To continue with exponential functions, I started the lesson by expanding on a problem from the textbook. (Sometimes commercial math texts can have good problems. Sometimes.)
I thought that this problem is fantastic but was presented in a poor way. I could get a whole lot of math discussion out of my kids with that single pattern. Or at least, I was hoping that I could.
I drew the pattern out by hand and then asked the students any patterns that they noticed or anything that they wondered about it. Although I love the strategy of asking students what they notice and wonder, I have not reinforced it enough. It isn’t a part of my classroom culture. (I am sorry, whichever #mtbos person came up with that. I don’t know who you are or I would give you credit for it.) I really need to do it more consistently. In fact as I write this now I think that will be a goal of mine for next year. A daily, or at least weekly, “wondering and noticing” session. My students are picking up on the idea that “mathematicians look for patterns” but I haven’t done enough to give them practice actually doing that.
Students came up with some great observations for most of the day and some of them gave excellent descriptions of how each next stage comes from the previous one. The blocks responded well and I did hear some good math conversation between students and between myself and students. But I have a lot of concerns and they feel jumbled in my head, so I hope I can iron them out and communicate them clearly here.
- I am still doing too much talking. They depend on me too much to explain the answer. And I enable that. I need to work on being comfortable with letting students be stumped as much as they need to work on it. Invariably at least one student in each box was able to verbalize that each box breaks down into 5 boxes in the next step. A few students in the last block called it “cutting out pieces” which totally made sense to me, but I am not sure if it made sense to the entire class. But I ended up addressing the whole class once that observation was made.
- Although I feel I am pretty good at allowing student notions and conceptions to guide discussion, I get too focused on the end goal and can cut off valuable side-tracks. (I often ask “Why did you answer that way?” to both correct and incorrect notions because I value the students explaining their thinking, even if it leads to the wrong answer.) For example, I took to calling them boxes today after my second class because several students felt “square” was ambigious and I totally agreed. Smaller square could make a bigger meta-square. I immediately explained that when I wrote square in the question I meant box, but in retrospect I should have drawn out that conversation more.
- They should be working in groups for tasks like this. What is the difference between a student telling the whole class the answer and me telling the whole class the answer? I feel that in either case the students can be sometimes robbed of working out the pattern. Now, I find value in a clear explanation and I think I am capable of very clear and helpful explanations when they are warranted. (And they ARE sometimes warranted.) One of my friends in graduate school said that I would sometimes go into “teacher mode” when explaining a problem. And also it can be good for students to hear a peer explain an answer rather than myself. But I should have put them in groups and had them work together while I walked around. The temptation for me to do whole class summary too early was extremely great and I prevented valuable mathematics talk with my choice of format.
- I need to figure out how to lead them to different representations better. There is a very cool connection between the pattern and the formula that describes the number of squares and I ended up giving direct instruction on it which I felt in this case was not the correct instructional method, but I also wasn’t sure how to help them see it without just telling them. How do you help students have genuine mathematical insights? I don’t consider telling them the insight the same as them having the insight. I believe the partial answer to this is choice of tasks and lines of leading questions, and I need to get better at both, but I also feel there are other tools there that I don’t know about yet. I would love to hear any that you have.
So the activity could have gone better, but it wasn’t a bust. I am sure there are other concerns that I have but I can’t remember now and it is getting late. After the pattern activity I did notes and direct instruction on exponential functions, followed by in-class practice. I will follow up tomorrow with more applications and practice.
Thanks for reading.