Alyssa Berger Post #8

This week, I chose to use a picture of the worksheet I gave my students with the Pizza Comparison Task on it for my lesson to analyze. I told the students that once they came up with one answer to the problem, they should try to come up with another possibility that way they didn't sit and become disruptive to the students who were still working out their first solution. A lot of students came up with more than one answer pretty quickly, but upon looking at and/or talking to their neighbors, they began working on more (even though they weren't supposed to be talking!!). This particular student chose to come up with his second answer so that it contradicted his first answer. He clearly was thinking in a very shallow mathematical way when he came up with the answer that Jose had the same amount as Ella because they both had half of a pizza. But then, when he thought about the problem more deeply, and attempted to figure out how it could be possible that Jose did eat more pizza, he discovered that if the pizzas were different sizes, Jose could be correct and he in fact ate more pizza than Ella. The thing I find most revealing in his example is when he draws the pictures of the different sized pizzas, then he drew a greater than sign between them to show that the pizza on the left has more than the pizza on the write, but he also chose to write next to the drawings "1/2 big" and "1/2 small." I found this very telling of his current mathematical understanding. It was a relatively common answer for the students to discover that if the pizzas were different sizes, Jose ate more pizza overall, but this student was the only one to add the description of "1/2 big" and "1/2 small." I think with some scaffolding, this student would be able to quickly understand area and how comparing the pizzas when adding area and/or diameter to the pizza problem, he would be able to much more clearly talk about how the bigger pizza is more. Similarly, I think this student is very close to being able to multiply fractions by whole numbers and understand how the numbers relate to each other in these problems (i.e. multiplying by a fraction has a result that is smaller than the initial number). This is a much more complex mathematical concept and I am curious if this student is already ready to move on to these problems or if he would need more help with these problems.

The three questions I would ask as a result of this piece of evidence are:
1. Does this student already have an understanding of area without explicitly knowing the term area and how to calculate it?
2. Has the student already experienced problems where he needs to multiply a whole number by a fraction to arrive at the understanding that the whole number would be smaller even though he was multiplying?
3. How could I modify this task to help answer my questions about his understanding?

To answer the first two questions, I would have to provide problems for the student in a one-on-one setting so that I could observe his thinking while he worked out the problems and see what he comes up with as an answer. I could find these problems in the 5 Practices book or in the CGI book (or at least a similar problem that I could modify). I think I would have to do a lot of thinking about what a meaningful addition to the problem would be that would also answer my questions about what the student is thinking. I could also do some research problems in curriculum books and on the internet to find similar problems that could help me modify the task in a meaningful way. I could also discuss with other math educators.