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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@ddmeyer https://t.co/tiZszn1l3Z "A liberal arts education makes a productive life in STEM possible." My experience as well. #MusicMajor

— David Wiley, PhD (@opencontent) March 11, 2016

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.

Related Reading

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.

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.