## Three Stories About The Same Thing

Apple’s count-up to 25 billion apps struck the same chord for several of us last week. Three of us tried to represent that moment for students with photos and video. I don’t find the question, “Who did a better or worse job?” as interesting as “What were the principles that organized each of our work?”

Mine

Dan Anderson

Sean Dardiss

Sean has a timer rolling in the foreground. Dan has superimposed his computer’s clock over the counter. Those design choices interest me. When he first saw the counter, I’m wagering Dan didn’t have his clock open. I’m positive Sean added that timer later in AfterEffects. What were they trying to accomplish with those additions?

A guess? They were trying to make the problem solvable, which is probably the most natural inclination for a math teacher designing a task for a math class.

I want to be explicit about my M.O. here without calling it better or worse than theirs. I’m still trying to figure this out.

1. I want to recreate as exactly as possible the moment when Apple’s counter perplexed me, when it dialed the pressure in my head up to eleven and made the question irresistible, “When should I start bombarding the app store with downloads if I want to win \$10,000?”
2. I want to separate the tools, information, and resources I used to answer that question from the perplexing moment itself.

The first point argues for a recording of the screen just the way it was when I first found the counter. Nothing extra.

The second point argues for postponing â€” not eliminating â€” a) data samples, b) a table, c) graphing paper, d) the slope formula, e) a lecture about the point-slope form of a line, etc., until after we’ve settled on a perplexing question that needs those tools, resources, and information.

This is a more accurate representation of how I solved the problem. (I had to decide that a table and a graph would be helpful. I had to decide that a linear equation would be the best model. No one gave me any of that.) It’s also more perplexing to see a problem as it exists in the wild, “posed simply and innocently, not flayed alive by terminology, labels, and notation.”

One More Example

I’ve created two visual prompts for the same question, “How many gumballs are in the machine?” (This picture is from Dan Anderson, also.) One version abstracts the problem at the same time that it tries to perplex students. The other postpones that abstraction for just one moment. Both will result in (more or less) the same mathematical analysis. I’m curious which one would perplex students more.

It would be nice to have a website to test out the difference a little more empirically.

Featured Comment

Eric:

I had kids staring at the screen with their iPhones out waiting to download at the beginning of class. It was hilarious. One of the most engaging problems weâ€™ve done all year!

I'm Dan and this is my blog. I'm a former high school math teacher and current head of teaching at Desmos. He / him. More here.

1. #### Dan Meyer

February 27, 2012 - 4:57 pm -

I say “more or less the same mathematical analysis” about the gumball machine examples but, really, the student who completes the problem on the left will do more mathematical analysis than the one on the right. The student on the left has to decide that a) the machine is a sphere, b) diameters define spheres, c) the gumballs are spheres, d) ditto on the diameters, and e) feet and inches are appropriate dimensions for this analysis. That work is done already for the student on the right.

2. #### Katrina Hamilton

February 27, 2012 - 6:01 pm -

It depends on what you want students to be perplexed about. The one on the left allows them to focus on the question “How many?” a bit longer, while the one on the right jumps them from “How many?” to “How do I use those numbers?” immediately.

3. #### Craig

February 27, 2012 - 7:03 pm -

The one on the left lets them come up with a variety of solutions and make a variety of assumptions. Their are more possibilities and different routes open for the students to think and experiment with the numbers. The one on the right (while still an awesome question with a ton of rich math in it) lets them get straight to the calculation (with a ton of problem solving skills would be required to solve this).

4. #### jas

February 27, 2012 - 7:57 pm -

About the apps: You could also just let the website run in class. No prepackaged video needed. Truly a problem “in the wild.”

5. #### Dan Meyer

February 27, 2012 - 8:07 pm -

@Katrina, my guess is that most of the students who are perplexed by the right version would also find the left version perplexing. I doubt the same is true in reverse. So why not start with the left and perplex a few extra learners. What’s the downside?

@jas, for the next few days, yeah. Once the countup finishes, some kind of record will be necessary.

6. #### Mr. K

February 27, 2012 - 8:17 pm -

>It would be nice to have a website to test out the difference a little more empirically.

Well, you’ve got a website. It seems that a bit of server side coding would allow you to serve up different anyqs to different visitors. Client side stuff would let you track mouse stuff, or just time on page, or other analytics type stuff. Oryou cold just have a survey at the end of it. Put a single URL up on your blog, and watch how people respond to the different versions.

7. #### Sean

February 27, 2012 - 8:31 pm -

I’m especially interested in the last line of this post: “It would be nice to have a website to test out the difference a little more empirically.”

More broadly, are there any plans for 3Acts to roll out with something like an NSF grant? And, if so, what outcomes would you measure? I would imagine achievement, but that seems beside the point to a degree. I think what you’re getting at here are almost non-cognitive factors. It’s tough to measure how a student views the world differently, especially in the long-term. How do you measure perplexity, or a newfound propensity to investigate curiosities like the gumball machine?

So much to look at. You’ve mentioned some informal evidence that is strongly suggestive that it’s made teachers view their own pedagogy differently. I wonder if this is generalizable to the population of math teachers, or if there are some unobserved factors of those who’ve embraced it (e.g. more willing to engage in risk, more reflective by nature, follow you on twitter).

What fascinates me here is the low-cost of doing something like this once every 8 lessons or so. Even if the effects are extremely modest, this type of intervention costs next to nothing. Why not roll it out, for real, and get some empirics?

8. #### Dan Meyer

February 27, 2012 - 8:57 pm -

@Mr. K & @Sean, I’m being a little disingenuous with that last line. I’ve been working on that website since last fall. I have a couple hundred beta users and I’ll open the doors this spring. If you guys wanted to poke around and throw me some feedback, I’d be obliged. Let me know what userid you want at this form and I’ll get you credentials pronto.

The data is kind of interesting. Here’s a screengrab.

The perplexity score on Sean Dardiss’ video is 75%. Mine is 100%. Both of our videos will get seen by 100 people total, so those scores will bounce around a lot. Each of those people will have have the opportunity to either skip our videos (because they’re bored) or ask a short question about them (because they’re perplexed).

perplexity = totalQuestions / (totalQuestions + totalSkips).

I would like to know if a given user’s perplexity score increases over time and if that increase has any kind of observable effect on how she approaches the time she has with her students. (I know it’s done a lot for me.) I’d like to know if we can draw any conclusions about the top ten most perplexing “first acts” (the video or photo that kicks everything off). What makes a first act perplexing? Et cetera.

9. #### BrianM707

February 27, 2012 - 10:15 pm -

I have been torn by this too Dan. I feel like your video, as well as the gumball picture on the left, are more interesting. But I think confusions hold on perplexity can be long lasting. And by that I mean if the intial reaction to the left gumball picture is confusion, then perplexity is going to be a much less probable reaction to the right gumball picture. And I’m not sure I have enough students buying into the left picture, and to your video, to make it worth it for me. I may start with Sean’s video just because the extra scaffolding will keep more students in their seats.

I have been trying to figure out ways to frontload (or scaffold I suppose) your 3Act problems so I can give them the same way you do, and still get the intial buy-in that I want. For example, with your Bone Collector lesson, I showed the video, but then before having them work the problem I told them “We need to find the killer, but since I realize you are not trained investigators, I am going to give you some investigator training before we work this case”, and I gave them ‘Investigator Training’, where I was able to walk them through similar investigations. Then when we were done with investigator training, I went back to the video and showed it again and followed your template, withholding all necessary information until it was requested by the students. I had the majority of the class setting up the correct proportions, which was a major success for me. And then on the chapter test my students did statistically better on the proportion problems than any other problems.

10. #### Debbie

February 28, 2012 - 2:43 am -

Brian’s #9 is similar to my thoughts on curriculum design. I think there needs to be content which could be called application or problem-solving or functional but does there also need to be the new information or transition teaching or drill in between? I don’t mean the teacher does three very similar investigations just before launching the biggie, I mean at some point in the past the teacher will have to have taught, for example, estimating measures and calculating the volume of spheres.

On the App counter it was Dan A’s that got me hooked. It wasn’t until I saw his that I thought, “when would I, personally have to buy an app. to be in with a chance of getting the prize”.

11. #### James C.

February 28, 2012 - 3:56 am -

Personally, I think clicking on the live website is the best choice. It’s as authentic as authentic gets, it’s real-time. Failing that, I like the idea of including an initial date and even time. For example, if on your video it had a sub-title pop up at the beginning, Monday, February 27: 9:03am PST. I think it would open itself up to a richer question space. Students could start by wondering “has it reached a billion yet?”, and then start thinking about “how long will it take to reach a billion?”, and then start questioning how fast it is increasing.

I guess the balance as some of you said, is balancing the amount of scaffolding you provide. You want to give your students as little as possible, but enough that they will be hooked. It might come down to knowing your group of students and what works for them and what doesn’t.

12. #### Dan Anderson

February 28, 2012 - 4:51 am -

Neat. I like the different approaches, makes for some great self-reflection.

My look is incomplete because I would certainly show something like your act 1 as an intro (the live webpage would be better yet) in class.

When the class had the proper questions primed (when is it going to hit 25 billion?), then for act 2 we’d measure the rate on that day with the live information. I’d ask them what they need, and how we can measure it.

My image would be more of a follow up to the original problem. (are the rates the same every day? How does that affect when we should buy our app to get the \$10,000 gift card?)

13. #### monika hardy

February 28, 2012 - 5:35 am -

a space for experimenting.. prototyping..
perhaps a space.. already populated by (all kinds of) people who really are wanting to be there… not just for a grade, et al.

14. #### Elaine Watson

February 28, 2012 - 6:32 am -

This task, in its many forms, is a great example of a real world task that (1) students can relate to and (2) has real world applications, and (3) is open-ended.

I plan to use it (citing the source!) in my work with middle and high school teachers introducing the Common Core (CCSS). It can be accessed in greater depth depending upon the grade.

Several of the the CCSS Standards for Mathematical Practice are exhibited by students as they solve this problem:
1. Make sense of problems and persevere in solving them.
2. Reason abstractly and quantitatively.
3. Construct viable arguments and critique the reasoning of others.
4. Model with mathematics.
5. Use appropriate tools strategically.
6. Attend to precision.

Thank you for sharing!

15. #### Tony

February 28, 2012 - 6:48 am -

Whenever Apple posts their counter for their next milestone I get all giddy. We’ve been chewing on this one for a good 2 days. I got data from the past 4 years (since the app store opened) from a business analyst and we looked at how that graph is different from the graph of our data collection over a span of 10 minutes. Then we’re going to predict the next milestone.

I know that opening up my browser to that website asked all of the questions that I needed it to without me speaking a word.

16. #### Ben

February 28, 2012 - 7:13 am -

I’m glad that you’re honestly asking questions and reflecting on this entire experience, because there’s certainly value in all of the examples you’ve shown. True, your original screencast of the rolling app counter does capture the moment of “curiosity” very well, but it also relies on you being there physically in the room to prompt the open ended questions. I could see how using Sean’s video with the counter superimposed would take away from the curiosity, and lead someone to a more directed answer and approach that isn’t as open ended as a traditional #anyqs, but it seems as though Sean’s approach might make it easier for students to encounter the artifact and have some direction without being prompted. I could be wrong though, as it’s been awhile since I taught math.

As for the gumball machine images, I’m of two minds. While it’s always nice to have the unfettered “un-textbook like” example, at some point students are really going to want to solve their questions, and having some numbers that seem reasonably accurate to actually compute with might be warranted. HOWEVER, that doesn’t mean they have to be placed on the graphic itself. I’m just curious if it’s necessarily a bad thing to have those numbers placed there so learners could attempt some calculations on their own without having to “guess” or ask the teacher what the measurements are.

In the end, bringing the real world into the classroom is going to be your “holy grail”, and that can be difficult as you also want to have numbers, measurements, etc. so students have something to work with after talking through setting up the problem.

17. #### Dan Meyer

February 28, 2012 - 7:18 am -

@Brian, the question I’ll encourage you to focus on isn’t “Should I scaffold or not?” but “Where and when should I put the scaffolds?” The second act of a problem can include exploration, data collection, and lecture. But it’s truer to life and more enjoyable for many of your kids not to experience that abstraction at the same time that you’re trying to perplex them.

18. #### Alex

February 28, 2012 - 7:52 am -

Just a note about those ‘perplexity’ scores you mentioned above –

It seems in general that questions vs skips is a fair enough metric. But where your video came out 2 days before Sean’s, that’s going to drastically increase the number of skips his receive (because I assume this website is small, so some people will already have seen yours).

I don’t know whether you already have some way of tracking whether users have already seen other videos with the same subject, and removing their scores from the perplexity stats (or at least showing them separately) – but otherwise, it might bear thinking about.

19. #### Marie

February 28, 2012 - 8:37 am -

The question I have is will Apple Apps continue to be downloaded at a linear rate (since that is the assumption), especially as it approaches the 25 billion mark and as more people know of the prize?

20. #### Belinda Thompson

February 28, 2012 - 9:56 am -

To Brian #9 and Debbie #10: What you describe seems to be a big decision. Is the purpose of the task for students to recognize an opportunity to apply something they already know or for creating a need-to-learn opportunity. Seems like the teacher would need to decide which of these to pursue.

Based on Marie’s #20 point. I like your question. When would we be satisfied that we had enough info to predict the sweet spot? If the learning goal is for students to understand that there is a linear relationship between the number of downloads and time, then we should verify that it’s linear.

About these tasks in general: It seems that there are a few steps involved in incorporating tasks such as this into your curriculum. This is a stab at a list that might be transferable to what we want students to experience, as well. (1) Learning to recognize potential tasks and generating a prompt or question. (2) Applying appropriate mathematics to answer the question. (3) Then there is some PCK involved: What should students come to understand as a result of investing effort in solving this problem? Will I preview the concepts or let the concepts arise at some tipping point in the problem?

Regarding which presentation of the situation is better: The one that most closely matches the goal of using the task in the first place. I am always left wanting to know more about commenters’ takes on what students should come to understand as a result of working on these tasks. Several commenters mention pedagogical moves as motivation, but what about the underlying math concepts?

21. #### Carl Malartre

February 28, 2012 - 10:54 am -

Both gumballs images are provoking a question in me – they are both perplexing me.

Choosing the best is maybe about the “creativity opportunity”:
LEFT GUMBALLS: the abstraction process, the quest to solve something feels creative to me. I feel like I need to discover.
RIGHT GUMBALLS: getting a part of a plan and executing it feels less creative to me. I feel like I need to validate and process.

This old experiment seems related:
http://en.wikipedia.org/wiki/Candle_problem
“The empty-boxes condition was found to be easier than the filled-boxes condition: more subjects solved the problem, and those who did solve the problem solved it faster. Within the filled-boxes condition, high-drive subjects performed worse than low-drive subjects.”

http://www.ted.com/talks/dan_pink_on_motivation.html?quote=526

Retesting both gumballs machine with and without rewards would be nice! ;-)

Carl

22. #### Dan Meyer

February 28, 2012 - 11:31 am -

Ben:

True, your original screencast of the rolling app counter does capture the moment of â€œcuriosityâ€ very well, but it also relies on you being there physically in the room to prompt the open ended questions.

I don’t understand the rationale for planning classroom tasks for classrooms that don’t have teachers. The teacher can and should mediate the problem solving process: asking questions, dispensing information when students know they need it, explaining new tools when students decide their old tools suck. Why load all of that on a worksheet or video in advance and hand it to the student?

Iâ€™m just curious if itâ€™s necessarily a bad thing to have those numbers placed there so learners could attempt some calculations on their own without having to â€œguessâ€ or ask the teacher what the measurements are.

I’m not asking the question, “Should a teacher ever give information in applied math problems?” The teacher will give the information. My question is “When and how?” If you think about a time you applied math to your own life, you likely had a question first. You engaged that question at a very concrete level with intuition and guesswork. Then you decided what tools, information, and resources you needed to solve it. You didn’t encounter all of that on the problem itself, drawn on the world by some unknown deity who knew in advance the question you’d have. Given how weird that’d be, what’s the argument for doing that with our students? (There may be one. We do lots of weird, unnatural things in our classroom all the time for the learner’s sake.)

23. #### Ben

February 28, 2012 - 12:03 pm -

Perhaps I miscommunicated my sentiments by unnecessarily adding the piece about having to physically be present in the learning environment.

As someone who has done a lot of problem-based learning in the classroom I should have said when the teacher “isn’t available” to the learners because they’re currently indisposed. I didn’t intend to try and make a point that the material has to be ready to go online or work in a blended learning environment, and I certainly didn’t mean to imply that a teacher should create a “digital worksheet”. Perhaps I’m reading too much of my classroom management style into this example, but I could see how it might be of benefit to have at least some of those tools and information “locked” at loaded to give students a mental “foot hold” while I’m busy working with another group across the room.

I totally hear you on “when and how” teachers should give students information. Again, I must have read your OP with something coloring my thoughts, as I didn’t intend to question whether or not the teacher should give the information to students.

Interestingly enough, there’s actually a LOT of information already printed, stamped, and put there by the deity of the packaging manufacturer. I know that may sound like nit picking, but I do indeed experience a lot of curiosity about the world around me, and while a lot of those situations don’t have information readily available for me, there are actually quite a lot of read-world curiosities and questions that do present me with the information I need (packaging on items at the grocery store, mile markers on the highway, etc.). Perhaps I’m just looking through a more elementary lens focusing more on base level skills like number sense. Again, I haven’t taught math in quite some time, so please take all of my comments with a grain of salt :)

24. #### BrianM707

February 28, 2012 - 1:04 pm -

@Dan#17 – “the question Iâ€™ll encourage you to focus on isnâ€™t â€œShould I scaffold or not?â€ but â€œWhere and when should I put the scaffolds?â€ The second act of a problem can include exploration, data collection, and lecture.”

Great point, and I’ll keep it in mind during the next month because I will be doing “Ticket to Ride” and “Super Bear”.

The problem I get is that if the students are met with confusion, then they are afraid of being perplexed. I have been trying to figure out ways to soften the transistion between perplexity and problem solving. If they are confused initially, then the intrigue of the problem gets shutout completely. Even if I do after-the-fact explorations, and lectures, that perplexity never remerges in the student.

@Belinda#21
“Is the purpose of the task for students to recognize an opportunity to apply something they already know or for creating a need-to-learn opportunity”

I agree that is a critical decision in designing the lesson. Sometimes if the hook is interesting enough to keep students engaged in a need-to-learn, then that’s great for me. Ideally I would like to use 3Act problems as an opportunity to apply what they already know.

25. #### mr bombastic

February 28, 2012 - 3:02 pm -

I liked the gumball on the left & Dan’s apps counter better. The other version are much more immediately identifiable as math problems. For most people, once they view something as a math problem, the creativity in them seems to disappear – they either use a procedure or ask for someone to tell them the procedure.

For the same reason, I think these are not the best problems if your class requires a lot of support to get them done. It kind of wrecks the mood if it is obvious to the students that you have been prepping them (warmups involving spheres or rates etc).

As far as creativity goes, I would hope that students would think about ways to measure the rate the apps are purchased, whether the rate the apps are purchased is constant, ways to estimate the size of a gumball and the machine, etc.

26. #### Bryan

February 28, 2012 - 7:36 pm -

Debbie (#10) mentions: “I mean at some point in the past the teacher will have to have taught, for example, estimating measures and calculating the volume of spheres.”

This, actually, is what I am most interested in. Without a doubt, the picture is engaging and perplexing. But how do we handle “teaching” students to calculate the volume of a sphere? To me, this is where constructivism meets application (and maybe satisfies both “camps” in math reform). The problem has hooked students and necessitated the use of volume, which now gives us the perfect opportunity to help students investigate and construct an understanding of these concepts.

27. #### David Wagner

February 28, 2012 - 10:03 pm -

You wrote: “It would be nice to have a website to test out the difference a little more empirically.”

Have you considered using Mechanical Turk? It is an amazing piece of technology. I bet you could prepare something useful pretty fast: a Turk task that just shows a picture and asks the viewer “What question does this picture make you want to know the answer to?” — and then compare results for a variety of variants upon the theme.

28. #### a different eric

February 29, 2012 - 8:37 am -

thanks again…
we’ll be doing this today!

the question i have been struggling with for my 8th graders.
our problems are awesome.
they have tons of buy in and love my class.

how do i get them to care about taking a problem like this and writing an equation from it. or graph it. who cares?

i have answers that satisfy me… not sure if they care as much as i do.

29. #### Dan Meyer

February 29, 2012 - 12:42 pm -

David Wagner:

Have you considered using Mechanical Turk? It is an amazing piece of technology. I bet you could prepare something useful pretty fast: a Turk task that just shows a picture and asks the viewer â€œWhat question does this picture make you want to know the answer to?â€ â€” and then compare results for a variety of variants upon the theme.

Ugh. Such a good idea. Bums me out I didn’t think of it first. I could probably get a question or a skip out of each of them for a couple pennies a piece. Thanks for the thought.

30. #### Dan Henrikson

March 1, 2012 - 5:39 am -

This really has some of my students interested. Lets try to get screen shot of it hitting 25 billion so that we have an act 3.

31. #### Dan Meyer

March 1, 2012 - 8:17 am -

You’re on my frequency, Dan. My math says it’ll happen tomorrow night at around 11:00PM Pacific Time. Feel free to give it a shot. I’ll be out of pocket. My next best hope was to snag Apple’s press release announcing the winner and hope they list the date and time.

32. #### a different eric

March 1, 2012 - 9:04 am -

i had kids staring at the screen with their iPhones out waiting to download at the beginning of class. it was hilarious. one of the most engaging problems we’ve done all year!

33. #### Peter

March 8, 2012 - 11:58 am -