…and why somebody might try to convince you they are.
It’s even in the Core Standards. This is taken out of context—but not very far:
A model can be very simple, such as writing total cost as a product of unit price and number bought… (Common Core, p 72)
Seriously?
Okay, I could make a case for it, but I won’t.
I’m becoming more convinced that the real hallmark of modeling is simplification (see this post for more).
Modeling is not simply using math on real-world problems, though that is a Good Thing; you can model to help with pure math as well. And I bet we could find good real-world problems that don’t involve modeling.
But back to simplification. The key element (I believe this afternoon anyway) is taking something and using math in a way that makes it simpler, less complicated than the thing itself. We model to make things tractable. We can handle the model even when the thing it represents it too complicated. If it’s a good model, it captures the essence of what we’re looking at; and exactly what that means may depend on the specific context.
- We might model a hexnut as a hexagonal prism with a cylindrical hole, and use that geometrical model to find a volume. We avoid the threads and the easing on the corners: they’re too complicated—but we hope our model captures the essence of the hexnut.
- We might model some messy data as a line or a curve. We can’t make a reasonable prediction from the mess, but we can with a function: just plug in a value and calculate.
- We might explore the behavior of a system of linked differential equations by creating a numerical model, a system of difference equations we can evaluate on a computer. It’s conceptually simpler (for the computer at least), so we sacrifice some precision for tractability.
- We might even take all the complexity of Americans and do a Census. When we do, we create a data model: the structure for the information we will collect. We have only approximately captured the people’s information (this is the Census, right, not the NSA). We hope our data has the essence that we need to know—but there is a huge amount of detail that we have ignored. Like the threads, like the deviations in the scatter plot, like the inaccuracies in the numerical model.
What does this have to do with word problems?
Suppose we ask, if Eduardo buys four cans of orange juice for $2.49 a can, how much does he pay altogether?
There is math here, no question. We can argue whether it’s real life.
But it doesn’t involve simplification. All of the information is present. There is no model, and no need for one.
A Best-Case Scenario 
>the real hallmark of modeling is simplification
Just when you thought it was safe to go back in the water…
…here’s a link to Common Core’s authors’ latest (obtuse) treatise on modeling, for what it’s worth:
“http://commoncoretools.me/wp-content/uploads/2013/07/ccss_progression_modeling_2013_07_02.pdf”
Great link! The document is worth a careful read, but in a short skim I can tell that (a) I agree with a lot of it and (b) there are some things that trouble me. (All along the lines of my ongoing concern that if we don’t watch out, everything will be modeling.)
And if it’s obtuse, well, we’ll see. Modeling is so ill-defined, any attempt to be comprehensive is perilous.
You say “There is no model, and no need for one”. I disagree. There is no modeling (that is, no creation of a new model), but there is a familiar old model: that the price of a set of items is the sum of the prices of the items. This is not a given—look at the weird pricing schemes of wholesale sales, where the price of a set of items is not necessarily monotone in the number of items—it is just such a familiar model that you apply it without thinking about it.
The multiplication problem could be a modeling problem for 3rd graders (though it is not phrased as a modeling problem, but as one that assumes students already know the model), but not for high schoolers.
I think we need to distinguish between “models” (which we use all the time already) and “modeling” (which is rarely taught except in some physics classes).
You make a good point: just as what’s a problem for one student is merely an exercise for another, there are levels of modeling and using models that change as we get more experience.
Even more important is the distinction between using and making models.
As to my “no model” quip, I’ll stick by it in this sense: the problem is stated in the universe of “math problems,” which has different rules from reality. When you say in a math problem that orange juice is $2.49 a can, that’s not a simplifying assumption—ignoring crazy pricing schemes, sales, asking the grocer for a discount because a can is dented or leaky—it’s irrevocable reality. So (in my book, today) the calculation is not a model for the pricing scheme, it’s just one way to represent it efficiently.
Now, suppose we gave the third-graders a table of prices showing the total cost for various numbers of cans. Or we let them ask a computer for the total price of any number they wanted. Then multiplication would be a model, and coming up with the idea to use multiplication would be modeling. (I think. This is such fragile territory, I’m grateful to have this opportunity to think about it.)
A model is a way to represent something efficiently. You are again confusing the construction of the model with the use of the model.
For a third grader, figuring out how to use the model of multiplication of unit prices to get total prices can be challenging, as they don’t have either the notion of multiplication nor the model of sums of unit prices well established.
I agree that the problem is not given as a modeling problem, but assumes that the reader already knows the appropriate model to use (even though it is more a math textbook model than a real-world one).
I’m trying to very cautious and careful about my logic here; if you’re saying that an efficient representation is necessarily a model, I disagree. I’m not sure that all models are efficient, though I’m more likely to buy that. In any case, this thread illustrates the perils of defining modeling. Well-meaning and thoughtful people disagree, especially around the edges.
My purpose throughout this series of posts has been to explore these perils, partly to clarify my thinking in advance of what I fear will happen as CCSS get implemented.
I’m not sure exactly what you mean by “an efficient representation”. I see a model as a simplification or abstraction. Representing a can of OJ by its unit price is a model, albeit a fairly simple one.
I think that you are trying to make models into something more complicated than they are, by making a binary model/not-model distinction. Many things are models without being useful instances of learning modeling. Students need to learn to use models, choose models, and build models. Simple word problems tend to be about using models, intro physics problems about choosing models, and the data analysis you favor about building models. All of these are useful skills, but the differences are in the verbs, not the nouns.