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.

Fully in agreement, MPG. Most of my students spend more time explaining to each other than me, and those conversations pay off massively.

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.

If both sides think they won then BOTH sides are wrong.

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.

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.

@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.

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

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.

@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.

@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.

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,
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.