I used to think that the best thing I had to offer as a teacher was really good, clear explanations. I prided myself on my board work. My student feedback forms always said I made everything really easy to understand, so I thought I was really doing a great job.
But in the past few weeks, I’ve been doing so much reading and I realized that I came to my really good explanations through a lot of deep thinking about the topic, connections with other pieces of math, and applications to the real world. And that my really good explanations as a product of my deep thought were robbing the students of the most crucial thing I want to teach them: how to think their own deep thoughts. After doing Taco Cart I was more convinced of the power of students making their own sense of something than ever.
This week in MP4 (9th grade) we moved on to a new unit, Algebraic Fractions, and I decided that I wouldn’t do any serious direct instruction. Instead I made a worksheet that the students would tackle in groups for 75 minutes. It was super interesting to watch their process as they developed their own understanding of the topic. Here was the first section of their investigation into simplifying algebraic fractions. I explained that an equivalence relationship was something that was always true (though we should leave out the strange case of x being zero).
It was interesting to see how the students approached this. Some cross-multiplied and checked that the two sides were equal. Others plugged in values of x to check whether it worked or not. Some of the rules they came up with were great. I particularly liked Mahima’s: “If the variable is bonded to the number through multiplication, it can be simplified. If the variable is bonded to the number through addition, it can’t.” I lovedthat word “bonded” — it makes it sound like a chemistry lesson.
Some students struggled with the idea of writing a “more complex fraction” for themselves, so I suggested they could just write any fraction that they could think of. Some came up with nice things involving powers of x.
Then it was on to the more complex stuff:
Woah was this a doozy! First of all, I did not foresee that a few students would write a perfect equivalence relationship for the first one: ! In fact two of my six groups came up with this. I had to make a blanket statement that while this was in fact perfectly simplified, doing it would hide what I hoped they’d find, so I wanted them to only accept “nice looking” answers.
Pretty quickly groups zeroed in on 2 & 5 as ones that could be nicely simplified. Daniel said to his group, “you can only simplify it if the thing on the bottom is a common factor of both the pieces on the top,” which I was really excited to hear. The frustrating thing about this was that although Daniel and most his group completely saw that, one student in the group of 5 (my biggest group) was toiling away on his own, methodically reducing to 2, in apparent disregard both for the work he’d done on the first page and what the rest of his team was doing. I asked him to prove that his equivalence relationship was correct and he seemed a bit at a loss. Is this a danger of no direct instruction? How was one member of the group so far off from the rest? More to think about here.
Finding 4 & 7 as other ones that factorized was extremely tricky for the groups. When I told the groups that they’d missed some, most realized that something had to happen with 4, but they really weren’t sure what. Mathilde tried plugging in numbers for #4 and pronounced to her group, “it’s always 3, so it must equal 3.”
For whatever reason, this fell on deaf ears as they desperately tried to find a way to reduce it. Each time Sonia (the defacto leader of the group) would call me over and present me with an even more complex answer, I’d say, “Mathilde, what does it simplify to?” and she’d say “3,” to which Sonia would exasperatedly say, “But how??”
Meanwhile in another group Addison became convinced that it could simplify because the top was just the bottom multiplied by 3. But honestly, I can’t tell you how long they wrestled with this in their groups. The biggest hint I gave was to suggest to students who saw that the top was just the bottom multiplied by 3 to write it like that so that others could understand what they saw. That was enough for most of them and soon they were simplifying number 7 to .
Except Gianluca’s group said the most brilliant (and again unexpected thing): “Number 7 simplifies to for every number for x except 3, because if it’s 3, it equals 0!” I said that they were absolutely right, that they’d have to qualify their answer. I asked them if there was a similar restriction on number 4 and they quickly came up with -5. I had not planned to talk to them about restricting the domain at all, since it’s a topic we cover next year. I love these unplanned moments! The students do not think in the same ways that we do!
Thankfully most students said that you needed to factorize to look for common factors and cancel them out. And then they were off and running to the final page of the worksheet:
I have to say that it was extremely satisfying to see the groups do these two problems without ever having seen a similar problem modeled for them. They reasoned their own way to the method and to the answers. They did their own deep thinking, not simply mimicking what I showed them. That felt good. When they finished, David said, “Wow, my face is hot from so much thinking.” This is a direct quote and it is awesome!
I wanted to make sure that everyone really got it the second day, but again, I didn’t want to codify anything for them. I did want to try to find their misconceptions though, so in the second day we started with this:
Number 2 is so crucial, because I had students tell me that a. was 2 and that b. was x – 2! And interestingly lots of students had a correct as 1/2 but still had b wrong. Asking the right questions is so important to weed out those wrong thinkings. When we went over it, this gave me the opportunity to ask them what were we actually doing when we canceled out? What were we really left with? I also got asked if you could still cancel x + 1 with (x +1) or did it need to have brackets? Student questions tell you so much about their thinking, it’s a shame to let direct instruction rob them of that!
I was really pleased with number 3:
Well of course the students instantly took to the graphing calculators. The new color TI-84s let you see the hole at x = 0 for the first one really clearly. The kids got a real kick out of turning off the axes to see it even better. This also reinforced the concept that y = a number is a horizontal line.
But they were really excited by the second one, with its bizarre shape. They struggled a lot to come up with values that x couldn’t be. Most agreed that x = 2 looked impossible on the graph. I instructed them to go to the table, where they were all excited to see that both x = 2 and x = -1 produced an undefined y. And Robbert, a usually quiet boy, explained to the whole class why that was.
It was really fun to take this detour because I got to talk to them about things they’ll do next year, touched on the connection between an equation, a graph, and a table, reinforced the impossibility of dividing by zero, and gave credit to Gianluca’s team for making me think of showing them this problem in the first place. This gave the class ownership of the material and the feeling that I am willing to make a diversion for something they think is interesting, even if it’s not technically in the curriculum. It was an unplanned detour that came out of a student discovery.
Finally, the last question:
And then they were off to work on complicated problems from the textbook, which was full of super tricky things like factorizing out the negative and more. In all, this turned what I normally do in one day into two days (tough in my school since I only see them twice a week), but again I really think it’s worth it, because they are creating the math and thus doing the thinking. Really excited about where things are going in this class (and need to somehow import this to my Dutch classes. One step at a time, I suppose)!