“You Get What You Assess”: another couple cents about the Common Core

common_core_logoIn discussions about the Common Core, I often hear (and often say), “you get what you assess.”

Sometimes, this is a backhandedly snarky dig at real classroom teachers: we’re saying, essentially, “we don’t trust teachers to do the right thing on their own, so we have to implement some sort of stick, in the form of an assessment, so that when they inevitably teach to the test, at least they’ll teach the right stuff.”

But it has another meaning: the Common Core Standards are, like the constitution, subject to interpretation. And the interpretation that will get implemented is in the hands of the assessors.

I don’t think there’s a way around this second point. We can’t write standards unambiguously and completely—and get them agreed on. This way, if we’re feeling charitable, we can see in the Standards what we want to see, enough, at least, so we can move forward. When I first read the grade 7 standards on stats and probability, my reaction was, OMG this is way too early! This is the college course! but when I calmed down, I realized that if we interpret the words of the Standards in a humane way, the 7.SP standards make sense and are the right thing at the right time.

But that does put a lot of power in the hands of assessment developers. They don’t ask me what I think we ought to ask seventh-graders about statistics—so I’m bound to think they’ve gotten it wrong 🙂

More importantly, (as was recently pointed out by an NCTM officer but I can’t find the reference) much of the giant brouhaha about Core Standards, corporate control of education, lack of teachers on the committee, and so forth, misses the point: the Standards themselves are probably OK; but the devil is in the assessment: what will students actually be asked to do, what do we need to do to prepare teachers to get students ready, and what are the consequences for teachers, schools, and students?

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

Beating the Modeling Drum

RemoteRadiusModel2Hoping desperately it’s not also a dead horse…

We just did a three-post sequence about “Chord Stars,” finishing up with how we could use insights from data to find radii of curvature remotely, that is, without ever finding the center of the circle. There’s a lot to discuss about that process; this post is part of that discussion.

In particular, it’s an interesting example of modeling. Quite a while ago I was worrying about the definition of modeling, not simply to get it “right”—many people model in different ways—but rather to try to identify things that we were pretty sure demonstrated modeling. Part of my anxiety, as the Core Standards lumber into classrooms, is that people will carelessly define modeling as “real-world” (or something equally weak) and we will lose a great opportunity to improve math education.

I often think of modeling in terms of using functions to model data. That’s partly because some of the coolest, most wonderful math experiences I’ve had have revolved around finding a function that was a good approximation to data. The process of measurement, improving those measurements, finding a suitable function, getting insight about the function as I wrangled it, and getting insight into the situation and the data from the function—all that together is an intoxicating cocktail of mathy-worldy wonderfulness.

But it’s not all there is to modeling, so I want to pause to point out another modeling genre (one of the ones I listed in this old post) that just appeared in Chord Star 3, namely, modeling real-world stuff with geometrical objects.

In fact, here are a curb with tools, and the relevant part of a Sketchpad sketch:


They clearly resemble each other, but I want to make two observations: Continue reading Beating the Modeling Drum

Isosceles EGADs: Functions, Geometry, and Modeling

ImageIn trying to come up with more activities for EGADs (Exploring (or maybe Enriching) Geometry and Algebra though Data), the following dropped into my lap. Because it’s so simple and so interesting, I’d better write it down…

Everybody get a sheet of paper and draw an isosceles triangle. Try to make your triangle big enough to kinda fill the page, but also try to make it different from those around you. Make your triangle pretty carefully, but don’t measure and don’t use a straightedge.

Individuals can do this too, but I’m writing this as if it’s a class activity. The idea is to get a wide variety of shapes. It is not vital that these just be sketched, but (a) I think that makes the data more interesting, (b) it opens the possibility to drawing more carefully later, and (c) it’s much faster.

Measure the base angles and the vertex angle, and write them on the page.

If you need to introduce vocabulary, do it here. By the way, we don’t assume that these students know that the base angles should be the same. Also, we all know that measuring angles is hard, right?

We’re going to plot the measurements from the whole class. So write your angle measurements on the board.

You may need to help organize this. Will we plot both base angles? Up to you. If so, consider having each kid make two entries in the T-table or whatever.

Now make a graph. Put vertex angle on the horizontal axis and base angle on the vertical. Think about the range of values before you make your axes!

You may want to discuss what goes on which axis. Without having done this with kids, I bet most of us think of the vertex angle as the independent variable and base angle as the dependent. I, at least, think of the vertex angle as the defining angle in an isosceles triangle. This also has the happy consequence of requiring a change of axes in order to get the coolest version of the formula.

At any rate, the graph should look linear. Address outliers (probably due to bad measurement).

Draw the line you think best approximates the data. Find its equation.

Be ready to present your data and line, and explain as much as you can about the line. In particular, why does it have that slope and intercept?

In the spirit of SERP “Poster problems” this could be a poster-plus-gallery-walk event.

Continue reading Isosceles EGADs: Functions, Geometry, and Modeling

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)


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.

Comments on the California Modeling Appendix

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:

First, be sure to read Lesh on Model-Eliciting Activities (MEAs). You will find references here and a longer summary here.

Continue reading Comments on the California Modeling Appendix

The Modeling Cycle (and troubles therein)

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…):

Modeling Cycle (Core standards)

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:

Modeling Cycle after Blum

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 The Modeling Cycle (and troubles therein)

What’s Modeling Good For?

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.

(So this is the third in a series on modeling. We began with some definitions, then proceeded to look at “genres” of modeling.)

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.

Continue reading What’s Modeling Good For?

Genres of Modeling

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.

Functions modeling data

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 Genres of Modeling

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