ryan grover realistic mathematics education · hypothetical learning trajectory begins with...

An Orientation toRealistic Mathematics

Education

Raymond JohnsonFrederick Peck

William CampbellRyan GroverSusan Miller

Ashley ScrogginsDavid C. Webb

What is Realistic Mathematics Education?

Plenary Preview

History and Origins of RME

Principles and Design Heuristics of RME

RME Examples and Exploration

AN ORIENTATION TO RME #RME5 18 SEPTEMBER 2015

History and Origins

Nicolaus Copernicus

1473-1543


Hans Freudenthal (1905-1990)


(Model of Hans Freudenthal as a baby)

● Mathematical achievements

● Early math education influences

Sputnik and Math Education Reform


● Direct impact was limited, but the New Math influenced a number of other math reform movements.

Utrecht, Wiskobas, and Origins of RME

● 1968: Wiskobas (“mathematics in primary school”)○ Edu Wijdeveld & Fred Goffree; later Adrian Treffers

● 1968: Educational Studies in Mathematics● 1971: IOWO (now FI) founded

○ Freudenthal first director, Wijdeveld general manager○ Purpose: to formally develop math curriculum

● 1973: Wiskivon (“mathematics in secondary school”)


Mathematics as an Educational Task (1973)

● 680 page compilation● Chapters on:

○ Re-invention

○ Organization of a Field by

Mathematizing


Organization of a Field by Mathematizing

“It can hardly be understood why mathematicians would prescribe the exclusion of mathematizing from instruction, because it is considered a scribbler’s activity. Mathematizing should be the business of the adult mathematician, they claim, not of the learner. Contrary to this tendency there is no doubt that pupils should learn mathematizing, too, and certainly on the lowest level where it applies to unmathematical matter, to guarantee the applicability of mathematics, but not much less on the next level where mathematical matter is organized at least locally.” (Freudenthal, 1973, p. 134)


Wiskobas and Treffers’s Three Dimensions

● 1978 (Dutch)● 1987 (English)● Describes the work of

the Wiskobas project○ Published program in 1976

○ Gave complete materials to

commercial publishers to

use and edit freely


HEWET and the OW&OC

● 1978-1980: HEWET Commission (Ministry of Education)

● 1981-1986: Hewet Project (OW&OC)○ Upper secondary

curriculum and PD

● 1981: IOWO becomes OW&OC


1980s: Math Education in the U.S.

● 1986-1995: NCTM Standards Project

● Tom Romberg (UW-Madison)

● 1988: Pilot study with FI


Mathematics in Context

● 1991-1996: NSF funding for initial MiC development

● Combines:○ NCTM Curriculum and Evaluation

Standards

○ Research on problem-based

approaches○ Realistic Mathematics Education

● 2003-2005: NSF-funded revision


RME Comes to the U.S., Part 2

● 1988: Cobb read Three Dimensions ○ Later visited OW&OC

● 1990: Cobb invited Gravemeijer to Purdue


Paul Cobb Koeno Gravemeijer

Cobb & Gravemeijer, et al.

● A 10-year collaboration● Significant contributions:

○ Design research○ Learning sciences


Fosnot and FI

● 1995-2002: Summermath in the City

● 2000-2006: Mathematics in the City

● Contexts for Learning Mathematics


Freudenthal Institute US

● FIUS Founded in 2003


What We’ve Missed

A lot.


RME Timeline and Q&A


Principles and Design Heuristics of RME

Six Instructional Principles

● Activity principle● Reality principle● Guided reinvention principle● Level principle● Intertwinement principle● Interaction principle


Treffers, 1987; van den Heuvel-Panhuizen & Wijers, 2005

Activity Principle

“What humans have to learn is not mathematics as a closed system, but rather as an activity, the process of mathematizing reality and if possible even that of mathematizing mathematics” (Freudenthal, 1968, p. 7)

● Historic and evolving human problems and explorations = historic and evolving human mathematical activities

● Mathematical organizing of lived in world & mathematical matter (horizontal & vertical components; Treffers, 1987)

● Students participate in process that yields product (Gravemeijer & Terwel,

2000)

● Activity follows Hypothetical Learning Trajectories (Gravemeijer, 1998; Simon,

1995)


Activity Principle: Example

Making sense of percentage (Mathematics in Context)

Contexts: Flowers, parking lots, game attendance, marathon completion rates, sale prices, compounding interest

● Hypothetical learning trajectory begins with exploratory activities and a qualitative way of working (percentages used as descriptors of ‘so many out of so many’ situations)

● Activities emphasizing a quantitative way of working (percentages used as operators) are introduced later (Streefland & van den Heuvel-Panhuizen, 1992)


Reality Principle

“I prefer to apply the term reality to what common sense experiences as real at a certain stage. Reality is understood as a mixture of interpretation and sensual experience, which implies that mathematics, too, can become part of a person’s reality. Reality and what a person counts as common sense are not static but grow, and are affected by the individual’s learning process.” (Freudenthal, 1991, p. 17)

● Common-sense and reality construed from the viewpoint of the actor (Gravemeijer & Terwel, 2000, p. 783)

● Realistic = imaginable, realizable, not just “real-world” van den

(Freudenthal, 1987; Heuvel-Panhuizen & Wijers, 2005)


Reality Principle: Example

Mathematics in Context 7th grade algebra unit

Context: an urban planning debate between parties advocating different density levels for future housing

Activities: develop, graph, and compare feasible plans, make sense of data, develop a logical argument for your party’s point of view


Guided Reinvention Principle

“Guiding reinvention means striking a subtle balance between the freedom of inventing and the force of guiding, between allowing the learner to please himself and asking him to please the teacher. Moreover, the learner’s free choice is already restricted by the “re” of “reinvention”. The learner shall invent something that is new to him [sic] but well-known to the guide” (Freudenthal, 1991, p. 48)

● Student’s perspective: Invention comes to the fore

● Teacher’s perspective: “re” and “guided” come to the fore,

while balancing with creativity inherent in invention(Freudenthal, 1991)


Guided Reinvention Principle: Example


Kindt et al, 2006, p. 3

Level Principle

Gradually moving from informal, context-specific, to formal mathematics through lived activity and reflection. (van den Heuvel-Panhuizen & Wijers, 2005)

Emphasis on models (Treffers, 1987)


● Emergent Models (Gravemeijer,

1999)

● Progressive Formalization (The Iceberg Metaphor; Webb et

al., 2008)

Level Principle: Example

Formal

^

|

|

|

v

Informal


Intertwinement Principle

Integration of mathematical concepts

Essential for realism in problems

Enables students to mathematize across strands


Treffers, 1987; van den Heuvel-Panhuizen & Wijers, 2005

Intertwinement Principle: Example

Calculating the size of a tree from its shadow

Measurement

Geometry

Proportions

Ratios


Interaction Principle

Individualized Trajectories

Discussions allow for

● justification● strategy comparison● reflection (Treffers, 1987; van den Heuvel-Panhuizen & Wijers, 2005)

Cobb et al (2008) adaptations

● Classroom as an activity system● From individual trajectories to collective

development


Interaction Principle: Example

● Urban Planning Lesson● Prime Numbers


ADD REFERENCE

Design Heuristics for Problem Contexts

● Contexts must be realizable by students. ● Contexts should present a situation that is

“begging to be organized” (Gravemeijer & Terwel, 2000, p.

787) from the students’ perspective. ● Contexts should be designed such that, by

mathematizing the situation, students will have a high likelihood of reinventing the desired mathematics.


Freudenthal, 1983; Gravemeijer & Terwel, 2000

Design Heuristics: Example

A class traveled on a field trip in four separate cars. The school provided a lunch of submarine sandwiches for each group. When they stopped for lunch, the subs were cut and shared as follows:

● The first group had 4 people and shared 3 subs equally.● The second group had 5 people and shared 4 subs equally.● The third group had 8 people and shared 7 subs equally.● The last group had 5 people and shared 3 subs equally.

When they returned from the field trip, the children began to argue that the distribution of sandwiches had not been equal, that some children got more to eat than the others. Were they right? Or did everyone get the same amount?


● Realizable?● “Begging to be organized”?● High likelihood of reinventing bar model and “partition and

distribute” strategy?

Fosnot, 2008

Examples of RME:Stations and Small Group Discussions

Example Stations


● Primary school● Middle school● Secondary school● Undergraduate mathematics● Comparing RME principles to the

Common Core “Standards for Mathemathematical Practice”

References

Cobb, P., Zhao, Q., & Visnovska, J. (2008). Learning from and adapting the theory of Realistic Mathematics Education. Éducation et Didactique, 2(1). Retrieved from http://educationdidactique.revues.org/276

de Lange Jzn, J. (1987). Mathematics, insight and meaning: Teaching, learning and testing of mathematics for the life and social sciences. University of Utrecht.Fosnot, C. T. (2008). Field trips and fundraising: Introducing fractions. Portsmouth, NH: Heinemann.Fosnot, C. T., & Dolk, M. (2001a). Young mathematicians at work: Constructing early number sense, addition, and subtraction. Portsmouth, NH: Heinemann.Fosnot, C. T., & Dolk, M. (2001b). Young mathematicians at work: Constructing multiplication and division. Portsmouth, NH: Heinemann.Fosnot, C. T., & Dolk, M. (2002). Young mathematicians at work: Constructing fractions, decimals, and percents. Portsmouth, NH: Heinemann.Fosnot, C. T., & Jacob, B. (2010). Young mathematicians at work: Constructing algebra. Portsmouth, NH: Heinemann.Freudenthal, H. (1968). Why to teach mathematics so as to be useful. Educational Studies in Mathematics, 1, 3–8.Freudenthal, H. (1973). Mathematics as an educational task. Dordrecht, The Netherlands: D. Reidel.Freudenthal, H. (1983). Didactical phenomenology of mathematical structures. Dordrecht, The Netherlands: Reidel.Freudenthal, H. (1987). Mathematics starting and staying in reality. In Proceedings of the USCMP Conference on Mathematics Education on Development in School

Mathematics around the World (pp. 279–295). Reston, VA: NCTM.Freudenthal, H. (1991). Revisiting mathematics education: China lectures. Dordrecht, The Netherlands: Kluwer.Gravemeijer, K. (1999). How emergent models may foster the constitution of formal mathematics. Mathematical Thinking and Learning, 1(2), 155–177. Gravemeijer, K., & Terwel, J. (2000). Hans Freudenthal: A mathematician on didactics and curriculum theory. Journal of Curriculum Studies, 32(6), 777–796. http:

//doi.org/10.1080/00220270050167170Kindt, M., Abels, M., Dekker, T., Meyer, M. R., Pligge, M. A., & Burrill, G. (2010). Comparing quantities. In Wisconsin Center for Education Research & Freudenthal

Institute (Eds.), Mathematics in context. Chicago, IL: Encyclopædia Britannica.Simon, M. A. (1995). Reconstructing mathematics pedagogy from a constructivist perspective. Journal for Research in Mathematics Education, 26(2), 114–145. Streefland, L., & van den Heuvel-Panhuizen, M. (1992). Evoking pupils’ informal knowledge on percents. In W. Geeslin & K. Graham (Eds.), Proceedings of the

Sixteenth PME Conference, Volume III (pp. 51–57). Durham, NH: Program Committee of the 16th PME Conference, USA. Retrieved from http://eric.ed.gov/?q=ED383538

Treffers, A. (1987). Three dimensions: A model of goal and theory description in mathematics instruction -- The Wiskobas Project. Dordrecht, The Netherlands: D. Reidel.

Webb, D. C., Boswinkel, N., & Dekker, T. (2008). Beneath the tip of the iceberg: Using representations to support student understanding. Mathematics Teaching in the Middle School, 14(2), 110–113.

van den Heuvel-Panhuizen, M., & Wijers, M. (2005). Mathematics standards and curricula in the Netherlands. Zdm, 37(4), 287–307. Wisconsin Center for Education Research & Freudenthal Institute (2010), Mathematics in context. Chicago, IL: Encyclopædia Britannica.Yackel, E., Gravemeijer, K., & Sfard, A. (Eds.) (2011). A Journey in mathematics education research: Insights from the work of Paul Cobb. Dordrecht, The

Netherlands: Springer.


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