Mathematical Interview

          I think I learned the most information about a third graders thinking by facilitating the mathematical interview. The problems in the interview packet had division and fraction concepts. The third graders had not talked about these concepts before, but they still had background knowledge to help them get started on the problems. I worked with Lorrie (pseudonym) that filled out a mathematical survey that said she was not very confident in math and that she needed a lot of practice to be good at math skills, but her thinking was just the opposite to what she said about herself. It was so fun listening to her ideas because they really got me thinking that I didn’t know that students knew so much about mathematical concepts without being taught how to use those concepts.

          The first problem that helped me learn the most about Lorrie’s mathematical thinking was the third problem in the packet, the cookie problem. The first thing she did after I read the problem was that she restated the important information. She said, “6 children and 10 cookies. How much will each child get?” After this she started counting the circles on the page. She said “1, 2, 3, 4, 5, 6. There are 4 extra cookies, so I think if I split 2 cookies (then she paused and was thinking). She then said “that won’t work though because then 4 people will be getting 1 and ½ cookies and then the 2 other people will get 2 whole cookies. So I think I am going to start by cutting them all in half.” At this point she goes to the first circle and cuts all of the cookies in half. She said, “There’s 20 pieces of cookie here now.” Then I asked her, “Can 6 people split 20 pieces of cookie evenly?” So she counts each half and said “there will be 1 cookie left over,” so then she looks at the last cookie and said “if I cut this cookie into 4’s” (she then cuts the cookie into fours, but looks puzzled, so she tries a new strategy). Since she knows there are 6 people she draws 6 circles below the box of circles. She starts putting tally marks below the 6 circles and I ask her, “What do these lines represent?” She said, “These lines are tallies and these tally marks represent each piece of cookies that each child gets.” Then I asked, “How big are those pieces of cookie?” She said, “They are as big as a half.” Once she has used all of the half size pieces from her original diagram she says that she is going to cut the last cookie apart so there will be 6 pieces. Then she states what she did to solve the problem, “So you cut 9 of the cookies in half and then you cut the 10^th cookie (she paused here so I repeated what she had said then she stated it again and continued on) 3 times, 2 times diagonally and then 1…actually I am going to change that. You cut all 10 cookies in half and then cut 1 cookie with an X. Then each kid gets four pieces of cookie, so each kid gets 3 halves of cookie and then they get one triangle.” Then to clarify what she was saying, I asked her “How many pieces are in cookie number ten?” She said, “There are 6 pieces.” Then I asked, “So how many is each child getting?” She said, “One triangle.” Then to get her to repeat her answer I said, “So how much cookie will on child get?” Her final answer was, “3 halves and a skinny pizza sliced size piece.”

This is what Lorrie's paper looked like after finishing this problem.

          After hearing Lorrie’s explanation of this problem I believe that she is between direct modeling and the counting on stage of the trajectory. Lorrie directly counted each half of the cookie and used her pencil to show which piece she was counting, so this is why I believe she is in the counting on stage. She did this a couple of times throughout this problem. I also believe that she is in the direct modeling stage because she physically drew 6 circles to represent the 6 people and then she put tally marks under each circle to represent the number of cookie pieces that each child would get. She used the picture given to her as a helpful map to the final answer for the problem. She also used some trial and error methods by starting the problem one way and deciding that it wouldn’t work so she began the problem again another way. I do not know where this information would fit on the trajectory, but I thought it was interesting. With this information, I would say that she is somewhere in the middle of direct modeling and the counting stage.

          Being able to look at a students’ thinking in depth has made me understand that all students have something to contribute even if they are not giving the right answer. There is something to notice about the students that are understanding the basic concepts of equal distribution of cookies.