Prospective elementary teachers use of representation to reason algebraically

Abstract

a b s t r a c tWe used a teaching experiment to evaluate the preparation of preservice teachers to teach early algebra concepts in the elementary school with the goal of improving their ability to generalize and justify algebraic rules when using pattern-finding tasks. Nearly all of the elementary preservice teachers generalized explicit rules using symbolic notation but had trouble with justifications early in the experiment. The use of isomorphic tasks promoted their ability to justify their generalizations and to understand the relationship of the coefficient and y-intercept to the models constructed with pattern blocks. Based on critical events in the teaching experiment, we developed a scale to map changes in preservice teachers' understanding. Features of the tasks emerged that contributed to this understanding.

Student Strategies Suggesting Emergence of Mental Structures Supporting Logical and Abstract Thinking: Multiplicative Reasoning

“…Teacher observations and informal assessment techniques can be used in conjunction with the list of indicators generated in Table . In effect, this list is another method for teachers to assist students' progress toward understanding multiplication and in determining the trajectories of students' multiplicative reasoning so that teachers can remediate and develop steps toward additional understanding (Ball, ; Richardson et al., ).…”

“…Advanced mathematical thinking is viewed as beginning in elementary school with multiplicative reasoning (Clark & Kamii, ; Harel & Sowder, ; Piaget, ). Richardson, Berenson, and Staley () in addition to Ball () acknowledge the requirement for additional research concerning preservice as well as in‐service teachers' understanding of multiplicative reasoning.…”

Algebraic Reasoning in Solving Mathematical Problem Based on Learning Style

“…The problem solving strategy among pupils is different, this is influenced by the learning style that each pupil has [8]. Learning styles are a complex way in which pupils perceive and feel most effective and efficient in processing, storing, and recalling what they have learned [9]. Learning styles are classified into 3 types; visual, auditory, and kinesthetic [10].…”

Preservice Secondary Mathematics Teachers' Knowledge of Generalization and Justification on Geometric-Numerical Patterning Tasks

“…Radford (1996) argued that "the logical base underlying generalization is that of justifying the conclusion" (p. 111). Unfortunately, few researchers have studied how pre-service teachers' knowledge of generalization and justification are associated (e.g., Richardson, Berenson, & Staley, 2009). This study seeks to gain an understanding of each construct and the interaction between them within the context of quadratic relationships.…”

“…The researchers observed that some participants justified their rule by aligning their rules to figures presented in the task, whereas other participants aligned their rules to numerical values from the task to argue the appropriateness of their rule. Another example can be seen in the work of Richardson et al (2009) where the researchers studied elementary preservice teachers' ability to justify their rules to linear geometric-numerical patterning tasks. The researchers found that in the beginning of the study the preservice teachers had difficulty justifying their developed rules, often being able to start their justifications, but unable to complete them.…”