I imagine, as pressures came from within and without the classroom, teachers felt pressure to continue to move forward. Maybe you didn't have enough time to spend with the slower students to bring them up to where the others in the class are. Maybe you felt that you needed to march onward, ever onward to even mention all the things your state curriculum mandate. Maybe a number of other things could come into play.

At some point, though, you get to a frustrating point and think, "What's the least you can 'get' and still be able to continue?"

Even if a student cannot get motivated or understand the background or whatever,

*at least*he can be expected to memorize the times tables, plug in values to the quadratic formula, or use his calculator to solve an equation. And so we teach there.

Derek Bruff, quoting Mazur in a recent tweet reminded me, "A problem is when you know where you want to get but don't know how to get there. We usually assign just the opposite."

Of course, we'd love to tell our students to do what they can to build a bookshelf and let them figure out all the stuff they need to do it, find directions, and then start taking steps to make it happen. In reality, though, a large number of students will ask why we're wanting them to build it, another percentage will see the wide expanse of possibility in front of them and freak out not knowing where to start, and most of them will just ask you to tell them how to do it. So, we let them fumble for a bit, but eventually, we get to the point where we say, "Ok, if you can't do it on your own, here is box with all the pieces you'll need and here are some instructions."

We tell ourselves that the lowest common denominator should at least be able to put the pieces together when we hand them exactly the correct parts they'll need and even give them a map for how to put them together. Even if they don't understand the Swedish words, they can look at the pictures and put slot A into tab B. And if you can't even do that, then I don't know how to help you.

Plus, if students can't think on their own, they will likely end up in jobs where they are meant to just follow orders from their boss. So, we're teaching them the life skills they'll need for that level of job.

This turns our classrooms into an Idiocracy. The students get so accustomed to being handed all the pieces and the map that they don't know how to think about problems when they are left to figure it out on their own. In some cases, the

*teacher*may not even remember how to grow plants without using Brawndo.

Glad to know my Mazur tweet made sense--and resonated with you. And your point about teachers not remembering how to grow plants is an emperor-has-no-clothes moment. If I don't know how to solve a problem, I'm probably afraid of what will happen if I turn my students loose on it. It's much less risky to have them memorize and follow recipes.

ReplyDeleteOne of the main changes I'm doing this year is purposefully aiming my teaching at the next-to-highest quarter of every class. I read some research somewhere (don't ask me to dig it up) which said teachers usually aim at the next-to-lowest quarter of students, and I worry that this approach not only dumbs down math into this horrid boring collection of meaningless procedures - it also communicates low expectations which make students set low standards for themselves and therefore work less.

ReplyDeleteSo I haven't found raising expectations to be a problem. What IS a problem, however, is finding the time and competence needed to create rich problems suitable for all or at least most students in class. I teach psychology as well as math, and so in statistics it's easy to get student interest with a "do men or women have higher IQ?" problem which opens into standard deviation, correlation (husband and wife?) and pretty much whatever else you'd like. How do I do that in calculus?

If I had a background in natural science, it wouldn't be difficult - I suppose - but can/should we expect mathematics teachers to be well-versed in a variety of fields outside mathematics?

Meyer's WCYDWT stuff is very good, but am I mistaken or is it mostly aimed at "lower" math such as linear equations and systems of equations? It also works by creating a quite superficial curiosity (who REALLY cares how fast it takes to run down an up-escalator?). In contrast to Cornally's approach using fascinating problems from physics (http://101studiostreet.com/wordpress/?p=1271 for example) Dan's stuff is entertaining and open but little else. In both cases however, you need the kind of time and skill on your hands that few teachers possess.

My solution so far is to focus on rich mathematical problems - with less emphasis on real-world context. This is working out OK except some students simply don't care about math that way. I don't know how to reach them.

So ultimately I'm saying that it's not just low expectations - it's also lack of teacher time and skill - which are the big obstacles to how teaching math in a creative and rich way. And that's a hard nut to crack.