Marbleslides is embodied in the way Polygraph is not. Marbleslides evokes physical movements and physical actions: the veeery satisfying, bouncy marbles. Polygraph evokes beautiful visualizations.

Both are lovely for modeling, and knowing which is embodied has meaningful teaching implications. Yay?

Quoting Dan M repeatedly here:

We can and should teach the (fill-in the blank with your desired topic) the same way. Do we want self-identified math modelers or the modeling “establishment” to “apply their skills across all of teaching”? Yes!

“The tasks that mathematical modelers often enjoy and Polygraph should be taught the same way. That’d be great for teachers!” Yes!

The thought “The distinction between the “real” and “not real” world doesn’t exist” is debatable. “Insisting on it makes everyone’s job harder” is also debatable. Are modelers “committed to the real / not real distinction”? ¯\_(ツ)_/¯

Do “we just need them (modelers) to drop this meaningless distinction between the real and non-real world” to pursue this work? I don’t believe so.

Talking with students about distinctions along the continuum of the “real” and “not real” (feel like I’ve been watching the Matrix movie series here) can advance the work of “eliciting early ideas, provoking students to determine their limits, and helping students develop their ideas further”.

Across time and context, what appears “real” and “not” to a person can vary, and “understanding” something often makes something “real.” There is a spectrum of the “real” as has been noted. Yes, a set of blocks is “real” (or a picture of those blocks). They’re also both potentially not real at all, just as the Polygraph can be real (or not).

Graphs or anything on the spectrum of “contextual complexity” are not necessarily real in the same way that a question about a change in the price of a life-saving drug might be, or a decision on which candy bar is a better deal. (Or in recent news, the velocity data of two crashed airplanes.) These are all real in different ways, and all have the potential to be “not real”. GAIMME and other examples push us in a good direction, toward ideas to notice.

As Dan M has argued in his pseudocontext series, the students know when we’re pretending. Let’s continue to talk to students about what they think is real.

Will read Dan M’s lit review in detail and be back. Thank you!

It sounds like the other members of the panel require a context in order for it to be modeling. It also sounds like Dan says once you have decontextualized the math then it’s the same pedagogically.

Right! Same pedagogy in both cases.

But I’m also am saying, along with other folks before me, that “contextual” isn’t a binary variable (where “decontextual” and “contextual” are the only settings) but a continuum. Decontextualizing (or “abstracting”) is the act of removing detail. You do that everywhere. When you drive, you don’t pay attention to every detail around you. You abstract away the details that aren’t important for driving. You have decontextualized the world. You do the exact same act when you’re trying to describe parabolas! You decontextualize the set of parabolas to a set of verbal descriptions.

Yes, that’s correct, “most people” do not find math intrinsically valuable. Yes, mathematical work has been done solely for the joy of figuring it out….by those relatively few people who find it intrinsically valuable.

Mathematical modeling (my definition here) gives the students more room to move and gives them ownership of the problem.

No doubt.

I am excited to see “more room,” though, which implies a continuous variable rather than a discrete one. That’s largely my point here, that the binary distinction between real / not real & modeling / not modeling is illusory and creates unfortunate pedagogical binaries.

:_(

But I think that’s why Dan is making his point. Students get excited about solving problems–if well crafted. Historically, so much mathematical work has been done solely for the joy of figuring it out. Let them find intrinsic joy in driving the double angle formula.

What about the joy one feels learning or solving problems about parabolas? I’m reading a book right now, The History of Algebra, for the sole intrinsic value that it brings me joy.