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
Preciseness
Programmability
Rich Perspective
Motivating
Examples
Generalizible
Concepts
Contents
1.
2.
3.
4.
5.
6.
Our Questionnaire Survey
Previous Researches
Use of Graphics in Textbooks
Utilization of Graphics Drawn with KETpic
Students’ Interview
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.) at
23 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 Grounding
Fischbein’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
1-3
R^2-R^3 model
Number of
Figures
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 Correct
of
answer
students
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
Vector space and
Matrix
Linear maps and
Matrix
Number of Comments
figures
28
R^2-R^3 figure
12
General theory context
with R^2 figure
Inner products
1
R^2 figure
Determinants
10
R^2-R^3 figure
Eigenvalues and
Eigenvectors
Convexity
2
R^2 figure
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 selfevident 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 Question
Is 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
Aim
To 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.
Object
To 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!!