This week, I've tried a few things with my two math classes (honors juniors and standard seniors). One, a scaffolded proof of the Sine and Cosine rules, went horribly for some very mysterious reasons. The other, a Binder check procedure I've copied from another blogger, is going very well.
What didn't work: First, on Monday, the seniors were learning about the Sine and Cosine rules. I hate how their textbook presents these, like they are some inscriptions on stone-tablets, and even thinking about proving or questioning them is heresy. My students typically have the confused and bewildered "wtf" expression when they see such things, and last year I had a successful class with students in groups proving these rules with some little scaffolding from me along the way. Scaffolding such as "maybe pythagoras' rule would help?" was enough, but the students took a full 90 minute class to complete the proofs, and this year I wanted to save time by providing more (much more) scaffolding.
So I made these worksheets. I'm trying to figure out how to include them in a post such as this, so bear with me while I experiment with different options.
Here is the file with the two worksheets, in pdf format.
I was particularly happy with having the students first try a specific case with rulers and all. Especially when it turned out that trying a specific case wasn't all that easy for them. I think by actually measuring and trying it out, they gained a good understanding of what the rules actually say. I will keep doing this type of "physically trying the rule" for other trig or geometrical rules.
The rest of the worksheets however, went not as good, to put it very mildly.
Above all, the students (even the stronger students) immediately complained they didn't understand what to do. I explained, and they moved one step along. Even though they were sitting in groups and usually have no difficulty discussing with each other, this time they didn't discuss, they kept just calling me over and asking questions, and I had the impression they just wanted to be spoon-fed stuff. The breaking point was when one student wrote the Sine rule and when I asked how he arrived at it, he just said that it was obviously what I wanted them to arrive at so he figured why not just write it already?
Oh well. That's when I gave up. The activity was taking too long time anyway and so I interrupted their "work" and, after I cursed at them for being so passive and not thinking for themselves (I'm not proud of this, but my relationship to these kids is good enough that it can withstand occasional lapses in judgement on my part), I quickly went over the proofs by myself on the whiteboard. The remaining ten minutes of class students worked on exercises, very quietly, and I demonstratively sat and played on my iPhone.
If you're wondering, the following class (starting sequences and series) went great, and so I feel no lasting damage was done.
Now, this experience has me a bit disappointed and confused. I often work with this type of scaffolded investigative material, though rarely with proofs with this class, and most often it works very well - students enjoy the process and feel ownership over the formulas they are then asked to apply in various problems. Why didn't this work? Was it too difficult, as my colleague J claims? Was it poorly formulated? Was it just a bad day for my students (and me, obviously)? Maybe it's just the students' expectations and interests? They are not taking math because math is fun, rather because they have to and this is the lowest level IB lets them get away with.
I'm going to try this same activity with my honors juniors in a month or two, and will write more about it then.
What did work, or is starting to, is the Binder check procedure Sam wrote about a while back. I use it with my honors juniors, and yesterday was the second check (I plan on checking every two weeks or so). Results dramatically increased from the first check, as Sam said happened in his class as well, and I feel that this way of checking homework is giving me some great information about the students' work between classes.
An important difference is that I do NOT grade homework. Because my students will be assessed ONLY on the final exam (in May 2012) and two investigative portfolios, neither the homeworks nor the test they'll have on Monday actually counts for their final grades. I'm happy to say that the students know this, and that still they take both the binder checks and the tests very seriously indeed.
Some proofs flow in an intuitive way. I've never found these proofs intuitive. I wonder if that's a factor.
ReplyDeleteNCTM's Illuminations website has some worksheets that walk students through these proofs. They are pretty good, although I seem to remember there were a couple of typos on them. My students struggled with rearranging the equations and applying the transitive property in a proof, but those are things with which they hadn't had much previous experience.
ReplyDeleteI wonder if the students have ever persevered to accomplish something academically? I don't think it's uncommon for them to have a binary approach to mathematics where they either know what formula to apply or they just give up. Most students are probably unwilling to spend/"waste" time playing around with algebraic expressions to prove something, regardless of its novelty. If they've never been properly challenged in the past, it wouldn't be surprising if they expected to be spoon-fed the steps to a proof. I do hear stories about people who memorize proofs step-by-step for exams...
ReplyDeleteHmm, I like your worksheets. They complained immediately, i.e. they couldn't even write down the expressions for sin(A) and sin(C)? That says to me that, actually, they don't understand what sin is - which would then explain the rest. I don't find it plausible that the problem was that they didn't have experience playing around looking for proofs, because if you were using these sheets, you weren't asking them to do that - you were giving them instructions for every step and all they had to do was follow.
ReplyDeleteRe: Perdita
ReplyDeleteI did assume that the students were familiar with sine and cosine. Still, if that were the case, it's also a problem that they did not ask "I forgot what sin(A) means, can you remind us?"
I guess I am trying to get across the idea of a more general problem of students shutting down when faced with unfamiliar situations (i.e. doing a proof via a worksheet as opposed to a worksheet with exercises full of problems of a type they have seen before).
Yeesss... Of course, this is easily testable in principle: suppose you gave students a worksheet which didn't mention trying to prove anything, but just gave them a diagram like your first one for the sine rule, and asked them the first set of questions (to write down expressions for sin(A) and sin(C), based on the ABC triangle with the perpendicular drawn in): would they then be able to write down those expressions with no problems, or would they still be unable to do that? If they could do that, it'd suggest that, as you think, something about the aim of the worksheet put them off; if they still couldn't do it, then as I hypothesised they have somehow only understood sin as something to plug into the calculator or something to use in a different context. Maybe before you do this with your honours juniors it might be interesting to give them a sheet that was structured like a standard worksheet but that got them to do very simple applications of sin and cos in a variety of different shaped contexts? Then if they still had problems when you took them on to this proof worksheet, at least you'd have ruled out one possible cause of problems. (My guess, fwiw, is that your honours juniors will have no trouble with your proof worksheet, whether or not you introduce it differently.)
ReplyDelete@Everyone - The students were familiar with sine and cosine (and tangent) in right-angle trigonometry problems, and had successfully followed a unit circle introduction (needed for the cosine rule derivation I chose). So it wasn't lack of prior knowledge. I think it was, like Hao said, more unfamiliarity with this type of tasks, and the unwillingness to approach mathematics from this angle (rather than "show me what to do on an exam" angle).
ReplyDeletePerdita - I think you're onto something about introducting this type of thing without mentioning it as a proof. I wonder how to structure something like this without making it too structured and thereby inadvertently hinting that they're doing something different than normal.
I think the trig sheets went badly because they're too easy.
ReplyDeleteNow let me explain! I don't mean your students could do them in ten seconds - all the "uncomfortable" algebra-ry stuff is still in there, like re-arranging the equation. But all the "magical" brain-taxing stuff is done for them.
I contend that many students reading those sheets will see them as complete proofs, with fill-in-the-blanks. That will encourage them to wait and see what the answers are, rather than figuring it out for themselves. It's also inherently boring - you've made it clear there's only one thing you want them to write, so why bother fussing over it themselves?
My solution? Well, solutions are always harder than criticism. But I think if you want the students to derive proofs themselves, you *need* to give them a lot of time - as you say you did the year before. If that's too much of a commitment, how about presenting the proof of the sine rule yourself, and then challenging them to derive the cosine rule? Hopefully if they've just got the one to work on, you could get it done in a bit less time without stifling them with a prescriptive scaffold.
(Hao mentioned, over at Kate's, that this discussion and the discussion over there are similar.
ReplyDeleteJulia, what country are you in?)
Sue, I'm in Sweden, teaching IB courses though so I guess I'm in International-land. :)
ReplyDeleteAlex, you're right! I've toyed with this uncomfortable idea as well, that I made the proofs uninteresting by giving it all away and making it "fill in the blanks". Next year, I'll maybe do what you say - derive one and have them derive the other - though I'll do the difficult one myself.
I was thinking the same as Alex when I first read the post. When asking students to prove proofs of anything I like to give just pictures as much as possible and and them to explain. It usually takes a very long time and I have never actually done it as a class project but as a differentiation for individual students.
ReplyDelete