Last week, I blogged about an experience I had when I was forced to figure out a different way to factor than I had ever seen before. I had to really struggle before it clicked for me and I saw how the problem was supposed to be taught.
As I wrote last week, I could work backwards from the answers (given and pictures below) and I could figure out the math, but I couldn’t initially figure out how to work forwards from the beginning AND explain it to the students in a way that would be meaningful.
In my blog last week, I explained how I finally understood problem c as finding groups of four. Thinking that I had figured it out, I moved on to the next problem in the series, f, where I got watch the deep structure of the Eureka Math program in progress in this next problem:
I started like I had before, thinking I was in good shape–rewriting multiplication as repeated addition:
Then I realized it was a “Eureka” moment (ha ha), where it wasn’t just “do more of the same” like the last problem–just undoing the multiplication as repeated addition wasn’t enough. I had to regroup, multiply, and factor differently to get groups of four:
Sigh. Every time I think I’ve figured it out, it changes again! This is why I struggle to write my own problems to augment Eureka, because I’m not deep enough of a thinker in the math to be able to make those shifts on my own. This problem, shown in the final Hint below, challenged me to not just do “more of the same,” not even as a teacher:
Doing this work for ASSISTments gave me a clearer sense of the scope of the Eureka Math program, a clearer sense of how each problem builds on the one before and the one coming. In the first example that I shared last week, I didn’t initially see the repeated addition until I was forced to put it into words to create a hint for students, In this example, which builds on the one from last weeks and takes it further, I had to find more than one way to find the repeated addition before I could use the process from the problem c from before. If I hadn’t been making these hints, I wouldn’t have seen that growth in complexity across the problem set.