Math Ability

I ended up in a fascinating discussion with several smart and thoughtful people who disagreed with this tweet from Rochelle Gutierrez’s ShadowCon talk:

It also happened to be the most retweeted of everything I blabbed during #ShadowCon16 last night, so it seems to have touched a nerve.

If I were to be so bold as to summarize the arguments that math ability exists:

• If we deny the existence of math ability, we are then denying that all talent exists. What about football?
• Experience shows that some kids have a larger ability to learn math.
• Saying math ability doesn’t exist sounds fishy to students and is unlikely to be meaningful.
• Believing in math ability doesn’t mean we’re setting kids up to fail; it just acknowledges that something exists.
• If math ability doesn’t exist, then everyone must have the same cognitive limit in math, and we know that’s not true.

A significant element of the argument that math ability exists is based on teachers’ intuition and experience. It’s certainly my experience as well; my time teaching a range of students suggests that some have more ability than others. But it has become a value of mine not to trust my intuition. Teaching is not intuitive, and my intuition is often wrong.

Here’s an example that strikes me as both counterintuitive and relevant to what we’re talking about: In Anders Ericsson’s recent book Peak: Secrets from the New Science of Expertise, he discusses “perfect pitch”. One in ten thousand people has this skill, the ability to hear a note on a musical instrument and name it, for instance the A-sharp in the second octave above middle C. It would seem, in popular opinion, to be a great example of a talent. There was some evidence that it wasn’t a pure talent — for instance, every person who has perfect pitch received some music training at a young age, and it is much more common in people who speak tonal languages like Mandarin, Vietnamese, and other Asian languages. Then, this:

We now know that this isn’t the case, either. The true character of perfect pitch was revealed in 2014, thanks to a beautiful experiment carried out at the Ichionkai Music School in Tokyo and reported in the scientific journal Psychology of Music. The Japanese psychologist Ayako Sakakibara recruited twenty-four children between the ages of two and six and put them through a months-long training course designed to teach them to identify, simply by their sound, various chords played on the piano. The chords were all major chords with three notes, such as a C-major chord with middle C and the E and G notes immediately above middle C. The children were given four or five short training sessions per day, each lasting just a few minutes, and each child continued training until he or she could identify all fourteen of the target chords the Sakakibara had selected. Some of the children completed the training in less than a year, while others took as long as a year and a half. Then, once a child had learned to identify the fourteen chords, Sakakibara tested that child to see if he or she could correctly name the individual notes. After completing training every one of the children in the study had developed perfect pitch and could identify individual notes played on the piano.

This is an astonishing result. While in normal circumstances only one in every ten thousand people develops perfect pitch, every single one of Sakakibara’s students did. The clear implication is that perfect pitch, far from being a gift bestowed upon only a lucky few, is an ability that pretty much anyone can develop with the right exposure and training. The study has completely rewritten our understanding of perfect pitch (xiv-xv).

Obviously music is not the same as math, but this is a pretty striking result. More important to me, it is enormously counterintuitive. Something that I believed to be a talent, that surface evidence suggests is a talent, seems not to be. This doesn’t prove that every student has the same ability in math any more than that study about hippocampus growth in London black cab drivers proves it. I don’t mean to argue that the cognitive science is settled here.

But Ericsson has studied a wide variety of skills — memory, music, chess, vision, athletics, medicine, and more. Another snip from his book:

Whenever you’re trying to improve at something, you will run into such obstacles — points at which it seems impossible to progress, or at least where you have no idea what you should do in order to improve. This is natural. What is not natural is a true dead-stop obstacle, one that is impossible to get around, over, or through. In all of my years of research, I have found it surprisingly rare to get clear evidence in any field that a person has reached some immutable limit on performance. Instead, I’ve found that people more often just give up and stop trying to improve.

One caveat here is that while it is always possible to keep going and keep improving, it is not always easy. Maintaining the focus and the effort required by purposeful practice is hard work, and it is generally not fun. So the issue of motivation inevitably comes up: Why do some people engage in this sort of practice? What keeps them going? (21).

My intuition tells me that some students have greater ability in math, and it’s hard to swallow, in the face of that intuition, that math ability doesn’t exist. But it also squares with my experience that there’s pretty enormous variance in motivation to think about and do math. The youngest age I’ve taught is 5th grade in a student teaching placement. Those students had already had five years of math instruction, and many more years where thinking about number and size and shape was (or wasn’t) a daily part of their lives.

I don’t mean to be intransigent. I realize that arguing math ability doesn’t exist is an extreme position. But there are lots of other things that absolutely do exist. Motivation exists. Much of it is a mystery to me, but I also know that my actions as a teacher impact students’ motivation in profound ways. Deliberate practice exists, and it’s pretty challenging to get students to be deliberate in their practice on a regular basis. Growth mindsets exist, and have a huge impact on students’ present and future disposition to learn. These seem like much more worthy areas to focus, where we know students vary enormously, and that we can do something about.

It’s worth clarifying my position here. I’ve seen no evidence that convinces me of the existence of math ability, and some evidence that suggests to me greater potential for all students than I had previously thought. Maybe that will change in the future, and if the evidence is convincing, I’m willing to change my view. I’m also not going to go around trying to convince every student there’s no such thing as math ability. I’ve tried before, by way of Jo Boaler’s videos, and found that they didn’t land with students. Similarly, my experience suggests that it’s not worth the effort trying to convince parents, or even teachers of other subjects who have entrenched negative views of math. But I think that Rochelle Gutierrez was right, in the sense that believing in math ability, and acting based on that belief, can facilitate traumatic experiences with math for some students. If something squares with my knowledge of the evidence, and has the potential to positively impact students, it seems like it’s worth believing to me.