Last spring, Mathematics Teacher published my paper on mathematical modeling. In this month’s issue, they’ve published a response from Albert Goetz [$].
Goetz worries that our collective interest in mathematical modeling risks granting the premise of the question, “When will we use this?” Math doesn’t have to be useful, argues Goetz. It’s beautiful on its own terms.
An emphasis on modeling–seeing mathematics as a tool to help us understand the real world–needs to be tempered by an approach that gives some prominence to the beauty that abounds in our subject. I want my students to understand how mathematics can explain the world–there is beauty in that notion itself–but also to see the inherent beauty and magic that is mathematics.
Agreed. But I no longer find adjectives helpful in planning classroom experiences, whether the adjective is “beautiful” or “useful,” “real” or “fake,” each of which is only in the eye of the beholder. Instead I focus on the verbs.
Mathematical modeling comprises a huge set of verbs that range from the very informal (noticing, questioning, estimating, comparing, describing the solution space, thinking about useful information, etc.) to the very formal (recalling, calculating, solving, validating, generalizing, etc.). One of the most productive realizations I’ve ever had in this job is that all of those verbs are always available to us, whether we’re in the real world or the math world.
Existence Proofs
“Math world” is the only adjective you could use to describe these experiences. When students find them interesting it’s because the verbs are varied and run the entire field from informal to formal.
- Which One Doesn’t Belong?. Students notice, name, argue, etc.
- Malcolm Swan’s Area v. Perimeter Problem. Students choose their own rectangle, draw it, graph it, imagine other solutions, imagine impossible solutions, construct an argument about them, generalize to other shapes, etc.
- Polygraph. Students notice, name, communicate, eliminate, etc.
Trick your brain into ignoring adjectives like “real-world” and “math-world.” Those adjectives may not be completely meaningless, but they’re close, and they mean so much less than the mental work your students do in those worlds. Focus on those verbs instead.
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Featured Comment
We shouldn’t overlook the usefulness of using this part of math to model that part of math. I see calculus as a way of describing and analyzing curves, including their curvature. I see analytical geometry as a way of representing “pure” geometry. I even see algebra as a way of modeling numerical patterns. Modeling is not just about the real world.
10 Comments
Howard Phillips
March 6, 2016 - 5:50 pm -We shouldn’t overlook the “usefulness” of using “this part of math” to model “that part of math. I see calculus as a way of describing and analyzing curves, including their curvature. I see analytical geometry as a way of representing “pure” geometry. I even see algebra as a way of modelling numerical patterns. Modelling is not just about the real world.
Rene G
March 6, 2016 - 10:40 pm -Thank you for this! I haven’t read your paper so I don’t know full context of the response by Goetz. However, this struck me as a bit off:
“An emphasis on modeling–seeing mathematics as a tool to help us understand the real world–needs to be tempered…”
Math was created to help us understand the real world. I am not sure it should be tempered because that is its purpose: a tool to be used. The act of “doing/using” and even the outcome, certainly, is/can be beautiful. True, one could argue there is some validity in doing something because the act of doing it is beautiful…to the person doing it and perhaps a percentage of onlookers. But for a good percentage of the world, math won’t ever be “beautiful” – Useful and necessary? Certainly!
I happen to love math. For me, it is the problem solving aspect. I don’t love math because it’s useful. I take math for-granted. I need it–it’s there–I use it. But, it becomes (a verb) beautiful when I find a problem that stumps me and I work through the problem; or even when I see a model that explains something in the real world that intrigues me. There, that’s when I developed a relationship with math. (Hang with me here because this part is still fuzzy in my thinking…)
I think adjectives are only helpful once we have a relationship with math; and that relationship can only be developed through the experience of understanding why and how we use math BY using math in contexts that are meaningful to us (all verbs). (Is this “relationship” stuff crazy talk?)
Ok, here is an example that happened this week. I guess it is a lie to say I love math because I hate statistics. I mean, I REALLY hate statistics. It’s not just because my brother is a statistician researcher who has always loved complex math and tortured me ruthlessly every time I asked for his help with anything related to math and we had the same math teachers who would always compare his genius to my not-so-genius. (Yes, I know that’s a run-on!) And, it’s not the models, or the complex problem solving I hate; its the whole probability thing! I don’t like probability, I like certainty. Too bad, so sad, stinks to be me…certainty is just not…well…all that certain. I don’t like that. I also don’t like the way that bias can be, and often is, used in research simply by the type of statistical analyses we choose. And, then there’s the whole correlation issue. I want causation… But, oh, yeah, that’s not how the real world works either. So I am stuck.
Nonetheless, I have to use statistics so I WANT to like statistics. That’s why I started reading “The Lady Tasting Tea” by David Salsburg. NOW, I am developing a (still emerging) relationship with statistics. NOW different methods make more sense to me; that is their USE makes more sense to me. I am not in love with statistics, yet; and I might not ever be, but it is becoming my friend…and in that there is beauty.
So, yes, adjectives can be useful to describe how we feel or see the usefulness (or not) of math; but only because we have used (verb of some kind) math in context.
Gregory L. Taylor
March 7, 2016 - 5:55 am -I did quite a stint as both a math and a science teacher. General Math through Calculus and beyond, Biology, Earth Science and Physics as well, Pre Engineering sequence (PLTW) and Architectural Drafting. This happened with great overlap at a number of schools and districts.
I had a colleague when I was working at SLPS who was a Moscow trained physicist… he used to say math “has no reason to exist without science” (and of course he meant physics :-)). I used to HATE when he said that to my core!
Unfortunately, now I must agree. Problem solving and pattern recognition are offshoots of the real world in some way or another; and while math can be beautiful, it loses its meaning without context. Context provides understanding and utilitarian motivations for remembering what we “learn”.
At some point one might be absorbed by the beauty of mathematics, but math itself would not exist without application.
Sadly, I once thought, Mr. Physics was correct on this one. But now I realize it is not sad at all… I am taken by the PBL approach and the discourse model for making understanding out of struggle and process. Students “find” the value and usability of mathematics if given enough exploratory time, while we as teachers “hold back” on our desires to rush to scaffold them, or confirm their responses.
If we peel the cocoon a butterfly will not thrive… the same is true for our students, and brain science is finally proving this one out.
You just cannot “tell” knowledge.
Scott Leverentz
March 7, 2016 - 6:59 am -Does one of these avenues beget a reason for caring about the other?
And do we need to consider the age and brain development of the clientele we’re trying to convince with these arguments?
As a preteen I appreciated the structure and rigidity of much of the mathematics I was taught, but for entirely different reasons than I do now as an adult.
The uncertainly of estimation and the unknown appealed to me in a different way, but also in a way that has carried more consistency in my experience of it into adulthood.
I’m not sure what that all means, but I’m suspicious that we need to keep in mind WHO we’re talking about these ideas with.
mike
March 8, 2016 - 1:05 pm -Speaking as a mathematical and mechanical engineer I have to take umbrage with the preceding arguments that mathematics:
– was created
– is for
– cannot exist without
– etc etc
… the ‘real world’.
Keep in mind that this criticism is coming from someone who, of all people, should instantly see, use, prefer, and perhaps even be biased for the sheer existence of mathematics AS service to application and real life.
math is created all the time that has zero immediate connexion with any application or context to the real world. it may very well become useful later, but that is not the job, purpose, or raison d’etre of math. its truth does not rely on physical reality or even reference reality as a measure by which it may be considered or judged.
when applications ARE made, they are invariably done through those who see how it can be used to model things in the real world. but it doesn’t have to mean that.
Riemann’s theory of manifolds were a purely theoretical geometry that was later applied to everything from relativity to group theory and complex analysis.
Number theory is strictly about numbers, no real world application. Later this turned out to be quite handy for coding theory.
The list goes on and on.
It’s INTERESTING that math can model the physical universe so well, and certainly many of math’s important developments were based on physical modeling, but this is not what math is.
Art can be beautiful. So can math. Beauty if subjective; it is human. The physical world exists with or without humans. Math? It needs to be invented. Is it beautiful? You decide.
Kronecker’s infamous quote about integers comes to mind here.
Yours,
A dedicated and unabashed mathematical engineer.
Howard Phillips
March 8, 2016 - 5:57 pm -L I K E
Joshua
March 10, 2016 - 7:13 am -Completely buy the arguments for mathematical modeling. Of course, modeling vs beauty is a false dichotomy. However, the beauty/fun camp seems to need an algorithm to make their feelings useful.
I think trying to understand the source of the feeling can be helpful for planning classroom experiences. For example, for me, the sources of delight in math generally come from:
(1) sudden epiphany: struggle with something, then it suddenly becomes clear
(2) deep understanding: to really understand what something means
(3) connections: links between seemingly unrelated things, often related to previous 2 points
(4) rewarded laziness: where there’s a clear boring brute force method, but a better way makes things so much easier
How does this help with planning? Here’s a simple example. For a concept where the beauty depends on connections, the students need to have some familiarity with both ideas before they have a chance to see or appreciate the connection. If they haven’t, then maybe delay that unit or find some other driver of interest.
Dan Meyer
March 10, 2016 - 1:54 pm -Joshua:
These four sources of delight are weeeeirrdlly similar to Harel’s five kinds of need for math. You should really read that paper. And then let us know how his thinking intersects with yours.
Elizabeth (@cheesemonkeysf)
March 19, 2016 - 3:38 pm -I agree. Verbs are pretty much everything.
– Elizabeth (@cheesemonkeysf)