# The key didactical ideas on mathematical modelling?

This question concerns teaching teachers who often already teach mathematics, but are now studying to get a formal qualification for it, and hopefully some more competency as well.

What are the key ideas about (1) mathematical modelling and (2) teaching it to pupils, who are often in middle school, but might be in primary school or even the latter part of secondary education (videregånde in Norway)?

My own understanding of mathematical modelling is that one takes some phenomenon or process from reality, simplifies it, captures it usually in a function or an equation (a much more complicated model is hardly relevant for these students, who are not physicists or engineers or even mathematicians), manipulates or solves that or deduces something from there, and then translates the solution back to the real life scenario, seeing what one has learned from there, and also checking whether the outcome makes any sense at all.

I would expect to teach something like this process. But what other key ideas are there in the didactics literature, especially about teaching modelling?

I'll talk about just one idea: quantitative reasoning is extremely important for mathematical modeling. This may seem obvious to some, especially to science educators, but I rarely hear mathematics educators discussing quantitative reasoning and the related literature, even though modeling is becoming an increasingly popular conversation topic.

In a 2022 article, "A bridging study analyzing mathematical model construction through a quantities-oriented lens," Czocher et al. describe various modeling competencies that align well with your description of modelling. "Modelling competencies include forming an understanding of a real-world scenario, simplifying, mathematizing (producing a representation), solving, interpreting, and validating."

In their introduction, Czocher et al. mention that the "simplifying" and "mathematizing" competencies are known to be particularly challenging for students. A goal of the study was to analyze the relationship between these competencies and students' quantitative reasoning abilities. The authors concluded that "with regard to mathematizing, the data support Thompson’s (2011) position that quantitative reasoning should be the basis of mathematical modelling."

In Thompson's (2011) article "Quantitative Reasoning and Mathematical Modeling," he describes a theoretical framework for quantitative reasoning, and there are several implications for teaching modeling that I've taken away. One is that I need to intentionally support students to conceive of quantities in productive or desired ways.

The point that quantities are mental constructions, and that their creation is often effortful, is central to mathematics education. Too often quantities, such as area and volume, are taken as obvious, and hence there is no attention given to student’s construction of quantity through the dialectic object-attribute-quantification. Instead, textbook writers and teachers just use them to teach. (Thompson, 2011, p. 34)

Lobato and Siebert (2002), in "Quantitative reasoning in a reconceived view of transfer," describe a teaching experiment in which students who had previously learned about slope were taught again with a focus on quantitative reasoning. They present the case of a student who, at the start of the experiment, explicitly rejected the use of the slope formula on a task involving the steepness of a ramp. According to the researchers, after "a lengthy process of reconstructing his understanding of the quantitative relationships involved," the student was able to employ proportional reasoning on the same task.

Another implication for teaching modeling that I've taken from the quantitative reasoning literature is that I need to attend closely to students' conceptions of "variation." In the same article, Thompson describes two students' differing perspectives on how quantities vary, a "chunky" perspective and a "continuous" perspective, from a teaching experiment by Castillo-Garsow.

In Tiffany’s thinking, the account’s earned interest did not pass through the 1/4-year amount on its way to the one-year amount. Rather, the 1/4-year amount was an amount to be computed ad hoc. Derek, who thought of time passing continuously and thought of an account having a value at each moment in time, thought more in terms of the account changing at the rate of a number of dollars per year for some number of years, and the number of years needn’t be a whole number. Earned interest accrued continuously. Tiffany could not see that the account's rate of change with respect to time was proportional to the account’s value. Derek saw this relationship intuitively. (p. 49)

There are other important ideas about teaching mathematical modeling in the literature, and Czocher et al. acknowledge several of them in their article. But I find the link to quantitative reasoning to be the most compelling, based on my personal experience with students' difficulties with modeling.

• Thanks, really good perspective. Sep 17 at 6:51

Opposing points of view to emphasizing modeling are mostly based on cognitive load theory, with John Sweller the most prominent voice. I would read some of that work and at least consider them before diving into a bunch of discovery/constructivist/IBL/etc. instruction.

I'm not even saying the counterargument is correct (but I lean in that direction), just that you should at least read both sides. Clearly the situation is not so simple, nor pedagogy as a field so clear cut, that people can't get contrasting views published. We are not talking about reporting spectral lines of atoms, when things get published as ed research.

Note1: A bit of a frame challenge. I would also carefully consider level of the kids and ability of the kids when thinking about modeling. What is best for older kids (with more models already in their head as schema to rely on) or stronger kids (with more native problem solving skills) may not be best for average or sub-average middle school kids. At least think about that variable in the equation. I'm worried about the experience factor in particular. 12 year olds don't have the same experience with chemistry and physics (many equations, models) as 18 year olds.

Note2: Another frame challenge. I would also consider that modeling activities may make more sense as part of a science course, not a math class.

P.s. If this sounds like I'm anti modeling or anti applied math, it's far from the case. If anything, I have a stronger general background in general technology and social science than most mathematicians and math educators. (And weaker math background than them!) And I love the gears and sweat and oil of the applied world more than the icy axioms of Mathland. But..."words problems are hard", as guest troll Barbie once said, so..."Danger, Will Robinson". ;-)

• I think it's helpful to distinguish instructional decisions from curricular decisions. I'd argue that it's possible to teach specific modeling competencies using a direct instruction approach, and it would involve explicitly teaching both the requisite mathematics and the requisite science. There's a course at UCLA which is doing exactly this, with promising results. That said, I agree that the approaches that many educators have in mind for "teaching modeling" are not like this, and those approaches can often be ineffective. Sep 17 at 20:59
• Modelling and didactics of it is one the subjects on the course that we are to teach, so at this point we would like to know how to teach it well. Critical perspectives are also appreciated, but slightly to the side of the question. Sep 18 at 5:10
• For both, if the text and course details are fixed, then just execute. Why even a question? If not (i.e. significant freedom on implementation), then, if you accept the critical views (and math/logic types, note this is a conditional statement), just do a fig's leaf of accommodation to the lofty course objective from some eduwonk standards writer. Like call the word problem stuff you already do "modeling". Don't let it derail you from more important objectives...like teaching middle school math. Sep 18 at 12:35