There is the classic misalignment of what they think math is and what I want them to be able to do. The makeup of the class is almost entirely people who are done with math after this course. In our department, we're trying to separate this from precalc, making a new course for people moving on in math and this course will be for people finishing their math. The goal is to make this course more conceptual, and prepare students for use of mathematics in other subjects. Previously this course had so many skill objectives that teachers were put into coverage mode. Tenure track faculty teach it occasionally, full time affiliate instructors sometimes and most frequently adjuncts.

The course started off on the wrong foot. Two problems that have been a smash in the past went awry. On the house painter problem they were unable to convince themselves of the answer. And on the fair pay problem a misconception was shared and caught on so that it became insoluble. When they came back, one student had a nice intuitive solution that he could not convince the class - even with my help - that it was correct. Finally someone asked me point blank if his answer was right. "Yes, but one of our goals is that you are able to decide for yourself."

Ay ay ay.

By nature, they are reticent to talk to each other. Despite my urging repeatedly, and sharing how math is best learned through discussion. Many don't engage in activities, they're waiting for me to tell them how, many are not doing the homework, and absenteeism is about 20%. Standards based grading has been a tougher sell than usual because this class, as a group, wants the math that was.

I've removed a lot of choice, I've been doing more demonstrations and spending time on the teacher half of gradual release of responsibility, and I'm super explicit about what problems show which standards on assessments. They still won't talk, and many won't engage in in-class inquiry. They will make up something rather than ask about a question they don't understand on an assessment.

Ay ay ay.

The other day I saw this image on Twitter, but stupidly didn't catch the source. Simon Gregg remembered - it was David Wees!)

Our College Algebra picks up with quadratics, but a lot of the work I do with students visualizing patterns is for linear content. (We did do the growth problems, though.) Doing some other work I realized that the students were not understanding symbolic representations as generalizing number patterns. (There are even such quadratic examples on my own blog!) They had been getting by on regression rather than representation. I though this problem would be a great introduction to this kind of generalizing. It did seems to be helping connections form. I wanted to extend this to cubic or higher, so I built this pattern.

First we discussed what was going on, what they noticed and what they wondered about. Very few students wondered about how many cubes for the next building. More assumed that the next building was to be built with exactly that many orange blocks. That's very different than my thinking, and emblematic of how difficult it is for me to anticipate how this class will respond to prompts.

I built a very scaffolded worksheet. I used to make stuff like this all the time, but have been moving away from it. But sometimes students need supports.

Another adjustment I'm trying to make is, instead of roaming the room to eavesdrop and do formative assessment, to roam and ask questions, try to encourage table conversation, and hover over students doing work for other classes or just sitting. I feel a little awkward promoting engagement by (what feels like) intimidation, but students need supports sometimes.

One of the more successful areas of class so far has been the math writing. They have six assignments over the course of the semester, they can count towards SBARs, they can revise for their final exemplars. Several people are writing about this problem for their current writing. That tells me this was at a good place for them, and I'm happy to see some of the sense making.

Meg writes:

Michelle:

The strength of the temptation to give in, to teach them how they think they'd like to be taught is way stronger than I would have guessed. But so is my stubbornness to not give in. What I am trying to be wary of is to keep my stubbornness from stopping me giving the support that students need and teaching the students in front of me, rather than some fantasy class.

"The strength of the temptation to give in, to teach them how they think they'd like to be taught is way stronger than I would have guessed." Thanks for your honesty! It's very tough sometimes to know what's right, because you do have to work with what they want and expect: if they tune out completely or stop showing up, nothing you do after that will help.

ReplyDeleteI've met you, so I imagine you are very encouraging and positive. I don't think things like using proximity to get someone to quit working on another class's assignment during your class are a problem: that's just reasonable. But with this crowd, you basically cannot oversell your confidence that they each, as individuals, can do math. (And if you don't have this confidence for your activities, then yes, you need to alter the activities.) Part of their passivity is probably having others tell them, implicitly or even explicitly, that they are not "math people" and SHOULD wait to be told the right way to do things.

On a more specific issue, by "getting by on regression rather than representation," do you mean they see a pattern as "adding 3 every time, so it's y = x + 3" rather than "x sets of 3 plus another 4, so y = 3x + 4" ? If so, the best way I've found to break through this (and I have found it is SOOOO much easier to break through with 6th graders, even the least confident mathematically, than with 8th graders, so it may take a lot with your age group) is to get them to think about Figure/Step/whatever #100. "We definitely don't want to draw all the figures up to 100, or to keep adding 3 a bunch of times, right? That would be a pain. Can we find a shortcut?" A lot of times they are able to describe what the parts of that would be and how big each would be. Then helping them move from doing that to doing it for a general x is not such a big step. That intermediate step helps a lot with the jump from the concrete to the abstract.

Thanks, Julie! I appreciate the general encouragement. It's probably wishful thinking that I could be intimidating. I am big on encouragement; I never say this is easy. I do say that it is hard, BUT I know they can do it. Our semester long theme is 'what do you do when you're stuck?' and we explicitly work on forming questions. Or, I try to. We're slowly getting some people to come to the board to share work, rather than answers.

DeleteI think that kind of linear generalizing is something they haven't done, which is why the block problems were so good for us. The regression is literal here. With a table of data, they don't think about getting the expression from the pattern, nor when they have an expression from regression, try to see it in the context. They treat equations like poorly understood magic, which require arcane handling. No sense making.

John,

ReplyDeleteOur project supports a whole bunch of schools with students with an extremely wide variety of mathematical understandings. We've been asking our teachers to use a couple of different instructional activities with their students and many of them have, and found that repeated use of these instructional activities helps with a bunch of different things including student confidence in mathematics and students capacity to talk to each other about mathematics.

Some sample tasks are here: http://math.newvisions.org/instructional-activities Each task has a link to the "how to guide" for running the instructional activity but for Contemplate then Calculate, this guide is the best I've seen: http://tedd.org/?tedd_activity=contemplate-calculate-submitted-bpes-boston-teacher-residency-program

Contemplate then Calculate asks students to first identify what they notice when an image is flashed. By flashing the image for just a moment, no one can notice everything and so students have to rely on each other. More importantly, there are some critical things to be noticed in a mathematical problem and typically we find that many students have not been taught to stop and see what they notice before attempting a mathematical problem. Once noticings are recorded for everyone to use, we ask students to try and find a strategy for solving the problem, efficiently, in their head. Some of the strategies that students come up with are shared and annotated so that every student is likely to understand the strategy being suggested and why it works. At the end of the activity we ask students to reflect on their work and see what thing they think can use again to solve a different problem in the future.

It's a thing to try if you have not yet.

David

And sorry, related the image you chose above that I created was for the second type of instructional activity we support called Connecting Representations.

ReplyDeleteI've used a number of new visions activities for my college algebra. It seems as though we are trying to teach it in a similar way and are seeing similar frustrations. Solidarity, John.

ReplyDelete