I said [Three Act problems] are most valuable to me before learning skills, or rather as the motivation for learning skills. I don’t expect that students will just figure everything out on their own, though. Act one helps generate the need for the tools I can offer them here in act two.
-“Teaching With Three Act Tasks: Act Two,” Dan Meyer
I’ve been thinking about it, and I think I disagree with Dan’s take here. I think there are important differences between providing instruction during, before or after a tough mathematical experience, and that instruction during a problem is often bound to be lost in the flood of ideas that a mind is awash in.
Here’s where I’m coming from. Over the past few class periods, my 4th Graders have been working on a lovely little activity. We watched a short video showing Andrew papering his cabinet with sticky notes. How many sticky notes would it take to cover the entire thing?
I showed this video, and was disappointed by the tepid response from my students. Then I asked my students to estimate the number of stickies it would take to cover the cabinet. More blahs. And then I clarified that we’re trying to figure out how many stickies would cover the entire cabinet, and my kids exploded with ideas and excitement: “Wait, can you give us time to figure this out?”
Really, really great stuff.
While walking around, I noticed some kids getting lost in their calculations. Lots of great ideas, but constantly losing the thread.
Other kids, though, used diagrams to preserve their line of thought. These kids, even if they were less computationally sophisticated than other students in class, were finding relatively more success in the problem.
When I noticed this, I realized that this sort of diagramming was an important mathematical idea that I should make explicit to everyone. When pairs called me over to help them make sense of their confusing calculations, I made the suggestion: here’s a diagram, here’s how you can use it, this could help with where you’re stuck.
No dice, so I decided to pause class and say it to everyone: hey all! I noticed that the tricky thing isn’t just the calculations, but trying to keep track of what you’ve figured out and what you still need to work on. Diagrams can help, here’s a diagram, here’s how you can use it, you might try this.
As I walked around some more, I poked around to see if pairs had adopted my suggestion. No dice, still.
Bell rings, kids hand in their work, that’s that for the day.
The next day, I start class by saying, “I noticed a lot of us got stuck on the problem yesterday. We’re going to keep on working today, but here’s something that might help: here’s a diagram, etc.”
Dice!
What happened? Hard to know, of course, but here’s what I’m thinking: the first time around, my kids had a million mental distractions. Some were wondering if their calculations were right. Others were just trying to get a grip on a plan of attack for the problem. Others were trying to remember where on their page they had written their current tally of the stickies on the front and back.
In other words, these kids had a lot to think about during this problem, and they weren’t really able to dedicate the brain space needed to understand a new and unfamiliar strategy.
This is also how I make sense of something I’ve noticed in my Algebra 1 class. I haven’t yet given these kids activities that explicitly address the “cover-up” method for solving equations, but I keep trying to bring it up when kids ask me for help with equations in class. The thing is, it never seems to stick.
It seems to me that if we think “just-in-time” instruction works particularly well, my kids should be able to hold onto this method a bit better than they currently do. After all, they have a clearly felt need for some new bit of math (they called me over, right?) and they are getting the instruction during their felt moment of need. Super-duper effective setting for instruction, right?
But then it doesn’t stick. And I think it’s for the same reason that my 4th Graders didn’t take up the “draw a picture” suggestion: they’re too mentally distracted to really focus on the new idea and properly learn it. After all, learning a new idea in all its proper generality can be a pretty heady bit of work. When my kids call me over for help with their equations, they’re potentially thinking about many other mathematical things — where am I in the problem? did I make a mistake by subtracting? what’s 4 divided by 6? — and often can’t focus on the strategy itself.
This, then, is a sort of dissent against the Three Act model of instruction. New mathematical ideas are not best introduced in the middle of a problem if they’re going to get the mental real estate they deserve. Students are often productively distracted by a difficult problem, and unable to focus on the strategy or tool at hand.
The thing that works better, in my experience, is following up a tough experience with a new idea or tool. This seems to me closer to ideal. The students get to spend of time struggling with a tough problem, which I think is valuable all on its own; they thoroughly understand the problem context, since they spent careful time on it; when I introduce a new idea after this experience, they are in a strong position to focus on this tricky new idea itself rather than the million other things it takes to comprehend this new tool.
As Dan Schwartz writes:
This report is based upon work supported by the National Science Foundation under REC Grant 0196238.
OK fine, but he also writes:
Instruction that allows students to generate imperfect solutions can be effective for future learning.
But instruction that comes in the heat of the moment is not looking towards the future — it’s coming during the chaotic present, a time when the student’s mind is being bombarded with many tricky ideas that are specific to a particular problem context. I don’t think that’s a great time to introduce a new idea, but tomorrow might be.
Teaching With Problems 
I guess I don’t see just-in-time instruction as necessary part of the second act. The next day is almost as just in time. What this post made me recall, though, is how often when coming back a second day, many times the interest is long gone. Exacerbated in college, for sure, where it’s 2 days later. I do like the idea of creating space for attention.
Always grist for the mill. Thanks.
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I think it is a fun self-experiment for teachers to try out interventions, both different levels and different stages into the problem. Here is a really nice collection of problems to use: Jewish Problems By TK and AR
The collection is great b/c there are 21 problems spanning several mathematical areas, a section of hints (4. Ideas), then a section of solutions.
I’d suggest teachers try the following or variants:
(a) look at a problem and then go for the intervention
(b) Set a timer (you choose 3, 5, 10 minutes), work on the problem, then look at the hint immediately when the timer rings
(c) set a stop-watch, work on a problem, then stop the watch when you feel you want to see the hint
(d) work on a problem, put it aside for 2 days, then look at the hint.
You could do something similar with the full solutions (or parts of the solutions).
My own, typical, experiences:
a) I haven’t gotten sufficiently invested, so I don’t really care
b) not ready to engage with the hint, I still have my own thoughts about how to attack the puzzle. I have to mentally pause, then engage with the hint. I don’t think this would work if someone where telling me the hint.
c) Depends on the problem. I have a lot of patience for being stuck.
d) Usually have lost track of where I was; needs some effort and backing up to get back into the problem. Note: also depends on how much I have going on in my life; if not much, then I won’t really set the problem aside, just continue nudging it mentally from time to time.
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I’m not going to argue with you about providing instruction when kids have the attention for it, and not when they’re coordinating too many things at once (I might also argue for more diagnosing during the in-the-moment conversations, and less instruction). I would like to ask whether 3-Act Math is inherently based on instruction during the 2nd act? I read 3-Act Math as mostly (and best) focused on mathematical modeling like this: 1) get kids wondering, 2) force kids to do some modeling thinking by only providing quantitative information needed to solve the problem on demand, 3) provide data/measurements/information that kids can compare their answers to.
Dan has, in the past, argued that 3-Act Math is also good for promoting headaches, along the lines of the Productive Struggle folks (like Manu Kapur), which then motivate instruction. I think that Kapur and co propose a situation along the lines of: suggest a situation about which kids can reason but for which they do not have efficient, expert procedures — a task that is non-routine but which should someday become routine. Get kids to a point of grappling with and identifying what makes this problem non-routine. Then *resolve the situation*. Then provide instruction on that which you did to resolve the situation. So a 3-Act Math Task for Instruction (as opposed to a 3-Act Math Task for Application or a 3-Act Math Task for Learning to Model) protocol might be: 1) Act 1, 2) Act 2, 3) Act 3, 4) Instruction in the tool that makes getting from Act 2 to Act 3 super efficient and which is based on kids’ informal strategies but recognizes emerging order and organizes them into efficient procedures?
I think what you’re pointing out here, importantly, is that kids have to be ready to learn in 2 senses: they have to have the question in mind that instruction will answer AND they have to have attention and focus on learning a thing… which is different than attention and focus on solving a problem.
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This is a great example of not blaming the student. Something doesn’t work, what can I do differently? That’s how we get better.
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I totally agree with Max‘s comment. Here are my favorite parts:
This is somewhat in tension with the line of thinking that instruction is most effective at the time of highest felt need for that instruction. But after that moment of intense need ends? Those experiences of struggle can set the stage for instruction. If you see someone drowning, don’t try to teach them to swim just then. But maybe tomorrow.
For sure. If I am going to give an explanation or lecture or notes on a procedure, it would come in Act 4, not Act 2.
Well said!
To draw a line between this and a metaphor Dan has used, the moment when you have a headache is the best time to take Asprin, sure, but it’s not the best time to start thinking about why I got a headache, and could it be related to my diet, and should I call a doctor about this, and maybe it’s the weather. I’ll look into all that, but after I get rid of this damn headache.
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I agree so much that I never do anything like 3 Act problems, even though I often give my kids longer problems.
Here are tasks they get:
1) Use the knowledge of the parametric function, the unit circle, to graph each of the sine and cosine functions individually. (trig)
2) Modeling Exponential Growth/Decay. (notice the difficulty with percentages, which is old information that they forget.) (algebra 2)
3) Determine if the geometric mean is always larger than the arithmetic mean. (trig)
In case 1, they have the values, but have never thought of them as individual graphs. All they are doing is finding the shape of the graphs with values they already know.
In case 2, they are asked questions of x given y (how long will it take to reach 1500 cells), and asked only to notice that hey, they can’t do it. (And I have learned how to give them percentage instruction the day before, with a worksheet, so it doesn’t get in the way.)
In case 3, they have already figured out the geometric mean, but never really considered the question, which involves algebra they are more than capable of.
In my view, you do one or the other–new process they can figure out for themselves with no additional information, or larger problem they’ve never considered but can do with existing skills. Not both.
By the way, I don’t call it the “cover up” method, but I find kids do very well with that method if they don’t look at an equation. I have $24, power bars cost $3 and sodas cost $1. If I bought 6 sodas, how many power bars can I buy? THen, after a while, when they are stuck when looking at an equation, I remind them to verbalize it, rather than try to solve it algebraically. (For weak students, of course. Strong students I beat them about the head and neck until they do the work.)
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A meta-comment, if you will permit me:
Beware of over-generalizing. There are situations where students can absorb the right hint in Act 2, and there are situations where the three acts set the stage for Act 4. And sometimes during the same lesson, the same Act 2 hint that will get one kid unstuck, will overwhelm another. Context is everything. We need not develop a universal theory out of any one experience or any one teaching strategy.
For a given teacher, adopting “what do you notice?”, or three acts, or Nix the Tricks, or etc. is a significant advance in their teaching. But after a while, they develop a need to balance this out with complementary strategies, owing to the complexity of teaching, the variability of students, the unpredictability of classroom dynamics.
Or, as I often put it, nothing works.
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I agree with the thrust of your comment entirely! In general, I completely agree with the caution against over-generalizing. (though sometimes over-generalizing might be appropriate har har)
But part of how we build knowledge is by building on the ideas of others. I like Dan’s (and others’) lessons, I like the Three Act breakdown of the modeling process. Building on their ideas, though, my argument is some of what Dan suggests works best in Act 2 would in fact work best after experiencing the problem.
In other words, my inclination isn’t to walk away after saying “nothing works.” It’s a helpful caution, but I think it’s appropriate to say more. What works? When does it work? Can we describe the contexts in which Act 4 is best? When instruction during Act 2 is best?
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Yes, exactly! Your post enriches the conversation beyond a dogmatic take on three acts, because you paid attention to the specifics of what was happening. That’s the sort of flexibility we all need.
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Such good thinking here. Clarifying question. When you paused the class, were the kids still sitting with their work in front of them? Or did you have them get up, move away, and join you somewhere where they could focus? I’m asking because, at PCMI weekend, I missed a whole bunch of useful instruction because I was actually still working when we were supposed to pause. It was right in front of me and I was in the middle of a thought when they started talking. It was irresistible. I should have stopped, flipped it over, and listened, though. So, I guess I wonder if you had their full attention when you thought you had their full attention?
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Great question, and they definitely were still sitting with their work in front of them! It’s interesting to wonder how much better this could have been if I had asked kids to formally pause their thinking as you’re suggesting. I’m quite sure, in fact, that I didn’t have their full attention, exactly as you’re describing. Another great coach’s catch, Tracy.
I also suspect, though, that these kids weren’t ready to give this new idea their full attention. Even if I paused their thinking, they would still be in the middle of understanding this problem fully. They were still noticing new features of the context, the relationship between area and stickies and the number of rows of stickies on each side.
This didn’t make it into the piece, but another concern: will kids see this strategy as locally relevant, or will they see it as a strategy that works in general? I suspect that instruction that happens during the problem-solving process is less likely to be seen as a globally applicable idea.
So I suspect that your suggestion would have helped, and I hope to incorporate it into my repertoire. But I’m not sure it changes my overall thinking, which is that instruction during the problem is very often not the right time for teaching.
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Michael, this video came to mind: https://www.teachingchannel.org/videos/skip-counting-with-kindergarteners Stephanie does an amazing job launching with a “Here’s something I’ve noticed” or “Here’s something you might try” mini-lesson. She has as full attention as she can get in K right there. They’re all on the rug facing her, not yet thinking about their problem. But then, watch at 4:12. She does a mid-workshop pause to introduce another “Here’s something you might try.” She has kids stop counting, stand, and come where they can see. Not as full attention as the opening of the lesson, but definitely more than if the kids still had their hands in their pom poms and puzzle pieces.
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For sure. There’s a lot of room for improvement in the way I handled this situation in class.
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Two conflicting thoughts:
1. Timing is important. You’re trying to figure out when is the optimal time to give them some aspirin. This is a great thing to think about.
2. While I think you should try to optimize timing, know that you can’t solve it perfectly. You have all these unique minds in the room, each thinking about the same thing and very different things. What’s the right timing for one kid will be the wrong timing for another.
I think what you’ll come to is that there are good, better, and best times and you’re going to need all three. Kids often need multiple exposures to an idea at different times for it to stick. This idea of diagrams and when they’re useful or not is a big one and you’ll need lots of passes at it in different contexts for them to really own the idea and use it.
Of course, there are also bad, worse, and worst times too. I think 3-Acts and the aspirin/headache story arcs help us avoid those times (e.g., the very beginning, before kids care at all).
I think your first couple of passes at diagrams were good, and the next day turned out to be best. We don’t know if the next day would have been best if it had been the first exposure, though. It’s entirely possible that you planted the seed the first day and they were ready to water it by the second, at least for some kids.
And then some still aren’t there yet, while others had figured it out on their own before any instruction about it.
It’s a big part of why teaching is so hard! There’s tension between what’s good timing for “them” as a group, and what’s good timing for each kid.
Thanks for getting me thinking!
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What I appreciate about the post is that it reminds me that any system, or model of instruction, should not be the ultimate determiner of instructional choices. Act 1, Act 2, Act 3 might be more effective in a situation if used as Act 1, Act 2, Act 1, Act 2, Act 3, Act 1, or any other such sequence. As I search for effective instructional choices in particular situations, I don’t want to invest in and persist with an instructional model if it’s not working in that situation. Too often, I’ve fallen into the “press on regardless” trap. It’s an issue, for me, of holding my beliefs as opinions so I can be instructionally nimble and make adjustments. It’s never easy.
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Marilyn — your comment, Tracy’s and Henri’s all seem somewhat along the same lines: teaching is complex, and there might be times when new instruction is best delivered at different (or multiple) points in this sequence. However helpful instructional routines are as paradigms of instruction, it’s important to be willing to swap things around depending on the circumstance.
I wonder if we can all agree with these:
* Classes learn a new idea best when they are able to give that new idea as much attention as possible.
* There is not one single moment when instruction should happen. Classes usually need multiple exposures to mathematical ideas.
* Sometimes, while in the middle of working on a problem, classes are too mentally occupied by their work to devote their best attention to the new idea. There are various teaching moves that can help minimize this distraction, sometimes entirely.
* Sometimes, classes are best able to devote their best attention to a new idea when they are already very familiar with a problem and its context. In this situation, it is often best to introduce new math after students have had time to work on their own in the problem. Kids might just need time to get used to a new context and all its details before seeing a new, relevant idea.
* Specifically in the context of Three Act problems, the above situations will often come up.
* It is probably wrong to talk about the “best” time to deliver instruction, and so it’s wrong to say that “instruction should happen during Act 2” or “new ideas should happen after resolving the problem” or anything as general as that.
And here’s one that I believe to be true, that maybe others aren’t willing to sign on to yet:
* But the problem is not just the problem of making generalizations about teaching. There are specific issues that arise when teaching new mathematical ideas in the middle of working on a problem that, in some ways, make instruction during Act Two a trickier thing to pull off than instruction before or after the Three Act sequence.
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I pretty much agree with this summary, including the last point.
I might distinguish giving a hint during Act 2 from providing instruction at that time. A hint will typically build on existing knowledge, while instruction will typically extend it. A hint will typically be addressed to an individual student, and be tailored to their need, while instruction will typically be addressed to the whole class, or a large fraction of it.
Kudos for thinking deeply about your teaching practice!
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Max:
I’m fried at the end of a day, and not as attentive to everything I should attend to, but this is going to stick with me. Thanks for hosting the conversation, Michael.
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It seems to me that, when you gave a hint the next day, you WERE still in Act 2. You hadn’t given the answer yet (Act 3), right? There’s nothing inherent in Three Act Math that says that Act 2 has to be a single day, or a single session of work. It is simply the time spent solving the problem.
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Interesting point, James!
So, with your observation, would I formulate my point as: Act 2 should come when kids aren’t in the middle of thinking about Acts 1 or 3? Hmm…that doesn’t sound quite right, but I’m not sure how to patch that up.
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I think I would phrase it as ” Teaching content/giving hints in Act 2 should only come when they have fully digested Act 1.”
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I’m trying to be careful not to overload the three-act thing. It’s a metaphor and not one that’s robust enough to support the weight of this conversation. Essentially it’s a separation of concerns. Separate the part where we conjure up curiosity and contextual knowledge from the part where we mathematize the context from the part where we witness the result of that mathematization. The metaphor has less to say about what teachers do during that second act mathematization.
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