Today is the last day I can treat myself to the luxury of sitting in a coffeeshop on a weekday morning/afternoon. I came here to think about what I wanted to do in the early days of my 9th grade class. In practice, this has translated into my spending most of the time solving and thinking about the “Tinker” problems. This has worked remarkably well to help me set priorities.
I wanted to teach 9th grade this year because I realized that I was not doing nearly as much as I could be to teach students how to be learners. Assigning students nonroutine problems has its drawbacks: though we have great class discussions and kids learn to see math for the open book that it is, students also have the perfect excuse to say, “I just didn’t know how to do this homework problem”, and teachers have the perfect excuse to give large hints that don’t empower kids to feel that they could have solved the problems themselves. 9th grade seems like a good opportunity to focus on changing some of my practices. We’re all making a fresh start.
Here are the things I most want to work on with my students this year, things that we need to establish in the earliest days.
What it looks like to have tried a problem. The message of our first habits lesson, “Tinker”, is conveniently relevant to this question. It means that you’ve written down something, anything, that you think might be relevant to the problem. And if you’re a good tinkerer, you’re probably also looking at simpler cases, trying to find patterns, bootstrapping on your work so far by rereading your work and seeing if that brings new thoughts to light. More concretely, it also means making the effort to look up words you don’t know. I’ll continue to let students know that I don’t expect them to solve all of the homework problems, but I also need to let them know that I still have expectations for the type of work they’ll do on all problems.
What counts as a good written explanation of a problem. Students have plenty of opportunity in class to explain their work orally, and they get lots of natural feedback from their peers about how clear they are being. However — and shame on me — I‘m generally not giving kids much feedback on their writing until there is some kind of graded assignment. Meanwhile, all sorts of bad habits are going on in their homework notebooks. I plan to have students swap papers early in the year and give each other feedback. The principle I hope they’ll discover: we don’t need to stick to arbitrary standards like complete sentences or the formatting of the page, but we do need to be able to communicate our ideas in a way that is simple to read.
What to do if you don’t understand a problem. It’s Polya’s step 1, but how do you get there? I’m always reflexively rephrasing questions for students, sometimes before they ask. This is bad, bad, bad! Instead, I need to give this type of advice. The habits of mind are useful in getting a handle on problems, even before you go looking for a solution. As you read the problem: draw a picture or make up an example of the type of thing the problem deals with. Add details as you read further. Read the problem phrase by phrase. What is the first phrase you don’t understand? Can you ask a specific question that will help you understand the problem? If all else fails, ask a classmate before you ask me. Maybe you’ll be able to work it out together.
There are reasons why I’m not already focusing heavily on these issues: they take time, and I’m always eager to dive into discussions of problems, often at the expense of stepping back. However, looking at my list, it’s woefully obvious that students who develop these skills will be well-placed to learn mathematics… or really any topic. They’re what we should be teaching. I’m going to try. Hold me accountable, please!
What are the habits you want to instill in your students from the earliest days of class?
acohen77
acohenher
Anand
Angela
btabrisky
Mimi
Tony
Park School Math 
5 Comments
Excellent. I’ll be forwarding this to teachers I coach. I think mathematical habits of mind, general work habits, writing and communication habits, and conversational habits (accountable talk in math class) are all very important. There’s little doubt that if you let things slack until there’s a summative assessment, it becomes hard for many students, particularly those that don’t come to your classroom already having assimilated earlier most of the habits you value and want to spread, to ramp up their game for the “bit test.” While being completely compulsive about things like complete sentences might be overkill, I believe it’s well worth the effort (and not just for math class) to insist on clear written communication. Complete sentences, coherent paragraphs of explanation, illustration and/or proof, are much easier for understanding (and proofreading) than are fragments with so much left to the reader that it’s simply unfair to claim that the student/author has actually said what we wish s/he had. You can relax any absolute as you see fit: if a non-sentence nonetheless is clear and helps make the point satisfactorily (to the reader, that is), that’s one thing. If essentially the reader has to do the heavy lifting? Well, I’d not want to do that as a learner, a fellow student, OR as a teacher trying to evaluate student understanding.
Great post! In teaching these mathematical habits, do you just refer to them orally or do you have some sort of poster or chart you’ve found helpful as a written reference for students?
Michael: I suppose it depends on the problem, but I probably wouldn’t want a kid to write a paragraph explaining how they did something like graphing a quadratic equation. (At least, not more than once.) In their day-to-day work, what I’d actually like are clear labels, tags, and annotations to make their work understandable to somebody (like me!) trying to follow it. Something like “set y=0 to find the x-intercepts:” followed by the algebra. Likewise with more playful, “habits” problems. I’d like some guidance in how to read their attempts: “Trying <> now:”, followed by whatever work the student is doing. What I’d really like is for students to have the impulse to “signpost” their work.
People often point to the use of symbolic notation as the reason they find math to be scary and confusing, but as reading an older text like Euclid shows, math can be pretty difficult to read when written out in words!
borschtwithanna: Unfortunately, I can’t say that I have focused systematically enough on the three goals I mentioned in order to know what short, poster-worthy phrases might be helpful reminders of them. Maybe I will discover something this year. But I have a feeling the only way I’ll really be able to communicate these things to kids will be through push-and-pull feedback, both oral and written.
When I initially commented I clicked the “Notify me when new comments are added” checkbox and now
each time a comment is added I get three e-mails with the same comment.
Is there any way you can remove me from that service?
Appreciate it!