|
Helsinki University of Technology
Institute of Mathematics |
|
Last update 29.6.2000
|
AMS-Scandianavian 2000 meeting, Odense June 13 - 16
Special Session 11: Mathematics Education
Abstracts of this
special session
Computer lab as a mathematics classroom
http://www.math.hut.fi/~apiola/odense2000/
Heikki Apiola
June 11, 2000
Abstract
I will consider examples of various teaching experiments at the Helsinki
University of Technology and the University of Helsinki I am or have been
involved during the past several years.
The use of software, especially Maple and/or Matlab and the possibilities
offered by the web will be discussed. The implications of compuer lab
activity to the teaching/learning process will be considered and several
examples of course material, pieces of software developed or found, student
projects, successes and failures will be given.
The courses referred to will cover large basic mathematics courses for
engineering students, some small special courses for advanced students
on special topics like Mathematical modelling, computational PDE's or
numeric and symbolic computing. The most recent experiment is a
basic course on mathematics tailored for a small group of students at a
newly established "information networks" degree programme at the Helsinki
University of Technlogy. The experiences from the first such course will be
communicated.
Outline
My transparencies were picked for the most part out of the material found on
these pages. I will do some post prosessing and more accurate indication
of examples I considered, the ones I just circulated and those that I would have
liked to include plus some general thoughts inspired by discussions, other
presentations etc.
I just have to postpone it until the latter half of July, (or even early August)
as I will spend
the first 2 weeks by the seaside and the summer is short...
General thoughts about reforms
My focus is on engineering mathematics, background in pure mathematics.
"Mathematics is difficult and not so useful"
(some engineering students, perhaps even
some professors) |
"In the computer age the role of mathematics is more important than ever" (Nokia, official statements, us)
|
NSF goal: "Increasing understanding and usefulness"
|
Curriculum, goals
What is the right balance between
analysis, linear algebra, logic, algebra,
discrete mathematics, statistics, numerical methods, etc.
Many methods have become obsolete. For instance finding formulas
for zeros of polynomials was one of the biggest parts of Reneissance
mathematics in Italy.
With the computer at hand we hardly care whether a problem can be
solved in the "Renessaince sense".
Both numeric and symbolic software changes our emphasis.
For instance software systems like Matlab, Maple,
Mathematica
support matrix algebra, which makes
it appealing to
- use matrix methods from the very beginning.
They also lead us to
- turning mathematical concepts and ideas
into working algorithms, symbolic and/or numeric.
- seeing some mechanical manipulations as CAS-routines while
giving more room for conceptual thinking.
Example: teaching differential equations
-
Emphasis on qualitative methods and numerical methods, phase portraits,
dynamical systems approach
-
Analytic methods: Some basic techniques like handling linear
systems are still important. Some of the traditional material
has become or is becoming obsolete.
(Some care has to be taken not to be too quick in such desicions, though.)
-
Linear algebra and differential equations support each other nicely
Demands from applications
Solving problems of growing complexity and size. Demands may result
from extreme physical conditions (eg.
superconductivity, nanotechnology, space technology) or entirely
new tasks like robotics, computer vision, neural networks, biological
or economical (or ecological) processes,
cryptography, etc, etc.
Students need (in the idealized world)
- Solid knowledge of basic principles and methods
- Realize that mathematics is systematically built on relatively
few basic concepts involving powerful unifying principles.
- Skills for both modelling and interpreting the results.
- Algorithmic and software skills.
How much effort should be spent on trying to teach basic concepts
like real numbers, functions, convergence, continuity, compactness,
linear independence, etc.
The ideal might be:
- Leave routine tasks to the computer.
- Use human brain capacity for increasing the level of understanding and
abstraction
Reality, student motivation
Many engineering students find mathematics a subject that has to be
endured rather than enjoyed. This makes it very hard for the teacher
to have them spend their time and effort especially on learning abstract
and difficult mathematical concepts. The limited time resources the
student has make quite often mathematics one of the topics
to suffer. (The student and teacher suffer afterwards in the
effort of pretending the student has some mastery of mathematics before
the engineer's degree can be granted.)
Students' response to
computer labs, project work, CAS work
-
Get very excited and are willing to spend much of
their time in planning algorithms, writing well documented worksheets
discussing and making an effort for understanding the mathematics also.
-
Press ENTER on a well-prepared Maple-workspace template, understand
nothing or very little.
-
Take programming challences too seriously
leaving no or little time for the mathematics.
(Typical example: Fractal images)
A large group of "conservative" students is a reality in basic
courses. They very carefully count the credits to decide whether
it is worth their while to participate the computer-lab activity.
The special courses are another story, well-motivated, bright students
Individual/team work, PBL, labs, tutoring
Pros | Cons |
More interesting exercises |
Time spent on syntax |
Team work skills |
Serious learning occurs in privacy |
Teacher-student interaction |
Academic freedom destroyed |
Deeper understanding of concepts |
Time consuming ("need some sleep also") |
Learn to use tools useful in other courses, too |
The right to just pass by minimal effort |
- "Pure" PBL is not suitable, mathematics is a systematic subject.
- Information networks: Too much PBL and project work in
almost all topics. "Why do we have to use computers even in mathematics?" "Project work again, frustration".
"My program"
Principles:
- Study the mathematical concepts in the traditional way
- Illustrate (if useful) with Maple (or Matlab) on a small-scale example
- Present some "real scale problem/example" with Maple or Matlab
- Projects (large or small) where some parts can be left as
black boxes to be considered later or just explained using some
heuristics (or left totally black).
- What the user of math. software should be aware of.
- Sources of error
- Validity ...
- Other computational aspects
Course material
Some of the links may be out of date (or out of order). (7.12.2009)
|
- Math K3
Fall -98Dept. of Mechanical engineering, Civil eng., Chemical technology + some others (300 students)
- Example on linear systems, temperatures
-
- Eigenvalues
- Linear mappings acting on vectors and figures (Maple and Matlab), "Strang's house, "Arnold's cat, conic sections
- Does a rotation have eigenvalues? Real/complex
- Simulate hand-calculation of eigenvalues/vectors
- Use black-box linalg[eigenvectors] , learn to interpret
output, algebraic and geometric multiplicity.
- Matlab's eig is very convenient for diagonalization.
- Differential equations
- Linear systems, matrix exponential function (Maple/Matlab)
linalg[exponential] (Maple), expm +
own piece of code linsys , Matlab.
- Student project on oscillating strings, beats, resonance.
Some very nice worksheets were returned.
- Phase planes, qualitative theory, Maple's DEplot ,
Matlab: DiffeqLab (programmed by Simo Kivelä at HUT), Golubitzky-
Dellniz-tools.
- Numerical methods. Both Maple and Matlab. Also some student
programming (like vector version of Euler/Runge-Kutta in Maple/Matlab)
Stability questions, stiffness, etc.
- Laplace transforms
- Maple worksheets on basic properties
- Return to integration, notice the use of partial fractions.
- Maple worksheet templates for solving linear ODE's
- Fourier series
- Maple worksheet templates
- Some useful functions like PeriodicExtender
- Math V2 Spring 2000
Lots of mainly Maple-based material (some Matlab-material as well) on
-
-
- Math V3 Fall 2000,
in progress
- SVD and image compression
- Maple ws for separation of variables
in the 2-dim. Laplace equation on a rectangle.
-
Computational PDE Spring -99
-
Temperature matrix, motivation for solving linear systems at basic course
-- 1st step in introduction to Difference method for Laplace equation for more advanced course.
- Student solution of exercises nr. 1
- Final project
- Numeric and symbolic
computing Spring -98
-
Introduction to application projects, Univ. of Helsinki