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“Who Cares?”

Joe Henderson:

Dan, I’m totally with you on this line of thinking but only up until a point. I’d add a further layer to your line of questioning: who cares? Who cares how fast you can fill it up? Ultimately, the motivation to learn needs to be intrinsic from the student. Just making something more flashy (as much as I think technology can help here) isn’t enough. We need to also make it relevant.

Maybe someone can help me find some sense in Joe’s objection.

As a starting point for a rebuttal, WCYDWT has less to do with “flashy technology” than it does with a template for talking about math. Some technologies enable that conversation better than others. That’s all.

And then there’s this: “who cares?”

If we’re planning on paring math curriculum down to the material about which students instinctively care, Joe needs to prepare himself to trim Algebra 1 down to somewhere near the size of a Scientology tract, the stuff that explicitly involves video games and cell phones. On the upside, we’d finish the school year somewhere in October. On the downside, you’d lose the entirety of pure math.

There’s a lot packed into Joe’s question concerning the role of the teacher, concerning the purpose of school, that I don’t have the energy or time to unpack. I’ll let it suffice to say that I find Joe’s question uncomfortably proximate to “who’s entertained?”

A closing anecdote and then a wager.

The Anecdote

I posted a video series not long ago where I walked down two flights of stairs. Nothing more than that. It’s hard to say why anyone should care about a guy walking down stairs. It’s a totally mundane scene. It’s harder still to say why anyone would care about a graph of that scene. Mundane + Math = More Mundane. Right?

The Wager

My career leads me to this prediction, upon which I’ll stake my credibility and whatever else you want:

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?!”

BTW: a blogger took up my wager. The entire post is well-written classroom anecdote but here’s the relevant excerpt:

One day I introduced a lesson based on a challenge presented by Dan Meyer (The Wager from “Who Cares?”). I played about 20 seconds of a video clip created by Dan Meyer of a transparent tank filling up with water at the beginning of class. I asked each group of students to predict how long it would take to fill the tank. Their guesses ranged from ten minutes to a few hours. The fill/drain a water tank problem is a pretty easy, if unexciting, application of linear equations to a real-world situation. Yet, by allowing the students to actually see the tank and having them make a prediction, the plan was to engage the students in the process of discovering the answer. 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 used the engagement of the students to invest them in solving a problem that involved some critical thinking. When we revisited the problem, I avoided the temptation to hold the students’ hands through writing a linear equation to represent the situation. Instead, I simply asked them to come up with a justified answer to the question of how long it would take to fill up the tank. Some students’ answers included a rate of change and/or linear equation. Most of the answers didn’t. All of them experienced the struggle, including failure, that it takes to solve a real problem, and the first group was able to make a connection between the content covered earlier in the year and a problem they had not seen before.

46 Responses to ““Who Cares?””

  1. on 25 Feb 2010 at 9:08 pmJoe Henderson

    Dan, let me first preface this comment with a great deal of respect for your work and some humility on my part. It is not my intention to belittle here or in my previous comment. Instead, I only want to push you on something that I have noticed over and over on your blog (of which I am mostly a lurker). So, in the interest of learning together, here goes.

    While I think the methodologies you write so eloquently about on this blog are really novel and a breath of fresh air, I worry that even though they are more open-ended and student-centered, they still end up being thoughts that originate in your head about the math in your life. Do you have your kids develop WCYDWTs? Does the authority ever flow the other way?

    The technology is largely irrelevant. It’s not about being “entertained” or about what technology does what. That’s the real danger with technologies of any kind, and I think you know that. How many teachers have we seen turn a Smartboard into a fancy version of teacher delivered notes? It’s about the quality of the question being asked (as you obviously understand), AND it’s about who gets to ask the question. I think my original comment may have been muddled due to the cavalier nature in which I presented it. Apologies.

    For me, it’s about who gets to come up with the learning experience. You just wrote this:

    “There’s a lot packed into Joe’s question concerning the role of the teacher, concerning the purpose of school, that I don’t have the energy or time to unpack.”

    That’s my point. We need to unpack that stuff. Until we fully do that, it sounds like we (and I definitely include my own practice here) are talking at the students, not with them.

  2. on 26 Feb 2010 at 3:06 amDoug Belshaw

    It’s not technology that affects learning positively – in fact indiscriminate use can affect it negatively. It’s the creative use of technology. And that’s what you showed there. :-)

  3. on 26 Feb 2010 at 3:09 amDoug Belshaw

    I’m a bit of a (OK, a lot of a) fan of both Google and Apple. We only run Apple stuff in our house and both at home and at school I use Google stuff online.

    But I’ve got my concerns about them both.

    I don’t like Apple’s monopoly and how the iPad won’t have Flash. And I’ve posted my thoughts on the amount of data Google must have on me now.

    Certainly food for thought, but then you always have to sacrifice something to have the free or the best – don’t you? :-)

  4. on 26 Feb 2010 at 3:49 amMike

    Dan,

    I share your concerns about Joe’s response. It is very difficult to teach things that the students find intrinsically motivating. I would argue that if we only taught math that the students found interesting, then we wouldn’t teach them much of anything at all. A major criticism of a wholly-student driven curriculum is that the students don’t know (and can’t know) what is interesting in a subject because they don’t know the subject! It is incumbent on the teacher to show students about the interesting parts of their chosen discipline (perhaps with interesting questions) and gradually increase their involvement in the choice of question and/or topics to pursue.

    However, Joe is right that we tend to talk at kids instead of with kids. There are constraints built into the education system that force some teacher-directed content where some of us would rather have intelligent dialogue. The challenge (and I think Dan is attempting to meet it head-on) is to engage students in as much interesting content as possible given a not-perfect world.

    I mean, seriously, where else would these elaborate math problems originate if not in the head of the teacher? Especially if the students do not have a background in looking for math in their surroundings? If Dan had these kids for 10 years and was still creating the problems for them, then there would be an issue to address. But since that’s not the case, then why not have a talented teacher show students where math could be applicable?

  5. on 26 Feb 2010 at 4:38 amJason Dyer

    This probably deserves a full blog post rather than a little comment, but I want to point out that the “open for guesses” technique works more effectively in some situations than others, and occasionally one can amp up the interest at that stage with the right technique.

    I recently presented your glass-rolling WCYDWT to a group of teachers, and did the prompted question “which one will roll the largest circle?”

    I got answers, but honestly, they didn’t seem too invested yet.

    In front of each group of teachers I had a set of actual plastic cups of various sorts; entirely different than the picture, but enough to experiment. So I told the group that each table needed a unanimous decision, and they had 10 minutes to work it out using what they had in front of them.

    Lots of discussion, experiment, back-of-the-napkin calculations ensue.

    Teachers come back. Everyone is decided … on what turns out to be the wrong answer.

    I show the answer. Everyone is _completely_ hooked.

    I am fairly certain just “ok you made your guesses let’s do the math and only show the answer at the very end” would have actually been less effective at getting the crowd interested in the math.

    I guess all I’m saying is you’re falling into a template here I don’t think works with everything. (In fact, I’ve had classes before where students were _not_ at the edge of the seats to find out who was right. This usually happened when they didn’t have a good intuition of the problem yet and were genuinely making total guesses.)

  6. on 26 Feb 2010 at 4:40 amJason Dyer

    (Sorry, just to make really clear, they knew the answer, but now really really wanted to know why. That turned out to be much more gripping than just “B or C?”)

  7. on 26 Feb 2010 at 6:21 amJoe Henderson

    Derek, while I largely agree with you about using hooks to engage kids and have done similar things in my own teaching, aren’t you providing an example of Dan’s “who’s entertained” concern?

    Also, the real world isn’t compelling? Do you really believe that?

  8. on 26 Feb 2010 at 6:31 amDan Meyer
    Joe: Do you have your kids develop WCYDWTs? Does the authority ever flow the other way?

    That’s where I’d like to be, but I don’t think it’s as simple as me, I don’t know, assigning a WCYDWT project where students go out and find math like I do here. If it were that simple, I would have done it.

    Because it turns out that even math teachers are remarkably unskilled at conceptualizing math in this way, as something that exists meaningfully outside of a textbook, and at structuring inquiry in the way I described in my last post. I don’t know, then, what I should reasonably expect of my students.

    Mike: I mean, seriously, where else would these elaborate math problems originate if not in the head of the teacher? Especially if the students do not have a background in looking for math in their surroundings? If Dan had these kids for 10 years and was still creating the problems for them, then there would be an issue to address. But since that’s not the case, then why not have a talented teacher show students where math could be applicable?

    I agree with this, though I’d probably take a little heat off that first sentence. This is how I’m useful to my students. Or put another way:

    Oliver Wendell Holmes: The young man knows the rules but the old man knows the exceptions.

  9. on 26 Feb 2010 at 6:44 amMrW

    You’re on. The video will be the beginning of my math block this afternoon.

  10. on 26 Feb 2010 at 6:48 amSteve Phelps

    These problems and subsequent discussion remind me of a few of George Polya’s Ten Commandments for Teachers

    1. Be interested in your subject.

    4. Try to read the faces of your students, try to see their expectations and difficulties, put yourself in their place.

    6. Let them learn guessing.

    9. Do not give away your whole secret at once — let the students guess before you tell it — let them find out for themselves as much as feasible.

    10. Suggest it; do not force it down their throats.

    Then I am reminded of a research study on problem solving that found if students can’t solve a problem in something like two minutes, they think it can’t be solved.

    I think these problems address these five commandments, as well as the result of the study.

  11. on 26 Feb 2010 at 6:51 amqandnotq

    Don’t sweat the universal laws of instruction. I think the real focus should always be on outcomes. (As if it were that easy!) The context is an engager and distracter subject to the affect of the students, but ultimately up to you… in your acquired wisdom and personality.

  12. on 26 Feb 2010 at 6:54 amqandnotq

    By the way, I think your style is incredibly engaging (and not just for students). Thank you for these really organic examples of critical thinking and ‘problem’ solving!

  13. on 26 Feb 2010 at 7:05 amJason Dyer

    Derek, apparently *this* is the clip you should be using:

    Fear Factor Junior

  14. on 26 Feb 2010 at 7:28 amDerek

    @Joe – sure the real world is plenty compelling…especially for physics. I do auto-accident deconstruction problems, golf, basketball, pretty much anything with physics – which is pretty much anything.

    But there are times when using something completely comical (and outrageous) will pull the group together.

    I find the variety important. When trying to help students find patterns and understand relationships in the world around them it is sometimes useful to do ‘what if’ scenarios.

    …what if g=-4.5 m/s2 instead of -9.8 m/s2.
    …what if the electrostatic force were weaker?
    …what if the force of friction were independent of weight?

    @Steve – I tend to be wary of lists. This is interesting but is there anything on there that is surprising to any of us?

  15. on 26 Feb 2010 at 7:33 amjosh g.

    “Do you have your kids develop WCYDWTs? Does the authority ever flow the other way?”

    For whatever it’s worth, as a newbie straight out of a relatively progressive teacher ed program, I sympathize with Joe’s comment but feel like I need to mediate for it or something. (Warning: I’m getting all philosophical here rather than talking from useful classroom experience.)

    I agree that it’s important to share respect and (in some ways) authority with students, to find ways for them to ask questions that can be brought into the class discussion, and generally to empower rather than oppress students. I want to re-read Pedagogy of the Oppressed every few years and let it re-challenge my teaching. (I cringe now every time I give in and separate a noisy student from his/her friends.)

    That said, Joe, I think you’re overshadowing what is already a pretty big revolution going on here. Dan may still be asking the questions, but he is deliberately removing the authority of the “right answer” off of both the teacher and the answer key and placing it back on reality where it belongs. I think that’s pretty huge.

    For whatever it’s worth, even Freire’s practice was driven by learning about his (adult) students’ lives, and then initiating dialogue by presenting some opening media for them to discuss, which the teacher then guided into lessons. (Wow, until I actually wrote that out I didn’t even realize how close that sounds to WCYDWT.)

  16. on 26 Feb 2010 at 7:43 amDerek

    Oh and I forgot to mention.

    Dan (and those in the commenting on here), I appreciate the dialogue and recurring shot in the arm! Keep it up!

  17. on 26 Feb 2010 at 7:55 amRiley

    I agree with Derek that it can be hard to find compelling situations in the real world (I mean the environment around the kids) that illustrate the need for that quadratic formula or that composition of functions, or whatever.

    The whole-wide-world is too big for unskilled people to find mathematics in, but small subsets with careful rules established (e.g. you only have this paper and these scissors) can give kids a tactile environment to play in. And this can make it compelling. I don’t know a way to make my kids like math, or find it interesting, but they ALREADY find _knowing_ things interesting. Their skill level in math is often too low to feel like they know things about “math,” but making it about a real water cooler or a real sculpture or a “real” computer model hooks into the innate desire to figure stuff out.

  18. on 26 Feb 2010 at 10:41 amJoe Henderson

    @Josh: I think you bring a set of important points into the discourse at this point. And yes, they are philosophical, which is intimately related to practice. Both are important and cannot be separated into disparate parts, as Friere himself articulated. And it is for this reason that we need to unpack all this stuff, I think…

    Let me step back for a minute and introduce the time factor. In my own science classroom I had to scaffold activities that moved generally from less ownership over learning towards more ownership as the year progressed. My students were trained in the ways of learned helplessness and correct-answer-grubbing over their many preceding years of schooling that by the time they got to me, it wasn’t enough to just think that I could throw inquiry science at them and that they would pick it up without careful scaffolds on my end. Having said that, the more I challenged them by “being less helpful”, the more intrinsically they were able to learn. This is where Dan, and you, are correct. I think that Dan could throw the WCYDWT to the students and allow them to organically create the experience. But that’s a HUGE leap of faith that isn’t generally rewarded given the intensely controlled nature of schools under the accountability movement.

    For me, it’s the interplay between the teacher-centeredness and the student-centeredness that is really interesting. Dan’s Oliver Wendell-Holmes quotes get right to the heart of that point. I think this needs to change over time towards autonomy. Sadly, I think schools do the exact opposite as the students ages.

    @Dan: Perhaps here’s a different challenge. Ask your students to come up with a WCYDWT. You know the standards in CA. Pick some that are coming up and see if your students can find it somewhere in their world. I bet they can, and I would love to read about that experience here.

    PS – In the interest of full disclosure, I should note that I just had a class the other night with some parents that homeschool their children for exactly the reasons that we’re talking about here. But that’s a huge rabbit-hole that we probably shouldn’t go too much further into. Still, it’s dominating my thinking, especially since I have a lot invested in the model of public schooling.

  19. on 26 Feb 2010 at 4:13 pmDale Basler

    You know, I see your side of this but I also think Joe has a point- relevance IS important. I can’t see this activity as one of those shining star lessons that make students really want to explore. But maybe, It’s because I’m not a math teacher.

    Joe later writes: “they [the WCYDWTs] still end up being thoughts that originate in your head about the math in your life”

    I sometimes feel that way about this blog. I never feel like diving into (and replying to) your post because: 1) the challenges never really grab me and make me scream, “oh, I have to figure this out” and 2) I feel like you’ll just roll out another beautiful post showing us the “right way” to teach a few days later and 3) if we get it wrong you’ll scold us

    It just seems safer for me to save my energy and time by doing more lurking at this site instead of sharing.

    I worry that students might feel the same way when a WCYWT is introduced.

  20. on 28 Feb 2010 at 5:06 amBrian

    In order to encourage students to care about what they are learning, why not ask them to accomplish something with their new knowledge?

    In the last few years, I’ve begun to adopt a modeling approach to teaching my physics classes. (Full disclosure: The following is not my idea, but one I saw in a modeling workshop.) A lesson starts out in much the same way as Dan described the water tank activity. Show them a pendulum, ask the students what is relevant about it’s motion and what they could change which might in turn change that motion. Once a number of reasonable ideas have been suggested (length, mass, angle, etc.), turn them lose in the lab to determine how these variables relate to how quickly the pendulum swings. After numerous stumbles, they develop a model for the period of the pendulum vs. length that they feel confident in. And then I challenge that confidence.

    I ask them to build a pendulum that will keep time with the beat of their favorite song. Everyone brings in iPods, selects a song and uses the model that they developed to construct their pendulum. They’re nervous and unsure of themselves as the moment comes to test it, but when it works (and it always does) they begin to see the usefulness and power of learning about the world around them.

    For constant velocity, they predict where a collision between carts occurs ala the classic two trains problem. For accelerated motion, they determine the get away time for a constant velocity cart vs. a rolling cart on a ramp. Each time, they are asked to use their knowledge to accomplish something. In my experience, this extra step starts to answer the “Who cares?” question.

  21. on 28 Feb 2010 at 10:06 amDan Meyer

    My apologies for neglecting a good thread.

    Jason: I am fairly certain just “ok you made your guesses let’s do the math and only show the answer at the very end” would have actually been less effective at getting the crowd interested in the math. I guess all I’m saying is you’re falling into a template here I don’t think works with everything.

    I suppose it should be said explicitly that students need intuition to draw on if we’re going to ask them to guess intuitively. Rolling cups is much less universal an experience than filling some kind of vessel with water. You have to let them mess around a bit first.

    josh g.: For whatever it’s worth, even Freire’s practice was driven by learning about his (adult) students’ lives, and then initiating dialogue by presenting some opening media for them to discuss, which the teacher then guided into lessons.

    I’ve tacked a lot of entries to my reading list off the comments here. Can you direct me, Josh, to the specific Freire text that addresses this?

    Riley: I don’t know a way to make my kids like math, or find it interesting, but they ALREADY find _knowing_ things interesting.

    This is a huge part of “the purpose of school that I don’t have the time or energy to unpack.” Entertainment, it seems we all agree, isn’t the highest to which we should aspire in our business. I’d argue the same could be said of “engagement,” which is too often synonymous with entertainment and subtly frames the teacher’s job as an entertainer. But I can offer my students a degree of empowerment over challenging material, a degree of empowerment over the world they live in, and a certain vocabulary for the intuition they already possess. Especially, for my particular crowd, I can offer them empowerment over material that has disempowered and bedeviled them for years.

    Joe, I found myself frustrated and a bit clenched up whenever I thought back on your comments this weekend. I’m old enough, thankfully, to recognize the value in this kind of frustration.

    Joe: I think that Dan could throw the WCYDWT to the students and allow them to organically create the experience.

    I realize “that would never work in my class,” is kind of a catch-all excuse for stubborn teachers, but don’t you feel the least bit weird issuing that kind of prescription without knowing my student demographics at all. Can you not conceive of a study body that so lacks a framework for mathematics and scholarship that an open-ended assignment of this degree (“We’re talking next week about linear inequalities. How about you guys keep your eyes open, your digital cameras on, and bring back an example to show us all. Be sure to capture the example in such a way that you can position it as a question. Be sure to take measurements so that we can answer that question. Be sure to take a photo, also, of the answer to that question so we can compare it our own answers.”) would be disempowering and self-defeating?

    Joe: But that’s a HUGE leap of faith that isn’t generally rewarded given the intensely controlled nature of schools under the accountability movement.

    I’m grateful you’re not the kind of hardline constructivist who reflexively assumes that anyone who engages a class in direct instruction is simply drunk on power, obsessed with her role as the “sage on the stage,” etc. I’m also grateful you seem to acknowledge that your prescriptions would be a lot easier to fill in a homeschool (or even private school) setting. There’s a lot I could in say (in affirmation of your point) about my personal frustration matriculating to a public high school after eight years of home schooling.

    Still and all, you assume it’s a lack of faith that prevents me and others from turning over vast autonomy to students who aren’t ready for it within a system that doesn’t just leave that gesture unrewarded but actively stacks the deck against it. It isn’t an article of faith. It’s numbers.

    1 class.
    28 students.
    2 criminal records.
    3 years behind grade-level, on average.
    25 standards and sub-standards.
    182 hours.
    1 homework problem per night that goes uncompleted by …
    80% of the class.

    Whenever I take my WCYDWT show on the road, I have to take great pains to point out to skeptical teachers that, even though you’re investing extra time on these kind of investigations, that investment into rigorous mathematical thought processes pays off huge dividends in direct instruction where you find students less impatient, more tenacious, and quicker to learn new skills by evaluating them against their existing intuition. I point quickly to the fact that my classes last year consistently outpaced and outperformed my entire department, even though those teachers took a more linear route through the same material.

    Suffice it all to say, I need your kind of feedback, Joe. Bad. But I don’t see you submitting the same due diligence here. I don’t know, constructively speaking, what I can do with “have more faith” within the constraints of public education in 2010.

    Brian: After numerous stumbles, they develop a model for the period of the pendulum vs. length that they feel confident in. And then I challenge that confidence.

    This, to me, gets to the essence of the Holmes quote above (“The young man knows the rules but the old man knows the exceptions.”) and, in my opinion, allocates the teacher’s talent and experience and balances constructivism and instructivism exceptionally well. Not for nothing, Brian, that’s a killer activity you describe. I encourage you to get a blog going.

  22. on 28 Feb 2010 at 5:56 pmJoe Henderson

    But that’s the thing Dan, when I say “faith”, I mean faith in the students, not the system or your abilities as an educator. I think all your readers, myself included, have faith in your abilities as an educator. And well, the system is…the system. I’m not sure I have much faith in that…but I digress.

    I still feel like you’re dodging my original point though. I was critiquing the locus of the constructivist activity you presented in the WCYDWT. The content locus still rests with you in that activity, and ultimately not with the students. I think you know this, but you still seem unwilling to give up some control here. Am I wrong on this?

    I definitely sympathize with your situation, as I’ve been there myself, including the students with the criminal backgrounds. I worry though that we (and I include myself here) get stuck thinking that “the kids aren’t ready” for this type of learning, which can then be used to justify the self-fulfilling prophecy of not experimenting with a true student-centered pedagogy. This is the reason I wrote so much about having to scaffold these experiences into the school-year. You cannot just throw linear equations at them without any scaffolds. But I bet they will surprise you. Why not give them some bare-bones, high-expectation instructions and then see what they come up with? Do this enough and it will become your classroom culture. I know this from personal experience.

    And let’s definitely not kid ourselves, not all kids are going to be able to do this. It’s a spectrum like anything else. Have faith in them, and provide lots of scaffolds along the way.

    Give it a shot man. Then write about it here.

    PS – Freire – Pedagogy of the Oppressed, Chapters 2 and 3 are money on this topic.

  23. on 01 Mar 2010 at 10:56 amDan Meyer
    Joe: But that’s the thing Dan, when I say “faith”, I mean faith in the students, not the system or your abilities as an educator.

    I think you misread me. I’m saying: faith in my students is insufficient within this system. Faith is only one component of a comprehensive structure of student support. There are practical concerns here that aren’t simply articles of faith.

    But never mind. Help me out here.

    We’ve just finished linear fun with airplanes and stacking cups. Students are by no means experts in linear modeling but a lot of kids get that when data looks linear, we can draw a line on top of it, “own” the pattern, and then use it to draw larger conclusions. Like “how long would it take to fly around the world nonstop?”

    So the leap of faith you describe would be to:

    Briefly review those linear exercises. “This is what you’ve done so far.”

    Briefly redefine what makes a real-life phenomenon “linear.”

    Give them four days to bring back a linear data source from the world around them. They need to bring back either the data itself (which they measured) or the location of the data set on the Internet (which they found). They need to analyze it in the same way that we already have — what does the slope mean, what does the y-intercept signify?

    Do I provide them with a rubric? Do I make it mandatory? Extra credit?

    This is what I mean by due diligence. Let me know how you’d approach this. The WCYDWT exercises, frankly, require twenty times the advance preparation and content area knowledge but I don’t mind starting here.

  24. on 01 Mar 2010 at 12:03 pmJoe Henderson

    As I see it, there are two routes you can go here. They already know about linear equations, so what you describe would be them just going out and finding something they already know. You could do that, but that’s a simple application activity. Might be useful.

    Instead, I would have them take their knowledge of linear relationships and use it to explore something they find in the world that requires the use of their prior knowledge to get at some larger unknown (and a higher level curriculum standard). Math isn’t my bag (I’m a science teacher), but I’m sure you can think of some high-level process thing that is coming up in your curriculum that would require the use of linear relationship knowledge. They already have the scaffold of understanding linear relationships. Now have them use it to learning something much more complex. You’ll have to figure out what that expectation is (and yes a scored rubric is probably a good idea here so that everything is transparent).

    The point is that student exploration of whatever your next concept is should precede the formal instruction of that concept. Then use what they produce to teach whatever that next concept is. Scaffold along the way as needed. You’ll have to gauge the kids as you go along. Does that make sense?

    On a related note, you might be interested in this in terms of teacher development/training. We use a modified version of this protocol with our science student teachers:

    http://physicsed.buffalostate.edu/AZTEC/RTOP/RTOP_full/

    The actual observation form can be found here:

    http://physicsed.buffalostate.edu/AZTEC/rtop/RTOP_full/PDF/RTOPform_IN001.pdf

    I can email you a more concise version if you are interested. That might be helpful for evaluating practice.

  25. on 02 Mar 2010 at 8:39 amJason Dyer

    The thing (from Dan’s context) that makes tossing the WCYDWT back at the students difficult is the 20% homework completion rate. I’ve taught such classes; I’m taught one where as high as 20% would be a miracle day. So with this theoretical

    Give them four days to bring back a linear data source from the world around them.

    I can say it likely there will be very few students who actually do what is described above.

    I have a class for students who seniors who haven’t passed the standardized test. After the test is over I have them work on a statistics project where they can choose nearly any topic they want. Ultimate fighting? Sure. Shopping for clothes? Ok! World of Warcraft? Why not? We do everything in class, with lots of computer lab visits and 4 weeks worth of time.

    Even with those conditions about 20% of the students do nothing at all, and another 25%-35% do what I’d call glancing off the surface.

    Given the same crowd, “find and bring something mathematical” will result in very few takers.

    So such a lesson is stuck with in-class, which means a lot of time. Maybe it’d work as an after-the-big-test substitute for Feltron, I dunno otherwise.

  26. on 02 Mar 2010 at 10:57 amJason Dyer

    Jason: I can say it likely there will be very few students who actually do what is described above.

    Sounds like a lack of faith to me.

    Just “experience with a very similar thing knowing that this is going to have trouble working as a take-home”. That one educational structure will not work and another might be needed has nothing to do with faith or lack theoreof in students.

    While I haven’t done the exact assignment before, I have done a similar one with 2 takers out of a class of 32.

    I’m sure it’s possible to structure things different for a different result. I’m just not sure yet what that different would be.

    Best I can think of is to be pretty light and forgiving on what consitutes a mathematical artifact; make it low-risk, make it personal. Let students bring in pictures of their dogs and figure out what the WCYDWT might be in class. Maybe there won’t be one for a particular artifact, but in the process students will have a better idea what a artifact to bring in is.

    Maybe after a low-risk run, students will get a better idea of what actually works as a WCYDWT, and the difficulty could be bumped up a bit.

  27. on 02 Mar 2010 at 11:11 amJoe Henderson

    And again with the self-fulfilling prophecy. Aim low, get low. Expect low, get low. And so we continue to avoid the really difficult work and continue to appease a culture that preferences the decoupling of student ownership from the school experience. And then we wonder why, by the time they are seniors, only 20% are intrinsically motivated.

    Look, I’m a realist on this. I know that it’s not going to work for everyone, but do it enough, and it works for most. That’s been my own personal experience.

    I think Jason is correct here with this starting point:

    “Best I can think of is to be pretty light and forgiving on what constitutes a mathematical artifact; make it low-risk, make it personal. Let students bring in pictures of their dogs and figure out what the WCYDWT might be in class. Maybe there won’t be one for a particular artifact, but in the process students will have a better idea what a artifact to bring in is.”

  28. on 02 Mar 2010 at 11:26 amDerek

    @Joe – You beat me to the punch. Must be someone lunch time for someone else.

    It is hard to expect great results and be disappointed but I think that too is important. Show disappointment. I’ve had students respond quite positively to me just acting surprised/disappointed that they did not complete an assignment.

    Avoid the message of “…it doesn’t matter to me one way or another whether you do work, I’m just punching a time clock. Plus the less work you do, the less grading I have to do.”

    This is essentially a quote from a past colleague (has since retired).

    What sort of results to suppose he got in return?

  29. on 02 Mar 2010 at 12:15 pmDan Meyer
    Derek: Avoid the message of “…it doesn’t matter to me one way or another whether you do work, I’m just punching a time clock. Plus the less work you do, the less grading I have to do.”

    Kind of a low bar, innit? I mean, I’ll really try.

    Joe: And again with the self-fulfilling prophecy. Aim low, get low. Expect low, get low. And so we continue to avoid the really difficult work and continue to appease a culture that preferences the decoupling of student ownership from the school experience.

    And again with the refrain that the only variable here is a teacher’s will.

    If I were you and I had two not-incompetent and not-recalcitrant teachers like me and Jason here scratching their heads over how your ideas are going to work, I might — if only for the sake of my own evangelism — try to brainstorm a few limiting factors outside the teacher’s locus of control. That way, my only response to the question “how?” wouldn’t have to be “I’m sure you can think of some[thing].”

    My struggle here is real. And as much as I have some faith issues to work out, it’s mostly an external confrontation between man and nature.

  30. on 02 Mar 2010 at 12:31 pmJoe Henderson

    Dan, before I respond more fully, I’m wondering if you can say more about “it’s mostly an external confrontation between man and nature.”

  31. on 02 Mar 2010 at 1:52 pmDan Meyer

    Sure. My best constructivist impulses (which I admit are probably still more restrained than yours) are at odds with the environment in which I teach. To cite one such constraint, my remedial classroom size doubled from 2008 to 2009, owing to California’s budget shortfall. This has constrained my ability to have a conversation with each student during investigatory work, which kind of hobbles the whole enterprise.

  32. on 02 Mar 2010 at 6:50 pmJoe Henderson

    So, if I’m hearing you correctly, there are all sorts of barriers that stand in the way of the implementation of more student-centered pedagogy and the creation of their own WCYDWT. Everything from student demotivation and apathy to increased class size and budgetary violence. I will completely agree with you here. Those dilemmas are very real.

    But I can’t sit here and say that I have easy answers or silver bullets that claim to know “what works” in generalizable situations. I don’t know your kids or the unique situation that you are in. But that’s the whole point of the constructivist pedagogy. It’s going to look different for your classroom and with your kids. And that pretty much goes against the official line being sold to us about standardization and cuts right to the heart of mass-production schooling.

    All that I am asking is that you try. Those limiting factors aren’t going away anytime soon, barring some sort of major change in policy/funding or the construction of different models of schooling.

    So really, I guess it is up to us then, isn’t it? If not us, who?

  33. on 03 Mar 2010 at 8:24 amJason Dyer

    I’ve been running a game of SimTeacher ™ in my head and I still think take-home will have little response even when being very open about what’s allowed. (This is given the low-homework-rate population.) Here’s a hybrid solution:

    Ask students to bring a magazine of their choice (as long as it is *cough* school appropriate).

    On the day of the project, have a bunch of magazines there waiting for students who don’t bring one (if your library is generous with their discards, that helps here).

    Get students to find a WCYDWT from the magazine they bring or of their choice from the collection. Some random experiments indicate there usually is a plasuable choice somewhere.

    (You can use computers for the equivalent, but prepared for a big multiplier on the time spent on phase 1.)

  34. on 03 Mar 2010 at 3:10 pmJeremy Grisbee

    This is my first time posting, although I follow this blog very closely and have followed this conversation as it’s been building. I teach high school math, so this dialogue is quite interesting

    I have a hard time imagining that ideas brought in by students would be interesting enough to really maintain their curiosity, despite being applicable to the world they see around them. And, even if they did find an example of WCYDWT that would motivate them to pursue a question, I don’t know if the questions they would ask themselves or would try to think of to ask the class as a whole would be of the caliber that Dan has really pushed for in the whole structure of WCYDWT.

    Think of the many posts that Dan has strongly urged the commenters to think of what question to ask, and has even complained that the questions they are asking are too simplistic or not the point or just boring.

    Students can recognize the applications and bring them in, but I think the argument that Dan and Jason are making is that they would not consistently or naturally bring in material that would allow you to build a classroom built around a routine like that.

    All in all, it seems like idealistic and could have a its beautiful moments, but does not seem sustainable to have a classroom centered around students ideas of WCYDWT.

  35. on 03 Mar 2010 at 4:55 pmjosh g.

    @Dan: I’ve only read Freire’s ‘Pedagogy of the Oppressed’ so I think it’s in there.

    Just to give you the heads up, though: Pedagogy of the Oppressed is written in really, REALLY thick revolutionary language that is hard to bring back down to reality. Then you get to the section where he gives a concrete example from his own work – somewhere near the end of the book – and it makes a little more sense. His actual process doesn’t throw away the role of teacher as a guide to learning, even if his philosophy leading up to it starts to sound that way.

    I don’t own the book, so unfortunately I can’t dig up an exact reference to what I’m talking about.

    Joe: Math isn’t my bag (I’m a science teacher), but I’m sure you can think of some high-level process thing that is coming up in your curriculum that would require the use of linear relationship knowledge. They already have the scaffold of understanding linear relationships. Now have them use it to learning something much more complex.

    Again I’m hearing what you’re saying and there’s some truth to it but the “I’m not a math teacher, but surely you can find something” approach isn’t going to get your point very far. At least give a concrete example of how it’s worked in a science class?

    I can totally imagine doing something like this, but it would almost certainly be started by asking them a specific question. I have experienced some fantastic math-class inquiry learning, but it was almost entirely led by a framing question.

    Honestly, I don’t think “Let students create a WCYDWT” makes sense; the awesome of WCYDWT lessons is in how they present a problem in an interesting and challenging way. They work because they’re designed to work, which means they were designed, not discovered. Getting students to ask questions is a great idea, but it’s not really the same as “produce this piece of media that serves as a great motivator for some interesting question that fits well into the scaffolding of what we’ve been covering so far in the course”.

    Now, encouraging students to ask questions and bring them to class is another thing; I think that sounds awesome and would love to encourage it. But that’s something that I think needs to built up over time, and is a bit opportunistic – when a student asks a great question, listen to it and work with it, but you can’t force great questions to just happen.

  36. on 03 Mar 2010 at 7:10 pmJoe Henderson

    Here’s a procedure that I cobbled together from the comments section of this thread:

    1. Develop a safe classroom culture where kids feel okay to make mistakes and not get punished. If you don’t have this, don’t even try.
    2. Determine the mandated state standard that you would like to teach/assess/worship.
    3. Model a WCYDWT and the thinking that goes into planning such an activity. I’m assuming that this has already been done. Brainstorm a list of things that makes it a high-functioning and high-value activity.
    4. Assign your students (probably with a rubric to maintain transparency in grading/expectations) a mission to find an example in the world of whatever standard you have chosen.
    4a. There are two ways to do this, as I see it. If you are courageous, lightly scaffold the experience so that the kids are in charge of finding an example of something meaningful to them (I’m assuming that they mostly have access to some sort of digital imagery technology via cellphone and cameras). Anyway, I think you need to frame a question/problem. So, for science, find an example of erosion in your world and explain how it works. Personally, I would just ask that question straight-up. Then, have faith in the kids to actually go do it. If they don’t, be disappointed. Continue the practice. OR, for the less courageous, precede the activity with direct instruction. For me, this is less interesting and less constructivist, but I can understand the fear that some might have in turning over some of the instruction to the kids.
    5. Now here is where it gets interesting. Have the kids bring whatever they have found into the classroom. Show it to everyone. Craft the meaning together. And this is tricky on the teachers end, as you have to balance both what the student is explaining to you with the “official” knowledge of the standards. This fundamentally changes your role from deliverer of knowledge to co-constructor of meaning.
    5a. PS – This is how I did notes in my science room. Exploration always preceded formal instruction. Yes it takes time. You will have to balance this with the dilemmas of all the structures that are stacked against you.
    6. Assess learning via your rubric or however else you decide appropriate for the given standard.

    I’m probably forgetting something from this thread, but that’s roughly how I would do it. You might say we “co-constructed” the meaning here.

    @Jeremy – You say this:

    “I have a hard time imagining that ideas brought in by students would be interesting enough to really maintain their curiosity, despite being applicable to the world they see around them. And, even if they did find an example of WCYDWT that would motivate them to pursue a question, I don’t know if the questions they would ask themselves or would try to think of to ask the class as a whole would be of the caliber that Dan has really pushed for in the whole structure of WCYDWT.”

    That’s really sad man. That’s what I mean by faith, or lack thereof.

  37. on 04 Mar 2010 at 6:41 amjosh g.

    Okay, that’s something we can work with.

    I could see myself giving this a try. However, I can also see the need to have a backup plan in case students don’t bring you much of anything. (I hope for the best in my students, but blind faith in other mere mortals is dangerous business.)

    I would be seriously surprised if students frequently came up with examples that went from “Good” to “Best” in Dan’s curriculum rubric. Not because I don’t have faith in students, but because I’m respecting just how big a design challenge that last step can be. Heck, we’re here as educators trying to master this for a living and still struggling with it. I’d have a hard time placing that expectation (ie. assessment) on students; which means that teacher-originated WCYDWT’s fill a role that this student-created media approach doesn’t.

    I can also see this working a lot better with some areas of mathematics than others. Sending students out to find examples where you need to figure out the surface area of something could work; asking students to take videos of factoring quadratics or logarithmic functions would be nuts. (Maybe not impossible, but not worth gambling on unless your kids have a long track record of success at this.)

    In that sense, this is something where science fits this approach much easier than math. Science is all about observing the world; math is all about abstractions we can use to understand … the world, or whatever, or just math itself. There’s an extra degree of separation in there.

    On the other hand, you can do advance exploration in more abstract realms with software or calculators. I’ve seen this approach tossed around a lot; heck, even in textbooks. I tend to swing between hating and loving these kinds of approaches depending on how they construct meaning (and/or my snarkiness levels for the day).

    All that said:

    Joe: Do you have your kids develop WCYDWTs? Does the authority ever flow the other way?

    Assigning students to finding their own examples of math in action gets bonus points for building connections between math and reality, for sure. And students get empowered as media creators rather than media consumers, so the digital media side of me is happy about that. But you’re still assigning homework. You’re still fundamentally the one asking the question, and you’re probably losing some of the value of well-crafted WCYDWT lessons. This seems to me like a trade-off rather than a moral-high-ground improvement when I look at my student-centered-pedagogy checklist.

    I still like the idea of empowering students to ask their own WCYDWT questions, but I think structuring it as an assignment is missing what goes on when Dan and others hit on something like this. You can’t assign someone to get inspired by the WCYDWT fairy. At best, you can ask them to produce results that are superficially similar to what happens when the WCYDWT fairy drops by. But my hyper-idealist side tells me that’s no substitute for the real thing. If I want my students inspired to see math in things, they need real inspiration. I can’t just assign authentic inspiration experiences to students and expect them to appear. I need to model what they look like and try to be ready to help when they show up. As far as that goes, Dan’s WCYDWT lessons have got the modeling-inspiration part in spades.

  38. on 04 Mar 2010 at 10:18 amJeremy Grisbee

    Joe,

    I can definitely understand where you’re coming from. And especially now that you have a structure outlined, the ideas that you have been spouting are starting to come full circle. I can see your reasoning.

    Although I resent being summed up to having no faith in the abilities of my students, I can understand where that one quotation of mine you chose to focus on could be interpreted as such. My main point was along the same lines that josh g is taking.

    You may have a handful of students that could be really good at finding real-life situations, stories, or stills that would apply to the concept being taught, but it wouldn’t come naturally, nor would it come consistently enough to build a class around the theory, mainly because you would be assigning them a task to perform. So the motivation is not truly intrinsic, because it is assigned, even if they find something in which they are interested. My ending paragraph had that summed up in fewer words, along with the idea that it would not be sustainable or consistent enough to have students not only bring in a WCYDWT, but make it enough to build a lesson or driving question around every day, or even once a week.

    I can imagine right now a student bringing in a very true example of slope found in real life: two planks going from the ground up to the tail-bed of a truck to get their four-wheelers up into the truck. They would take a picture of it, assuming correctly that this could be interpreted as slope and feel like they completed the homework assignment. That would be the minimalist student, doing only the little he needs to get by.

    Then you have another similar student that would see the purpose of WCYDWT, not only taking a picture of the planks, but also measure the horizontal and vertical distances needed to measure slope. I can imagine this student asking “what is the slope?” to the class, leading the class on the hunt to find the slope in, umm, a minute?

    This example is a fairly true example of slope in real-life (kind of), but the driving question is boring. And most of all, who cares what the slope of it is. They just want their four-wheelers in. As a teacher, you may come back with “Well what would happen if the planks were shorter/longer?” or “How short can the planks get, yet still get the four-wheeler into the truck?”, but at this point, you have nothing concrete to show, demonstrate, or test with the students. It’s reduced down to the boredom found in a textbook problem with a picture of a truck and planks next to it.

    Maybe the students I am imagining are sub-par to the ones you are imagining or wishing for or actually have, but these personalities and the amount of investment put into homework are real to me at this point. To the other teachers out there, am I crazy or faithless to assume that there would be many well-intending students that would fall into this category, and not just be in the 80% of students not doing the homework as stated in other comments?

    Another hole that I see in your theory (or application) is that these students would be finding real-life applications of how the concept at hand really applies to the world around them, or even better, something they are actually interested in. I have to ask, when would a student ever come up with a “Stacking Cups” activity to model linear relationships without ever knowing what linear relationships are? When would they come up with an activity to discuss the concepts of slope and y-intercept without even knowing the words? The purpose of these activities are to pose questions that are structured and designed to get you from point a to point f or g or x through scaffolding in a way that can be talked about in escalating amounts of math terminology. The point is to get them to think of the situation first, then think about a way to solve it, and then “oh crap! We’ve been talking about math this whole time without realizing it”.

    If your point is to make the student-led applications at the end of a unit or concept as a project of how to apply it, then I can definitely see it working pretty well for some. I cannot, however, imagine a whole year centered around this student-led pedagogy.

  39. on 04 Mar 2010 at 11:28 amDan Meyer

    One, the average WCYDWT activity takes me ten hours to complete. Maybe I flatter myself, but this form of curriculum development involves something more than just a few snapshots.

    Two, if your curriculum defies real-world exploration, to what extent can you realistically expect real-world exploration from your students.

    [Sidebar: to what extent should all curricula apply to a student's daily life?]

    Three, were I to somehow summon up enough faith and courage and high expectations and all that so that my remedial students brought in digital artifacts expressing their understanding of the real-world applications of, let’s say, “factoring quadratic equations with leading coefficients greater than one,” I imagine the process of presenting and collectively winnowing those presentations down to a working definition (again, for the real-world application of factoring quadratic equations with leading coefficients greater than one) would take hours longer than the alternate route through direct instruction. Just guessing. And, for the record, I would prefer it to direct instruction. But that long-term strategy would have us through 50% of California’s Algebra I standards at the end of the year, courageously kicking the can down the hall to their next teacher.

    I mean, four, case in point: everyone guessed at the number of jelly beans in a jar yesterday. I gave them laptops, a spreadsheet with everyone’s answer, and the exact answer of jelly beans. I asked the students what they wanted to know. They wanted to know who was the best guesser, who was the worst guesser, and a ranking of the people in between.

    I knew we needed to get through absolute value, percent error, and average percent error. I had some amazing, truly satisfying conversations with pairs of remedial students who strained for a definition of “correctness” and strained to operationalize it in Excel. I had to chomp off my tongue repeatedly to allow them to come up with the definition unassisted. We needed more time. In a two-hour block half of the groups made it through half of the day’s content standards. Half hit absolute value. Nobody hit percent error.

    [Sidebar: does anyone yet have a fix on where these kinds of impediments fit into Joe's framework of faith, high expectations, and courage? I'm at a loss here.]

    Five, I’m not looking for someone to spell out what, by Joe’s definition, can only be spelled out by teachers according to the needs of their students. But it would be helpful to see a few case studies of teachers making that happen within California’s Algebra 1 curriculum. Shouldn’t be that hard to find a few courageous teachers among thousands. Right?

  40. on 04 Mar 2010 at 1:04 pmDavid Cox

    Joe
    It seems to me that you might be setting up a bit of a straw man. I am not sure this has much to do with faith, high expectations or courage because we have to consider the students in front of us. Dan and I are teaching the exact same standards to a completely different set of students. What may be high expectations for high school kids in a remedial algebra class may be the floor for a group of precocious 7th and 8th graders.

    I can give my kids an assignment like the one you outline and they’ll do it. From the outside, it may look like there is some authentic learning taking place, but in reality, many of these kids will simply be jumping through another hoop (albeit a high one) whereas Dan’s kids may say, “screw your hoop, I’m tired of jumping.” Which kid really “get’s it?”

    It’s one thing to ask a kid to find an example of erosion in the real world but another to ask a kid to find an example of linear relationships, rates of change, etc. because of the working knowledge a kid may need in order to even recognize it. Heck, there are times Dan throws up a WCYDWT and I even scratch my head at what the actual objective may be. But in order for a teacher to do this kind of teaching, the concept has to be understood inside and out. Otherwise what is intended to be an inquiry based lesson turns out to be a half baked investigation where no one really knows where it’s going to end up.

  41. on 04 Mar 2010 at 6:37 pmMr. K

    6) Most Math teachers myself included, can’t pull off a decent WKYDWT, even after we’ve been playing along and making the effort for years now. How do you get the average kid to do this, rather than just dump off a cheap facsimile?

    Or, IOW, The WKYDWT problems are in the very top strata of the top layer of the top level of bloom’s taxonomy, not just for the subject matter, but for pedagogy as well. You’re not going to get a kid with little to no teaching experience beyond tutoring the kids next to them to be able to deliver something worthwhile.

  42. on 05 Mar 2010 at 7:52 amJoe Henderson

    Well, I think I’m basically done here. I’m not so sure there’s much more I can add beyond the process considerations that I have already put forth numerous times. I get the sense that people are looking for silver bullets, and I just don’t have any that are broadly generalizable to all of your unique situations. My original point and objection was that the WCYDWT needs to involve more student ownership. I still believe that. Dan asks:

    To what extent should all curricula apply to a student’s daily life?

    I would argue that all of it should, at least as a starting point. Yes, we should have standards. Yes, teachers should navigate those with the students. Yes, we should assess understanding in a transparent way. But education fundamentally isn’t about us, it’s about the students. This is NOT to remove teachers from the equation, quite the opposite. Imagining education as student-centered makes our job that much more difficult and I empathize with the struggles articulated over the course of this thread. But, at a very basic level, you cannot experience these struggles unless you cede some authority over the learning process to the students, and that is something that people seem very unwilling to do as this thread has evolved. I find that interesting and am not really sure what to say to that.

    So, guys, I’m out. I’ll leave you with a little Dewey from 1900:

    “From the standpoint of the child, the great waste in the school comes from his inability to utilize the experiences he gets outside the school in any complete and free way within the school itself; while on the other hand, he is unable to apply in daily life what he is learning in school. That is the isolation of the school–its isolation from life. When the child gets into the schoolroom he has to put out of his mind a large part of the ideas, interests and activities that predominate in his home and neighborhood. So the school being unable to utilize this everyday experience, sets painfully to work on another tack and by a variety of [artificial] means, to arouse in the child an interest in school studies …. [Thus there remains a] gap existing between the everyday experiences of the child and the isolated material supplied in such large measure in the school.” – John Dewey, The School and Society

  43. on 05 Mar 2010 at 11:34 amJason Dyer

    Joe, I still don’t think you’re getting nobody is arguing against big-picture student centered learning here, just making the exact format of WCYDWT student-centered.

    Your rhetoric reminds me of ed seminars I’ve been in that talk only of “staying positive” and “setting a high bar for the students” without getting into any details on how these things are done (“actionable”, to be jargonlike). Finding actionable methods is not a trivial task, not something that can’t be waved away with “I’m sure you’ll figure it out”.

    I’ll take it for granted there must be at least some aspect of WCYDWT made simpler, WCYDWT-lite so to speak, for it to work with students. The boiling down for students may make it more appropriate to call the thing something else.

    I’m willing to try my magazine example above some time and see what happens, but I’d love to get a more concrete structure down to raise the probability of student success.

  44. on 01 Apr 2010 at 5:00 pmblaw003

    I know I’m late in the game commenting, but I just came across the conversation today, and later stumbled across this book , possibly providing some requested examples of students generating their own study of mathematics. I don’t know the book at all, and if someone has read it, maybe they could comment.

    Second, as what may be called a “committed constructivist”, I want to remind all that constructivism is a theory for how people learn, or better stated–a theory for knowing. It very certainly says nothing as a theory for teaching; at best it may suggest what might be more powerful. In very brief summary, as a theory for learning constructivism states that no matter what method of instruction used by the classroom teacher, the student–or better the students’ minds–“construct” meaning (a way of knowing) from the experience. Piaget: “The mind organizes the world by organizing itself.”

  45. on 01 Apr 2010 at 5:05 pmblaw003

    Sorry, apparently I cannot include a link. The book I referenced is titled, “Mathematics as a Constructive Activity: Learners Generating Examples”.

  46. on 06 Apr 2010 at 7:12 amLaura

    I have for 8 years asked students to do this kind of “bring the math concept home” activity. And the students enjoy it, they learn how to get a project done by a deadline, they learn how to accept peer critique without feeling criticized, but in the end I wind up with a drawer of ACTIVITIES. If I can truly use 1 the following year it is rare.
    Constructing “good questions” is the main focus of lesson study and many international models that the US is so hot to model. I look forward to your site with searchable lessons as a repository of something that we do well in the US imagine. And isn’t that what all your lessons are structured around Dan? Imagine this situation!!!