Whatâ€™s so compelling about the three-act math project isnâ€™t that it does a better job of teaching the body of knowledge of mathematics; itâ€™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 â€” the IB, the Common Core, the modeling cycle, anything. They all define modeling in similar terms. Here’s the Common Core. It’s scary:

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

**Featured Comment**

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

## 14 Comments

## Jim Doherty

March 7, 2013 - 5:16 am -Reading the problem above I cannot help but recall the Peanuts animated cartoons where we heard what adults sound like there. As I read the problem, I imagine my students hearing ‘blah blah’ lines, ‘blah blah blah’ where do they intersect.

Depressing

## William

March 7, 2013 - 6:58 am -Dan – that example is pretty powerful. Math textbooks tend to focus on #3 (and so much obscure #3!) to the utter shafting of #s 1, 2, 4, 5, and 6.

I had a baseball coach who told us that, “he wasn’t here to teach us to play baseball, he was here to teach us to be baseball players.” Took me a long time to realize, but he was talking about the cultural practice of baseball. The skills of the game are ephemeral, but the culture of the game, individual responsibility in the context of shared goals, is permanent. (Big shout out to Coach Douglass, wherever he is.)

The big battle in all education right now is between the idea that teaching is the cultivation of cultural practice and the (terribly reductive) idea that teaching is the transmission of a body of knowledge. Three Act Math is a powerful argument in favor of teaching as the cultivation of cultural practice.

If we think about teaching math is the cultivation of a cultural practice, we can look to other disciplines to see how *they* cultivate their cultural practice in their students. What if we have students produce drafts of their work for critique by their peers? What if we ask students to evaluate and explain a beautiful mathematical model that we provide them? What if we have students read and emulate the masters of the craft, like Euler, Wilf, Shannon, Newton, or Archimedes? Do our students even know that there *are* mathematicians worthy of emulation?

Teaching mathematics is about seating the cultural practice of modeling (and a few others) in a context that’s meaningful and inspiring. Three Act Math tackles that well from one end. I think the other avenue of attack is to produce graded readers of the classics of the genera aimed at students. I’m working on that (perhaps too quietly and slowly, alas; I should start a blog).

In passing, it also turns out that the cultivation of cultural practice has a long history in education. In particular, the medieval university had a curriculum structured around it. The trivium[http://en.wikipedia.org/wiki/Trivium] breaks teaching down in to three components: grammar, logic, and rhetoric. From the common core’s definitions, #s 1, 4, and 6 would be mostly rhetoric, #s 2 and 5 would be logic, and #3 would be grammar. In the last twenty years, a pedagogy has grown up around the trivium that teaches grammar, logic, and rhetoric differently. It’s mostly been applied to the humanities, but gradually a few of us are brining it to math instruction.

## David Wees

March 7, 2013 - 8:57 am -I’m not sure #1 always applies in its current form. For example, if you begin to analyze the Four Colour Problem, you soon realize that the shapes of the countries doesn’t matter, it’s the connections between them that matters most which leads to using Graph theory to model various arrangements of countries, and then being able to apply some relationships which exist in graph theory to come some conclusions about the Four colour problem.

I’m not at all clear what I would call a variable in this case. Does this not count as mathematical modeling because I don’t have variables? It sure feels like modeling to me.

## Phil @liketeaching

March 7, 2013 - 11:24 am -I agree that it’s very much #3 in the list which textbooks seem to focus on, and its both down to the medium of print and to the difficulty of teaching the other parts.

To be fair though, I’m not sure that three act maths does this particularly well either; not explicitly anyway. Where I think that three act maths excels is in motivating a genuine, transparent need for modelling, and rewarding the work you do on the problem. There is still a big gap before the ‘give them the information they need’ step which needs to be filled by the skill and the discretion of the individual teacher. Three act maths is a way to set up a modelling lesson rather than a way to teach it.

The act of modelling, though, is very dependent on the students’ prior knowledge, confidence level and familiarity with modelling. It may therefore be a good thing that these steps are left so loosely defined, but I can envisage the possibility of a lesson plan which explicity teaches the skills in that list, even if I have not yet been able to write one.

## Michael Pershan

March 7, 2013 - 3:32 pm -The following comment might not be coherent, and it might be based on a misinterpretation of your work. So, apologies.

I think there’s a distinction between three-act lessons and the three-act format.

Your three-act lessons exemplify pretty much all the standards for modeling. But the three-act structure itself don’t say much about good modeling problems.*

*

Unless I’m wrong! I have no desire to tell you what your thing says.If I get the structure, then Act One sets up the conflict, Act Two makes resolving the conflict possible, and Act Three resolves the conflict (and sets up another).

Bad modeling problems within those guidelines is very, very possible.

This is a long way of saying that I think both the 3act structure and the 3act lessons are great contributions, but they’re very different ones. Your 3act lessons are great modeling problems. But your 3act structure is truly brilliant, I think, because it serves as a great, memorable and intuitive checklist that helps me with everything that I do in the classroom, modeling or not.

The great contribution of the 3act structure — I think — is that it draws attention to under-appreciated aspects of lesson design. Act One reminds me to ask an interesting question, and make sure that everyone understands it, and to do that first. It also reminds me that this should be such an interesting question that it requires little exposition on my part. Act Two reminds me to NOT do certain things while presenting the initial question. Act Three reminds me that it’s more satisfying (and authentic?) for students to test their work against the world than to trust me.

It’s obvious to me that the 3act structure fits well with good modeling problems. But I think they’re separate insights.

## Dan Meyer

March 8, 2013 - 8:12 am -David WeesAs you direct your attention to certain features of the map and not others you are inevitably selecting essential variables. You said it yourself. You don’t care if the border is jagged and curved like California or if the sides are straight like Wyoming. You care how many states could possibly border it. So you’ve identified essential (and inessential) variables.

PhilMy own lessons are pretty agnostic about when students learn how to perform the operations. Sometimes the task has so many toeholds (like Pixel Pattern, Pyramid of Pennies) that students can start anywhere and learn skills

throughmodeling. In other instances, we pause after students are engaged by the first act and learn some operations.The point is that nowhere in the three-act structure are students

deniedthe opportunity to do #3. It just isn’t all that specific abouthowthey do #3. Meanwhile, the textbook’s modelingdeniesstudents the opportunity to do #1 (by identifying and naming the variables) and #2 (by offering the models already). That’s an interesting difference to me.@Michael, if you’re going to tell me this three-act thing applies to more classes of math problems than I’m allowing, I’m not going to stop you. Certainly I find useful elements to pull out of it in other tasks. I just think it fits modeling best.Quoting you here: “Act Three reminds me that itâ€™s more satisfying (and authentic?) for students to test their work against the world than to trust me.” In what kind of problem outside of modeling do you test your work against the world?

## Michael Pershan

March 8, 2013 - 9:01 am -“In what kind of problem outside of modeling do you test your work against the world?”

I mean, depends what we call “the world.” But yesterday I asked a bunch of Alg2 students to turn their calculators off and figure out everything they could about the graph of y = cosx * sinx. Then we checked the graph.

## Phil @liketeaching

March 8, 2013 - 11:35 am -agreed: Three acts sets up a modelling lesson, textbook lessons such as the example above effectively shut out the possibility of (many of the important steps of) a modelling approach

## Bowen Kerins

March 8, 2013 - 1:30 pm -I’m no fan of the PARCC task as written, but some of that is due to a goal of having the majority of the test be computer-scorable. Parts (a) and (b) of the task are computer-scored, and part (c) is hand-scored. My biggest beef is that the numbers 1.16, 1.2, and 1.3 are available for the rate, right after the word “about” — based on this all three answers should be correct, but they’re not (only 1.2 is). That makes this task unacceptable for a high-stakes exam.

I hate that everything is a defined relationship between x and y, and I’d rather have students type in their trend line instead of dragging options. That could still reasonably be computer-scored.

The “modifications” question (part b) is pretty good, and asking conceptually about what will happen when the jar is narrower (part c) is pretty good too. The task also targets specific high school content standards and tests whether or not students know them — and that is a very important thing for an exam to do correctly.

I like how open the writing prompt is for the SBAC item. But who in their right mind would have this many data points to go from in decision-making? Someone with this many data points should already be able to draw the same conclusions without the fitting lines. One big plus to the SBAC item is its scoring — 20 or 60 is correct if it’s got the proper reasoning. (See the scoring rubric at http://www.smarterbalanced.org/wordpress/wp-content/uploads/2012/09/math-rubrics/43028Rubric.pdf).

But there are some really bad details here. The SBAC item is marked as a high school item but it tests no high school content standards (8.SP.1, 8.SP.3, 8.F.5). The SBAC item’s scoring rubric also makes a serious mathematical error, giving the incorrect earnings in the $60-140 range. The lack of on-grade standards being tested makes this item unacceptable for use in a high-school exam, and the lack of accurate math (in a sample item!) is really troubling.

PARCC sample items sometimes have similar major issues. For example, their item about cellular growth (http://ccsstoolbox.agilemind.com/parcc/highschool_3836_2.html) uses subscript notation for the recursive rule, when the Common Core and the progressions explicitly say not to do this.

Short version: I’m worried, but at least all these problems are better than the Pearson Algebra 1 sample you give. That problem reminds me of a Phil Daro quote, “If you see a textbook and it has items labeled as the Common Core items, they’re doing it wrong.”

## William

March 8, 2013 - 2:25 pm -> Meanwhile, the textbookâ€™s modeling denies students the opportunity to do #1 (by identifying and naming the variables) and #2 (by offering the models already). Thatâ€™s an interesting difference to me.

Affording kids the chance to do #s 1 and 2 is a huge win. One other place I think the three act structure is a step forward compared to regular textbooks is the back end. If the students have to work out things of consequence, they have a chance to communicate things of consequence too. I often wrestle with how to do #s 5 and 6 well. When students are done with the second act, how do y’all have them students report on their work? Orally? In writing? Posters?

## l hodge

March 8, 2013 - 3:55 pm -The SBAC is nicely presented, but is one dimensional (interpret slope as a rate). The PARC question covers a lot of ground (vocabulary, displacement, modeling, systems of equations), but is really wordy. It would be nice to see an animation of the balls being dropped into the cup of water instead of a table with the measurements.

The SBAC question really doesnâ€™t seem to be about modeling in my mind. The situation is fixed, and the model is provided (plot & lines of best fit). Also, this sort of question is just pretending to be open ended. Yes, there are a range of answers and approaches, but they have steered the student towards one approach. The intuitive approach would be to calculate and compare the profit corresponding to a few points. The practical approach would be to add a profit column in the table that must have been used to create the plot. Apparently these approaches are undesirable to SBAC, probably because they are grade school/middle school level techniques, so they put in the lines of best fit in an attempt to steer the student towards a more mathematically sophisticated approach.

PARC seems to be assessing a number of different things. Part a) is largely vocabulary and basic knowledge of the slope intercept form of a line. I like the questions on b) & c) that get at whether the student really sees how a change in the experiment would change the model (line) – to me that is modelling. Though, who on earth labels an item â€œquestion c) part c)â€?

## Eric Jablow

March 9, 2013 - 1:34 pm -So, in 2015 no one ever uses travel agents, and in 2010 and afterwards, everyone uses the Internet to plan for travel. Thrilling.

## Christian

March 24, 2013 - 11:21 am -I like Blum and Leiss’s cycle for modelling.