Guess My Rule Creates Common Cognitive Challenges for Yonny and Brandon, Ariel and James

This VMCAnalytic was created as the second “video story” in a set of four. The four VMCAnalytics were used as the central tool of professional development in the Teachers Algebra Workshop (TAW), August, 2016. The TAW was created for a study that examined the use of video stories in the professional development of algebra teachers of minority students with low socio-economic status (Leslie, 2019). In this study, teacher participants’ responses to this video story were audio-recorded. The teacher participants discussed the mathematical reasoning, student engagement and pedagogy that they noticed in the video. The teacher participants were guided by questions designed to focus the discussion on these three areas. The discussions were audio-recorded and analyzed in the author’s dissertation (Leslie, 2019). The source video for this analytic was produced by an NSF Study (Informal Mathematics Learning (IML), Award REC-0309062). In this NSF study Teacher/Researcher (T/R) Arthur Powell adapted an algebra game, “Guess My Rule,” that was originally pioneered by Robert B. Davis. Clips from the original NSF study were selected for inclusion in this video story/Analytic. The students 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 the after school, 3-year NSF study. This Analytic presents two stories: In the first “component” story, Yonny and Brandon solve two Guess My Rule problems and they use inductive reasoning to find y-values for new x-values. When the new x-value is too large for their inductive rule, they choose to use proportional reasoning to find the new y-value. In the second component story, Ariel and James solve a different Guess My Rule problem, and they also fall back on proportional reasoning to find new y-values. In each case, the rule being sought is linear and proportional reasoning does not provide a valid solution. (For example, if y = ax + b, (and b is not equal to zero), when the input is doubled, the output is not doubled; it is NOT 2(ax+b) or 2ax+2b. It is y = 2ax +b.) These component stories are placed side by side in this video story for two reasons. They both illustrate how algebra students use proportional reasoning in search of y-values for new, perhaps large, x-values. The second reason is that the Ariel story, in this analytic, is the preface to the video story that will be presented on the third day of the TAW. On day three of the workshop the video story will focus on Ariel’s solution to the Ladder Problem. (The Ladder Problem has an explicit rule of y = 3x + 2.) In the first Guess My Rule problem, T/R Arthur Powell guides Yonny and Brandon as they work to discover a rule to generate the (x, y) points in each of two examples, y = 2x + 1 and y = 2x + 5. Note that in each case the students are given a table of points. Yonny and Brandon are given this set of (x,y) points: (1,3), (2, 5), (3, 7), (4, 9), (5, 11), (6,13). (In the video, these points are shown in a table format.) The explicit rule y = 2x + 1 generates these points. Brandon and Yonny analyze the points inductively and explain the that x values go up by 1 and the y values go up by 2. Yonny explains a version of this inductive rule to T/R Powell, “plus two, plus three, plus four, plus five.” We can see that this reasoning applies to the points as follows: When x = 1, y = 1+2 = 3 When x = 2, y = 2+3 = 5 When x = 3, y = 3+4 = 7 When x = 4, y = 4+5 = 9 When T/R Powell asks them to find the y-value for x = 20, Brandon individually works on finding all the y-values from x = 6 to x = 20, using his inductive pattern. When he is finished, he announces his finding that y = 41. Then T/R Powell challenges Yonny and Brandon to find the y-value for x = 100. The boys express concern about finding the y-value for x = 100. Yonny says softly, “You can’t make us do that.” When T/R Powell asks them if they “can find another way to get the answer,” (a way that doesn’t require them to find all the y-values between x = 20 and x = 100), they resort to proportional reasoning. Brandon explains, with visible hesitation, that “since twenty times five = one hundred, we can do forty-one times five equals two hundred and five.” (The actual y-value is 201.) Forty-one was the y-value for x = 20, and Brandon multiplied it by five to get the y-value for x = 100. At almost the last minute we hear Yonny say, “It’s two-hundred and one.” T/R Powell then gives Yonny and Brandon another Guess My Rule problem that is similar to the first. This list of (x,y) points was presented to Yonny and Brandon in table form: (1,7), (2, 9), (3, 11), (4, 13), (5, 15). The boys proceed to solve this problem using the inductive patterns that they observed. When asked to find the y-value for x = 100, Yonny and Brandon, again, decide to use proportional reasoning. The patience that Yonny and Brandon have for their work and the patience that T/R Powell displays for the students is worthy of note. Even when the boys are discouraged by the amount of work they will have to do to find y when x = 100, they do not demand an answer from T/R Powell. T/R Powell does not rush the boys to an explicit solution. At every step he asks the boys to simply explain their solution(s). The third math problem is called The Ladder Problem. T/R John Francisco guides Ariel and James as they build ladders with green rods. Ariel constructs the ladders to find out how many rods a 1-step, 2-step or 10-step ladder needs. The solution to this Guess My Rule task is y = 3x + 2. For example, when x = 5 steps, y = 3(5) + 2 = 17 rods. When Ariel starts to build a 10-step ladder, he stops at 5 steps and decides to use a proportional solution. He decides that a10-step ladder should use twice as many rods as a 5-step ladder. The video stories in this analytic present two different groups of students, each looking at different Guess My Rule problems. Both groups make a choice to use proportional reasoning when they have to deal with large x-values. Once this decision is made they create an incorrect solution that they are asked to explain or justify. This video story shows how the students persist in their work even when the questions they are given are challenging and even when they find evidence of mistakes in their own assumptions. This is further evidence of how pedagogical choices can create deep engagement and persistent problem-solving among their algebra students. Teachers may consider the reasons why a teacher would let students explore their assumptions and their mistakes in detail. How might these explorations help students build a strong understanding of linear functions? The video events in this analytic were used in the original TAW that was analyzed in the author’s dissertation (Leslie, 2019). REFERENCES Agnew, G., Mills, C., & Maher, C. M. (2010). VMCAnalytic: Developing a collaborative video analysis tool for education faculty and practicing educators. Proceedings of the 43rd Hawaii International Conference on System Sciences, Honolulu, HI. Boaler, J., (2016). Designing mathematics classes to promote equity and engagement. The Journal of Mathematical Behavior, 41, 172-178. https://doi.org/10.1016/j.jmathb.2015.01.002 Borko, H., Koellner, K., Jacobs, J., & Seago, N. (2011). Using video representations of teaching in practice-based professional development programs. ZDM Mathematics Education, 43, 175–187. Davis, R. B. (1992). Understanding ‘Understanding.’ The Journal of Mathematical Behavior,11, 225- 241. Leslie, Joyce (2019). Investigating a model using video stories for professional development for algebra teachers of low SES minority students. 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