I’m excited to see Dan’s reply!

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 didn’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.

Dan- good luck on Finals. Many, many thanks for starting this dialogue.

Josh’s comment: ‘Of course the ‘regular’ method’s more efficient — they’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?

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, What CAN you 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.

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.

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

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?”

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 ‘‘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 repetition’’. 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 ‘‘cognitive load theory’’ 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, ‘‘should always be used
if available’’.

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.’

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.

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!)

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.

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.

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.

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!

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, we’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 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.’

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.

@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.

{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.

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?