1. Computing in the undergraduate mathematics curriculum?
Knut Mørken
Department of Mathematics
Centre of Computing in Science Education
Faculty of Mathematics and Natural Sciences
University of Oslo
Årsmøte, Norsk matematikkråd
27. september 2017

2. operations per second

3. More than 1012 operations per second.
Would have been the world’s fastest computer around year 2000.

4. Computers change the rules of the game!

5. Computers change the rules of the game!
We have extremely powerful tools for doing calculations that have radically changed the work of the professional scientist …
and the problems that can be tackled
Has the core, undergraduate curriculum in mathematics changed accordingly?
What about the sciences?
Or do we just use computers to teach the classical curriculum?

6. Traditional course in mechanics
Topics, examples and exercises are chosen so that all equations encountered are solvable by pencil and paper, possibly after some standard simplification

7. Traditional course in mechanics
Topics, examples and exercises are chosen so that all equations encountered are solvable by pencil and paper, possibly after some standard simplification
Why this constraint?
Makes it difficult to include practical and realistic examples
Not the primary methods used in research and industry

8. Centre for computing in Science Education
Centre of Excellence in Education (SFU)
Research: How to teach science with computing to enhance learning?
Development of new teaching materials
National and international resource for computing in university science education

9. What kind of mathematics should the students learn?

10. Questions
Are the students going to be controlled by the computer (calculator, Maple, Mathematica, …)?
Or should they learn to control (program) the computer?
What does it mean to solve an equation?
Is the concept of continuity relevant in computing?
What are the limitations of computing?
???

11. Differentiation
Definition
Approximation
Programming

12.
numbers

13. Differentiation of sound

14. Differentiation of sound –

15. Differentiation of sound –

16. Differentiation of sound – – – – –

17. Differentiation
Definition
Approximation
Programming
Error

18. Solving equations What does it mean to solve an equation f (x) = 0?
Find an exact expression (number) for x:

19. Solving equations
with Mathematica

20. Solving equations
usually more useful

21. Solving equations When can we find a formula for the zero?
Equations of degree one, two, three and four, some exponential and trigonometric equations
In other words, some very special equations
Using tricks that are difficult to generalise
We may derive properties of zeros in more general situations

22. How to find a zero of a general function?
Intermediate value theorem. For a continuous function f defined on [a,b], with opposite signs at a and b, there is some c∊(a,b) such that f(c) = 0. f

28. What about splitting in three?
Can we exploit the size of f at the ends?
What about the error?

29. Bisection method i = 0; m = (a + b)/2; while i ≤ N
if f (m) == 0 a = b = m;
if f(a)f(m) < 0 b = m;
else
a = m;
i = i + 1;
m = (a + b)/2;

30. Is continuity important for computing?

31. Continuity and computing
Suppose we are to compute f(a) where a is a real number
In a computer a is replaced by the nearest machine number a + e
In other words, we compute f(a+e) instead of f(a)
Then f had better be continuous!

32. ε-δ definition of continuity
Let ε > 0 be given. A function f is continuous at x if there is a δ > 0 such that |f(y) – f(x)| < ε for all y such that |y – x| < δ.
Can computing illuminate this definition?
For a given f and x, write a program that computes δ when ε is given.

33. What about algebra? Computing for large x gives large round-off errors
But
which does not give large round-off error

34. Programming and creativity
A core part of mathematics is to derive new algorithms — very demanding in pure mathematics
Programming is all about deriving precise algoritms, also for solving simple problems — systematising problem solving
Even programming a given algorithm is experienced by most students as very inspiring — gives a feeling of mastery
Creativity in mathematics becomes more accessible with programming

35. Mathematics and computing
From mathematics to computing
From computing to mathematics?
Discrete modelling
Limitations

36. Basic questions Common question:
What is digital competency in mathematics?
More basic:
May computers contribute to making mathematics more relevant?
May computers make possible a more creative approach to mathematics?
In which way is mathematics being changed by the computer?

37. Bachelor program in physics
Semester 6
Specialisation
Semester 5
FYS2160
ExPhil
Semester 4
FYS2140
FYS2130
Semester 3
MAT1120
FYS1120
AST2000
Semester 2
MAT1110
MEK1100
FYS-MEK1110
Semester 1
MAT1100
MAT-INF1100
IN1900
10 stp

38. What is mathematics?
What happens when algorithms can be performed 1015 times faster than before?
Radically new framework for doing mathematics, and therefore science!
Algorithms may replace some of the need for structure etc

39. What is mathematics?
What happens when algorithms can be performed 1015 times faster than before?
Radically new framework for doing mathematics, and therefore science!
Algorithms may replace some of the need for structure etc
Other algorithms become interesting

40. Changed mathematics
The framework for doing mathematics has changed, what is the ‘new’ mathematics?
The computer not just as a black box to do ‘classical’ mathematics
If you want to control the computer you must be able to program
Programming may illuminate mathematical concepts
Relevance and creativity?
Maple & Mathematica?

41. Greatest challenge
Coherent change in mathematics, computing, physics, statistics,…
We generally prefer the classical way of doing things, including building curricula…
And education is generally privatised…

42. Final words
Everybody should learn to program because it teaches you how to think!
Steve Jobs