And we’re back. In the last several posts, I’ve been trying to take a careful look at modeling. Today, I want to take a look at the “Modeling Cycle” and record what I like about it and where I have some nagging doubts.
The Core Standards presents this diagram (you can see I’ve snipped it out of the actual document, page 72…):
I actually prefer one I’ve used, based on work by Blum. It amounts to the same thing, but labels the arrows, rather than the boxes, with the verbs:
Whichever you prefer, I think a diagram is useful. And both diagrams have what I think is most important here: the distinction between mathematics and reality, which I have highlighted in my rendition of the Blum diagram. Modeling handles the transition across the divide from reality into mathematics, and then re-interprets the mathematical result back into reality.
An important consequence is realizing that traditional math instruction focuses only on the orange part: the bottom two boxes and the arrow connecting them.
So far, so good. What’s not to like?
This: I’m having trouble mapping modeling activities into these diagrams. I’d like to bring them into alignment. Maybe it’s not a giant problem. Maybe other considerations will make my uneasiness go away. For example, we have yet to talk much about how things we might do in the classroom represent only part of the modeling cycle (and therefore make the mapping hard). But I have to enter this discussion somewhere. So let’s begin.
Applying the diagram: the snowplow problem
I was first introduced to the Blum diagram with a problem about some villages in winter. Roads need to be plowed. What is the minimum number of miles we have to plow in order to connect all the villages? If we fast-forward, this is a discrete-math problem about a spanning tree; the student is supposed to use the Dijkstra algorithm or its equivalent (e.g., inspection) to solve it.
How does this map onto the diagram?
- The real problem is villages and roads.
- In the real model, we’ve recognized that only the lengths of the roads and which villages are connected to which are relevant. “Structuring” the problem has simplified it.
- In the mathematical model, we have a diagram with dots and labeled lines—a representation of a graph—and the problem clearly specified: find the subgraph, with minimum sum of edge lengths, that also connects all the vertices.
- The mathematical result is that subgraph; the “process” was the algorithm.
- The real result is the list of road segments corresponding to the edges in the subgraph.
- The validation step is when we try to plow those roads; if we succeed, hooray! The villages are connected. But we may find out that one segment has a turn the plow can’t negotiate; if that’s the case, we have to iterate. Our model was insufficient.
I like this, mostly, but we should be really critical. For me, my “lameness detector” raises a flag early on when we make a distinction between the real model and the mathematical model. Yes, we simplify the real situation by focusing only on the connections and lengths, but is it useful to think of this as a separate step from making the diagram? In reality, we may start making a diagram before we really know what’s essential; there is a sort of negotiation as we look for the mathematical representation.
Maybe the modeling cycle diagram is a retrospective analysis of the process, identifying what must have happened in a successful modeling effort rather than how it happened.
(Why should we make this distinction? It’s all about how teachers and developers implement modeling. I really have doubts about a “modeling cycle worksheet”—you can just imagine it—where the kid has to write down the real model and the mathematical model separately. I mean, I’ve thought about modeling for years. It has taken me a couple of days to understand what I’m after in this post, and I’m figuring it out still even as I type, and if somebody comments on how wrong I am, I’m ready, eager, to change.)
Next application: paragraphs
- First, it’s not clear what the real problem is. I’ll ignore this and hope to spend more time on it in a future post. For now, let’s just say it’s “understand the sizes of paragraphs.”
- The real model might then be the simplification, the decision to study height and width, and the decisions involved in measuring the heights and widths of the paragraphs on the sheet.
- Then the mathematical model would be the measurements, especially as represented in the graph. Does it also include the function? If modeling clearly includes finding the function, then the model must include it, right? But it feels as if function-finding should be part of the mathematical process.
- The mathematical result certainly includes the function—the relationship between height and width.
- When we interpret the model function, we see that the parameter represents a constant area. We make sense of that in the real result, since the text and font size are the same.
- Validation may let us make a good prediction about a future paragraph. But there is no future paragraph in this incarnation of the activity.
My big question here is, where exactly is the modeling? And where’s the mathematization? My gut says, you mathematize the paragraphs to get measurements and a scatter plot. The model is the function, the smooth graph, the graph that represents a simplification of the relationship between height and width. The modeling is finding the function.
On the other hand, just measuring height and width, in a slightly stretchy sense, is modeling as well: measurement simplifies the paragraphs, converting the detailed, word-filled non-rectangles into simple geometrical shapes. It’s as much modeling (we could argue) as modeling the peanut butter cup as a pair of cylinders.
So do we have two instances of modeling here? Will teachers really believe that interpreting a paragraph as a rectangle is modeling in the same way that fitting an inverse function to a set of points is modeling?
You see my problem?
I don’t want to get into a big philosophical brouhaha about definitions, or where some process happens in a frigging diagram. But if we want teachers and developers to use modeling activities, and we’re about to put a diagram in a thousand slideshows, it ought to be easier to connect the diagram to the activities they can use.
I think I “get” modeling. I think the paragraphs activity (despite its lack of real-world context) reeks of modeling. But it doesn’t even fit the coolest part of the diagram, namely, the notion that traditional classroom math only happens between the two orange boxes. Why not? Because we don’t do any of the math in the paragraphs activity in the traditional class. Part of what makes modeling great is that it opens up the class to doing this sort of thing.
I think the diagram is perfect for the kind of problem where, when you model, you get a mathematical situation to which you can apply an algorithm. Like if you come up with an equation you need to solve, or a quantity to optimize as in the snowplow problem. But something like the paragraphs activity—or any of a large number of real-world, interesting, rich situations in which you get data and need to find a relationship—those may demand, in their modeling diagram, a different set of boxes involving measurement and inference. And these may smell more like science than Blum’s (or Core Standards’s) diagram.
Coda: California Grade 4
The California Framework gives this example of modeling:
For example, at a very basic level, fourth grade students might be asked to find a way to organize a kitchen schedule to serve a large family holiday meal based on such factors as cooking times, oven availability, clean up times, equipment use, etc. (English 2007). They are engaging in modeling when they construct their schedule based on non-overlapping time periods for equipment, paying attention to time constraints.
This is a terrific task, but how is “paying attention to time constraints” making or using a mathematical model? Perhaps the model is just the idea that use of resources doesn’t overlap, and we construct a representation (for example, a physical chart with slips of paper to move around), and use it to solve the problem, which we then translate back into reality (“Sylvia, you better put in the turkey by 9 AM.”).
I could buy that, but again, I can only imagine the workshop where the district’s fourth-grade teachers are challenged to invent and assess modeling problems.