Volume in Calculus: Conceptualizing before Formalizing

One of our PD sessions in the past was about how to support students with learning differences. One of the points that the presenter made was that most pedagogical tools that you would use the better serve these students are great tools to reach all learners. This struck me especially because I teach almost entirely students for whom English is their second language, and sometimes when I do something specifically to help students with the language of mathematics I come to larger and more general pedagogical understandings.

For example, this past week, I introduced solids of known cross-section in AP Calculus in a way that I thought would ease my students understanding of the tricky language involved in the problems, but what I ended up doing was really effectively let them develop their own conception of how these solids are formed and THEN interpret the AP problem language and integral notation in those terms. Conceptualize and then add mathematical formality to their own conceptual framework.

Here’s how it worked. I put 4 of these solids out around the room:

  1. First, I gave them 1-2 minutes to SILENTLY write down in bullet points how they would describe to someone else how the solid was formed.
  2. Then I gave them 2 minutes to share ideas in groups.
  3. Then I cold called on 7 or 8 students via a deck small cards with their names on them (which is by far my new favorite teaching tool). After I called on some students, I called for volunteers with any other ideas.
  4. LAST, I asked them to flip to the back of the paper and read the actual description.

During the “share” part, students said some of the craziest, random stuff, but most of the important parts of the description were said by various students. When it came time for them to read the description, at first they were like “whoa” because the language is still a bit daunting. But after a minute or so of close reading, they connected everything in that description with things that they themselves had said. So when it was time to do the actual integral, the intermediate notation I use made 100% sense:


So general pedagogical moral of the story? Letting students conceptualize something on their own before bringing in mathematical language and notation makes it more likely that the notation will aid in their understanding rather than provide another hurdle in learning. 

A teaching fellow (a first year teacher) was observing my class (and has been observing periodically throughout the year). Afterwards, she remarked that she felt this was one of the most effective 10 minutes of the year, and I agree! And I think 10 minutes on this (instead of just 1 minute reading the question) will save lots of time in the future. Next week, I hope to try the same strategy with solids of revolution!!



Posted on April 13, 2013, in Calculus and tagged , . Bookmark the permalink. 14 Comments.

  1. Did you make those shapes yourself? Can you help me get started making some?

    I’ll be doing these topics in about a month, and would like to try doing it the way you suggest. (I usually skip over the ‘solids of known cross sections’ and go straight to the volumes of revolution. But this looks great. They need help visualizing. This activity looks like it speaks to that need well.)

  2. yeah sure! it was pretty easy but did take an hour or so. i just printed off a region and then took some colored paper and then made a bunch of the same shape of varying sizes (not worrying about where they go). i just went down by about half a centimeter each time.Then, you can just slide a shape along the region until it fits in the right spot. For each shape, I just left a little rectangle of paper on the bottom of the shape and that was the tab that I used to attach the shape to the paper. I cut the tab in half, folded one part forward and the other backward so that each cross section stood up well, and I just taped them down with scotch tape. sorry, that’s hard to type, i’m happy to send some pictures of what i mean if you want!

    so this year was the firs that i’m doing cross sections before revolutions and i think it’s a really good switch. i think they need to conceptualize the accumulation before anything else, and the problem with the rotation ones is that they are using brain space to visualize the revolution, which doesn’t really have anything to do with how you set up the integral (I had kids last year who couldn’t understand which way the circles went). So i’m hoping by doing the cross sections first they can visualize the accumulation before adding in the step of the revolution.

    oh, and the revolution solids were made with an iPad/iPhone app called “123D Make”. You just trace the shape you want to rotate, rotate it, and then it prints off blueprints for you! So easy, but also kind of time consuming. But I’m DEFINITELY going to use the app in class too for showing how the rotation works!

  3. I think I get it. Thanks. Can you tell me what functions you used, with what cross sections? And did any work better or worse than the others?

    • X^2, y=0 and x=1. I did semicircles, squares, rectangles with height half their length and isosceles right triangles with one leg in the plane. The shorter cross sections worked better (squares got a bit floppy), but the squares were easiest to construct

  4. Brilliant beyond brilliant. As usual. I’m teaching calc again next year!! Soooo…I’m stealing all your stuff, fyi.

  5. I did something similar this year, although after reading this post I think I’ll tweak it for next year. In past years I’ve had students do a project where they had to build a model, write its description and show the calculus to find the volume. This year as we started the volume unit, I had my kids analyze past projects that I kept: can you see how the description is related to the model, what is going on in the problem that leads to the integrands, etc. It made for my best volumes unit ever.

    After looking at this post, I’m thinking that I may separate the models, verbal descriptions and calculus, then try to get the kids to match them up. From there, go into how to do a volume problem from scratch.

  6. A few years ago, I also switched to cross-sections before revolution and I prefer it. With revolution the cross-section was always circular, and I found the students struggled when switching to a variety of cross-section shapes.

    I do have a question. I like how you start your integral notation with the shape and the area formula written inside. Do you do this for every cross-section: triangle, circle, semicircle, washer, etc…?

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