Symbol Sense Framework

The proponents of the symbol sense framework are Fey (1990) and Arcavi (1994).

Symbol sense is considered as the heart of algebraic competency (Arcavi, 1994). It is difficult to define symbol sense because it interacts with other senses like number sense, function sense, and graphical sense in problem-solving situations. Arcavi (1994) made a remarkable attempt to characterise symbol sense through a rich variety of examples and illustrations of mathematical behaviours (Zehavi, 2002). Kinzel (2001) describes symbol sense as the combination of notational awareness of expressions and the skill to manipulate and interpret these expressions. Boero (2001) uses the terms “transformation and anticipation” to analyse behaviours in algebraic problem solving. He refers to the continuous tension between “foreseeing and applying” as a dialectic relationship. Zorn (2002) viewed symbol sense as the ability to extract mathematical meaning and structure from symbols, to encode meaning efficiently by symbols, and to manipulate symbols effectively to discover new mathematical meaning and structure. In order to be proficient, mathematics learners must acquire an understanding of letters, variables and objects (Arcavi, 2005).

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Arcavi (2005) argues that having ‘symbol sense’ is central to mathematics learning and good teaching aims to achieve ‘symbol sense’. Symbol sense is an essential prerequisite for advanced mathematics and science and is the primary purpose of algebra (Sullivan, 2013). Some keys themes for teaching symbol sense were suggested by Fey (1990) and Arcarvi (1994). Arcavi (1994) modified the list proposed by Fey (1990) and considers that the symbol sense must include, among others, an understanding of and an aesthetic feel for the power of symbols, which brings the idea of visual salience (Kirshner & Awtry (2004); an ability to manipulate and to "read" symbolic expressions as two complimentary aspects of solving algebraic problems. Arcavi (1994) further asset that knowing the algebraic manipulations to solve problems it is not enough, instead it is necessary to understand the meaning of the symbols. He identified four key behaviours: reading instead of manipulation of the symbols; reading and manipulation; reading as the goal for manipulation, reading for reasonableness.

Goldin (2002) explains that communication in mathematics is viable if symbolic systems are understood and relations between systems could be used to enhance symbolic understanding. Holmqvist et al, (2011) define symbol sense as a complex and multifaceted "feel" for symbols. Zehavi (2002), like Arcavi (2005) conceded that it is difficult to define symbol sense because it interacts with other senses like number, function and graphical in problem-solving situations. In an attempt to define symbol sense, Hawkins and Allen (1991) described it as an accurate choice of symbols to represent a mathematical situation or concept. Pope and Sharma (2001) provided a comprehensive definition in which they defined symbol sense as the ability to appreciate the power of symbols, to know when the use of symbols is appropriate, and to manipulate and make sense of symbols in a range of contexts. Thus, there is no concise definition of symbol sense but descriptions of behaviours that illustrate whether a learner has symbol sense or not.

102 Arcavi (1994) characterised symbol sense as an:

a) understanding of situation and stage where symbols can be and should be used in order to display relationships;

b) ability to abandon certain symbols in favour of other approaches in order to make progress in solving a problem;

c) ability to carry out mathematical processes and to “read” symbolic expression as complementary aspects of solving algebraic problems;

d) awareness that one can initiate symbolic relationships that express the verbal or graphical information needed to make progress in solving a problem; or

e) ability to select a possible symbolic representation of a problem.

Learning algebra requires learners to have symbol sense (Naidoo, 2009). Algebra involves much more than mastering basic skills; it also involves choosing sensible strategies to tackle problems, maintaining an overview of the solution process, creating a model, taking a global view of expressions, wisely choosing subsequent steps, distinguishing between relevant and less relevant characteristics and interpreting results in meaningful ways. Symbol sense is regarded as a type of meta-knowledge in algebra. Symbol sense involves the flexible algebraic expertise or algebraic literacy that often operates in the background without our conscious awareness. Based on insight into the underlying concepts, it directs the implementation of the basic routines. It plays a role in planning, coordinating and interpreting basic operations and consists of three interrelated skills:

i. The strategic skills and heuristics to approach a problem; the capacity to maintain an overview of this process, to make effective choices within the approach, or if a strategy falls short, to seek another approach.

ii. The ability to view expressions and formulas globally, to understand the meaning of symbols in the context and to formulate expressions in another way. Process- object duality plays a role in that skill.

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iii. The capacity for algebraic reasoning. This often involves qualitative reflections on terms and factors in expressions, symmetry considerations or reasoning with particular or extreme cases.

Arcavi (1994) states that many learners fail to see symbols as tools for understanding, communicating, and making connections, even after several years of study. He sees development of symbol sense as a necessary component of sense-making in mathematics. He argues that having ‘symbol sense’ is a fundamental requirement for the study of mathematics especially algebra.

Bergsten (2000) describe symbol sense as an appreciation for the power of symbolic thinking, an understanding of when and why to apply it, and a feel for mathematical structure. Adams, Pegg and Case (2015) compared symbol sense with number sense and found it to be a higher level of mathematical literacy. Wu (2009) explains that communication in mathematics is viable if symbolic systems are understood and relations between systems could be used to enhance symbolic understanding. Arzarello, Ferrara, Robutti and Sabena (2009) urged learners to acquire skills in manipulating various symbols in order to solve a mathematical problem or to prove a formula. Research has revealed how learners interpret and make use of mathematical symbols, a facet of the work on symbol sense. Arcavi (1994) described it as “making friends with symbols” (p. 25), including an understanding and feel for symbols, how to use and read them. While solving a mathematical problem, the learner is required to analyse, identify and recognise the relevance of critical areas of a mathematical representation. Kenney (2008) adopted a symbol sense framework constructed using the work of Pierce and Stacey (2001, 2002) and Arcavi (1994, 2005), to investigate learners’ reasoning with mathematical symbols at different problem solving stages. She identified the following components of symbol sense: