We’re continuing to host commenters from across a vast philosophical divide (including the co-authors of The Atlantic article under discussion) commenters who are unlikely to share the same physical space any time soon. People have largely kept it together and you’d have to be a committed ideologue not to walk away with a better understanding of the people who disagree with you.

I haven’t been able to shake one particular exchange, though.

Halfway through the comments, two people who disagree with each other as completely as anyone could each made a precise and articulate case for their diametrically opposite theories of learning.

Ze’ev Wurman, a longtime advocate of traditional math instruction:

I thought the purpose of a problem in a classroom is to check whether a student knows sufficient math to solve it, rather than learn bout the nature of human thinking processes. If it is the latter, Dan is completely right, except it belongs to cognitive science experiment rather than a classroom.

Brett Gilland, an infrequent blog commenter who should comment more frequently:

I can not disagree with this enough. The purpose of a problem in my classroom is almost always to understand the nature of that humanâ€™s thinking processes. This allows for amplification, further investigation into how the student is able to navigate similar problems with subtle variations and complications, and attempts to draw student mental models into internal conflict to create pressure for remediation and revision of said mental models.

I suspect that Gilland’s employer, and certainly the parents of his students, would also disagree. Some quite strongly. The primary purpose of school is to educate the kids at hand, not to train the teacher. This doesnâ€™t mean that teachers do not learn from experience, but if gaining experience and insight is the primary reason for what the teacher does, heâ€™d better get approval from an IRB and a waiver from each individual student or parent that attend his class.

Funny thing, that. My employer, my parents, my students, my district, the state evaluator for my school, etc. all support my teaching. Most quite strongly. This might be due to the fact that when most people hear â€œI work really hard to understand your childâ€™s thought processes so that I can better guide their thinking and draw out subtleties and conflicting mental models,â€ they donâ€™t think â€œOh my God, that man is performing experiments on my child to improve his educational practice.â€ Instead, they think â€œOh my God, that man really cares about what is going on inside my childâ€™s head and is attempting to tailor instruction to what he finds there. Thank goodness he isnâ€™t stuck with a teacher who believes that teaching is just lectures interspersed with quizzes to determine if my child gets it or needs to be droned at more with another utterly useless generic explanation that takes no account of what my particular child is thinking!â€

I’m sure that everyone walked away feeling like their side won, but one side is wrong about that.

**Featured Comments**

â€œEvery school should be organized so that the teachers are just as much learners as the students are.â€ (Adding It Up, 2001, pg. 13)

Alas, I feel that Mathematics is reaching a junction â€“ in which the traditionalists and the progressives must come to a head and work together to forge a stronger future for our young mathematicians! Whilst todayâ€™s world demands an ability to think and to use available resources to find new meaning, we must not forget those who generated those resources in the first place. a fine craftsman needs to learn the tools of his trade before he or she can produce the creative thinking in his head. A computer programmer must use efficient logic before we can play those game or use those apps to be progressive learners. To what degree should thinking, reasoning, and problem solving come before skills acquisition , or vice versa? Sound like the chicken and the egg to me.

My biggest fear for us as teachers is that we are robbing our young people of the beauty and passion for maths through repetitive, applied drill, just so that they can demonstrate a high level skill skill that they will never use. or, we are not providing enough technical skills to enough of our students to ensure quality craftsmanship I am saddened every single time that my best mathematics students tell me they want to study medicine or dentistry instead of using their mathematical ability to grow our field.

Let us first and foremost provide our students with mathematical challenge- that requires both the creative solution and advanced skills acquisition. Whether the challenge be abstract or modelling a real situation need not matter. What is important is bringing back the passion for mathematics that we, as mathematics educators share, passion- through intrinsic motivational challenge and drive.

## 30 Comments

## Keith Devlin

November 14, 2015 - 7:51 pm -Wurman’s view made sense before we had readily available software like Mathematica, Maple, Wolfram Alpha, etc. It’s how I and everyone in my generation were taught. Those days are gone. In today’s world, what mathematicians provide is a unique and valuable way of thinking. Gilland seems much more it tune with the world today’s students live in and will work in.

## Michael Paul Goldenberg

November 14, 2015 - 8:26 pm -@Keith Devlin: Hear, hear! It’s good to know that some of us 60-somethings are able to see past our own educational experiences and appreciate positive aspects of contemporary thinking, technology, etc.

I just wanted to tag onto the second quotation from Brett that it’s not just that teachers gain from probing student thinking but that if we educators do things effectively, that information feeds back into student self-reflection so that eventually we are needed less and less for students to advance more and more.

## Brett Gilland

November 14, 2015 - 10:17 pm -Fully in agreement, MPG. Most of my students spend more time explaining to each other than me, and those conversations pay off massively.

## Rene Grothmann

November 15, 2015 - 1:58 am -Teachers teach the best way students should think about a problem. Or rather, the best way to sort out and solve a problem. If students handle a problem in inefficient or even wrong ways, teachers should know that. So, yes, they should discuss the thinking process with their students.

But, math is math. It is not psychology. Having the right attitude helps. But you need all the little tricks that are needed to solve real life or classroom style problems. You need to learn a language too, the language of math. Again, obviously, you need to check if your students are on the right track. In my opinion and experience, you need to guide on that track too.

## Stan

November 15, 2015 - 3:37 am -No, At least one person, myself, walked away thinking both of these guys need practice in logic and rhetoric and each was a terrible spokesperson for their side of the debate.

Wurman is wrong that the only use of a problem is to check knowledge. He is also wrong that an attempt to understand another’s thinking amounts to a cognitive science experiment along the way to providing an answer to Searle’s Chinese Room problem.

It is perfectly legitimate to explore a students understanding. The question raised in the Atlantic article is about how you do this. As it says perhaps asking them to solve more problems will be a better approach than others.

But Gilland also gets a fail. Has he really never been to an interesting lecture? He should try some of Devlin’s online ones. And Gilland ignores a major item in the Atlantic article. A teacher over emphasizing checking for understanding through student explanation work can make math very tedious for otherwise talented students.

Gilland should also worry that the sort of confidence he expresses in himself and all those telling him he is right is the same we hear from those saying buy just before a stock market bubble bursts. If he is really so clever he should know ad-populum is a poor argument even if the other guy started that line.

## Andy Shores

November 15, 2015 - 4:35 am -@Stan: I don’t think Gilland isn’t saying, “People like what I do so I must be awesome.” Rather, he is countering the claim that the parents of his students are concerned that he is experimenting on their children.

While I agree that there can be too much of a good thing like student explanations, I would rather err on the side of too much searching for the level of student understanding.

## David Griswold

November 15, 2015 - 8:05 am -If both sides think they won then BOTH sides are wrong.

## Maria

November 15, 2015 - 8:17 am -@Dan:

“Iâ€™m sure that everyone walked away feeling like their side won, but one side is wrong about that.”

I don’t understand why one side is wrong, I strongly believe that BOTH won, both are right!!

I understand and agree with all the arguments for both theories, this is why, since long ago, I use both methodologies in my classroom: I teach basic math trough traditional master classes, and then I put questions to my students so that they need to use their reasoning and what they’ve learned.

Why does it work so well, at least to me? Because my students with less autonomy need traditional teaching to have a guide, and my “adventurous” students need an “extra incentive” to test themselves.

So why keep arguing and not take the chance to learn the best of both strategies?

## Chris H

November 15, 2015 - 10:47 am -## jennifer potier

November 15, 2015 - 3:25 pm -My biggest fear for us as teachers is that we are robbing our young people of the beauty and passion for maths through repetitive, applied drill, just so that they can demonstrate a high level skill skill that they will never use. or, we are not providing enough technical skills to enough of our students to ensure quality craftsmanship I am saddened every single time that my best mathematics students tell me they want to study medicine or dentistry instead of using their mathematical ability to grow our field.

Let us first and foremost provide our students with mathematical challenge- that requires both the creative solution and advanced skills acquisition. Whether the challenge be abstract or modelling a real situation need not matter. What is important is bringing back the passion for mathematics that we, as mathematics educators share,passion- through intrinsic motivational challenge and drive.

## mike

November 15, 2015 - 3:51 pm -Does any side have a claim to authority of truth in this? Is not the best way to teach simply the best way that works, regardless of how one might feel him or herself about what should work?

I think it’s rather telling that while hosting both sides of an argument, Dan concludes that one is just plain wrong. Interesting, and unfortunate.

## Dan Meyer

November 15, 2015 - 4:24 pm -I added comments from

Chris Handjennifer potierto the main post. Thanks for the contributions, everybody.mike:Even if neutrality were possible, it isn’t one of my obligations as host.

## Mike

November 16, 2015 - 4:20 am -As you might well say yourself Dan, if you’re going to be dismissive of a line of thought, you might at least provide justification.

“Show your thinking” and all that.

But your not obligated to do this either.

Just says something when you don’t.

## Michael Paul Goldenberg

November 16, 2015 - 4:47 am -@Mike: Dan’s been showing his thinking for all the years he’s been writing this blog. It’s not up to him to repeat it for you now, is it?

## Stan

November 16, 2015 - 6:47 am -@Andy,

I can see how you might see it that way but it is still an over confident ad-populum response that would be better argued by giving evidence that what he is doing is in line with the stated expectations of the various stakeholders.

Depending on how you ask the question you would get two answers. Sure trying to understand my child is great. But making experiments in cognitive science to get insight into what is going on in the student’s head the primary purpose of a classroom is going to get the opposite answer.

It shows a complete lack of mathematical awareness for people to keep falling into: the extreme of my opponents idea is obviously bad therefore it follows (because all things are linear) that the alternative extreme is a great idea.

Even if people think they are not taking the extreme alternative position the argument that more of X is better and I do more so I am right is a fallacy. Unless the issue is linear more is better is only true on one side of an optimum.

Do people think any of these tradeoffs have a linear benefit curve? Time spent instruction verses time spent discovering, Time spent explaining verses time spent practicing more problems.

## Matthew

November 16, 2015 - 8:40 am -I think both sides have their moments, and both sides are off-base. It’s in the combination of the two that the sweet spot lies.

Learners need to know their basic facts – yes, from memory. Some things don’t necessarily need to be explained, but applying that knowledge to problems will definitely need that explanation in order to heighten a learner’s rigor.

It’s about the thought process behind solving problems that we as teachers want to know, so we can help them work through it.

## Dan Meyer

November 16, 2015 - 9:13 am -Clarifying: the argument in this post has nothing to do with procedures, application, memorization, lecture, or any number of other flash points in our ongoing discussion about math education. The argument here strictly relates to our motives for assigning math problems.

## Kenneth Tilton

November 16, 2015 - 9:44 am -Maybe I am wrong, but the only place I see “someone is wrong” is in Mr. Wurman’s takeaway from what Mr. Gilland wrote.

Mr. Gilland does not seem to be doing cognitive science on students, he just seems to be doing what any good teacher does: throw kids a curve to test their understanding himself or so they discover themselves if something is wrong with their mental models.

Call it “Benny Detection”, perhaps? http://www.wou.edu/~girodm/library/benny.pdf

## Kevin Moore

November 16, 2015 - 10:28 am -If one is charged with teaching, learning,then, must be deeply understood. And as learning is a complex process, constant, keen observations of learners is necessary. Since truly learning mathematics involves an investigation of ideas, teachers must be committed to developing understanding. How can a student’s understanding be determined without thoroughly observing his/her thinking?

## Michael Paul Goldenberg

November 16, 2015 - 10:40 am -@Stan: I believe that if you look at the history of US mathematics education, you’ll be hard-pressed to find an era in which many teachers “erred” on the side of “discovery,” concepts, student explanations, etc., over procedures, lecturing/teacher explanations of everything, computation.

But mathematics education is not a zero-sum game. There is room for a wide spectrum of teaching and learning – and this is key – even in the same classroom (and even at the same time in the same classroom).

I suspect anyone who argues that there is a single best way that suits everyone all the time, or who acts or teaches as if that were the case. That means that extremists are not trustworthy. They tend to be fanatics, lazy, ignorant, or a combination thereof. That’s not pointing a finger at anyone here; it’s a reflection of my own teaching practice and my practice as a student teacher field supervisor, professional development instructor, and especially as a content coach.

So once again I point back to the NCTM Standards volumes from 1989-93, not as models of a perfect set of standards for teaching, assessing, training teachers, etc., but as a model of a reasonable proposal for change: less emphasis on X, more emphasis on Y. If only that had been honored more in the observance than the breach, we might not be having the same (mostly) tiresome, pointless argument more than a quarter century after the first volume appeared.

## Mike

November 16, 2015 - 2:46 pm -@Michael

No it’s certainly not up to Dan to repeat arguments to me.

And it’s certainly not up to Dan to do so for anyone who reads this blog, when he makes a gigantic claim with no reason provided.

Especially for new readers, nor those who might wonder what aspect of reason he’s referring to, nor those who might wonder if his reasoning has evolved at all, nor for those who simply share a different opinion, nor for those who do research in this field and arrive at different conclusions than he. Most especially not them.

Because this is, after all, just Dan’s blog.

You’re right.

## Michael Paul Goldenberg

November 16, 2015 - 2:54 pm -@mike: Glad we see eye to eye on that. [an example of Sarkham’s Razor]

## Michael Paul Goldenberg

November 16, 2015 - 3:09 pm -@Dan: the “why do we assign problems?” reminds me instantly of Michael de Villiers’ Rethinking Proof With the Geometer’s Sketchpad in which he offers about half a dozen reasons for doing proofs, only two of which would fit most folks’ routine thinking.

Since I’m an atheist when it comes to traditional grades and assessment, at least one of the standard justifications for assigning problems cuts little ice with me. Instead, I’d list, off the top of my head:

1) Giving students the opportunity to check their understanding of taught (and, one hopes, learned methods, concepts, procedures, etc.);

2) Same as above but with the emphasis on pushing beyond strictly what’s been directly taught to at least some degree of extending to unfamiliar but related notions and examples;

3) Giving the instructor information on 1 & 2 for purposes of providing individual constructive specific formative feedback: a) what did you do that appears to have worked? b) where might you need to think more, practice more, try something else; and c) what might you try to take things to a next level of depth?

4) Providing challenges for students to really stretch well beyond what they already know well (their safest zone of proximal development), including making connections among more than one idea, method, concept, etc., making connections between computational problems and the underlying mathematics and something outside of pure mathematics [What can someone do outside of math class with this?]

I believe there are other good reasons, but I’m getting sleepy. I’m sure others will add their own. My point is that summative grading is not the only or even the best reason, and it’s quite possible to teach well and meaningfully with little or no emphasis on summative assessment, grades, and the rest of the rewards that many of us have be miseducated into believing are THE prize instead of the booby prize.

## Dev Sinha

November 16, 2015 - 3:41 pm -Thanks, Dan. This does crystallize the difference. Some people care about what goes on in students’ heads, and some don’t.

The moment it became clear to me that what is in students’ heads really matters was when I learned about the Wason Selection Task. See:

http://pages.uoregon.edu/dps/cognitive.php

At first I didn’t believe it – I told the cognitive scientist who brought it to my attention that it must be flawed. If our brains are “math machines” (an over-simplification of my model at the time), how could they respond differently when presented logically equivalent problems?

That moment has led to almost 20 years now of considering the interplay between cognition and mathematics content, including at the research mathematics level, where for example I structure research seminars for graduate students with some of the same things in mind as other posters here do for their K-14 students. When it comes to quality of pedagogy, I don’t see two valid sides here, just as I don’t see two sides in the evolution or climate change “debates.”

Look, in one approach to multi digit multiplication, students through area models are also given tools to extend the method to multiplication of polynomials (or other quantities/ functions which come as sums), and then also have a useful picture they’ve worked with to build on to understand the product rule in calculus. In another approach, they’re given none of that, and typically have to make such connections on their own, which they typically don’t.

At the end of the day, I’ve gone so far as to think of mathematics as more of a human practice rather than a subject, even at the research level. What matters is less what is printed in our libraries (though I value that) and much more a community of people who can understand, use, and extend what is there. Take some topic in math from 120 years or so ago which is no longer researched (some study of special determinants or…). If all of the articles on that topic went up in flames, would it be “lost”? Absolutely not (or, I’d allow, what’s lost would be an understanding of math history) – we have people who know either deeper theorems which can reproduce much of that, as well as a much wider community (say at the College Math Journal level) who could “figure out” just about anything at the level done at that time.

I admit that models for mathematics in philosophy which may be independent of human thought is an open topic up for debate. But when one cares about learning and starts at all to account for literature on cognition, then one is forced to include student thinking as a central concern.

## Stan

November 17, 2015 - 6:01 am -@Micheal,

Unless you believe that what choices teachers make has no effect on student outcomes it is exactly a zero sum game.

You have 60 minutes (or whatever) to spend with a class. If you spend it on using problems to gain understanding of students thinking by having them explain in detail what they do and then spend the time to review each of those explanations you don’t have time to spend on another problem where all you do is provide the correct answer to the student and then have the student do more problems.

One approach has doing problems for the benefit of understanding the student’s thinking.

The other approach has doing more problems for the benefit of the student getting better at doing problems by practice.

It is unlikely that either extreme – work through and review all problems with explanations or do more problems but with just answers is optimum. It is likely the optimum varies with the student and topic.

You might argue that both approaches have some benefit so neither produces zero output but in terms of the available time if you spend more time on one you take exactly that time from the other – zero sum of available minutes for the other approach.

To your other point for an individual class the answer that in the past most teaching has been done one way doesn’t make a difference. Unless you don’t care about individual student’s outcomes you should worry that a teacher getting it wrong and going too far in either direction is inferior for at least one year of a class’s math education.

## Jason Dyer

November 17, 2015 - 7:54 am -If our brains are â€œmath machinesâ€ (an over-simplification of my model at the time), how could they respond differently when presented logically equivalent problems?@Dev:

I have a story that might interest you.

Two identical (yet not identical) Pythagorean theorem word problems

## Michael Paul Goldenberg

November 17, 2015 - 8:07 am -Stan, please. This isn’t my first rodeo in mathematics education. I’ve been working as an educator since 1973, and in mathematics education since the late 1980s. My point is that there’s no absolute reason that teachers must go to one extreme or the other. But in fact, the probability of finding a teacher who is very close to 100% of class time doing teacher-centered instruction with zero time spent on examining student thinking in class is far closer to 1.0 than is the opposite: finding a teacher who spends close to 100% of class doing student-centered learning/instruction with nearly all the time spent on examining and sharing student thinking, which is in fact close to 0.0.

The best teachers I’ve seen are not at either extreme, but as suggested by Brett, “err” on the side of spending time not lecturing. That said, how the time IS used can vary from day to day. Some folks are sold on the so-called “flipped” classroom model. Some do mostly group work. Some are engaged in what might be called a Socratic model. And there are many other approaches which may be part of a mix or a main focus. On my view, the particular blend varies because teachers and students vary, and because despite suggestions to the contrary, teaching is not a science, even if it can be informed and improved by science. We’re looking at systems of interactions among people that are enormously complex. If you want a useful metaphor, perhaps the many-body problem comes close: you’re just not going to find any simple way to predict what’s going to happen in any given situation such that a teacher can even come close to knowing the “right” next move. On my view, there are many good moves in a given situation, many less good ones, and teachers who select frequently from the first group are ahead of the curve. Better than that? I think you’re dreaming. Reducing the choices to two? That has nothing to do with the reality of any classroom I’ve ever been in under any circumstances and in any role.

I’m unwilling to engage in conversations about education that hinge on my accepting mechanistic or reductionist models that might sound fabulous to engineers, but make no sense to experienced, reflective educational practitioners. And if you insist that what’s happened in the past in mathematics education in this country has no bearing on what’s happening now or should happen tomorrow, I’m not sure that it’s possible for us to communicate meaningfully.

## Michael Paul Goldenberg

November 17, 2015 - 8:14 am -Jason Dyer: I just left this comment on your blog:

“In the real world, the problems are not the same. In an idealized mathematics world, they essentially are. So in the first case (real world) I’d say that neither has an answer of exactly 5. In the second (idealized world), both have an answer of 5. Thus, I’m not so sure that it’s reasonable to judge whether those students really got either problem right or wrong. But the tasks would be very useful in mathematics and mathematics education classes. The conversations they engender would likely be extremely useful if the teacher has developed a classroom culture that promotes meaningful, respectful discourse.”

## Stan

November 17, 2015 - 10:00 am -@Michael,

I am not asking you to accept a particular view. I am simply trying to make the point that there is a tradeoff and whatever the uncertainty there are likely discoverable bounds to what optimal and a danger that people won’t be aware of that.

My example of the tradeoff between two purposes for problems used just two as I was just trying to illustrate that even with two purposes there is a tradeoff in available time.

As you point out there are more than two and so this makes the problem more complex.

But there is a tradeoff. The more you do of one thing the less time you have for others.

I would easily agree that there is no way to determine the perfect tradeoff. But is it possible to bound the problem? Would you agree it is possible to say that in particular cases more emphasis on one priority will produce a worse outcome.

My point about the past is that some people see an over correction as the current problem. There is no way to argue with them that because it was too far in another direction in the past the pendulum has not swung too far. The only way to address a complaint about too much emphasis on -using problems for understanding thinking through explanations- is to look at what people argue for and do today.

Keep in mind children are not taught just by highly experienced teachers. They are taught by novice teachers and those that just follow what they are told and that education boards might jump onto new ideas without always ensuring everyone is aware of any problems of over doing it.