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.