using analogy in generalization and conceptual learning in cal of geometry jiří vaníček...
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Using analogy in generalization
and conceptual learning in CAL of geometry
Jiří Vaníček
University of South Bohemia
CADGME Conference
Hluboká nad Vltavou, June 30-th, 2010
Generalization
Mathematics: a playing field for generalization from numbers to formulas from objects to classes from invariant to concept from comparison of objects to analysis of feature as itself
Sayer: „generalization ... an approximate quantitative measure of the numbers of objects belonging to some class or a statement about certain common properties of objects“
Sayer: importance of question posing
Analogy
analogy - a method of generalization cognitive process
transfer a character from object to the next one verbal description of this process
application of analogy problem solving, decision making, creatibility, memory,
explanation, communication tasks of recognition of places, objects, people, faces analogy – fundamental of knowledge (Hofstadter) learning analogy in mathematics – Polya, Richland,
Holyoak ... analogy in logic, law, legislature, biology, linguistics
Training of analogical thinking
constructivistic and constructionistic statement Papert: learning as building knowledge structures Papert: it happens in a context where learner is engaged in
constructing a public entity relative young children
(but description) problem of correctness of used analogy
Dell: list of most common analogies synonyms, part and whole, measure and unit,
arithmetical relations, ...
Main research objectives
Possibilities of mathematical software to teach analogy, analogical thinking constructive geometry – algebra plane – space geometry
Questions: How we can teach analogy using DGE software Which tasks are appropriate teachers (in-service, pre-service) in this field
Methodology searching for usable tasks
- Geogebra Cabri II Plus, Cabri 3D
using in lessons CAL of math for teachers series of graduated tasks participant observation analysis of students’ work
Analogy geometry - algebra1. Analogy centroid – arithmetic mean
triangle: S = (A+B+C)/3, segment, tetragon
2. How to construct centroid S of tetragon
knowledge of the formula did not help construct prove
Construction in plane and space
Triangle given three sides - SSS
Draw two vertices. Find the other vertex as an intersection point of circles of centre in drawn vertices and radiuses equal given lengths.
Tetrahedron given six edges – EEEEEE
Draw two vertices. Find the other vertices as intersection points of spheres of centres in drawn vertices and radiuses equal given lengths.
Circumscribed circle of a triangle.Solution: Circumcenter is an intersection point of
axes of triangle sides.
Circumscribed sphere of a tetrahedron.Solution: Circumcenter lays on an intersection of
planes of symmetry of tetrahedron edges.
Centroid of a triangle.Solution: Centroid is an intersection of medians.
Median is a line joining a vertex to the midpoint of the opposite side.
Centroid of a tetrahedron.Solution: Centroid is an intersection of „medians
of a solid“. Median is a line joining a vertex to the midpoint (centroid) of the opposite face.
Tangent of a circle passing through a given point.Solution: Draw a segment from centre of a circle
to a given point. Draw Thales’ circle with its centre in centre of the segment and passing through given point. A tangent point lays on an intersection of both circles. Tangent line passes through a tangent point and the given point.
Tangent of a sphere passing through given point.Solution: Draw a segment from centre of a sphere
to a given point. Draw Thales’ sphere with its centre in centre of the segment and passing through given point. A tangent point lays on an intersection of both spheres. Tangent line passes through a tangent point and the given point.
Square with given lenght of a side AB.Solution: Find another vertex as intersection of
suitably chosen perpendicular line and a circle with centre in A and radius AB. Construct remaining vertex as an image of chosen vertex in translation by a vector given by two adjacent vertices.
Cube with given lenght of an edge AB.Solution: Find another vertex on an intersection of
suitably chosen perpendicular plane and a sphere with centre in A and radius AB. Construct remaining vertices as images of chosen vertices in translations by vectors given by two adjacent vertices.
2D 3D
Oldknow, A.
Dimensions - video 2D -> 3D
3D -> 4D
Abbott, E. A.: Flatland
teachers:
very nice, but
not interactive
Leys, J., Ghys, É., Alvarez, A.
Dimensions – models in Cabri 3D example 1 example 2
Towards 4D after brush-up of 4D solids (terminology) students to create analogical settings of
constructive tasks in 4D to discover analogical construction
Circumscribed sphere of a given tetrahedron.Solution: Circumcenter lays on
an intersection of planes of symmetry of tetrahedron edges.
Circumscribed hypersphere of a given simplex (pentatope)Solution: Circumcenter lays on
an intersection of hyperplanes of symmetry of simplex edges.
3D 4D
Results and conclusion step 2D -> 3D
teacher students were able to find analogical settings difficulties: find analogical construction procedure preferred plane construction steps (even more complicated)
step 3D -> 4D did not find appropriate analogical settings create analogical procedure – very hard searching for solution in 4D: only grammatical
exercise, or mathematical imagination? missing modelling environment example 3
Thank you
References ABBOTT, E. A. Flatland, a romance of many dimensions. [online]. Sealy a Co., 1884. Retrieved
from: < http://www.ibiblio.org/eldritch/eaa/FL.HTM > BOURKE, P. Regular Polytopes (Platonic solids) in 4D. [online] University of Western Australia,
2003. Retrieved from: < http://local.wasp.uwa.edu.au/~pbourke/geometry/platonic4d/> DELL, D. Analogies Lesson by Diana Dell, Ed.S. [online] In: Dell, D. (ed.): Teaching and Learning
with Technology. Retrieved from : <http://mrsdell.org/analogies> HOFSTADTER, D. R. Epilogue: Analogy as the Core of Cognition. In: Gentner, D., Holyoak, K.J.,
Kokinov, B. (eds.). The Analogical Mind: Perspectives from Cognitive Science. Cambridge, MA, MIT Press, 2001. ISBN 0-262-57139-0
LEYS, J., GHYS, É., ALVAREZ, A. Dimensions. Educational video [online]. 2008. Retrieved from: <http://www.dimensions-math.org>
OLDKNOW, A. Solid geometry and Cabri 3D [online]. In: CY maths ICT. Chartwell-Yorke Ltd., 2004. Retrieved from : <http://www.chartwellyorke.com/cabri3d/oldknow/solidgeometry.doc>
POLYA, G. Induction and Analogy in Mathematics (Mathematics and Plausible Reasoning, Vol I). Princeton (NJ): Princeton University Press, 1954.
RICHLAND, L. E., HOLYOAK, K. J., & STIGLER, J. W. The role of analogy in teaching middleschool mathematics. Cognition and Instruction, 22, 37-60, 2004.
SAYER, A. Method in social science: A realist approach. London and New York: Routledge, 1992.