Math is not a spectator sport…

… so why do I teach it like it is?

My classroom is mainly me standing in front, talking. A typical day goes like this:

Me: Check your home enjoyment answers with the answers on the board. Be sure to correct your work, and talk with your partner if you get something wrong but you don’t know why it’s wrong.

[Hand raised, and I walk over there: Hello child. What’s going on? Oh you want help? Did you ask your partner? No? Oh. Okay. Bye.]

Me: Okay, are there any questions?

[A couple minutes pass when we go over unresolved questions.]

Me: So today we’re going to build off of what we did yesterday…

Then I start teaching with a back and forth: me, student, me, student, me, student. Blarf. (I know, I know, I complain about this a lot, because it’s something I need to really work on. ) I usually introduce a new topic, ask a few questions, work through a sample problem with student input, and then have students work with their partners on a similar “check yo’self before you wreck yo’self” problem. Then we move on. [1]

I’m the authority in the classroom. The kids don’t see each other as authorities. Not really, not in any meaningful way. [2] That’s my fault. I don’t let them be authorities.

A couple weeks ago, in one of my Calculus classes and my only Algebra II class, I was ahead of schedule. So I introduced the material in the same sort of way, but then instead of me talking, us doing a problem, them doing a problem… me talking, us doing a problem, them doing a problem… I streamlined it. I talked and we did a few problems together, and then I let them at it. I give them their homework and let them start tearing through it.

I loved it. I mean, they were doing math. They were having trouble. But because they were in class, and not at home, they turned to their partners and talk. [3] I wasn’t the sole authority when they got stuck. They felt more comfortable talking with each other in this informal situation, instead of talking in The Big Scary Everyone Is Looking At Me back and forth we have when I introduce new material. And what I loved watching is that the kids themselves are seeing each other as authorities.

(They were simply happy that they get to actually spend time in class doing their homework.) (And I was happy they can have less on their plates for when they get home.)

Most of you out there are probably horrified that I’m just latching onto this idea now: give them time to work in class. Trust me. It’s not a new idea for me. I mean, obviously I always try to have my kids working independently in class. But I tell ya, it isn’t easy to do:

1) I suspect that most new teachers feel guilty letting kids work on homework or problems in class. They feel like they need to be “teaching” (which means: lecturing). I know when I first started teaching this was chronic. I would plan my 50 minutes to a T. (Whatever that means.) Even to this day I have remnants of that fear. A week ago, I gave my Calculus kids 20 minutes (out of 50) to work on their homework problems. Part of me still felt like I was doing something wrong. Like I was wasting valuable class time. Like I hadn’t prepared enough. Which wasn’t true. I had planned those 20 minutes. I have to catch myself.

2) More significantly, it’s hard to get through jam-packed curricula and have much class time to work on problems. I mean, let’s say you wanted to teach Absolute Value Inequalities in a single day. (Which, in some years, we have to do.) To get kids to the point they can work the problems, they either need:

a. 50 minutes talking through the concept and building their understanding so that they can conceptually understand the problems and the solutions, or
b. 25 minute lecture on the procedure to solve Absolute Value Inequality problems.

The first plan involves a lot of student thinking and discussion and a little doing. Mainly watching. Then the student will go home and practice problems with the knowledge they’ve gained from class. They’ll be alone at home struggling through if they run into problems. The second plan leads to a good amount of time for students to work out problems. But most wouldn’t know what’s going on — and they would memorize a bunch of rules. However they would get to work out problems in class, so that if they have trouble, they can find out what they’re doing wrong before they go home and struggle.

Clearly I tend to opt for the 1st. I could get through so much more if I dove straight for the method/procedure. But that’s not the way I roll.

I know this setup is a false dichotemy — plan a or plan b. There are probably lots of alternatives that I just haven’t yet seen.

My name is Sam and some days I feel I could be doing infinitely better at my job.

[1] Okay so this isn’t totally true. There are days where we deviate. But I’m illustrating a point here, so I’m going to gloss over nuances.

[2] Getting kids to see each other as authorities in the classroom was one of my goals this year. I feel like it has actually happened to some degree in calculus. My kids are helping the heck out of each other inside and outside of class. And I ask ’em to talk to each other before coming to talk to me.

[3] If they were at home, I’d hope that they use some of the strategies we talked about for when they get stuck. But I know that for many of them, it is wishful thinking. I’m trying to teach my students to be students. To learn how to learn well. But it’s hard and you can’t force it.



  1. That’s me! Did I write this?

    There are a few of the more intuitive lessons where the students can have a lot more input. But they’re not used to challenging or going for deeper understanding (at least not at my school).

    Like any kind of math or science research papers you may read, there’s not much time/space/whatever for following the dead-end paths. I’d love to field ideas for how to do this or that and pursue any feasible thought to see why it might or might not work. But for some things we’d end up spending a month following a path that doesn’t work and they’d get frustrated and throw up their hands.

    Anyhow, I totally do what you’re wrote about. I generally rush through the lecture as quick as I think possible and then throw questions at them to work on in groups or as a whole class. I try to step up the difficulty a little at a time and throw wrinkles in here or there.

    I’m trying to figure out a way of teaching calculus by only doing this. I took a topology class in grad school where the entire class was the Moore Method. We’d get a sheet of definition and problems and work through them as a class. The teacher mostly just sat there through class.

    I feel like carefully crafted questions in a certain order could do that in calculus. It may need to be supplemented with short discussions/lectures about definitions or theorems, but I think it could be done. (But, like the New Year’s diet, it’s always something I’ll get to tomorrow.)

  2. Yes – I just had this talk with the other ms math teacher at my school. I am really into them understanding where the math comes from, and pretty much anti-formula and rules as much as I can be. Thus, they seem to understand and “get it” but never have enough time to practice on their own in class. I’m not convinced that they do the homework thoughtfully (not eating, watching tv, texting, …) while doing it. So, I am not even convinced that they get any really good practice on some days. How do we fit it all in? I just taught a unit where they really got it in class (concepts, concepts, concepts). I was so impressed! But then, they bombed the test! If they understood in class and “got it” but then bombed the test, then they must not be getting enough practice. That is when I feel like I have totally failed my students. So then I make my pendulum swing the other way. White boards and practice, practice, practice in class. Why can’t I find a happy medium?

  3. Same struggles, although on a science teacher continuum. I had written longer but it sounded whiny. I’ll just add that I get a TON of mileage out of posting problems + answers and letting students figure out the method. It’s got low overhead, reasonably efficient, and they get good conversations out of it.

  4. First, I can’t comment without saying that I like your snow!

    Thank you for admitting that you talk. In all of the reading that I do, I feel like I’m the only one who spends lots of time talking in the classroom. It is not that I want to do that way all the time, nor do I, but it is efficient given the curricular scope. Not every concept can be investigated within reasonable time constraints (like @CalcDave said.)

    I am better at plying different instructional methods in Geometry than in Pre-calc Honors because the curricular pressure is not as great. However, when I tried this approach recently in PCH for verification of trig identities, I similarly found that the kids were on top of it. Even in their struggles, they were bouncing ideas off each other. One group would set off one way with the identity and the other group would try something different. They were owning the math. It was great.

    I think that structure needs to vary and is often topic dependent. And hey, I also think that most days I could be doing way better, too. But really, we balance a lot. If we can get kids to be nice to each other and to think, then we’re doing something right.

  5. The class/HW/test thing might be more about genre. They are three very different situations in which to be solving problems.

    The Look Who’s Talking measure of teaching carries a lot of weight for me. It’s incredibly hard to shake when it’s what students want and expect, parents demand, administrators look for, and all you’ve experienced. The Teaching Gap has a great part about how teaching is a cultural activity, which makes it hard to change.

  6. In our elementary school we have an hour for math 5 times a week. I frequently get the kids going on their homework. If not directly working homework problems, they are working a different problem set, either problems I write or (gasp) problems from the textbook! I know the kids appreciate a jump on the homework and I can reinforce good work habits; kids who mess around have more homework than the efficient, task-oriented kids. I roam the room checking work, answering questions and sitting with kids who need help. The kids are allowed to work in small groups as long as the group is productive. Disruptive groupings are not permitted and I have some students who must work solo. They stink at group work and will have to earn the right down the road.

    My classes tend to also follow Sam’s pattern; I do feel I talk a lot but I think my kids get plenty of air time. With middle school kids I find that getting them to stay on topic (and not launch into why they did the wrong homework assignment; I’ve told them I don’t want to hear it and neither do their classmates!) is vital. I will bluntly interrupt a student who deliberately moves off topic. Older kids do this less, but 5th graders are pros!

    So I am all for getting work started in class.

  7. The third way you were thinking about is the inverted classroom model. I’ve blogged about this a lot at my place and you’ve probably seen my tweets about it. Basic premise: You outsource the lecture component of your course to video podcasts, audio podcasts, and/or reading assignments and fortify student’s engagement with those things with a small set of highly targeted, very simple practice problems. Then you let them essentially do “homework” in class the whole time — though you can throw harder problems at them if you want, and then instead of lecturing you are just helping them work through their problems.

    In the usual model of instruction, we spend class time getting students involved in the easiest activities from a cognitive standpoint — listening and taking notes — but put them totally on their own for the hardest cognitive activity, namely assimilating the lecture info through working on problems. The inverted classroom model reverses this situation.

    There is a small but growing body of research on this mode of instruction at the university level that suggests it is very good at promoting deep learning and the kinds of skills needed for lifelong learning such as the ability to locate and evaluate resources for a problem without external cues.

    The main hitch with this is that it violates people’s notion of what a teacher is supposed to be doing. Parents, administrators, and lots of students have a hard time buying in because for a lot of people, teaching means lecturing, and if you aren’t lecturing then you’re just sitting back while students work. I’ve had students flat-out tell me that they CANNOT learn unless somebody is telling them what to do, and it’s hard work to convince them otherwise.

    1. I remember learning about this a few summers ago, and I totally forgot about it. Yes, yes, yes! Thank you for reminding me about it. When I first learned about it, it struck me as something I wanted to experiment with. It will take me some effort to do this, but I am going to try it for a week in one of my classes — maybe in the 3rd or 4th quarter. It makes SO much sense for many kinds of learners.

  8. Balance between “teacher in charge” time and “students in charge” time is the toughest to achieve.
    Last month, I presented an AP Calculus workshop for College Board and shared many student-centered activities (sorting and matching, Rule of 4 Link Sheets, brainstorming webs, etc). The toughest questions I got from the 30+ participants included how to accomplish the balance and, what was my %age of teacher talk to student work time.

    My answer was that originally, as a traditional teacher, I’d say it was me 80% and my students 20%. After years of working to create a student-centered classroom, I’d say it became me 35-40% and students 60-65%. The results on classroom, district and AP exams were the same as or better than they were before. And, students “liked” learning this way much more.

    Lastly, based on 25+ years of teaching, I’ve come to learn that some topics require more teacher talk than others.

  9. Sam,

    I now have time to breathe and to comment since winter break is on us…

    My high school Calculus teacher, the great Barry Felps, rarely spoke more than 15 minutes per class. He would answer one or two HW questions, he would introduce or extend some idea we were to engage with and then he sat and did homework while we were expected to do the same. While he worked alone, we were encouraged to work together and this was one of the most powerful learning experiences I ever had. However, I have spent my career teaching at independent schools where many of the students feel more stressed about achievement than my peers and I did. A generalization, I know this, but I have found it to be more true than not. This is usually a good thing in that my students are very diligent and conscientious. It is a bad thing in that they often bristle at an approach that feels different of feels like they might be making mistakes rather than learning the ‘right way’ to do things. We all know those mistakes are gold, but students need to be won over to that point of view. In my AP classes, the majority of the courses I teach, where the benefits of real reflection and conversation would be the greatest, I encounter the most anxiety about this approach. I WHOLEHEARTEDLY agree that I need not be the ‘voice of authority’ in class, I just need to keep working on making it work right.

  10. Some of us tell our students wrong stuff, and let the students correct us. It’s important that our students already recognize that we have a whole lot of mathematical authority — then they can argue the math against us without thinking we are idiots.

    But insist that 1/x keeps going down as x gets big, so it has to eventually get negative, and say it like you mean it. Just an example, but it is fine. Let the kids decide that you are usually the authority, but that the math is always the authority. (I would not convince them that they are the authorities — dangerous with know-it-all kids, let’s them think that having a good guess is as good as having a correct answer. Their opinions are important in composition and history, not here)


  11. You know, you could’ve saved that entire day by deciding that absolute value inequalities aren’t useful enough to be worth the class’s time … ;)

    Also, white snow + white background = thoughts of hallucination.

  12. Sam,
    In geometry, my last unit was on quadrilaterals. We spent a day taking notes on properties of squares, rectangles, parallelograms, and rhombus. The next day I broke them into groups with a sheet of 8 problems. Two of those were easy and the other 6 were pretty tough, problems I never would have given them for homework. Honestly, I didn’t think they could do it. I had each team start with the easy problem and gave them 1 minute no talking, to attempt their problem on their own. Then I gave them a few minutes to talk as a group and figure it out. I told them to make sure everyone in their group knew the answer and could explain how they got it. I then randomly chose a team and again randomly chose a person from the group. That person had to stand and explain to the class what to do to get the answer. Since the problem was easy, they felt more confident and more motivated to start. Because I was randomly choosing, students were more likely to participate and not feel like I was trying to single them out. They worked together, found creative ways to solve that I hadn’t notice, they discussed math, they taught each other, they presented before the class. The next day we added in our notes about trapezoids, isosceles trapezoids, and kites. Then we set off doing more problems. I find there is less to prepare because I’m just preparing problems instead of lecture notes for me to talk all hour. Also, now the students are doing the work and I just check and correct.

    I’m not smart enough to know how this could work for Calculus but it is a ‘group work’ structure that I really like and that I think really helps the students.

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