My classroom is a game that my students play. I set the rules by how I allow them to succeed or fail in my class. If I’ve done it right, then the rules I set should motivate genuine learning and reflect that knowledge in the form of a ‘grade’. –Daniel Schneider
I’ve proposed a session for #TMC13 on “A map of problem-based class designs”.
Last year in pre-calculus, all my students had passed algebra 2, but some had passed with flying colors while others had passed with a D- (and a D- in geometry before that, and in algebra 1 before that). Everyone needed a different course. So, the rules were:
- Problems spanned a range of difficulty from pre-alg to pre-calc.
- You got credit for solving and presenting a solution, if you convinced the class you were correct.
- You got credit for writing up what you presented and revising until I accepted it.
- You couldn’t present something someone else solved, or write up someone else’s solution.
Result:
- Almost everyone participated wholeheartedly and solved problems on their own that were challenging for them.
- They all got better at writing.
- Not much pre-calc was learned, as even the strongest mathematicians dropped in and out of the main precalc problem sequence in favor of brainteasers.
- Presentations were a little desultory, since most students were totally unfamiliar with the problem the current student was presenting.
This year, we reorganized sections so that all pre-calc students had done reasonably well in algebra 2 last year. To get everyone learning the pre-calc content, I reorganized the course. The new rules:
- Problems almost entirely required grappling with and mastering precalc concepts.
- You got credit for solving a problem and presenting it.
- You got credit for writing up someone else’s solution—but half as much as presenting an original solution.
Result:
- People are learning pre-calc; some of them are mastering it.
- They are all getting better at writing.
- Presentations are better than last year and the discussions are more interesting.
- Some students have earned good grades by reporting their classmates results, without ever solving a problem on their own.
Compare these examples to Exeter’s nightly problem sets, or College Preparatory Math’s groupwork, or any other problem-based course. All make different choices about what the problems are, who solves what on what schedule, and how assessment is done. I’m convinced the rules defining a problem-based course are the main factor in determining what students actually do, and therefore, what they learn. I’d like to discuss the feasible and infeasible regions in this choice space. What choices do you go back and forth about, and why? What results are you trying to drive, or avoid? What choices are you excited about? Worried about? Why? If “it depends on the kids”, what exactly about them does it depend on, and how?
I’d love to start hearing people’s questions and answers about this now. We can summarize and continue the discussion in Philadelphia.
This is fabulous! Please, someone take notes for me during your session!
How do you get kids in shape for proof reading/writing? Where do you get problems from? What does class time look like? Do kids get frustrated? How large are your sections? How robust are your class designs?
I have been using problem-based techniques for over 8 years now – initially just adapting the Exeter problems, and eventually creating my own (and finally, with colleague support). Would love to hear the results of this conversation! Wish I could be there.
Very much interested in this talk. My school’s trig/precalc classes definitely need a makeover. I look forward to Philly!
@Sue: Maybe you should come! I’m also maybe going to be at Notre Dame with the Math Circle folks; maybe see you there?
@Michael: Proof writing: They hack out a draft with little direction, I give them feedback and they revise as many times necessary until the paper makes sense. They tend to need fewer and fewer revisions as the year progresses.
Problem sources: See comments near the bottom here. I’ve also been making them up.
Class time: 5×75 minutes. MWF we start with student presentations. Any time remaining, and all day Tues and Thurs, is independent work time. Kids work on whatever problems they are working on or on papers. Work time is a little boistrous as kids tend to work with / argue with each other, or get help on a paper from the person who presented it, or whatever.
Frustrated: Yes. Yes! (Remember this?) During work time my job is to circulate, check in on what people are working on, help them choose problems, encourage them when they’re stuck (and help normalize ‘stuck’ as the most common status of problem solving), sometimes suggest avenues they might try, occasionally help them see they’ve actually finished. There was a guy in Duluth, MN years ago who ran summer research experiences for high school students, and I remember he wrote “they don’t know how to start, and they don’t know how to finish.” I’ve seen that, too. I also have 3-4 students who have totally disengaged, for different personal reasons. So it’s not an unqualfied success.
Section size: one section of 29. My other classes are more like 20.
Robust: I could guess at what you’re asking here, but I’d rather just ask, can you clarify this question?
@Jim: Wish you could be there, too! You can participate in spirit though – would you mind posting here or emailing me (see ‘about’ page) some info about how you run the class, what the rules are, how it succeeds for you and what some of the unintended consequences are? Maybe how you’ve evolved the rules and why? More territory for our map to cover!
@Glenn – see you there. Maybe you’d be willing to take notes for Sue? 🙂
I will absolutely take great notes for Sue! I will take notes for myself and share them on my blog for sure. I am really looking forward to the thoughts and ideas for precalc.
Thanks, Glenn!
And for all the non-precalc-teaching folks out there, I’m hoping the session will be general enough to be useful for thinking about any problem-based course.
This sounds fantastic! I love how you are differentiating while at the same time holding everybody to the same generalizable standards of success.
I too have a couple of kids who have consciously and willfully made the choice to disengage, and I think there is a practical limit to what we can do about this at any given moment. Some kids are learning that exercising autonomy means living with the consequences of your choices, which is an experiential life lesson that everybody needs to go through at some point. Like @delta_dc, I believe that our job is to create the conditions under which learning is possible.
Can’t wait for your talk!
– Elizabeth (@cheesemonkeysf)
Wow. I love the way you’ve defined the questions (especially the way “feasible vs infeasible” highlights the tradeoffs). The one question I am most torn about these days is about readiness for future work. How do you make sure that the people who pass are ready for the next class? Or is that important? If it is, what does “ready” look like? How ready do they have to be? Especially where the students have a lot of autonomy in choosing topics, what happens when someone does a good job of writing up and presenting to the class on some topics, but by the end of the semester they are missing foundational skills that will be difficult, unlikely, or impossible to pick up in their next course? Or does the problem selection make that impossible?
What do you do if the class accepts some contradictory reasoning, or reasoning where the conclusion doesn’t follow from the premises? If someone’s writing is too brief, vague, jumbled, or copied for you to understand their reasoning, and it doesn’t improve with feedback, what do you do? I’ve had several instances of students using source material from the internet that they believe they understand but can’t restate in their own words, or even orient themselves to the questions I ask them about it. It results in students repeating things while getting increasingly exasperated… in that situation, should they just move on to another problem?
Can your students recognize the difference between when they don’t know how to explain something, and when they don’t know what it is?
How do you teach students to distinguish a conclusion that follows from its premises vs. one that doesn’t?
@Elizabeth – Yes, I agree … and yet, …. on our map we’ll have to ask about reactions to failure or to nonparticipation.
@Mylène – These are great questions of yours, definitely sprouting new axes in our choice space. Here are answers from my course at the moment:
I’m right with you on preparation. I’ve got some kids clearly prepared for the next step (which might be calc with me, so I’ll reap what I’ve sown), and others clearly not…need to figure a way to make this clear, to /them/. Also, my deepest values are around people experiencing figuring something out themselves, and I’m offering that opportunity but things aren’t structured to /require/ it, because of the tradeoffs I’m imagining. Much to ponder here.
To address flawed reasoning, I have a few arrows in the quiver, including (a) ask the speaker to clear up something “I’m confused about”, (b) let it slide, but then when people write it up, ask about it during the review process, or (c) pretend to go along, but then pose another problem that requires the first in a way that shows up the problem. (c) is best but hardest to pull off; I do (a) most often but wish I didn’t ’cause I want them to own the logic not look to me to arbitrate it.
I haven’t had too much trouble with things not improving with feedback, but when a second draft is just as screwy as the first, I’ll generally take the clue and sit down to do some one-on-one tutoring with that student on that topic. It’s about the only circumstance when I direct-instruct a whole concept. The draft after that is usually better. Copying from outside sources isn’t generally permitted. I sometimes let it slide for ELL’s if it’s embedded in logic and writing that is clearly their own and if the source is cited.
I’m not sure they can distinguish between don’t know and can’t explain, but one-on-one I can usually tell, I will sometimes use that moment to just hand them some vocab or a convention to help them express what they’re trying to get across.
I haven’t been teaching deduction directly. It’s funny – I observe that sometimes, a student makes a claim and it’s clearly not supported and the class is all over them, and sometimes if they are making an algebraic argument and don’t show the steps, they get called on it. Where it tends to break down is in modeling: if a student uses the data in a nonsensical way, they can often get away with it. The class is starting to ask “Wait, why did you divide by 5 instead of multiply or add or something else?” because I’ve modeled that. When the approach is nonsense, that question usually brings it to a screeching halt. I also model sanity-checking the answer by comparing it to some more-easiy-generated estimate, but that skill takes longer to master. I haven’t seen them doing it much. (yet?)
Sarah Langer is bumping into the kinds of questions we’ll be talking about as she considers how to structure her year-long project: choice tasks.
Dan, Yes(!), if you come, we’ll see each other there! I am coming!
Ah, Sue, it turns out I can’t get there this summer. I am bummed – you’ll have to represent me.
Ahh, I’ll have to get some lessons in DanGoldnerishness to represent you. May our paths cross soon.
Hi Dan,
I’m hoping to be able to attend your session; it sounds fantastic. You are listed on the official schedule, but not on the form Lisa asked us to fill out with our session preferences, so you may want to bug Lisa about that.
See you in July.
David
Hi David,
See you there! (And thanks for the heads-up; I just filled out the form and the session was listed, so I think we’re set….)
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