"Practical Math"

Unlike many members of the educational commentariat, Andrew Hacker, writing the the New York Times yesterday, is more concerned with finding a solution to mathematical under-performance than wringing his hands about the problem.  In my book, this automatically makes him about five times more worth listening to than your ordinary educrat.  Dr. Hacker suggests that the formal math standard in American schools is largely not useful either to most workers or to the man as citizen.  He proposes that math courses should instead of the discipline of formal mathematics focus on logical and quantitative reasoning and application.

While he has a point - "useful" mathematical knowledge is far from as common as it should be - I think he misunderstands the real problem.  He begins by citing the number of students failing and doing poorly across the United States in these traditional classes.  He observes that most teachers are dedicated and competent.  He concludes that the problem must therefore be the material.  "Algebra is a stumbling block..."

As a teacher myself, I believe Hacker has conflated two problems.  Many students, I think, find algebra difficult because they have not mastered arithmetic.  I inherit seventh graders, half of whom do not know how many feet in a yard, and the other half are too unsure to volunteer the answer.  Tenth grade students new to me have forgotten or never learned their perfect squares.  Students at all levels are incapable of or unwilling to do unit conversions.

Algebra I would define loosely as the art of manipulating and finding the values of unknown numbers.  Two critical ideas in carrying out algebraic operations (by which I mean mathematical manipulation performed on expressions - "clauses", if you will - with variables), can be taught simply and directly from basic arithmetic techniques.

The first is the concept of the variable itself.  A student familiar with measurements and conversions can be introduced to the idea easily: he is used to answering questions like, "How many meters are in 2400 millimeters?"  Arithmetically, we set the problem up in stages: find the starting amount, and then from the final units set up the conversion factors - which should be memorized by 4th or 5th grade but in any case are easy to look up.  Algebraically, we introduce the idea of equivalence, and mathematical symbols as a language.  The student already knows that, for instance, meters in "math" are m.  Now he learns that "question words" are represented by a symbol: a box, a question mark, or an x.  He learns that "are" and "equals" are (in basic algebra) equivalent.  So we reach the algebraic statement "x m = 2400 mm".  The conversion factor still needs to be reintroduced in its algebraic place, of course, and enough practice done to learn the methods.

The second technique which is an easy extrapolation from basic arithmetic is that of variable operations.  Polynomials, which are (simple version) expressions with multiple variable powers, can be added, subtracted, and multiplied (and therefore exponentiated) almost exactly like ordinary numbers, but with the different powers indicating "place value".  (Division, while also analogous, is slightly different and therefore harder.)  So a student who knows why his arithmetic works the way it does, who has been taught the decimal system, will recognize "place value" and make the connection in most cases without significant difficulty.

In short, I believe the root cause of high school failure is far more likely to be inadequate elementary education than difficulty with the material.

While I think Hacker's reasoning as to cause is flawed, his main concern - is this worth teaching anyway? - is a question worth asking.  I think the answer is still "yes", but with caveats.  I have blogged before about the state of mathematics textbooks in the modern United States: torn between traditional American mathematics and the "unified" approach of much of the rest of the world, having to teach or re-teach concepts not taught successfully before high school, fascinated with fiddly properties at the expense of overview, and so forth.

Algebra or Geometry as disciplines, though, are branches of a formal mathematics which exists - as people realized thousands of years ago - not only for practical purposes but as a training tool for the mind and as an art.  Algebra is valuable for much the same reason that a foreign language is useful, or music, or memory: to cultivate the human spirit in all its facilities.  Certainly most people will not use hyperbolic equations or non-Euclidean axioms in their day-to-day life, any more than most people will use French or German.  Certainly you can enjoy the original Hugo novels - you can also dabble in Newton.

The practical mathematics which Dr. Hacker champions is of course still valuable.  It is found in applied form in physics, chemistry, economics - all things which we say should be normally studied as useful.  But its basic tools can and should be provided before high school.