OK, I came up with an investigation in my class that I thought was really awesome. (See previous post.) It was simple, engaging, and collaborative, and the students actually argued about it and then rolled up their figurative sleeves and worked it out with only occasional intervention from me. In addition, there were multiple ways to solve it, and there was room for extension at the end.
But I felt slightly less cool when I realized that the next section of the textbook had a homework problem that was almost exactly like it. (It was couched in more pseudo-context about a greeting card manufacturer who for some unfathomable reason cared whether their card, when folded, was similar to the unfolded state….) So I’m not as blindingly original as I felt in the moment.
Nevertheless, I assigned the problem as part of the next night’s homework assignment. That’s when the real ego smack-down happened.
They had no idea how to solve it. I don’t think a single student could do it. (At least, no one would admit to solving it, which is another problem in classroom culture I need to deal with.) This was almost EXACTLY the problem they did as groups THE DAY BEFORE.
Here’s me.
So what is the deal? Does group work feel good to us as teachers, while not really teaching the students anything? Does it make us think they are learning individually, when really they are just leveraging each other’s partial understanding?
What can I do as part of group work to help each student to consolidate the whole process? (I am thinking journalling about the problem, but that’s something for next year, because there’s not time in the schedule to establish that as a classroom habit at this point.) Is there something not as time consuming I can do (or have them do) as a wrap-up?
Help me, math twitter blogosphere. You are my only hope.
beginnersmindmath 
Dawn, I’m not sure how to answer you, but I had a slightly similar situation in that I accidentally put a the same problem on a review sheet as on an exam. Not a single student commented, and many made silly mistakes – which tells me how much they actually studied. My feeling is that many students do not know how to (or want to) dig in and apply themselves to figuring out how to do something that they don’t immediately recognize. This is not a new problem – I remember being the same way as an adolescent, when I would routinely leave the ‘hard’ problems at the end of the homework assignment over because I couldn’t answer them quickly. I think this is a habit that takes a long time to develop, and perhaps it has to be part of the classroom culture from day 1. And I still don’t think you [generically, not you specifically] will succeed with everyone. But don’t beat yourself up – it’s part of YOUR process of learning.
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I feel like this is a chance to explicitly refer to the math practice standards. I’m thinking this is “CCSS.MATH.PRACTICE.MP8 Look for and express regularity in repeated reasoning.” Is that an appropriate interpretation do you think?
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This is a fantastic example of “Nothing Works”! Trusting any one strategy is sure to lead to disappointment. Your question about group work is spot on. For all the good things about it, it *always* runs this risk. Always! The culture we need to develop is that the group work is not about answer-getting: that is only the first step. It is about learning the underlying math. One of my colleagues frequently says to his students: “we work together today so we can work alone tomorrow”, or some version of that.
Thought experiment: you reverse the sequence. They fail to figure out the homework, and then they struggle with it in class. That *may* yield a better outcome, because their focus might be on learning the math, as they had previously faced frustration.
I would break up your reflection about this into two parts: on the one hand the general issue about group work, how to complement it with whole-class discussion, writing, and repeated exposure to important ideas in different forms. I suspect that the solution you seek lies not within group work, but in complementing it. And on the other hand the specifics of what made this particular problem so difficult, so you can pick up the pieces this year, and do a better job next year.
To be honest, I believe you should feel good about this episode: they didn’t learn as much as you thought, but you definitely did, and you have more students in your future than you do in this class.
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By the way, here’s a lesson on this topic. Too late for this year, but you might get ideas from it for next year. http://www.mathedpage.org/attc/lessons/ch.14/14.01-rectangle-ratios.pdf
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Thanks, Henri, this looks a lot more thorough. I will save it for next year. I particularly like the international paper standard bit.
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