My introduction to the GeT community was attending the Teaching GeT working group a week before I presented a seminar to the group in November of 2020. I wanted to get to know the community before I presented my NSF-sponsored DRK-12 grant (DRL 1907745), Using Animated Contrasting Cases to Improve Procedural and Conceptual Knowledge in Geometry (AC²inG), which had just completed its first 15 months. During that presentation, I set the theoretical framework, grounded in research from cognitive science and extended to mathematics education, for using contrasting cases in learning eighth grade geometry content. 

In midst of the pandemic my research team had only been able to work on the curriculum development, so the presentation focused on grounding our curricular design. One thing that makes the AC²inG materials (Krupa et al., 2019) unique is that we created the first web-based contrasting cases in mathematics, and we harness the visual nature of geometry; the cases are animated to highlight key geometric concepts. As we created our materials, we considered several design features: animations and colors to draw student’s attention to the geometric content in meaningful ways, characters’ methods purposefully selected to spark comparisons, geometric thinking of fictitious characters, and diversity of characters throughout the units.

In short, our digital curricular materials place two fictitious students’ voices at the center of mathematics learning, and each lesson includes five unique features: a page for the first fictitious student’s solution strategy on a given geometry task, a page for the second fictitious student’s solution to a geometry task (which could be the same or different task shown on first student’s page), a page with both students’ strategies side-by-side, a discussion sheet with four questions for the students to answer, and a thought bubble page summarizing the key mathematical concepts in the problem. The side-by-side pages are where students really focus on comparing and contrasting the solution strategies. The discussion sheet and thought bubble page are designed to make the instructional goal of each Worked Example Pair (WEP) more explicit and to scaffold discussions among students as they summarize their work from the WEPs (Star et al., 2015). 

We were unable to test our interventions in schools in the spring of 2020 and throughout the 2020-2021 school year since schools were closed, and virtual learning stressed the educational system. We had to pivot from our original plan of implementing our digital materials with students in classrooms. It was very tough to let go of the original goal—one that had been conceived meticulously, approved by the NSF, confirmed by external reviews, and vetted by our advisory board. Unable to conduct our randomized-control design, what I needed was student feedback from using the materials. So, we transitioned to conducting virtual think alouds with students across the United States. 

Last spring, we conducted 56 hour-long open-ended semi-structured clinical interviews (Piaget, 1976; Opper, 1977) in the form of think alouds with individual participants (n=42). Our goal was to elicit student thinking as participants engaged with the materials and discussion questions, not to get them to a “correct” response (Opper, 1977). In order to engage participants in each phase of the WEP during the interviews, we followed a detailed protocol: examine the first method, examine the second method, horizontally compare the two methods, solve the problems on the discussion page, and read the thought bubble at the end. If needed, we had questions for each phase of the protocol to probe student thinking. In all, there were 3,249 turns that were coded. 

Of the 3,249 instances, 1,354 (41.67%) were coded as geometric thinking of the student, 756 (23.27%) were students making comparisons between the WEP characters, 621 (19.11%) were instances of students analyzing the geometric thinking of the WEP characters, and the rest fell into smaller categories. It will take us additional time to unpack the geometric thinking the students displayed during the think alouds, but we have begun to document the types of comparisons students made during think aloud interviews regarding the fictitious student methods to mathematics problems. When students were making comparisons between the characters, most often they were discussing differences between the characters (n=484), but they also noted similarities (n=267) and used both WEP characters’ strategies to verify a mathematical idea (n=5). In addition, regardless of whether students were pointing out a similarity or difference in the two strategies, students often referred directly to the method they were using to solve a problem.

Specifically, when pointing out differences, students most often described differences in the methods the characters used to solve a problem (n=380). For example, when analyzing strategies related to translating a figure, one student stated, “Jaxon is more plotting it out, while Maxine is subtracting the values to go left or down. They both had it in the same spot, which is good; I think that’s the idea.” This student realized Jackson is using a visual geometric method, while Maxine is using an algebraic approach, yet they arrive at the same answer. This student was attending to the visual/algebraic aspects of Jaxon and Maxine’s approaches. Students noted differences in the students’ methods regarding WEP specific content. For example, in a WEP designed to have students understand why the interior angle sum in a triangle is 180 degrees, one student said, “Alex, like, ripped his triangle apart and… what did Morgan do? … Morgan, just drew the line and just used, like, the parallel cut by transversal stuff to figure everything out. To figure out that it was 180 degrees.” Here the student is attending to specific mathematics content in the WEP. 

This research is the very beginning of showing a viable scientific basis for using comparisons to explore multiple solution strategies of students in geometry, as students were able to note similarities and differences in the strategies. Given critiquing reasoning is important to deepening mathematical understanding, these findings are a step towards documenting the ways in which contrasting cases can be used in geometry. Currently, we just completed our first classroom-based implementation of the materials, a randomized control experiment with 102 students engaging with the AC²inG materials. The main difference between the treatment and control groups was that the control group only engaged in one student solution at a time without the comparison page. An analysis of these data will be forthcoming after we have caught our breath from teaching middle school geometry for 14 days! 

References

Krupa, E. E., Bentley, B., Mannix, J. P., & Star, J. R. (2019) Animated Contrasting Cases in Geometry: 8th Grade Supplemental Materials. Retrieved from: https://acinggeometry.org/

Opper, S. (1977). Piaget‘s clinical method. Journal of children’s Mathematical Behavior, 5, 90-107.

Piaget, J. (1976). The child’s conception of the world (J. Tomlinson and A. Tomlinson, Trans.). Littlefield, Adams & Co. (1926).Star, J. R., Pollack, C., Durkin, K., Rittle-Johnson, B., Lynch, K., Newton, K., & Gogolen, C. (2015). Learning from comparison in algebra. Contemporary Educational Psychology, 40, 41-54.