Marilyn Burns and the number 11.

Recently I read an interesting post by Marilyn Burns about a pleasant discovery she made. It niggled at my mind (in a good way). It was not the discovery itself that struck me, but rather the reservations she had about how she reported it. This is what she said:

What I’ve done is given an explanation that falls into the category of "teaching by telling," which I avoid in the classroom when I want students to "uncover" knowledge that’s based in understanding relationships.

She had written a book about the number 11 for her grandson, and in it she had mentioned that 11 could be expressed both as 6 + 5 and as 6² – 5².

Marilyn's Problem 

Sometime after completing the book, she became curious about whether the same thing worked for other pairs of numbers that differ by one, and she found that indeed it did: for example, 

7 + 6 = 13  and   7² – 6² = 13.

He question was about how to generalize this. She explained how she did it both graphically and algebraically, but she was uneasy about the "teaching by telling" approach that she used. This lingered in my mind for some days because it reminded me that some of us (math educators) have not yet sorted out the relationship between "exploring the math" and "direct instruction," or, if you wish, between "learning mathematical reasons" and "applying the math trick."

Moreover, her two approaches (plus another that I have added) reflect the increasing distance between math via “discovery” and math via “direct instruction.” I think of the former as not requiring an extensive math background, and the latter as depending on already acquired knowledge. Perhaps I am being naive about this, but let me explain:

Her graphical approach, via (1) above, is related to what I think of as being "discovery based." You don’t need a whole bunch of content knowledge to understand it. The 11 blue squares visually illustrate the difference in areas between a 6 x 6 square and a 5 x 5 square, and the process can be extended to show the difference between 7² and  6² and so on.

Her approach, via (2) above, is a very natural transposition of the problem to the algebraic domain: compare the sum of the two numbers with the difference of the squares of the two numbers.

A third approach, via (3) above, is also a transposition to the algebraic domain. It is clean, and it leads very quickly to a solution (because, for consecutive numbers a - b = 1),  But, to me, it is a lot less natural than (2), for it depends upon having memorized a particular algebraic fact and having it always at the ready. For this particular problem, using (3) covers up the thinking about the math involved. It verges on being a mathematical trick.

Or does it? Do I really believe that it is a trick?

Using (3) certainly bolsters the argument that content knowledge helps you do mathematics. No mathematician would ever deny that content knowledge is important. An awful lot of mathematics takes knowledge from one area and applies it elsewhere, and one hopes students learn how to do this. 

I’m not sure how students can acquire content knowledge and learn to apply it without some teaching by telling. Yet, "telling" may lead to mimicry rather than understanding. (That may explain why teaching problem solving is difficult! And that’s maybe why we get a zing when we do solve a problem.)

Now, I think (or rather, I hope) fewer and fewer people are still invested in the total primacy of teaching content over everything else. An episode from my past suggests that content knowledge alone is insufficient.

Jim's Problem

In high school, my friend Jim would occasionally bug me with puzzles that exposed my poor abilities with mental arithmetic. He was fond of asking me things like 

Without using a pencil and paper, what’s 48 squared?  or  What’s 61 squared? 

(BTW, I existed as an entity long before calculators did, so that’s the pencil and paper reference.) 

But before I could even get started he would tell me the answer. 

48 squared is 2304.   61 squared is 3721.

How could he get it so fast? When pressed, he explained how he did it:

48 squared has got to be close to 50 times 46.  (I just added and subtracted 2.) The algebra goes like this:
50 × 46 = (48 + 2)(48 - 2) = 48²  -  4.
And 50 × 46 is easily seen to be 2300, so 48 squared is  2304.

Somehow, I didn’t see the trick until he showed it to me, even though by that time in my life factoring a² – b² had become an automatic reflex. If you're reading this, I’ll bet it’s an automatic reflex for you as well.

I find it interesting that both Marilyn’s problem and Jim’s problem can be resolved use exactly the same content knowledge. For Marilyn’s problem, that knowledge led me immediately to an answer, yet for Jim’s problem I did not make the connection.

Are you like me, or did you immediately see the connection with Jim’s problem?