I also wanted to archive the random ideas I gathered from the Klingons, before they got lost in the ether:
- Keep a physical toolbox somewhere in the room. And when kids are stuck, make a dramatic point of walking to the toolbox, taking it out, and loudly plopping it on the desk. “What tools are in our toolbox?”
- Bring a construction helmet to class. When you need to get things settled and move on, put it on. “This is a work zone, people, a work zone.”
- Play “Math Taboo” where you have kids evidence their understanding of concepts. Have notecards with things like “Coordinate plane” and have them try to explain to their team what it is, but without using other words on the card, like “x-axis” “y-axis” “graph” etc.
- Ask a lot of what if questions. So, if you are in geometry and have covered that triangles have 180 degrees, ask: “What if we didn’t have a triangle, but a quadrilateral or pentagon? Would this still work? How many degrees do those have?” (This is very much under Polya’s art of problem solving philosophy.)
- On the top of every homework page, students need to write a list of problems they had difficulty with and circle it. If they didn’t have any difficulties, they can write the null set and circle that. On that vein, don’t put up the solutions to the homework problems that weren’t from the book (or the even ones from the book) until 2 minutes into class. Students need to be talking with their partner and comparing answers and asking questions first. Then halfway through “homework check time” project answers. (This is only for classes where you check homework.)
- Have practice tests (call them “scrimmage quizzes”) before tests, asking students to solve problems to assess their own understanding. But do NOT make them exactly like the summative assessment. They need to learn how to do problems without having the numbers be slightly changed. But make sure they cover the same ideas / understandings.
- When you’re in a zany mood, use phonetic punctuation (http://www.youtube.com/watch?v=lF4qii8S3gw). You know, just for fun.
- Have the class, at the start of the year, come up with a collective list of classroom norms. Make sure to refer back to that list throughout the year, and enforce it. These norms should be enacted each and every day. And students have ownership on them. (Add to the norms too, when need be.) Frame the norms positively. Also, collectively make a list of attitudes shared by good math students (e.g. tenacity, willingness to ask questions, etc.) and refer to those.
- Change language. Don’t call problems “problems” but “challenges.” Don’t call tests “tests” but “celebrations of learning.” Don’t write the number of points off, write the number of points earned.
- When students are asked to show their work to the class, don’t tell them to “show their work” or “show their solution.” Tell them to “teach the problem.”
- If a student shows up late, say to them “I’m so glad you’re here. Thanks for joining. We value your thoughts.”
- Keep a stack of postcards/little notes in your desk drawer. If a teacher does something really nice, or well, write a short note to the teacher telling “I appreciate…” and leave it in their mailbox.
Throw in other things below, if you want!
Continuous Everywhere but Differentiable Nowhere 
I need to hear more about the “Work Zone”. I love to dress up in class (hat) so it sounds like something I could pull off with beaucoup swag.
I have a few versions of the “Math Taboo” that I found online. One variation is that you make up the terms only. Then, the groups come up with the “taboo” words that the other group cannot say and write them on the card. Coming up with the taboo words is the hardest part and requires more thinking than even “playing” the game. But no, I haven’t tried it yet.
I did play a “Who Am I” where I taped a pic on each back and they gave each other clues. They loved it and even ended up announcing what they were in a “fashion show” style by sashaying down the center isle and doing the 3-point turn to display the card on their backs. <3 6th grade!
If calling problems “challenges” seems too dorky and try-hard, a subtler way to rephrase those is as “exercises”. In fact I think this is an even better word than “challenges”. I wrote a blog post on this back in 2009: http://www.xamuel.com/problems-in-mathematics/
Sam, “exercise” and “problem” are often used to distinguish between two very different sorts of practice for students. An “exercise” reinforces what they have just learned by repetition of what should be simple for them—it is intended *not* to be challenging. Some students need a lot of this sort of practice to reach automaticity, some need very little. Most elementary school math in the US is exercises. A “problem” or “challenge” requires thinking, not automatic application of simple skills. A good problem engages students’ attention for a long time and provides an “I did it!” reward on completion.
The same assignment may be an exercise for some of the class and a problem for others.
Giving only exercises leads to boredom. Giving only challenges will frustrate most of the class (but may be the only way to teach anything to the top 5%). Finding the right balance is tough.
The idea about homework is interesting and ties into your drive for metacognition. But I wonder if at some level, “you don’t know what you don’t know.” How much can a student judge about his/her own level of understanding? Isn’t the teacher more skilled at detecting problems and providing feedback?
I imagine that when you go to the doctor’s office, “Where does it hurt?” is a fair question and provides a good first level of information for further diagnosis. But in the math class, everyone is being rounded up for a check-up. How many would say, “Actually, I feel fine” ?
As a last point, this is potentially weird on a cultural level. If I’m turning in a problem set as a math major, no way do I want to identify the problems that contain errors. If I come from a culture that expects perfection on homework, then the act of stating the problems I struggled with might be a form of self-contradiction.
Hi! I think you’ve hit things exactly on the head. It is a cultural thing – and we’ve built systems all around us to lead students into thinking math is about perfection. Our normal grading system is exactly based on that idea. We don’t respond positively to errors in class, as opportunities to learn. The idea of perfection is dangerous, because it goes against the heart of what it is to really be grappling with math. So what I’m currently thinking about his how to concretely turn an environment/culture which is based on perfection into one where students AREN’T as concerned about perfection. It really does mean upending traditional notions of the classroom. And I don’t know how to do it. But I think there are lots of small things that can be done that can help.
As for the “you don’t know what you don’t know” that’s true. But kids DO know when they are stuck, or where they struggle. So writing that list of problems down can be helpful. And then when checking them all over, they can see if there is something they’re getting wrong that they DIDN’T think was wrong. That is going to help them identify the most serious errors — the things that they didn’t know they didn’t know.
Yes, definitely, there needs to be right answers in the math classroom. But I would not say that has to happen ALL the time.
As you can probably see, I’m still struggling through this.
Always,
Sam