Computing in the undergraduate mathematics curriculum?

Transcription

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 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

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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 Specialisation Specialisation Semester 5 Specialisation FYS2160 ExPhil Semester 4 FYS2140 FYS2130 Specialisation Semester 3 MAT1120 FYS1120 AST2000 Semester 2 MAT1110 MEK1100 FYS-MEK1110 Semester 1 MAT1100 MAT-INF1100 IN stp 10 stp 10 stp

38 What is mathematics? What happens when algorithms can be performed times faster than before? Radically new framework for doing mathematics, and therefore science! Algorit hms Struct ures Algorithms may replace some of the need for structure etc Patter ns Relati ons

39 What is mathematics? What happens when algorithms can be performed times faster than before? Radically new framework for doing mathematics, and therefore science! Algorit hms Struct ures Algorithms may replace some of the need for structure etc Other algorithms become interesting Patter ns Relati ons

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