Beginning to Understand Linear Functions: Guess My Rule

The purpose of this analytic is to show how mathematics learning occurs when children are encouraged to "create their own way of understanding" [RB Davis, 1992]. The patience and guidance exhibited by Teacher/Researcher in this analytic is a pedagogic model for teachers. The way the students explore and struggle with the first "guess my rule question", and the way they explore with success that excites them on the second problem, provide a rich model of student math learning. This analytic shows 6 boys: Ariel, Duwad, Brandon, Christian, Yonny and James, learning about linear functions through a game called “Guess My Rule†. In this game, the students are given points on a line and asked to guess the equation. Points are expressed as ( box, triangle) pairs instead of the traditional (x,y) values. The rule they are guessing is a linear equation. Teacher/Researcher Arthur Powell uses a chart to show each pair of values: | box |triangle | | 5 | 13 | | 3 | 7 | The boys are told what the rule does to each “box†number, but they have to figure out how the rule works. The first rule they work on is y=3x -2 (or triangle = 3box - 2) . The boys begin to work on this rule by looking at the difference between the values. As they work, they begin to see a pattern, and Ariel is the student who figures out that if we add 8 to 5, and 10 to 6, then we would add 12 to 7 (not shown in the video) and 14 to 8. This is a recursive solution that is not clearly articulated or used by all of the boys. Once Powell puts zero in the box (or x) column, even Ariel is not sure of his recursive solution. (0, -2) is what the rule tells them. Note that Powell always asks them if they know the rule - but he doesn’t tell them the answer even though they are struggling. Instead, Powell decides they should move on to another rule. The second rule was created by Yonny, one of the 6 boys. Yonny charts the information and the boys start out by giving Yonny the number 1. Yonny puts (1, 15) on the chart. With this second rule the boys are energized and quickly call out answers, incomplete and with errors at first, but quickly they figure out that the rule is y=10x+5. It is notable that they first express the rule as a concatenation of symbols: 1-> 15, 2-> 25, 3-> 35. Christian says to take the number (box) and put a 5 at the end to produce the triangle value. Powell has to tell them to express the rule as a (mathematical) operation. This game was pioneered by Dr. Robert B. Davis, and adapted for use by Dr. Arthur Powell for this study. The boys in the source videos for this analytic were part of a group of 7th graders in the Frank J. Hubbard Middle School in Plainfield who participated in an after school, 3 year NSF study called IML (Informal Mathematics Learning, Award REC-0309062). References Davis, R.B., (1992). Understanding Understanding, Journal of Mathematical Behavior 11, 225-24