A mathematical algorithm is a procedure for performing a computation. At the heart of the discipline of mathematics is a set of the most efficient — and most elegant and powerful — algorithms for specific operations. The most efficient algorithm for addition, for instance, involves stacking numbers to be added with their place values aligned, successively adding single digits beginning with the ones place column, and “carrying” any extra place values leftward.
What is striking about reform math is that the standard algorithms are either de-emphasized to students or withheld from them entirely. In one widely used and very representative math program — TERC Investigations — second grade students are repeatedly given specific addition problems and asked to explore a variety of procedures for arriving at a solution. The standard algorithm is absent from the procedures they are offered. Students in this program don’t encounter the standard algorithm until fourth grade, and even then they are not asked to regard it as a privileged method
It is easy to see why the mantle of progressivism is often taken to belong to advocates of reform math. But it doesn’t follow that this take on the math wars is correct. We could make a powerful case for putting the progressivist shoe on the other foot if we could show that reformists are wrong to deny that algorithm-based calculation involves an important kind of thinking.
What seems to speak for denying this? To begin with, it is true that algorithm-based math is not creative reasoning. Yet the same is true of many disciplines that have good claims to be taught in our schools. Children need to master bodies of fact, and not merely reason independently, in, for instance, biology and history. Does it follow that in offering these subjects schools are stunting their students’ growth and preventing them from thinking for themselves? There are admittedly reform movements in education that call for de-emphasizing the factual content of subjects like biology and history and instead stressing special kinds of reasoning. But it’s not clear that these trends are defensible. They only seem laudable if we assume that facts don’t contribute to a person’s grasp of the logical space in which reason operates.
In other words, reform movements are largely based on the rejection of a “reality-based” concept of education. We couldn’t possibly have anything as piddling as facts interfering with the joy and beauty of learning. If a child wants to believe that 2+2 =5, shouldn’t they be praised for thinking independently?
In all seriousness, though, there’s something borderline sadistic about schools refusing to teach actual, well-established knowledge, knowledge that makes learning easier. Not every student is genius capable of re-deriving the Pythagorean theorem on their own. Yes, by all means, teach students to understand why things are true – I’ve heard from math tutors who constantly encounter kids who do just fine in calculus because they’ve learned when to plug in about four formulas but who fall down on comparatively basic SAT math because they don’t really understand why things work the way they do, or how to apply simple formulas when they’re presented in unfamiliar ways. The point is, teach them something, don’t just let them flail around trying to figure it out on their own.
What’s the point in all those centuries of accumulated knowledge if schools are just going to toss it out the window?