The Home Plate Area Mystery

A major-league home plate. Dimensions in inches. From Math World, http://mathworld.wolfram.com/HomePlate.html
A major-league home plate. Dimensions in inches. From Math World. Click the image to go there.

The illustration shows the dimensions, in inches, of a major league home plate, according to the official rules of baseball.

What’s the area of the plate?

Another way to present this question is to note that major-league bases are 15 inches square—and wonder which is bigger.

In either case, the problem of figuring out the area of this pentagon involves taking shapes apart or sticking them together. This skill, of dissecting or composing shapes, is important for students; they need to visualize the easy shapes that are inside (or outside) the hard ones. It even appears in the core standards, obliquely, in 7.G.6. It’s also wonderful that there are different ways to do it.

In this case, it’s not too hard. Students who know how to find the area of a right triangle can be successful without much teacher intervention.

Continue reading The Home Plate Area Mystery

Why (most) word problems are not modeling

…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.

Modeling: Looking for definitions

Modeling is at the center of what I love about math and math education, so I’m thrilled that the Core Standards highlight modeling and that it figures in our latest drafty California framework.

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:

  • What is modeling?
  • What isn’t modeling?
  • How much do we care whether we can come up with a definition?

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 Modeling: Looking for definitions