Bill Carey:

What’s so compelling about the three-act math project isn’t that it does a better job of teaching the body of knowledge of mathematics; it’s that it reshapes the cultural practice of mathematics in a way that more closely reflects how grown-ups engage in mathematical inquiry.

That’s the goal anyway, particularly w/r/t mathematical modeling. Pick any definition of modeling you want – the IB, the Common Core, the modeling cycle, anything. They all define modeling in similar terms. Here’s the Common Core. It’s scary:

1. identifying variables in the situation and selecting those that represent essential features,
2. formulating a model by creating and selecting geometric, graphical, tabular, algebraic, or statistical representations that describe relationships between the variables,
3. analyzing and performing operations on these relationships to draw conclusions,
4. interpreting the results of the mathematics in terms of the original situation,
5. validating the conclusions by comparing them with the situation, and then either improving the model or, if it is acceptable,
6. reporting on the conclusions and the reasoning behind them.

That is a huge list of important, valuable skills. The scary part is how little our curriculum helps students develop those skills. Here’s a task from Pearson’s Algebra I text, which is pretty typical in this regard:

That brave little icon indicating the “Modeling” practice begs the question: Is this modeling? Who is doing the modeling? Try to locate each of the six parts of modeling in that textbook problem:

• Who is identifying essential variables? Where?
• Who is formulating the model for those variables? Where?
• Etc.

Then do the same for any arbitrary three-act lesson plan.

The three-act structure isn’t the only worthwhile approach to modeling and it’s still a work in progress. But we should all stop pretending that including some real, physical, made-from-atoms item in a word problem does justice on its own to the complicated, exhilarating stew of skills we call “modeling.”

BTW. While you’re at it, feel free to compare the Common Core modeling standard against the Common Core modeling assessments. As you may know, there are two consortia developing the assessments. Here is an item from SBAC and an item from PARCC. They are more different than they are alike.

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Bowen Kerins offers up a useful analysis of the SBAC and PARCC tasks.

Which of these drinks has the strongest caffeine concentration? Can you rank them from strongest to weakest? I couldn’t. What information would you need to know to find out?

Two options here:

One, have students talk about the information they’d need and how they’d get it. (Two questions central to mathematical modeling.) Then just give them that information.

Two, have students talk about the information they’d need and how they’d get it. Give them the names of the drinks and have them research the information themselves.

I’m not too uptight about the difference. Option two has the students practicing their Google-fu. Option one costs less time.

Here are the goods. I’m pretty sure caffeine doesn’t work the way I illustrate it in the third act, so you may want to skip that clip.

I also included the cost of each drink in case you’d like to ask students to calculate the “bang per buck” ratio as a follow-up, a cool improper fraction that reads “milligrams per ounce per dollar.”