First, Do No Harm

Wouldn’t that be a great stricture for Math Education to adopt, just like medical professionals do?  Whatever else you do, make sure you do no harm.

You’d think that this would represent a really low standard of math education: that it is better to nothing at all in third grade than to do stuff that destroys the third-grader’s ability and interest in thinking mathematically.

And of course it is a very low standard.  And yet it is also one that we are failing.  You watch a bunch of kindergartners who are excited and enthusiastic about learning about numbers and who are counting their blocks or their money in total concentration, confident that they can make sense of what they are doing, and working hard to make sense of things even without a teacher anywhere nearby.

I remember my daughter, at maybe four years old, shouting to me: “Look, Daddy, I can count money!”  She had four pennies and a dime in front of her.  “One penny, two pennies, three pennies, four pennies, five monies.”   If this strikes you as strange, I would agree.  If this strikes you as wrong, I would strongly disagree.  It struck me as wonderfully inventive, she wasn’t just reciting counting words, she was actually keeping track of what she was counting in the best way she knew how.  After counting four pennies, she understood she couldn’t continue by saying “five pennies,” because the next thing she was counting was a dime.  She didn’t do the thing I would have done, which is to convert the dime into pennies.  But what she did do was a completely sensible thing: she found a common denomination, which she called “monies.”   Clearly, no one had taught her to count money like that – it would never have occurred to an adult to teach that particular way of counting money.  She invented it in the moment, clearly driven by a need that what she did make sense.

I wonder for how many kids the experience of K-12 school in the United States has been that of systematically learning how to suppress their sense of curiosity about math, and instead settle into learning what the teacher/textbook wants and expects.

If the textbook introduces a runner who continues to run at sprinter speeds for hours on end, without break and without slowing down, the students will readily accept that as the way things are done in school, knowing quite well that no real runner would run that way.
Just like Wile E Coyote can survive a series of crashes that would debilitate or kill a lesser animal, kids will accept that as the law of the cartoons and they are usually smart enough to know that this has nothing to do with the law of coyotes or the law of gravity.
Doesn’t Kenny in South Park get killed in episode after episode and then shows up like new in the next episode?  We accept that as the law of South Park and don’t confuse it with the laws of how the real world operates.
Kids will accept the law of School Math and don’t confuse it with the real world either.  They will accept that you can add numbers in arbitrary order and get the same result, and be ready to shout “commutative law” in unison, and equally accept that if you write 2x+3 you get full points but get marked down if you write 3+2x.  If Mr Johnson tells them that -2x+1 is “simpler” than 1-2x they will readily accept that as the law of Mr Johnson, and give up any notion of what “simpler” means in the real world.
If Ms Jameson tells them in math class that “time is always the independent variable, because time marches on and doesn’t depend on anything,” students will readily accept that as the law of Ms Jameson, and not see any conflict with Ms Jung’s physics class where they vary heights of inclined planes and then measure how long it takes for a ball to hit the bottom.  After all, in Ms Jung’s class, we pay attention to Ms Jung’s law, and not Ms Jameson’s law.  Of course, Ms Jameson isn’t being arbitrary in what she says, she is beholden to the textbook and the answer key.
If the answer key gives 2.5 as the number of buses that will transport 125 kids if each bus can hold 50 kids, then what is Ms Jameson going to do?  Tell the kids that half a bus makes no sense?  In a unit about decimal numbers and decimal division?  And what if the textbook says that buying 1000 toilet paper rolls for $200 is a better buy than 10 toilet paper rolls for $3?  Sure, if I buy 1,000 toilet paper rolls all at once I pay less per roll – but is it really a good idea to commit that much money for something I might not even have room for in my apartment?  Teachers and students learn very quickly to ignore such practical concerns, and accept the law of the textbook.

Should we be surprised that students come out of school thinking that math has nothing to do with any real world they live in?  Should we be surprised if a student solves a school problem and blithely writes down an answer of $200/cookie and doesn’t stop for a moment to consider that these are really really expensive cookies?  Why would they?  It is math class after all.  In math class, cookies can cost anything, and in math class you can divide those cookies into 17 equal pieces.  What does math class has to do with the world away from school?  For most students, I submit, the answer is very little or nothing at all.  And that, I would say, is doing harm.

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2 Responses to First, Do No Harm

  1. xiousgeonz says:

    I think this problem extends much further than things that *actually* make no sense.

    There are many times when things make perfect sense to the teacher and four students – or, perhaps, one. I remember talking about fractions and realizing that it was only Pat O’Brien who knew what 1/3 was. Every single other student could break things into halves and talk about it, but couldn’t connect the process “1/3 x 12” to the real world. They got all the problems right, but only Pat knew what it meant. Fortunately, when I prevailed upon him to give others time to think it through, he complied and at least a few more students got it when they left for lunch.

    I see textbooks making enormous efforts to “Make sense,” but if the student has already figured out that Math is a Different Planet with Different Rules, the *student* doesn’t make the connection. The infamous “Jack is on a boat sailing east with the current…” kind of problem that stand-up comics can depend on for sympathetic humor actually makes sense … to the author.

  2. Bert Speelpenning says:

    I think your comment is right on: we tend to assume that “making sense” is located in the problem or in the exposition or in the explanation or in the textbook. Ultimately, making sense is located in the learner. If it doesn’t make sense to Pat’s buddies, that’s what’s relevant to Pat’s buddies.

    For a learner, learning will always and forever have to start from where the learner is. A textbook, at best, can organize and present material in a way that will connect to where many of the students find themselves. Most learning is interactive, involving grappling with a problem and grappling with articulating what it is you see and what it is you don’t see. A textbook can only get you so far.

    The role of the teacher in facilitating learning is a complex one, and my post only deals with one aspect of it: to look at places where the traditional approach to teaching math has the effect of systematically filtering out students’ drive to make sense of math. You call it “connecting the process to the real world,” I might call it “connecting the problem/process to the world of the student.” For sure, it is all about connecting and connections.

    The role of students in each other’s learning is also complex, and an underappreciated resource in traditional math teaching. It is great that you have a classroom where some of your students learn from Pat – or at the very least one where Pat doesn’t get in the way of their learning. In too many classrooms one student will blurt out the (expected) answer, and the teacher will take that as a sign it is OK to move on.

    It is not my intention in this blog to suggest easy answers to questions of how to teach or how to organize a classroom. In this particular post, I’m attempting to address the very issue, a systematic issue, that has students predictably conclude that – as you put it – “Math is a different Planet with Different Rules”.

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