Response to a NYT editorial

A recent editorial in the New York Times:

presents a plan to “fix math education.”

I’m disappointed.

The arguments are variations on themes that have surfaced (and been debunked) many times over the past century.

First of all, the authors set up a straw man in their characterization of Common Core: a codification of school mathematics as a highly abstract program, devoid of context, that introduces “the mysterious variable x, which many students struggle to understand.” In fact, a major difference between Common Core and the state standards that have plagued us for at least a generation is that Common Core strives for meaningful mathematics, where algebra is understood as a way of expressing generality and precision. The very first standard in the Common Core states that “Mathematically proficient students start by explaining to themselves the meaning of a problem and looking for entry points to its solution.” For one example, and there are many, the Common Core says, “Reading an expression with comprehension involves analysis of its underlying structure. This may suggest a different but equivalent way of writing the expression that exhibits some different aspect of its meaning. For example, p + 0.05p can be interpreted as the addition of a 5% tax to a price p. Rewriting p + 0.05p as 1.05p shows that adding a tax is the same as multiplying the price by a constant factor.” Understanding that different recipes for calculating the same thing can have different uses (in this case, 1.05p is much more efficient than p + 0.05p, something that’s important in computer calculations) shows the power of abstract reasoning in calculation systems.

This simple example also debunks another claim of the editorial: “There is a world of difference between teaching `pure’ math, with no context, and teaching relevant problems that will lead students to appreciate how a mathematical formula models and clarifies real-world situations.” Common Core is full of examples of ways to apply algebra and geometry to all kinds of situations.

What the authors really object to is the way Common Core organizes its standards around mathematical themes and conceptual categories. But this is precisely what makes the standards so compelling. Common Core describes an approach to precollege mathematics based on mathematical coherence, showing how a small number of general-purpose ideas can be used to build a textured and intricate edifice of results and methods that have utility all over mathematics, science, and everyday life. The editorial asks us to “Imagine replacing the sequence of algebra, geometry and calculus with a sequence of finance, data and basic engineering.” I just did the exercise and concluded that the result would be a program organized around a set of special-purpose techniques that would likely be out of date by the time students graduate college. What’s more, to get them to use the specialized tools, we’d have to include some shallow and formulaic versions of classical topics like geometric series, expected value, and recursive algorithms.

This stance about organizing programs around mathematical themes isn’t just based on the experience of my colleagues and me. Schmidt and others have analyzed curricula in “high performing” countries and they describe the curricula there as “organized around the ways that ideas are organized within the discipline.” That’s what one finds internationally.

The editorial ignores another distinguishing objective of Common Core: Equipping students with what the CCSS calls standards for mathematical practice, and what we at EDC call mathematical habits of mind. This gives them, as it has for centuries, the mental wherewithal to deal with problems that don’t yet exist.

When I started teaching high school, I thought that mathematics was an ever-growing body of knowledge. Algebra was about equations, geometry was about space, arithmetic was about numbers; every branch of mathematics was about some particular set of mathematical objects. Gradually, I came to realize that what my students (some of them, anyway) were really taking away from my classes was a style of work that manifested itself between the lines in our discussions about triangles and polynomials and sample spaces. I began to see my discipline not only as a collection of results and conjectures, but also as a collection of habits of mind.

This realization first became a conscious one for me when my family and I were building a house at the same time that I was researching a problem in number theory. Now, pounding nails seems nothing like proving theorems, but I began to notice a remarkable similarity between the two projects. It’s not that house-building requires applications of results from elementary mathematics (it does, by the way), but that the two projects required the same kinds of thinking. In both theorem-proving and house-building, you perform thought experiments to visualize things that don’t (yet) exist, to predict results of experiments that would be impossible to actually carry out, to tease out efficient algorithms from seemingly ad-hoc actions, to deal with complexity, and to find similarities among seemingly different phenomena.

This is not to say that other facets of mathematics should be neglected; questions of content, applications, cultural significance, and connections are all essential in the design of a mathematics program. But reorganizing a mathematics program around this or that set of topics or applications is just the wrong way to do it.

Then there’s the question of student engagement. I get nervous when grown-ups try to predict what will hook kids. The only general rule is that students, like all people, get great satisfaction from figuring things out. My colleague Paul Goldenberg talks about the popular puzzle books that one finds for sale at the checkouts of many supermarkets. Deborah Schifter, another colleague, likes to say that people take delight in their own mathematical thinking. At the high school level, I’ve seen over and over how students—all kinds of students—get hooked on something just because it engages their intellect.

In the field tests of early versions of our precalculus course, we held an advisory board meeting of high school juniors and seniors (many of them “very weak” in terms of traditional measures). This was at the end of the first term; up to that point, students had been experimenting with recursively defined functions, modeled in a CAS on their calculators, finding closed forms for such functions, proving that their closed forms and the recursive models were equal on the non-negative integers by mathematical induction, and then doing a bit with Lagrange interpolation. At the meeting, Wayne Harvey, another project member, asked the question, “How is this mathematics course different from others you have taken?” Four kids answered, almost in unison, “It’s more realistic.” That response was startling, even to us, because “realistic” is usually taken to imply everyday or other “real world” contexts, and the activities the kids were talking about were purely mathematical. But what the students meant was that it felt more like real work, more like the kind of thinking they must do when they are solving a real problem. What mattered was that they got a chance to exercise their own creativity. What mattered was how, not where, their mathematics was used.

This issue of viable and engaging contexts is complicated for a couple reasons.  Many of the students in my high school classes came from situations that many of us would find hard to imagine; the last thing they cared about was how to balance a checkbook or figure the balance on a savings account. But they loved solving problems. For another thing, reality is relative. The authors claim that “it is through real-life applications that mathematics emerged in the past, has flourished for centuries and connects to our culture now,” and I agree. But the best mathematicians and scientists I know, and the students in my classes who really got it (and these were not necessarily the “good students”)—see the power and satisfaction one can derive from doing mathematics—all see mathematics as part of their real world.


Counting with Polynomials

This is to expand on a recent comment I made to “Factoring non-monic quadratics.”

We hear a lot these days about “modeling with mathematics.”  One aspect of this is the use of formal calculations as a modeling device.  An example, pretty common in many high school programs, is the use of the binomial theorem to compute combinations.  So, you can use (t+h)^{10} to compute the number of ways that you can get, say, 4 heads and 6 tails when you toss a fair coins 10 times.   Another example is when you use powers of a matrix to get Fibonacci numbers.  The calculations not only give you answers, they let you derive properties of the phenomena they model.

One of my favorite uses of formal calculations comes from a project that my daughter gave to her sixth grade class some year ago: compute the most likely sum (and the distribution of sums) when several dice are thrown.  For two dice, the problem is fairly standard, and most kids make a 2 by 2 table of all possibilities.  For three dice, her students invented all kinds of clever representations, some very algebraic — not in notation, but in spirit.  This triggered an idea that has become a recurring theme in our high school program.  It goes like this:

If you use expansion boxes to multiply polynomials, you see that the expansion of


contains exactly the same numbers as the 2-dimensional table that records the possible outcomes when you roll two dice.  In other words, the number of ways of rolling a 5 when you throw two dice is the coefficient of x^5 in


It’s the number of ways you can make 5 as a sum of two integers, each between 1 and 6.  The same reasoning shows that the coefficient of x^k in


gives the number of ways you can roll a k when m dice are thrown.

From here, you can use the structure of the expression (x+x^2+x^3+x^4+x^5+x^6)^m
to get results about the distribution of sums.  For example:

  • There are 6^m possible sums, because this is the sum of the coefficients, and the sum of the coefficients comes from putting x=1
  • There are as many even sums as odd sums (replace x by -1).
  • The distribution of sums on three dice whose faces are labeled \{0, 2, 3, 4, 5, 5\}, \{0, 1, 1, 2, 2, 2\}, \text{and} \{1, 2, 3, 6, 6, 6\} can be read off from the coefficients of the product of three different polynomials.
  • Various factorizations of (x+x^2+x^3+x^4+x^5+x^6)^m give you different information about the distributions.  One interesting thing to try is to rearrange the factors of (x+x^2+x^3+x^4+x^5+x^6)^2 to try and produce two dice, different from the standard ones, whose distribution is the same as the standard distribution.  (This may or may not be possible, we’re not telling.)

All this is a preview, accessible to high school students, of the incredibly useful field of algebraic combinatorics and generating functions.