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Peter Gerdes's avatar

I mostly agree with your points, but in practice my experience teaching mathematics is that often by *trying* to teach more kids math we make sure only the gifted really understand it. The problem is the incentives favor moving more to rote learning when you try to get more students to succeed and the end result is often only the very gifted actually can extract real conceptual understanding from the class.

The problem is that even if you're intelligent and conscientiousness if you hate math and thinking about it makes you feel frustrated and miserable it's exceptionally difficult to engage in the kind of hypothesis generation and testing (ohh maybe it works like this) needed to get a real conceptual understanding. If the teacher asks questions that can't be done by rote and test real conceptual understanding those students just can't do them.

OTOH those students can memorize rote rules with enough effort and pain. The net result is that the incentives for the school, teachers and most students are to just learn those rote rules. The very gifted students can still extract understanding but you've lost all the kids in the middle who could have excelled in a class that asked conceptual questions on exams and required real understanding. But I fear that in order to reach those students your incentives have to allow for the fact that you can't give a backup rote mechanism for doing well (so some motivated students ..even with pushy parents.. won't succeed).

So I sorta agree with you in principle but I fear that when you give this as a policy goal the way it trickles down into the classroom is to incentivize teachers to increase the fraction of students who succeed in meeting some testing threshold and that often means presenting the class with a list of rote tricks. So instead of getting say (in say a high achieving suburban class) 70% of kids who leave with real understanding you get 20% with real understanding and 90% who can manage to the computations (and forget them once the class ends).

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Steven Klaiber-Noble's avatar

I think this perspective, combined with the research around deliberate practice, does result in policy recommendations that will have a lot of enemies. Far more hours of the school day need to be used for guided practicing of math.

The way more people learn math is to have more time doing exercises in a deliberate practice framework (eg JUMP math is one example, khan academy another). But homework is a bad tool for this because real time correction and redirection is essential. The goal is to be practicing at the edge of ability, which means it’s easy to get blocked without a guide.

This does mean less time for other topics. From my perspective, there does not need to be as much time spent on study that is mostly memorizing names and facts.

But I expect this recommendation to provoke a lot of disagreement.

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