fredag 8 november 2013

Computational vs Analytical Calculus (at Chalmers)

In connection with the apparent return to Chalmers of (a bit of) the BodyandSoul mathematics education reform project, which was run at Chalmers 1998 - 2006 but was disrupted in 2007 when I moved to KTH, it may be of some interest to compare BodyandSoul as Computational or Constructive Calculus with the current standard of Analytical Calculus as presented by e.g. the standard text book Calculus: A Complete Course by Adams and Essex, a standard which found its form 100 years ago.

Standard Analytical Calculus:
  • symbolic analytical solution of a limited variety of specific algebraic and differential allowing analytical solution
  • as many analytical formulas and tricks as possible = thick book with many examples
  • limited generality as large collection of special cases
  • elementary functions picked from a hat
  • solution work done by symbolic manipulation on paper
  • maximal number of formulas and tricks to convince student of generality, which is illusion,
  • mastery of the many analytical tricks difficult for the ordinary student
  • the general problem cannot be tackled, not even by teacher
  • conceptual difficulty: tricky symbolic manipulations, existence unclear and properties magic
  • in short: many difficult special cases = powerless mathematics.  
Computational or Constructive Calculus:
  • constructive solution of general algebraic and differential equations (by time stepping)
  • focus on a few basic analytical formulas and tricks
  • focus on a few basic constructive algorithms (fixed point iteration, Newton's method)
  • generality by computation
  • elementary functions constructed by solving differential equations by time stepping
  • solution work done by computer
  • few basic analytical formulas and computational algorithms can be mastered by the ordinary student
  • student can tackle the general problem
  • conceptual simplicity:  construction by basic algorithms gives existence and properties without magics
  • in short: a few simple general cases = powerful mathematics.    
As you can see, the two approaches give very different weights to Complete Calculus as the union of Analytical and Computational or Constructive Calculus.

The fact the basic text at Chalmers is Adams, shows that the weights still are those of a standard course, and that BS in more complete form is still waiting to return to Chalmers.  As long as the text book is Adams, the standard since 100 years will continue to control (and limit) young minds, at Chalmers and elsewhere. It is a tragedy.

PS For more comparison between analytical and computational math, see next post.

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