What's the Difference?!? Applied Math vs. Doing Math

06/06/2012

I hear people talk a lot about the importance of context in mathematics....usually, I think, for all the wrong reasons. There is a common misconception that if we show students how they can "use math" that they will find it more enjoyable and see the value in learning it in school. I'm pretty sure the problem isn't that students really want to know how this stuff is valuable in usage. The problem, I think, is that when the subject is relegated to procedures/facts/tricks it often disrupts the way a student naturally thinks.

Applied Math

By it's very name, this approach implies that we are "applying" something. Usually, there is a mathematical topic presented and then problems are selected in which students have to apply what they have learned to solve these problems. Sometimes, the problem might come first and then topics are presented (as facts) to provide students with the "tools" necessary to solve a problem ("necessitating" content). The focus here is on math as a tool to solve a problem. The assumption is that the mathematics exists before, and inspires, the problem.

Doing Math

Doing math is an act of creation by students. Usually, there is a problem/task/situation that requires students to think in the form of reflective abstraction. Mathematical "facts" arise as generalizations students make by looking for patterns and consistencies. These facts might, then, be explored in the abstract. The focus here is on problems as an entry to creating mathematics. The assumption is that the problem exists before, and inspires, some mathematics (I say "some" because although that may be anticipated by the teacher, it is ultimately defined by the students).

As I see it, yes, context is important but not to show students how they can use math. The context is important to show students how their mathematics is a natural extension of how they think and live in the world. Constance Kamii says it best:

"Most math educators think about verbal problems (word problems) as applications of computational 'skills,' rather than as the beginning point that eventually leads to generalized computation, without content, context, or practical purpose."

Kamii, C. (1985). Young children reinvent arithmetic: Implications of Piaget's theory. New York, NY: Teachers College Press.

Comments

Christopher Danielson link

06/07/2012 6:16am

Kamii...the woman is quotable, huh?

I have colleagues who do a lot of kvetching about Fosnot and Dolk's imperfections in their "Young Mathematicians at Work" series, but their writing about "context" is spot on and completely defensible. They use the term in your sense of "Doing mathematics" and are very careful to distinguish it from what you're calling "Applied mathematics". Indeed, one of their imperfections (in my view) is the introduction of the painful-to-hear term "mathematizing" to refer to the process of abstraction you're writing about here.

They're worth a read if you haven't already. Also keep an eye out for their "Landscape of learning" metaphor.

Bryan Meyer link

06/07/2012 9:15am

@Christopher Thanks for the recommendation...I'll be sure to check it out. Can you elaborate on your distaste for "mathematizing?" You have me curious...

Gerry

06/07/2012 6:22am

Interesting post, Bryan... I agree with your premise, and might be able to add a litt bit.

One of the big 'red herrings' in current educational philosophy is the notion of 'authenticity'. We try to make everything 'authentic', try to find context (as you mention), try to make math more 'like the real world'.

The problem is, that's not the real world. That is just more math problems, morphed into pseudo-contexts so they sort of look 'authentic'. (which you also correctly point out)

Here's a line from a recent graduation speech I gave:

"Sieze the hidden agenda. There is a reason for doing most things, and it isn’t always obvious. Math is a good example. You don’t do math to learn math. Only a very small fraction of people care about conic sections or trigonometry. You do math to learn how to think, to solve problems, to consider alternatives. That’s the hidden agenda, and lots of people get rich doing those things."

If that doesn't look like math class, then we're doing it wrong.

Math is about learning to use tools, yes. It's also about learning and memorizing theorems, proofs, facts and formulas.

But math is really about learning to think, to analyze, to postulate, to problem-solve... if we get THAT part right, I think the rest just might take care of itself.

Christopher Danielson link

06/08/2012 7:40am

Re: "mathematizing". Just that I'm generally not in favor of the verbification of nouns. The phenomenon they use the term to describe is a-o-k with me. The grammar chafes.

blaw0013

06/08/2012 3:01pm

I just goggled the word to find out what it meant.

eek ;-)

blaw0013

06/08/2012 3:02pm

try to be snarky and then misspell googled (I blame it on the autocorrect "feature")

Jay Nihal

06/12/2012 8:20am

This reminds me of a discussion I had in college about whether or not numbers exist without the numberer. It seems to me that math is a non-verbal intellectualization of the structures of nature. The problem, it seems, as to the way it is taught, is that one is trying to verbally communicate a non-verbal system. That is a very hard thing to do. I think the approach you take, of leading kids to the question, the essence of the problem, is far more valuable than the traditional method.

I have been staring at this comment for 20 minutes having a discussion in my head...there's too much to say in a comment.

Bryan Meyer link

06/12/2012 3:50pm

@Jay "...whether or not numbers exist without the numberer." I...LOVE...this! This is something I have been thinking and reading a lot about lately. Even though I think most teachers have yet to ask themselves that question, the answer to it plays out every day in how we choose to teach. Sounds like you need to drop the whole "work from home" gig and become a math teacher!

Kelly

06/12/2012 3:40pm

YES! I totally agree with you in every way. I am currently taking a PBL course as part of a masters program, and the focus on application and "authenticity" is driving me nuts. For this class, we have to design a PBL unit in the context of biomass to biofuels. YUCK. To me, the proper context to presenting math content is one that allows students to make predictions, see patterns, and generalize; it has absolutely nothing to do with whether or not it's a "real-world" problem. The reason students hate math is that it doesn't make sense to them because it has been "relegated to procedures/facts/tricks"; not because they don't see the practical applications. It's going to be a tough couple of weeks for me..

Bryan Meyer link

06/12/2012 4:00pm

@Kelly I agree with SO much of what you write here. The PBL and PrBL approach gets a lot of positive press from people who know it at, what seems to me, a very cursory level. There are many variations of these approaches and, in my opinion, not all of them are valuable (however, some are incredibly valuable).

I have had similar experiences with courses like the one you describe. It would be awesome for you to use the opportunity to expose people to your thoughts on PBL and how it might be done in a way that offers students a genuine opportunity to DO mathematics and to think mathematically. It is so important that we push common thinking around this and challenge the definition of PBL. Design an awesome project....and keep me posted!

Anna link

06/19/2012 6:40pm

This post is giving me a lot to think about. Overall, I agree with your point that the context should be an entry point to creating mathematics as a way to describe/resolve/abstract the situation. Do you think that there is a valid and engaging context for any mathematical premise or conclusion? And if not, should the topics that don't have such an entry point be cut from the curriculum? I feel like I can come up with such a task maybe once or twice a week for a given class, but not every day and certainly not for every topic and often not in the time that's been alloted for that topic. Is this just a case of practice vs. theory or is there a bigger math ed issue here with the topics we are teaching or maybe with just the sheer quantity of them? I would love to hear your thoughts on this.

Bryan Meyer link

06/20/2012 9:15am

@Anna Thanks for your thoughts. I like the idea of starting with context because I think it mirrors the development and creation of mathematics. In this way, we can help students to see that math is something they create as part of a community of thinkers (which I try to create in the classroom). With that said, mathematics also ventures into the purely abstract. However, historically this development has also been in response to some question or problem...so I guess I wouldn't advocate that the question necessarily be "real world," only that it is perplexing and pointed in a way that presses on students' current understanding of math and the world. In other words, I like to lead with the question and leave students to sort things out.

I suspect I am in the minority when it comes to standards, but I think they are less beneficial than most people would believe. Could I live without them....I think so. In fact, they are highly under-emphasized at my school already. If they need to exist, I would prefer much broader standards....ones that get at big ideas rather than what might be considered specific skills.