A Collaborative Learning Environment in a Student Centered Math Classroom: A Focus on the Beneficial Effects Through the Lens of Pirie and Kieren’s Model of the Growth of Mathematical Understanding

In a traditional classroom setting, often the teacher is the head of the student body, orchestrating the conversation and sporadically incorporating the voice of the students in a teacher-to-student conversation. In a setting that is more collaborative and student-centered, students lead the discussion and converse with one another. This creates an environment where every student has a greater opportunity to an opinion, demonstrating that each student’s voice has value. Research has shown that as students interact through discussions, they become more efficient in developing critical thinking, reasoning, logic, and problem solving skills (Cho & Jonassen, 2003; Jonassen & Kim, 2010). Furthermore, students are able to broaden and deepen their understandings of the content while also developing processes such as self-elaboration, self-explaining, and rethinking through collaborative discussions (Pi-Sui, Van Dyke, & Yan, 2015). According to Pirie and Kieren (1994), the growth of mathematical understanding is a “recursive process” which is “leveled but non-linear”. This can best be shown in a diagram containing eight concentric circles similar to that of a bull’s eye. The diagram illustrates that every “level” includes the understanding from the previous levels while also noting that the growth of understanding can move fluidly back and forth through the circles. The levels of mathematical understanding are as follows: Primitive Knowledge, Image Making, Image Having, Property Noticing, Formalising, Observing, Structuring, and Inventising. The first four are considered the informal levels while the second four are considered to be formal (Pirie & Kieren, 1994). The following analytic combines both of the ideas noted, indicating the benefits of a more student-centered classroom through the lens of Pirie and Kieren’s (1994) model for studying growth in understanding. The first portion of this analytic focuses on twelve 5th grade students who are identifying the number of possible pizza combinations from two toppings: sausage and pepperoni. After each student receives a copy of the problem, they begin by verbally discussing possible combinations. As the students continue working, they propose organizing their work after finding duplicates and having some confusion with notation. The group considers drawing images of pizzas with the different combinations and eventually organize their solution into categories: pizzas with single toppings, pizzas with two halves with different single toppings, and pizzas with both sausage and pepperoni. Together, these events are indicative of Pirie and Kieren’s (1994) first four (informal) levels of growth of understanding as noted in the description of each event. The second portion of the analytic exhibit the outer three (formal) levels of Pirie and Kieren’s (1994) model of growth of understanding. During this section, the analytic turns to focus on four 8th grade students, Brian, Michelle, Mike, and Romina, using Cuisenaire Rods to determine which rod is half the length of the blue rod. After working only seconds, one member exclaims that there would be no exact match because of the length and sizing of the rods. The group continues to work with the rods on a new challenge, determining a formula to identify the volume of any Cuisenaire rod if the white rod had the volume of one unit. Each member of the group continues to work on the challenge, checking their own work for accuracy before presenting their formula to the researchers. Later in their work, a fourth group member mentions the necessity of incorporating the rod’s width and height in the formula, regardless if the value is one. This boosts the group to developing a formula for identifying a rod’s volume while also keeping in mind the importance of units. Together, these events are indicative of Pirie and Kieren’s (1994) outer three (formal) levels for studying growth in understanding, as noted in the description of each event. In the final portion of the analytic, Romina, a student seen in both previous portions, is interviewed during her junior and senior year of high school. In 11th grade, Romina reflects on her experiences, expressing the benefits of group work and student centered learning. She indicates that because of the style of teaching she was exposed to during her education, including a collaborative environment, she is more likely to explore alternate approaches to finding solutions. Romina continues, noting how the students worked with one another, building their understandings of a concept from the very beginning, and in turn, encouraging a more long-term understanding of the material. In 12th grade, Romina discusses the role of the researchers and how they became facilitators asking thought provoking questions, rather than simply being reassuring adults, encouraging the students through each and every step of a problem. Romina indicates that while the student will obtain the correct answer with an adult at his or her side, he or she will not fully understand how that answer is reached without exploration. Cho, K. L., & Jonassen, D. H. (2003). The effects of argumentation scaffolds on argumentation and problem solving. Educational Technology Research and Development, 50(3), 5-22. Jonassen, D. H., & Kim, B. (2010). Arguing to learn and learning to argue: Design justifications and guidelines. Educational Technology Research and Development, 55(4),439-457. Pi-Sui, H., Van Dyke, M., & Yan, C. (2015). Examining the Effect of Teacher Guidance on Collaborative Argumentation in Middle Level Classrooms. Research in Middle Level Education Online, 38(9), 1-11. Pirie, S., & Kieren, T. (1994). Growth in mathematical understanding: How can we characterise it and how can we represent it? Educational Studies in Mathematics, 26(2), 165-190