The four pillars of high-school mathematics

4 pillars

What do we teach in high-school math? If we look at the big picture —not at specific topics in specific courses — what do we teach? Numbers? Formulas? Algorithms? Taking standardized tests? All of the above? None of the above? There’s no simple answer to these questions, but formulas, algorithms, and taking standardized tests are way down on the list. Even numbers are not very high up.

High-school math actually rests on four pillars:

  • representation
  • abstraction
  • quantitative thinking
  • reasoning.

Let’s step back and get a big-picture view before we explore each of the pillars individually. First of all, notice what the pillars are not: they’re not algebra, geometry, trigonometry, and calculus. Although the American fashion is to divide mathematics into these arbitrary “subjects,” such separation is not common in the rest of the world, and even some American schools have abandoned it (for better or for worse). Second, as you may know, I teach a course in the summer that Harvard has dubbed “quantitative reasoning,” and therein lie two of the pillars. Not a coincidence. Mix in representation and abstraction — all intertwingled — and you get the true four pillars.

Is there any glue that ties all four pillars together? I don’t want to strain my metaphor too much and discuss cementing them to the platform on which they all rest, but that really is what I’m talking about. Perhaps the glue is problem solving. All of these big ideas — representation, abstraction, quantitative thinking, and reasoning — help us solve problems. Mere facts can let us solve exercises, but big ideas are needed if we want to solve problems. Math is a language that lets us do that. Using the language, we solve problems with skills and concepts (not skills versus concepts); we need knowledge of and knowledge how.

Math isn’t exactly a unique discipline in this regard. It connects to other fields. We want to build bridges, not walls — but what are we bridging? Interdisciplinary teaching is a wonderful idea (using all of our four pillars), but it has some intrinsic problems. I still remember a workshop we held in a school where I taught in the ’70s; algebra teachers and earth science teachers got together to design some cross-disciplinary units. Great idea, no? The workshop pretty much collapsed when we discovered that we had very different concepts of our proposed cross-disciplinary unit on ratio and density: the algebra teachers wanted to spend six weeks on ratio and proportion, using density as a one-day example, whereas the earth science teachers wanted to spend six weeks on density, using ratio and proportion as a one-day introduction. Even if we’re on the same side of the building (literally and metaphorically), math and science teachers have very different sensibilities. So the ideas in this post may well relate to other disciplines, but they’re specifically just about mathematics. In particular, note that nothing here suggests that math is the servant of science, even though some people would like to think so. Mathematicians, in fact, would like to think that it’s the other way around:


purity

Now let’s move on to the four pillars, one at a time.

Representation is certain a key idea in mathematics. In Algebra II, for example, we spend a lot of time on different ways to represent the same function — as an algebraic rule written in symbols, as a table, as a graph, as function machines, as a computer program, and as a description in English that might show a transformation from a parent function. The twofold difficulty lies not only in seeing that all of the representations are more-or-less equivalent, but also in picking the best representation for a particular task. This idea extends far beyond pure mathematics. It’s crucial in applied math as well, and it is found in disciplines that are altogether different from math: maps, codes, and musical notation are all instances of representation (perhaps one of the reasons that I’m interested in all of these, at least according to the father of a former student). In databases and spreadsheets we’re concerned with different views and layouts of the same data. Even a webpage is a representation of other ideas. We can get all semiotic about representation — talking about signified vs. signifier, signs vs. symbols — or we can stick to math.

Then we get to abstraction. This idea trips up a lot of students. Arithmetic is fine, but algebra is hard — not only because it requires multiple representations but also because it’s no longer concrete. “Five minutes ago you said that was 12, and now you say that it’s 9 — how can that be?” We can easily observe that 2+3 equals 3+2, but as soon as we write it as a+b=b+a we have become abstract. Statements like this relate to quantification: we really mean something like “for all values of and b, a+b=b+a.” But as soon as we represent that in symbols — perhaps as ∀a∀b | a, b ∈ ℜ, a+b=b+a — your eyes glaze over with symbol fatigue. So we want to abstract — we need to abstract — but we can’t overwhelm our students with too much symbolic representation. If we don’t abstract, we run into the trap that Robert Pirsig outlines in Zen and the Art of Motorcycle Maintenance: “data without generalization is just gossip.” But when we abstract, things can become difficult. Almost everyone can cope with the fact that our sketch of a triangle isn’t really a triangle — and we read Plato’s Allegory of the Cave to provide a framework for discussing this — but it’s harder when we tell students that they can’t just draw a right triangle when the problem refers to “triangle.” A right triangle is “a triangle,” isn’t it? And it’s worse when we deal with the so-called real world. Recall Mandelbrot’s famous observation about Euclidean geometry:

Clouds are not spheres, mountains are not cones, coastlines are not circles, and bark is not smooth, nor does lightning travel in a straight line.

Things still have meaning when we abstract, but the meanings can be hard to see. Worse yet, symbols can have multiple meanings. No wonder students get confused about expressions like a(b+5); if is a number, this expression means to multiply the value of by five more than the value of b, but if is a function, it means to apply the function to the results of adding 5 to b. And why do science teachers tell students that they must use units religiously, whereas math textbooks and teachers use them only occasionally, mostly in applied problems? There’s a really good reason for this: almost all formulas in science are meaningless without units, whereas a statement like “3, 4, and 5 can be sides of a right triangle” gains its power from its unitless abstraction. But it seems difficult and arbitrary to many students.

And what about variables? We said that algebra is abstracted arithmetic, and that’s mostly because it represents relationships between variables. Yet it can seem very confusing. The sentence y = mx appears to contain four variables, but two of them are really parameters: the values of and can still vary, but they’re constant when describing any particular line. Confusion sets in! It’s all so abstract.

But abstraction is necessary and crucial. Marianne Moore famously described poetry as “imaginary gardens with real toads in them,” but she could just as well have been describing mathematics. Too many unsuccessful students are so concrete that they see only the toads and never the gardens.

Abstraction requires moving from the specific to the general. Even numbers are actually abstract. You can’t go out and find an instance of the number 4; you can only find collections of four objects. And then what about fractions? More abstract still. And negative numbers?? And irrational numbers??? And complex numbers???? Yikes!

Even maps (which I mentioned above when discussing representation) are truly abstract. Straight lines aren’t really straight. The width of roads is almost never drawn to scale. All details (of necessity) are omitted. Subway maps are the best example of this principle, where distances and even directions can diverge from reality. In fact, all maps are distortions: the famous Mercator projection distorts areas in order to preserve directions. If you prefer, you can distort directions in order to preserve areas, but you can’t win.

OK, enough of abstraction. Now on to quantitative thinking. Part of the difficulty lies in the fact that numbers tell the truth, but statistics can lie. Grasping this duality is truly important; a well-informed citizen needs to understand statistics from the government and from business. Numbers can be reliable evidence — as long as they aren’t distorted or simply made up. At Weston we have a wonderful Risk Assessment unit in our course called “Applied Discrete Math Concepts”; people are notoriously bad at estimating relative risks, and this course helps students get better at it. Because people have seen so many badly constructed statistics, both intentional and unintentional, they tend to distrust all statistics and refer (in Mark Twain’s famous words) to “lies, damned lies, and statistics.” Quantitative thinking at least seems like a familiar idea to people — after all, high-school math ought to be about numbers — but in fact most people are bad at all aspects of it, not just estimating risks. People believe what they want to believe, no matter what the numbers show.

Finally, we get to reasoning, a.k.a. logic. I have come across logic in a remarkable variety of disciplines: not just math, but also philosophy, computer science, and even psychology. Of course logic is used as a tool in almost any discipline — for instance, in evaluating evidence in history, English, and linguistics — but it’s an object of study in and of itself only in math and philosophy. In many universities, in fact, courses on logic carry credit in both those departments.

Logical reasoning comes in two flavors: inductive and deductive. Many people claim that science is inductive and math is deductive. There’s some truth to that, but it’s a vast oversimplification. What’s true is that only deductive reasoning is accepted as evidence when presenting a mathematical proof. Deductive reasoning can take many forms, including the famous Sherlock Holmes dictum (“when you have eliminated the impossible, whatever remains, however improbable, must be the truth”) which represents a form of indirect proof. We also have the misleadingly named mathematical induction, which is actually a form of deduction —  but it’s called induction because the steps that begin the proof are indeed inductive. And then we have analytic geometry proofs, which are so algebraic that they may not feel like proofs at all. In precalculus we prove trigonometric identities in ways that don’t look at all like geometry proofs, but they still use deductive reasoning. Underlying all of these is the big idea that a conditional is equivalent to its contrapositive but not to its converse — a deep and subtle point that often trips students up.

So is logic part of math? Or perhaps, as Bertrand Russell believed, math is part of logic! In any case, the trouble with all sorts of related disciplines is that we want to build bridges, not fences. Math and science are different disciplines, but sometimes we want to bridge the differences. Math and philosophy are different disciplines, but logic is a bridge between them. What about statistics? Is that part of math? Well…sort of. It is almost always included with high-school math departments (and often in universities as well), even though, as one of my students observed, it isn’t really math. (“Why is this formula correct?” “Because it seems to work.”) Statistics definitely relies on inductive reasoning more than deductive. And yet it’s appropriate to think of it as applied mathematics. The same holds with cryptography, which Weston classifies as part of Algebra II. It’s not exactly math, but (like statistics) it’s a branch of applied math. As Ted Nelson observed, “everything is deeply intertwingled.”



Categories: Math, Teaching & Learning, Weston