## More Thinking About Thinking

*Published on August 8 2011*

I want to elaborate on the July 11 post about the importance of understanding how children think about math. In that post I said that, “Paying attention to how children think about math is important. It provides insight into the child’s developmental level of understanding of a concept as well as clues on how to proceed.” Here’s an example of what I mean.

I was working with three first graders on counting, specifically, the number before a number and the number after. The number they were given as an example was 24, with ‘23’ written before and ‘25’ after.

Samson studied the numbers for a few minutes and then quickly filled in the blanks. Here are the first three from his worksheet. (The numbers in bold are the students’ responses):

*5** 46 7*

*7** 68 9*

*8** 79 10*

Yuko reasoned out the assignment in a completely different way. Here are her first three responses:

*26 **46 27*

*28** 68 29*

*30** 60 31*

Diana found a third way to think about the numbers:

*61** 60 62*

*32** 31 33*

*94** 93 95*

Each child had received the same instruction yet each one approached it in a unique manner. True, not one of them completed the assignment correctly and their work didn’t tell me whether they could identify preceding and succeeding numbers for a given number, the lesson objective. Nonetheless, I got valuable information from how they thought about the problem and the underlying math (counting). For one thing, it told me they had each engaged in some remarkable mathematical reasoning. See if you can discern how each child thought about the task before reading on.

Samson looked at each two-digit number, decided to treat them as two separate one-digit numbers, and wrote the number preceding the tens digit and the number following the ones digit.

Yuko ignored the number in the middle and simply continued counting sequentially from the numbers in the example: 23, 24, 25.

Diana, too, counted sequentially forward but she chose to start with the middle number in each set and write the two succeeding numbers, one to the left, one to the right.

Each of these students analyzed the numbers and came up with a solution that made sense to them. When they explained it, their thinking made sense to me, too, even though the responses were incorrect. But their solutions told me that they were willing and able to reason mathematically, an ability that I could capitalize on in subsequent lessons.

The following day I reviewed the lesson with them and they were successful. Equally important, I learned a lot about how children think about math.

Dave Gardner

Mathematician in Residence, Explorations in Math