lördag 8 augusti 2009

The Tower of Mathematics Education

In a critical analysis of school mathematics, Sverker Lundin identifies the gap between what school mathematics claims to offer the students, and what is actually offered, a gap which I will argue originates in university mathematics education. A gap which today can be bridged by the computer.

Primary school mathematics is a bleak simplified version of secondary school mathematics, which is a bleak simplified version of college and university mathematics. The ideology of mathematics education is formed on the highest level, which in bleak simplified versions is propagated down to secondary and primary levels. 

The ideology of university mathematics as presented by mathematics departments offering mathematics education, is based on the following postulates:
  • mathematics is the language and basic tool of science
  • mathematics is a prime expression of human intelligence and creativity 
  • the World is governed by mathematical laws
rooted in Greek mathematics and boosted in the scientific revolution initiated by 
  • Newton's law of gravitation
  • formula for solution of differential equation of two-body problem,
expressing the power of 
  • Calculus: derivatives, integrals and differential equations
  • linear algebra: arithmetization of geometry
which has formed the basis of university mathematics for more 100 years. 

The goal of mathematics education is to show that mathematics is a universal tool with universal applicability. In Calculus/linear algebra this is demonstrated by
  • generality of the concepts of limit, continuity and derivative
  • list of primitive functions of elementary functions
  • formula for solutions of simple differential equations
  • generality of linear algebra,
aimed at expressing that all problems can be described by algebraic/differential equations which can be solved analytically in terms of elementary functions. A gap results from the fact that in reality few differential equations can be solved analytically. 

In primary education the universality can be expressed in
  • regula de tri: proportionality
  • Euclidean geometry
  • formula for solution of 2nd order algebraic equation
where regula de tri represents a universal method for solving a wealth of problems of proportionality. A gap results form the fact that few problems are simple problems of proportionality or can be formulated as a second order algebraic equation.

The gap between fiction and reality of mathematics education comes from the fact that analytical mathematics is not universal: Already the three-body problem lacks an analytical solution formula. This gap undermines the crediblity of mathematics education with the effect that most students fail to get the main message that mathematics is universal with a potential to solve a universe of problems.

However, computational Calculus/linear algebra obtained by adjoining the computational power of  computer, has universal capabilities, which opens new possibilities of closing the gap by bringing ideology and practice together in a modern meaningful mathematics education ...the goal of the Bod&Soul project...see knols on mathematics/science education...

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