[LOA] Hypothesis #2: Paper Is A Problem

#2. Print-based tasks often obscure the process of abstraction.

I advanced a hypothesis in my last post that we don’t clue students into the everyday abstractions that come so easily and subconsciously to their teachers. We find it easy to represent contexts (applied or pure) with symbols, tables, line drawings, and coordinates, so we often glide over those processes, obscuring them in the process.

So it’s helpful to give students a concrete context and explicitly show them how to climb the ladder of abstraction at every rung.

For example, when I worked with teachers on Popcorn Picker last week, a task that starts without any mathematical abstraction whatsoever, just a video, I marveled at different times at our work on the board and on their papers. “There’s no popcorn here,” I’d say. “Where’s the popcorn? You took that video and said, ‘The color of the wall doesn’t matter. The actual items filling the cylinders doesn’t matter. The guy filling the cylinders doesn’t matter. This is all that matters.”

It should go without saying that if the contexts in your textbook are predigested with those symbols, tables, line drawings, and coordinates, we’re already in trouble. The context has already been abstracted and we can only hope that every student already understood how to apply that abstraction.

My hypotheses here is that this predigestion is a fundamental condition of print-based curricula and very hard to counteract. For example, here again is Pearson’s cell phone tower problem with a presentation that conceals the ladder of abstraction.

Let me offer a presentation that would reveal the ladder. We would start with the satellite view of the cities, a low level of abstraction.

Then we’d move up to the ladder to a road map, clear-cutting forests, damming streams, getting rid of information that isn’t relevant to our question.

Then we’d abstract away most of that information, leaving behind three points on a plane with their labels.

To talk about the location of those points, we’d put a coordinate plane beneath them.

We’d consider each of those four frames separately. We’d move on to the next frame only after we had discussed the abstraction required to get us there.

With digital media, those four frames cost nothing but a few extra bits on a hard drive. But if I print each of those frames out on its own page and then bind those pages into a book and then mass produce that book, those four frames become very expensive and very heavy very quickly.

So instead, print-based curricula compress all those frames into one. They default to a very high level of abstraction and hope that everyone is already comfortable working at that level. It’s an expensive problem to fix in print.

This isn’t to say that print-based curricula isn’t great for a lot of things. This is just to say that making the ladder of abstraction clear to students isn’t one of those things.

2012 Sep 20. Brian Stockus starts a series on the issue.

Featured Comment

Brian Stockus, who works in the industry:

The other issue with print-based materials is that they can’t control the release of information very well. Generally a problem and any accompanying pictures or sample work are printed right next to each other. What if the person who wrote that problem wants the students to look at the sample work *after* they solve the problem for themselves? Well, it’s hard to make students do that when a sample solution is mere centimeters away from the problem.

Or in the case of the four images you suggest to use to help students abstract the cell tower problem. If they’re printed on the same page, the students can see all four steps at once which is inherently different than revealing one at a time and discussing each in turn.

This makes a case for the added value of digital tools in education. Not only is a video more dynamic than text, but you also have the ability to pause and rewind. Even using software like Powerpoint is more powerful than print in this regard because you can control when information is revealed. The same holds true for asking students questions on the computer using some sort of educational software. The instructional designer can control student movement so that an abstraction can be revealed in steps or after students have had the opportunity to think it through on their own first.

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

18 Comments

  1. Wouldn’t it make more sense for the origin to be located at one of the cities? (Of course, with the right tech you could have the students lay down the grid wherever they want. Maybe they prefer positive numbers.)

  2. This series of posts is really helping me to put my own head around why I am such a proponent of the 3-Act approach to problem solving.

    Your metaphor of predigestion is an apt one. In the Pearson example, all of the abstraction has been done for them. There is no “problem”, there is just a template set in a pseudo-real-world scenario, with the sole purpose of practicing the distance formula (which should be taught as an application of the Pythagorean Theorem)

    I’m looking forward to the next three posts!

  3. I think it is amazing how similar the reform is in other subjects to the reform I see trying to happen in my subject, foreign language. You discuss here the problem of print textbooks and how for mathematics they make it impossible for students to do their own abstractions. In foreign language, a print textbook makes a story or passage permanent. But what we are finding in our subject is that giving the students control over the story increases their engagement and comprehension of the “text.” This kind of control is not possible if stories are in ink, and it thus becomes impossible to “retell” a story or make it one’s own.

  4. The other issue with print-based materials is that they can’t control the release of information very well. Generally a problem and any accompanying pictures or sample work are printed right next to each other. What if the person who wrote that problem wants the students to look at the sample work *after* they solve the problem for themselves? Well, it’s hard to make students do that when a sample solution is mere centimeters away from the problem.

    Or in the case of the four images you suggest to use to help students abstract the cell tower problem. If they’re printed on the same page, the students can see all four steps at once which is inherently different than revealing one at a time and discussing each in turn.

    This makes a case for the added value of digital tools in education. Not only is a video more dynamic than text, but you also have the ability to pause and rewind. Even using software like Powerpoint is more powerful than print in this regard because you can control when information is revealed. The same holds true for asking students questions on the computer using some sort of educational software. The instructional designer can control student movement so that an abstraction can be revealed in steps or after students have had the opportunity to think it through on their own first.

  5. Hi Dan…

    I’ve been thinking a lot about this since you started blogging about the limitations of print. I’m wondering if the problem is the print itself, or the fact that (in many math classes) the print itself serves as the lesson. Many of the great lessons I have seen start with one question/problem. As that question unravels, the teacher helps create discussion by making certain student work visible or by probing with appropriate questions.

    In other words, it seems when we think of text/curricula we tend to think of it as independent of teacher and classroom community. This sort of “teacher-proofing” devalues the art of teaching, creates a more shallow mathematical experience for students, and has led to the rise of online instruction and classroom “flipping.” The text/curricula/video/image, I think, needs only be a starting place. From there, it is up to the teacher to respond to the thinking of the community of students they are with (assuming we are talking about problems based in sense-making and not merely exercises). I’m curious about your thoughts on this. Do you think the problem is print curricula or is it the role that curricula has been assigned?

  6. Bryan,

    There’s so much to what you’re talking about that I don’t think I can do it justice in a comment. I might have to write about it myself! All I can say is I totally get what you’re talking about. I’ve been an instructional designer with a small digital curriculum company for the past three years. It’s one thing to design a lesson for yourself or for your teammates, which I did many times as an elementary school teacher for 8 years. It’s a whole different ballgame to design lessons that have to be usable by people you’ll never meet or talk to. These materials are made because a market exists for them, but there are definite trade offs when you teach with materials designed for general use rather than materials designed within and for a specific context.

    To quickly hit your question in a few points:
    * Print is not inherently bad. Some is bad designed and some is well designed. However, it does have trade offs. Once example is in my previous comment about how print materials have a difficult time controlling the presentation of content.

    * Curriculum materials are designed to be usable by as many people as possible. If they are too difficult to use, then people will not use them, they will not be bought, and the company producing the materials will hurt financially. As a result, decisions are made that do not always align with best practices. On one hand I fault companies, but on the other, if teachers aren’t well versed and/or comfortable in best practices, then can you blame them for not wanting curriculum materials that they don’t *want* to use?

    Anyway, I could go on, but I’ll stop there for now. I hope it does give you a bit of insight into what you were asking about.

  7. Brian…

    Thanks for your response, thoughts, and insight into curriculum development. I don’t want to impose myself too much on Dan’s blog here, but I think the conversation is interesting. I’m just not sure that the release of pieces of a lesson should be in print (or video, or otherwise). Possible trajectories that students might take with a question/task/problem can be hypothesized, but never predicted with certainty. For this reason, I don’t see how we could ever prepare a scripted curriculum in this sense….it should always unfold in response to students and their ways of thinking.

    I have the benefit of working at a school where students don’t use textbooks and teachers are curriculum designers for their classes. I know that not everyone is in the same position. But, it seems that improving curricular materials is only a small part of improving teaching.

  8. Interesting points by Bryan & Brian.

    Usually lessons in books or pre-packaged worksheets (reform or traditional) severly limit the ability of the teacher to control the release of information. I think that has less to do with paper and more to do with the way U.S. math teachers want to teach. I agree with Bryan that this also has to do with some folks wanting to teacher proof lessons.

    Japanese lessons are paper/chalk board based, and designed to take a very specific path, but the information is released in a way that keeps the path hidden for much of the lesson.

    Japanese students also are much better at taking notes, which allows for a little more flexibility than handouts.

  9. Dan, are you reading Mark Guzdial? I keep finding overlap in the topics you are writing about. See especially the Abstraction Transition Taxonomy and Contexts and Problems Come First, as well as some posts about trying to solve some of the problems with paper but having a hard time getting student uptake.

    For an interesting experiment in how to abstract in a way that makes the thought process visible, I’ve been enjoying the blog What If, by the people who brought you XKCD.

  10. I buy the criticism of the paper texts, and I see how your digital presentation is an improvement, but it doesn’t seem to be comparing kind with kind.

    My understanding — correct me if I’m wrong here — is that a fundamental constraint on textbooks is that they should serve both as tools for the classroom and references for later. These texts are supposed to make sense to a kid who is studying them on her own. The homework problems should also make sense to a kid on her own.

    If what we’re talking about here is classroom use of curricula, then the problem isn’t print, because no one is forcing teachers to toss up Q21 up on the board during class time. Sure, there are better things for the teacher to be doing, but that’s hardly print’s fault. It’s more, like, you want curricula to provide things other than print, like slide decks and videos.

    But how do you see digital curricula transforming a question like the cell-phone one for homework or assessments? Do you want to provide narration to your pictures and state the problem in a video? How big of an improvement, LOA-wise, would that be over the print version? (That last question is sincere, not rhetorical.)

  11. One more thought: I think that there’s something that is worth making explicit here.

    Here’s a text-based version of the cell phone tower problem that, I claim, does not already do the abstracting for the students:

    Three towns are buying a cell phone tower, but arguing over where to place it. Where is the fairest place to put the tower?

    This problem, as stated, is wicked hard. Absolutely none of the abstraction has been done for the student. Now, we could put this in front of a kid, but we won’t. They’d freak out, have no idea what to do, throw a fit, etc. So we want to give them more guidance.

    The issue isn’t just “How do you get text-based curricula to leave off the abstraction?” It’s more like, “How do you provide some guidance towards abstraction without giving the whole thing away?”

    I think it’s worth teasing this out because text-based curricula have another option. They could systematically provide students with the tools for moving themselves up and down the ladder of abstraction. When students encounter a problem that offers no abstraction they can learn to try various things: taking guesses, finding different ways to visualize the scenario, tinkering, being playful with notation, etc.

    For various reasons that might not be a live option. But I think it’s true that print gets into trouble because kids can’t move up and down the ladder themselves. But there are strategies for negotiating the ladder, and kids might spend time learning them.

    Here’s a challenge: The lesson above helps a student move up and down the ladder of abstraction for the cell phone tower problem. What reasons do we have to believe that this student will be better at moving up and down the ladder on their own? (Is it just because we’re explicit when we make the moves up and down the ladder?)

  12. Bryan Meyer:

    I have the benefit of working at a school where students don’t use textbooks and teachers are curriculum designers for their classes. I know that not everyone is in the same position. But, it seems that improving curricular materials is only a small part of improving teaching.

    I’m glad you’re happy at your job but I don’t think we should expect every teacher to teach without curricula. Their efforts would be incoherent and duplicative in many cases. The Exeter and Park Math curricula makes us all better off, etc.

    And absolutely, it’s possible to take problems that have been fully abstracted by a textbook (or not abstracted at all, in your example) and then facilitate that abstraction with our students. But I think we should expect curricula to help teachers facilitate that abstraction or, at the very least, not hinder it.

    Michael Pershan points out that the cell phone problem isn’t intended for an interaction between a teacher and a student. It’s in the practice set.

    He’s right about that, but the material that’s meant for student / teacher interaction looks exactly the same only with all the steps worked out. Even in the student edition, the worked examples that precede the practice problems start at a single (high) rung on the ladder. The only difference is they show you how to perform all the operations at that rung.

    I don’t think textbooks should exclude those worked examples, FWIW. But I want to see a) more rungs, and b) the opportunity for a student to do something at every rung.

    Michael seems to be envisioning some kind of feedback system that helps the student up the ladder. I don’t think computers are good enough for that kind of help yet. The teacher should be that help. What computers can do is just show a context, ask a question, and then aggregate the class’ responses for the teacher. The teacher can do something with those responses and then tell the computer to move up to a higher level of abstraction. Repeat as desired.

  13. @ddmeyer

    it’s possible to take problems that have been fully abstracted by a textbook… and then facilitate that abstraction with our students. But I think we should expect curricula to help teachers facilitate that abstraction

    I see that as you are coming to make meaning of this concept abstraction, it is a notion that is defined within a mind-free mathematics, not within the realm of people’s ways of knowing. That may be a way to think productively of curriculum authoring.

    @bmeyer
    It strikes me that you would say the curriculum could not be named until after-the fact. I think this reflects the constructivist’s view on curriculum (c.f. Steffe, 1990, “Mathematics curriculum design: A constructivist’s perspective”).

    @michael p
    My first thought for getting beyond the printed text is to lose that constraint that it should serve as both a classroom tool and a reference. Similar to what I think Dan imagines, I tentatively suggest that the ideal textbook would be a sequence of mathematical problems that follow a research-based trajectory of student learning. Of course, this trajectory does not match the potential for every student, but has shown to be viable in many classrooms.
    Leaving print, could allow for the teacher to sequence the problems posed based on the last lesson — a “choose your-own adventure” story if you will. Some options, but still structured. More powerful would be if these sequence of prompts were held together by a powerful storyline, or even more ideally a larger problem–what many define as a problem-based curriculum.
    While Dan Meyers task ideas might effectively engage kids on a Friday, he makes quite clear that in his view, teacher’s should not be asked to create curriculum; an unwieldy job–as I am sure he is learning. I am uncertain that one person’s text or image-based or video-based problems would be that of another, whether it be enactable by all teachers, or strike a chord with all learners.
    I suspect Polya, who greatly valued problem-posing, recognized this dilemma when he began to focus on encouraging questioning. Dan Meyer recognizes this as well. Maybe a good text is characterized by the potential to generate worthwhile questions in learners? I know Paulo Freire, Ubi D’Ambrosio, and Rico Gutstein value precisely this quality in mathematics education.

    @Brian

    Curriculum materials are designed to be usable by as many people as possible. If they are too difficult to use, then people will not use them, they will not be bought, and the company producing the materials will hurt financially.</blockquote.
    Sigh, the words of a corporation, not of an educator.

  14. Awwwwwwwwwwwww…but we don’t feel THAT sparsely populated! lol

    Well, if you ever come up here, I expect to feed you dinner and have you go through my kids’ math assignments! Most of what you say goes OVER my head, because I focus a lot more on the Developmental Math prior to Algebra with my kids. But I still learn a lot from your posts.