I’m having a really interesting conversation with Anna and Dylan about whether we should be trying to help students understand the Standards for Mathematical Practice. (It’s on twitter here.)

My position is a work in progress, but I’m trying to stake out a position that favors teaching kids about math, but doesn’t seek to connect these to the Standards for Mathematical Practice. I’ve been yapping on about how I think the concepts (not just the language) of these SMPs depend on k-12 math knowledge.

It occurs to me that it would be helpful for me to make a quick list of things that I think *are *attainable (and important) to teach students about math. I’m making this list so that each of my practices are related to the SMPs, but I want to be clear: this is not a kidified version of the SMPs. I don’t know if that’s possible. Instead, this is a kiddified list of things k-12 kids should know about the nature of mathematical work.

**Big Math Idea 1:** People often make important contributions, even when they fail to solve a problem.

**Big Math Idea 2: **In math, understanding what a problem is asking is often really hard.

**Big Math Idea 3: **Giving reasons and explanations is an incredibly important part of what mathematicians do.

**Big Math Idea 4: **Math is used by lots of different people in a lot of different ways to understand the world.

**Big Math Idea 5: **Despite what people think, math actually involves a lot of messy choices.

**Big Math Idea 6: **A lot of math involves coming up with a definition or name for something that’s hard to describe.

**Big Math Idea 7: **Despite what people think, math actually involves a lot of creativity.

**Big Math Idea 8: **There’s an important sense in which math is the study of patterns.

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*Related*

The Reasons for Mathematical Practice.

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Your list made me think of all the different flavors of “mathematical work”: trying to understand and strategize a new-to-the-student operation like 1/2 of 3/5, or 3 to the -3 power; mathematizing the world, as in a typical 3-act problem (how boxes of cookies to fit in the back of a car); solving a challenging geometry problem or construction; finding the generalized function for a visual pattern; investigating the patterns in pythagorean triples or consecutive sums… and how I might be more explicit in highlighting for the students how each of those feels as they are engaged in them. I also started making a parallel list in action form: Idea 1 – Highlight the contribution of failed lines of inquiry, Idea 2 – Engage the students with problems in which they have to find the question, Idea 3 – Make reasons and explanations a regular and essential part of daily classwork, etc.

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I love the idea of highlighting things I can do in class to bring out these ideas about what math is like. Did you publish these anywhere? Do you have any thoughts on how what we can do with these during class?

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No, I haven’t got anything written out yet, but I will take a stab at starting something and let you know when I post it. I was imagining how students (and much of the world) probably think of math monochromatically (and with very low – and mostly useless – resolution): get the answer. Those who feel successful could probably add: memorize the formula and then get the answer. They don’t even realize the implicit step in there: memorize a bunch of formulas, *recognize which one is useful*, get the answer. And of course I was also thinking about all of the different kinds of work we are trying to engage students in these days, and how different their brains would look and sound if we could see and hear them like the Visualizer on iTunes as they switch from solving problems to generalizing to listening and justifying to practicing to investigating…

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Good stuff. Makes me think of a book I read by Reuben Hersh (“Experiencing Mathematics”) in which he essentially defines proof as any explanation sufficient to convince a fellow mathematician that you’ve thought the problem all the way through.

In other words, mathematics is a social construct. We can do it in isolation, flipping numbers over in our minds, observing patterns, jotting down interesting discoveries, but in order for that work (or play) to qualify as quote unquote mathematics it has to pass muster with other members of the overall mathematical community.

Anyway, before I move too far sideways, I’m wondering if there might be a big idea hiding in between the items on your list. A Big Idea 1.5 or Big Idea 0: What looks like math to you might not look like math to a mathematician (and vice versa). Learning how to talk in common language without sacrificing intellectual freedom is what makes mathematics challenging and enlightening at the same time.

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