Questions on the Board

July 4, 2010

The Place: the reading room at Elliott Bay Books. Large but with no natural light, and imperfect lighting.
The Time: this afternoon at 1:30.
The Crew: 7 kids, in the 2-4th grade range.

If art requires inspiration, and math is an art, then my job is, in part, to provide inspiration.
I brought in a chessboard.

We warmed up with the handshake problem, and then I presented the board.
What are some questions we could ask about it? Here’s what the kids came up with:

  • How many squares are there of any size?
  • What pictures of letters could you make out of squares (the same color)?
  • How many edges on all the squares on the chessboard?
  • How many diagonals?
  • How many corners?
  • Is the perimeter of a square different from the inside of a square?
  • How many sides if the squares are cut out?
  • How many sets of chess pieces could we fit on the board?
  • How many letters would fit inside the chessboard?
  • How many squares are there if non of the same size are allowed to overlap?

It’s actually a very rich list of questions, and this group had no trouble digging in.

Spoilers follow.

The first problem is a very natural and interesting question to ask, and one I would have brought up if the kids hadn’t. Two kids, worked on it together, oblivious to the outside world, until — breakthrough! — they noticed a very cool symmetry:

the number of 1 by 1 squares is 64 (or 8 x 8);
the number of 2 by 2 squares is 49 (or 7 x 7);
the number of 3 by 3 squares is 36 (or 6 x 6);

and so on till

the number of 8 by 8 squares is 1 (or 1 x 1).
So the answer, interestingly, is just a sum of squares numbers!

Another student found the number of diagonals on the board, and then went back to find a pattern, starting with smaller board sizes and working his way up to 8 by 8. A very tidy pattern, in the end. The last group (of 4) tackled the last problem, which is more technical and involves extra cases. What’s the pattern, though? Two of them went home determined to figure it out.

Here are some pros and cons of letting the kids ask their questions and choose what to work on:


  • Time is set aside specifically ask questions, one of the most important parts of doing math (or anything).
  • Every kid owns their work, from start to finish.
  • Every kid gets to choose what to work on, and every single person in the group is engaged pretty much the entire class (something that is essentially impossible if you lecture, no matter how compelling an orator you are).
  • Kids get to present their own findings to the class at the end.
  • This is a math class where the kids do what real mathematicians do.


  • It’s much harder to create an arc for the class, since I don’t choose the problems ahead of time, and I can’t guarantee as many “wow” moments. It takes a subtler hand on the tiller.
  • Since not everyone is working on the same problem, there can be less overall class cohesion. Sometimes students aren’t that interested or tuned in to what other kids have figured out.

In my opinion, the pros outweigh the cons. However, I think there’s even a way to eliminate the cons altogether. The first meeting was great; as we keep meeting I think it will get even smoother.

What’s next? Math games, I hope. Infinity, as well, possibly via Zeno and infinite divisibility.
I’ll be thinking about it.

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