FLM 38(2) - July 2018 - text_FLM 2018-06-13 5:52 PM Page 34 CONTEXTUAL MEANINGS OF THE EQUALS SIGN AS CONCEPTUAL BLENDS GEORGE BOGGS, IAN WHITACRE, JENNIFER SCHELLINGER, ZACHARY CHAMPAGNE, ROBERT SCHOEN Students today may more readily recognize the symbol # as problem from the standpoint of algebra learning. Framing the hashtag used on Twitter and other social-media plat- the problem in terms of future roadblocks to high-school forms than as a symbol for weight, musical notation for a mathematics achievement, equals sign interventions have sharp note, the pound key on a telephone, or as shorthand for introduced typologies and made instructional recommenda- number. Digital media add to the many streams of symbol- tions focussed on reasoning relationally. Some seek to shape rich information children experience, so recognizing can be the discursive landscape in which children encounter the seen as an important and active interpretive process affected equals sign, for instance, by adopting the language of same- by experience, awareness, and sense of ownership. In social ness (e.g., reading the equals sign as ‘is the same as’) media, mathematical language and symbols are being appro- (Saenz-Ludlow & Walgamuth, 1998), using true-false ques- priated in ways that stretch their meaning and appear to tions (Molina & Ambrose, 2006), or introducing the create new combinations. This process, which children metaphor of the balance scale or seesaw (Mann, 2004). experience simultaneously with formal instruction, presents Efforts to steer children’s understanding of the equals sign mathematics educators and researchers with both challenges through the use of particular semiotic tools depend on recog- and opportunities. nition that differing views of the sign may coexist in the Signs and texts are subject to reinterpretation and contex- mind of the child. In other words, children’s understandings tualized translation into new contexts. Mathematics of the equals sign are negotiated, not simply sequenced education research faces the challenge of embracing multi- developmentally. Light (1980) identified six meanings of the literate mathematical thinking to support students’ adaptive equals sign within mathematics; Baroody and Ginsburg production and consumption of mathematical texts. We (1983) raised the possibility that multiple views coexist; and argue that the blending of mathematical symbols, expres- Jones and Pratt (2012) found that some students used two sions, and thinking from formal to informal uses and from distinct meanings of the equals sign, a relational meaning school to non-school contexts represents a normal multiliter- and a substituting meaning, to create arithmetic puzzles. ate feature of 21st century life, a process that children can Documented cases of children flexibly recruiting different become aware of and use as they expand their communica- meanings to successfully complete tasks point to the impor- tive repertoires both in and out of school. We are interested tance of the child as a negotiator of meaning in relation to in how a ubiquitous symbol—the equals sign—is appropri- signs and context. ated and used contextually in advertising and social media. Our analysis explores how people use and see the equals In cross-cultural contexts, ‘distance’ between disciplinary sign in contexts outside of mathematics, especially social norms for mathematical communication can clash with local media and advertising. We consider the variety of meanings linguistic and cultural histories (Berry, 1985, p. 19). And that these ways of using the equals sign evoke as evidence of ambiguity between everyday and specialized meanings of the multiliterate environments that people navigate on a mathematical words and symbols can be a source of mathe- daily basis. We propose that these media-saturated environ- matical ‘misinformation’ (Durkin & Shire, 1991, p. 73). ments call for pedagogical approaches to children’s Diverse appropriations of signs within mathematics dis- mathematical understanding that take into account the lin- courses point to deep polysemy, so strong ‘awareness’ of guistic and cultural contexts in which equals signs are used. context may be seen as an antidote to misinformation This perspective has implications for efforts to assess human (Zazkis, 1998, p. 30). These issues can be viewed as central understanding of words and symbols, because understanding to young learners’ experience, as ‘new’ contexts suggest and language use occur in a variety of contexts. meanings that contrast with ‘familiar’ ones (Mamolo, 2010). Conceptual blending Views of the equals sign We frame our analysis of contextual meanings of the equals Progression from simpler to more complex mathematical sign in terms of linguistic pragmatics, specifically how con- thinking is a vital concern for educators and researchers. ceptual blending (Fauconnier & Turner, 2002) supports sign Developing awareness in research of the need among teach- users’ interpretation of abstract symbolic meaning (Peirce, ers and students to manage ambiguity contrasts to some 1903). Conceptual blending is a process by which elements degree with research and teaching goals having to do with from different source domains, or input spaces, are comin- the equals sign. gled to create a unique, blended space, resulting in both An operational view of the equals sign among school chil- gains and losses in terms of what people appear to under- dren (e.g., Knuth, Stephens, McNeil & Alibali, 2006) is a stand (Zandieh, Roh & Knapp, 2014). A blend involves a 34 For the Learning of Mathematics 38, 2 (July, 2018) FLM Publishing Association, Fredericton, New Brunswick, Canada FLM 38(2) - July 2018 - text_FLM 2018-06-13 5:52 PM Page 35 combination of elements from each source domain or hybrid Ways of using the equals sign in advertising elements influenced by both source domains. Conceptual and social media blends are ubiquitous and may be found in everyday lan- Below, we consider ways in which the equals sign is being guage, politics, literature, technology, mathematics, and used in advertising and social media, sorted into categories advertising. In mathematics education research, conceptual based on the sorts of conceptual blends we created as we blending has been used to investigate students’ mathematical read the expressions. We analyze several examples by fram- reasoning (Zandieh et al., 2014), as well as preservice teach- ing them as conceptual blends that bring together distinct ers’ feelings about mathematics (Zazkis, 2015). Here, we use domains. The categories presented in this article exhibit the it to illustrate the existence of divergent meaning-making diversity of conceptual blends, not a comprehensive or pathways. authoritative sorting of possible blends. Nor do we attempt We diagram a variety of conceptual blends we created as to quantify the relative frequencies of these ways of using we recognized the equals sign in social media and advertise- the equals sign in the world. Our purpose is to demonstrate ments. Figure 1 shows how we illustrate potential source the context-dependent variability of the equals sign’s mean- domains, corresponding elements, and a blended space [1]. ing and to consider the implications of conceptual blending Within the circle representing Source Domain 1, we present for contemporary mathematics education. We describe the phenomena belonging to that source domain relevant to the complex thinking involved in these blends as manifestations blend. We then specifically identify the elements that are of often invisible sense-making processes children and involved in our blend. We do likewise for our Source adults readily perform. This inter-psychological work neces- Domain 2, and we use dotted line segments to illustrate the sitates adaptive understanding of formal mathematical fusion of elements into the blended space. To further clarify language appropriated in unpredictable but intelligible ways. our conceptualization of a given blend, we also describe the categorical nature of each element in the source domains and Plus Result then the categorical nature of the elements in the blended Figure 2 is an advertisement warning against drunk driving, space. The latter may derive from one or the other of the especially the driving of motorcycles. The statement in this source domains or involve a fusion of meanings deriving case is ‘DRINK + RIDE = LOSE’ with the further dramatic from the two domains. specification that ‘FREEDOM!’ is what will be lost [2]. Of special importance to our inquiry is the pragmatic lin- Plus Result statements specify two or more inputs or condi- guistic principle that symbols acquire their potential to mean tions that together yield some result or consequence. through a variety of cultural processes, with the result that Another example is ‘Sweat + Sacrifice = Success’. In such symbols can have competing meanings by virtue of their dif- statements, we infer the meaning for the equals sign as fering histories (Peirce, 1903). The construction of relations ‘yields’ or ‘results in’ or ‘causes’. Plus Result statements are between symbolic signs and their objects, in this case a con- one-directional and meant to be read from left to right. Play- cept or perhaps multiple concepts, sets them apart from other ful manipulations (based on the rules of arithmetic) yield signs, resulting in layers of potential meaning for symbols as entertaining but nonsensical statements, such as ‘LOSE – they are imported into new contexts. DRINK = RIDE’. Plus Result