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

34 000 000 000 000 000 operations per second

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

Computers change the rules of the game!

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?

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

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

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

What kind of mathematics should the students learn?

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? ???

Differentiation Definition Approximation Programming

0.237976 0.196136 0.125732 0.474854 0.294189 0.469055 0.433350 0.470581 0.875549 288 000 numbers

Differentiation of sound 0.237976 0.196136 0.125732 0.474854 0.294189 0.469055 0.433350 0.470581 0.875549

Differentiation of sound 0.237976 0.196136 – 0.237976 0.196136 0.125732 0.474854 0.294189 0.469055 0.433350 0.470581 0.875549

Differentiation of sound 0.237976 – 0.0418396 0.196136 0.125732 0.474854 0.294189 0.469055 0.433350 0.470581 0.875549

Differentiation of sound 0.237976 – 0.0418396 0.196136 – 0.0704041 0.125732 – 0.0782471 0.474854 – 0.0180664 0.294189 0.0174866 0.469055 – 0.00357056 0.433350 0.00372314 0.470581 0.0404968 0.875549 0.0445557

Differentiation Definition Approximation Programming Error

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

Solving equations with Mathematica

Solving equations usually more useful

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

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

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

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;

Is continuity important for computing?

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!

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.

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

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

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

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?

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

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

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

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?

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…

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