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Masataka Kaneko, Satoshi Yamashita (Kisarazu National College of Technology)

The Effective Use of LaTeX Drawing in Linear Algebra -- Utilization of Graphics Drawn with KETpic --. Masataka Kaneko, Satoshi Yamashita (Kisarazu National College of Technology) Hiroaki Koshikawa, Kiyoshi Kitahara (keiai University) (Kogakuin University)

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Masataka Kaneko, Satoshi Yamashita (Kisarazu National College of Technology)

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  1. The Effective Use of LaTeX Drawing in Linear Algebra-- Utilization of Graphics Drawn with KETpic -- Masataka Kaneko, Satoshi Yamashita(Kisarazu National College of Technology) Hiroaki Koshikawa, Kiyoshi Kitahara (keiai University) (Kogakuin University) Setsuo Takato (Toho University)

  2. This work is supported by Grant-in-Aid for Scientific Research (C) 20500818

  3. Typical Issue in LA Necessary condition for teaching and learning mathematical concepts (G. Harel2000) Concreteness(Reality) Necessity(Motivation) Generalizibility(Formalization) Use of Graphics Illuminative Application Axiomatic Approach

  4. Typical Issue in LA Necessary condition for teaching and learning mathematical concepts (G. Harel 2000) Concreteness(Reality) Necessity(Motivation) Generalizibility(Formalization) Use of Graphics Illuminative Application Axiomatic Approach Self evident in Matrix Oriented LA (R^2-R^3)

  5. Typical Issue in LA Necessary condition for teaching and learning mathematical concepts (G. Harel 2000) Concreteness(Reality) Necessity(Motivation) Generalizibility(Formalization) Use of Graphics Illuminative Application Axiomatic Approach High Quality Graphics are needed

  6. Typical Issue in LA Necessary conditions for teaching and learning mathematical concepts (G. Harel 2000) Concreteness(Reality) Necessity(Motivation) Generalizibility(Formalization) Use of Graphics Illuminative Application Axiomatic Approach Abstractness is overemphasized

  7. Our Proposal LaTeX Graphics Drawn with KETpic Preciseness Programmability Rich Perspective MotivatingExamples Generalizible Concepts

  8. Contents • Our Questionnaire Survey • Previous Researches • Use of Graphics in Textbooks • Utilization of Graphics Drawn with KETpic • Students’ Interview • Future Works

  9. 1. Our Questionnaire Survey

  10. Focuses of the Survey (I) The methodshow teachers produce and use graphical class materials (II) The needs of teachers for using graphical class materials • Terms 2008.9.1 – 2008.12.31

  11. Posted to Teachers at Universities and College of Technologies in JAPAN↓667 teachers (of mathematics, computer science, physics, technology, etc.) at23 universities and 56 college of technologies answered • Here we analyze only the answers of 378 mathematics teachers (which are the large majority)

  12. Questions and Results

  13. Subject (multiple answer allowed)

  14. Method to make class materials or exams

  15. Frequency of displaying graphics on printed matters, with a video projector, or others (excluding "on blackboard")

  16. In positive case

  17. Subject in which you frequently display graphics

  18. The topics which you think the display of graphics is effective for 　　　　　　　　　　　　↓ The answers are classified to two categories.

  19. Topics needing precise figures Graph of one or two variable functions (77) Taylor series expansions (26) Quadratic curves and surfaces (4) Solution curves for differential equation (9) • Topics needing conceptual figures which are difficult to be drawn on blackboard Differential and Integral (27) Partial differential and Total differential (24) Double integral and Repeated integral (23)Vectors and Linear transformations (14)

  20. Method to display graphics

  21. In negative case

  22. Frequency of drawing figures on blackboard

  23. To positive respondents we asked The reason why you do not show graphics to students excluding on blackboard

  24. To negative respondents we asked The reason why you do not show graphics to students

  25. No need (29) Explanation by words is sufficient. Explanation through visualization is not appropriate (sometimes misleading or even contradicts to the abstract notion like “orthogonality” in two dimensional space). Students’ reasoning may be captured by figures. • Referring to figures in textbook is sufficient (8) • No skill (6) • No time (5)

  26. Graphics in JAPANESE popular textbooks (with many figures)

  27. 2. Previous Researches

  28. Is it true that using graphics is not necessary (or not effective) in the teaching and learning of linear algebra? ↓ This problem was studied comprehensively by Ghislaine Gueudet-Chartier. “Using Geometry to Teach and Learn Linear Algebra” (CBMS Issue in Math. Ed., Vol.13, 2006, AMS)

  29. Research GroundingFischbein’s theory of “intuition” “Credible Reality” is needed to productive reasoning. “Models” are a central factor of intuition. Abstract models Analogical models Intuitive models Paradigmatic models

  30. Research Questions(1) How do mathematicians and students use geometric (and associated figural) models? (2) What are the possible uses of geometric models in linear algebra? (3) What are the consequences of the observed uses of models on students’ practices and thinking processes?

  31. Research Methods and Results

  32. (1) Questionnaire survey to 31 mathematicians concerning the use of drawings in LA Used the following figures or not? If “Yes”, for what purpose? Used any other figures?

  33. Teachers do not use many drawings. Most of the drawings were used to illustrate situations occurring in only R^2 or R^3. (which is used not as paradigmatic model but as analogical geometric model)

  34. (2) Textbook survey concerning the possibilities and limitations of the R^2-R^3 model Banchoff-Wermer: “Linear Algebra through Geometry” (Springer)

  35. Basis of R^n Orthogonal projection

  36. Historical analysis of LA (Dorier) suggests “LA is a general theory designed to unify several branches of mathematics” Thanks to using coordinates, R^2-R^3 serves an excellent paradigmatic model for R^n. Due to structural isomorphism, R^n does not serve as a paradigmatic model for general LA.

  37. (3) Interview to students Since the interview described in the paper is related only the extension from R^2-R^3 to R^n, we review the following example given in her another paper “Geometrical and Figural Models in Linear Algebra”Test (with justifications): Is there any linear transformation sending the left parallelogram (spanned by u and v) onto the right?

  38. Though LA can not be taught nor learned as a mere generalization of geometry, geometric model can be helpful. Geometric models must be used carefully in LA courses. Especially, geometrical model for general LA requires additional research.

  39. 3. Use of Graphics in Textbooks

  40. Almost all of the textbooks (in English) which we investigated use few graphics. ↓ Exception: S. Lang: “Linear Algebra”

  41. The use of graphics in Lang’s text has a similar feature as Banchoff-Wermer’sone.

  42. As another example of aggressive use of figures in the general theory context, we pick up the text written by G.Strang. “Introduction to Linear Algebra”. ↓ Even in this text, only 3 figures are used.

  43. Kernel and Cokernel in case of R^n

  44. Orthogonal complement in case of R^n

  45. Linear transformation in case of R^n

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