Response to a NYT editorial

A recent editorial in the New York Times:

http://www.nytimes.com/2011/08/25/opinion/how-to-fix-our-math-education.html

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.

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Factoring Non-Monic Quadratics

Regardless of the course, non-monic factoring was always a thorny issue in my teaching.  My students never seemed to get “good” at it, even though they seemed alright at monic factoring (“monic” just means the first coefficient is 1, like x^2 + 14x + 48).  This topic made me really question why I was teaching it, for several reasons:

  • The very next thing was the quadratic formula, and for most of the things non-monic factoring could be useful for, I felt the quadratic formula would be just as good.
  • I couldn’t find many places later in my curriculum where non-monic factoring was being used, so it seemed like a topic taught for a single purpose.
  • The methods I saw for factoring non-monic quadratics had little or nothing to do with the methods for factoring monic quadratics.
  • The method I learned and first taught amounted to trial-and-error.

And maybe you know this method too: to factor 6x^2 + 31x + 35, you write down all the factors of 6, separately write down all the factors of 35, and start making pairs.  Eventually you either find the pair that works, or you run out of pairs:

(x+1)(6x + 35) = 6x^2 + 41x + 35, nope

(x+5)(6x + 7) = 6x^2 + 37x + 35, nope

(x+7)(6x + 5) = 6x^2 + 47x + 35, nope, keep trying…

And I was polite in picking 6 and 35 here, two numbers with only two prime factors each!  I feel this method is a mathematical nightmare.  Keep testing, keep checking.  And don’t give the ones that aren’t factorable, since the only way to know it’s unfactorable is to test all the possibilities, and that’s just mean.

A year or two later, I learned and taught the “key number method”: multiply the coefficients of “a” and “c” (6 \cdot 35 = 210).  Then you break up the middle term (31x) into two pieces whose coefficients multiply to 210:

6x^2 + 31x + 35 = 6x^2 + 10x + 21x + 35

Then “group” in pairs and a miracle occurs:

(6x^2 + 10x) + (21x + 35) = 2x(3x + 5) + 7(3x + 5)

= (3x + 5)(2x + 7)

This worked a lot better for my students, by which I mean they got correct answers faster and with greater accuracy.  But the core of this method is the “miracle” that splitting the 31x in this exact, specific way will do great things.  It works because it works.  (There are better explanations, but my students just memorized what to do.)

One advantage of the key number method is it can be applied to monics, too, visualizing the “sum and product” concept:

x^2 + 14x + 48 = x^2 + 6x + 8x + 48

= x(x+6) + 8(x+6)

= (x+6)(x+8)

But this generally comes after the fact: I didn’t teach students to factor monics in this way.

While working on CME Project, I learned (through Al Cuoco and Jeremy Kahan, a field test teacher) about a “scaling” method that uses monic factoring as the core of non-monic factoring.  It feels a lot more natural, cements monic factoring, and fits tightly with Mathematical Practice #7, “Look for and make use of structure.”  It starts with specific non-monics like this one:

25x^2 + 70x + 48

Try factoring that for a second using either of the methods presented above.  It’s messy!  But, would you believe this is actually a monic quadratic?  It’s just got a different variable: 5x.

25x^2 + 70x + 48 = (5x)^2 + 14(5x) + 48

Now cover your finger over each 5x: it reads finger^2 + 14 \cdot finger + 48.  Doesn’t matter what’s under the finger: it factors!

F^2 + 14F + 48 = (F+6)(F+8)

And you’re done when you lift your finger, remembering that F = 5x.  In teaching, I used capital letters for these substitutions, to remind students that there was more work to be done later.

(5x+6)(5x+8)

How fast was that?  And understandable, too!  The core concept of a replacement of variable (the book calls this “chunking”) plays forward deeply into later topics and courses: when I say I used capital letters for substitutions, I generally was doing that with Precalculus or Calculus students, but the concept can be seen much, much earlier.  By using it frequently, it becomes a tool students actively look to use when they see something complicated.

But I fudged the example: it’s got 25x^2.  How about that original one, 6x^2 + 31x + 35?  It doesn’t have a perfect square term, but … wishful thinking … we can make one by multiplying through by 6, then paying it back later.

6(6x^2 + 31x + 35) = (6x)^2 + 31(6x) + 210

\mathbf{= F^2 + 31F + 210}

\mathbf{= (F+10)(F+21)}

Note that this method includes the step that was part of the “key number method”: the 210 is produced by multiplying the coefficients of “a” and “c”, but this time there is a more mathematical reason for doing so.  And the payoff is the same, since we then need two numbers that add to 31 and multiply to 210 — but we use the monic factoring method to perform that step.  This cements monic factoring skills, as it becomes part of the process in the later topic.  And now the miracle, as we have common factors in the two right-side terms:

6(6x^2 + 31x + 35) = (6x+10)(6x+21)

= 2(3x+5) \cdot 3(2x + 7)

= 6(3x+5)(2x+7)

And now you zap the 6 from each side and you’re done:

6x^2 + 31x + 35 = (3x+5)(2x+7)

It was shocking to me that this method works at all, and especially shocking that it always works: any factorable non-monic quadratic can be dealt with using this method.  And variable substitution is a natural method used in other places: completing the square is a variable replacement using \left(x - \frac b 2\right) as the variable … x^4 - 1 is a difference of squares … circles and ellipses all relate to the unit circle x^2 + y^2 = 1… trigonometric equations are just regular equations when you cover your hand over the “\sin x” part … a z-score is a linear substitution … and others.

The biggest benefit of presenting substitution methods as early as possible is that students learn a general-purpose tool they can apply repeatedly across grades and topics.  I also think it makes quadratic factoring easier and faster to teach.  What do you think?

Next: how this method can be used to develop the quadratic formula…

(If you know how to better display equations easily in places like this, let me know.  The LaTeX equations look pretty bad in the vertical alignment category, and I had to force a white background on each equation.  As long as it’s readable, I guess, but somehow I think it could be better.  Thanks to Mark Betnel for the pointer to the LaTeX commands available.)