Taxonomy for Assessing Conceptual Understanding in Algebra Using Multiple Representations
Algebra students may often demonstrate a certain degree of proficiency when manipulating algebraic expressions and verbalizing their behaviors. Do these abilities imply conceptual understanding? What is a reliable indicator that would provide educators with a relatively trustworthy and consistent measure to identify whether students learn algebraic concepts beyond procedures? The article introduces taxonomy for assessing middle school algebra students' levels of understanding of linear equations with one unknown. The taxonomy is rooted in ideas related to the degree of abstraction and reducing level of abstraction, theory of operational and structural conception, and the concept of representation in mathematics education. Introduction The concept of unknown and operations with unknowns are central to teaching and learning middle school algebra. Students often reveal a certain degree of proficiency manipulating algebraic symbols, and when encouraged, can verbalize and explain the steps they perform, thereby demonstrating awareness of procedures according to fixed rules. It is well documented however that correct and seemingly fluent demonstration of a procedure does not necessarily indicate conceptual understanding (Herscovics, 1996; Herscovics & Linchevski, 1994; Kieran, 1992; Kieran & Chalouh, 1993; Langrall & Swafford, 1997). To objectively analyze and assess students' thinking and behaviours, and make informed decisions about learning algebraic concepts beyond procedures, educators need reliable indicators to refer to, and simple, yet structured language to communicate. This paper describes a segment of a large scale longitudinal study on representations in algebra and suggests the taxonomy for assessing conceptual understanding of linear relationship with one unknown. Three key ideas serve as the foundation of this research study: a) the role of multiple representations in probing understanding of mathematics learning, with a particular focus on the interpretation, connection and translation among representations of the structurally same relationship, b) the idea of adaptation to abstraction and the development of conceptual understanding as a process of growth of the degree of abstraction, and c) the idea of reducing the level of abstraction as a mental process of coping with abstraction level of a given concept or task. These ideas guided the researcher's observations, analysis, and formulation of the taxonomy. Foundation This section briefly outlines major theoretical positions and research that pertained to and served as a background for this research study. Algebraic reasoning and conceptual understanding in algebra Algebraic reasoning has been described as a process of generalizing a pattern and modeling problems with various representations (Driscoll, 1999; Herbert & Brown, 1997). Langrall and Swafford (1997) defined algebraic reasoning as the "ability to operate on an unknown quantity as if the quantity is known" (p. 2). Understanding in general and in mathematics in particular is a logical power manifested by abstract thought. Conceptual understanding in algebra is demonstrated by the ability to recognize functional relationships between known and unknown, independent and dependent variables, and to discern between and interpret different representations of the algebraic concepts. It is exemplified by competency in reading, writing, and manipulating both number symbols and algebraic symbols used in formulas, expressions, equations, and inequalities. Fluency in the language of algebra demonstrated by confident use of its vocabulary and meanings as well as flexible operation upon its grammar rules (i.e. mathematical properties and conventions) are also indicative of conceptual understanding in algebra. Representations Symbol systems and representations are essential to mathematics as a discipline since mathematics is "inherently representational in its intentions and methods" (Kaput, 1989, p. …