## onsdag 14 oktober 2009

### Mathematics Education in the IT-Society: What, Why and How?

Here is a draft of my upcoming talk at Seminar on New Perspectives in Teaching Mathematics
at Helsinki University of Technology. This type of seminar can not take place at KTH.

We compare
• OLD: before the IT-age: without computer
• OLD: unreasonable effectiveness of analytical mathematics in sciences: FALSE
• OLD: not understandable + not useful
• OLD =  math education of today
• NEW: in the IT-age: with computer
• NEW: reasonable effectiveness of computational mathematics in sciences: TRUE.
• NEW: understandable + useful
• NEW = reform math of tomorrow
• Death and Rebirth of the Computing Engineer 1,  2,  3,  4.
We consider one major area of technology: fluid mechanics = experimental + mathematical:
• mathematical fluid mechanics = Navier-Stokes equations: turbulent solutions.
Facts:
• OLD: Cannot solve NS by analytical mathematics: turbulence
• NEW: Can solve NS using computational mathematics: turbulence.
Result:
• OLD: Incorrect cook-book formulas: useless: lift/drag mystery: flight mystery
• NEW: Correct computational solutions: useful: no mystery.
What:Why:How:Value
• OLD: analytical formulas: solve NS: pen/paper: useless
• NEW: simple analytical formulas + computation : solve NS : pen/paper + computer: useful.
NEW: Solve

(DE)                                      du/dt = f(u,t)     (NS)

by time-stepping. To learn:
• calculus: derivative
• integral = solution to du/dt = f(t)
• construct by time-stepping: u(new) = u(old) + k f(old)
• elementary functions = solutions of simple DE: exp, log, sin, ...
• linear algebra: solve AX=B
• discretization: Galerkin finite elements
• programming.
Mathematical fluid mechanics part of NEW mathematics education.

NEW mathematics education:
• condensation: fewer formulas
• expansion: general differential equations: richer applications
• effectivization: Ockhams Razor
• understanding: principles
• proof by construction step by step
• ability to simulate complex systems
• = simulation technology = technology for/with simulation
• = solve equations: convection/diffusion/reaction/wave...
• = computational fluid/solid/quantum/electromagnetics...
• construct - compute - analyze - understand - construct -
NEW calculus:
• constructive -- computational
• Lipschitz continuity: | u(x) - u(y) | <  L | x - y |
• Derivative du/dx by linearization: | u(x) - u(y) - du/dx (x-y) | <  C| x - y | | x - y |
• definitions of continuity + derivative without using limits
• bisection: limit
• fixed point iteration  x = g(x): limit
• Newton f(x) = 0: limit
• Solve DE: du/dt = f(u,t) by time-stepping: universal
NEW linear algebra:
• vector, matrix, linear combination, basis
• analytic geometry
• Gaussian elimination
• Jacobi iteration
• eigenvalue
• orthogonality.
Mathematics = ??
• NEW: formulate + solve equation = simulation technology = engineering mathematics
• OLD :mathematics = art + beauty + logic + theorem + proof = ?
Theorem -- proof  vs  constructive proof  -  construction = theorem:
• NEW: proof by construction: theorem
• OLD: theorem: proof by ??
Constructive mathematics = computer game:
• engineering mathematics = simulation technology
• simulation technology = computer games
• math education = how to construct computer games.
Mathematical computer games:
• bouncing ball: geometry in 2d and 3d + visualization + interaction
• spring-mass system: Newton + Hooke law
• fluid mechanics: Newton law + Eulerian description
• solid mechanics: Hooke's law + Lagrangian description
• fluid-structure interaction: action games, sports.
• science games: quantum ....  cosmology.
Summary:
• global competition requires engineering education reform: NOKIA?
• computer game math: active, interactive, think, do, learn, understand, open
• strong resistance to math education reform: passive, sit, look, confusion, closed
• now: collapse of traditional math education+  birth of reform math
• horrible time for classical analytical mathematics
• wonderful time for constructive computational mathematics!
• What: calculus + linear algebra + computer + programming
• Why: computer simulation of real world
• How: construct computer games.
Realization:

#### 2 kommentarer:

1. Professor,

I've had your Computational Differential Equations text on my desk for nearly a decade - with the Body and Soul books added more recently. Probably my favorite mathematical texts ever. (John Boyd's text on spectral methods comes in close behind.)

Every day I learn to appreciate further the general application of numerical and analytial mathematics to attack problems of mathematical physics in a generalized way rather than the narrow cook-book methods still taught in university today. (I think of this as I dread a viscous fluid flow lecture I will attend this afternoon where we are scheduled to review a handful of analytical solutions to the NS equations...)