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Collaborative processes when reasoning creatively about functions
Ellen Kristine Solbrekke Hansen
To cite this version:
Ellen Kristine Solbrekke Hansen. Collaborative processes when reasoning creatively about functions.
Eleventh Congress of the European Society for Research in Mathematics Education, Utrecht Univer- sity, Feb 2019, Utrecht, Netherlands. �hal-02429067�
Collaborative processes when reasoning creatively about functions
Ellen Kristine Solbrekke Hansen
Norwegian University of Life Sciences, Faculty of Science and Technology, Norway;
Keywords: Collaborative processes, creative reasoning, function problems.
As observed for the last 50 years, students in upper secondary school struggle with the function concept (Dubinsky & Wilson, 2013). The function concept can be represented in different ways, for example as a graph, a table, or an algebraic expression. Translating from one representation to another, transforming a situation into a function, and discovering the function notation are important for students’ ability to “describe relationships of change between variables, explain parameter changes, and interpret and analyze graphs” (Clement, 2001, p. 745). Students’ exploration and engagement in making connections, for instance among different function representations, is important for understanding mathematical concepts and for learning to apply it as a tool in problem solving (Francisco & Maher, 2005; NCTM, 2014). In a student-centered environment with focus on collaboration, students have the opportunity to investigate, share and evaluate each other ideas.
Research on mathematical communication, supporting students’ collaborative activity and reasoning, is crucial for students’ mathematical understanding (Maher, Sigley, Sullivan, & Wilkinson, 2018;
Mueller, Yankelewitz, & Maher, 2012).
Lithner (2017) defines reasoning as “the line of thought adopted to produce assertions and reach conclusion in task solving” (p. 3), and proposes two main types of reasoning while solving a mathematical problem: imitative and creative. Students’ reasoning do not have to be formal, a mathematical proof or a high-quality argument, it simply has to make sense to the student himself (Lithner, 2015). Creative mathematical reasoning (CMR) sequences are created (or re-created) by the student, plausible to him, and anchored in intrinsic mathematical properties (Lithner, 2015). Imitative reasoning, on the other hand, refers to imitating a solution procedure or memorizing facts (Lithner, 2015). Imitative reasoning is closely linked to rote learning, whereas creative reasoning to a larger extent will promote conceptual understanding (Lithner, 2017). Roschelle and Teasley (1994) define collaboration as a “coordinated, synchronous activity that is the result of a continued attempt to construct and maintain a shared conception of a problem” (p. 70). If mutually engaging in problem solving “students share ideas and ways of solving problems; thus individual understanding becomes shared” (Mueller et al., 2012, p. 372). Creating a shared understanding between collaborating students, is what Roschelle and Teasley (1994) call a Joint Problem Space (JPS). Constructing JPS is processes of building (suggesting and agreeing upon ideas), monitoring (questioning and explaining ideas) and repairing (negotiations and corrections of conflicting ideas), through the use of language and the situation (Roschelle & Teasley, 1994).
There are many theoretical frameworks for analyzing students’ difficulties with the function concept, but not corresponding literature providing pedagogical strategies helping students overcome these difficulties (Dubinsky & Wilson, 2013). The current ongoing research project focuses on both student-student interaction, as well as the interaction between the student and the teacher. In line with Design-Based Research (DBR) methodology, three teachers and their mathematics students (age 16)
from a Norwegian upper secondary school participated in an iterative cycle for testing and developing an intervening design (Juuti & Lavonen, 2006). In the student-student interaction, the aim is to provide insight in the collaborative processes when reasoning about a function problem. Combining such insight with perspective on teacher-student(s) interaction, could give further insight to pedagogical strategies for promoting students’ reasoning in collaboration, for deeper understanding of mathematical concepts.
This paper poster will illustrate a student-student interaction in a pair creating their JPS through processes of building, monitoring and repairing, when engaging in creative mathematical reasoning.
Particularly, different aspects in their argumentation connected to the process of repairing their shared understanding. Some periods in their conversation were identified as CMR-sequences, characterized by turn-taking and creative mathematical reasoning solving a part of the function problem. However, not every uttering in the turn-taking was identified as creative. In situations where students had conflicting ideas, they often argued for their thoughts. Students used superficial arguments to support ideas, as well as arguments built on mathematical properties. If a counter-suggestion was made, without any support, and conflicting the other students’ idea, the following process was significant.
If only accepting the counter-suggestion, they were not likely to explore different representations and deeper meaning of the function concept together. Thus, experiencing ownership in the problem solving process, or over the mathematical content, was not likely to happen. If not agreeing on the counter-suggestion, the peer could ask questions about the idea, or make a counter-suggestion.
Anchoring the counter-suggestion, question or suggestion in properties regarding the function concept, were important for the process of creating the JPS. Hence, repairing of the shared conception of the problem, or sub-problem, indicated to a greater extent mutual ownership.
REFERENCES
Clement, L. L. (2001). What do students really know about functions? The Mathematics Teacher, 94(9), 745–748.
Dubinsky, E., & Wilson, R. T. (2013). High school students’ understanding of the function concept.
The Journal of Mathematical Behavior, 32(1), 83–101.
Francisco, J. M., & Maher, C. A. (2005). Conditions for promoting reasoning in problem solving:
Insights from a longitudinal study. The Journal of Mathematical Behavior, 24(3-4), 361–372.
Juuti, K., & Lavonen, J. (2006). Design-Based Research in Science Education: One Step Towards Methodology. NorDiNa, 4, 54–68.
Lithner, J. (2015). Learning mathematics by creative or imitative reasoning. In S. Cho (Ed.), Selected Regular Lectures from the 12th International Congress on Mathematical Education (pp. 487–
506). Cham, Switzerland: Springer.
Lithner, J. (2017). Principles for designing mathematical tasks that enhance imitative and creative reasoning. ZDM Mathematics Education, 49(6), 937–949.
Maher, C. A., Sigley, R., Sullivan, P., & Wilkinson, L. C. (2018). An international perspective on knowledge in teaching mathematics. The Journal of Mathematical Behavior, 51, 71–79.
Mueller, M., Yankelewitz, D., & Maher, C. (2012). A framework for analyzing the collaborative construction of arguments and its interplay with agency. Educational Studies in Mathematics, 80(3), 369–387.
NCTM. (2014). Principles to actions: Ensuring mathematical success for all. Reston, VA: NCTM.
Roschelle, J., & Teasley, S. D. (1994). The construction of shared knowledge in collaborative problem solving. NATO ASI Series F Computer and Systems Sciences, 128, 69–69.
