Coincidence? Sure hope so. But, NCTM, how could you not notice?
Many data-oriented lessons in math and science ask students to come up with the question they want to investigate. The idea behind this is great. It ought to promote buy-in: students will want to use data to explore something they’re interested in—so we get them to come up with a question they want an answer for.
I have no trouble with that.
But I recently saw a lesson which was, essentially,
The teacher materials even had a section for what do do if students have trouble coming up with a question, with suggestions such as asking, “how do you think the variables might be related?” Fine prompt. But how does it actually elicit a question about a table of data the student has only just seen?
Since this is in a tech-rich environment, it might be better to approach (numerical) data like this:
At this point, you could retrospectively come up with a question, but chances are good that the question—since it really wasn’t a question you were interested in to begin with—will be lame.
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)
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.
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.
The draft Framework has a place where you can submit comments, but each box is limited to 500 characters. I know I go on too long, but seriously, I can’t fit what I have to say in that space!
So, California Modeling Persons, I’m really interested in what gets said about modeling in the Framework. I’d love to help this important topic make its way into the schools in a way that will not make teachers, parents, or Mathematically Correct roll their eyes in despair. I’m worried that if we proceed without great care and thoughtfulness, modeling will become a word-du-jour about which an army of consultants will make powerpoint slides, and which will then fade away by the time the next set of standards or frameworks emerges.
Comments close in two days (and I have no actual job that will justify my writing about this) so I apologize that I will not be creating a fully-formed and coherent narrative suitable for cutting and pasting, but will only be making comments. Besides, Usiskin and others have written well on this subject.
So here we go:
Part of the motivation for the recent posts on modeling is that I’m writing a paper for a friend. It takes off from the hexnut weight data and cubic model that appear in The (still unpublished, sigh) Model Shop. More on that soon, because I’ve found some interesting features in hexnut data. But first, another friend sent me a link to this post on doghousediaries I now share with you. Check it out.
Because I had just been doing hexnuts, I immediately thought about coins as an alternative: how do you suppose the weight of coins (Physics people: I will be reporting weight in grams. We both know that’s mass, not weight. We can handle that, right?) is related to their size?
I mean, hexnuts look more or less geometrically similar (they’re not, but that’s the other story), so you’d think their mass is cubic in the linear size: a nut that fits a half-inch bolt ought to be eight times the weight of one that fits a quarter-inch bolt. And that model fits pretty well.
So as teachers, we look for situations that stretch that understanding. EGADs has you cut out cardboard squares or circles and weigh them; since the difference in size is two-dimensional, the weight-size relationship is quadratic.
But what about coins? They look kind of two-dimensional, but the big ones tend to be thicker. So are they cubic or quadratic or something in between? Well. Whenever I come back from some other country, I usually have leftover coinage, which gets saved in that jar in the top drawer. And I have a good scale and (I know, I’m a measurement geek) a decent micrometer. So it’s an empirical question.
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? Continue reading
What’s the purpose of mathematical modeling? The easy answer is something like, to understand the real world. When I look more deeply, however, I see distinct reasons to model—and to model in the classroom. I hope that trying to define these will help me clarify my thinking and shed light on some of the worries I have about how modeling might be portrayed.
Let’s look at a few purposes and try to distinguish them. To save the casual reader time, I’ll talk about prediction, finding parameter values, and finding insight. I think the last is the most subtle and the one most likely to be missed or misused by future developers.
Maybe I’ll post more about each of these in detail later, but for now I’ll move quickly and not give extended examples.
I’m still trying to clarify my own thinking about modeling. Maybe trying to write it down will help.
Yesterday, we explored some definitions. Now let’s describe different types of mathematical modeling. I’ll call them “genres” here, but that may be too fussy. The point is that people mean different things when they use the term “modeling.” Sometimes it’s a difference in definition (is the key thing real-world application, or simplification, or does that matter?) and sometimes it’s using a different set of tools.
Here, then, are several different kinds of activities that all seem to me to be more-or-less clearly modeling.
This is the clearest to me, and the most obviously useful. And fun. It’s one of the things I like best about experimental science. It’s really cool to find the function that characterizes a set of data, and see how it fits (or doesn’t fit) the theory that gave rise to it. (You can also use the data and function to get insight into the phenomenon—use the data to figure out the theory—but that’s for another post.)
This type of modeling fits with our current math curriculum as well, the one that hurtles students towards calculus, because it’s all about functions. Students who fit curves to data find a use for all those functions they’ve been learning about. This is the impetus behind (my books!) EGADs and The Model Shop in math and A Den of Inquiry in physics.
The paragraphs activities are good prototypes for this kind of thing.
In such an activity, you have data you can represent on a scatter plot (this means the data are continuous as opposed to categorical), and you try to find a function that “fits” the data as well as possible.
This function is imperfect. It’s a simplification of reality, but possibly a useful one. You can use this function to make predictions (more on this another time; I think it’s an important purpose of modeling), but, because the data don’t lie exactly on the function, there is some uncertainty in any prediction. The function’s precise shape and location are governed by parameters (e.g., slope and intercept for a line). Modeling reveals the values (or range of plausible values) for these parameters; often these values have a meaning in the original context (e.g., speed for the slope on a distance-time graph). That is, with modeling, you can learn things about the situation that gave rise to the data. Continue reading
But I’m worried about definition creep. I’m worried that, in two years, when they’re trying to come up with modeling curriculum, people in schools doing the hard day-to-day work will be tempted to say that practically anything is modeling and come up with plausible rationalizations. That, in turn, will dilute the importance of including modeling in policy documents, and result in students who can’t model.
To forestall this, it’s important to know what modeling is and isn’t. So it’s with some embarrassment that I, modeling maven and aficionado, have trouble drawing the lines. So consider this a first step in clarifying these questions for myself:
Let’s start with the Framework (Modeling appendix, April 2013 review draft, lines 13–14):
Put simply, mathematical modeling is the process of using mathematical tools and methods to ask and answer questions about real world situations (Abrams, 2012).
Of course they go on at length, but the key is a connection to the real world. Here is another definition that I have used recently:
A model is an abstract, simplified, and idealized representation of a real object, a system of relations, or an evolutionary process, within a description of reality. (Henry, 2001, p. 151; quoted in Chaput et al., 2008)
Here, the key ingredients are abstraction and simplification.
Another distinction worth noting is that the first is a definition of modeling—a process—and the second defines a model—a representation. I have no clue whether that matters much.
What do the Core Standards themselves say? First of all, the document identifies modeling as one of eight Mathematical Practices, a great list I have mentioned before. Here is the one called Model with Mathematics, and it’s worth quoting in its entirety:
Mathematically proficient students can apply the mathematics they know to solve problems arising in everyday life, society, and the workplace. In early grades, this might be as simple as writing an addition equation to describe a situation. In middle grades, a student might apply proportional reasoning to plan a school event or analyze a problem in the community. By high school, a student might use geometry to solve a design problem or use a function to describe how one quantity of interest depends on another. Mathematically proficient students who can apply what they know are comfortable making assumptions and approximations to simplify a complicated situation, realizing that these may need revision later. They are able to identify important quantities in a practical situation and map their relationships using such tools as diagrams, two-way tables, graphs, flowcharts and formulas. They can analyze those relationships mathematically to draw conclusions. They routinely interpret their mathematical results in the context of the situation and reflect on whether the results make sense, possibly improving the model if it has not served its purpose.
This is great, but it’s easy to imagine this lofty overarching idea getting lost when you’re designing a curriculum—or an assessment—and you have a chart of content to fill in. Fortunately, the Core Standards promote Modeling to the level of a content standard at high school. What do they say? Here’s a quote I find chilling:
Modeling is best interpreted not as a collection of isolated topics but in relation to other standards. Making mathematical models is a Standard for Mathematical Practice, and specific modeling standards appear throughout the high school standards indicated by a star symbol.
That is, you should find it everywhere, so we won’t list very many actual skills and goals. One could see this as a good thing: we’re celebrating the ubiquity of modeling. But I’m less sanguine about our ability to keep “overarching ideas” in mind, especially as we design assessments. Continue reading
Last month, in Falmouth High School in Maine, some Honors Physics students were estimating the period of a mass hanging from a spring. They used InquirySpace/Data Games software and Vernier motion sensors, and got data that looks like this (Reading is in meters; Time in seconds):
To do their investigations, they needed the period of this wave.
But this post is about an alternative to all of these techniques—one that uses all the data and gives a much more precise result than the first two.
You can read about this is excruciating detail in a paper I wrote. And I made one particularly careful group of students this (awkward and quickly-made) video describing the technique. So I will be brief here. Continue reading