Matrices Aren't Only for Algebra II

02/28/2012

There seems to be a lot of effort recently to teach math through "real world" contexts. You may scoff at this a bit because, as we all know, this is NOT a new argument. If you stop and think for a bit though, you might realize that this is still the "heart" of the reform movement in math. Project-based learning, interdisciplinary classes, and even "WCYDWT" (which have all helped to better education) have, at their core, the desire to show students that math really does exist outside of a classroom. To be clear, in many ways I think this is a HUGE improvement but, and maybe I'm asking too much here, I think we can do more than that (as I alluded to in a previous post).

I'm lucky to have a wonderful advisor, Stacey Callier, in my grad school program. I work in a project-based school and, as she knows well, I often push back against the hidden assumption in PBL that math is a "tool" that helps us solve real-world problems (I think the terminology I used today was that math seems to always be the "servant of science"). We started talking a lot about the constructivist philosophy that is central to my work and we (mostly she) came up with this matrix:

Is it better if we situate mathematics in a "real" context that students find engaging? Of course! I just think we can do that AND STILL honor a student's way of understanding and knowing, give them opportunities to author and create their own mathematics, and help them construct their own meaning for ideas that help them solve a problem. Call me an optimist (read: delusional), but I think its possible.

UPDATE 3/1

I have been thinking about this a bit since I posted it yesterday and I'm not too sure how I feel about it still. There are a few things that are bothering me:

1. The "HOLY GRAIL" label implies that this is where math education "should" be…which I'm not sure everyone would agree with (in fact, I'm not sure I can say that this is where I think it should be.

2. The top (applied vs. non) seems to get at "why teach math" while the right (constructive vs. non) seems to get at "how teach math." Is it ok to compare these two things in a matrix? Not sure.

Mostly, I labeled the top left "HOLY GRAIL" because I strongly believe that math should be taught in a constructivist fashion. If we can do this AND situate the math in a "relevant/applied" context shouldn't we do that?!?

Comments

Stacey

02/29/2012 7:35am

Way cool Bryan! This is a great place to build from as you craft your Understandings - your whole approach and goal can be illustrated with this. I can't wait for you to flush out what each of these quadrants looks like in practice! Tell us more!

Also, this blog is awesome. You've got to add it somehow to your GSE DP!

blaw0013

03/01/2012 9:44pm

As a learning theory (rather than teaching method, as you imply here), constructivism recognizes that anything said is by that of an observer. Why I mention that is, rather than evaluate teaching, or the classroom, or students experience of instruction through your eyes, what if you asked your question doing your best to imagine the student/learner experiencing of the interaction. In what ways would the developing mathematical mind be growing in "pure" ways versus "applied" ways? I suggest there is no difference, OR if there is it is how conscious the knower is of the connections in the ways of mathematical knowing to some "applied" context. [not sure what I mean by this all, expect to say that I think the goal for application is that of the teacher, and only a complaint of the child who sees no personal interest/relevance in crappy teaching/telling efforts--that is they were never puzzled (placed into disequilibrium)]

Bryan Meyer link

03/02/2012 9:14am

"In what ways would the developing mathematical mind be growing in 'pure' ways versus 'applied' ways? I suggest there is no difference…"

I would definitely agree and I think this is where some of my dissatisfaction with the matrix lies. For me, the goal of math education should always be about developing ways of thinking and helping advance students in response to their ways of knowing and understanding. My argument is that if we can use an applied context to get at this, why not? For example, you could use the grains of rice on a chessboard problem or you could ask students what the global population will be at a specific date and time. Similar trajectory for mathematical understandings, different contexts.

"Not sure what I mean by all this, except to say that I think the goal for application is that of the teacher, and only a complaint of the child who sees no personal interest/relevance in crappy teaching/telling efforts - that is they were never puzzled (placed into disequilibrium)"

The last line really caught my attention (maybe a conversation for a later date?). I'm not sure if the goal for application is solely for the teacher?? As much as I would love all students to realize that mathematical thinking and reasoning is a process that we all involve ourselves in on a daily basis (and is, thus, the 'purpose' of math education), this is not always an easy sell. "Math helps us understand and describe our world." Does it mean as a way of thinking? As a way of understanding applied phenomena? Are those the same thing? Do students understand that? Can it be both? Questions and more questions….I'm getting there.

blaw0013

03/02/2012 9:38am

"(M)ath helps us understand and describe our world" places an emphasis on an external mathematics, some "shared" conception of this thing we agree to call (M)athematics, the discipline of (M)athematics.

I am more comfortable with your statement, and interested in it, when it reads "(m)athematics is a part of our ways of knowing and thinking that helps us understand and describe our experiential reality."

Bryan Meyer link

03/02/2012 9:54am

"mathematics is a part of our ways of knowing and thinking that helps us understand and describe our experiential reality"

LOVE IT! Thanks! I had always thought about my previous statement in this way but never stopped to think about the hidden implications in the way it was written/said. Your clarification is a huge help.