On Thursday, May 21, 2009, ** Chris O’Neal ** brought a colleague from Fluvanna to observe my 3rd grade math students work with their iPods. The day before, I had discussed with my kids what they thought we should show and how the class should be organized. I WISH I had videotaped that conversation, as it was simply amazing. However, I didn’t, so this is take 2, the description of Station 2. (See previous post for description of station 1.)

In Station 2, the kids were to use the WhiteBoard app to drill each other. They decided they wanted to show this to our visitors to show how the iPods could connect over WI-Fi. When we were establishing the boundaries for the problems, the initial rule suggested was “no plus less than 12.” You can imagine the conversation that ensued over the meaning of that. . . it was a perfect reinforcement of my prior lessons on the need for precise language in describing mathematical situations, (though I chose not to mention that in this lesson.) After we established what that meant, I wrote “No + <12” on the board. (I take every chance I can to reinforce the “greater than” and “less than” sign, as kids typically confuse those or don’t even name them, preferring instead to use the alligator trick.) Someone then suggested no multiplication over 100. I probed as to how many digits could they use, did that mean we couldn’t ask anything beyond 10 X 10, and the agreement became 1 digit by 2 digit problems up to 99 as the highest number, so I added that to our list. I should have added n x nn, with n being 0-9 and nn=<100, to have that chance to reinforce a *variable*, the *less than* sign and *algebraic thinking, but I didn’t think of it at the time.* We stopped there, as I clearly got the impression they WANTED to practice multiplication facts, so I didn’t even address the other arithmetic operations.

Then I asked about checking correct answers. How were they to do that? They decided that first, the person who developed the problem would also work it, and they would compare answers. THEN they would check it on a calculator–and they wanted a separate calculator, NOT the one on the iPod. (I didn’t ask specifically why, but they wanted a separate one rather than move back and forth between apps on the iPod.)

When I went to this table, the problem they had created was 8 X 16. The kid working it, one of my best problem solvers, was doing something in his head , so I asked him to think out loud. He said he’d added 16 and 16 and gotten 32. I had taught the strategy of “doubling and halving” where if you double one of the numbers, you can halve the other one and get the same answer (i.e., 4 X 32= 8 X 16), so I asked him what he was going to do to the 8 now. He stared at me blankly. (**HMMM. . .I may have taught it, but he, at least, didn’t get it–and neither did his partner, cause she didn’t jump at the chance to answer that one!**) He then said, “128” and I asked him how he got it. He repeated that he had added 16 and 16 and gotten 32 and just kept adding. His partner volunteered, “I did it another way” so I asked how she did it. She described adding 16 and 16 to get 32 and then 32 and 32 to get 64 and then 64 + 64 to get 128. I didn’t probe as to how hers was different from his, (I think she figured he was continuing to add 16s) and I just wanted them to move to another problem to practice more without me interfering. I left them using the calculator to check, but realized several things about my teaching from their work.

- I had taught doubling and halving, but needed to work more with it.
- NEITHER of them went to splitting the problem into smaller parts such as 8 X 10 and 8 X 6. I need to help them shore up the various strategies they use, so they don’t rely on the same one all of the time, and learn when to use which to be more efficient.
- When I tried to suggest splitting the number, working 8X6 was not efficient for the boy, so he obviously didn’t know that fact. That tells me I need to work on fact mastery some more. This is a kid with an incredible memory, so I need to provide him some opps to practice the facts. Him not knowing that says I haven’t provided enough times to practice it.

I think that whole process of them deciding what kinds of fact “boundaries” to use was fantastic. How often do students get the chance to think about that and, even if they don’t say it outloud, in their heads they’re pondering about WHY they want to use 2 or 3 digit boundaries, etc. YOU didn’t set the limit. A worksheet didn’t. They consciously thought and talked about what would make the most sense, and I really think they made it more challenging for themselves. That whole discussion was very brief, but so meaningful. It was the perfect “argument” against the things I sometimes hear like “well, I don’t have time to do all that.” What did it take – 3 minutes?

By the way, thanks to you and them – I went home and spent HOURS digging through more ipod touch apps. There goes my weekend! 😉