Open Set

I was surprised by some of the questions I got the other day in my 10th grade class. It’s May, and if you teach, you know that there aren’t too many academic surprises by May. 

On this day, students were factoring quadratic trinomials, a review from 9th grade. They were doing boardwork, with their Visibly Randomized Groups and Vertical Non-Permanent Surfaces. (I am not a BTC groupie, but I do boardwork about once a week or so to mix up modalities and get students more engaged.) The questions I got were great! “Can I simplify x² + 3x because they both have an x?” And, “I have to do –2x + 5x. Is that 3x or –3x? Because a negative and a positive make a negative, right?” Students were looking at the details of how to solve these problems. Instead of assuming that they knew the answers, or just not caring and putting down any old answer, they were questioning. And the questions were very specific, not just, “Miss, what am I supposed to do?” Not that I never got questions like this earlier in the year, but there were more of them.

A few weeks ago, I got my copy of the new Building Thinking Classrooms problem sets for secondary school. What most surprised me when looking at the problem sets they offer was how slowly they built up. There could be eight or ten questions that were almost identical before offering a new wrinkle to the mix. They talk about thin slicing, but really, how thin can you slice it? Well, I decided to try making a more repetitive problem set once, and see where it went. The results? Most students were relaxed as they worked through the problems, the majority stayed on task, and I started to hear these questions that showed they were thinking more deeply about the foundations of algebra than usual. I covered less ground than I would have otherwise, but I have students who are more open to the work we are doing, and who are actively filling in the gaps that have plagued them in math class for years. 

What makes a question generate thinking, and what makes it feel like practice? When I was a kids, all we had were worksheets full of “ten problems just like this.” I hated them because they were too easy. Some people never understood any of it, but just repeated what the teacher did over and over. Some people enjoyed the relaxing, repetitive nature, and few might have learned something. Did those worksheets generate questions about missing skills, and lead to many students staying on task? My memory says no. So what’s going on here? 

Over the years, I have taught more than my fair share of honors and AP classes. I also taught at a school for gifted students for a while. The key to gifted education is that you need to let them create, synthesize, pull together ideas and do something novel with them. At the gifted school, one way I frequently planned a lesson was to pick a problem that they should be able to solve after the hour was up, and present it towards the beginning of the class. Then I would help the students break it into parts, or see how it connected to other things they already knew, and figure out how to solve it. By the end of the class, they developed that understanding for themselves, synthesized how to worked, created new methods, and understood more math. If I presented the topics linearly, bit by bit, I might have had their attention, I might not. But when I gave them a big chunk of an idea, they were excited to delve in and make sense of it. 

On the other hand, I think that all students need to have experiences like that. All students should take a big problem, see the component parts, connect it to prior knowledge, and figure out how to synthesize their knowledge to solve it. The biggest difference between the advanced students and the other students is just the size of the chunk they are ready to work with. Maybe for some students, that size might sometimes just be, “exactly the same as the last problem.” 

This reminds me of what happens in a humanities classroom. I’ve observed a few classes, from 9th Grade English to AP European History, where teachers are getting students engaged in a discussion about a topic. If the students are really engaged, they will work through an idea collectively. The teacher introduces an idea, a few students have new ideas to add, but a lot of the conversation involves students repeating what someone else already said. And often, I think that they think that it was their own idea. Maybe some students are just trying to get points with the teacher by talking without having anything to say. But you can definitely come across students who hear something talked about three ways, and realize how it works, and that realization feels the same to them as coming up with their own unique idea. 

If it takes students half an hour of discussion to internally develop an idea that the teacher introduced at the start of class, what does that mean for math class? I don’t think that our ideas are any less complex than humanities ideas – in fact, they can be more complex, with more moving parts and systems. (Math is hard!!) Maybe we can take a move from humanities discussions, and give our students a chance to listen to a group of peers saying the same thing, trying it out on their own, and then think that they came up with it independently. Maybe a list of “do ten just like this,” can sometimes let students feel like they are creating that knowledge for themselves as they get comfortable with it. 

The danger is that they approach the problem as ONLY a list of steps. Too much repetition might make them go into robot mode, trying to execute a list of simple procedures without noticing the problem and its components. And yet, going too fast can do the same thing. 

The text book our school adopted this year is a perfect example of this. The “Learn” section has definitions and procedures. Example 1 takes it up three notches, often reaching the highest level of complexity you would expect from an honors student, with nothing in between. Example 2 usually pulls in a new concept or technique, and Example 3 is yet another thing to memorize. The message seems to be, “If we tell you the information you need to solve something, then you can solve any problem presented that needs that knowledge.” As a result, students give up, and just wait for the teacher to tell them what to do. But boy, it does a great job of covering all the standards in the CCSSM, and more to boot! 

I’ve tried this extra-thin slicing twice now, and both times worked well. I’m curious if it will hold up over the next few weeks, and what happens if I try it again in September.