Yeah! Thx for doing this blog series. I thought I was the only one who wasn’t exactly sure how to handle these 3act videos. I have tried one, when I asked for questions (from a room full of 16 year olds), I get endless Qs like “why would anyone build a pyramid like this”. Eventually 1 or 2 quiet kids will offer up a suitably mathematical question.
Looking forward to the rest!

“I assume that the pedagogy around these tasks has been self-evident”

Nooope – I think after a teacher encounters you, one of the following happens:
1. They dismiss the idea as too weird and unfamiliar. These people might be beyond saving, but could be swayed if they see kids/participants excited and engaged in meaningful stuff and learning.
2. Too scared to try because they don’t know what it looks like in a classroom
3. Try it but execute badly because they don’t know what it looks like in a classroom, lesson fails, they give up or go though a trial and error process trying to make it work
4. It aligns with how they already conduct lessons anyway and they absorb all your great ideas and beautiful materials into their arsenal of things they’ve collected already.

Video of instruction will go a long way enabling the #2 people to give it a try, and put the #3 people in a position to go kill it much more quickly. And maaaybe sway some #1 people. And give the #4 people ideas about moves to try and tweak. Basically, there is only upside.

So not that you need it, but this is me validating that this is so, so valuable. The potential to affect behavior and effect change results from ways of communicating that live on a spectrum that goes from talking about a thing, to showing a thing, to experiencing a thing. I implied (possibly wrongly) misgivings from your parenthetical “(You don’t have to do any of this of course.)” and want to say: stop that, this is awesome.

Feedback:
add to the 00:43 “I’m curious about what questions this brings about for you.” That’s important – an instruction, even though very general, about what they’re supposed to be doing while watching. Doing something like this with kids who aren’t used to it — it’s important to set an expectation about how we’re learning something here, we’re not screwing around watching youtube.

The next three timestamped elements are think-pair-share. Just an observation, thought it was worth acknowledging — this is what TSP is for and what it looks like in context, done with a purpose.

At 4:43, it’s important that you played the video again.

7:23 — I’m curious about the pros/cons of writing names next to lowest highest guess. On one hand, woo, my name’s on the board, I’m famous. On the other hand, I’m famous for being at an extreme, I might feel bad about that if I am 14. Would it be equally effective to just write the numbers? with the attitude “as a class our estimate is somewhere in here” giving it a more “we’re all in this together” feeling instead of a “who was more right and who was way, way wrong” feeling. Maybe, maybe not, maybe it depends on how supportive/antagonistic the dynamic in your class is, but curious what people think.

Other notes:
I like the choices you made for where to edit out chunks where there is just crowd noise going on. You didn’t edit out all of it, and I appreciate that I still got a sense of the pace.

This is fun.

Thanks for posting this, I appreciate it. I do have one question/clarification. Because of the break in the video, I wasn’t sure if you asked students to share out their too high/too low guesses. I’ve always had students share them out, but I’ve struggled to get the discourse to be as meaningful as I’ve hoped. I feel like the gut-level estimations of my students have improved dramatically, but I haven’t quite been able to reign in the too high/low responses to be, for example, too low but not ridiculously such (like 3).
I could probably have asked that question much more succinctly: what else do you do with the too high/low guesses? (I hope you haven’t already answered this in the rest of the video…I haven’t watched it because I don’t want to cheat).
Thanks again, looking forward to tomorrow’s post.

My own observations…

I’ve done penny pyramid with three different classes, and I feel that this particular task has a pretty obvious question (how many pennies?). Most students had this question or some variation of it (how much is it worth? how much does it weight?). I try to pick a question that has some degree of popularity. If most students want to find the weight, I go with that, because I know that it will be just as challenging as my desired question. The math involved will be more or less the same. But what do you do if none of the students come up with your desired question? I hate it when this happens, and I hate having to say, “Hey, those are all great questions…but what I’m really interested in is this.” I think that can be very deflating for students.

I was never a fan of the high/low questions. I always felt like a guess was enough. I suppose it helps with estimation skills, but I feel like I’m already getting that with the guess. Am I missing something here?

To address Kate’s concerns about the guesses, I too never felt comfortable assigning a name to a guess. I also feel bad when a student provides a really bad guess to the class. If just one student criticizes him or her, they’ll never do that again.

I prefer having students write their guesses down on a piece of paper, and I collect that info as I monitor their progress. Later we can look at the data through a box-and-whisker plot or stem-and-leaf or histogram or whatever (no pictographs).

Would it be worthwhile for students to justify their guesses? For example, if a student’s “too high” guess was 100 million pennies, a possible teacher response might be:

“How do you know 100 million pennies is too many?”

Possible student responses:

“If they had 100 million pennies, that would be $1million and there is no way someone would save that much money in pennies.”

or

“There are 50 pennies in a roll of pennies and there is no way there are 2 million rolls of pennies in that pyramid.”

It might be helpful for students to hear other student’s thought process for how they make estimations.

Funny side note… I lured my English teacher wife (who is not on speaking terms with math) into watching the stacking video at the beginning. She stood over my shoulder and watched all the way until you presented the total number of pennies.

I am inspired by the math problems you present. I have looked through many of your 3-act math problems and have had struggled with how to implement it. This video has cleared up a few things, thank you.

How long does it take the you to work through the entire problem? Say a 56min period? Or does this take more of a 90min period? Yaacov’s question is important as well. When/how do you introduce these problems?

Fantastic stuff Dan! I think Kate has already nailed the ‘pedagogy isn’t necessarily obvious’ thing, so thanks for making it explicit.

I think the question of “what happens if your students don’t ask any of the questions you think are important” is an interesting one. Three comments:

1. I guess at some point you need to make the call whether the question(s) you want answered need answering. If so, you might have to go down the, “What about (teacher question)? Who else finds (teacher question) interesting?”

2. The questions you get will give you a good idea if this context is a ‘hook’ for them. If they all say, “Why would anyone care to do that?”, or no-one else finds (teacher question) interesting, then you can choose pull the pin on that example.

3. If your students ask interesting questions that you don’t have data/answer then that’s still ok. You can entertain the question, but using assumptions rather than data. The assumptions can be evaluated at the end. You can then record this as useful data for the next time you use it.

I also find the ‘guess’ thing interesting. I’m guessing (heh) that the ‘guess’ aspect becomes more useful over time, as students hone their estimation skills? I would also be tempted to make more of the ‘guessing process’ at the end, to evaluate the different strategies used by the class to estimate the upper and lower bounds in order to develop these skills.

When collecting the estimates I like to have the students put them on a number line. I ask for the highest guess, put marks on the board and have them write their estimate on a sticky note and place it on the number line. This provides a nice visual of the estimates. A couple of extensions: Create a box plot of the data to review concepts of median, quartiles, etc. Have the students explain how located their point on the number line.

James, I feel the same way. I often feel like I have to keep,what I’m doing in my class a “secret” almost bc it’s not the way most of the others teach.

When I first started doing three-acts, I found that students were generally ok diving right into the problem solving process when the task has a low entry point. For instance, I’m doing Dan’s toothpick problem with 7th graders. Dan’s problem is fairly easy for them to solve. You then build in sequels that gradually make things more difficult. (What if Dan’s toothpicks were doubled? How much would it cost to make that? How long would it take?) I think these are all interesting questions and it wouldn’t take much buy-in for the students. Some kids are going to get this stuff done pretty quickly…that’s where you make it tougher. How large of a structure can we fit in this classroom? How many toothpicks? How much would that cost? How long would it take? What if we worked together?

I’ve had plenty of experiences giving a task that was very difficult for the students. This can be very frustrating as students have no idea how to get started. You try to go from group to group and give a little bit of guidance, but with a large class, you can’t get to everyone. Then some students get off-task and start poking each other with pencils and you start freaking out. That’s when you need to get their attention and talk as a class about the first few steps.

I think the timing issue is interesting. My classes are generally 38 minutes long. It is rare for me to get this done in one period. I could easily spend a week on some tasks. And for some students this is frustrating. And for some, this is the first time in their lives where they were really challenged to problem solve and they see that it can be a pain in the ass. But in the end, it’s worth it and it’s rewarding.

If you’re not convinced that this is the thing to do, try Andrew Stadel’s File Cabinet (http://mr-stadel.blogspot.com/2012/04/file-cabinet.html) and watch your entire class get excited as they watch the third act. I still get goosebumps watching them and feeling the energy in the room.

Thanks so much for this guidance! It was great to be a student as we did 3-Act activity with you when you came to Reading, MA. I can’t wait to try this one. The challenge for me is definitely in choosing the right questions to ask and when to ask them, without giving away something that they can discover. I think I’m getting better at it, thanks to your modeling.

For the sake of accuracy:

287,820 pennies = $2,878.20, not $28,782… :)

http://youtu.be/YG9oqlQdVp0?t=22m03s

Speaking of sincerity and honoring kids’ questions and how to circle around and follow up, here are some things I’ve noticed:

* When we use real problem contexts and can answer kids’ questions about “Why would someone do that?” or “What happened next?” with honest answers, they’re pleased.

* When we use contrived problem contexts and answer kids’ questions with “I’m not sure. Some of these questions we can’t answer without getting Charlie on the phone. Which questions can we answer just using our brains?” and following up on their “why?” or “what next?” questions with “what would you do if you were Charlie?” they are still usually pretty pleased. The task is still meaningful if we’re answering reasonable questions and the kids can imagine what they would do if they were Charlie.

* If students come up with questions that you can’t address in the last few minutes of class, you can hook them for life by being willing to think about their question after school (in that mysterious miasma between school days when students are shocked to learn that you do things like have a life, go to the store, sleep, etc.). Coming back with a response to a kid question you hadn’t anticipated proves your sincere interest in their thinking and about math, showing by your actions that their math ideas are worth thinking about outside of school.

@Kathy Sierra, can you say more about how justifying guesses changes them for the worse? I don’t have them justify when doing 3-Acts, but I do for Estimation180, and I haven’t noticed a decline.

Thanks for this series on 3-Acts. I was fortunate to see you facilitate the Penny Pyramid in Palo Alto last summer. However, reading your comments now helps clarify some of the teacher moves that I have forgotten.

Reading the comments from others, I notice that many of their issues are similar to my own issues of facilitating a 3-Act. I have introduced them to teachers over the past year (I’m a math consultant). I’ve always been uncomfortable and a little bit manipulative when I know what question I want them to ask and for which I have the information that they need. I remember you saying about the Penny Pyramid that someone ALWAYS asks how many pennies. In all of times I have done that problem with teachers, it has been one of the questions. I also remember that someone in our Palo Alto class asked, “Why did they build a pyramid”, to which you answered “It was fundraiser” and had a press release about the fundraiser. To other questions that you couldn’t answer, you admitted that you either couldn’t answer it, or that it would be answered after we worked through the problem.

I don’t think that there is such a thing as a “real world problem” in K — 12 education. The students are still using training wheels. In my opinion, the 3-Act process is the closest we are going to get before we remove the training wheels and throw them out into the real world that they will be running when we are, hopefully, collecting Social Security. So we want to train them as well as we can to run a prosperous and productive world, which will, alas, involve non-routine problem solving.

I don’t know if you recall, but after you worked through the Penny Pyramid with our class, I said to you, “The teacher moves you made elicited all 8 of the Common Core Math Practice Standards.” You said, “Well, That’s nice of you to say, but I’m sure I missed some.” I went back to the Palo Alto Quality Inn and wrote up the following summary. I’ve shared this with many teachers as I try to encourage them to include more non-routine problem solving in their classroom. Thanks to you and Andrew Stadel, teachers have a rich treasure trove of engaging problems to access. Hopefully, these problems will encourage teachers to be on the lookout for 3-Act situations of their own.

Standards for Mathematical Practice Task Alignment
Task: Dan Meyer 3-Acts “Penny Pyramid”
Teacher: Dan Meyer
Date: 7-30-12

1. Make sense of problems and persevere in solving them.
– Asked students to brainstorm question “I saw this clip and I had TONS of questions. What was the first question that came to mind?”
– Wrote all questions on board, ranked them, and weighted by interest
– Narrowed down to one question.
– Asked students what information they needed to answer that question. (The teacher had anticipated ahead of time what questions would be asked and what information would be needed.)
– Only gave out information as it was asked for.
– Listened in to groups working and guided them as necessary using questions such as “Have you thought about?” By doing this, the teacher made sure that the students stayed on task and that they were on the right track, but did not use leading or scaffolding questions. The teacher created an environment of trust and safety that encouraged perseverance.
– Noticed that some groups had the answer and then asked an extending question: “How tall would a pyramid be that contained a total of one billion pennies?” – Students brainstormed. They had input into developing questions.
– Once question was decided, students had to determine what information was needed in order to answer the question.
– In order to figure out the information they needed, the students were forced to make sense of the problem.
– When teacher asked, “Have you thought about…”, the students were more comfortable persevering rather than giving up. Also, working within a group helped students to persevere.

2. Reason abstractly and quantitatively.
– Asked students to make a guess…then low and high
– Students reasoned quantitatively about the possible magnitude of the answers in order to develop an estimate. By guessing a high and low estimate, they were forced to reason further about their estimate.

3. Construct viable arguments and critique the reasoning of others.
– Asked students to write down question and share with table (think-pair-share)
– Asked “How many people are wondering that now?” This encouraged students to critique the ideas that others had come up with.
– After writing down their own question, students discussed their question with tablemates, creating the opportunity to construct the argument of why they chose their question, as well as critiquing the questions that others came up with.

4. Model with mathematics.
– The teacher chose an engaging problem that could be modeled mathematically.
– The teacher monitored the students developing the model, but allowed them to develop the model on their own.
– Once the given information was communicated, the students used that information to develop a mathematical model.
– The model involved creating a sum of terms that expressed the value of each layer of the pyramid.

5. Use appropriate tools strategically.
– The teacher monitored the use of tools, but did not interfere.
– In the “debrief”, the teacher discussed the different use of tools. By doing this, students are provided with more tools in their toolbox for future problem solving. Once the model was developed, different tools were used to find the answer:
– Numerical expression
– Summation
– Excel spreadsheet
– BASIC program

6. Attend to precision.
– As students were asking for information about the dimensions of the pyramid, the teacher recorded what they were asking and encouraged them to be very specific in their vocabulary. The teacher, in turn, was very specific about how he labeled the information given.
– Students needed to differentiate between names of the variables that were quantified. For example “stacks” indicated the towers of 13 pennies each, whereas “horizontal layer” indicated the whole layer that was composed of the stacks of 13 pennies. Without this precision,

7. Look for and make use of structure.
– During the “debrief “of the problem, the teacher facilitated a discussion about the structure of the arithmetic expression. What numbers were constant in each term of the sum? Can we use the distributive property? Do we see a pattern?
– The students had to develop an understanding of the physical structure in order to develop a mathematical model that had a numerical structure of its own. The student had to make the connection between the physical structure and the numerical structure of the mathematical model.

8. Look for and express regularity in repeated reasoning. – Early in the discussion of the problem, the teacher helped students develop an understanding of the physical structure of the pyramid. He asked how many stacks of 13 pennies were there on each side of the second lowest horizontal layer. As a discussion ensued, the teacher guided, via strategic questioning and showing strategic images, that there were 39 and not 38.
– Once the base of 40 by 40 was given, the students had to determine how many horizontal layers were involved. There was a regularity of each layer having one fewer set of stacks of 13 pennies, and the top layer had 1 stack. This repeated reasoning allowed them to create their mathematical model.

Dan

I ask for the “first” question because I’m trying to access their rawest, most unmediated perplexity – the first thing they think about before they start wondering, “What does the teacher want me to think about?”

Great, thank you.

I’m thinking of “patient problem-solving,” and wondering if there’s an analogy with “patient inquiry.” I watch this and my immediate question was the same as the 61+ folks in that room: how many coins are there?

But then I wondered: how would the questions (and their corresponding popularity) be different if the prompt was “Take a minute and think of a question.” Doubtless you’d have some kids thinking: “What does the teacher want me to think about?” I’m not sure that would be true in your class, though. Or a lot of the teachers in the math blogosphere. And that minute might incent something unusual, something provocative, etc.

I’m interested in the relative advantages that “mediated perplexity” or “patient perplexity” might offer. e.g. If you had a student with an initial question of “how many” that, after a minute, was “not only how many, but let’s convert this thing to euros.”

Is there a way to have the students derive the equation at 18:42? Or, would this activity be more appropriate after having learned summation?

One thing I do when I ask students to guess some of the given information (like the fact that each stack is 13 pennies) is to have each student write their guess on the whiteboard and then have everyone simultaneously show one student, “Bryan.” Then Bryan is tries to ball-park an average of the numbers everyone showed him. It takes about 45 s., but they seem to enjoy the process.

When I started doing 3acts with my pupils (12/13 years old) it struck me that they posed “word problems” such as: “If there are 250k pennies and every penny weights 3 grams, how much does the pyramid weigh?”.

They had so deeply carved that maths was about solving problems with arbitrary numbers that in many occasions they were surprised that the actual number of pennies could be calculated using the actual data!. They expected the pyramid (or the car spiral, or the picture of the lighthouse…) to be the initial point of the problem. They expected a bunch of absurd made-up data and a question at the end…

It took them some time to accept that I was actually interested in their “rawest, most inmediate” question. The basic “how big/tall/large/cool is that?”.

I don’t dare to say that they love these kind of tasks now… but they certainly prefer them to the classical textbook problems.

On your spreadsheet of lessons (which I greatly appreciate!!) many of them have the Lesson Plan box checked. I am able to download the videos and use them but I am not able to read or download the lesson plans. I am sure I am missing some important points that could help my students and/or steer them in different directions.

Any suggestions on how to view the lesson plans also?

Thanks so much!

I like the ideas you bring up, but how do you “test” your students?

My tests would still look like those dry word problems, right?