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A math class is a place where knowledge is shared, not handed down. By insisting on relying on personal experience, we limit ourselves as teachers, and this makes us more likely to listen, and less likely to talk, for how much of what we say has come from personal experience, as opposed to what we have read or were told somewhere?
The Math Forum hosts six different Problem of the Week projects, featuring non-routine word problems for the elementary and middle school levels, and for geometry, algebra, discrete math, and calculus/trigonometry. Students earn credit only if they provide both correct answers and detailed explanations of the strategies used in their solutions. A version of the cylinder problem was showcased as The Math Forum's October 25, 1999 Middle School Problem of the Week. Problem of the Week mentors reply to students with individualized feedback via email. These replies are designed to engage students in conversations that will improve their solutions or clarify their explanations. Students have an opportunity to revise their answers in the event they initially get the problem wrong, or if they simply want to improve an answer based on the mentor's feedback. An ongoing investigation into the effect of students' participation in Problems of the Week over a ten-month period suggests student gains in the ability to: (a) identify what the problem asks, (b) generate strategies for working with the problem, and (c) be more independent in problem-solving (Renninger, Farra, & Feldman-Riordan, 2000). In the cylinder problem, students were asked to predict whether the cylinders would have the same volume or whether one would hold more than the other, and to explain the reasoning behind their predictions. They were then to experiment by building the two cylinders, placing the tall one (cylinder A) inside the short one (cylinder B), filling cylinder A with some loose material, and then lifting away cylinder A and observing: Is there enough material to fill cylinder B? Does the material fill cylinder B exactly? Does the fill material overflow? They were to conclude by stating which cylinder turned out to hold more, and whether or not this result agreed with their initial hypotheses. As in a discourse class, where students are asked to verbalize the reasoning behind their predictions, here students were asked to write down their reasoning. Explaining predictions is an important part of the process, as it eliminates thoughtless guessing and helps students think about what might happen and the mathematical relations inherent in the situation. In a sense, it primes them for perceiving the mathematical implications of the experimental results. Furthermore, our experience suggests that the initial struggle students go through in formulating their reasons for a prediction -- whether naive or correct -- enriches their experience and motivates them to continue.
In all, there were 667 responses to this Problem of the Week. Students made
four types of predictions. These included: a) same volume for both cylinders,
b) tall cylinder holds more, and c) short cylinder holds more. Some students
gave more than one prediction, had unclear predictions, or provided no
prediction.
Now let's take a closer look at the reasoning behind students' predictions.
"Same Volume" Prediction "I predict that the two cylinders will hold the same amount because the two pieces used to construct the cylinders were the same size to begin with, and all the surface area of each is still exposed."Other students who predicted that the volumes would be the same said that the stout cylinder and the tall cylinder "balance out." Still others said "equal out" or used similar vocabulary, some thought the width and height would balance out, and some said that the cylinders were nearly the same size or looked the same. Interesting explanations included: "I thought that the cylinders would hold the same amount of Cheerios. I thought this because I looked at the cylinders and I tried to visualize making cylinder B the same height as cylinder A while making cylinder B thinner." "Tall Cylinder Holding More" Prediction Only 22 students (3.3%) thought that the taller cylinder would hold more. These students reasoned that tallness would add more to the volume than other factors. For example, "I predict that cylinder A will hold more than cylinder B because it looks sturdier and it also looks a bit bigger." "Short Cylinder Holding More" Prediction A total of 171 students (26.5%) thought that the short cylinder would hold more, because the greater height of the taller cylinder would not make up for the greater diameter of cylinder B. A few students showed some knowledge of the formula for cylinder volume by saying that the radius gets squared, which gives it more importance than the height. Others referred back to previous problems, where area was maximized by squares, not rectangles, and reasoned that the shorter cylinder is closer to the shape of a square. Below are sample of responses from students who predicted the shorter cylinder would have a greater volume than the taller cylinder: "I think cylinder b will have a larger volume because its base surface area is much larger. Even though cylinder a has a greater height I still think cylinder b will be larger."One student connected a prediction to prior knowledge: "I based my hypothesis on a problem I worked last year or the year before that. It was about which neighbor had a bigger fenced in area for their dog to run around in. One neighbor had a 4x4 plot of land, and another had a 2x6 plot of land. They both used the same amount of fence, but the square plot had a bigger area. Cylinder B is more like a square than cylinder A, because its sides are close to equal, so I concluded that it would have more volume."Other sample responses for this prediction: "Prediction: The shorter, wider cylinder (B) will hold more because it will have a greater volume. I think this is true because in the formula for the volume of a cylinder (height x area of the base), the area involves squaring the radius (pi x r^2). Since the radius comes from the circumference (the length of the paper rolled into a circle), having the largest possible number **squared** instead of just multiplied once (the way it would be in the height part of the equation) would give a bigger volume. If this is confusing, think of the formula: height x pi x r^2. Since the r (radius) comes from the circumference, or the length of the paper that is NOT the height, it would be better to get the longest possible r so the larger number can be squared instead of just multiplied in once." Student Reflections Other responses featured reflections specifically on the subtext of word problems and changes of mind. "My prediction before I conducted the experiment was that the wider, shorter barrel would hold more. I thought this because all of these math problems that make you think are usually trick questions. They want you to think that they are the same because the sheets of paper are the same." Of the 667 students, 406 or 60.8% wrote accurate and detailed justifications of their predictions, which meant explaining their predictions and drawing the right conclusions after the experiment. A few of the students noted the conclusion with some surprise, saying that maybe surface area and volume weren't as related as they had thought. One even wrote, "One thing that I would like to find out is if the rule of 'surface area doesn't necessarily relate to volume' works on everything, not just this version of the experiment." That student is clearly ready to go to the next level, to study the formula for volume and to begin to understand how it differs from the formula for surface area.
As with the experiences our teachers had in their classrooms, this
problem shows that surprise at an outcome can engender increased
interest in students and prepare them to learn more about a topic
(Berlyne, 1960; Piaget, 1950; Strauss, 1998).
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