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 principles and efforts
Some examples for illustrating the principles
Successes/failures
Links, literature, sources

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

They also lead us to

Example: teaching differential equations

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)
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:

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

  1. 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.
  2. Press ENTER on a well-prepared Maple-workspace template, understand nothing or very little.
  3. 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

My principles and efforts

"My program" Principles:

Some examples for illustrating the principles

Successes/failures

Links, literature, sources