When I saw the tape of Judy K's class I realized that I was part of the problem. I realized that I tended to react with a head nod or a verbal "good" or "what about this...?" to most student comments. What I observed Judy doing was eliciting all the possible student responses before she made any comment other than, "Any more ideas?" It was clear that the expectation (culture) in her classroom was that all student ideas would be put on the table before any were discussed. It was powerful. As soon as I started to do the same thing, the discussion dynamic shifted.
As is apparent from our lesson plans and implementations, there are many ways of presenting this problem. Following are highlights from our reflections that focus on important moments in teaching the lesson.
In Resnick's problem-solving sessions (1988), children frequently failed because
they lacked sufficient prior knowledge of relevant mathematical content. We have
found that in order for this cylinder lesson to be effective, introducing it at
an appropriate point in the curriculum is critical. Unfortunately, because
of the constraints of our project, some of us had to teach the lesson at an
artificial point in the curriculum, without having time to cover all the
needed prerequisites. Depending on the level, these mathematical prerequisites
can be introduced in prior lessons or activities. In these clips,
Susan Boone elicits her students' prior knowledge
through questioning [view clip],
and Susan Stein asks her students to write down what
they know about circles and related concepts
[view clip] and then has the
class review by sharing what they have written.
In our workshop we tackled the problem of maximizing the volume of the cylinders over a family of rectangles having constant perimeter. Most of us, however, felt that it was best to defer the study of constant perimeter, starting instead with constant area and two-dimensional sheets of paper, and hence working with the constant lateral surface of three-dimensional cylinders.
In general, we began by showing students the two pieces of paper, rolling them from different sides to form a tall, narrow cylinder and a short, stout cylinder, and posing a variation of the following question: "Which cylinder would you predict will hold more, and why?" Student predictions are important, since they engage students more fully in the problem, give them a stake in what happens, and make them more likely to care about the results.
Again, this lesson lends itself well to moments that stimulate conversation. Predictions offer one such moment, and we found it particularly important in discussing these hypotheses to stop and let students verbalize their thinking. Most students thought that the two cylinders would have the same volume, but regardless of the prediction, the reasoning behind it provided a powerful connection to the remainder of the lesson. Some students struggled to come up with explanations for their hypotheses, but it is exactly this struggle that advances their mathematical thinking (Stigler & Hiebert, 1998) [view clip].
Sharing the reasoning behind students' predictions created a rich climate for
mathematical discussion in which every explanation could be valued. Susan Boone
reassured her students that no matter what prediction and reasoning they came up
with, "you can't be wrong" [view clip].
At this stage in an inquiry, the key is for students to be able to form opinions
about the problem based on their reasoning and prior knowledge, as depicted in this
clip from Susan Stein's class [view clip].
Art Mabbott used this moment in the lesson to paraphrase a student's prediction,
providing the mathematical term "wider diameter."
Some teachers chose to ask students to write out their thoughts and explanations. This can help students to be more reflective and to create more coherent reasoning. In our experience, students felt more confident in explaining their results with the entire group if they first shared them with their own team members.
Even after the demonstration, some students were still baffled by the outcome, especially if they had originally predicted the cylinders would have the same volume. Such surprise and perplexity are a sign of readiness to grow, and provide the kind of discrepancy that can focus a discussion (Atkins, 1999). During such a discussion, teachers can assess students' learning, and students have an opportunity to use their mathematical vocabulary to construct complex arguments.
Other students were intrigued by the result, and generated other questions: "Why is this happening?" "If we use the same lateral surface area, can we make other cylinders that will follow the same pattern?" "Is there an upper limit to the volume -- a biggest volume?" "How do we know?" Students can be led to formulate such questions by making physical models of several more cylinders with the same constant lateral surface area, and conducting the experiment with these new models, as Jon Basden illustrates [view clip]. In addition, building the family of physical models -- the original two cylinders and at least two more sets -- can give students a tactile experience, letting them see and feel the connections among the dimensions of the rectangles, the sizes of the cylinders, and the cylinder volumes. In cutting the original paper in different places to make cylinders of different sizes, students grasp the notion that the area of the paper is kept constant the entire time [view clip]. Ideally students will ask how to generalize the problem to other cylinders, experiencing ownership and creating a powerful motivation to pursue the problem further, although the teacher often reformulates it to spur the next phase of investigation.
Most students organized their data by making a chart. They chose several values for length and width, in each case keeping the surface area of the paper, and hence the lateral surface area of the cylinders, constant. The next step involved calculating the volume for the different numbers created in the chart. When Cynthia Lanius presented this problem at our first workshop, we responded to this part of the problem in ways that every teacher hopes to see in the classroom: almost all of us spontaneously picked up a tool of choice to determine which of the cylinders would have the greatest volume. One of us chose a drawing tool and started making visual models so as to "see" the problem better. Others pulled out graphing calculators to observe the behavior of the function. Still others began entering data into spreadsheets to generate the graphs, or constructed the problem using the Geometer's Sketchpad. Annie Fetter created two sketches that connect the changing rectangle dimensions to the graphs of the respective volumes. Much discussion ensued, with people moving back and forth among a variety of tools and problem-solving methods.
At this stage of the lesson, technology can help with the number-crunching, creating more time for discourse in the classroom (Goldenberg, 1996), as for example in Art Mabbott's classroom.
Students might use a graphing calculator to create the chart, enter formulas to find the needed values, and graph the numbers, as in Susan Boone's class. Other tools and software, such as spreadsheets or the Geometer's Sketchpad, may be substituted if they are available.
Finally, the interpretation of the graphs and the explanation of the behavior observed offered other important opportunities for discourse. In Judith Koenig's class, connecting the graphs and the problem helped students solidify their knowledge and make connections among mathematical ideas [view clip]. Art Mabbott ended his class by reviewing the questions the class was investigating [view clip].
A version of this cylinder problem was showcased as the Math Forum's
October 25, 1999 Middle School
Problem of the Week.