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:

Screenshot 2016-04-16 at 7.25.38 AM

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.

12 thoughts on “Math Ability

  1. N C

    Found your blog by way of #MTBoS and #NCTMannual.

    Thanks for getting this conversation started! I’m curious what is meant by “ability” (by Rochelle, by the general public, by people who disagreed with your tweet).

    I interpret Rochelle’s quote about “mathematics ability” as “being good at/able to do mathematics” and I would push back against the idea that some students have more “mathematics ability” for reasons similar to the perfect pitch example – some people have more practice and experience with mathematics, which allows them to more quickly see patterns, make connections, make sense of quantities, but they are not inherently “better” at mathematics – they are “more able” to do mathematics because they have more (or stronger) background knowledge and confidence. If other students who haven’t had those opportunities and practice did have those opportunities, would they be in the same place? Maybe not, but I’m guessing they would be in a stronger place than they currently are.

    Also, correct me if I’m wrong – it sounds like much of the contention was around whether ability exists (as opposed to mathematics trauma existing), yes?

    (For context, I teach at a small, district school with 100% emerging bilinguals/English Language Learners. Quite a few of our students have gaps in their formal education and it’s FASCINATING to see who “catches up” to where we should be quickest).

    Thanks again for getting this conversation started!

    1. dkane47 Post author

      I think you’re right that an important part of this conversation is defining what we mean by ability (that was the part that was disagreed with, not the part about trauma). I wonder if we could usefully break down different things that get conflated together — learning something faster, engaging more deeply with mathematical thinking, having a growth mindset about math learning, and more. I wonder if breaking down those different pieces and avoiding the word ability would help to resolve some of these arguments that tend to get lost in semantics.

      You offer a useful thought experiment, and I don’t know the answers to those questions — but I agree with Rochelle that the trauma associated with certain beliefs is real, and something that I can work to avoid.

  2. Mark Pettyjohn

    Nature vs. Nurture
    That’s what is being debated. The word ability is a red herring. Abilities can be developed. So of course, someone like James Tanton has much more math ability than I do. The question is, did he come out of the womb destined to create exploding dots, or did he work his ass of to create them or a mixture of both?

    Nature or nurture?

    Sports ability is easy to see. If you have the genetics where you end up with a body like LeBron James, then nature is in your corner. We don’t have simple, reliable measuring sticks like that for cognitive abilities. If it’s an equation, nature + nurture = ability then for measuring cognition we’re operating nearly blind. To try to unravel the makings of James Tanton’s math abilities that are on display in his presentations and in his work would be impossible. In all but fringe cases, extremely high functioning (think: 2 year old composing symphonies), and extremely low functioning, determining nature’s responsibility for an ability is beyond our current means.

    Picture a pendulum. Maybe statements like Rochelle’s are inaccurate. Yet she’s pushing the pendulum to the opposite extreme of using nature and the idea of innate math ability as a cop out for us educators to use. Each approach is bound to create incentives with second and third order effects. I’m much more inclined to work on the end of the pendulum Rochelle is pushing towards and keeping an eye open for second and third order effects. It’s a hopeful, nurturing mentality and if we want to be in the business of education is there really an alternative?

    1. dkane47 Post author

      Thanks, Mark — I think two of those perspectives, the pendulum and the second and third order effects, are particularly useful ways to frame this debate that is likely to be more productive than arguing about “ability” in the abstract. There’s a lot that we don’t know, but despite not knowing, the way we frame those things matters.

      The point about extremely high and low functioning students is an interesting one, and a subtlety that often gets lost in this debate. It’s not one I feel knowledgeable enough about to speculate on.

  3. Simon Gregg

    French was the one subject I failed in the exams I took at sixteen. In fact, a lot of us failed it. There were a few kids who were really good at languages. My friend Nigel seemed to home in on Scandinavian people and miraculously seemed to soak up the languages. Maybe the rest of us just weren’t very good at languages?

    But in 2003 I came to France, and discovered something – everyone speaks French here! My son started going to French school and soon was speaking English and French like a native, cringing at my mistakes in French!

    We switch perspective, from how it was done at my school, to how it happens in the best circumstances… and “ability” vanishes. Everyone speaks it!

    In terms of sports players, I think of the research Malcolm Gladwell reported in Outliers: how for instance 40% of Canadian hockey players are born between January and March, and only 10% between October and December. Something big apart from “nature” is at work here. The “nurture”, according to Gladwell, and it figures, is that those people were the oldest in their year group and so got picked for their teams because they were the “best”.

    1. dkane47 Post author

      I had a similar experience with singing. Didn’t think of it as a skill I could get better at for a long time. I’m still not a very good singer because I don’t practice much, but I’m better than I was.

  4. cheesemonkeysf

    Some ramblings and reflections here.

    I think there is too much conflation between and among the ideas of “ability,” proficiency, and mastery.

    “Ability” feels like a raw capability. Just because you have it doesn’t mean you have developed it. And conversely, proficiency or mastery in something can be developed even with very little native “ability.” There are plenty of people in the world who are very proficient musicians but who might not have the same level of deep, inherent “ability” as others. Some people can play the piano fluently with very little training or practice. Others achieve the same level of fluency with a moderate or high level of training or practice.

    The same is true of math, writing, drawing, basketball, cooking.

    It seems to me that we need to get away from the idea of “ability” because it is just not that helpful. The categories of proficiency and mastery, however, are far more helpful to me both as a teacher and as a learner. I definitely have less “ability” in mathematics than many of my peers. But what I lack in “ability,” I more than make up for in motivation and in my long-cultivated skills as a learner. Being someone who is good at learning how to be good at things has benefited me tremendously as a learner and teacher of mathematics.

    I see a lot of confusion about whether a student has been labeled as a “high-ability learner” as opposed to “a learner who has attained proficiency or mastery of a given set of skills and concepts.” This feels like a much more productive distinction to me because as movie mogul Samuel Goldwyn said, “The harder I work, the luckier I get.”

    – Elizabeth (@cheesemonkeysf)

    1. dkane47 Post author

      I definitely agree on the confusion piece. The more conversations I’ve had as a result of this post, the more I think that a better argument would just be: Whether or not ability exists, talking about ability isn’t useful — there are so many more things that are within our control and easy to mistake for ability, and it’s unproductive to do so.

  5. Pingback: Why Students Don’t Learn | Five Twelve Thirteen

  6. Tooting Hustle

    I agree that there is a talent involved with maths ability and it can go to great levels. I do also feel like there is a trauma involved. I did maths A level and would panic when a certain topic or question type would come up. My mind would blank despite being a capable student.

    I find it a very interesting concept, thank you for the great read. Check out my blog too for some interesting or comical reads.

    Happy blogging x


Leave a Reply

Fill in your details below or click an icon to log in: Logo

You are commenting using your account. Log Out /  Change )

Google photo

You are commenting using your Google account. Log Out /  Change )

Twitter picture

You are commenting using your Twitter account. Log Out /  Change )

Facebook photo

You are commenting using your Facebook account. Log Out /  Change )

Connecting to %s