GeT: A Pencil

Author: Erin Krupa

  • Technology in GeT Courses

    Technology is integral to modern education, supporting exploration, computation, assessment, communication, and motivation. National curriculum and teacher preparation standards, such as the Common Core State Standards for Mathematics (NGA & CCSSO, 2010) and Standards for Preparing Teachers of Mathematics (AMTE, 2017), emphasize the strategic use of technology in teaching mathematics across grade levels. At the college level, the Committee on the Undergraduate Programs in Mathematics recommends increasing the sophistication of technological tools used by mathematics major students (Zorn, 2015). The Mathematical Association of America Instructional Practices Guide views technology as a constant theme across instructional practices, promoting student engagement and learning (Abell et al., 2018).

    As GeT instructors, our collaborative interest in technology emerged from our exploration of technologies such as Dynamic Geometry Environments (DGEs) and Digital Proof Tools (DPTs) in our teaching of geometry. We believe providing GeT students with initial exposure to technology tools can help them develop a positive attitude and appreciation for these tools and thus inspire them to incorporate such tools in their future classrooms. Our initial work on technology was the development of the narrative of SLO 6 – Technologies, calling for students to effectively use technology in GeT courses to explore, conjecture, and strengthen their understanding of geometric concepts and relationships. Working with DGE tools like GeoGebra, we observed how students naturally discovered geometric relationships through hands-on digital manipulation, while DPTs, like FullProof, could support both proof construction and proof evaluation with instant feedback (Baccaglini-Frank, 2011; Buchbinder et al., 2023; Bülbül & Güler, 2022). Combining research-informed practices and our own teaching experiences, we contributed a chapter on the topic of the importance and application of technology in GeT courses to the upcoming book entitled GeT Courses: Resources and Objectives for the Geometry Courses for Teachers. In the chapter, we further elaborated on SLO 6 and shared how DGE and DPT-incorporated activities could cultivate students’ geometric habits of minds (Driscoll, 2007) in various geometry contexts, such as proof writing, constructions, transformations, Euclidian and Non-Euclidian geometries. 

    Looking forward, we see tremendous potential in emerging technologies (e.g., generative AI and augmented reality). GeT instructors should stay updated on the latest technology trends and developments, be aware of potential challenges and solutions, and reflect on the use of technology in their teaching, in order to create supportive and engaging learning environments to foster students’ mathematical thinking.

    References

    Abell, M., Braddy, L., Ensley, D., Ludwig, L., & Soto-Johnson, H. (Eds.). (2017). MAA instructional practices guide. Mathematical Association of America. https://maa.org/resource/instructional-practices-guide/

    Association of Mathematics Teacher Educators [AMTE]. (2017). Standards for preparing teachers of mathematics. https://amte.net/standards

    Baccaglini-Frank, A. (2011). Abduction in generating conjectures in dynamic geometry through maintaining dragging. In Proceedings the 7th Conference on European Research in Mathematics Education (pp. 110-119).

    Buchbinder, O., Vestal, S., & An, T. (2023). Lessons learned using FullProof, a digital proof platform, in a geometry for teachers course. Proceedings of the 25th meeting of the MAA special interest group on research in undergraduate mathematics education. Omaha: RUME.

    Bülbül, B. Ö., & Güler, M. (2022). Examining the effect of dynamic geometry software on supporting geometric habits of mind: A qualitative inquiry. E-Learning and Digital Media, 20427530221107776.

    Driscoll, M. J., DiMatteo, R. W., Nikula, J., & Egan, M. (2007). Fostering geometric thinking: A guide for teachers, grades 5-10. Heinemann.

    National Governors Association Center for Best Practices [NGA] & Council of Chief State School Officers [CCSSO]. (2010). Common core state standards for mathematics. http://www.corestandards.org/Math/Zorn, P. (Ed.). (2015). 2015 CUPM curriculum guide to majors in the mathematical sciences. Mathematical Association of America.

  • Transformation of AC2inG  Classroom-Based Research Due to COVID-19

    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.