# Thinking like a mathematician…or someone like one

This article makes the point that the emphasis on getting students to “understand” by using alternatives to standard algorithms is a subterfuge. The purpose, the article contends, is to make students look smarter than they are.  They reason as follows:

The problem is this: “number bonds” is a counterfeit of the way kids who are genuinely good at math act by the time they get into elementary. While the other kids are counting on their fingers, kids who’ve been playing with numbers in their heads since they were two or three have figured out all the relationships and will take numbers apart to make it easier to solve. Not something stupid like seven plus seven, of course. More something like 115 + 115.

Having figured out that number-gifted children will do this as 100+100=200, 15+15=30, so 115+115= 230. This is quite nifty for a first-grader, but the left thinks it can skip all the work getting there. If they just teach perfectly normal, average children to think in terms of taking numbers apart, voila! Everyone will be a math genius!

I don’t know that it’s “the left” who thinks this way–I have run across plenty of apolitical math reformers who seem to be on this wavelength– but I take the article’s point.   The thinking amongst these reformers is that one indication of “understanding” is whether a student can solve a problem in multiple ways. Reformers then insist on having students come up with more than one way to solve a problem. In doing so, they are confusing cause and effect. Forcing students to think of multiple ways or using “number bonds” does not in and of itself cause understanding.  They are in effect saying  “If we can just get them to do things that LOOK like what we imagine a mathematician does … then they will be real mathematicians.”

Robert Craigen, a math professor at University of Manitoba sees this type of thinking as wrong-headed: “Mathematicians don’t think that the ephemeral truths of higher reasoning have any validity when disconnected from the basic, mechanical foundation on which they are built.”

Maybe if we call the traditional approach something that sounds more appealing, we can get on with teaching math properly.  Something like “alternative math”.  What that alternative is (and to what it is an alternative) will be our little secret.

## 5 thoughts on “Thinking like a mathematician…or someone like one”

1. SteveH

K-6 pedagogues claim that they love the balance of understanding and skills, but CCSS does not effectively isolate and test the skills portion. K-6 math classes are set up to trust the process and assume that, like thematic learning, facts and skills will be learned automatically, but they only check skills in the context of fuzzy problems or at very low NON-STEM levels. I don’t have any hope that we can find some way to make them see the understanding embedded in mastery of basic skills. I learned all about splitting and combining numbers in different ways in my traditional math classes. I think our goal is to get them to separate and test the skills they claim to care about. Separating and testing these skills is easy to do and gives teachers and parents clear feedback and simple goals to work on rather than just improving “problem solving.” In addition, all K-8 schools should publish exactly what it takes for students to get on the high math track to a proper algebra class in 8th grade. Report cards should show exactly how kids are progressing to this tracking point – not just any vague sort of low CCSS “distinguished” level. In my son’s school, many kids and parents were stunned by that end-of-6th grade surprise. By then, it’s too late.

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2. I have long been an advocate for US conversion to the metric system; it is simpler, faster, more efficient, etc.. I have heard the argument, however, that our kids must be smarter because they know two systems. The key probably is in the word “know”. I doubt that our kids know either system–metric or inch-pound–as well as kids in other countries know metric.

Multiple ways of solving problems is radical egalitarian diversity applied to math. As if each solution method was a different ethnic or cultural group. We mustn’t slight any, and none is superior to others, just different. In US education, there is no trade-off considered between equity and efficiency. Equity must trump efficiency …always.

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3. And thoughtful leadership of poor or vulnerable groups has no interest in this type of education. These leaders want schools to teach all children early and well.
And they do not expect education to solve the myriad of social, economic and cultural issues some children face; they simply want children taught and taught well.

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4. In England, when there was a switch to greater didactic teaching and more mental arithmetic in the 90s, it was called “whole class, interactive teaching” by researchers and “the National Numeracy Strategy” by schools and was treated as something completely new.

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• RIght; thanks for the comment. I recall Jeremy Hodgson talking about the National Numeracy Strategy in his talk at researchED at Oxford last year. Someone gave me the link to the NNS but I’ve lost it, so if you have it, I would greatly appreciate it!

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