Math Like A Summer Blockbuster

A student in my online course did not appreciate the water tank lesson:

The [water tank video] is simply boring. I do not think middle school students would sit and be engaged throughout the entire video. [..] All we know is that someone is filling the tank; we don’t know its shape or dimensions, we don’t know the rate of the water flow, and, in the end, we don’t care. The video is too long and, quite frankly, uninteresting. I couldn’t watch the whole video (although I left it playing to listen for any changes) and I certainly can’t see a group of middle schoolers watching and being engaged.

This response gave me a good angle on three return volleys:

  1. What if we decided not to show the entire video? What could we do with that?
  2. The student mentioned that “we don’t know its shape or dimensions, we don’t know the rate of the water flow ….” Do you realize how much conceptual skill that critique requires? How many of your students can answer the question, “What do you need to know in order to solve this problem?”
  3. This.

That third bullet is a response to a wager I made back in February:

Twenty seconds into watching the hose dribble water into the tank, ask “how long do you think this is gonna take?” Ask [your students] for guesses. Just guesses. Write them on the board next to the guessers’ names. Whenever anyone raises the maximum or lowers the minimum, point it out. Then turn the clip off. Turn off the projector and proceed to whatever else you had planned for the period.

I wagered that students would riot. I found the whole classroom anecdote worthwhile, but here’s the vindicating part:

After the students made their predictions, I abandoned the water tank problem and moved on to something completely different. In each one of my classes, eventually a few of the students made a comment along the lines of “you never told us how long it took to fill the tank.” Sometimes the comment came only a few minutes after we had moved on. Other times, it came much later. More convincing evidence of the students’ level of engagement in the exercise came at the end of the lesson when I played the rest of the video. With their focus on the screen, you would have thought they were watching a summer blockbuster at the movie theater, not a tank filling with water in a classroom.

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. I’m very excited to have discovered your blog.

    I have recognized three (3) distinct educational methods:
    1) Presentation: A single leader lecturing and demonstrating a topic,
    2) Master/Apprentice: a one on one hands-on internship with feedback,
    3) Performance Interaction: many people seemingly doing their own thing until one of them does something someone else finds worth noting, which often results briefly in #1 or #2, and then a reversion to individual action.

    #3 is the type of spontaneous educational experience that happens when someone has a new phone with interesting features, or when basketballers, skateboarders, or jugglers gather and practice in a single location.

    The problem with #1 especially, and #2 somewhat is that often the students do not believe that there is ‘real-world’ application for the received information. That is, they have a prejudice about the utility based on how information came to them.

    I once tended bar and my barback (assistant who brings ice, washes glasses, etc) could never remember the correct order of the three sinks. He was a chemistry major at University and so I reminded him that the cleanser was acid and the sanitizer base and that if you didn’t rinse the glass between the other two sinks, you would end up with white condensate on the glasses. When I pointed that out, he was amazed. He had never tried to apply what he learned in the classroom to events outside a laboratory.

    One great thing about the water-tank video and discussion, is that it helps to bridge that gap.

  2. Kris Kramer

    May 24, 2010 - 6:32 am -

    Currently I’m a student (4th career, I’m almost 46 years old) with the intention of teaching math grades 6-12 by fall of 2011. Having worked as an engineer many years ago, having an aptitude for math, loving education, and feeling a deep passion about students and how public schools operate, I decided to go for this career. As much as I can engross myself in a math problem, I truly struggled with how am I going to answer that infamous question, I’ve heard so many students ask, “Why do we need to know this?” Numerous responses come from the teachers I’ve seen and not a single one of them has managed to satisfy your average student.

    To say math helps you learn how to think analytically or will help in a future job (which in some cases is simply not true!) gets eye rolls and even sleeping students–so I’ve seen during my field work.

    For the past year +, I’ve been searching for applications that students would really use now or in their future. The WHY has been the driving force–as the online student in this post seemed to be wondering. In some cases the ‘why learn someting’ may only apply to certain careers or just for the sake of education. Not very motivating to students.

    Along comes Dan Meyer–who I learned about not even a full week ago on TED– and offers a new focus: ENGAGEMENT.

    Such a simple change in focus and yet so powerful! Even if a student buys the WHY, if they’re not ENGAGED it doesn’t really matter. But when we’re truly ENGAGED, we don’t even think to ask WHY.

    That’s what’s so great about this water tank; if presented in an ENGAGING fashion, no one cares WHY. Learning for the sake of learning. How cool is that?

  3. I get a similar kind of reaction from teachers in my online classes when I discuss standards-based grading.

    And I did show the water tank clip…BIG hit with my 10th grade geometry students when studying volume.

  4. Why would anyone even think of having the kids watch the whole video? I cannot watch the whole video at once. I have to take a break for a trip to the bathroom.

  5. I think this case illustrates quite clearly that simply having a video or some other newfangled digital content doesn’t mean that it will improve the quality of problems or of the learning.

    Let’s face it, the video is boring. What makes it engaging isn’t the fact that it’s a video. What makes it engaging is how the video is presented. Not that it’s novel to point out that it’s not technology that improves instruction, but rather how the technology is used…

    @Garth: That just leads to a follow-up activity where you determine whether the number of students who take bathroom breaks on the day you watch the video is statistically significantly larger than other days. :-)

    @Dan: It’s been fun to watch you pick up steam and all the recent media attention. I think it says volumes about the quality and depth of your thinking on education. Let me know when the “I read dy/dan before he was famous” merch goes on sale…

  6. Dave Barnes

    May 24, 2010 - 9:43 am -

    I was thinking that with the recent events in the Gulf of Mexico this activity is more relevant. I wonder how students’ estimations compare to those of the oil company and experts with respect to range. Seems like you could run a lot of different ways in trying to determine how to make better estimates with less information. Interesting to student? I don’t know.

  7. @Brian, nice catch, there. Yeah, wow. That’s a great visual motivator for a fairly traditional quadratics problem. I’ve got to figure this out. “When do you drop the ball?” Is that the question? Or is it, “From how far back to you drop the ball?” It needs a supplemental video but I can’t figure it yet.

  8. I am wrapping up my second year as a Bio teacher and I have been trying to figure out how to do just this kind of teaching in Bio. As it turns out, it will be a moot point because I have been informed that I have been assigned to teach Integrated Physics and Chemistry next year.

    I was, as a student, a “why” kid. I wanted to know why I cared. I don’t know that this would give me that answer but it would certainly have helped. I was one of those kids that struggled (to be polite) with algebra and trig and excelled at geometry. I liked your explanation in a different post about why that might have been the case.

    When it came time for me to do a masters degree I took a few courses to fill in my weak (took all the field ecology courses I could and avoided the math based science like the plague) science back ground. One of these courses was calculus. I started the calculus course and my immediate reaction was “so that’s</i< the point of algebra, why didn't they just say so?"

    Any how, I hope, if I am teaching IPC next year, to use some of these techniques.

  9. @Brian…WOW. I love those videos, and reading up on those guys was inspiring beyond math applications.

    @Dan…I’ve been a lurker since seeing your TED video and have enjoyed digging through your archives. I teach 2nd grade, and I’m in my 2nd year of teaching. Since teaching summer school math intervention last year (a setting where I had free reign and little curriculum) I’ve been restless and constantly wondering how I can give my classroom kids more problem-solving opportunities. Your “WCYDWT” posts, and your TED video, give awesome ideas AND you do a great job of modeling the preparation part, too…teachers are always musing about the nuts and bolts of “…so how do I do this?!” Your modeling is so successful that I’ve already taken bits of my own curriculum (just bits, mind you) and created problems out of the material.

    In sum…thanks. Though I don’t have the same content to cover as you do in math, nor the exact same management issues (7 year olds come with a different set of joys and challenges than those double their age), I’ve gotten so much out of this blog. I believe I can do a lot to set these kids up to be critical thinkers (who see failures as learning opportunities) at such an impressionable age. I’ve always believed in being less helpful, and it’s nice to connect with other people who subscribe to the same mantra.

    Keep up the good work :)

  10. I’d like to see some scores comparison on math tests, your kids versus theirs, to see if your methods are resulting in anything positive other than attitudes and classroom involvement. Sure, classroom involvement is fun but does it actually result in these kids being able to solve problems the other kids can? Can they bring their skill levels up to pass state tests, factor, do trig, etc?

    I like your enthusiasm and I WANT TO BELIEVE, MAN…but I gotta see the numbers, not just the rhetoric. Maybe you already posted it on your blog and I just haven’t looked for/seen it.

  11. @Chris, fair comment. I appreciate a guy who keeps his hand on his wallet, etc. A couple of notes:

    One, in our most recent comparison of final exam scores, my two Algebra 1 classes outperformed and outpaced the rest of the department, and not by any slender margin. I wish I had more data than that, but I teach a class presently that no one else teaches, which makes it tough to scratch up a control group.

    Two, I’m not proposing anything particularly radical. In this particular instance, I’m doing little more than take the status quo textbook application problem and a) visualize it better and b) involve students in its creation. In the end, though, it’s the same problem that other classes are doing. We do fewer of them, true, because it takes time to do them right, but it isn’t like I’m substituting skill practice for experiential journaling or something where you’d really want to say, “Whoa whoa, show me some data here.”

  12. Hmm, I like the sound of the first point. Helping kids score well on tests is always rewarding and ensures they understand the material on some level.

    I was hoping for some more data with state tests, though, or PSATs or what have you. So I think tests administered by a third party might be convincing?

    I also do agree with you that you aren’t using radical methods, and they certainly do appear engaging.

    It just seems that the claims and implications you make about curriculum and teaching methods are very strong and that they should be implemented across the board. To me, that treads into the area of standards, not individual teaching styles. It would radically alter how questions in text books are written! Instead of having all the data thrust into a complex paragraph in front of them and parsing it, they’d be forced to extrapolate, guess, and figure most of it. I mean, I think many teachers like the sound of that, but does it actually WORK?

    And it does seem like it would take a lot more time; it’s almost like a science or a physics class instead of a math class. Most traditional math classes aren’t taught in a hands-on way. (Maybe that’s part of the problem!) However, I’d need proof that hands-on math gives better results than hands-off, traditional, abstract ones. Also, you teach remedial students, so there’s no guarantee the methods you use would result in higher learning curves for advanced students. Are you advocating that hands-on, be-less-helpful math is the way you should teach pre-calc, advanced algebra, factoring, quadratic solutions, derivatives?

    This actually sounds like it boils down to the experiential/experimental/discovery learning versus…the contrasting philosophy (I haven’t gotten my teaching degree yet :) ), and so I don’t know if there is definitive proof for either side.

    Therefore, I’d be happy to view state-administered testing data when considering claims about changing curriculum, text books and question styles.

  13. Upon further reading, I see in “And Like That…” that you did approach an abstract subject in a discovery manner. I like it. :) But does it result in better problem solving? And does it result in better retention? (It seems initially that it might.) Or is that a separate issue?

    Ah, upon reading the Wikipedia entry for “new math”, I found this, which seems to sum up your argument nicely, and I would tend to agree with:

    ‘Furthermore, noting the trend to abstraction in New Math, Kline says “abstraction is not the first stage but the last stage in a mathematical development” (p. 98).’

    Thanks for all the work you do, both for the students and on stirring the pedagogical pot.

    However, I still don’t think it’s the right path for everyone. Even if you aren’t saying that, the implication is there. And abstract concepts should be able to be taught for there own value to those who appreciate them. Didn’t Euclid say “give him a coin, since he must make gain out of what he learns”? But especially at the initial stages with uninterested high schoolers (which wasn’t me), your approach is appealing.

    Sorry to spam your comment board, but this is a fascinating problem, and I’m enjoying trying to pick out the threads. Hey, abstraction! :)

  14. I think he missed the point. That video isn’t supposed to be compelling or engaging on its own. Heck, I haven’t even watched the whole thing (although I did go a little farther to hear the conversation with your neighbor when another poster mentioned it). I wouldn’t expect middle school kids to sit through the whole thing without complaining. That’s the point. They grumble, and as soon as one of them mentions anything involving time, you stop it and then invest them in it. After they’ve guessed, suddenly they’re more interested in seeing how it all comes out.

  15. @Chris, I hear that argument often from teachers I work with. “I can’t teach that way because I have to get them ready for a diploma exam (our standardized test).” A teacher once said to me during one of my sessions, “I get that kids are going to have fun and play with math, but when do they actually learn?” I almost cried. The problem is that teachers who push through a curriculum and check off outcomes don’t allow students to truly understand the material. If you teach for understanding (and Dan’s methods do that), then the curriculum will come along for the ride. Kids will do as well, or better than their peers on those standardized tests. I would argue, also, that they will do better in University because they are better thinkers.

  16. I can understand the idea that fun discovery learning is very, very important, especially at the early stages of getting them hooked on math.

    However, I wonder if it will slow down their progress on the amount of subjects they learn, if they spend more time solving problems.

    I guess that boils down to ANOTHER endless education battle, breadth versus depth…

    I mean, if we really want those kids to learn those subjects intimately, there’s a chance we will a) alienate those kids who learn at different rates and grasp the subjects before others and b) fall behind on covering what else they need to know.

    (Also, I’d be irritated if I were a student and were forced to sit through exploratory learning every single time I came to class. If someone’s already figured out the rules to the game, I want to know about them! I don’t want to waste time inventing a chainsaw every time I want to cut down a tree…)

    Hmm, maybe I’m seeing a pattern?

    Those who usually score low on traditional tests work better with hands-on practice, that gradually lead up to abstraction. “Just let me do it”, etc.

    Those who usually score high on traditional tetst can handle the abstraction sooner, and actually would prefer it to initial hands-on practice. “Just tell me the rules”, etc.

    (And of course, the third group, the uninvolved students. “Just let me go to sleep”. ;) )

    So, to sum (thanks for the comment, thescamdog), I think discovery learning is fairly important but time-consuming (yeah, I know…) and can serve to alienate those who are ready for abstraction and instruction. (Different approaches for different students, I guess…) I also think it slows down the pace of covering new subjects. However, I think it does offer a key piece of learning something new AND retaining it; I would just prefer more abstraction and instruction over application, but that’s why I’m a math and not a physics major. :)

  17. Brian
    Looks like I now have a video to go along with this applet. I think I’m going to have to rework it so that it’s a basket ball beign dropped into a hoop.

    Thanks for sharing.

  18. tl;dr is at the bottom.

    Dan: We’re talking about splitting every problem into four or more layers and constructing them separately, in dialogue.

    And that sounds like it’s almost focused more on how to solve problems than the actual material. A lot of time is spent breaking down the problem into graspable bits because we want to show them how to abstract the important bits and solve problems. This is actually a separate skill than the material being taught.

    Perhaps we could abstract the actual problem solving itself into a class, so they have the instruction on solving problems and then can focus more on digesting word problems. There’s a textbook that focuses on just problem solving tools, “Crossing the River with Dogs”. Maybe that could help them. I know I want to actually create physical representations of tools with words written on them; a paper cut-out of a wrench with the words “What do you HAVE (facts, equations, definitions) and what do you WANT?” written on it, a paper cut-out of a screwdriver with the words “Make a list of results”…something like that to crystallize the fact that these are TOOLS. Then put those tools on the board in a collage format or something. Heavy-handed, but hopefully it gets the message across. Or maybe I’ll have exploratory learning experiences for their daily warm-ups, the end goal being they have discovered and learned how to use a new “tool”…

    Anyways, tangent…

    tl;dr Problem solving and abstraction skills should be taught in a separate class, and the math word problems left as they are. Maybe.

  19. @thescamdog

    I agree, teaching for understanding is important. Hell, it’s why we’re here! :)

    I think Dan’s approach is helpful for some but not all. It’s much more applied math than pure math. That’s not good or bad, it’s just, what appears to me, a fact.

    I would hope the curriculum would come along for the ride. I think the pacing is up to both the class’s mastery level and the teacher in most cases.

    I’m not sure I agree that those kids will do as well or better than their peers on the standardized tests after learning this approach. I’m ALL FOR IT if it’s true, but I’d like to see experiments done. I mean, at one time New Math made logical sense too…

    A side note: if I were presented with a slowly filling water tank, I wouldn’t necessarily be invested either. I’d probably take out my textbook and start reading the chapter, or start on the homework, but that’s just me; I learn better by reading the rules first, generally, and enjoy it more.

  20. @Chris, you’re dropping some solid objections here and I’m going to roll with them for as long as you’ll tolerate my rebuttals. (Except for your objection that I don’t have specific standardized assessment data to back up my classroom practice. Nothing I can do about that.)

    There’s the kind of discovery learning that’s sometimes turgid for advanced learners, the kind where students cut out the angles of a triangle to demonstrate that they really do form a straight line. I think some students would rather just receive the axiom and then get to work applying it. Fair enough. That’s not what I’m after, though.

    My best WCYDWT questions appeal first to intuition. “Make a guess. Let’s see who gets closest.” I have yet to find a student population that doesn’t vibe to that competition.

    Then, with no information yet available, I ask them to solve it. The advanced learner loves to mention “it’s impossible,” at which point I ask her “why?” and she scoffs that she doesn’t have this, this, and that data. At which point, I provide those data and they solve the problem.

    Bracket for a second how much easier (I find, at least) it is to challenge an advanced learner than it is to offer access to a remedial learner. At what point is the advanced student bored by the pace of that exchange? Is the problem that the student has solved radically different from what was already in the textbook? Have they developed skills through that exchange that the same problem in the textbook didn’t offer them?

    My answers? Never, no, and yes.

    In a lot of ways this kind of curriculum doesn’t deserve the labels thrown at it by its detractors. Constructivists and the promoters of discovery learning are courageous folks. Meanwhile, I’m proposing an incremental improvement on the status quo. Little more than that.

  21. In addition to “Crossing the River…” I’d recommend this reading on mathematical habits of mind.

    Regarding problem solving and abstraction skills, I believe (and this is just my humble opinion) that if you remove problem solving and abstraction skills all you’re left with is the equivalent of getting students to memorize hundreds of digits of pi. Sure, it’s sort of applicable to life and might be fun for some but it’s also pointless. Students who end up liking math like it only because they are good at it. And even those kids won’t retain the information we are teaching them (how many college educated adults do you think remember or can derive the half angle formulas). Instead, the subject is just used as a gatekeeper, allowing “good” students to go on to become STEM majors and future teachers who down the road find themselves in front of their own students wondering why their students aren’t as excited as they were. Even conceptual understanding is somewhat vacuous if you aren’t interested in exploring unfamiliar problems. I see problem solving and abstraction skills as the whole point. What you are calling content is just the medium (unfortunately, it’s a lot easier to assess procedural skills). Our hK-12 math curriculum could just as easily be teaching kids alternate base systems, truth tables, and circuitry logic which some might argue would be more applicable to the “real world”. We haven’t devolved into anarchy because this isn’t being taught in K-12 and we won’t devolve into anarchy if we eliminated or changed most of what is currently taught.

    Ok. Now for an argument that is less value based and opinionated. When students are just learning the steps necessary to solve exercises, even the kids who are successful with this have difficulty extending these ideas to new realms and problems involving higher critical thinking (this is backed by research such as Jones & Tarr, 2007). Even if the teacher focuses on why things work, they are still reinforcing the idea that math is a subject where you should know how to solve the problem at the outset (leading to disturbing results such as the belief that someone “smart” should be able to solve any math problem in less than 10 minutes). This leads to a subject that is fragmented, irrelevant, and frankly disliked by too many people.

    Exploration takes time…lots of it. It’s also really hard to assess improvement/development in exploration (ie mathematical habits of mind/problem solving skills/heuristics, etc).

    “If someone’s already figured out the rules to the game, I want to know about them! I don’t want to waste time inventing a chainsaw every time I want to cut down a tree…” This is back to just my opinion, but I don’t think this is ALWAYS true. People enjoy doing puzzles even though they’ve already been solved by someone. People still watch movies even though others have seen the ending. It’s about my own personal journey. We just need to find a balance so that by the time you’ve invented the chainsaw you still have some interest in cutting down the tree (and make sure you had any interest to begin with).

  22. Bill Bradley

    May 26, 2010 - 10:15 am -

    @Avery If you think it’s bad that kids just like Math because they are good at it, you should see what happens with Science. Many students come in thinking that they are good at Science, when it’s really Algebra and identifying which numbers to plug into the formulas (gee, just like the Math textbook problems). If you expect them to interpret formulas or develop them from experimental data, they’re lost. So techniques like Dan’s are about more than just getting them interested in Math, they often prepare students to be able to “do” science!

  23. @Bill: I couldn’t agree more. You are asking students to collect data and explore problems that might be messy.

    Reminds me of a GB Shaw quote:

    “You see things; and you ask ‘Why?’
    But I dream things that never were; and I say, ‘Why not?’”

    It’s an oversimplification, but I think of science as the subject of the first line (formulating laws to explain/predict phenomena in the world) and math as the subject of the second (creating a system of rules and exploring the implications).

  24. Dan
    Forget the cost of the books, can a problem actually be broken down the way you propose in a hard copy? I think CPM does a really good job of bringing students through the process of problem formulation. Problem is that no matter what, the problem’s static and there is no way to withhold information until the students see the need for it. No matter what, they try to skip all the preliminary reading and get to finding the answer.

    In my opionion, that’s the real strength of what you’re doing here. The images/videos are really cool, but not everyone has that skill. Any teacher with a projector, can break a problem down and simply hold back some info.

  25. I think maybe Chris is missing the point a little, kind of going sideways. Maybe?

    This is my take on it, about the purpose of math being to discover our environment. (Its original purpose)

    It reminds me of a calculus problem, how much work it takes to fill an ice cream cone-shaped tower with water. The curve and maybe the 3 factors make this calculus. To extrapolate downwards, how big must a building be to hold 500,000 shoeboxes? With 18 inches left to the ceiling for the fire marshal? Or even, the amount of shoeboxes Nike makes in one year? Go on down to the elementary, primer level – ‘We have an ice cream tray, and we’re making pops out of orange juice. How many popsicle sticks will me need?

    I think the process of starting with nothing, and having to get to a goal, is more creative, more interesting, and, ultimately and more importantly, more likely to cultivate the type of student who can solve the problem, ‘How much oxygen must be created for 2 humans to survive in a spaceship going to Pluto and back?’ ‘How large do pipes have to be to let water flow freely from a sink to the sewer?’ ‘How deep do the girders have to be to support a 12-story building?’

    I think that students who can turn problems inside out like that, from concept to detailed numbers rather than from numbers to concept, can’t fail to pass any standardized test, but I really don’t care whether they do or not.

  26. Dan,

    I’m a big fan of your WCYDWT content, but the big problem I have with this one is that when I click Play on the video the students can see the length. Could we have a longer video, with the last 8 minutes or so being a blank screen?


  27. Dan and other commenters,

    Great talking with you. Wish we all could do this in person to come to a decision more quickly. I have to get back to my Euler line lesson plan, but I’ll jump back in later tonight.

    Sure thing, Dan. :) I always appreciate a good discussion, and your ideas and plans of attack spark just that.

  28. Sue: If it’s just incremental, Dan, why do you think it’s generated so much attention?

    This particular increment has been so nettlesome for lack of a particular enabling technology. Like David says, projectors are cheap(er) and (more) accessible than ever before and that enables patient problem solving in a way that paper can only dream about at night.

    David The images/videos are really cool, but not everyone has that skill. Any teacher with a projector, can break a problem down and simply hold back some info.

    Can you elaborate? How do you see these problems broken down without images? What is this projector projecting anyway? What do you think you’re projecting in this post?

    Michelle: I think that students who can turn problems inside out like that, from concept to detailed numbers rather than from numbers to concept, can’t fail to pass any standardized test, but I really don’t care whether they do or not.

    Bravo. This is exactly right.

    Pete: The big problem I have with this one is that when I click Play on the video the students can see the length. Could we have a longer video, with the last 8 minutes or so being a blank screen?

    We need a different solution than that. If the timer reads “15:00,” we won’t get some of the awesome hour-long guesses referenced in this post. The best option, I think, is to toggle the projector’s input to a different device while you put the video into full screen mode and start playing it. Then you toggle the projector back with no one the wiser.

  29. Dan
    That post is exactly what I was thinking about when I wrote my comment. I was simply trying to say that the images don’t have to be so elaborate as to scare the average technophobe away. I built that lesson in about 10 minutes. I’m on your side here. The power of that lesson was my ability to hold back info until students saw the need for it. A book can’t do that.

  30. Hi Dan,
    Like Sue, I’ve been really impressed by your incremental method of “patient” problem solving and have been thinking about why it’s so cool even though it’s such a simple concept (and, as you said, one that’s simple to execute now that projectors are cheap). I think one reason is that by just removing information you are helping students develop basic research skills. It’s really cool that by doing something so simple you’re getting kids to practice information gathering skills that we traditionally reserve for special data gathering projects (that usually have all the steps outlined anyway) or non-math classes.

  31. Four words: SHOW ME THE MONEY.

    As soon as I see some proof that this works, I’ll be a much happy camper. After all, the currency of math and science themselves is evidence and proof. And to quote Michelle, math is “intended to discover our environment” (although a discussion could be had about whether or not studying water tanks is interesting to everyone; not everyone is a future engineer, and so the theme of the problem does matter).

    Everything hinges upon whether it works or not. So, we could cut a bunch of wasted time if it was proven.

    Don’t get me wrong, I love enthusiasm, passion and rhetoric as much as the next fellow. Hell, that’s what hooks kids into a subject and teachers into a new pedagogy most times (it seems): a teacher who is passionate about their subject and able to communicate well.

    And that, I suspect, could be playing a large role in the “success” this method claims to have. How to control for that, well, that’s part of the challenge of drafting up an experiment.

    I know you claim to see the results of it every day. But if YOU weren’t teaching it, if someone else was handed your lesson plan for a term, would the method still hold? And would they retain it?

    But seriously, Dan, thanks for your work and theories. As long as it doesn’t become religious doctrine…

  32. I hope that last comment didn’t come across as too rude or brash, Dan. Not my intention.

  33. @Chris writes: “Don’t get me wrong, I love enthusiasm, passion and rhetoric as much as the next fellow. Hell, that’s what hooks kids into a subject and teachers into a new pedagogy most times (it seems): a teacher who is passionate about their subject and able to communicate well.

    And that, I suspect, could be playing a large role in the “success” this method claims to have. How to control for that, well, that’s part of the challenge of drafting up an experiment.”

    Control for that, as in, remove it and see what happens? I hope no Human Subject committee approves any such attempt!

    I would like to reference “Deschooling Society” and “Medical Nemesis” here. What works in the loving environment is INCREDIBLY different from what works in sterile labs. Mechanistic metaphors of inputs, outputs and controlled experiments are dangerous here. We got to talk in terms of love and ecosystems. I don’t suggest discarding the scientific method, of course, but using its branches suitable for studying complex, living systems.

    Yes, this works because Dan is a passionate, caring and involved person. Will it work for teachers who aren’t? Why would we want to know in the first place? Whose kids would such an experiment be conducted on? Because I am not volunteering my daughter or any of my students for that…

  34. @ Chris & Maria

    That’s why I mentioned scripting, which is used a lot in elementary ed.

    I may have a different viewpoint because I am thinking about younger students, elementary and middle school age. That’s why I don’t think about ‘pure’ mathematics, although all math started because of some natural problem, like geometry starting because lands needed to be measured after the Nile River floodings in Egypt, where the Greeks went and learned about it; and even the theory of relativity, etc., started out as an original problem rather than the rote ‘Given that the diameter of the circle is 12 inches…’
    Anyway, with scripting, a person thinks about the words they say to get their meaning and emotion across, and WRITE IT ALL DOWN. A teacher using the text then has the option to teach word-for-word, or at least know where the author is coming from.

    This is useful in elem. ed. because the pedagogical knowledge of the teachers varies greatly, (some have just barely passed exams after many tries, and some went to Harvard) and is a way of making certain teaching methods universal. It seems to me that math ed. has the same problem, esp. in elem. ed and middle school.

  35. @Maria Maybe I’m not being clear. I’m not advocating performing brain surgery on kids. ;) When I say perform experiments, I’m talking about kid-safe ones; ones where the item being tested is the technique. I didn’t mean in the test to have kids taught by someone who doesn’t care; bad teaching seems like a bad element to include in an experiment because it will always result in poor learning. Therefore, you’d have a teacher that exhibited caring and empathy, but a different teaching style.

    I’m saying if this is a new method, try it out with different instructors and see if it works.

    Isn’t “I do, we do, you do” a teaching technique? How is that different from seeing WCYDWT as a tool/technique that should be tested? If it’s a tool that works independently of the teacher who started it, then great!

    The reason I’m not so concerned about “I do, we do, you do” so much is that it doesn’t affect the curriculum at all; WCYDWT would! It would change how word problems are presented and ultimately how the problem solving process is taught with math problems. It flies in the face of the instructional, traditional lecture method. Sure, we all like that; FEELS good, doesn’t it? Proof, please…

    I’ve experienced textbook learning techniques that I felt were just wrong for factoring, but we were still expected to teach them. What if I felt the same way about Dan’s technique? If it were badly implemented at the school when brought into the curriculum (and something is usually lost in translation) would THAT affect the kids? Imagine a teacher improperly using Dan’s technique; they put up a video of a water tank, then just sit there for 7 minutes, then give them the answer.

    Seems like this tool comes down on another dividing philosophy; which is the best way to learn abstract problem solving? Guidance, instruction and emulation, or figuring it out yourself?

    If you believe in the second method, you need to ensure they are highly motivated. Sure, this method claims to inspire that, but with a less charismatic teacher and enthusiastic teacher who’s not a natural leader, WOULD IT WORK? Or would the kids lose focus and get frustrated and still hate math? If it just depends on the teacher’s natural enthusiasm and ability to keep kids focused, then it’s not a tool that works for everyone, and should be looked at long and hard (ie, proven) before it starts to shape curriculum.

  36. Chris wrote:
    After all, the currency of math and science themselves is evidence and proof.

    Chris, this isn’t evidence for Dan’s method, and it’s not a math class, but it is evidence based, and it’s physics, which is pretty close to math.

    I know I can’t do anything like what Dan does with the media. But I’m a good storyteller, and I hope to use these ideas with my own twist when I get back in the classroom in the fall.

  37. @Michelle: Scripting is often enforced, which is a terrible disrespect of teachers. Even when it’s not abused that way, I can’t see it helping much, if any.

    There’s no way you can predict the questions the kids will ask, and the directions they’ll want to take a discussion. You need to know which are fruitful and which aren’t. You need to know that some crazy-seeming alternate way of thinking about the problem is actually right on, and exciting.

    Kids need teachers who understand math. Yes, most elementary teachers are not comfortable with math, and that’s a serious problem. Scripting is not a useful answer.

  38. Sue wrote:

    Chris, this isn’t evidence for Dan’s method, and it’s not a math class, but it is evidence based, and it’s physics, which is pretty close to math.

    I’m not sure what you are referring to by “this” and “it”. The water tank video? I thought he was teaching a math class with that video. And how can the teaching method itself not be used for evidence? Are you saying that the water tank video is not evidence for his method?

    Not being snarky, just trying to understand the statement.

  39. @Sue
    Some of the things I saw in these lessons, I think can and should be scripted. If you can ask the right questions, in the right manner, the differences in the responses of the students can be anticipated to some degree. (often, examples of responses are included)

    The thing is, even if you do hate scripting, sometimes this is the only way to accomplish the task. Particularly in math. You might say that all teachers should know math, and you might be right, but they WON’T. They just won’t. They won’t. You can’t make them.

    So then what? Only the 10 or so people writing on this site can ever incorporate these ideas, that we feel are beneficial for all?

    If this is the case, then everybody is just wasting their time reading and writing these things. Don’t bother. Just go read a book. Watch Two and a Half Men. Because right after the students receive great ideas in your class, they will go to another classroom where a teacher has never heard of these things, doesn’t care, and teaches in the traditional manner. Will they retain what you’ve taught? Hopefully, to some degree.

    At least with scripting, you can have a type of continuity of learning. Then, maybe the NEXT generation won’t be full of mathphobes.

    This being said, I am mainly concerned with elem. ed. and middle school, where the teachers have almost never majored in math, or even taken very much of it. This is because I personally believe that this is where the need is. If students aren’t reached here, they arrive at high school believing that math is hard and boring, and they can’t do it. Then they proceed to avoid it and self-fulfill that prophecy.

  40. Mr. Meyer, Mr. Meyer, can you help us?? *waves hand* ;) :D (How meta is THAT…)

    I’ve changed my mind about the howling “PROVE IT” refrain. After all, this is really just a teaching style and I don’t see how it can harm anyone’s learning. Also, it should be more widely adopted to see if other teachers get the same results; THEN it can be the subject of a study if it gets enough support.

    Sorry about that. You’re right; it’s better to get new ideas out there and try them out in the field and see if they work.

    The reason I changed my opinion is because I was thinking about my own planned styles of teaching and lesson plans, and how counter-productive it would be if I were limited to only given and proven teaching methods. No room for creativity, ya know? How am I going to prove my Clockmaker’s problem solving method(now that has a nice ring to it…the Clockmaker’s Problem…) if I don’t try it out? And having to prove it before it’s fully shaped is just too early in the development cycle.

  41. “There is at least one field, containing at least one sheep, of which at least one side is black.”

    @Chris: I think experimentation will happen naturally as more people use the method.

    @Michelle: We have to be extremely careful with any intervention that undermines individual autonomy and power. In math, anything that undermines problem-solving is just scary. There is a tool that is somewhat similar to scripted lessons, but does not undermine problem-solving and autonomy. I am talking about edited dialogues (or videos) captured as different teachers work on the same topic. These can be powerful for a new teacher, because they provide multiple voices – multiple examples of how the same topic can be done. Multiple examples invite experimentation and help newbies find their own voice.

    As for the sheep anecdote…

    There is at least one teacher for whom the WCYDWT method works at least part of the time: a case study.

    There are also data about some other cases of success. People used WCYDWT and reported that it worked for them.

    There are also data about cases of failure, much of them coming from Dan himself.

    Do we need to put some conscious effort into clustering these data into categories, or will it happen over time anyway, through the natural social mechanisms?

    The other day we were discussing math curricula with a group of family educators who are free to try as many as they want. Some try a new one literally every day till they find what they like. When a newbie mom asked what she should try first, others requested descriptions of what she and her kid like to do. Based on these descriptions, they quickly recommended four of five likely candidates. Collectively, this group has the knowledge about different approaches – Montessori, Singapore, Living Math, Waldorf – that we simply don’t have, yet, about WCYDWT.

    When WCYDWT accumulates more examples of use, we will be able to categorize them, and to give recommendations for different situations.

  42. I’m hesitant to self-promote, especially since the reading/writing of blogs is quite new for me being an old fogy of 33, but I will be addressing some of the research questions that have been raised here in a classroom next year. I’d love for this community to be a part of that and, who knows, maybe I’ll even be able to show you the money.

  43. Sorry, Chris (and everyone). I thought I had a link there. (The word this was supposed to be highlighted for it…) I wanted to show you Eric Mazur’s talk on how he was converted away from lecturing.

    Link to video:

    Link to my post about it:

    The evidence isn’t on Dan’s method, but it is on why we have to get away from lecture.

  44. @Sue

    That is a brilliant video. I’m excited to try PI.

    I’ve heard that quote before and I still like it: “the plural of anecdote is not data”.

    Awesome link, thanks.

  45. I watched your TED talk and then two days later, watched the reality show on Bravo about artists…unrelated you’re thinking. BUT…on the artist show, one of the commentators said “Your work is too literal. It leaves nothing for the viewer to imagine or think about. It’s all filled in for the audience.” And, I thought, wow, that’s what Dan’s doing. He is leaving lots of emptiness for kids to fill with their own thoughts. HOw cool is that? So I usually shot this video about noodle making to intro exponential functions….but next time….no sound kiddos! Leave some “empty” spaces! Go Dan!

  46. Susan: And, I thought, wow, that’s what Dan’s doing. He is leaving lots of emptiness for kids to fill with their own thoughts.

    You have math, which is a discipline older than the stars, right? And then there’s high school math, which is a subset of math notable mostly for being completely resolved, no open questions.

    So how do you give students a sense of discovery, a sense that they are still bringing something of themselves to an ancient discipline, and not just guessing after the footprints of the ancients themselves.

    That’s an open question.

  47. I’m a newcomer. I’m just leaving an impression of the video.

    How well this technique will work depends on your audience. For above average students or even average students, I would be careful in using it. However, for struggling students, there are benefits. Its difficult to get them engaged, and these examples of math applications may do just that. I’m for the technique, as long as its used for the right audience.

    For example the boredom of the water tank movie could actually be a benefit. It might provoke the students to wake up and work systematically to answer the question: How long is this going to take?, lest they have to watch the entire movie to find out. Sometimes suffering can be a good motivator.

    However, I would not use this approach for mathematically inclined students.

  48. Dan,
    First of all, I want to underline that I would use this approach for teaching math averse students.
    Now that I’ve thought more about it, I might use it for math inclined students as well. It would depend on the temperment of the class. These kids are often anxious to get to the core material, so I might just describe the problem. But, again, depending on class temperment other approaches could work better: one might describe the problem leaving out some details; or one could show a still picture from the video, or one could show a brief excerpt from the video, or just leave the video run, or some other approach.

  49. Dan,
    As an ESL algebra teacher my biggest problem is finding ways to get students interested in the discussion. Since videos appeal to visual learners and they have the added benefit of transcending language I believe that they can be of significant value as a teaching tool. Unfortunately sometimes you still have to rely on hard work and repetition to truly master basic skills. Herein lies the rub, math will always require a level of commitment and students must understand that they will only succeed if they truly challenge themselves.

  50. Bob –

    Good point here! Getting students engaged and involved is difficult, especially where there is peer pressure and lack of basic skills.


  51. Ethan-

    I have to respectfully disagree with the idea that this is not for above average students. What I like about Dan’s ideas is that I believe they would show how math is applied in the world, and the resulting math would be creative, not just the rote ‘I know how to do the math problem very well.’
    What if there is something to be measured, somehow, that has never had math applied to it before? How do you create? I believe that minds trained in this manner will be able to ultimately take a situation and design a math problem for it that will ask and answer new questions, not just follow the textbook. And I think that’s what we need right now – a LOT of people who can do that, not just a few.

  52. K, went to Dan’s #3 on top, that guy’s reference to expeditionary learning, thought about it mathematically (tried to).

    Came up with this project for someone to mesh out if they want.

    May be good for at-risk schools? Where I hope to teach.

    Thinking about homelessness mathematically:

    I guess this is stats class –
    Incorporating soc. stud., etc.

    How many homeless were there during the great depression compared to now?

    Interview some, research some.

    Why do people become homeless? What’s the graph, probability of each type? (ex. lost job, medical, psychiatric, drug-related, maybe families) Graph out how long each type stays homeless.

    Are there more minorities homeless today? Were there during the G.D.? Graph it. How long do they stay homeless? Compare this to previous decades. Using demographics of the concerned decades, do factors such as racism before the civil rights movement, use of drugs, have any relevance to the statistics?

    Graph out homelessness / geographical areas in U.S.? Difference in probabilities?

    Given this research, what do you think is the number, length and type of homelessness in your area today?

    Were you right?

    (Okay, can this be made into an algebraic equation?)

  53. I agree with a lot of you. We DO fill in so much for the students and do a lot of the “drill and kill” type memorization that can get extremely boring for the kids! I teach lower elementary in a under performing urban school and we are pressured with getting lessons done..almost at the expense of the students understanding the work! It’s all about exposing the kids to more and more and not allowing time for self-discovery. I feel that it is so important for the kids to experience issues, ask questions and come to conclusions on their own. However, I feel each year we get further and further away from this and closer to the scripted lessons that Michelle was talking about above.

  54. Michelle,
    I wasn’t clear in my original post. When I was talking about technique, I was mulling over the specific technique of using a video of a water tank filling in order to get the class prepared for doing a math problem on the same subject.

    The general point that it’s advantageous to interact with applications in the real world: You won’t get any argument from me there.

  55. Bob,
    Yes, That’s a good point.
    While videos and other motivational ideas can make good segues to initiating discussion of a problem, but how do we promote the sustained focusd reasoning that’s necessary to solve these problems in a math averse class?

  56. Regarding comments #67, #61

    I run into a problem of “sustained focus” with some kids in my math clubs. Dan wrote about “math averse” students not even wanting to engage with media in the first place – not finding these choice tidbits interesting

    My kids usually do find the introductory cool media, hands-on activities, jokes and explorations interesting. We do math at the level of mild engagement, and then… some aren’t willing to go any deeper than that. Actually, quite a few.

    We all filter media all the time, only pursuing a few of the paths beyond the initial interest sufficient to lightly check them out. How do we support kids who similarly want to filter math content we offer?!

  57. Bob, Ethan, Maria and Amy –

    As a student who used to consider myself bad at math, I wonder if the children are really ‘math-averse’ or, rather, ‘failure and confusion-averse’.

    When I took a standardized test and realized I had a good score, that I really did know some math, my confidence and interest in it rose. When I understand the math, I really like it, and with the new confidence, when I don’t understand it I think it’s challenging but that I can eventually get it.

    Before I used to think it was frustrating and I would never be able to get it, so why try? Looking back, I can see how my brain actually actively disengaged and I distanced myself from the problem.

    So, I’m thinking, the students need to really have a good background in mult, div, fract and percents before they can do well in higher math, feel really comfortable with it. So, so many don’t and it ruins math for them.

    I’m really worried about the person who said the students move on too fast, because I believe her and I think it’s highly detrimental. Somehow, the students need to get the extra practice, maybe for homework, some incentive if they do extra work (remedial and/or challenger) and turn it in, come in early or stay after school, but it has to be done so that they can understand the material.
    I’m thinking that the lecture about using points like in gaming might be a good idea to try.

  58. In addition, having taken child development, I’m wondering if decimals and percents are being taught at a time when many adolescents’ brains haven’t developed enough capacity for the abstract thought involved.

    I’m wondering about this because of the universality of problems by students in this area – almost everyone I meet and talk to, had problems in this subject and I recall how difficult it was for me then, yet how easy it is now.

  59. Susan’s link totally exemplified what I was talking about! I guess it really is universal. One point I really liked was

    ‘Math’s power to hurt is based on the perverse culturally taught belief that accomplishment in math is a manifestation of some important inborn intellectual attribute and struggle to understand is evidence you don’t have it.’

    This is so true, and that thought is so harmful. I want to come from a premise that every child can do well at math. Pursuing a math/science-related career is a matter of choice, not math ability. I strongly feel that if I can pass Calculus, anyone can. :) (At night school, working full time.)

    During a Teacher’s Geometry class where I had the opportunity to write a paper about ancient mathematicians, I can see how math came to be such an erudite subject – they meant it to! They were very into the special, secret abiltities of mathematics then – but now, that’s just bad for our nation.

  60. Hey, for those who commented on scripts, this is a script, one like what I meant – it tells you what to do, how to do it, things you might say, reponses a student might have – is this the same as the kind of script you’re thinking of?

    “Play the water tank video clip in front of a class of students of any age between 12- and 30-years-old. The video is best served cold. No music. No text. No introduction.

    Your students will start to inventory their surroundings. They’ll identify the crucial elements of the scene. Again, you shouldn’t say anything.

    Twenty seconds into watching the hose dribble water into the tank, ask “how long do you think this is gonna take?” Ask for guesses. Just guesses. Write them on the board next to the guessers’ names. Whenever anyone raises the maximum or lowers the minimum, point it out.

    Then turn the clip off. Turn off the projector and proceed to whatever else you had planned for the period.

    At least one student will ask what 90% of the class will be thinking:

    “Well … who was right?!””

  61. I thought Dan’s presentation was very enlightening. However, the comments received showed how important it is to know your students and to be adaptable to maximize learning.