 Alternating rows strategy 9 0 5 1 0 Brick wall strategy 2 0 0 1 0 Units in rows strategy 0 0 1 1 0 Complex strategy 1 1 3 3 1 Fit one at a time strategy 21 1 0 3 4 Grid strategy 1 0 0 0 0 Growing strategy 8 6 2 3 5 Random strategy 1 2 1 12 0 Semi-random strategy 10 6 0 2 1 Random units strategy 1 0 1 0 0 Simple rows strategy 9 9 4 8 3
305
Table 11. cont.
Spiral strategy 1 0 2 0 0 Units strategy 5 3 14 5 0
Uncategorized codes did show some change. The two students who traced tiles
both did so only in the first interview. There are more instances of ―Abstractness of
drawing‖ and ―Concern for accuracy‖ in Interviews #1 and #3. These would indicate
more awareness of the limitations of hand drawings in those interviews. This is likely
task related because Interviews #2, #4, and #5 used less freehand drawing.
Understanding (accuracy of fit and tiling) showed a small decrease in accuracy
in the later tasks, probably due to the more complex nature of the those tasks. There is no
clear trend towards greater inaccuracy as the interviews progress. There are more codes
of all kinds for the first interview, which used four different basic shapes. Most codes
were done during the drawing phases, so there are few codes for the tasks that did not use
paper and pencil, such as in the fifth interview.
Confidence was more heavily coded in the first interview. There were few
expressions of confidence (strong or weak) during the fifth interview. There is no clear
tendency towards greater or weaker confidence across the interviews, only fewer
expressions of either kind.
Symmetry is closely related to the prototile and therefore depends on the task.
(See page 213.) For example, the parallelogram in Interview #1 naturally created rotation
symmetry of order 2. The fifth interview was more complex and resulted in some higher
order symmetry. Codes for ―No symmetry‖ tended to occur during the first interview
(Basic Tilings) and the fourth interview (Polyominoes). There was no tendency to use
more or less symmetry as the interviews progress.
Reasoning (infinity and tiling justifications) did not change across the
interviews, with the possible exception of ―Limited infinity,‖ which was only evident
during discussion of the basic tilings in the first interview. This may indicate that students
began to accept the idea of an infinite tiling after the first interview, or it may indicate
that limited conceptions of infinity were only clearly revealed in simple tilings. In fact, all
of the data concerning students‘ limited ideas of infinity came from only two tasks: the
equilateral triangle task and the rhombus task, both in the first interview. In other tasks,
students‘ actions were sometimes coded as ―functional infinity,‖ which meant that they
recognized the infinite nature of the tiling, but still struggled with it in some way, such as
Marie‘s creation of temporary walls to scaffold her tiling. Other high frequency codes
seem related to the task. For example, many students were familiar with the basic tilings
in Interview #1, but not the tiling tasks in the other interviews. The codes for ―Experience
justification‖ are therefore all in the first interview.
306
Strategy codes showed almost no change across the interviews. Codes that seem
to show change are probably all task related. For example, the ―Fit one at a time‖ strategy
is closely related to the ―Random‖ strategy; it is applied only when students begin placing
tiles like a puzzle in hopes that a non-random pattern will quickly emerge. This mostly
occurred for the hexagon task in the first interview and the regular polygon task in the
fifth interview. Alternating rows was a common strategy for triangles, and was therefore
used mainly in the first and third interviews.
In conclusion, with very few exceptions, I can find no evidence for significant
change in children‘s thinking across the first five interviews.
307
Appendix G: Analyzing Greater Symmetry
In order to analyze whether children preferred tessellations with greater
symmetry, I analyzed the symmetry of the tessellations in the Other Tilings tasks of the
sixth interview and then studied children‘s preferences between pairs of tilings. There is
not a single clear way to analyze the symmetry of these tilings. The most rigorously
mathematical method would be to analyze their symmetry groups.
The set of symmetries in a pattern, together with the operation of composition,
creates a symmetry group for any pattern, finite or infinite. We can therefore analyze the
symmetry of any tessellation by describing its symmetry group. There are currently
several different notations for describing the various symmetry groups of tessellations,
the most useful of which is orbifold notation. Very briefly and at the risk of
oversimplifying slightly, orbifold notation lists all orders of rotation symmetry about
distinct points (without reflection symmetry), followed by a ―star‖ (*) if there is
reflection symmetry, followed by orders of symmetry that have reflection and rotation
combined at a distinct point. By ―distinct‖ point I mean that we do not count points
equivalent by a translation symmetry more than once. Glide reflection symmetry is
indicated by a ―cross‖ (×). If the pattern has no translation symmetry, so that all
symmetries fix a point, the point symbol (•) is added. For frieze groups with translation
symmetry in one direction only, the symmetry order is considered infinite and ∞ is used.
For periodic patterns, no further symbols are needed, unless the tessellation has
translation symmetry only, in which case a ―wonder-ring‖ (o) is used to describe the
symmetries.
For example, the infinite tessellation in Figure 92 below has the orbifold notation
*632 because it has reflection and rotation symmetries of orders 6, 3, and 2. The notation
is such that it describes the symmetries of any pattern completely. I will not go into
further detail because orbifold notation did not provide a conclusive analysis for this
study. For more details, and much greater mathematical depth behind the notation, see
(Conway, Burgiel, & Goodman-Strauss, 2008).
Figure 92: Tessellation with orbifold notation *632
308
Did students prefer tilings with greater symmetry? In order to carry out such an
analysis, it was necessary to analyze each tiling for symmetry so that they could be
compared. There is not one clear way to do this. Three methods were tried:
1. Tilings were analyzed according to orbifold notation. Numbers were paired
between tilings to be compared as far as possible. Larger numbers represented
richer symmetry; numbers after a star represented more symmetry than numbers
before a star. If one tiling had more symmetry for each symbol than another tiling,
then it was considered to have richer symmetry. For example, *442 is richer in
symmetry than 442; 2*22 is richer than 22*; and so on. *632 and *442 are not
comparable because 6 > 4, but 3 < 4. Dilation symmetry could not be compared
with other symmetries using this method.
2. Tilings were analyzed by simply listing all types of symmetry: reflection, rotation,
dilation, glide reflection, etc. A tiling with more types of symmetry, or higher
order symmetry, than another tiling was considered to have richer symmetry.
Only the highest order rotation or reflection symmetry was considered.
3. Tilings were analyzed according to simple reflection and rotation symmetry only,
under the idea that these are the most important symmetries to children and
perhaps the only ones they noticed. A tiling with n-fold symmetry was considered
to have richer symmetry than a tiling with order-n rotation symmetry because n-
fold symmetry contained both reflection and rotation symmetry. Only the highest
order symmetry was considered.
In any method that was tried, there was the problem that many tilings were not
clearly comparable. Dilation symmetry is neither richer nor poorer in symmetry than
other types of symmetry. Simple reflection symmetry cannot be directly compared to
simple rotation symmetry. And so on. A further problem is that these methods can only
be carried out for the Other Tilings task, in which data were probably insufficient for
drawing any conclusions.
Each of the three methods revealed a slight or moderate (but not overwhelming)
preference for more symmetric tilings. Other aesthetic criteria frequently intervened to
lead children to choose the less symmetric tiling. Children also frequently failed to agree
among themselves concerning which was the favorite tiling as each child applied
different aesthetic criteria to the evaluation. It would appear that children sometimes
preferred greater symmetry and sometimes turned to other aesthetic criteria, but no firm
conclusions can be reached from these data.
309
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Vita
In 1981 Robert Scott Eberle received the degree of Bachelor of Arts in French and
mathematics from Rice University in Houston, Texas. He received the degree of Master
of Arts in mathematics from Columbia University in New York in 1987 and the degree of
Master of Education in mathematics education from the University of Texas at Austin in
1989. He has worked as a teacher at the secondary school or college level in Texas, South
Carolina, Zaïre, and Niger. He has served as a Christian missionary with SIM in Niger
since 1992. In August, 2006, he entered the Graduate School at the University of Texas at
Austin.
Permanent email address: [email protected]
This dissertation was typed by the author.