Is Algebra Necessary?

Algebra, as remembered by many,  seems to be  nothing more than a set of meaningless formal tasks, and if that memory is accurate, then indeed it has no right to be included as a requirement of everyone's basic education. [ Some who see it that way try to justify the subject  as an exercise in mental discipline, but that argument is as vacuous as it was when applied to teaching Latin grammar (without the literature) in centuries recently past. So if algebra is really as people claim to remember it then it certainly is not "necessary".] But such memories are either false, or are evidence of incompetent teaching.  And they are not by any means a justification for abandoning the subject as a universal requirement.

 

 

In fact high school "algebra", properly taught, is not meaningless at all, and it is a crucial tool, as fundamental as reading and writing, for the understanding and communication of how things work in the world around us.  What "algebra" is really about is the analysis of relationships between variable quantities.  And the capacity for addressing such relationships, without reference to the specific values (or even the applied interpretations) of the quantities in question, is of vital importance in many areas.  It can certainly be learned in lots of contexts other than an "algebra" class, but for anyone who has learned those ideas, algebra is so trivial that the requirement is not a burden.

To anyone who understands the properties of basic arithmetic and the use of symbols as names of variables an exercise such as proving that

(x^2 + y^2)^2 = (x^2 - y^2)^2 + (2xy)^2

is no more daunting than a three or four digit long multiplication (a bit tedious to have to do too many of perhaps, but a single example certainly should not be seen as much of a challenge).

For anyone to whom that is not obvious (who does not want to pass through life as an intellectual cripple), some instruction is necessary - along with whatever amount of practice is required in order to develop facility with the concepts.

The main weakness of Is Algebra Necessary? - NYTimes.com is demonstrated by its author's use of simple numerical relationships as examples of how the essentials can be learned in the context of other disciplines. His failure to come up with appropriate examples of algebraic thinking shows that, for him at least, those other disciplines did not in fact lead to an understanding of what is important about "algebraic" thinking.

In addition to citing the above trivial problem as something too hard to expect the average student to understand, he actually cites long division as something that should(!) be learned by all, and cites "Fermat's dilemma" in a failed attempt to appear familiar with what he dismisses.

The article has drawn critical responses also from a philosopher

and a mathematician

and scientific american also chimes in.

[Update 2013.02.10 The responses keep coming - eg this from Slate, and a search for "Fermat's dilemma" yields many others full of scorn for that lapse as evidence of the NYT author's failed attempt to pretend familiarity with what he declaims about.]

To those who argue for Statistics as a more appropriate choice I would ask how we can expect to understand statistical relationships between variable quantities without understanding deterministic ones first, and to those who advocate discrete math I would argue that although this includes many interesting examples of applied mathematical thinking these tend (at the elementary level at least) to be in silos between which there is little transfer of general concepts (except for a bit of combinatorics perhaps). The understanding of a general linear relationship y = mx+b with variable parameters, on the other hand, allows one to see the analogy between slope of a roof and speed of a car - not to mention being essential so far as I(*) can see for making any serious use of the much-touted alternative subject of statistics.

* - along with the majority of college Statistics instructors according to a survey by the BCcupms

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