In the early 1990s I was working at ARCO Chemical Company in the suburbs of Philadelphia. Times were good at ARCO, although that would change. We were making money, dress was formal, and we devoted time and resources to supporting the local community. As part of that effort, I was asked to host a group of high school science and math teachers for a day. I gave them a tour, fed them lunch, and provided a description of the types of science and math skills which were used in my department. At the end of the day one of the teachers asked me this: “We are told by our administrators that we need to train our science and math students to work on collaborative problem solving and team work techniques. Is this important to you?” I paused for a moment and responded, “Really what we need is for you to teach them to get the right answer. You can leave the teamwork lessons to us.” The room erupted into whoops and cheers. It was kinda nice.
If anything, the question of how best to educate our budding scientists and engineers is even more critical today than it was during that long ago session with those teachers. And at least with regard to educating our strongest math students, I think we are on the wrong track.
All of us are born with different subsets of aptitudes. Just like some people are born with a talent for dancing (1) some people are naturally good at math. Math aptitude can be predicted at an early age using some simple tests. For example, if you ask a preschool child look at five coins sitting on a table top and ask her how many there are, a high-performing math student will look at them, see the pattern, and say five. An average-performing math student will count them. Psychologists call these pattern-perceiving skills number sense.
If you want to know if your child has good number sense, play a game of monopoly. If, after rolling the dice, your child moves her playing piece around the board based on the recurring patterns of corners and railroads (it’s all fives and tens) there is strong possibility that attending math olympiads is part of your future. If instead she counts the spaces off one at a time, some different sort of time-consuming parental obligation awaits.
While all students need an appropriate and challenging math education to prepare them for a full and prosperous life and career, the remainder of this column will focus on the education of the high-performing math students who may go on to be scientists and engineers. As I have foreshadowed, I am concerned about how this is being managed.
In order to keep this column at some sort of reasonable length I am going to focus on just one critique(2), the current obsession with making the student “explain how she got the answer”. If you happen to be born with good number sense, this question can be particularly vexing. For example, let’s say I asked you to solve two plus two. You know the answer is four. Now try to explain how you got the answer. Difficult, isn’t it? It’s tempting to say “because it’s four”. For high-performing math students, problems which may sound complex to a general audience, read just like “what is two plus two?” Forcing these students to construct a sentence or two to describe the process that occurred in their head is a frustrating waste of time.
As an analogy, let us again consider someone with a natural ability to dance, someone who can feel the music, anticipate where the music going, and then move his body in a way that evokes the essence of the piece. Imagine now that in the middle of a particularly moving segment of choreography we turn off the music, make the dancer stop, and require him to write down why he chose to move his left foot in front of his right rather than behind. I suspect that it would be difficult for him to explain, detract from his joy of the dance and in no way help him to be a better dancer. It’s no different for the mathematician. You can’t see it, but during the calculations something beautiful is happening in her head. Making her stop to reduce the intuitive to prose is just as disruptive and non-productive as interrupting the dancer.
Unfortunately the authors of the Common Core seem not to agree with me. Here is a passage that appears in the introduction of the new math standards.
There is a world of difference between a student who can summon a mnemonic device to expand a product such as (a + b)(x + y) and a student who can explain where the mnemonic comes from. The student who can explain the rule understands the mathematics, and may have a better chance to succeed at a less familiar task such as expanding (a + b + c)(x + y). Mathematical understanding and procedural skill are equally important, and both are assessable using mathematical tasks of sufficient richness.
The mnemonic being impugned here is FOIL (first, outside, inside, last) which describes which pairs of variables are to be multiplied in (a + b)(x + y). I know how to do this problem but if asked to “explain” this rule, frankly, I would have no idea what to write.
Instead, let’s talk about what “mathematical understanding” means. It is my firm position that it has a lot more to do with the pattern recognitions involved in determining number sense than in being able to write a sentence about the mechanics of division. Further, I believe that math education for those with good number sense should strive to help grow and improve these skills. Let me use the problem above from the Common Core as an example of a pattern recognition approach to math.
At some point early in your math education someone taught you to do the problem below.
a(x +y) = ax + ay
Not long after you learned the one below and were taught the mnemonic FOIL.
(a + b)(x + y) = ax + ay + bx + by
Let me pause to make something clear at this point. In addition to being opposed to making students write a paragraph about how to solve this problem, I am also no fan of mnemonics. The problem above is the solution for how to multiple two binomials. If you can see the underlying pattern, then it is broadly applicable to other situations. This ability to see patterns and apply them to other problems is the essence of mathematical understanding.
Look at the patterns in the two solutions above and now consider the problem (a + b + c)(x + y) which the Common Core holds out as difficult. But if you’ve learned to do the two problems in the previous paragraph, the answer is obvious, whether you have good number sense or not. Furthermore, you also know now how to do (a + b + c +d + e)(x + y + z) or any other problems like this with any number of terms within the parentheses whether or not you’ve ever written a paragraph about it.
Here in the Chapel Hill Carrboro City School District we have a mantra of reaching each child at his or her own level. For our students with good number sense (3) we need to find a way push the Common Core aside and let them get back to dancing.
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(1) Sadly your author was born without this aptitude.
(2) In an earlier, much longer version, of this column I included a lengthy diatribe about math vocabulary. It seems as though over the last several decades text book authors have decided that every step in a mathematical solution should have a unique name. To me this just clutters things up and detracts from math education. It also included a suggestion that calculators be banned from all grades below ninth.
(3) Because I am a person with good number sense, I feel more confident in critiquing the applicability of the Common Core to others with good number sense. However, I would like to share with you my suspicion that its approach is a misfire for all students. Maybe a bit more focus on procedural skill would be a better pathway towards mathematical skill for all.