Nuggets III: Problem Design

I have to make this one quick because I have a cold and am trying to pack up my life and move uptown. But I wanted to finish my little “Nuggets” series with a thought inspired by Catherine Twomey-Fosnot and Maarten Dolk’s awesome Young Mathematicians at Work books. Twomey-Fosnot runs Math in the City, a math education think tank and professional development center run out of City College. She is writing a new book about algebra with the research mathematician Bill Jacob. I’m excited.

Anyway, people often talk like it’s a choice between developing students’ understanding of concepts and developing their technical ability. My experience is that everybody agrees that the two goals support each other, but there are major differences when push comes to shove in terms of where people believe the emphasis belongs. Maybe you all saw everything clearly from the jump, but I feel I spent a lot of years locked in this dichotomous framework. (Partly because of some experience with curricula like IMP that do a very good job with one of the goals and not the other.) What Young Mathematicians at Work did for me was to abolish the framework.

Nugget: You can develop conceptual understanding and technical ability (for example, computational ability) with the exact same lesson. The secret is to embed the technical instruction in the design of the problems you assign.

It’s necessary to take great care in designing the problems so that they support the development of skills over time. According to Fosnot (and I’ll take her word for it), very few American curricula have given adequate care to the sequence of problems and how it supports this development. My own consciousness was definitely raised by reading Young Mathematicians at Work about the extent to which (for example) the choice of the numbers matters.

For example, consider the following pair of questions:

At Sweet Virgo Desserts, a small chocolate cake costs $7.00. An apple dumpling costs $3.50.

1) How many chocolate cakes can you get for $49?

2) How many apple dumplings can you get for $49?

In 2007, a few months after reading Young Mathematicians at Work, I gave this pair of problems to a classroom of very-weak-skilled 6th graders, who would have balked at #2 (“you want me to divide by a decimal without a calculator?!”) if it had come first. They answered it easily and without any help after being asked and answering #1 first.

The two problems are formally identical. The only difference is the numbers. The important thing is not that #2 is harder; it’s that the way the numbers are chosen makes #1 a hint for #2. It’s also a hint with an applicability far beyond this problem: if n is hard to divide by, would 2n be easier? Pretty soon, the same class was using the technique to solve straight division problems accurately in their heads. (I’ve unfortunately lost the followup worksheet so I can’t tell you what problems; but they were things like 60 / 7.5 and 15 / 1.25.) This is a piece of computational technique; and teaching it this way supported the development of the students’ conceptual understanding of division at the same time that their proficiency with certain division computations was improving. The goals don’t have to be addressed separately.

Maybe you all think this is obvious. But I’m still constantly hearing folks (most recently, a college professor, a former high school principal, and the parent of a mathematically precocious 7-year-old) say things like “but at some point, they just have to memorize those times tables.” Meaning, “all this talk about understanding is really wonderful but you have to admit that there are some things you just have to bang into your head.” I used to be plagued by doubts of this form. Now I’m not. Yeah, you have to learn the times tables, but there’s never a reason to bang something into your head. Can’t remember 6×7? Great, do you know 6×6? How are they related? You go thru that a few times and not only will you remember 6×7 but you’ll be building the groundwork so that later it’ll seem intuitive that 6(x+1) = 6x+6.


17 thoughts on “Nuggets III: Problem Design

  1. A few nifty problems don’t bridge the gap between teaching for skill and teaching for understanding, although there are certainly many places where good problem design can help.

    The curriculum is crucial. Are these embedded in a unit on division? Or a unit on bakeries?

    And teacher skill/proficiency matters, a lot.

    Despite the best of intentions, I know districts that send smart kids to high school knowing strategies and meanings, but who cannot consistently multiply. And other districts, despite their intentions (I hope), whose kids look at a question and think “oh, I know that formula.”

    Anyway, I do appreciate the importance of problem design, but let’s not think that they can bridge the gulf.


    1. jd2718, I guess I wasn’t that articulate – I think you’re missing the point.

      I’m not talking about a few cool problems here and there. In principle I’m talking about designing the entire curriculum to embed as much of the skill instruction as possible in the choice of the details, constraints and numbers in the problems. Ideally, we’d have curricula into which this detailed kind of attention had already been put. Unfortunately, no major textbook series in the US that I am aware of does this adequately. The curricula that are interested in the development of understanding just aren’t paying enough attention to skills.

      I don’t have enough knowledge of foreign curricula to recommend any specific ones but I think that looking at the curricula published in countries that have better math performance would be a place to start. (Maybe I’ll try to do a review later on this blog.) Also, Fosnot has herself recently published some curriculum units. I haven’t looked through them but they might be awesome.

      In the meantime, I have been thinking a lot, and want to get other people thinking a lot, about how to develop skills and understanding at the same time through careful problem design. Everybody (at least in the US) acts like there’s this cosmic rift between teaching for understanding and teaching for skill – your use of the word “gulf” is an example. And because we all think there’s this rift, we produce it. (Hence the districts you were mentioning.) And since lack of skill is an impediment to understanding while lack of understanding is an impediment to skill, everybody focusing on one thing or the other loses some of both. Meanwhile, in Bulgaria, Japan and Holland, they are not talking at all about this great dilemma we are supposed to have, and instead quietly graduating students who are superior to ours in skill and understanding.

      (Fosnot’s co-author in the Young Mathematicians at Work series is Maarten Dolk, who is Dutch, and Fosnot sees what she’s doing as a collaboration with Dutch math education.)

      1. Embedding skill development in carefully constructed problems, that’s what’s being advocated, is wonderful.

        But the large scale debate has not been over individual problems, but over overall curricular conception and design. And not at the level of units, but curriculum. That’s where the choice is.

        Ultimately, choosing IMP or choosing a traditional algebra will determine how we need to design problems – but those of us who care enough to discuss, we make the fundamental choice at the level of new vs traditional curricula, not at the level of teaching for meaning vs teaching for skill.

        And the divide between the two types of curricula is unbridgeable. So tell us which type you’d like to see these types of problems embedded in, and I think there might be some interesting discussion.


  2. (I notice that I adopted the meaning vs skill language in my first comment – that is unfortunate shorthand for what I intended. I hope the second comment clears that up)

    1. It does, thanks.

      The public debate you refer to is played out. My personal experience of it is that it just creates an impasse that makes it impossible to actually talk about teaching and learning. I began my career as a teacher at a big old public high school where the math department was deeply fractured along the lines of this debate. Four years of math department meetings and it was the exact same conversation every time.

      I refuse to believe we are stuck with this dilemma in this form. As I mentioned above in a slightly different context, the countries that seem to be most effectively educating their math students do not seem to be obsessing about this supposed dilemma at all. What’s wrong with making students think for themselves and practice?

      The unbridgeable divide I do perceive as a real and serious obstacle is the one between, on the one hand, the insane incoherent laundry lists of topics that state frameworks ask us to teach in a year, and on the other hand, the logical coherency, and the sense of expansiveness about time, that it takes to give mathematical content its due. Cultivating the combination of understanding, resourcefulness, and technical skill requires an unhurried rhythm. (Such as the one you described in your post about the sums of consecutive integers.) A rhythm I like a lot, if I can think of the right problems to make it work, is:
      1. provocative problem
      2. reflect on the lesson of the problem
      3. use the lesson of the problem to solve lots more related problems to consolidate the new skills and insights
      4. one of these new problems is provocative in a new way
      and repeat.
      But under time pressure, and lack of logical development in the ordering of topics, this can easily give way to either
      1. “here’s how you do it” lesson
      2. boring practice
      and repeat,
      which leads maybe to some skill development but no understanding, or to
      1. provocative problem
      2. cool conversation
      3. new provocative problem but no practice of the insights and skills from the old one, leading maybe to some understanding of some ideas but to no skill development.

  3. Oh, jd2718, I just realized a possible meaning to your question that I hadn’t gotten before. When I talk about problem design, I am talking about the problems that drive the curriculum. I’m not talking about problems 5,6 and 7 on tonight’s homework; I’m talking about the problem or question that opens class. So what I have in mind does involve having the instruction be problem-driven.

    I student taught from the IMP curriculum, which is problem-driven but the problems are mostly focused on developing ideas and not skills. What made Fosnot’s work such a revelation to me was that she is talking about problem-driven instruction where the problems are being carefully constructed to develop skills and not just ideas.

    1. And I am understanding better.

      However, the question of how the curriculum is organized is real. What curriculum are we “driving?’

      The math wars have not played out. (Had they, we could talk about the resolution. We cannot.) The battles don’t make the front page, but they continue to erupt; the advocates of the newer stuff have not quit.

      Early on in my involvement I decided that traditional curricula could be made tolerant of innovation, of – and you’ll forgive me – teaching for meaning.

      For ten years I have advocated using traditional curricular organization, old-fashioned courses, but choosing exercises, problems, doing off-topic problem-solving, etc. I can see the sort of problem you are underlining fitting quite well into that sort of framework. I am not sure, however, that I can see that sort of problem guiding the curricular development.

      I certainly cannot see any of the new curricula (IMP, but much more than IMP) being part of that.


      1. jd2718, thanks for your continued thoughtful engagement! I kind of feel self-conscious responding at this point – I feel like we should be doing this over email by now. That said –

        By “played out” I meant “tired,” not “over.” You’re right, they still rage. A teacher I work with just forwarded me this article from only ten days ago, for example. I just mean I’m done with it.

        I have a lot of respect for both sides of the “math wars” and I am very frustrated that conversations about math education so easily crystallize into these polarized camps and then nobody can say anything without infuriating everyone else. Like anything so hotly debated, both sides have a point. Traditionalists: practice is important! Reformists: thinking for yourself is important! Both sides think the other side is literally out of their mind because they are not banging a drum for the same thing. But sane effective math teaching clearly requires both points. You are identifying yourself as a traditionalist (if you had to pick a side) and yet it’s obvious from your blog and your comments here that you go to great lengths to make sure students are thinking for themselves. I would probably be identified as a reformist by what I bang a drum about, but anyone I’ve worked with can tell you the value I put on practice.

        I disagree that the only way to make this balance work is to start from a traditional curriculum and supplement with well-chosen exercizes and problems and off-topic problem solving. I think you can start from the other side too; as long as you’re beholden to the twin goals: a) make them think for themselves, and b) make them practice. Now that you’ve got me excited about reviewing some foreign curricula, I actually expect that they will provide starting points that are even more amenable to the twin goals.

        I actually think the IMP curriculum is brilliant, though highly flawed. I wouldn’t want to teach from it but I love to mine it for problems and inspiration about how to develop various content. I also really like TERC – it has some brilliant ideas for embedding skill practice in games. And, on the other side, I love the Dolciani textbooks because they have such large, awesome sequences of exercizes, even though their content exposition practically begs for students to turn their brains off. The only textbooks I have no respect for are the ones like Prentice-Hall that don’t have a coherent vision of any kind and are trying to be everything to everyone.

  4. Ben, I am loving your posts!

    You said you hadn’t seen any curricula that take both concerns into account. I have trouble seeing the big picture when I look at actual curricula, so I’m curious what you think of the Singapore curricula. I’ve used it some, but not in the way it’s meant to be used.

  5. Ben, what do you think of Everyday Math, the series that drives the anti-reform folks crazy? (Maybe it’s just elementary level, and you don’t know it well?)

    I haven’t had enough experience with any of these curricula to know what I think of them. I see myself as leaning toward the reform end of this issue, but agree with just about everything the two of you have said.

    And my experience with college-level calculus texts is that one of the most reform (long time ago, I think it was out of North Carolina?) looked hard to teach from. I want the textbook to back me up, not to tell me how to teach. I’ve never been in a department that loved reform, so I’ve taught mainly from middle-of-the-road texts, I think.

    I find it very difficult to decide what makes a good text before using it.

    We were using Triola for statistics, and I didn’t like it. I kept finding good bits in the Sullivan text and pushed for us to adopt that. We did, and other teachers are hating it, for reasons that made sense to me. Years ago, I picked a book that looked good, and hated it myself. I wonder if there are ways to doing a better job at this.

    1. Sue, I hear what you are saying about adopting a text and the difficulties in doing so. Each teacher has their own style of teaching and their own ideas of what their students need.

      I have often been at the center of this debate with my department. What I keep trying to tell my peers is that as much as we like to do it and easy as it makes our jobs seem, the textbook is not the curriculum. We, the teachers, need to decide what suits the needs of our students and if they are having difficulty with skill level problems, we need to give them more practice before moving on.

      I’m starting to move towards the idea of teaching independently of any book, just to help myself focus on what the students need and where they want to go next. I’m doing a bit of this in my Linear Algebra class. Using online texts makes it easier for me to see what the students need and where their thoughts lead them is much more important than the topic covered in next section.

      1. I definitely want to move away from the textbook too. I’ve started working on how I want to teach beginning algebra.

        But I needed to work from the book (for years) until I could see the big picture. We’ll always need to select what we hope is a good text, as a starting point, at least for newer teachers.

  6. Ben, I just discovered your blog. Stimulating discussion. I don’t read many blogs, but you’re on my list now.

    Are you familiar with Vygotsky’s work? The use of diagrams as psychological tools helps students represent mathematical relationships, see the mathematical structure of a problem and logically solve it. The computational steps required to solve the problem flow naturally out of the logical analysis. Students still need to practice, but there is a blending of concept and computation. The Russian curriculum of Davydov, based on Vygotsky’s theory, only covers grades 1 to 3, but it is very effective. One reason for that is the curriculum design; another is the teachers are good. I don’t think we can adopt a good curriculum and get good results without teachers who know math well. My own view is that the teachers are more important than the curriculum, and the evidence is that the majority of elementary school teachers don’t understand math concepts.

    Davydov’s curriculum was brought to New York by Jean Schmittau, but she seems to be very protective of it. It is also used in the University of Hawaii Lab School. You might find this article of interest: Jean Schmittau, Vygotsky Theory and Mathematics Education: Resolving the Conceptual-Procedural Dichotomy, in European Journal of Psychology of Education, 2004, vol. XIX.

    Singapore and Japan score high on the TIMSS math tests. Both countries use diagrams as psychological tools in the Vygotskian sense, although the diagrams are different from the ones used in Davydov’s curriculum. Singapore math relies heavily on block diagrams. You can get a good sense of their use here. The Japanese use tape diagrams which are essentially the same as block diagrams, but they also use the tape diagrams with the number line, and double number lines for proportional relationships. Tad Watanabe has written articles on Japanese math education, and will have an article in the 2010 NCTM Yearbook on visual representations from Japanese textbooks.

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