Encountering Proportional Reasoning During a Single Algebra Lesson: A Microgenetic Analysis

Abstract

This case study explores how 12-13-year-old students encounter proportional reasoning while working with geometric patterning tasks using concrete materials. The focus is on the students' use of spontaneous concepts when first dealing with such patterns in the context of collaborative work. Based on video recordings of a single lesson, a microgenetic analysis was performed to identify students' learning trajectories, starting with students familiarizing themselves with pattern structure, followed by engagement in proportional reasoning, and ending with students perceiving a new technique to handle a situation where proportional reasoning did not suffice. While some student groups were able to move along the whole trajectory, most groups, when facing challenges, regressed to simpler techniques. The results provide new insights into students' learning trajectories, which can be used to support students' progress in the context of student-teacher interaction.

The Effect of Error Analysis-Based Learning on Proportional Reasoning Ability of Seventh-Grade Students

“…Moreover, the study gains the importance of dealing with proportion and related topics that are difficult for students, as there are many errors and wrong perceptions of their basic concepts, in addition to their importance in mathematics, and their applications in daily life and different sciences. In addition to the study importance in the improvement and development of proportional reasoning, which is the cornerstone of mathematics in the lower and upper grades, as being one of the challenges that teachers face in teaching it, and students face in learning it, besides being an indicator for moving toward higher order thinking patterns (Van Dooren et al, 2003;Lundberg, 2022).…”

“…Many aspects of life proceed according to the proportional rules that make the ability to proportional reasoning necessary to explain the different phenomena around us, and it is the cornerstone of mathematics in the lower and upper grades, and an indicator of movement toward higher thought patterns (Van Dooren et al, 2003;Holzrichter, 2016;Lundberg, 2022). Proportional reasoning is also considered an important way to develop algebraic thinking and to understand the meaning of function (Cai and Sun, 2002).…”

“…Proportional reasoning is defined as the reasoning in a system of two variables between which there is a linear functional relationship (Holzrichter, 2016;Lundberg, 2022). According to Lamon (2007), proportional reasoning is the discovery, expression, analysis, interpretation, and presentation of evidence about proportional relationships.…”

“…Reys et al (2009) regarded proportional reasoning as understanding proportional relationships and working with these relationships. Arican (2019b) and Lundberg (2022) considered the development of proportional reasoning as one of the most challenging aspects facing the individual in the ability on mathematical reasoning, as it requires the learner to be able to deal with multiple concepts and skills related to rational numbers, comparison, simplification, and operations on them, especially multiplication and division. These concepts and skills are learned over a long period by placing the learner in various life situations that he models, to be used in proportional reasoning.…”

“…These concepts and skills are learned over a long period by placing the learner in various life situations that he models, to be used in proportional reasoning. Lundberg (2022) has gone further in considering proportional reasoning to be the application of mathematics to real-life.…”