Researching the relation between language and mathematics
The themes collected in this section evolve around those questions about mathematics that regard the inner workings of this special discourse and its relation to other human activities. More specifically, I will be inquiring about linguistic mechanisms that a central role in its formation and about factors that set these mechanisms in motion.
Theme 1: Linguistic mechanisms that generate mathematical objects
At this point, it seems proper to return to one of the claims made above and still waiting for some elaboration: objects of mathematical narratives, rather than being ‘out there’ in the world, are in fact discursive constructs that emerge in the course of telling stories about them.The question to be asked now is why and how we use language to build the very universe we are talking about.
One of the main forces fueling the constant development of mathematics is the mathematician’s deeply felt need to make their storytelling as effective as possible, that is, to say more with less. This need is typically provided for with the help of recursive linguistic procedures in which new objects are built from those that have already been created before. There are at least three linguistic mechanisms that may be used for this purpose, thus helping in the discursive “compactification” (Sfard, 2008).They can be seen as techniques of objectifying (Sfard, 2008), as they all result in new mathematical objects. First, there is saming, the operation of giving one name to objects that, so far, seemed unrelated. This is what happens, for instance, when the noun quadrilateral is introduced in order to tell stories that hold simultaneously for parallelograms, trapezoids and kites. Another discourse-squeezing device is encapsulation - the technique of using a noun to talk in singular about properties that emerge when many objects are taken together.Thus, when one encapsulates all ordered pairs of numbers of the form
Let me conclude with just two examples of related questions for linguistic research that mathematics educators may find relevant to their concerns. First, more work is necessary to fathom how language is used in the gradual “squeezing” of mathematical discourse, thereby increasing its complexity without compromising its manageability. Second, there are questions about the function of different linguistic forms, that is, about ways in which people utilize these forms for their purposes.5 To give just one example, cleansing mathematical narratives from human subject, which often comes together with the three objectifying operations mentioned above, renders the resulting objects as-if mind independent and makes the stories about them sound as undebatable truths told by the world itself. This “alienation” of the object from its human creator can be seen in the example with which reification was illustrated in the previous paragraph. Alienation is this special linguistic transformation from which mathematicians, as proxies of the superhuman narrator, draw their position of power and authority. Those concerned about teaching and learning mathematics can certainly identify more linguistic techniques likely to have impact on the positioning of those who participate toward each other and toward the mathematical discourse itself.
Theme 2: The role of language in the historical emergence of mathematical discourses
In the light of what was said above, emergence and subsequent development of mathematics involve changes in both the vocabulary and syntax of this discourse. Some of the syntactical transformations - for instance, those responsible for objectification - have been listed in the previous section. In this one, the focus will be on the other, lexical change. More specifically, the question to be addressed now is that of what triggers the appearance of new mathematical terms and where these terms typically come from. I believe the answer one gives to this query may have important consequences for our understanding of how people learn mathematics. Whereas a thorough historical linguistic investigation is indispensable to find such an answer,6 some theoretically grounded ideas as to the possible responses may be formulated already now. Below, I present those of them that have also been partially corroborated in anthropological studies.
In the search after historical origins of different types of mathematical discourses let me consider one paradigmatic case: the discourse of mathematical objects that in the globalized school mathematics taught today around the world are known as natural numbers.7 Two properties make this discourse particularly appropriate for my present purpose: first, this discourse is a basis for all other mathematical discourses and second, it can be found in one form or another in almost any culture. Based on theoretical speculation as well as on some anthropological findings, the activity of counting, in which the numerical discourse begins, was invented in the context of the practical activity of choosing a larger of two sets. More specifically, it resulted from an attempt to extend this practice beyond those cases in which sets could be compared physically, for instance by being mapped one into the other (Lavie & Sfard, 2019). Invention of counting meant the introduction of a sequence of words that had to be recited in a constant order whenever one counted. As shown in cross- cultural studies, the numerical vocabulary could be composed of terms introduced specifically for counting and dedicated to their role of number words; or it could be borrowed from the existing linguistic repertoire. One of the best-known examples of this latter option was found in Papua New Guinea among Oksapmin people, who traditionally counted using names of body parts (Saxe, 2012). Unlike in this latter case, the sequences of number words to be found in many other languages are practically unbounded. This latter property is ensured by the possibility of creating ever new number words by combining the existing ones in certain well-defined ways. An interesting question that has already attracted considerable attention of mathematics education researchers is how a given system of number words, with these words and relations between them reflected to differing degree in these words’ composition and structure, impacts the numerical thinking of their users (Kilpatrick et al., 2001)
A similar question can be asked within the context of more advanced mathematics: What are the sources of words chosen by mathematicians to signify newly created mathematical objects? The striking feature of these signifiers is that, more often than not, they are by no means arbitrary hitherto-meaningless sounds. Rather, mathematicians turn to words that have already been in use, although not necessarily in mathematics. Often, the names for new objects come from a colloquial discourse, one that does not naturally bring any mathematical association.This was the case, for instance, when mathematicians introduced the objects we now call functions, or those we know as slopes, derivatives, matrices, fields, groups or negative numbers. Why this practice of calling new things old names? What are its benefits? The answer imposes itself on us the moment we recognize this mechanism as one of metaphorizing. Indeed, words imported to a new discursive context from their “native” discourse often bring with them old routines and endorsed narratives that prove helpful within the new context. Of course, as any other metaphor, they are a double-edged sword and, as already observed (see e.g. Pimm, 1987; Pirie & Kieren, 1994), their entailments may lead the user astray. I will say more about it when proposing agenda for research on language in the learning of mathematics (see Theme 5 below). For now, suffice to say that it may be useful to study metaphorical roots of mathematical terms, while also reflecting on both edges of the metaphorical sword - on the metaphors’ potentially helpful entailments and on those that must be deemed unhelpful.
Theme 3: Linguistic relativity of mathematics
The claim about cultural practices as the primary sources of mathematics harks back to the issue of the universality of this discourse. More specifically, one may wonder about how - and whether - the language in which one mathematizes impacts the process and product of this activity. This question takes us back to Sapir-Whorf Hypothesis (SWH), which generated much research in the 1940s, and whose popularity since then has known multiple falls and rises. Today, SWH seems to be making its comeback in a slightly revised form, under the name of theory of linguistic relativity.
In the eight decades of its existence, SWH has been presented in many different versions, sending varying messages about, first, the nature of the entity supposed to be dependent on language - from thinking, to perception, to one’s worldview; and second, about the strength of this dependence - from deterministic to merely possible. It seems that the inconclusiveness of the attempts at corroborating SWH - the direct reason for the vacillations of its popularity - stemmed from its being inherently ambiguous. Researchers were likely to interpret SWH in differing ways and may thus be looking for different kinds of evidence.
Discursive approach allows for operationalization. To do this, let me introduce the idea of isomorphism of discourses: two discourses in different languages count as isomorphic if the set of all the endorsed narratives in one of them, when translated to the language of the other, yields the set of all the endorsed narratives in this other discourse. In less technical words, isomorphic discourses, although differing in their language, can be seen as telling the same stories about the same sets of objects. In the case of mathematics, SWH may now be expressed as follows: Mathematical discourses in different languages do not have to be fully isomorphic. In contrast to the original versions of the thesis, this new operational formulation does not imply causal relations, grounded in problematic dichotomies. As such, it is fully researchable.
Cross-cultural research has already brought findings that can be seen as corroborating the thesis of linguistic relativity. To give just one example, studies have shown that some languages people use while going about their everyday affairs lack lexical tools for dealing with mathematical objects existing in other colloquial languages. This kind of phenomenon, known as semantic void, has been found e.g. in the traditional Tongan8 language, which did not have counterparts for the English terms for parts or fractions' (Morris, 2014).10 Attempts to explain this seemingly surprising absence led the researchers to the realization that, traditionally, Tongan people avoided dealing with parts of things and their cultural practices involved only whole objects. In culture in which there was no use for anything less than the unbroken whole, there was no need for a discourse of fractions.This, and many similar examples known from cross-cultural and ethnomathematical research, give the final blow to the myth of the universality of mathematics. And yet the absence of discursive isomorphism and the underlying cultural differences are not always readily visible. It is thus the LiME researchers important mission to sensitize teachers to the possibility that imperceptible semantic voids hinder mathematics learning of some students in multilingual classroom.