Everything should be as simple as possible – and no simpler.

Albert Einstein

Remedial Mathematics

Usually, a remedial course in any subject is a euphemism for a catch-up course to help those students who've failed to keep up with the course as normally taught. The pages I intend to write here have a quite different goal: to remedy some confusions and misconceptions with which the students who did keep up have commonly been saddled. Those familiar with the Charlie Brown cartoon strip may remember some character remarking that young Linus van Pelt was doomed to spend years unlearning things his big syster, Lucy, had taught him; this course is like that, only with mathematics students as Linus and orthodox teaching as Lucy. Here, remedial is no euphemism: my intent is to remedy some (apparently) common errors of teaching.

head-bang The various mathematicsl topics I'll cover are taught in ways that can be (indeed, are) made to work but that conflate things that should be distinguished, mangle notation to the point that it obscures the underlying mathematics or otherwise impede the student's understanding, where a clearer approach to the subject matter could illuminate it better. The cause of orthodoxy's deficiencies is, in each case, an accident of history (propagated from generation to generation by teachers teaching it the way they were taught it): when the subject was first explored, those doing so cobbled together notations and modes of discourse that worked to the extent of their grasp of the subject; those were later refined as folk got a clearer grasp of the subject, but some fossils of the initial approach remain, that we would (IMO) do well to expunge. A common theme of several cases is the concealing of important geometric operations in notation, rather than overtly exposing a relevant entity, alongside those on which it acts.

I shall thus aim to deal with each topic by, first, outlining the things that are mis-taught; then explainig what's wrong with the orthodox treatment of them and providing a clearer teaching; finally, explaining why the old way of doing things worked and how to live politely with those you don't have time or opportunity to lead away from using the old way. (These may include your examiners, if you're a student: to pass the exams, you have to do wrong things in the right ways. Hopefully, a later generation's examiners shall know better.) The topics I intend to cover (for most of which I still need to complete a good clean write-up of my own) are:

Contracting vectors
The use of yTx (or similar with a big bold dot in the space between y and x, in place of the T above the gap) to take the innter product of two vectors.
Matrix multiplication and transposition
The general idea that T just means reflection in a diagonal; and the confusions that arise from conflating four kinds of linear map into one kind of matrix.
The outer vector product and ∇
A convenient feature of three-dimensional space obscures a richer (and more general) algebraic structure and hides the metric's rôle in integration over volumes.
No, they're not scalars. The most common case can naturally be encoded as a scalar, but it's the only case where this is faithful.
Partial differentiation
… or: why differentiation should first be taught in the context of linear spaces and only later shown to be elegantly simple on the underlying reals.
Component-centric tensor notation
Physicists still use a notation for the study of smooth manifolds that dates from the early twentieth century; mathematics can now do it better.

Once I've actually written any of these, I'll make them links.

See also

This broken futon metaphor is one way to understand the problem with the broken teachings I'm discussing here: although, in some sense, they let folk work with a class of problem, after a fashion, the misrepresentation leads to habits of thought that get in the way of further development.

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