Academic journal article Education

Three Phase Ranking Framework for Assessing Conceptual Understanding in Algebra Using Multiple Representations

Academic journal article Education

Three Phase Ranking Framework for Assessing Conceptual Understanding in Algebra Using Multiple Representations

Article excerpt


The concept of unknown and operations with unknowns are central to teaching and learning middle school algebra. Learning about linear relationship with one unknown, students may often demonstrate a certain degree of proficiency manipulating algebraic symbols. When encouraged, they can often verbalize and explain the steps they performed, thereby demonstrating awareness of well-known procedures with symbols according to fixed rules. These students show a certain level of "operational conception" (Sfard, 1991, p. 4) or "process conception" (Dubinsky, 1991; Dubinsky & McDonald, 1991, p. 3). It is well known and documented (Herscovics, 1996; Herscovics & Linchevski, 1994; Hiebert, 1988; Hiebert & Carpenter, 1992; Kieran, 1989, 1990, 1992; Kieran & Chalouh, 1993; Langrall & Swafford, 1997) that correct and seemingly fluent demonstration of a procedure does not predictably indicate conceptual understanding or as Skemp (1976) suggested "relational understanding" (p. 21).

In order to make informed decisions and avoid relying on subjective perceptions and feelings whether students learn algebraic concepts beyond procedures, educators are searching for consistent and dependable tools to objectively assess student thinking and behaviours. What would be a reliable indicator that could serve as relatively trustworthy and consistent measure of students' conceptual understanding? When does the transition from 'operational' or 'process conception' to 'structural' or 'object conception' take place? To answer these questions and to extend our knowledge of the processes related to developing conceptual understanding in algebra, the researcher launched a longitudinal study which led to the formation of the three phase ranking framework of conceptual understanding of linear relationship with one unknown described in this paper.


Three key ideas served as the foundation of this research study: a) the role of multiple representations in probing understanding of mathematics learning, b) the theory of reducing level of abstraction as a mental process of coping with abstraction level of a given concept or task, and c) the idea of adaptation to abstraction and the development of conceptual understanding as one's ability to cope with higher levels of abstraction. These ideas guided the researcher's observations, analysis, and generation of the language that helped communicating some features of the development of conceptual understanding of linear relationship with one unknown. This section briefly outlines major theoretical positions and research that pertain to and serve as a background for this research study.

i) Algebraic reasoning and conceptual understanding in algebra

The term algebraic reasoning has been used to describe mathematical processes of generalizing a pattern and modeling problems with various representations (Driscoll, 1999; Herbert & Brown, 1997; NCTM, 2000). Driscoll (1999) defined algebraic reasoning as the "capacity to represent quantitative situations so that relations among variables become apparent" (p. 1). For Langrall and Swafford (1997) algebraic reasoning is "the ability to operate on an unknown quantity as if the quantity is known" (p. 2). Vance (1998) characterized algebraic reasoning as a way of reasoning involving variables, generalizations, different modes of representation, and abstracting from computations. Kaput (1993) viewed algebraic reasoning as a process of construction and representation of patterns and regularities, deliberate generalization, and active exploration and conjecture. These definitions will serve as a basis to explain conceptual understanding in algebra in this paper.

Understanding is a logical power manifested by abstract thought. Piaget (1995) suggested that understanding in general and in mathematics in particular is a highly complex process of abstraction. He proposed the term reflective abstraction (Piaget, 1970, p. …

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