Making sense of math

Math makes sense. Not only to mathematicians, it turns out. Math just makes sense. It’s internally coherent, and shows you so when probed. All the rules in math that seem like “just because”–you can think of them probably pretty quickly, like don’t ever divide by zero, or a number raised to the zeroth power equals one, or to divide fractions you multiply by the reciprocal–have very good, very sensible reasons behind them*. So sensible, in fact, that when you really see these reasons, the rigor and meaning of the mathematical framework strikes one as almost impossibly beautiful.

Math makes sense, but it doesn’t always make sense to us, as math doers, right away. However, the fact that it is internally coherent means there is a task for us, which is to uncover the purpose that underlies the definitions, rules, patterns, and behaviors we see. As mathematicians, when we sit down to work, we are in essence sitting down to a science of deep reason. Things have a place, a reason, a logical flow.

So there are two things happening here. One is the sense the subject itself makes–the internal consistencies that were built into the subject over the centuries as its practitioners added more ideas. The other is the sense we must make of it–the sifting and mining we do to uncover the reason of the mathematics. They are deeply related, both to each other, and to the practice of doing and thinking about mathematics. To miss the fact that math makes sense is to misunderstand one of the most essential ideas math was built around. Sometimes you have to work for it, though, patiently and persistently.

Many people go through the entirety of their math lives without ever understanding that math actually makes sense–indeed, is more firmly dedicated to sense than any other subject or area of study–because we aren’t teaching it in a way that illustrates this, and we aren’t teaching it in a way that encourages students to pursue the sense that’s there.

How do we teach math in a way that highlights its inherent sense, and pushes our students to dig into problems in an effort to make sense of them? Though there are many useful answers to this, here are a few:

  • In order to learn how to make sense of problems, students need to be given problems that resonate with deep, internal coherence.
  • They also need to be given problems that require their effort to uncover the reason behind the problem. These problems need to be hard enough to put students into productive struggle, but not so hard that they give up. Well-crafted math problems should be like adventures, with the opportunity for real diversions, points of interest along the way, and rewarding views after periods of sustained struggle.
  • Students’ questions should be used to find where the math isn’t making sense on a conceptual level.
  • Internally motivated math experiences can help students develop real persistence in sense-making. When students care about answering a question, and care about the answers they get, they work longer, harder, and also object when their work or the problem itself doesn’t seem to make sense.
  • Aesthetics help! Go for beauty!

As Common Core picks up steam around the country, there are going to be more conversations about making sense of math. Common Core includes a list of math habits of mind, or standards of practice, all students should be developing. Making sense of problems and persevering in solving them is the first of these practices, and for good reason. Because one of the most important things to know about math is that it makes sense, and that we can see it if we try.

*For example: a number raised to the zeroth power equals one. Why? Maybe I’ll let you think about why it makes sense, but here’s a framework to help—what happens as you go down this list of powers?

5^4=5 x 5 x 5 x 5 = 625
5^3 = 5 x 5 x 5 = 125
5^2 = 5 x 5 = 25
5^1 = 5
5^0 = ?

Do you see a pattern? To make that pattern hold, what is the “natural” value that 5^0 should have, that would make the most sense?

And what should 5^(-1) be?

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2 Responses to Making sense of math

  1. Kalid says:

    Insightful post. I’m of the opinion that math makes sense, but it’s our job to “uncover” the sense behind it.

    For example, Roman numerals are extremely cumbersome, but “numbers” (as a concept) are very simple and make sense. It turns out we were representing them clumsily.

    For things like exponents, we write 5^2 but forget that it’s in *reference* to a unit of 1. So I see 5^2 as “1 times 5 times 5″ or “start with 1, then multiply by 5 twice”.

    For 5^0 to make sense internally, I have to shift my reference to “start with 1, then multiply by 5 zero times”. We were already at 1, and decided to do nothing (remember, we aren’t multiplying by zero! We’re multiplying by 5, “zero times”).

    Part of the fun is aligning our internal metaphors with the built-in “common sense” that seems inherent to math.

  2. Great post. It leads me to wonder what it means that something “makes sense”. Is there a cultural component to this sensibility? Are we limited in our exploration of mathematics by some circumscribed parameters inherent to the human brain? Does mathematics exist without human thought? A very compelling discussion on the origins of mathematics can by read here: http://www.kavlifoundation.org/science-spotlights/kavli-origins-of-math

    Also, if you have a chance, please check out my response to the discussion at: fightthemonsters.blogspot.com

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