Wednesday, September 22, 2010

Confessions of a Math Teacher

I believe that my job as a math teacher is not to make things as easy as possible for my students.  I do not give them the shortest explanation with the most straightforward formula and ask them to solve different versions of the same exercise for homework and tests.  I do not tell kids how to do problems from the next section in the textbook so that their 1-29 odd homework will be as painless as possible.  I will not become impatient and compromise a deep conceptual understanding and let kids get away with just knowing the procedure.

Or at least I try really hard not to do these things.

I always let kids explore an idea and try to build an understanding themselves before I tell them anything.  I will give homework problems that are unlike any problem we've talked about in class.  I do think "play with it...look for patterns...make observations" is a legitimate homework assignment.  I do ask students to work on messy problems, problems that I don't know the answer to, and open problems.  I have let kids leave the classroom with a misconception that I know about and did not address.

Or at least I try really hard to do these things.

I see the need to teach math as a verb.  I teach the activity of doing math as much as I teach the subject of math.  This means teaching kids to enjoy, embrace, and feel confident working with hard problems that they won't solve in thirty seconds, five minutes, one sitting, one week, or maybe ever.  This means caring about beliefs, perceptions, and attitudes as much as I care about fractions.  I want kids to enjoy the process of thinking, playing, trying, failing, and lots of other -ing words.

But this post isn't my call for you to see the wisdom of my ways and throw out your lesson plan for tomorrow.  No, it is a plea for help.  Tonight I don't need help with fractions.  I need help with teaching the activity.

This student has been on my radar after an email from the father about frustration at home and "disliking math for the first time."  A conversation was had, but I now know that the attitude equivalent of a deep conceptual understanding has not been made.  The one positive note is that the student is talking to me about this instead of repressing these feelings that will develop into yet another adult who hates math.

Via email: "  I just want you to know that even though you said [all] the [homework] problems aren't mean't [sic] to be finished sometimes when you hand in unfinished work it can make you feel stupid and like you failed. I thought that that was something you should know."

So any ideas on how to teach "the activity of not finishing problems in five minutes" to a super smart sixth grader who still participates actively in class but is clearly not enjoying math on the home/homework end?  If it helps, here and here are the homework assignments that sparked this frustration.

It also might be helpful to know that my policy is that students should spend no more than 30 minutes on homework.  If they spend any more than 30 minutes it should be because they'd rather be doing this than playing, reading, running around outside, talking with friends, or whatever else they're allowed to do.  For these two assignments, the first 2 problems gave me all the information I needed to know about their understanding of the content at hand.  Aside from the students who tried to compute 2 to the 57th power, everyone made it past #2 but only a handful finished the sheet(s).


  1. This is a constant problem, especially in math, where so much of what goes on is invisible. And not to go all pop psych on you, this definitely seems like a growth/fixed mindset thing. Ben has a good post here that's related:

    At your school you've got extra battles because of it being a school of high achieving wunderkinds whose parents are high achieving wunderadults. It's going to take a lot of culture shift. Specifically, I think you do a lot of the stuff you need to be doing. Celebrate the failures as much as the successes. Model your own struggles. It helps when they see it's not "easy" for anyone. Like Bree said in her post, the whole "easy" thing is a killer.

    As for the actual homework, if I were your student I'd probably say the main problem is that it's homework. Struggling alone SUCKS. You've got math circle type problems, without the circle. So perhaps switch it around so these problems they're working on together in the class, maybe in groups on self-selected problems/problem sets like the PCMI way that Sam Shah blogged about.

  2. How about putting a line under Question 2 (or 3, but not too much later - you want to be pretty sure students can do the stuff you don't label as optional in the time available), and then an explicit instruction, like "Only go on to Q3 and later if EITHER you haven't yet spent 30 minutes on this homework, OR you really want to." It may seem logically unnecessary if you've said this, perhaps repeatedly, in class, but writing down your intention might really help students (and their parents!) to believe you mean it. I assume they don't have grades that are affected by how much they do? (If they must, could you make the grade depend on the first 2-3 questions only, and of course tell them this?)

    Something I do with university students is to invite them (not require them) to tell me in their submissions how long something took them. This can be illuminating.

  3. I like what Perdita said, especially the part about making sure that parents understand that you do not expect students to complete all of the problems. You might consider numbering them as #1 and #2, then put the line and put Challenge Problem #1, etc.

    Or perhaps you feel strongly that they shouldn't be seeing these as separate?

    In The Teaching Gap, Stigler talks about the classroom culture in each of the countries he looked at (from TIMSS), and how hard it is to change that. You are getting pushback from a student who liked it the way it was. Often even students who do badly with the traditional math classroom (that's most of my students) hate to see it change.

    I do give them procedures and answers some of the time, to soothe them. I wish I could be as clear as you are, but I want to make sure my students are willing to follow me into the wilderness. ;^)

  4. My suggestion is to give this student the same problems to do a week or two from now, after whatever long-term goal you had in mind for the problems. The student may be shocked to find they can do these things that baffled them so thoroughly at the beginning.

    Problems like these that don't get solved in one day (or, more accurately, 30 seconds) are the most satisfying ones, so anything you can do to give this student that level of satisfaction, do it. You'll get the kid hooked again. I agree the pushback comes mainly here from wanting it the way it was, because it was easier and because you're dealing with what I called a "speed demon".

    Another way to deal with speed demons is to give several problems in a row that are related and have the same answer. Speed demons may not even notice this happening, and the result is they're more likely to "look around" a little more before and after working a problem.

    Good luck! You are the future of education in America apparently! Keep up the great, thoughtful work.

  5. This might sound overly simplistic, but what if you just made the homework assignment shorter? I think that struggling with and only finishing 2 of 3 problems might seem less worrisome than struggling with and only finishing 3 of 7 problems.

    And, to follow up on what Jason said about Math Circle problems without the Circle, it's important to realize that homework is a different beast from in-class work. The environment is different from class and will be different for each kid. It's hard to find work that will meet each student's needs. I think having clear expectations about homework is key, as well as having students be super aware of what the goal is for this particular set of problems.

    I know you were at the Deborah Ball talk at NCTM. Maybe channel her a little with the next few homeworks.

  6. I just read the comments on another of your posts and realized I cited the same post of Ben's you cited earlier. Whoops.

    @bowen - re: giving same problems. I like this idea some of the time, but it requires careful picking of problems. I think if I didn't care about solving the problem in the first place, I certainly wouldn't care the second time. Even if I could do it now. So perhaps this is where self-selection comes in and it seems similar to just-in-time project based learning where you have a problem/project and learn different things as needed to solve it.

    @betweenthenumbers - Can you elaborate on your Deborah Ball comment? Thanks.

  7. I agree with the desire to give problems students struggle with. I also agree it is hard to turn in incomplete work. One thing I do is to ask them to attempt the problems at home, but not turn it in until they have had an opportunity to go over it in small groups, and as a class (if the small groups can't get them). If you have them use a different color pencil or pen at each step, you get the initial attempt and lots of the information you want and the student gets to turn in completed and possibly correct work. I only grade on what is handed in, but I comment on all three stages. You can use this method with almost any level or set of problems.

  8. @Jason: What? You weren't at NCTM to hear Deborah Ball? Oh, that's right, you're a *science* teacher. :)

    The way Deborah Ball did homework with her summer lab classroom was (this is from memory, so I might miss some of the details) to break it up into three sections, each of which was labelled and had the instructions written at the top. The first section, which contained pretty standard looking problems, was practice on skills the students had been doing in class; they were supposed to work on these alone. The second section was also on things they had been working on in class, but these problems leaned more toward the math circle type problems; students were supposed to work on these with an adult and explain their work to said adult--DB said the goal of this section was to communicate with families. The third section was new types of problems, which were similar to what the students had been doing in class, but that previewed what they would be doing later and that pushed students to expand their thinking; students were instructed to work on these problems alone.

    Just what the goals are for each section might vary depending on what level you teach or what you want students to get out of the homework. What I meant by 'channeling DB' was the idea of having goals, including the possiblity of having multiple goals, and writing these goals out explicitly on the homework assignment.

  9. My first thought is to label certain problems as 'advanced.' However, I believe doing so creates a hierarchy in the classroom and can serve to reinforce perceived ability tracking. I am curious if there is a way to call a problem advanced without implying that students who struggle are not advanced. One thing you could try is to preface different problems with comments like "warning: this problem is known to cause a head-spinning sensation in some problem solvers" or "this problem is not for the faint of heart" or (when a problem has a historical significance), "don't fret if you find this problem difficult - it stumped some of the greatest mathematicians in history." This is all speculation, but I think there must be some way to warn students about the difficulty of some problems without implying anything about their problem solving ability.

  10. I commented on gmail buzz but I am really intrigued by this because your idea is exactly the opposite of the philosophy of the really great math teacher with whom I work now. Her philosophy is that homework should always reinforce a feeling of success for students. The struggles can come in class. She does give different levels of homework and allows students to choose whether they want the "challenge" homework. I think this echoes what another commenter wrote about home not necessarily being the place for frustration to be productive for some students. It also comes out of working with a population of students who have had a very "traditional" math education and haven't been pushed towards much exploration. I guess I would hope to start with the success and then slowly push towards the productive frustration over time. If that makes any sense.

  11. This post reminds me of my first year teaching at my current school. I had a mother call me about her daughter, who was in tears over the physics homework I had assigned. To be honest, I was thrilled that the student was interacting with the homework to that extent, though having gone through the physics homework crying episodes myself in college I also know that while it is happening it feels terrible.

    The student came into class the next day and was an excellent participant in our class discussion about the homework and went on to become one of my top AP physics students the following year. She got full credit on the homework for attempting every problem, regardless of whether she kept her same answers after the class discussion.

  12. I don't know much about teaching, but I know a thing or two about shame. For anyone who feels pressure to be perfect, an assignment like that would be hell on earth.

    I suggest letting the student know that the homework is for his benefit, not yours. If it's making his life worse, he shouldn't do it.

    If I'm right about what he's feeling, this concept will completely blow his mind.

    While I'm on a soapbox: If I had my way, grades would be abolished. They're a net negative to the learning process - what you gain in feedback, you lose in the needless stress of constantly having to second-guess the teacher's motivation.

    What matters in life is how well you live up to your own standards, not those of anybody else. Why do we constantly hammer home the opposite message every day of every kid's life? It's barbaric.

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