The one nagging question I continue to have about WCYDWT is…what exactly does it accomplish?

Yes, it appeals tremendously to our intuition. Students are looking at the world and designing questions about curiosities. This certainly appears to be the kind of skill we want to teach. But research is at best divided about the kinds of gains similar projects have had in the past.

Dan has pointed out that his students out-performed his entire department, that they showed real and true gains according to every measure we currently use.

But still I wonder. How much of those gains were attributable to the actual WCYDWT lessons? How much of it was attributable to his skill and highly developed craftsmanship as a manager, questioner, evaluator? WCYDWT doesn’t strike me as terribly efficient. And I know that’s not the point, but where is the research showing that these kinds of problems are effective?

I think Dan also mentioned that he would do these kind of lessons bi-weekly (about 1 in 10 days). To which I say, fair enough. He was efficient and skilled enough in those other nine days to experiment with something like this.

I don’t know, though. These problems appear to make students conceptually flexible, which is brilliant. To make them procedurally flexible- arguably more important and more difficult to teach- is probably what Dan was doing the other 90% of the time.

I know the focus of this blog is WCYDWT and debunking conventional textbook wisdom. But when there’s no coke or sprite, escalators, three-pointers, or cheese, how do you do the other stuff?

## 41 Comments

## Rick Lehman

March 12, 2011 - 8:13 am -I’m excited to see Dan’s reply!

## MathZombie

March 12, 2011 - 8:48 am -I have always wondered the same thing – how do you teach a non-WCYDWT lesson?

## Brian

March 12, 2011 - 9:02 am -Sean writes: “How much of it was attributable to his skill and highly developed craftsmanship as a manager, questioner, evaluator?”

I think the better question is how did Dan develop his skill and craft? I think we will find that it’s not how these lessons change his students, but how these lessons changed Dan. Dan has been transformed by the hard work he has put in, and how that hard work has allowed him to learn about his students in ways he could not have before.

On Dan’s transformation of teacher identity he writes:

“I don’t trust myself to be an effective inquiry-based teacher if I’m not living an inquiry-based life.”

On Dan’s hard work, he writes:

“For every WCYDWT lesson I spitshine and post on this blog there are dozens that never got off a list of ideas I keep and several more that got off the list and didnâ€™t pan out.”

That said, trying out and reflecting on these lessons may be a good first step for others teacher to transform themselves as well. But they will have to put in the same hard work. Dan invites us hear not just to borrow his lessons, but he invites us in, so we can transform ourselves with him.

## Josh Gates

March 12, 2011 - 10:17 am -Of course the ‘regular’ method’s more efficient – they’re used to learning by memorization and imitation! The more the learn to think for themselves, the faster they are at doing it. If they had practiced these sorts of skills earlier, then I bet they’d be more efficient with it now!

My kids have become remarkably more efficient with both WCYDWT? and “quick labs,” where I require them to conceive, design, execute, and present (on a whiteboard) a “lab” experiment. I’ve pretty much thrown out (at great personal pain) all of my old carefully tailored labs, because they engage with this more and learn more by being more independent thinkers. I don’t have FCI results from this year yet to compare to previous years, but we’ll see then!

## Dan Meyer

March 12, 2011 - 10:24 am -FWIW, I think Sean’s critiques are all very important and I’d like to compose some kind of response in this space as soon as possible. Unfortunately it’s finals week so, thanks, Sean, for the rotten timing on an excellent comment.

## Greg

March 12, 2011 - 10:26 am -Which is more efficient: a class that teaches the student one concept that they remember a year later, or a class that teaches the student ten concepts that they forget after the next test? (Sean alludes to this, although the commenters seem to be missing it.)

## Sean

March 12, 2011 - 11:14 am -Dan- good luck on Finals. Many, many thanks for starting this dialogue.

Josh’s comment: ‘Of course the â€˜regularâ€™ methodâ€™s more efficient â€“ theyâ€™re used to learning by memorization and imitation!

Hi Josh- I’m not sure what you mean by ‘regular’ method. I think reducing other pedagogical strategies to ‘memorization and imitation’ is unnecessarily dismissive. For example, a lot of research seems to debunk constructivist approach you imply has made your students more independent. That’s not to say that it’s ineffective, as your blog certainly indicates otherwise, but many other factors are at play.

In addition, I think students can learn to think for themselves in a variety of settings, some completely detached from real-world examples, quick labs, or whatever else. Giving students an abstract linear system, for example- boring and maybe as anti-WCYDWT as it gets- can become a ripe question if framed correctly. Listen to two students debate the relative merits of substitution and elimination while someone else precisely graphs it can be something to behold. Simply asking students to justify their thinking at every step- having a dialogue with them about why they do what they do- greatly reduces the ‘memorize and imitate’ classroom culture.

That said, Dan’s very elegant premise- that real-world problems should come from the real-world- is not lost on me. I want my students to invest time in the abundance of rich material on this site and I want it to lead to better outcomes for them as math students. In fact, I want it to be true so bad that I wish someone would prove it.

We are all trying, as math educators, to cultivate what we perceive to be important habits of mind. Like Dan, I agree that curiosity and creativity are paramount for math students and problem-solvers in general. They should be priorities in any math classroom with any modicum of soul. My bigger question is- what is the best and most efficient way to do this? If it’s WCYDWT, then I will be teaching WCYDWT and ravaging this site even more than usual. If it’s not, though, what is it?

## josh g.

March 12, 2011 - 11:17 am -Here’s my optimistic, theoretical response in defense:

WCYDWT does more than just build conceptual flexibility, although that’s part of it. It also builds up student confidence that what they know is *relevant* in this classroom. They can all take a guess at the answer. They can all see the results without needing a textbook to tell them if they’re right.

I want to believe that this makes a serious dent in the armor of students’ math anxiety. With the anxiety out of the way, even boring old lecturing is going to be more effective afterwards.

Disclaimer: I’m totally guessing here, but it sounds good.

## Karim

March 12, 2011 - 11:28 am -I spend most of my time writing math lessons/content, and Sean’s concerns are certainly one’s I’ve shared with respect to WCYDWT:

Can you structure an entire curriculum around this kind of open-ended exploration, or is it a sort-of mathematical dessert?I always found the prompts interesting, and certainly potentially rich mathematically, though in the back of my mind wondered, perhaps cynically, WhatCANyou do with this, really?In the end, though, I suppose that a lot of my cynicism had more to do with a certain insecurity in my own process than anything inherent in Dan’s approach, one that was predicated on the dualistic notion that there are only two options: scaffolding or free-fall; handouts or nothing at all. It wasn’t until Dan emailed me something to the effect of, “I’m not trying to rewrite the entire curriculum, but a very specific section within it” that I finally understood how valuable this kind of thing really is, and was able to (in the words of President Bartlet)

see the whole board.Math education is a varied landscape and, like any good construction project, requires a range of tools. WCYDWT is one of them, and an excellent one at that. Of course, just like you couldn’t build a piece of furniture with only a screwdriver, nor could you build one without it. Again, it’s part of a toolbox, and I think any frustration–or, for that matter, any sycophantic obsession–with this one tool is to miss a much larger picture…and indeed, to miss the point.

WCYDW-WCYDWT? It seems so meta, but even though I no longer teach, I’m personally grateful for Dan’s work. We’re finishing up a re-write of all of our content on Mathalicious in preparation for the new website, and the process has been deeply informed by what I’ve learned from this site: from its users and host. Even though our lessons look very different from Dan’s, they’re better because of them.

@Sean, perhaps you’re right: perhaps WCYDWT-style inquiries aren’t “terribly efficient.” But perhaps neither is pure efficiency. Anyway, a really cool conversation. Thanks for getting it started.

## David Wynn

March 12, 2011 - 11:39 am -Speaking only as a previous student, I’ll throw my hat into the ring on a potential benefit of WCYDWT lessons: improved retention.

Traditionally, math is taught in a very abstract way, with lines and graphs and planes and such. While this is useful in some contexts and works for some kids, it’s very difficult for others to grasp. By making math more concrete and more tied to the real world, you create triggers for students to think about math in their lives outside the classroom.

For example, I remember pretty clearly the calculus lesson where we talked about someone floating across a river landing downstream because of the river current… but I only remember it because I ended up on a boat in the Caribbean and we had to grapple with that exact problem. I didn’t even recognize it until I asked my friend why staring at the map of the Bahamas looked so familiar. “We did this problem in Calculus,” he said.

Like real-world flash cards, the concrete-ness of the WCYDWT lessons probably creates more chances to remember (and thus better imprint) the concepts in daily life.

## Michael

March 12, 2011 - 2:03 pm -In my experience, math education has become this set of facts students are to remember for some state test, at least that is what it seems to have become.

One reason why math is taught is not necessarily for the specific math content, but the logic and reasoning and thinking skills that are not quite taught as effectively in other content areas. It seems that WCYDWT problems require students to apply their logic, reasoning, and thinking skills to an out-of-the-box problem that perhaps “traditional” textbook problems fail at effectively requiring students to use.

Mathematics is a way to model the world. Adding, subtracting, Algebra, Geometry, Calculus and other tools of mathematics are just that, tools to model our world. Should we not be learning how to use these tools in the context of what these tools are used for? If I teach my son how to use a hammer, I am not going to just teach him to hammer nails into a static block of wood. Once he gets the nack of hammering, we will build a birdhouse, shed, or something practical to give purpose to learning how to effectively use that hammer. WCYDWT does just this with our math tools, in my opinion.

Perhaps the WCYDWT problems that Dan posts on this blog, at the very least, try to come up with practical and efficient methods for students to use their math tools in context. Perhaps there can be much criticism for the content of the WCYDWT problems, but I think the underlying premise of presenting these problems on this blog is to have a collaborative effort to expose the faults of such problems, fill in the holes, and collectively create a product worthy of every math classroom.

The question is, what are we preparing our students to do? Are we preparing them to apply their math in artificial ways for a test or apply their math to model the world to answer very tough questions? I think our modern world is demanding the later. Should we fail in preparing our students for the later, the world will look to others who can use math to model the world and America will be left in the dust. Tradition is not necessarily working. WCYDWT attempts to provide a part of the solution.

## MBP

March 12, 2011 - 4:49 pm -A perfectly sufficient justification for WCYDWT is that the real world application problems that we’re working with now are poisonous (See: Pseudocontext). To the extent that real-world application problems have a place in our curriculum, WCYDWT problems are a better way of doing them.

Someone here should correct me if I’m wrong, but as I understand it there are three separate insights that go into WCYDWTs. First, that when a person is asked an interesting question to which they have an intuitive response, that person becomes invested in what follows. Second, that technology can allow a teacher to ask way more such questions. Third, that WCYDWT problems more accurately reflect the way math and reality interact than textbook problems.

The first insight is an insight concerning presentation, and that’s generalizable. The second and third insights more narrowly concern the realm of real-world application problems, but these are already part of our curriculum. But that first insight is a good one, whether you’re doing a “WCYDWT” problem or not.

## Lisa

March 12, 2011 - 5:21 pm -I have started reading your blog in the last few months…after a colleague found a talk of yours on the web. I belong to “an escape the textbook” web community and have been pondering all of the real life/project-based philosophical movement in mathematics education. I personally teach (in N. CA) a very traditional Algebra I program, but with many scaffolds and as many entry points as possible. I work mostly with students that have taken Algebra I more than once…so I affectionately call it “recovery” algebra.

Thanks for your work!! I look forward to folding in some WCYDWT problems. I really need to know, however, what “WCYDWT” stands for – exactly – please.

Lisa

## MBP

March 12, 2011 - 5:39 pm ->I really need to know, however, what â€œWCYDWTâ€ stands for â€“ exactly â€“ please.

“What can you do with this!”

## R. Wright

March 12, 2011 - 8:31 pm -Sean’s questions about the “gains” of students and whether Dan’s methods are “effective” raise some huge issues that I think are usually overlooked in these sorts of debates — for starters, I don’t think it’s at all clear how the effectiveness of a math class should be measured. What should students get out of a math class? Why do we teach them what we do?

It seems to me that a great deal of the “math” that is traditionally forced upon students is of no value to them. But most people with any interest in math teaching seem only to ask “Are the students learning the material?” rather than “Why on earth would anyone want to learn this material?”

## Cathy

March 12, 2011 - 9:19 pm -Maybe I am way out in left field here….I recently read another blog post which popped in my mind as I read these comments. I know that the topic is not math-related but the question I think is similar to what we are posing here. Look at Dean Shareski’s blog here: http://goo.gl/12i6s

His second question is: “But what does that have to do with improving student learning?” and he asked readers which of the following options they would choose:

1. Do it because it’s not only fun but likely does address some cirriculuar outcomes but you might have to look them up later. Fingers crossed.

2. Do it and to heck with the outcomes, doing joyful things with students is important.

3. Do it but perhaps as an extra-curricular activity because you’re not sure where it fits with a robust curriculum but still think it’s important.

4. Not do it at all

I don’t think that option 3 would be appropriate here since problem solving is part of our curriculum, but what if we replace ‘fun and joy’ with ‘math and problem-solving’? Would some of the comments on his blog be equally important or relevant to WCYDWT?

## Dan Meyer

March 12, 2011 - 11:33 pm -I don’t know that I can do any better than this. Let me leave some dead air here, though, for someone to step in and argue against assigning applied problem solving to our students.

…

If no one is willing to make that case, then we all agree (to different extents) that applied problem solving is important. So what’s the best medium for representing those problems? Paper has made economic sense for a long time but it’s a disastrous representation of applied problem solving. Paper distorts both context (with low-resolution clip-art and stock photography) and the problem solving process itself (by including the verbalization, visualization, abstraction, and decomposition of a problem all in the same visual frame). Seriously, if paper were our only option for representing applied problem solving, I’d be on the front lines arguing for a curriculum focused exclusively on pure math. It does more harm than good.

WCYDWT improves the visualization of context (with high-resolution photos and video) which means students can more easily verbalize a question, abstract the relevant features of the context, estimate an answer, and then verify it â€” all from the same multimedia. The best WCYDWT comes with its own answer key. It also improves the representation of problem solving itself. Just for starters: how many times do you encounter a problem in your day-to-day that’s already been completely abstracted and decomposed?

(Again, this is only compelling to the extent you think we should be assigning applied problem solving in the first place.)

At its best, the representation of context is so realistic, compelling, and curious, the teacher doesn’t even have to ask a question. I suppose I’m proudest of that aspect. Publishers aren’t playing on that field. They aren’t even laced up yet. They’re stuck in traffic on the way to the stadium asking questions no one would think to ask about contexts that are impossible to film because they’re fake.

What does WCYDWT accomplish? It’s tough to say this early with such a small sample. There is a long and disappointing track record for the alternative, though.

Which leads to this:

Seems like some citations would be appropriate here, Sean. The research base around problem-based learning is frankly too complicated, too thorny, too full of caveats, questions of implementation, assumptions about the point of learning and the nature of problem solving to let either side of this debate dismiss the matter as settled without so much as a hyperlink. The final project of my winter quarter is a literature review of problem-based learning so I’m more than happy to get into the weeds on this one. I’m not sure what exactly you’re reading, though.

BTW, even if we all agreed on first premises, this is why curriculum interventions are notoriously difficult to stage, scale, and measure. We can’t compare my Algebra 1 class (which completes one WCYDWT problem once a week) to another teacher’s Algebra 1 class (which completes none) because, like you note, I could just be more effective in general. So we randomly assign my students to two classes, one which completes one WCYDWT per week and one which completes none. The WCYDWT class results in higher achievement gains at the end of the year. The criticism of this experimental design is obvious, right?

No disrespect for not taking my word on this, but my students’ skills abstracting applied problems made them more efficient at abstracting procedural problems, not the other way around.

I’m interested in the argument that procedural flexibility is more important and more difficult to teach. Nevertheless, even though I’m less motivated by “the other stuff,” I should probably balance the scales a little bit around here. I’ve had a post in draft for a long time. It was originally called “Pressure Points.” Then “Cognitive Conflicts.” Lately it’s titled “Who Am I Kidding â€” Vygotsky Had This Stuff Nailed Fifty Years Ago.”

## Brian

March 13, 2011 - 2:14 am -*Likes* “Who am I kidding-Vygotsky Had This Stuff Nailed Fifty Years Ago.”

## Sean

March 13, 2011 - 2:58 am -Some cherry picked research showing balanced and not-so-balanced reviews of discovery-based/constructivist learning.

From Star and Rittle Johnson’s review of lit (2008):

Discovery learning is viewed by many as the ideal learning context for supporting robust learning (e.g., Fuson et al.,

1997; von Glasersfeld, 1995; Hiebert et al., 1996; Kamii & Dominick, 1998). For example, Piaget (1973, p. 20) asserted

in his book To Understand is to Invent that â€˜â€˜to understand is to discover, or reconstruct by rediscovery, and such

conditions must be complied with if in the future individuals are to be formed who are capable of production and creativity

and not simply repetitionâ€™â€™. Typically, discovery occurs when students are encouraged to work out their own

problem-solving strategies and to reflect upon multiple strategies. In support of discovery learning, children who discover their own procedures often have better transfer and conceptual knowledge than children who only adopt instructed

procedures (e.g., Carpenter et al., 1998; Hiebert & Wearne, 1996; Kamii & Dominick, 1998).

On the other hand, there is a large literature that suggests that direct instruction is more conducive toward learning

than discovery (e.g., Chen & Klahr, 1999; Klahr & Nigam, 2004; Rittle-Johnson, 2006; Zhu & Simon, 1987). In particular,

information-processing theories such as â€˜â€˜cognitive load theoryâ€™â€™ propose that discovery conditions can overload

working-memory capacity (e.g., Kirschner, Sweller, & Clark, 2006; Sweller, 1988). Based on a large number of

empirical studies, Sweller (2003, p. 246) claims that direct instruction, rather than discovery, â€˜â€˜should always be used

if availableâ€™â€™.

Here’s this from Kirschner, Sweller, and Clark (2006):

‘After a half-century of advocacy associated with instruction

using minimal guidance, it appears that there is no body of

research supporting the technique. In so far as there is any

evidence from controlled studies, it almost uniformly supports

direct, strong instructional guidance rather than

constructivist-based minimal guidance during the instruction

of novice to intermediate learners.’

## Karl M

March 13, 2011 - 3:23 am -Obviously it isvalmost impossible to study the effects of WCYDWT on education, you have missed a massive point here. The enjoyment of the students. I use this style lesson for all my period 5 lessons in England. This is the last and hardest lesson of the day generally. These lessons thoroughly engage the students, they love them.

So whatever happens, the fact low attaining groups can be switched on to maths after a full day at school is breathtaking.

Further to this, Dans ideas are just a logical extension of the Dutch model. Which, if we excuse Hungary, Japan, China, Turkey and India as not being culturally similar, show the best results for knowledge and understanding of maths in the low to middle ability groups. Check out MMUs Maths In Contxt series for a reference of all the source materials.

In closing, anything that improves my practice is something I’d like to discuss, so please, decimate my reply!

## serafina

March 13, 2011 - 3:34 am -I started doing WCYDWT-esq lessons upon first reading Dan’s blog (October). While many students come and go in my setting (juvenile detention center), those that have been with me for extended periods of time have shown to a large degree a general interest in inquiry. They take more guesses… They interact more with the subject… THEY THINK. That is, I have done far less “thinking” [for them], and let them take the reigns in their own construction of understanding. Students no longer get frustrated and say “you teach a different topic every day!,” because they truly know that everything is interconnected.

My setting might be different (should I say… ‘captive audience’), but regardless, the effect on students has been profound. Even if I only have the average student for ten days, I truly believe (maybe I should somehow measure this) that they walk out of my classroom with a deeper understanding of the connectedness of math.

## Iain

March 13, 2011 - 4:53 am -Iâ€™m new to this Math teaching stuff; having spent 20 years in industry as a Systems Engineer, then took the plunge and went back to University to do a teaching diploma and have been teaching now for almost 4 years now.

Iâ€™m new to this Math teaching stuff; having spent 20 years in industry as a Systems Engineer, then took the plunge and went back to University to do a teaching diploma and have been teaching now for almost 4 years now.

I have found the â€œoldâ€ math curriculum in Scotland uninspiring.

It does not seem to produce thinking skills of the type that are needed to be useful in the â€˜real worldâ€™ and those which I used every day in my old life.

My students ask â€œWhen are we ever going to use this in real lifeâ€ when they are multiplying two mixed numbers, or simplifying surds with a conjugate. Never, for most of them?

Sean â€œThe one nagging question I continue to have about WCYDWT isâ€¦what exactly does it accomplish?â€

I think it produces thinking skills that are far more transferable to the problems the students will need to face beyond the classroom than the rote learned, exam passing, skills.

Sean â€œI don’t know, though. These problems appear to make students conceptually flexible, which is brilliant. To make them procedurally flexible- arguably more important and more difficult to teach- is probably what Dan was doing the other 90% of the time?â€

Teaching must have a variety of approaches and has to accomplish a number of things including knowledge acquisition. But I believe WCYDWT also allows students to think and apply their knowledge in a far more memorable way. I think WCYDWT produces better thinking skills and therefore students who are procedurally much more flexible.

A Speed Distance Time problem can be set a number of ways. But show the students â€œWCYDWT Pure Performanceâ€ and they will have learned how to decode most any SDT problem you can throw at them.

Higher order thinking skills massively cut down the time required for rote learning practice

Educational research is a quagmire so I have no proof and doubt others will ever be able to end up with any definitive research evidence for this.

Iâ€™m with Karl and Kathy â€¦..Above all the majority of students seem to have much so more FUN in class with things like WCYDWT which canâ€™t be a bad thing!

Ps

The good news in Scotland is a recent report on the new curriculum being developed in Math has WCYDWT as a key component

The report says

â€œTo ensure depth, young people must be able to apply the key mathematical skills and understanding they have acquired in new, nonâ€routine and relevant contexts. Central to this is the development of higherâ€order thinking skills that enable the learner to identify which particular mathematical techniques can be appropriately applied in order to progress towards a solution to a problem. When a new skill has been acquired, a degree of repeated practice and consolidation may be required in the short term but it is vital that learners are provided with a range of realistic opportunities and activities within which to apply the new skill in both familiar and unfamiliar contexts.â€

## Audrey

March 13, 2011 - 6:55 am -Dan, are there any of your actual students who might care to weigh in here? I know you’re pretty young, so likely most of your former students aren’t even 18 yet…Did you ever have a classblog going while you were doing WCYDWT activities once a week? (David Wynn’s response made me wonder about that – David at first I thought you meant you were one of Dan’s former students!)

## Matt W.

March 13, 2011 - 8:22 am -I just had a look at Kirschner, Sweller, & Clark (2006) – it’s available at http://www.google.ca/url?sa=t&source=web&cd=1&ved=0CBoQFjAA&url=http%3A%2F%2Fwww.cogtech.usc.edu%2Fpublications%2Fkirschner_Sweller_Clark.pdf&rct=j&q=kirschner%20sweller%20and%20clark%202006&ei=2ex8TZzbPJKBrQHGmr3NBQ&usg=AFQjCNG0OrVu9AN4XM1k7BjPTc0Ge-MUrg&cad=rja

The opening paragraph reveals what in my mind is a major flaw in the article’s point of view and an unfortunate feature of much of the debate that rages around the false dichotomy of direct instruction vs. discovery: It breaks the argument into 2 opposing sides.

Problem-based learning, inquiry, constructivism, or whatever you want to call it does not require turning the kids loose with no guidance and hoping they will somehow discover everything they need to know. The teacher must still assume a very strong guiding presence – it’s just a different kind of guiding presence. Nor do these methods require abandoning all forms of direct instruction.

Unfortunately, I’ve seen teachers interpret greater student ownership in exactly the same all-or-nothing way. Then – no surprise – their lessons bomb, and they swing back to the other equally harmful extreme.

## Erin Pletsch

March 13, 2011 - 9:11 am -Dan, I had the pleasure of seeing you this week at the charter conference in SD. I introduced myself in passing and my only regret is not better articulating what an impact your talk had on me. No exaggeration that it was the best session I attended (by a lot). You had me at Burning Man…

Reading through the comments, I can’t help but feel there is a void in this particular conversation regarding how kids learn. I’m taking liberties here in the absence of anecdotal evidence, but how many of Dan’s students have sat around a dinner table and actually EXPLAINED to some interested party what s/he learned that day in math? Lots, I bet. Not because they were eager to demonstrate content mastery, but because they made a connection between math and their every day life and they’re EXCITED about it.

I haven’t interpreted Dan’s approach as ‘Eat This, Not That’. It’s about debunking the myth our students have about the application of math in the real world; exposing it, shining a light on it and making it relevant. If we can’t make time for that, we may be winning the battle but we’re losing the war.

## Shari

March 13, 2011 - 11:19 am -If it is research you want, search for “anchored instruction” or “in-context learning”. I’ve seen these phrases used in research that I think aligns well with what Dan is doing. They are finding that students are more likely to retain what they learn if they have something of substance, something meaningful, to anchor what they learn. Here’s one link to get you started. One researcher you may want to search for specifically is Bottge.

http://www.cited.org/index.aspx?page_id=93

In grade school I remember math being the same every year, only the numbers got bigger. Textbooks have changed some since then, but not a lot. Even so, teachers often find that they need to spend the first month of the school year (at least) reviewing last year’s math skills. I think if more math skills were taught in context that is meaningful and relevant to students, we’d find it unnecessary to repeatedly reteach skills.

## Carol

March 13, 2011 - 5:01 pm -I have been reading this blog for a year. I teach high school biology. I have been inspired by this blog to constantly be evaluating the quality of my lessons.

All of the above comments include plenty of worth while teacher jargon. As a parent, I want to know a few things about my child’s teacher: 1)does (s)he LOVE the subject 2)does (s)he LOVE students 3)does (s)he understand that great teachers spend quite a bit of time preparing for class.

I don’t want to compare test scores at the end of the year. I want to interview former students ten years later.

I enjoy reading this blog and all of the comments. For the record, I have four children and I hope that by the time they start high school a few WCYDWT math teachers have migrated east.

## Michael Paul Goldenberg

March 13, 2011 - 5:41 pm -First, who gives a rat’s patootie what is the “most efficient” way to teach mathematics? I’m reasonably certain that “God’s pedagogy” isn’t going to be discovered by any human, and certainly not by anyone in my lifetime. Nor, frankly, if someone could show that pedagogical algorithm to me, am I convinced that it would be the best approach for ME to take or for me to use with every class of students.

Without wanting to go into a 20 page philosophical treatise, I’ll merely state that I’m not moved by the notion of any ultimate teaching strategy. Further, I’m highly suspicious of using that sort of criterion or the “most efficient” criterion to evaluate someone’s practice.

We are a long way from being to implement the so-called “gold standard” of double-blind, controlled research in public school classrooms (and I suspect we always will be). I’m not even sure it’s a bad thing that we are. Just as I despise the current push for national “Common Core Standards” (despite the fact that a number of folks I usually respect seem to either have honestly swallowed that Kool-Aid or have felt compelled to pay them lip-service for reasons I’m not quite clear on), I find the notion of some universal pedagogical approach repugnant.

What makes Dan’s open exploration of his work valuable is NOT that we get to decide whether it’s the pedagogical King of the Hill (at least until knocked off the throne), but rather that because it’s so open, we get to help him think about and craft it as it evolves, and we get to borrow and play with all the pieces.

Now, of course, if what he is doing weren’t enormously intriguing, the openness wouldn’t be enough. However, the combination of powerful innovation with his willingness to make the private public is irresistible to me. Coupled with his obvious passion for what he’s doing and his having his finger on many new tools on-line and off, there’s just an unprecedented (to my knowledge) opportunity here for individual and collective growth in classrooms.

That doesn’t mean that Dan or WCYDWT – in whole or particular – is above criticism. I’d just be more interested in criticism that strikes me as well-founded. Given the apparent premise of the “nagging question” that fueled the current critique, I’m not inclined to see a firm ground.

As to the claim of research alleged to have “debunked” constructivism: really? How, exactly, has research successfully challenged the actual claims of constructivist theory? I’ve seen some rather awful, painfully twisted research that claims, for example, to have shown that using concrete objects doesn’t help the transfer of ideas. If the research to which Sean refers is that sort of thing, I’ll happily ignore it, because every instance I’ve seen is clearly operating from a misunderstanding of constructivist theory coupled with enormous philosophical bias, then rolled together with flawed methodology in order to “debunk” something that was never claimed in the first place.

In order to “debunk” constructivism, of course, it would be necessary to try to show that it fails as a theory of learning (not as a theory of teaching, which of course it never has been). I’ve yet to see such research, but I’ll be happy to look at anything that ostensibly does so. What I won’t do is waste more time looking at biased nonsense that wants to show that manipulatives are inherently bad, or that doing inquiry in math class can’t work or is “inefficient,” or that Textbook E is simply awful but Textbook S is magical, hence, “constructivism is wrong.”

## Marty

March 13, 2011 - 6:00 pm -I’m using Problem-based Learning as the cornerstone of my PhD I’m doing right now. I’m bringing it over from Medicine and giving it a test-run in Teacher Education. We’ll see how it goes! :)

I have now done an extensive literature review and could always do more.

I know Sweller and co. well (fellow Aussie), and they are folks on the other end of the spectrum to most of the folks who visit this blog. They are cognitive load folks, who say if you give a novice a complex problem, they will froth at the mouth and their heads will explode. Too bad when they leave school and enter real life!

Their big oversight is there belief that that constructivist learning is FOFO learning (f%^k Off + Find Out). Obviously that is not how I’m saying it in my lit review LOL.

## Laura

March 14, 2011 - 3:13 am -I LOVE this discussion :) How invigorating! When educators take time to extricate sacred cows and really, truly get dirty trying to articulate WHY they do what they do, then everyone benefits.

@Cathy, I loved the blog you linked to (and the video therein). Great stuff. I emphatically agree with you, and others here that have opened this line, that joy and enthusiasm is not currently an academic standard, but needs a firm and steady planting in the center of what we do.

@MathZombie, me too! How DO you teach a non-WCYDWT lesson? I’d love to hear Dan’s perspective on that.

More than anything here I value the processes at work.

Sharing is caring, folks :D

## Colleen

March 14, 2011 - 6:42 am -The strand about whether or not it is important to teach our students applied problem solving reminds me of Conrad Wolfram’s TED talk, “Teaching Kids Real Math,” in which he presents a case for spending the majority of our efforts as teachers on just this. He is critical of the amount of class time that is spent teaching kids to hand-calculate, and and argues that we should spend the majority of our efforts teaching the parts of math that computers cannot do. I would love to hear other math teachers’ reactions to this!

## Colleen

March 14, 2011 - 6:44 am -Pardon my bad manners – here is the link to Wolfram’s TED talk:

http://tinyurl.com/2uu6frl

## Sean

March 14, 2011 - 11:03 am -Matt: ‘Problem-based learning, inquiry, constructivism, or whatever you want to call it does not require turning the kids loose with no guidance and hoping they will somehow discover everything they need to know.’

This made me hungry to seek a more nuanced view of constructivism and discovery-based learning. This study (http://tinyurl.com/6efpxzu) details three forms: pure discovery, guided discovery, and expository methods. My impressions are that WCYDWT falls almost entirely under guided discovery (‘give me a guess you know is too low…’, etc.) with some faint shades of pure (having students create the question/solution strategy). It’s noteworthy that the authors have ‘guided discovery’ as the most effective approach of the three.

Shari: ‘ I think if more math skills were taught in context that is meaningful and relevant to students, weâ€™d find it unnecessary to repeatedly re-teach skills.’

On a purely procedural level, though, are they more likely to remember how to solve a quadratic equation six months later if they had a really memorable problem that required a quadratic equation?

I really don’t know. I think exposure to- and a rigorous discussion about the efficiency of- a range of different solution strategies may be effective. Where WCYDWT fits in here I’ll leave to Dan.

Michael: ‘ First, who gives a ratâ€™s patootie what is the â€œmost efficientâ€ way to teach mathematics? Iâ€™m reasonably certain that â€œGodâ€™s pedagogyâ€ isnâ€™t going to be discovered by any human, and certainly not by anyone in my lifetime.’

Asking questions about the effectiveness of one part of a curriculum and seeking a Deified Pedagogy I think are different. I’m also not convinced efficiency is a bad word. Sweller and Kirschner’s critiques may not have been as nuanced as we’d like, but there’s something to be said that their findings indicate direct instruction and worked-out examples are clearly the most effective way of learning. Dan’s right in that the literature is thorny, and Marty seems to have a much better grasp of the competing viewpoints, but I think a dispassionate review is better than a dismissal of an idea based on how it makes us feel.

Lastly, a lot of commenters have rightly defended WCYDWT based on the high enjoyment levels their students have while engaged in a lesson. I have had similar experiences with students, particularly in a summer school class. My small concern about this is ‘enjoyment’ as a paramount criterion for the value of an activity. In Dr. Ferguson’s research in urban schools, students rated enjoyment as considerably less important than ‘does this teacher have control of the class?’ and ‘am I being challenged?’ Seems noteworthy.

## Dan Meyer

March 14, 2011 - 11:50 am -I agree. Here’s the thing, though: if a teacher asks the students to ask a question that interests them about a video, to predict the answer to the question, to set bounds on the answer, and to come up with a list of information that will be essential for solving the problem, I can’t bring myself to care, really, if the teacher just works out the problem explicitly (a la Sweller, Kirschner, and Clark) for the students to take down in their notes.

Personally, I want to give the students a moment to start in on a solution themselves, if only to identify the students who are so advanced they’ll experience a negative effect (according to Sweller et al) from worked examples. My working theory is that, even if I just start lecturing there, students will see gains over the status quo. Certainly in motivation (they care about the answer) if not ability also.

Agreed again. Enjoyment shouldn’t be paramount but it should figure somewhere into the cost-benefit analysis. I expand on this in the post I was writing when Sean’s comment came in, all of which has been one colossal diversion from the paper I’m supposed to be writing right now. (Which happens to be about PBL, though, so perhaps I’m less off task than I thought.)

Interesting. Can I get the reference on that? I would have expected a pretty strong interaction effect between “am I enjoying myself?” and “am I throwing a chair at the teacher?”

## Michael Paul Goldenberg

March 14, 2011 - 1:33 pm -@Sean: You really find the studies that “prove” that direct instruction and worked out examples are “the most effective way of learning” to be compelling? I’m so unmoved by what I saw in them (and have criticized them publicly), that I’m always surprised when someone seems to accept them.

Of course, when we’re told (by people from University of Oregon’s Direct Instruction cabal) that Project Follow-Through definitively settled this question a long time ago, it’s obvious that we’re hearing from folks who have a dog in the fight. The methods used to similarly “prove” the superiority of instruction in the studies that have appeared lately seem so contrived and absurd that it matters little whether the folks who did them also have a dog in the fight or merely are sloppy researchers.

However, the fact is that Sweller, Kirschner and Clark don’t do much, if anything, to hide their disdain for what is loosely (and, to my mind, inaccurately) termed “constructivist” methods of teaching, and it is obvious that their 2006 piece and the rest of their research in this arena is focused on discrediting something they dislike. Unfortunately, the manner in which they attempt to do so is yet another case of loading the dice to “prove” that one’s biases are “objectively true.”

Reading some of the comments found at http://halfanhour.blogspot.com/2007/11/kirschner-sweller-clark-2006-readings_12.html , I was struck by comments from Barak Rosenshine (University of Illinois) regarding what he alleges is “romanticism” on the part of those curriculum folks at universities who favor constructivism, vs. the allegedly “rational” views of the researchers in question. Of course, Rosenshine’s description of constructivism is completely off the mark, but even so, he loads the dice with his dichotomy so perfectly that any other nonsense he offers up is completely predictable.

There are some good critiques of the 2006 article included at the above-cited URL, so rather than go through the arguments for and against Kirschner, et al., I’ll let interested readers check for themselves.

I’m still highly skeptical that their work has anything important to say about Dan’s teaching. Indeed, I consider their work an enormous red herring, the pursuit of which will not prove enlightening. Your mileage may, of course, vary.

## Michael Paul Goldenberg

March 14, 2011 - 3:32 pm -A timely quotation from “The Daily Papert”:

â€œIf you need to know whether drug X reduces blood-pressure, you may fairly safely draw a negative conclusion from a â€œtreatment modelâ€ experiment in which hospitalized patients were given X and no change in blood-pressure was observed. On the other hand, you would not deduce that drug Y does not increase fertility from the simple fact that hospitalized patients who received it had no babies. You would want to know more about other conditions that are known to be necessary. Nor would you deduce that ice is a bad material for building dwellings if you heard that I tried to build an igloo in Boston in mid-summer and failed. The right environment and, I presume, a high degree of special skill are necessary. Such a failed experiment would say much more about me than about whether â€œigloos deliver what they promise.â€

Papert, S. (1987) Computer Criticism vs. Technocentric Thinking.

A version of this piece was published as â€œM.I.T. Media Lab Epistemology and Learning Memo No. 1â€³ (November 1990). Another version appeared in Educational Researcher (vol. 16, no. I) January/February 1987.

## tripst3r

March 21, 2011 - 4:55 am -{Drive-by meta-comment alert}

I’m not a regular reader of the blog, but I do check in from time to time. In particular, I think the dissent postings are some of the best things on the blog.

Along those lines, I’d like to point out how well Dan furthers the conversation by engaging with Sean rather than piling on the reflexive criticism. Implicitly, Dan assumes that he and Sean have the same goal, viz., improving math instruction. Many comments above seem to feel that any discussion of alternatives to WCYDWT or discussion of the spaces in between present theories are bricks thrown through the windows. My experience (which is therefore not universal or even provably extrapolatable) is that the best teachers are always teaching *and* learning.

## Brian

March 21, 2011 - 5:04 am -@tripst3r. I just want to double down on what you are saying above. Instead of dismissing dissent, Dan does a great job of welcoming dissent and engaging with critical perspectives in ways that further all of our thinking. Part of this skill is exactly what you are saying, Dan seems to assume (at least until proven wrong) that people are coming at him with good intentions.

## Dan Meyer

March 22, 2011 - 5:05 am -Thanks for the notes, fellas. Good critics seem hard to come by these days (whereas I have a surplus of boosters and lousy critics) so I’m trying to cultivate the ones I have with the dissents feature, which, I hope, at the same time signals to other lurking dissenters that this is a safe space for disagreement.

## James

March 22, 2011 - 6:21 am -We did not get to read everything above but if anyone wants to improve math comprehension, people beginning to learn about math (age range: 4 to ?) ought to be taught about fractions while they are learning to count. That would lead to a situation in which the learners would discover what we consider to be the ‘magic of mathematics’ sooner, and they would be more able and willing to help their colleagues. According to Steve Ehrmann (see TLTGroup.org) the best way to learn more about something is to teach it!

Hope this helps!

JM

PS: this is only about 1/10 of what I have to say on the subject

## James

April 6, 2011 - 3:38 am -since many people learn better with visuals, write up some diagrams to show how, well yes, 1 + 1 = 2 but besides that, 1 is half of 2 or 1/2…things like building blocks can also aid in comprehension…we have these two blocks but if we put them together and take them apart…

Also, 1+1+1 = 3 but 1 is one third of 3…and using building blocks…

Children will begin from the earliest of ages to understand fractional concepts more completely, and many will learn enough to appreciate the magic and will be eager to show their classmates/friends…

ok so now you’ve gotten the second of 2 tenths…is it too early to show that 2/10 = 1/5…can somebody make suggestions here…

If this can be shown, a whole new world of mathematicians might be born who may end up teaching their parents!!!

Can I get an AMEN?