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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

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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

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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

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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,...

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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

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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

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Analogy geometry - algebra 1. 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

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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.

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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. 2D3D Oldknow, A.

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Dimensions - video 2D -> 3D 3D -> 4D Abbott, E. A.: Flatland teachers: very nice, but not interactive Leys, J., Ghys, É., Alvarez, A.

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Dimensions – models in Cabri 3D example 1 example 2

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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

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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

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Thank you References ABBOTT, E. A. Flatland, a romance of many dimensions. [online]. Sealy a Co., Retrieved from: BOURKE, P. Regular Polytopes (Platonic solids) in 4D. [online] University of Western Australia, Retrieved from: DELL, D. Analogies Lesson by Diana Dell, Ed.S. [online] In: Dell, D. (ed.): Teaching and Learning with Technology. Retrieved from : 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, ISBN LEYS, J., GHYS, É., ALVAREZ, A. Dimensions. Educational video [online] Retrieved from: OLDKNOW, A. Solid geometry and Cabri 3D [online]. In: CY maths ICT. Chartwell-Yorke Ltd., Retrieved from : POLYA, G. Induction and Analogy in Mathematics (Mathematics and Plausible Reasoning, Vol I). Princeton (NJ): Princeton University Press, RICHLAND, L. E., HOLYOAK, K. J., & STIGLER, J. W. The role of analogy in teaching middleschool mathematics. Cognition and Instruction, 22, 37-60, SAYER, A. Method in social science: A realist approach. London and New York: Routledge, 1992.

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