Doug Lemov's field notes

Reflections on teaching, literacy, coaching, and practice.

12.04.14On the False Dichotomy in Math

Fascinating post from William Schmidt, Distinguished Professor at Michigan State University and co-director of the Education Policy Center.  Schmidt compares the differences between the PISA and TIMMS assessments.  In a nutshell, he notes the

Major distinction in what the two tests purport to measure: the TIMSS is focused on formal mathematical knowledge, whereas the PISA emphasizes the application of mathematics in the real world, what they term “mathematics literacy.”

In other words the TIMMS is closer to what many people criticize about assessment. It measures students ability to solve problems in discrete areas with proficiency.  This, skeptics note, is not like the real world where problems require intuition and mathematical literacy–“number sense.”  I’m glad there are assessments of both and I think we should always try to make our assessments more rigorous–rather than trying to use questions about the rigor of some questions as a rationale for not assessing, say, but  it’s hard to underestimate how simple and important what Schmidt observes is. (Or what it may be. See below for a major caveat).

Schmidt notes:

After analyzing the new PISA data, we discovered that the biggest predictor of how well a student did on the PISA test was exposure to formal mathematics. This is a notable finding, to be sure, since the PISA is designed to assess skill in applied rather than formal math. Exposure to applied mathematics has a weaker relationship to mathematics literacy, one with diminishing marginal returns. After a certain point, more work in applying math actually is related to lower levels of mathematics literacy.

In other words what Schmidt finds is that the thing that most enables students to succeed in applied math is the strength of their foundation in “formal math.”  Surely they should also get experience in applied “real world” math, but that experience ,must supplement and not supplant the formal foundation.   In other words, if you skip the formal and just do applied math you get very little math.  And if you try to make math more rigorous by over-weighting the applied thinking you may, in the end, get reduce outcomes.

Why is this so important?  Well, there’s a bit of a tendency in US math, you’ve probably noticed, to pit “higher order thinking” or “conceptual thinking” or “number sense” against formal math which, in such situations, is often caricatured as “rote learning” or, at its most pejorative, “drill and kill.” But the two “opposites” are actually not just synergistic but inter-reliant.  Everyone loves to admire a shining tower but the building depends on the foundation (and I should note, a foundation ain’t much without a building), so foundation versus superstructure arguments are counter-productive.  We should replace them with foundation AND superstructure arguments.

Further there may be something counter-intutitive about where the right balance is. We think we’re winning when we get as much of the abstraction in as we can but in fact we might get more value out of our abstraction if we did a little less of it, a little better informed by the concrete.  In a Japanese parable a student of the Zen tea ceremony asks, “Master, why do we do it this way?” To which the master replies: “Do it a thousand times and you will understand.” Wisdom and intuition come from deep formal knowledge.  At least that’s one possible interpretation.

Two final points. 1) a big caveat 2) an application note

The caveat.  Paul Bruno, who is consistently insightful in looking at research on teaching and learning, notes that the findings are undercut by their reliance on self-report.  In other words, the PISA results are correlated to students descriptions of how much formal math they got and that is highly unreliable.  So we have a pretty big methodological caveat and have to take this with a grain of salt… not sure how big but certainly not insignificant.

In the meantime, Paul Powell, principal of Troy Prep Middle School and who I am trying to convince to writing a book on math one day, has designed a program that appears to align to Schmidt’s findings. They go “high low.”  Lots and lots of fast practice automaticity with basics. Strong foundation in the formal. Supplemented by Olympiad Math type problems several times a week.  In other words, they build up constant formal knowledge for 90 or more minutes a day and then spend 20 minutes once a day or maybe every other day doing a problem that requires problem solving and math literacy–specifically where students not only have to solve but have to figure out how to solve and to use multiple tools to solve.  The questions are very demanding.  The kids get better and better at them as they practice. And their practice is constantly informed by a base of new foundational skills.  His students routinely score in the top 1% of schools in New York state in math and last year, even with a FRPL percentage of about 95%, his students passed the state test at two and a half times (74% versus 28%) the state average. The year before that they were among the top 20 schools in New York in their percentage of students not at level 3 but at level 4.

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2 Responses to “On the False Dichotomy in Math”

  1. Bill G.
    January 3, 2015 at 7:53 pm

    Hi Doug,

    Do you see similar “dichotomies” play out in your work on literacy? I imagine you encounter similar puzzles thinking about ways to develop both depth of vocabulary and fluency. Do you recommend approaches to developing literacy in your upcoming book that may translate to math and other areas?

    I ask primarily out of interest in these same issues in sports. Teaching motor skills and game sense, and integrating the two, seem like very similar problems. In hockey, even some highest-level coaches will tell you that you can’t teach instinct for the game. They do some of it, but because they are unable to articulate how they do it, they undermine it.

    Thank you for all of your contributions.


    Bill G.

    • Doug Lemov
      January 7, 2015 at 3:19 pm

      Thanks, Bill. Great question. Yes to literacy. For example, when students struggle to make sense of a complex text it’s often because of very fundamental thing. They read this section from Oliver Twist and fail to understand it because they don’t get what “it fell to Oliver” meant or “per diem” or that “held a council” is mock ironic.

      Oliver Twist and his companions suffered the tortures of slow starvation for three months: at last they got so voracious and wild with hunger, that one boy, who was tall for his age, and hadn’t been used to that sort of thing (for his father had kept a small cookshop), hinted darkly to his companions, that unless he had another basin of gruel per diem, he was afraid he might some night happen to eat the boy who slept next him, who happened to be a weakly youth of tender age. He had a wild, hungry eye; and they implicitly believed him. A council was held; lots were cast who should walk up to the master after supper that evening, and ask for more; and it fell to Oliver Twist.

      You could spend all day asking abstract questions but the lack of insight comes from not understanding the basics enough–and quickly enough without conscious thought–to be able to reflect on the larger themes.

      Similar with sports. One of my son’s soccer coaches said: “I don’t just want you to be able to receive the ball like this now. I want you to be able ot receive the ball like this every time. And i want you to be able to receive the ball like this every time without thinking about it so you can be reading the opposition and thinking about where the ball should go next.” Translation- if the basics aren’t fluid and automatic, you don’t have the bandwidth to problem solve. That was the day i knew i would never move my son from his team.

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