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Henry Pollak, in his essay, “What Is Mathematical Modeling?

Probably 40 years ago, I was an invited guest at a national summer conference whose purpose was to grade the AP Examinations in Calculus. When I arrived, I found myself in the middle of a debate occasioned by the need to evaluate a particular student’s solution of a problem. The problem was to find the volume of a particular solid which was inside a unit three-dimensional cube. The student had set up the relevant integrals correctly, but had made a computational error at the end and came up with an answer in the millions. (He multiplied instead of dividing by some power of 10.) The two sides of the debate had very different ideas about how to allocate the ten possible points. Side 1 argued, “He set everything up correctly, he knew what he was doing, he made a silly numerical error, let’s take off a point.” Side 2 argued, “He must have been sound asleep! How can a solid inside a unit cube have a volume in the millions?! It shows no judgment at all. Let’s give him a point.”

What a fantastic dilemma.

Pollak argues that the student’s error would merit a larger deduction in an applied context than in a pure context. In a real-world context, being wrong by a factor of one million means cities drown, atoms obliterate each other, and species go extinct. In a pure math context, that same error is a more trivial matter of miscomputation.

The trouble is that, to the math teachers in the room, a unit cube is a real-world object. They can hold a one-centimeter unit cube in their hands and, more importantly, they can hold it in their minds.

The AP graders aren’t arguing about grading. They’re trying to decide what is real.

What a fantastic dilemma.

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Shannon Alvarez:

Whenever I wanted to give students the most amount of partial credit, my coop teacher would ask me the poignant question, “What exactly are you assessing?” I found this was a great question to continue asking myself. So, in the example you gave, are you assessing students’ ability to perform mathematical functions correctly or are you assessing their ability to connect those math functions to the real world?

Benjamin Dickman alerts us that Pollak’s piece is online, free, along with a number of his modeling tasks.

Thanks for making the trip all the way out to the San Francisco Bay Area, team. It’d be a treat to run into any or all of you during conference week. Here are my whereabouts.

Association of State Supervisors of Mathematics

Math, Education, and Technology: Reasons for Pessimism and Optimism. Monday. 9:15AM. The Metropolitan Room.

My optimism and pessimism are currently balanced at about 50/50.

National Council of Supervisors of Mathematics

Beyond Relevance and Real World: Talking with Teachers About Engagement in Mathematics. Monday. 1:30PM. Grand Ballroom EFGH.

I’ll summarize some of what I’ve learned from five years of offering workshops to teachers on student engagement in mathematics.

National Council of Teachers of Mathematics

Task Makeover Techniques for Grade 6-12 Math. Friday. 11:00AM. Exhibit Hall.

This is one of those informal “Networking Lounge” sessions. I’ll bring some textbook tasks that are eager for our help. We’ll brainstorm improvements.

ShadowCon. Friday. 5:00PM. Yerba Buena 7.

Zak Champagne, Mike Flynn, and I recruited another set of six amazing speakers from across K-12: Burrill, Bushart, Fletcher, Gutierrez, Kaplinsky, and Turner. We’ll film and post all their talks. They’ll all be eager for your conversation afterwards.

Beyond Relevance & Real World: Stronger Strategies for Student Engagement. Saturday. 9:30AM. 134/135 (Moscone).

This is the last time I’ll give this talk. I’ve been polishing it over the last 15 months, sanding off rough bits, tightening its scope, and adding more practical strategies. I’m really happy with it. I will also record it and post it here shortly after NCTM, so I welcome you to enjoy any of the 57 awesome other sessions NCTM has wedged into that timeslot.

Looking for Some Extracurriculars?

  • Desmos and Mathalicious are hosting their fourth annual happy hour and trivia contest on Thursday at 6:30PM. First round is on us.
  • From 1:00PM to 5:00PM on Friday you’ll find the Desmos Teaching Faculty in the Desmos booth answering any questions you have about our activity builder or calculator. (My slot is 2:00PM.)

Whose sessions are you excited to see? (Here is the unofficial #MTBOS track.) Recommend some new names, fresh voices, or first-timers in the comments below.

Danny Brown:

Some of the other online modelling resources, such as Dan Meyer’s blog, don’t really fit what I would class as meaningful modelling, and can feel contrived, or of little relevance/import to students’ lives; if I am going to spend the time bringing modelling situations to my classroom, I want to address matters of importance, socially or politically.

Geoff Wake:

Yes, I’m interested see how Dan Meyer promotes a sort of pseudo-modelling that seems to be quite popular among certain teachers. I think one aspect that appeals is that he suggests a narrative that is immediately accessible. On the other hand some of the questions are not particularly meaningfully tackled using mathematics seriously.

I see two tacit questions.

One, should math be important?

And by “important,” I’m using Danny’s definition: relevant to a student’s life, either socially or politically.

See, there isn’t any one agreed-upon definition of “mathematics.” They’re all arbitrary, personal, and cultural. And given finite hours in a school year to spend learning math, they’re all political. They create winners and losers. Class time spent how you’d prefer is time not spent how someone else prefers.

So I help students learn math for one reason alone, and it doesn’t have to be your reason also. I want to help students learn to puzzle and unpuzzle themselves. Math offers us the opportunity not just to solve puzzles, but to generate them from scratch – just you and your brain and maybe something to write with.

Those puzzles may have sociopolitical importance, but that’s a higher standard than I choose to set for myself. So it’d make more sense for Geoff and Danny to criticize my standard than to assume I’m aiming at theirs and missing. I’m not.

Two, should modeling be important?

I suspect Danny, Geoff, and I would agree more about the point of mathematics than the point of modeling. Their criticisms specifically concern modeling, and the fact that I ask questions like “How many pennies are in the pyramid?” and “How long will it take the water tank to fill?” rather than questions like (I’m guessing here) “Is capital punishment sentencing just or unjust?” or “How should California manage its water supply?”

But there is much more consensus around the definition of “modeling” than “mathematics,” and that definition doesn’t specify culture, context, or importance. Modeling is mental work, work of a certain character, work that I think we’d all agree is uncommon in many classrooms and unfamiliar to many students.

Modeling asks questions about a context. It works to make those questions more precise and tractable. It nourishes those questions with data where none exists. It sets reasonable bounds on an answer before finding a solution. It solves questions mathematically and then tests those answers against the world’s answer.

Basically, “modeling” is a verb and it doesn’t help our understanding of the verb to attach it a priori to adjectives (like “important” and “relevant”) or to nouns (like “capital punishment” and “water supply”). If you want to understand modeling, ignore the adjectives and the nouns. Watch the verbs.

Featured Comments

Chris Evans:

Additionally, we have to remember (as math teachers) that we are not the only teachers and courses these students encounter. I teach mostly 11th and 12th graders, and they frequently tell me about the political conversations they are having in government class or the serious social topic they are writing about in English. I have observed that, although students seem to appreciate these connections to real-world problems, these topics are heavy, and at times students appreciate engaging in “lighter” application problems and activities.

Nick Hershman:

Except that when you watch students engaging with a task that they are motivated to understand they are doing all sorts of things that relate to the “way they view their place as a member of society“. I can’t imagine a situation in which a student isn’t both learning something about their place in society and simultaneously asserting some version of their belief about their place in society. It’s happening all the time.

John Mason:

So working on socially relevant issues is valuable. But ‘relevance to me’ means, ‘real to me’, and as the RMP project has shown, well as it has confirmed what has been known for ever such a long time, what can be real to someone has to do with what they can imagine, can grow to imagine, and is not confined to what they already do every day.

Test With Water

On Twitter, I remarked that Marilyn had summarized the entire modeling cycle in a single tweet. But the part of that cycle she summarizes best is the last: validating your answer.

With mathematical modeling, you don’t have to be the answer key. The world is.

If you have total faith in the perfect accuracy of your mathematical models, testing with water may sound unnecessary. For the 99% of your students who wonder if math has any power outside of their textbooks, test with water.

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Bryan Anderson:

It was one thing to manually figure that out [if some glue could hold up a human], and then another to try the same thing with a bowling ball experiment modeling the same thing. We were able to see if our answer actually held up in that situation, it was a moment that will stay with me forever.

Freddie deBoer’s latest post is your weekend must-read:

Yet on the level of thinking of our Silicon Valley overlords, aspects of my cognitive abilities that are absolutely central to my educational success are taken to have literally no value at all. In educational research, perhaps the greatest danger lies in thinking “that which I cannot measure is not real.” The disruption fetishists have amplified this danger, now evincing the attitude “teaching that cannot be said to lead to the immediate acquisition of rote, mechanical skills has no value.” But absolutely every aspect of my educational journey — as a student, as a teacher, and as a researcher — demonstrates the folly of this approach to learning.

I’ve said it many times, though people never seem to think I’m serious: years studying literary analysis, now widely assumed to be a pointless and wasteful activity, have helped me immensely in acquiring the quantitative, monetizable skills that ed reformers say they want.

I applied to film school out of high school and spent a large fraction of my university math education reading screenplays and writing about movies. The coffin eventually closed on those aspirations, but my interest in narrative and storytelling has permeated every aspect of my teaching, research, and current work in education technology.

Freddie deBoer’s argument, both as I read it and experience it, isn’t that a liberal arts education makes a productive life in STEM whole. It’s that a liberal arts education makes a productive life in STEM possible.

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