The SBAC question really doesn’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 “question c) part c)”?

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?

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.”

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.

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.

As 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.

Phil

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.

My 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 through modeling. 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 denied the opportunity to do #3. It just isn’t all that specific about how they do #3. Meanwhile, the textbook’s modeling denies students 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 it’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?

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.