Le Roux, K. (2008). Relevance and access in undergraduate mathematics: Using discourse analysis to study mathematics texts. In J. F. Matos , P. Valero, & K. Yasukawa (Eds.), Proceedings of the Fifth International Mathematics Education and Society Conference (pp. 340-351). Centro de Investigação em Educação, Universidade de Liboa and the Department of Education, Learning and Philosophy, Aalborg University.
- I report on the use of discourse analysis to study the text of a mathematics problem which is used in a first-year university Mathematics course designed to give students access to tertiary study in Science. I use the method and tools of Gee (2005) to identify and explain (a) how the text presents the activity of answering a mathematics problem, and (b) how the text may position the student. I argue that this analysis raises questions about the concepts of relevance and access in undergraduate mathematics. (340)
- language-use is a social practice (Gee, 2005). From this perspective, language is not value-free and simply a grammar and set of rules for how to use this grammar. Rather, language is linked to the context in which it is used, and language forms take on meaning in particular contexts. Consistent with this perspective on language is the view of Mathematics as social practice. Mathematics is not viewed as skills-based and divorced from contexts, but is learned and used in social contexts (Baker, 1996). (342)
- I then explore the enacted identities in the texts and how the activity may position the student. I link the texts, via the concept of intertextuality to other texts, and via situated meaning, social language and Discourse to wider social practices. I present the textual evidence for my claims (although space prevents me from providing all the detail).
In relating the three texts to wider social practices I have named certain Discourses, for example, the “Access Discourse” and the “Everyday Discourse”. This process of naming and classifying has the effect of fixing these Discourses in time and space and of setting up boundaries. Yet, by definition, Discourses are overlapping, changing, and may vary across communities. This dilemma is noted by Moschkovich (2007) in her discussion of the naming of what she calls “Discourse practices”. Furthermore, my classification is influenced by my understanding of the setting of the study, and unavoidably reflects some value judgements about this setting. (345)
- Discourse analytic frameworks have been used to study how texts construct roles for and position users of mathematics texts (Herbel-Eisenmann & Wagner, 2007; Herbel- Eisenmann, 2007) (342)
- The initial analysis of the features of the text is done in two ways, as suggested by Gee (2005, p.54-58). “Form-function” analysis allows me to focus on the meanings communicated by particular textual features, for example, the layout, repetition, naming, etc. “Language-context” analysis enables me to study the specific meanings that different language forms take on in a particular context. Gee (2005, p.59) notes that in a specific context, a language form will take on a particular meaning, called a “situated meaning”. Certain features associated with a language form will be grouped together in a pattern, a pattern that a specific group of people find significant. (343)
- Secondly, Gee’s (2005) concept of the “seven building tasks” provides me with a systematic way to investigate how the textual features give meaning to the text. He argues that when we use language we build a “reality” by building seven things, which I give in italics in the following description. Gee claims that a situation in which language is used will involve activities in which people take on certain identities, develop relationships with one another and use certain sign systems and forms of knowledge. In such a situation certain things are given status and people and things take on meaning and significance and are connected or not connected to one another. (343)
Thirdly, how does one explain this “reality” in the context of wider social practice? Gee (2005) claims that different people will have differential access to identities and activities, and different value and status will be assigned to these identities and activities. Gee provides a number of tools that can be used to study how this may be done through language, two of which I have already described, namely “Discourse” and “situated meaning”. In this analysis I also use the concept of “intertextuality”; Gee (2005) argues that when we use language our words may reference other texts, either directly by quoting or indirectly by alluding to them. Lastly, the term “social language” is used by Gee (2005) to refer to the language aspects of a Discourse. (343)
- What does the text indicate about the nature of the activity identified as solving “related rates problems”? Firstly, the classification of the group of problems as “related-rates problems” suggests that certain problems in this Discourse can be grouped together according to their characteristics. Secondly, the insistence that the student follow the given five steps when solving the “related rates problems” suggests (a) that a systematic problem-solving approach is valued, and (b) that all the problems can be and should be solved in the same way. Thirdly, the five steps in the textbox, together with the worked solutions for the car problem, make it possible to identify that certain types of activity are valued, for example, presenting a full, neat, and systematic written solution, converting flexibly between different mathematical representations, carrying out certain mathematical procedures like differentiating, and having some conceptual understanding of the notion of instantaneous rate of change. The emphasis on both conceptual understanding and procedural proficiency can be linked to what is valued in the Calculus Reform Discourse. (347)
Firstly, the successful student is positioned as a student who has access to the First-year Undergraduate Mathematics Discourse. Such a student will have access to the situated meanings in the text, for example, knowing the pattern associated with
“related rates problems” and hence being able to recognise these problems in the Course material. S/he will also have access to the situated meaning of terms such as “correctly”, “diagram”, “correct notation”. The use of terms with particular meanings in the social language of the First-year Undergraduate Mathematics Discourse, for example, “variables” and “differentiate” positions the student as someone who understands and can use this social language.
Secondly, in order to arrive at a correct answer the successful student is required to demonstrate both conceptual understanding and proficiency in mathematical procedures such as differentiation, which can be related to the Calculus Reform Discourse. Thirdly, the successful student is required to deal with the real-world context of the cars appropriately, that is, s/he must choose only those aspects of the Everyday Discourse that are appropriate for a problem that is located in the First-year Undergraduate Mathematics Discourse. The successful student thus needs access to the assumptions of the School Mathematical Word Problem Discourse.
Continuing on the theme of the successful student, one could possibly argue that the presence of the problem-solving steps in the textbox construct this student as a problem-solver in mathematics, and as a student who can solve a real-world problem by mathematising it and using appropriate mathematical tools. However, I argue that certain features of the text may construct an identity for the student that conflicts with the identity of the successful student described so far. Firstly, the student is instructed to solve the “related rates problem” in a certain way, when the method presented is not the only possible method for solving the car problem. Secondly, the student is repeatedly reminded (with textual features such as upper case letters, bold and underlining) to follow these steps. Thirdly the student is presented with a hint for starting out with the problem (sentence 4 of the car problem). I argue that these textual features construct the student (a) as someone who needs help solving the car problem, and (b) as a student who does not usually do what is required when instructed to follow given problem-solving steps. The student is thus positioned as an “access student”. (347-348)