Recent Student Interview and My Thoughts

 2/6/12

            I conducted a student interview with a student from Carlsbad High School last week. The student, lets call her Sara, is a 10^th grader whom is currently enrolled in Algebra II. I understand that she took Algebra I in 8^th grade and Geometry in 9^th grade (last year). She says that math is not her favorite subject (usually gets A’s in her other subjects vs. B’s and C’s in math). For the interview, I provided paper (blank, lined, as well as gridded), a ruler, protractor and compass for Sara to use. 

Note: The problem is the following area/perimeter problem one might see in a geometry textbook. My instructions as interviewer are to simply be the observer. I am there to try and examine the student's mathematical thinking & understanding, and to not play the teacher role.

You can see some of Sara's work on the sheet

            Sara seemed to be really stuck at the beginning. I asked her first to restate the problem for me. She said that she needs to figure out which area was biggest and which was smallest. I then asked her to try to verbalize to me what she was thinking. She didn’t have much of anything to say so I asked her “What do you know about area?” She contemplated about that for awhile, thinking that it meant length; I told her I thought length was a measure of distance, like the length from one point to another. She remembered, agreed with me and was able to come up with her own definition in saying that area was the amount of space something has.

            Now this dialogue of determining what our goal of the problem was resulted in Sara still being stuck. I asked her to reread the informational portion of problem statement and tell me what it means. She said “The pastures are made of half-circles. Pasture A is made of three half circles.” I said that was interesting and to show me these half circles because “I am having a hard time seeing them.” From the attached work, you can see that Sara drew the boundary lines of all the half circles in the figures. At this point I asked her if she felt she had all the tools for answering the problem and she said she wasn’t sure. This led to more questioning…

            She said she could remember the formula for area of a rectangle, length times width, but that she couldn’t remember the area formula for a circle (despite high marks on her geometry report card). Resisting the urge to verbally damn the teachers (or school system, I know it’s not always the teachers fault) of Sara’s past, I brought up similar problems to give to Sara that I was confident she could solve. I asked her an alternate area problem involving rectangles (see attached work). She knew right away that the area of a 2x3 rectangle was 6 and the area of a 2x4 rectangle was 8. I wanted to tell her the area formula so that she could solve this problem the most accurate way, but as veteran student interviewer, I kept my cool. I asked Sara instead how you can answer an area problem without using multiplication, there was a pause. So I drew the previous 2x3 rectangle in grid form and she told me right away that you can count the number of squares…

            I won’t go into the nitty gritty of the rest of the interview. I was much temped to be a teacher in the situation, especially watching the student try to count squares. But a method is a method. I did notice some important aspects of Sara’s mathematical understanding. The main one was that if two circles (or half-circles in this case) have the same radius, then they must have the same area. When Sara noticed that the three pastures all had a large semi-circle of radius 6, she decided not to try and count those squares. Similarly, she noticed that the 3 smaller half circles in Pasture B were of the same radius length and therefore of the same area, hence the equal 7 count written inside those semicircles in her work.

            What I also learned about Sara’s mathematical understanding is that her arsenal for solving problems is often maintained only for a short period of time, often forgotten soon after it will no longer be on the test. I asked her what problems she had done like this before and her response was that she did problems like this in geometry the school year before but could remember what she did to solve them (what the formula was). To her credit, memorizing the area and perimeter of a circle is really the only way kids are able to “understand” it at that point in their public school mathematics careers.

            It is in my opinion that one student in a public school like Carlsbad High could not grasp the reason of why the perimeter and area of a circle is “2 Pi r” and “Pi r-squared” (respectively) until they have learned trigonometry, without intrinsic motivation to understand “why.”

            I respect the subject of math in such a way that I see it unfair to teach rote memorization to kids as they do today. I have been able to learn a lot, in a little bit of time, in my observing of kids’ mathematical thinking. My challenge now is to absorb, analyze, and respond to this thinking that will effectively foster their mathematical minds.