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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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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.

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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Real Work v. Real World

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Howard Phillips:

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.

Treatment #1

A small rectangular prism measures 7 inches x 2.3 inches x 4.6 inches. How many times could it fit in a larger rectangular prism with a volume of 39.3 cubic feet?

Treatment #2

Nissan is going to stuff the trunk of a Nissan Rogue full of boxes of Girl Scout cookies. Nissan lists the Rogue’s trunk space as 39.3 cubic feet. A box of cookies measures 7 inches x 2.3 inches x 4.6 inches. How many boxes will they fit in the trunk?

Treatment #3

Show this video.

  1. Ask for questions.
  2. Ask for wrong answers.
  3. Ask for estimates.
  4. Ask for important information.
  5. Ask for estimates of the capacity of the trunk and the dimensions of the box of cookies.
  6. Show the answer.
  7. Ask for reasons why our mathematical answer differs from the actual answer.


Treatment #1 and Treatment #2 are as different from each other as Treatment #2 is from Treatment #3.

A layperson might claim that Treatment #2 has made Treatment #1 real world and relevant to student interests. But the real prize is Treatment #3, which doesn’t just add the world, but changes the work students do in that world, emphasizing formal and informal mathematisation.

“Real world” guarantees us very little if the work isn’t real also.

Design Notes

You can check out the original Act One and Act Three from Nissan.

I deleted this screen from Act One because I wanted students to think about the information that might be useful and to estimate that information. I can always add this information, but I can’t subtract it.


I added a ticker to the end of the video because that’s my house style.


I deleted a bunch of marketing copy because it was kind of corny and because it broke the flow of their awesome stop motion video.

I left the fine-print advisory that you should “never block your view while driving” because the youth are impressionable.

The Goods

Download the goods.

[via whoever runs the Bismarck Schools’ Twitter account]

Craig Roberts, writing in EdSurge:

Beginning in the 1960s psychologists began to find that delaying feedback could improve learning. An early lab experiment involved 3rd graders performing a task we can all remember doing: memorizing state capitols. The students were shown a state, and two possible capitols. One group was given feedback immediately after answering; the other group after a 10 second delay. When all students were tested a week later, those who received delayed feedback had the highest scores.

Will Thalheimer has a useful review of the literature, beginning on page 14. One might object that whether immediate or delayed feedback is more effective turns on the goals of the study and the design of the experiment.

To which I’d respond, yes, exactly!

Feedback is complicated, but to hear 99% of edtech companies talk, it’s simple. To them, the virtues of immediate feedback are received wisdom. The more immediate the better! Make the feedback immediater!

Dan’s Corollary to Begle’s Second Law applies. If someone says it’s simple, they’re selling you something.

Ed Begle:

  1. The validity of an idea about mathematics education and the plausibility of that idea are uncorrelated.
  2. Mathematics education is much more complicated than you expected even though you expected it to be more complicated than you expected.

Begle coined those two laws in the latter half of the School Mathematics Study Group, a multi-decade project to figure this mathematics education thing out. I’ve heard those laws before but I hadn’t tracked down the original source until today. He seems weary in the speech. His list of tried-and-failed innovations is lengthy and disturbingly current.

Over forty years after Begle’s work with SMSG ended, those laws still offer us lots of comfort and at least a little humility. Math education is hard. My gut is probably wrong. Anybody who says differently is selling something.


Begle, E.G. Research and evaluation in mathematics education. In School Mathematics Study Group, Report on a conference on responsibilities for school mathematics in the 70’s. Stanford, CA: SMSG, 1971.

2016 Feb 26. Bowen Kerins’ links to a better copy of the entire proceedings. That site also contains links to some of the SMSG “New Math” curriculum, which I’m excited to investigate.

2016 Feb 28. Raymond Johnson cautions us not to read Begle too pessimistically:

I really do love the history of my subject and posts like Dan’s send me into hours of searching through old papers and citations. But, I must be mindful of our tendency to underestimate change when we read from our wisest predecessors. It’s too easy for us to throw our hands up and say things like, “Dewey knew it all along!” or “We’re stuck in the same damned place we were 25/50/100 years ago.” Is Begle’s 2nd law (“Mathematics education is much more complicated than you expected even though you expected it to be more complicated than you expected”) still true? I would agree it is. But, as a field, we’ve made enormous progress since Begle gave this talk in 1971. The danger, as individuals, is to not learn from this progress. To avoid reaching the same conclusions as Begle, we need to avoid starting in the same place as Begle. When I browse the pages of Begle’s final book, Critical Variables in Mathematics Education: Findings From a Survey of the Empirical Literature, I’m struck by the sheer number of things Begle and the field knew little or nothing about compared to what we know now. Don’t we owe it to ourselves, as individuals and as a field, to push past prior conclusions by starting farther ahead and taking more seriously work already done?

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