Saturday, March 12, 2011

The Mis-Education of Mathematics Teachers?

In Vol. 58, No. 3 of the Notices of the American Mathematical Society, a very disturbing article, The Mis-Education of Mathematics Teachers, by H. Wu, appeared. WU began by posing the rhetorical question: "If we want to produce good French teachers in schools, should we require them to learn Latin in college, but not French?" Good point! Because it seems in training American math teachers, this is equivalent to what is being done!

Wu gives the example of teaching fractions, say in grades 5-7. And say one wishes to add:

1/2 + 3/4

Easy, no?

Not according to Wu. And the reason is that the maths educators seem to be required to take courses that focus on the abstractions as opposed to the mechanics and operations.

Thus, he notes that Math professors may wonder why teachers of math must be provided with a knowledge of fractions relevant to the classroom. They may ask:

'What's so hard about equivalence classes of ordered pairs of integers?'

In this approach, to remind readers who may be unaware, we let Z be the integers and S the subset of ordered pairs of integers: Z X Z, consisting of all the elements (x.y) so that y not equal zero.

One then introduces an equivalence relation (~) in S by defining:

(x,y) ~ (z, w) if xw = yz

Then one denotes the equivalence class of (x,y) in S by (x/y) so that y not = 0.

We call the set of all such (x/y) the rational numbers, Q, because they are designated by a RATIO.

One then identifies Z with the set of all elements of the fom: (x/ 1) and one has:

Z ( Q (Z a subset of Q)

Finally, we convert Q into a ring by defining addition and multiplication in Q by:

(x/y) + (z/w) = (xw + zy)/ yw


(x/y) * (z/w) = xz/ yw

Is this needed to teach fractional addition (or multiplication) to kids in grades 5-7? HELL NO!

But, as Wu notes, this is normally how we teach our math majors (in 2-3 lectures) to do it. Can you do this with middle school students? Well, if you try either get set to have all the ipods pulled out with you tuned out, or be looked at as some kind of freak.

As Wu points out, it's "totally consistent with the fundamental principles of mathematics" but not much use in pedagogy!

Why? A number of reasons:

1) It requires an understanding of the partition of S into equivalence classes

2) It requires the ability to consider each such class as one element.

3) It requires understanding yet another level of sophistication in terms of the identification of Z with {x/1: x in Z}.

Wu is correct when he notes that schools were in existence (and teaching fractions) long before their formal representation as equivalence classes of ordered pairs of integers, but this doesn't mean the teaching preceding that formal presentation is any less valid. It merely means that the teaching was devoid of the abstraction and proofs.

In this sense, I disagree with Wu that the informal teaching of fractions is not teaching mathematics but a pretense of such. No, it is teaching math, but not predicated on a formal abstract theory. In the same way, I can teach physics students (say at A-level) the principles of atomic behavior and how emission or absorption spectra are formed (including using experiments) without introducing all the quantum mechanics abstractions, to the effect atomic states really entail "probability waves" and one has Hilbert spaces to define the behavior of the wave functions.

Thus, cutoff points for pedagogical abstraction don't mean one is operating under "pretense" but more accurately, a lower conceptual domain. (This would be in accord for Jean Piaget's notable levels whereby a student may grasp a thing at one, but not another).

But to call the rudimentary mechanical teaching "pretense" is to not understand how teaching differs from mathematical research in the first place.

Wu is correct that "mathematics depends on precise and literal definitions"- but again, one must not let oneself become a hostage to technical formalisms and procedures or abstract pedantry. Physics also depends on precise and literal definitions, but I wouldn't subject an A-level physics student to the definition of a quantum superposition or even the Heisenberg Uncertainty Principle (or Principle of Complementarity) using Poisson brackets, before letting him examine radioactivity - say from a decaying alpha source near a Geiger-Müller counter.

It is here I believe that common sense in teaching must prevail, and yes, often details must be surrendered.

The trick, whether in math or physics, is knowing where the details end and the principles begin, and also if putting the principles in a very rudimentary form constitutes "pretense".

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