Stretching, sliding and strategy: indexicality in algebraic explanations

Abstract

In this study, we describe the linguistic expression of strategy in explanations of algebraic procedures. Stating the steps of an algebraic procedure does not require a student to indicate the relationship between different mathematical actions, but describing algebraic strategy does. The coordinated nature of strategic proficiency suggests that linguistic forms known as indexical language, ''pointing words'' that link speech to context, may be fundamental resources for expressing this type of competence. A class of first-year university mathematics students developed a habit of reporting procedures that we consider a speech genre. The classroom genre emphasised procedural explanations, but when students expressed strategic competence, they often relied on indexical language. Indexical verbs of motion like slide and drop proved to be a particularly efficient means of expressing algebraic strategies. This informal speech style extended the communicative capacity of the classroom speech genre, and allowed classmates to participate better in strategic mathematical reasoning.

Linguistic indexicality in algebra discussions

“…These verbs may represent what Talmy calls demonstrative paths in which a "fictively moving line functions to direct or guide someone's attention" (1999, p. 219). As we describe elsewhere, students in this class tended to offer algebraic procedures in terms of mathematical operations and resultant lines of text which the instructor wrote on the board (Staats & Batteen, 2009). The full class then evaluated the student conjecture until they reached a consensus.…”

“…In Talmy's perspsective, the verb slide may represent a targeting path, that is, "fictive motion (that) establishes a path along which the Agent further intends that a particular subsequent motion will travel" (Talmy, 1999, p. 220). We have suggested elsewhere that the verb slide may replace the operation subtract in this algebraic explanation (Staats & Batteen, 2009). While replacing the verb subtract with the verb of motion slide reduces the mathematical precision of the explanation, the directionality of slide illustrates the problem-solving strategy of isolating constants on the right side of the equation and variables on the left.…”

“…Talmy conceptualizes verbs of motion in terms of "fictive motion," descriptions of movement which does not actually take place, along various types of pathways (Talmy, 1999). Elsewhere, we have argued that verbs of motion occur in students' spontaneous classroom speech because the origo anchors attention in the representation of the problem, and the motion illustrates mathematical processes like addition or subtraction (Staats & Batteen, 2009).…”

Tense and aspect in word problems about motion: diagram, gesture, and the felt experience of time

“…Perhaps the complex grammar of verb forms is linked to these visual modalities in ways that contrast with the usual alphanumeric rendering of logical relations. Staats and Batteen (2009) have shown how word problems that entail motion verbs, like slide, drop, and pull, can be seen as indexical words because of the way they "anchor the motion of the verb" in the spatial context of the diagram rather than the actual spatial context where the "real" motion was occurring (p.59). This fact points to the complex ways in which language is fused with particular diagramming habits.…”

Metaphor and the Philosophical Implications of Embodied Mathematics

“…An extensive analysis of the embodiment of such abstract mathematical structures is pursued by Lakoff and Núñez (2000) in their book Where mathematics comes from. Speakers commonly refer to mathematical objects, such as numbers, in metaphorical terms, for example, in referring to "high numbers," "falling prices," and "rising taxes" (Lakoff and Johnson, 1980, p. 15-16), or when describing arithmetical operations in terms of such concrete actions as "sliding," "putting," or "taking away" (Staats and Batteen, 2009).…”