- I consider the content and function of mathematical text. The term ‘text’ refers to written mathematical text, spoken text and texts presented multi-modally. I explore both the reading/listening and the writing/speaking dimensions of mathematical text in this broader sense. (65)
- Texts exist only through their material utterances or representations, and hence via their specific social locations. (67)
. Typically advanced mathematics text books conceal the processes of knowledge construction by inverting or radically modifying the sequence of transformations used in mathematical invention, for presentational purposes. The outcome may be elegant texts meant for public consumption, but they also generate learning obstacles through this reformulation and inversion (68)
- A semiotic system is defined in terms of three components, as follows (Ernest, 2005b, 2006):
3. An underlying meaning structure, incorporating a set of relationships be-
tween these signs.
The semiotic system of school algebra at the lower secondary school
level has for its signs constants (numerals), variable letters (x, y, z, etc.), a 1- place function sign (–), 2-place functions signs (+, –, x, /), a 2-place relation sign (=), and punctuation signs (parentheses, comma, full stop).2 These are typically represented as textual inscriptions on the chalkboard, in printed texts or worksheets or in student written work. In practice, the set of signs changes over the course of schooling. Early on, in the introduction to alge- braic notions during the later primary or elementary school years, a blank space ‘ ’, empty line ‘_’ or empty box ‘’ may be used instead of a variable letter. Later, after the initial development of school algebra in secondary school including those listed above, further primitive function signs are in- troduced, including sin, cos, tan, xc (a 1-place function sign, x raised to the power of a constant c), xy (a 2-place function sign with variables x and y), etc. In school algebra at all levels, the formal signs may also be supple- mented with written language (for example, in English or German). (68)
- Semantic rules concern the dimension of sign interpretation and mean- ing(s). Thus, for example, deriving ‘2x = 4’ from ‘2x+3 = 7’ in a semiotic system incorporating school algebra can be justified in terms of the mean- ings of the signs ‘2’, ‘3’, ‘4’, ‘7’, ‘x’, ‘+’ and ‘=’. In terms of significance, the dominant sign in these expressions is the binary equality relation ‘=’ and this has an underpinning informal meaning of balance that must be respected to preserve truth. The import of this is that whatever operation is applied to one of the binary relation sign’s arguments (one of its ‘sides’) must also be applied to the other. Another feature at play in this example is an implicit heuristic of simplification. This seeks to reduce the complexity of terms in an equation en route to solution. (68)
Pragmatic rules include contingent and rhetorical rules and these are de- termined purely by social convention. Examples include classroom stipula- tions as to how answers to mathematical tasks should be presented. In the past, sample rules that have been observed in use include teacher require- ments that students label answers with the prefix ‘Ans. =’, and double under- line the answer. Likewise, in university and research mathematics the end of a proof is commonly signified with the Halmos bar ‘’ (analogous to the classical QED). Such pragmatic rules are socially imposed or agreed con- ventions which are immaterial to syntactic and semantic correctness. (68-69)
- The underlying meaning structure of a semiotic system is the most elusive and mysterious part, like the hidden bulk of an iceberg. It is the re- pository of meanings and intuitions concerning the semiotic system which support its creation, development, and utilitization. For individuals it can range from a collection of tenuous ideas and fleeting images, to something more well defined, akin to an informal mathematical theory. The meaning structure of a semiotic system can be described in three ways: as a set of mathematical contents, an informal mathematical theory, or a previously constructed semiotic system. In any of the three cases, the meaning structure intersects, by necessity, with the performance norms of the social context (69)
As described above, solving problems and working mathematical tasks is based on the transformation of texts. The transformations employed are sign-based processes that follow the rules of the semiotic system. These rules, whether explicit or implicit, are the key operative mechanisms and principles through which new signs are formed and composite texts are constructed and elaborated. The overt function of these rules is to provide a technology for the transformation of mathematical signs in a goal directed way3; that is, a means for bringing the signs closer to some desired (and sometimes locally defined) canonical state, and in so do- ing preserving key invariants within the meaning structure. The two best known types of transformations, corresponding to two dominant problem solving activity types, are numerical calculation and proof, in which the in- variants preserved are typically numerical value and truth value, respec- tively. (69)
- Trying to teach rules explicitly rather than through exemplification can lead to what I have termed the ‘General-Specific paradox’ (Ernest, 2006). If a teacher presents a rule explicitly as a general statement, often what is learned is precisely this specific statement, such as a definition or descriptive sentence, rather than what it is meant to embody: the ability to apply the rule to a range of signs. (70)
Instead, mathematics refers to a semiotic space, a socially constructed realm of signs and meanings. Human beings are sign using and sign making creatures, and most humans can participate in the semiotic space of mathe- matics, even if only to a limited extent. (71)
- Halliday (1985) has developed the theory of Sys- temic functional grammar which provides an illuminating tool for the analy- sis of mathematics as a sign system. He distinguishes three metafunctions of text in use that can usefully be applied to mathematical text. These are the ideational, interpersonal and textual functions.
1. The ideational or experiential function concerns the contents of the
universe of discourse referred to, the subject matter of the text, the propositional content. This includes the processes described and the objects or subject matters involved in the process. Morgan (1998)
relates this to mathematical questions such as “What does this mathe- matical text suggest mathematics is about? How is the mathematics brought about? What role do human mathematicians play in this?” (p. 78).
2. The interpersonal function concerns the position of the speaker, the interaction between speaker and addressees, and their social and per- sonal relations. The related mathematical questions suggested by Morgan (1998) include “Who are the author and the reader of this mathematical text? What is their relationship to each other and to the knowledge constructed in the text?” (p. 78).
3. The textual function is about how the text is created and structured, and how it uses signs, and so on. Morgan (1998) relates this to mathematical questions such as “What is the mathematical text at- tempting to do? Tell a story? Describe a process? Prove?” (p. 78)
- The meaning of the signs of mathematics resides primarily in their uses and functions. Some signs, such as numerals, appear to represent objects, numbers in the case of numerals. Some signs represent operations on signs, and through them appear to represent processes applied to objects. (72)
Classroom texts spoken by the teacher, often in conjunction with other supporting modes of repre- sentation, position the actors in a number of reciprocal and pairwise defined roles including task setter – task performer, work manager – productive worker, assessor – assessee, knowledge giver – knowledge applier, knowl- edge owner – knowledge requester. In each case the student is in the second of the two roles, the less powerful position. This reflects that the teacher has two interrelated roles, namely as director of the social organisation and in- teractions in the classroom, i.e., social controller, and as director of the mathematical tasks and work activity of the classroom, i.e., task controller. (74)
- Halliday has argued that positionings in the text become a surrogate for social regulation. They stand in for and reproduce social structures and power differentials as experienced by children. Thus there is a “chain of de- pendence such as: social order – transmission of the social order to the child – role of language in transmission of the social process – functions of language in relation to this role – meanings derived from these functions” (Halliday, 1975, p. 5). Thus social structures and power relations are embodied in lan- guage uses (in discursive practices), and in particular, in the uses of texts. (74)
There is currently a growth of interest in research on mathematical identity within the mathematics education research community (e.g., Boaler et al. 2000, Lerman 2005). Little of this work as yet, however, treats the role of mathematical text in positioning its readers (and writers) and the impact of this on identity construction. (75)
- This raises the question: how do mathematical texts impact on their readers, and what is specific about these texts (and readers)? Do these texts take for granted certain competencies and address the reader in modes that de- mand she enact those competencies? The deepest analysis that promises answers to these questions is the semiotic theory of mathematics due to Brian Rotman. As part of his project on the semiotics of mathematics Rotman (1988, 1993) analysed the language of published research mathematics texts. He identified sentences to be the main linguistic units, and these to be made up of symbols, terms (nouns) and verbs.9 Following the traditions of literary and grammatical analysis, he takes the type of verb case in mathematical sentences to be the main indicators of the roles of author and addressee. He finds these verb cases to be of two main sorts. (76)
Rotman. As part of his project on the semiotics of mathematics Rotman (1988, 1993) analysed the language of published research mathematics texts.
First, there are verbs in the indicative mood, concerning the communica- tion or indication of information. In this case, “the speaker of a clause which has selected the indicative plus declarative has selected for himself the role of informant and for his hearer the role of informed” (Berry, 1975, p. 166). Thus, the speaker/author asserts to the hearer/addressee some state of affairs that obtains (or more commonly in mathematics where texts describe mathematical actions and processes) the outcomes of these processes. Such sentences not only describe the outcome of past, contingent sequences of actions and procedures—a particular transformation of signs—but also claim that when operating within the rules of the language game, the outcome de- scribed is what always must happen. The descriptions of these outcomes re- semble logical predictions, taking place in a timeless realm but describing the logical outcomes of the processes involved. Thus, indicative propositions might be said to describe thought experiments which persuade us to accept the validity of their assertions. (76)
Second, there are verbs in the imperative mood, asking for an instruction or action to be carried out. There are two forms:
(i) the inclusive imperative (e.g. “Let us define … .”, “Consider a lan- guage L …”), in which the addressee is required to cooperate or collaborate in following the speaker or carrying out the instruction in some imposed shared realm of discourse jointly, and
(ii) the exclusive imperative (e.g. “Add…”, “Count the cases…”, “Integrate the function…”) which asks or demands that an action be carried out by the hearer alone in a presupposed shared frame.
In both cases “the speaker of a clause which has selected the imperative has selected for himself the role of controller and for his hearer the role of controlled. The speaker expects more than a purely verbal response. He ex- pects some form of action.” (Berry, 1975, p. 166)…These findings are echoed in Shuard and Rothery’s (1984) analysis of school mathematics texts. They found several types of text, each with its own purpose: Exposition, Instructions, Examples and exercises, Peripheral writing, and Signals. (76)
Another type of language in school mathematics texts utilizes the imperative mood. This includes instructions to the reader to write, draw or to perform some action, typically utilizing direct imperatives. It also includes examples and exercises for the reader to work on. Often these are routine problems involving the application of specific procedures to symbols, but they can also include word problems, non-routine problems and investigative work requiring the use of general heuristics to guide solutions. These tasks may be expressed with direct imperatives, but can also utilize implied im- peratives if they are in question form or contain the substance of imperatives without the appropriate verb. (77)
- Deconstructionist readings of domains like mathematics reveal that Huhs don’t flag errors; they don’t point to inter- pretations that need to be corrected or replaced. Rather, they announce interpretations that seem to be calling for elaboration—to be blended with a new image, a novel metaphor, a broader analogy. (82)
We knowers, and we who seek to prompt the knowing of others, are storytellers, poets, weavers. Our shame is in our vanity, in our eagerness to dis- play the artfulness of our finished weaves/texts. (84)