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)

This work is supported by

Grant-in-Aid for Scientific Research (C) 20500818

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

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)

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

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

Our Proposal

LaTeX Graphics Drawn with KETpic

Motivating Examples

Generalizible Concepts

Preciseness

Rich Perspective

Programmability

Contents1. Our Questionnaire Survey2. Previous Researches3. Use of Graphics in Textbooks4. Utilization of Graphics Drawn with KETpic5. Students’ Interview6. Future Works

1. Our Questionnaire Survey

• Focuses of the Survey (I) The methods how teachers produce and use graphical class materials (II) The needs of teachers for using graphical class materials

• Terms 2008.9.1 – 2008.12.31

• 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)

Questions and Results

Subject (multiple answer allowed)

Method to make class materials or exams

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

In positive case

Subject in which you frequently display graphics

The topics which you think the display of graphics is effective for

　　　　　　　　　　　　↓ The answers are classified to two categories.

1. 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)

2. 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)

Method to display graphics

In negative case

Frequency of drawing figures on blackboard

To positive respondents we asked

The reason why you do not show graphics to students excluding on blackboard

To negative respondents we asked

The reason why you do not show graphics to students

1. 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.

2. Referring to figures in textbook is sufficient (8)3. No skill (6)4. No time (5)

Graphics in JAPANESE popular textbooks (with many figures)

2. Previous Researches

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)

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

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?

Research Methods and Results

(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?

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)

(2) Textbook survey concerning the possibilities

and limitations of the R^2-R^3 model Banchoff-Wermer: “Linear Algebra through Geometry” (Springer)

Chapter Contents Number of Figures

1-3 R^2-R^3 model 85

4 R^n model 5

5 General Theory 2

Basis of R^n Orthogonal projection

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.

(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?

Reasoning process Number of students

Correct answer

Use geometric transformations in R^2 (for positive) and refer to dimensional properties (for negative)

13 1

Use algebraic properties that linear transformation is characterized by the image of basis vectors

14 2

Use intuitive model (preservation of parallelogram) for the algebraic property (stability of sums and scalar multiplications under linear transformation )

10 4

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.

3. Use of Graphics in Textbooks

Almost all of the textbooks (in English) which we investigated use few graphics.

↓ Exception: S. Lang: “Linear Algebra”

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

　Contents Number of figures

Comments

Vector space and Matrix

28 R^2-R^3 figure

Linear maps and Matrix

12 General theory context with R^2 figure

Inner products 1 R^2 figure

Determinants 10 R^2-R^3 figure

Eigenvalues and Eigenvectors

2 R^2 figure

Convexity 5 R^2 and R^n figure

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.

Kernel and Cokernel in case of R^n

Orthogonal complement in case of R^n

Linear transformation in case of R^n

All of them are used in R^n theory context. (Directly connected to the matrix oriented Linear Algebra) All of them are abstract figures. (Experience is needed for students to fully understand the meaning of them)

The reasons why the use of graphics are tend to be held off in LA textbooks seem to be (1) Since simple shape is used, it tends to be thought that using blackboard is sufficient.(2) Since high dimensional figure can not be drawn, graphics tends to be regarded as obstacle to learning general theory. (3) Since many concepts tends to become self- evident in R^2, figures in R^3 are desirable. However, it is not so easy to draw fine R^3 figures.

Explanation of eigenvectors and eigenvalues

4. Utilization of Graphics Drawn with KETpic

Our Research QuestionIs it possible to make effective figures which serve as good help in students’ learning general LA?

　　　　　　　　　　　　　↓ KETpic will be able to produce such figures

rank and dimension – stemmed from matrix oriented LA

parallel projection and basis -- related to matrix representation

composition of linear transformations -- related to conjugation

eigenvalues and eigenvectors – related to the change of basis

structure of linear transformation – canonical basis case

searching eigenvectors – related to complex eigenvalues

differential system – related to normal form

5. Students’ Interview

AimTo examine the effect of using (KETpic) graphics in teaching and learning general LA.

Subjects 8 students of KNCT in the last grade (all 20 years old) They are familiar with matrix-oriented LA (containing the calculation of eigenvalues and eigenvectors). They have some knowledge about general LA.

ObjectTo understand the meaning of basis change

Topic The structure of a linear transformation

which is not diagonalizable (Jordan Normal Form)

Flow of the Interview

(1) Students answer preliminary problem

(2) Teacher explain the solution by using the following figures drawn in the printed material

(3) Students answer the main problem

(4) Only two students draw appropriate figures and realize the structure by own efforts

(4) Other three students draw graphics but 　　 could not realize the structure for 30 minutes

(4) The remaining three students could not draw figures or persisted in matrix-oriented calculation

(5) Students are given the following figure and they are encouraged to re-try the problem

(6) All of the students could understand the structure in 10 minutes

(7) The following figure is distributed for further study

Students’ Response (sample)

(1) By seeing the arrow of v_2, it becomes easier for me to realize the relationship to the preliminary case.

(2) Adding the arrow of v_2+2v_3 is desirable.(3) More of the hidden line elimination is desirable.(4) In case when more than one eigen vectors in the

wider sense appear, how will the structure become?(5) Without seeing the figure, it should have taken

much more time for me to understand the structure.

6. Future Works

1. Accumulation of interesting examples

2. Web-tech based accumulation (Web application of KETpic is under construction)

Thank you very much for your attention!!