DescriptionThis Analytic was created for a professional development workshop for algebra teachers.
This Analytic was created as the first “video story” in a set of four; these Analytics 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-taped. 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-taped and analyzed in the author’s dissertation (Leslie, 2019).
The source video for this Analytic was produced by an NSF Study (Informal Mathematics Learning, Award REC-0309062). In this 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.
In this story, 6 boys - Ariel, Duwad, Brandon, Christian, Yonny and James - play “Guess My Rule.” The students are given points on a line, one at a time, and asked to guess the rule/equation. Points are expressed as (Box, Triangle) pairs instead of the traditional (x, y) notation. T/R Arthur Powell uses a chart to show each pair of values as they emerge; the first two points are (5, 13) and (3, 7).
The boys are told what the rule does to each “Box” (each input number), and they have to figure out how the rule works. The first rule they work on is Triangle = 3(Box) – 2. (The traditional teacher writes this equation as: y = 3x – 2.) The boys begin to work on this rule by looking at the differences between the values. As they work and accumulate more clues (more points), they begin to see patterns. Ariel is the student who figures out the pattern is increasing, and that a difference of 2 (between the numbers "added" to Box in the recursive solution) is relevant. He points out, "if we add 8 to 5, and 10 to 6, then we would add 14 to 8. (Ariel is only using Box values that have been used in the game: 5, 6 and 8 were used and 7 had not been used.)
Ariel’s recursive solution is not clearly articulated or used by all of the boys. Once T/R Powell puts zero in the Box column, even Ariel is not sure of his recursive solution. The fact that (0, -2) is the point, and that the Triangle value is negative may be a cognitive barrier for the boys.
Note that T/R Powell always asks them if they know the rule, but he doesn’t tell them the answer even though they are working hard and struggling. Instead, he decides that they should move on to another rule. Note also, the boys do not demand “the answer” from Powell either.
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, with errors at first; they quickly figure out that the rule is y = 10x + 5.
Christian explains the rule: take the number (Box) and put a 5 at the end to produce the Triangle value. Powell tells them to express the rule as a (mathematical) operation, which they do.
This story reveals the mathematics learning that happens when children are guided and encouraged to "create their own way of understanding" (Davis, 1992, p. 226). The patience and guidance exhibited by the Teacher/Researcher, Arthur Powell, is a pedagogic model for teachers.
The way the students persist in their exploration of the first "Guess My Rule” question, a question that challenges them, is evidence of how pedagogical choices can create deep engagement among their algebra students.
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).
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. Dissertation, Rutgers. New Brunswick, NJ.
Maher, C. A. (1998). Constructivism and constructivist teaching, Can they co-exist? In Ole Bjorkqvist (Ed.), Mathematics teaching from a constructivist point of view (pp. 29-42). Proceedings of Topic Group 6 at the International Congress on Mathematical Education [Report on Proceedings] (8th, Seville, Spain, July 14-21, 1996). Faculty of Education Report No. 3.
Steele, C. M. (1997). A threat in the air: How stereotypes shape intellectual identity and performance. American Psychologist, 52(6), 613-629.
GenreEffective teaching, Professional development activity, Student collaboration, Student engagement, Reasoning, Representation