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.

“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.”

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.

@Dev:

I have a story that might interest you.

Two identical (yet not identical) Pythagorean theorem word problems

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.

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.

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.

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.

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.