Historical Mathematics: Bridging the Perceptual Gap

| March 22, 2011

The way students perceive mathematics and their performance are related.  It is not clear based on current research whether a student sees mathematics as an informative means of comprehending the world because he excels in mathematics or vice versa. The perception that mathematics is eminently out of reach, that it is a discipline for the elite is evident in classrooms.

One of the reasons for this perception is the presentation of mathematics in the classroom.  Most modern mathematics textbooks present materials in the most logically tight manner where the learner can not see the thought processes of the mathematicians who discovered the connections. Textbooks present mathematics in its crystallized form, distilled of any historical impurities.  One effective way to bridge the gap between students and mathematics is to show the development of certain theory from its beginning. Seeing historical mathematics can expose some of the earliest challenges mathematicians faced, and show some of the most persistent cognitive obstacles that blocked the intuition of those who tackled these problems first.

For example, take Schrödinger’s wave equation that describes the quantum states of a physical system:

Schrödinger’s original inspiration was that particles behave as a string which only emit and absorb energies of certain unit. String behaves similar to a wave. Thus if an educator begins with a discussion of Newton’s wave equation, the cognitive gap between a student and Schrödinger’s wave equation can be bridged. Consider a string divided into n equal segments of length d with a point mass m attached at the junction between each segment. If the tension in the string is T, and we use subscript j to identify the point masses in order from left to right. If the initial displacement of the mass is

yj = sin (jkπ/n)

By Newton’s second law, the vertical displacement can be expressed as

where μ is the string’s mass per unit length and FT is the net force. In this case, if a lecture begins with Newton’s second law, Schrödinger’s equation becomes much more accessible.

References

Polya, G. How to Solve it. Princeton University Press; 2nd edition (1971) QA11 .P6 1973 c.2

Gasiorowicz, S. Quantum Physics. John Wiley & Sons; 2nd edition (July 1995) QC174.12 .G37