Great example for modeling. I wonder if part of the simplification in modeling lies in the assumption of treating the potato as a homogenous mass rather than heterogeneous mix of potato mass and water, both of which heat and cool at two different rates. I also wonder if the overall shape of both potatoes have to have similar shape since the heat dissipation is along the radial direction from the center and if one is elliptical and other spherical i.e. some ratio of the radii needs to taken into consideration. Now of course, one makes it a more complicated model and defeats the purpose of the lesson you were trying to convey.

Bottom line – I enjoyed reading it :- ). Excellent website. ]]>

I would argue though that building from data is only one approach to mathematical modeling. It’s what the Common Core calls “descriptive modeling.” The other approach, what CCSS calls “analytic modeling” builds on the application of physical law to construct models. For example, if I’m trying to model motion, I’m going to start with Newton’s Laws and leverage those to get a deeper understanding than I can get via data alone.

Love the point about connections between CCSS and NGSS! We should be teaching mathematical modeling in partnership with our science colleagues!

]]>Yes! And that actually makes sense because doing parts is kind of what has already been happening in the name of mathematical modeling. So they’re trying to push that we really really do modeling now – not the lame stuff that has been tagged as modeling in textbooks. Dan’s MT article actually highlights this well.

Given that even in the GAIMME report, the one task featured for elementary level modeling is once again the same old bigfoot problem (which is a nice problem!) that has been done and written about from researchers all around the world, I’ll admit that I’m a bit of a skeptic as to whether or not young children can really Model with Mathematics.

I liken this to proof in mathematics. Past standards documents tend to talk about Reasoning and Proof. In fact, the Sense Making document described reasoning on a continuum with proof being at the higher end of that continuum. With few exceptions, we generally tend to talk about young children explaining, reasoning, or justifying. I think that most people don’t expect young children to engage in proving. I wish that there was some other word or phrase, like reasoning, that we could similarly use to describe “baby modeling” or pre-modeling activities that are perfectly accessible and appropriate for younger children. Instead, the best that we’ve come up with (so far!) is scaffolding it by having students engage in parts of the modeling cycle. That’s not to say students cannot mathematize real-world phenomena – I’m just not sure what genuine Modeling with Mathematics looks like for young kids. And even for older students, it’s definitely challenging to engage in the entire cycle in an authentic way.

]]>I explored a little bit of this idea in:

http://modelwithmathematics.com/2015/08/what-exactly-is-a-thought-tool/

Michelle and I have spent a lot of time thinking about the thought tools that mathematical modelers wield and how one learns to wield them. Note, this notion of thought tool is exactly the same as thinking of having students experience parts of the process of mathematical modeling.

I think the GAIMME and COMAP folks, etc., are partially reacting to a fear that people will only do parts and then, only the easy parts. Plus, as far as I can tell from the literature, there is not a great consensus on what the core competencies really are for mathematical modeling. Folks in Germany have done some great work in this area, but still, feels incomplete. But, that is no excuse for not having students experience parts! Although, I too, always feel it necessary to add the caveat that it has to be building to the whole and that both teachers and students need to understand that as well. Also, I guess I’d add that we need to make sure that teachers understand that when they’re doing parts, they’re doing parts, not mathematical modeling. See, I sympathize a bit with the fear that comes through in GAIMME and COMAP etc.!

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