Should we require kids to produce math facts on demand?

What does it mean that something is “true,” “valid,” or “proven?” (Hint: they do not mean the same thing.) And upon what fundamental system is arithmetic based? (I’ll just tell you: place value.) These are the big questions that will drive our Math Circles this year. More specifically, ages 11 and up will explore statistics; teenagers will do proofs; ages 9-10 will explore place value, extending into different bases, and possibly infinity; ages 6-8 will explore truth and logic.

This summer, a few of you have contacted me with concerns that your children either don’t do well in math, or don’t like it. What to do?

First, let’s address what math is and what math isn’t. Mathematics does not mean the ability to do rapid calculations. Mathematics is a way to seek structure in our world. It involves philosophy. Logic. Why logic? Well, here’s one definition of math: a search for patterns with simultaneous attempts to break the patterns you’ve found. In order to refute a conjecture about a pattern, you need to understand how to refute things in general.

Dancer/mathematician Malke Rosenberg describes what math is well in her blog post “The Elephant in the Room.” Mathematician James Tanton does a nice job of telling us what math isn’t in his video “How to Think Like a School Math Genius.” It’s a long video, but the first four minutes shake out some of our assumptions.

So should we require kids to produce math facts on demand, or instruct them at all in procedures? Or should we heed the call of Paul Lockhart’s Mathematician’s Lament? This is not a new question. In 1929 L.P Benezet experimented and wrote extensively on the benefits of Delaying Arithmetic. Math educator Denise Gaskins delves into Benezet’s results in her wonderfully informative blog “Let’s Play Math.”

Ask yourself what the stakes are in your child’s math education. Can you take your time and provide rich sources of material to allow your child’s natural inclination to seek structure to manifest? A rare few of you can’t, as you are preparing your children for a transition from homeschooling to school math. Gaskins’ series’ “How Can I Teach Math if I Don’t Understand It?” and “PUFM” are excellent starting points for those of you who have asked me for advice on how to do math instruction (vs. exploration) at home.

Getting back to the question on how to encourage kids’ affection for or adeptness at mathematics: the obvious answer, of course, is to come to a Math Circle. But what if you can’t? In addition to the sources of inspiration above, you could play Math Lawyer. The child posits a conjecture she is pretty sure of. (“10 + 5 = 15.”) The parent argues against it. (“One 10-year-old child plus one 5-year-old child does not make a 15-year-old child.”) The child defends it. (“But a group of 10 children plus a group of 5 children does make a group of 15 children.”) Maybe the conjecture gets modified. Maybe the parent gets trapped. Maybe the jury is hung – for now. The conversation continues. And true math is a conversation.

Here’s another Math Lawyer debate that came up in our home: “Diameters are unnecessary.” We’re still arguing that one.

— Rodi