At the start of the problem I broke the first class up into groups of three and the second class into groups of two. I attempted to choose groups whose students' strengths would be complementary. In addition, I tried to avoid placing the most talkative students together. This step alone proved to be exceptionally problematic in the first class. Every student wanted to work with their best friend and would outright refuse to work with anyone else. Regardless of any encouragement on my part, they just would not work together and kept on trying to establish their own groups. In the second class, I had the students work with the person to their left as Mrs. Porter's seating assignments had done most of the work in keeping problematic students apart. This session went much better, and the students seemed much more obliged to work together (though there were still a number that didn't try particularly hard).
The problem at hand is particularly interesting because it uses fractions, inequalities, and (I believe) a great introduction to proofs. For this reason, I started by explaining the difference between definitions and theorems. Definitions being the most basic rules of math that we accept as true without proof and theorems being the subsequent rules built by applying definitions. I then had several students read the scenario out loud. While some students dislike being put on the spot and having to read, there a fair number that are excited to participate and I think that more students pay attention when one of their classmates is doing the talking. We then proceeded to Noticing and Wondering as has been done in the past. This section appeared to work much better than before, the students seemed more apt to work together in generating ideas than by themselves. After giving them about five minutes to compile some ideas, I started asking them for their observations and proceeded to write them on the board. Their willingness to provide feedback has increased compared to the first problem I did with them, and I got several good suggestions. However, I found that both classes concentrated on the numbers in the example pictured above rather than key phrases in the scenarion; "any proper positive fraction," "add 1 to the numerator," "add 1 to the denominator," etc. In addition, I had notable difficulty in convincing the class that 4/5 > 3/4. Even after putting the decimal equivalent on the board, some were reticent to accept the statement.
In light of the progress made on the first day, I was very excited to have them work on the proof. I thought it would be optimistic if one or two students figured out how to complete the problem, but I hoped that they would be able to start work on it and write the comparison in algebraic form. I ran into timing issues in the first class that prevent me from giving the students time to work on their own. Instead, I tried to turn the problem into a class exercise with me guiding them through the problem. I lost the class almost immediately when I wrote the comparison of the two fractions with two variables and few numbers. Despite having already gone through an entire chapter on solving equations, the students do not seem to have developed a lot of flexibility in their ability to solve problems. The inequalities alone seemed to confuse most of the students and several didn't even know why variables were being used. The concept of a "general solution" was much more difficult to grasp than I initially expected, and only the students I spoke with individually seemed to see the point. There were several different ways of solving the problem and I had wanted to show the ones that I had found to the students so that they might see there was more than one way of approaching things. Unfortunately, it was difficult enough to get them through the first solution, let alone the subsequent ones. As has become common, there were maybe 5 or 6 students altogether that had genuine interest in understanding the problem, but their classmates proved to be too effective at distractions.
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