The homework asked for factors of various two digit numbers. For 24, my kid put 2x12, 1x24 & 24x1.

I said, "Those last two are the same thing, what other factors can you figure out?"

The response was immediate and somewhat intense. She was convinced that 1x24 and 24x1 were different because that is what the teachers said.

I mentioned we had read about the commutative property in Beast Academy 3B but, sweetly, nothing could sway her loyalty to her teachers and her opinion about what she thought they had taught her.

It was at this point I thought back to all the things Christopher Danielson has written about Cognitively Guided Instruction and the wonderful modeling of his Talking Math with Your Kids project. These approaches show the worth of conversation around math with an emphasis on the adult really listening to what the child is thinking.

She continued. "See! One times twenty four is [pointedly counting] one, two, three, four, five, six, seven, eight, nine, ten, eleven, twelve...twenty four. Twenty four times one is...[pausing, then saying emphatically]

*twenty four*."

I nodded. "Oh, I see what you're thinking. The first way means you have to count by ones 24 times. The second way you just have to say 24 once."

In her mind it's the

*process*of getting to the final answer that makes the two facts different. Never mind that she gets the same answer both ways. Never mind that she knows all about the "twin facts" on multiplication chart. Never mind that we're having fun finding different ways to memorize multiplication facts including sneaky guerrilla tactics. Nope. This is her reality and it's not going to budge by quoting official definitions.

All I said was,"You can put those two facts on the paper, but your teachers may want you to put some others as well." In the end she found all the factors of 24, but wrote each combination twice (e.g. 6x4 and 4x6).

In the process of writing my new book, tentatively titled

*Meaning in the*

*Making: The Body Learning Math*, I've been doing a lot of reading and thinking about how the processes of doing and learning math are just as important as the product. In this case, she can easily figure out factors of two digit numbers, but it's by watching her process closely and engaging in conversation about her thinking where we really get a glimpse into what she knows and how she knows it. Specifically, we can see how she is literally

*making*and reasoning out her own meaning of how multiplication facts are combined.

We only get half of the picture if we look at the final product/answer (double facts). I know how to watch for and identify understanding through the processes of making math and dance at the same time, but now I'm really learning about how it works with numbers, too! Fun stuff.

Love this. I sent you some thoughts on Twitter and will surely write about it in more depth somewhere. In short, the commutative property assures us that 24*1 and 1*24 have the same value, but not that they have the same meaning. I think this probably extends to other properties as well. In fact, I know it does; I work hard with future elementary teachers to tease out the different meanings of (say) (2*3)*4 and 2*(3*4) (the associative property of multiplication).

ReplyDeleteI hadn't thought of this characterization of properties of numbers and operations before reading this piece today. Thanks for it.

Can't wait to read more of your thoughts on this! You might be interested to know that this year they're using TERC's Investigations for 3rd & 4th grade in her classroom. Heavy emphasis on conceptual understanding of multiplication which might explain how she came to her conclusion.

ReplyDeleteThe other thing is that before reading Sfard I might not have picked up on what my kid was saying, but Sfard has had me thinking about the processes of constructing mathematical notions/objects/concepts -- it was really fun to have an example of it show up in front of me. :-)

With factoring, I usually think of blocks (like in today's MTaP post), which make a rectangle the same any way you look at it. But if I think of chairs, instead, then one row of 24 chairs is definitely not the same as 24 rows of 1 chair each.

ReplyDelete