The Aesthetic

Michael Paul Goldenberg, WCYDWT spokesmodel:

When they want to see more methods, they’ll let you know. When they become discontented with their ideas of proof, they’ll let you know. And it WILL happen. Because there will always be kids who ask themselves and their peers: “Why does that work? Why does that make sense? How do you know?” And that’s all we need to nurture in them: their own natural curiosity, rather than suppress that and replace it with curiosity about only the following: What does the teacher think? What does the teacher want me to say or do? What do I need to do to get an A?

I like this.

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


  1. This is what education is all about. I love when my students question me and and what I am teaching them. As a history teacher this is paramount to my goal-students need to learn to question with intent and interest.

  2. I’ve got to add that I spent a lot of years trying to get my kids there, standing back and offering far more questions than answers. This year, toward the end of the year, I finally started to tell them what I think about what we’re reading, about their video idea, about their speech. Not a single kid regurgitated my ideas. They all took what I had to say and ran with it, producing something much better than they had before and feeling a lot more confident in their ideas, much more willing to explore their thoughts. We are experts in the classroom and keeping our voices down until the students ask us, questions that they might not even know to ask going unvoiced, we make a mistake and don’t let students in on everything we have to give them in order to encourage their own thoughts.

    Then again, I didn’t offer my ideas until I really saw they struggling. They didn’t necessarily ask for my advice, but I gave it anyway when I noticed that they needed something more. Maybe that’s the E/LA equivalent of kids letting me know they need more information.

    It’s a great idea (and ideal) to want to nurture curiosity and to let student questions drive the information you dole out. But sometimes that curiosity doesn’t manifest in the form of questions and sometimes students don’t know which questions to ask. We have to be ready to deal with that, too.

  3. Thanks, Todd. That’s what I would have guessed based on your comments.

    Keep in mind that my original post from which Dan quoted was primarily focused on what happens to elementary kids in mathematics classrooms in the early grades. By the time they reach high school, the damage has been done in terms of creating highly passive kids. You simply cannot lay creative, student-centered instruction out there and expect that they’ll get it. Most won’t. How could they? And why should they respond even if they intuit vaguely what you’re offering them? They’re so locked into passivity that self-motivated intellectual movement, risk-taking, hell – simply some minor degree of independent thinking – is pretty much beyond them. There’s simply way too much focus on either doing the minimum to pass, or doing the minimum to get a high grade. Actually thinking for the sake of what that might bring on its own just isn’t on their radar.

    I should note that I did my student teaching in English, not mathematics, and I taught freshman and lower division lit classes at University of Florida (a glorified high school as far as the vast majority of kids I taught went) in the mid-’70s for three and a half years. I’ve also taught some elective lit classes to at-risk high school kids (about a decade back).

    That said, there is a dynamic here that is hard to keep sight of, and at the high school level it’s almost too late. There is a constant tension in education between inculcating (or trying to) and literally “educating” (‘to lead forward”). Unfortunately, the dominant teaching paradigm in this country is the former one. And as a result, some people (occasionally under the name “radical constructivism,” but often without any formal theoretical considerations) swing towards such a strongly opposite extreme that they avoid doing anything that might be construed as direct instruction.

    The problem here should be obvious. The teacher, at least in theory, should have a mature, developed perspective on the requisite subject matter such that s/he can fit pieces of the content together in complex, sophisticated ways. S/he has critical/analytic tools at his/her disposal that the neophyte student has yet to develop, or has only done so at a rudimentary level. And the temptation is, of course, strong to simply try to hand those tools over to the student.

    But the constructivists aren’t wrong, I think, in believing that students can’t simply absorb the significance of that which is bestowed upon them by sage instructors. The dilemma, then, is how does the knowledgeable teacher act responsibly towards students without becoming simply a form of indoctrinator (often a less than fully effective one, at that)?

    The answer cannot be to utterly abandon students to their own devices. But neither can it be to use their need for structure as an excuse to return to utterly teacher-centered methods. As far as mathematics education goes (and that’s my complete professional focus), I’ve simply seen too much telling, very little “leading forward,” but where well-meaning teachers back off, they tend to fail to provide the sorts of frameworks and grounding that I alluded to in my post. (Keep in mind, too, that I was speaking about more than just academic scaffolding, but also of basic social behavior in a learning community. Kids who can’t talk to one another in pairs without fighting aren’t going to do a lot of effective mathematical problem solving in pairs or small groups).

    At present, I’m starting to take a serious look at an approach to early mathematics education that comes out of the work of Lev Vygotsky and V. V. Davydov. I don’t want to get into details yet, as I’m simply learning about it and have not formulated any definite judgments. On top of that, I’m finding it difficult to obtain any of the 3 elementary mathematics books Davydov and colleagues developed that were translated and published here around 1999 at SUNY Binghamton (anyone who knows where I can get them will be thrice-blessed). But from what little I know, Davydov believed that kids should learn abstraction FIRST and then learn the concrete. So the math books present algebraic ideas before arithmetic ones, as far as I understand. More than that I won’t say, as I may not even have that part right.

    What concerns me, however, from what I’ve read, is not the reordering of the standard curricular approach, but what appears to be a totally teacher-centered, instruction-by-telling-only approach. I’m skeptical that this sort of thing is necessary or ideal, but I need to find out a great deal more before pushing at that concern more seriously.

    In any event, there’s no doubt that there are mathematical structures and frameworks that students need to be exposed to. And similarly, there are such things in the study of English and literature. Asking students to write a decent essay about a piece of literature when they are truly clueless about what that might be is probably a losing proposition, if my experience as an English teacher is any indication (and mind you, I was a multiple award winning graduate teaching assistant at the University of Florida). I know that the vast majority were light years from knowing how to pick a decent topic, let alone develop a paper from one. And I believe I failed them in not doing a great deal more to help them learn how.

    In retrospect, I know I could have been far more effective. And I no doubt would have been doing some of what you’re trying or feel you must try to get them where you’d like them to be.

    However, I would not abandon the fundamental belief that our deepest, most profound job as teachers is to help students become independent thinkers who are reflective and self-motivated. I don’t have “the answer” or even a clear sense of “an answer” yet, but my guess is that there are some good ones that involve blending and synthesizing well-structured curricula with discovery learning and constructivist principles and methods.

  4. Actually thinking for the sake of what that might bring on its own just isn’t on their radar.

    Yeah, I think all of us would be hard pressed to find any evidence in the majority of students to suggest that education for education’s sake is good enough. If there’s no greater reason for the content other than improving their minds, there’s no reason for the average teenager to pick it up and play with it. I’d venture that this has always been true. There’s little you say that I disagree with. I wasn’t even disagreeing in the beginning, just pointing out that not all of our lessons can be structured this way. Probably not even the majority of them because we have students who don’t even realize what they need to know in order to solve the problem placed in front of them.

    I’m with you in your last paragraph and that’s what I was advocating for originally. It’s a blend that we’re after, not only one or the other. Really, it’s the ability to bounce between the two to fit the situation, assignment, or student. Simply the act of talking about what I would do if I were in their position lead a lot of students to a new understanding. Again, it wasn’t a “repeat what the teacher said” effort. They took what I had to offer, put their own spin on it (to which I added appropriate encouragements and questions as they worked it out in the class), and produced something new. For many students, when I did this on the final paper is when they actually got into what they wrote, trying to puzzle out what the author was doing and why. I had students in my room at the end of the day drawing graphic organizers on the board, things we hadn’t used since November crawling out of the woodwork. It was outrageous.

    We can’t remain silent when we see that there is information they should be asking for. Even if they don’t ask for it, there comes a time when we simply must provide it. I like the idea of not saying anything until students realize, “wait a minute, I need to know how many cups that is.” But if a student doesn’t even think to articulate that question, yet I know it’s what they need to figure it out, at some point I need to give that info.

    Your experience in the third-to-last paragraph, exposure to structures and frameworks, is one that I share and is really similar to this final paper/movie/speech I referred to. For a lot of students, they needed to hear some ideas about the stories they picked, how they relate, what the author’s message might be, how the descriptions are similar or different, and the like. Once I set them on that road, they had an idea of where to go. Without that, though, students would be lost and wouldn’t have thought to ask why.

    As for the cause of that, I suspect you’re right and by high school a lot of damage is done. So to the extent we can set up situations to dial back that damage, we should. When we can’t, we need to be prepared for direct instruction that still leads to independent thinking. And for some students, your subject area is too far out of their domain and they will need you to provide what you know they must have. They will never think of it on their own. I know that to be the case in my classroom. That doesn’t lessen the value of things like WCYDWT or your ideas, Michael, nor does it preclude discovery learning; it just means we need to be comfortable with both sides of this. I was never more comfortable with direct instruction than I was at the end of this year, when I used it in moderation along with everything else in that bag of tricks.

  5. Steve Phelps

    July 6, 2009 - 3:03 pm -

    “But from what little I know, Davydov believed that kids should learn abstraction FIRST and then learn the concrete. So the math books present algebraic ideas before arithmetic ones, as far as I understand. ”


    There is a similar kind of idea in geometry (Tatyana Ehrenfest that 3D Geometry should be taught before 2D Geometry. Makes sense…kids experience a 3D world every day.

  6. @MPG – you wrote “What concerns me, however, from what I’ve read, is… what appears to be a totally teacher-centered, instruction-by-telling-only approach.”

    Are you familiar with IMP? ( If not, I strongly recommend it. The series changed my experience as a teacher. Instead of teaching students how to *do* math, IMP helps me to teach my students how to *think* mathematically.

  7. Yes, Touzel, I know the IMP program and some of its authors. I’m talking when mentioning Davydov above about a program that teaches the first three years of math, not high school, however. And what I said that you quoted very specifically refers to that particular program for teaching elementary math. There are more student-centered programs that do not depend so heavily on direct instruction, but their approach to arithmetic itself is traditional from the perspective of what Davydov develped from Vygotsky’s ideas about scientific thinking.

    So then the question is: Has anyone developed a student-centered, less direct-instruction-based program grounded in Vygotsky/Davydov? It’s difficult to determine what’s actually IN the Davydov books unless one is fluent in Russian (I’m not fluent, but can scratch by) and has the Russian books (I do not), because it seems that the person who helped develop the translations of Davydov in 1999 is not cooperative at all about letting others see them. I just acquired a copy of a dissertation by one of her former graduate students that may prove useful, however.