Q: How did the chicken cross the road?
A: One step at a time.
In various Twitter conversations with @Foizym, @ddmeyer, @MathMinds, @gfletch, and @Simon_Gregg, the question of what mathematical modeling could or should look like at the elementary level has been a recurring theme. I want to explore this question a little bit in this post and offer a few ideas that may be helpful for those trying to incorporate mathematical modeling into their elementary math classrooms.
A good deal of the angst around teaching mathematical modeling at the elementary level seems to revolve around what it is that students already know. That is, they have a limited mathematical toolbox to draw from and a limited extra-mathematical toolbox to draw from. Put simply, there’s a lot they don’t yet know. And, I can’t argue with this as being a serious challenge. If the mathematics taught in K-8 was adequate for dealing with all the myriad problems that mathematical modelers attack, well, then we would never have needed to invent calculus, or differential equations, or algebraic topology, or… you get the idea.
And, it simply is true that the extent of the tools in your toolbox limits what projects you can successfully tackle. If all I have is a hammer and saw, it’s pretty hard to build a cathedral. If I really want to build a cathedral, I’ll probably spend a good deal of time using my hammer and saw to build other tools and then use those to build still other tools, and then get around to building a cathedral. The obvious analogy is that if I want to do mathematical modeling of fluid dynamics, and all I have is algebra, I would probably need to spend some time building calculus before I’d be able to get very far.
But, this, of course, is deeply unsatisfying. It almost kicks us back to the horrible answer to a students’ question of “what is this good for?” of “wait, you’ll see in a few years.” So, if we reject that answer, and yet realize that students at the elementary level do have a limited toolbox, what do we do? I’d like to offer two answers.
The first of these is that we can assiduously look for those problems that can be tackled with a limited toolbox. This is hard! Once you own a table saw, you’re not so likely to whip out your handsaw any more, so when we survey the scientific literature, we’re not so likely to find many problems that rely only on elementary mathematics. That doesn’t mean that they don’t exist nor that the problems we do find couldn’t also be attacked by elementary means, just that what we’re seeking won’t be sitting right on the surface for us to discover. It means that someone seeking to find good problems for elementary students will themselves have to be a pretty darn good modeler. They’ll have to not only be able to digest the modeling literature that uses advanced mathematics, but also be able to see how such problems could be attacked more simply, or how parts of those problems could be attacked using elementary math. That’s a tall order but I think this is a potentially useful approach. That is, I’m arguing that rather than try and construct elementary mathematical modeling projects from scratch, lets have elementary experts sit down with modelers, plow through some hard problems, and see what comes out the other side. Could such a team identify the accessible projects? I don’t know, but I think it is worth a try. I offer our previous discussion of “fairy circles,” albeit aimed at high school, as an example of what I mean here.
So, what’s the second answer? I’d argue that another approach we could take is to think really hard about why tools like calculus were developed in the first place. I’d argue that while much of the mathematical machinery used day-to-day by mathematical modelers is inaccessible to young students, a lot of the ideas behind that machinery are actually very accessible. The simplest and perhaps broadest example that occurs to me is the notion of iteration. Much of what we do in science and much of what we do as modelers is to build on a very simple idea that I’ll state like this – right now, the world looks exactly like it did a little bit ago, but with some tiny changes. That is, calculus, differential equations, and a whole bunch of the mathematical machinery that modelers rely on was built to be able to model change. And, the idea that is repeatedly used is the one we just stated, things change in small steps.
So, if I want to know how the chicken crossed the road, the useful answer is “one step at a time.” I then build up my understanding of “chicken crossing” from a chicken repeatedly executing single steps. If I do this over and over, I get my chicken where he wants to be. To see how we might follow this line of thinking to get from a “standard” calculus-based modeling topic down to the elementary level, let’s look at a simple, canonical mathematical model that you’ll see in any modeling textbook – population growth.
Typically, in studying population growth (or how the amount of any “stuff” increases or decreases with time), you’ll first examine linear growth. If is the amount of stuff at time , you’d write:
So, what are we really saying? Well, we’re saying that the rate of change of is a constant. We can view that differently if we approximate our derivative by it’s difference quotient:
Now, what does this really say? Well, it says the amount of stuff we’ll have in the future, , is equal to the amount of stuff we have now, , plus a tiny change, . That is, in a little bit, the world looks just like it does now, with a tiny change. We do this repeatedly, and we get a picture of how our system changes according to the underlying process of changing by constant small steps.
Well, that’s not really very hard to grasp at all! So, imagine we introduced this idea to young students. Imagine we showed them a simple table of data that gave the population of something for different times. Maybe it looks like this:
and so on… Often, we’ll give data like this and ask students to predict what comes next or to discover the pattern. Now, what we’re doing is asking them to guess at the underlying process leading to the data they see. If a student’s answer (model) looks like “the amount of stuff we have later is equal to the amount of stuff we have now plus a little bit,” that seems to me like a pretty powerful realization.
You can imagine then presenting students with different systems that change with time in different ways and have them engage in the modeling exercise of trying to understanding the key underlying process. Their model might be as simple as a written statement, but one they can test by doing repeated addition, subtraction, multiplication, and division. Heck, they might even start to wonder if their is a more convenient or powerful way to do this stuff…
I guess at the end of the day what we’re arguing is once again that mathematics is the science of pattern and mathematical modeling is about applying that science to patterns that occur in the real world. If we keep this in mind, and look for those patterns in the real world whose origin or dynamics can be explained by some iterative process that at its heart consists of addition, subtraction, multiplication, and division, we might indeed be able to engage students at all levels in genuine, interesting, engaging mathematical modeling investigations.