## Friday, October 10, 2014

### Equal Signs, Balance, and Timbits (#tmwyk)

Callum had to start taking the bus to Kindergarten.  When I picked him up the first day, I decided to get him some Timbits as a reward.

(official looking Timbit image from Tim Horton's website)

A few minutes of driving passed, and I looked in the rearview, and suggested he save some for his brother.  To which, Callum replied, "there's only one left."

Nobody intends their child to eat 9 Timbits in one sitting, but it seemed like a good time to talk some math.

The conversation went something like this:
-if 9 are in your stomach, how many are left for Alec?  (one)
-if you started with 10, and 1 is left for Alec, how many did you eat? (9)

I wanted to get him to add "9+1=10".  I was working on the concept of "one more than nine, one less than ten."

Here is where things get interesting.  I use "plus" or "add" as much as possible, because they are specific operational words he is going to need really soon.  But based on recent readings, and #tmwyk conversations, I restrained myself from summarizing the math we were discussing as "nine plus one equals ten".

We are doing some work on Marian Small's Uncomplicating Algebra book, and the Ontario "Paying Attention to Algebraic Reasoning" monograph.  One common stumbling block is seeing an equal sign as merely an indication to "get the answer", or, even worse, what you press when you're done pushing buttons on a calculator.  More properly (and foundational to algebraic reasoning), an equal sign should be a statement of equivalence, or balance.

So I tried this:  "9 Timbits in your belly, and 1 more for Alec, are the same as the 10 we started with".  I wondered if "the same as" could be a good substitute for "equals" in talking math with young children.  I think small things like one word choice can have large effects on how young children learn math.  The precision of our math language really matters when we are building the conceptual foundation.

Consider this example (this mistake happens all the time in upper elementary):

4 + 5 = __ + 3

Those with less of an understanding of balance will right exactly this:

4 + 5= 9 + 3= 12

In grades 6, and 7, we see this exact error all the time.

Here is a powerful Kindergarten example showing how you can build the concept of balance with young children:

Any thoughts on the uses and misuses of equal signs?