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.!

]]>I’m responding to Avery’s question above. He’s opened up that possibility.

> I think if you imagine holding up that model to the modeling cycle you’ll see where one need not engage in the process of mathematical modeling to construct or use such a model.

I appreciate the application / model distinction. I’ll probably mentally throw a bunch of problems at it over the next couple weeks and see if I can make it hold up in my head.

This may have no bearing on this conversation, but I think one way in which I depart from a lot of the modeling-friendly organizations like GAIMME, COMAP, and even the NCTM yearbook, is that I see value in students experiencing *parts* of the modeling cycle.

Constructing an algorithm for the “square-ness” of a rectangle, for example, allow students to formulate a model, even though there isn’t a lot of noisy data for them to sift through, discard, and retain.

Similarly, I’m happy for students just to answer the question, “If I’m planning a Thanksgiving dinner for my family, what information is important for me to know to figure out how to set a budget?” even if they don’t formulate a model for those data, calculate an answer, interpret it, or validate it.

The way a lot of modelers talk about modeling, teachers are either exposing students to the entire cycle of nothing at all. I don’t think that kind of high bar is helping modeling make inroads into classes.

]]>I think you are setting up a bit of a straw man here with your proposition. I don’t know of anyone who says “modeling questions can’t have correct known answers.” I think a much more nuanced reading of these statements is required, which is what I’ve tried to provide in the blog post above. The statements like the ones I mention above are trying to explain what the practice of mathematical modeling is really like, not draw hard and fast boundaries around the nature of modeling questions. I think that is the real point – if we want to really teach kids the art of mathematical modeling, we have to pay very close attention to what those who do mathematical modeling actually do.

Your mortgage example points to another issue – the word “model” is used in many ways, by many people, with a myriad of meanings. Yes, we might call the way you calculate mortgage payments a “model,” but it’s not one that is obtained via the process of mathematical modeling. That’s a very subtle point. I’d prefer to call examples like the one you mention “applications of mathematics” to draw a distinction. The difference is that in the mortgage case, the entire system is inherently mathematical, how you calculate mortgage payments is simply defined from the start in terms of mathematics. There are no assumptions, choices, or decisions to be made, it’s just follow the definition. I think if you imagine holding up that model to the modeling cycle you’ll see where one need not engage in the process of mathematical modeling to construct or use such a model. Now, there is certainly value in having students work with such models or do such problems, but it doesn’t teach them the process of mathematical modeling.

This points to one of the key difficulties in teaching modeling in K-12. Mathematical modeling really does require engagement with the real world and real world problems in ways that often push the knowledge needed well beyond the typical mathematics classroom. I think this often leads to a tendency to look for problems that don’t do that, i.e. problems like the mortgage problem that keep one comfortably out of the real world and really, totally in the realm of mathematics. This discomfort is hard for many teachers to deal with, but encouragingly, not as hard for students once you turn them loose.

Anyway, too bad you are on the west coast, be great to have this conversation over a beer! Certainly lots to discuss!

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