Mary J. Schleppegrell Synthesizes Literature on the Linguistic Challenges of Math

Posted on September 9, 2012 by

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Schleppegrell, Mary J. 2007. “The Linguistic Challenges of Mathematics Teaching and Learning: A Research Review.” Reading & Writing Quarterly 23 (2): 139–159. doi:10.1080/10573560601158461.
Methods:
  • This article synthesizes research by applied linguists and mathematics educators to highlight the linguistic challenges of mathematics learning and suggest some pedagogical practices that may help learners develop mathematical understanding. (139)
Findings:

  • Halliday defined ‘register’ as “a set of meanings that is appropriate to a particular function of language, together with the words and structures which express these meanings. We can refer to a ‘mathematics register’, in the sense of the meanings that belong to the language of mathematics (the mathematical use of natural language,that is: not mathematics itself ), and that a language must express if it is being used for mathematical purposes.” (p. 195) (140)
  • Features of the Classroom Math Register
    Multiple semiotic systems
    .mathematics symbolic notation
    . oral language
    . written language
    . graphs and visual displays
    Grammatical patterns
    . technical vocabulary
    . dense noun phrases
    . being and having verbs
    . conjunctions with technical meanings
    . implicit logical relationships
  • (O’Halloran, 1999, p. 24). Language provides the contextual information about the situation, the mathematics symbolism describes the pattern of relationships between the entities, and the diagram provides a connection between the material world (a cliff and a river) and the mathematical processes that are constructed in
    the problem, a connection that was formulated in oral language in the classroom. Thus, the written language, the mathematics symbolic statements, the visual representation, and the oral language work together to construct meaning as the teacher and students interact in discussing the problem. (142)
  • MacGregor found that ‘‘students who described a relation between numbers in an informal, unclear or immature way were unable to relate it to a mathematical operation’’ (2002, p. 1). This makes it crucial for teachers’ and students’ talk to apprentice students into the technical language of mathematics. (143)
  • Teachers typically recognize the technical vocabulary as a challenge, but may not be aware of the grammatical patterning that technical vocabulary brings with it. This grammatical patterning includes the use of long, dense noun phrases such as the volume of a rectangular prism with sides 8, 10, and 12 cm. Such noun phrases often have pre-numerative phrases that name an abstract, but quantifiable, mathematical attribute of the head noun (e.g., the volume of, the length of ), classifying adjectives that precede the noun (e.g., prime number; rectangular prism), and qualifiers that come after the noun (e.g., a number which can be divided by one and itself; prism with sides 8, 10, 12 cm; examples from Veel, 1999). (143)
  • These long noun phrases then participate in constructing complex meaning relationships in the problems students have to solve. Typically, they participate in relational processes constructed in clauses with be and have (Veel, 1999). These verbs construct different kinds of relational processes that are common in mathematics, attributive and identifying processes. An attributive process constructs information about membership in a class or part-whole relationship (e.g., A square is a quadrilateral or Three and four are factors of twelve). An identifying process, on the other hand, constructs relationships of identity and equality, as in A prime number is a number that can only be divided by one and itself or The mean, or average, score is the sum of the scores divided by the number of scores. What is key is that the attributive clause is non-reversible; it is not true to say that A quadrilateral is a square, or The factors of twelve are three and four, whereas the identifying clauses construct relationships of equality, and hence, are reversible. So we can say A number that can only be
    divided by one and itself is a prime number or The sum of the scores divided by the number of scores is the mean, or average, score (examples from Veel, 1999, p. 195). Attributive clauses classify objects and events, while identifying clauses define technical terms and provide a bridge between technical and less technical ways of presenting knowledge in mathematics by enabling two formulations to be presented as equivalent (for example, Sides of the triangle that are in the same positions are corresponding sides of the triangles). Relational processes are also a feature of the multiple choice questions that are often used to assess students’ mathematics knowledge on standardized tests, as they ask which of the following is correct=true=the best way, etc. (Veel, 1999). The verbs be and have and other related verbs (means, equals, etc.) are challenging in their grammatical features.
    Students with first languages other than English may be accustomed to constructing relationships of attribution and identity in different ways than English. In Spanish, for example, the verb is has two different forms, construing different meaning relationships. (143-144)
  • Another challenge of the mathematics register is the precise and technical meanings of conjunctions that may be used in different ways in ordinary everyday language. In word problems, and in developing theorems and proofs, conjunctions such as if, when, and therefore are used in precise ways, and constructions such as given and assume take on new roles. Conjunctions often link clauses in complex ways, with variations presenting similar meanings. In addition, the mathematical operations that are used to construct mathematical reasoning are sometimes left implicit (O’Halloran, 1999, 2000). O’Halloran shows how solving mathematics problems often involves ‘‘long chains of reasoning that provide little or no indication of the results, definitions, axioms, operational properties or laws that have been used’’ (2000, p. 377). For example, in geometry proofs, the various properties and postulates that underlie the argument made in the proof are not spelled out, but rather are assumed to have been already learned and internalized. Chapman’s (1995) linguistic analysis illustrates that even in mathematics textbooks, these underlying principles are often left implicit.(144-145)
  • The linguistic aspects of mathematics that distance it from ordinary use of language include the multiple semiotic systems that bring together symbolic representations and visual images that do not match up exactly with their ‘‘translation’’ into the oral and written language used to develop the meanings they present. In addition, the technical vocabulary and grammatical structuring associated with it make the oral and written language challenging in its own right. The grammatical patterning brings together long, dense noun phrases in clauses and sentences constructed with being and having verbs that present a variety of meaning relationships. In addition, mathematics problems often use conjunctions that have meanings different from their everyday uses, or include implicit logical relationships that are not spelled out (145)
  • O’Halloran (2003, p. 196) illustrates this. She shows how a mathematics problem that is presented in math symbolism requires dense nominal structure when translated into words. Here is the
    problem:
    a^2 + (a +2)^2 = 340
    In written language, this equation can be represented as: The sum of the squares of two consecutive positive even integers is 340. (145)
  • As O’Halloran points out, what the language encodes as one thing, in the dense noun phrase The sum of the squares of two consecutive positive even integers, is represented in mathematical symbolism as
    a series of processes; squaring a, squaring ab + 2, and adding those products together, and that this thing is then equated, using is, with 340. As we have seen, the grammatical patterning of mathematics often presents processes as if they were things by construing them as nouns and noun phrases. The distinctive mathematics operations, for example, such as addition, subtraction, and multiplication, are processes, but the grammar constructs them as things (in noun phrases). This makes mathematics a very objectified discourse (Sfard & Lavie, 2005), and students need to be able to recognize the relationship between the things of the grammar and the processes of the mathematics reasoning. (146)
  • In the classroom, the teacher uses oral language to discuss the equation and its solution, adding another layer of linguistic complexity, as the oral language does not exactly capture the relationships in the ways the written language or symbolic language does. O’Halloran (2000, p. 384) reports that in this case, the teacher said:
    “. . .and then you’ve got to add on the ‘a’ squareds because of the
    brackets and the squareds, add up the ‘a’ squareds so you get two
    ‘a’ squareds plus your four ‘a’”
  • The teacher’s oral language again presents the elements in the mathematical symbol statements as things to be manipulated, even though the notion of square root, for example, is not a thing, but a process or operation. Translating among all these semiotic resources and maintaining the technical register is a challenge for teachers and students. (146)
  • As with all language development, students need opportunities to use the mathematics register in interactive activities in which they construct meaningful discourse about mathematics. They need to practice the multisemiotic construction of meaning, drawing on all modalities. This section discusses research that suggests that teachers play an important role in helping students use language effectively in learning and demonstrating their mathematical knowledge. (147)
  • The explanations textbooks provide tend to be dense, so the teacher plays a key role in helping students learn to negotiate the symbols, diagrams, and technical language. On experimental tasks, Leung, Low, and Sweller (1997) found that students benefited from verbal explanations of mathematics problems, at least until they gained a greater facility with solving these problems (147)
  • Veel (1999) reports that teacher spoken language predominates in mathematics classes, and the
    teacher’s words are needed to interpret the meanings that the visual and symbolic representations construct, as ‘‘it is spoken language which provides the link between the symbolic and visual representations for students, and is therefore a powerful agent in the learning process’’ (p. 189) (147)
  • Working with experienced interlocutors is the only way to accomplish this. O’Halloran (2000)
    recommends that teachers use oral language to unpack and explain the meanings in mathematics symbolism as a way of using the multisemiotic nature of mathematics to help students draw on the different meaning-making modes for understanding. Explicitly focusing
    students’ attention on the linguistic features can help students explore and clarify the technical meanings. This does not mean just talk for talk’s sake; teachers need to give attention to when the technical talk can help students develop the mathematics register (Sfard et al., 1998). (148)
  • Students do not take up the technical language of mathematics merely by being exposed to it through the teacher’s talk or textbook. Veel (1999) compares the student use of mathematical language and
    teacher=textbook use of mathematical language and finds a major gap between them in features that distinguish mathematics language from everyday language. (148)  The teacher’s language has higher lexical
    density, a greater percentage of relational processes, and more long noun phrases, the language resources that construct the mathematics (148)
  • Huang et al. found that knowledge structures such as classification, principles, and evaluation
    were only used by the teacher. Students used only description, sequence, and choice knowledge structures, even when pushed by the teacher. The authors suggest that students need explicit instruction in articulating principles to move them beyond the practical aspects of math knowledge in their discussion. They recommend that students be asked to ‘‘talk their way into habits of expressing higher level knowledge structures’’ (Huang et al., 2005, pp. 44–45), and that teachers integrate thinking and talking at all levels. (149)
  • Two of these are explicitly constructing mathematical knowledge: procedural talk that lays out the steps taken to solve a problem, and conceptual talk about the reasons for calculating in particular ways or for using particular procedures. In addition, teachers use regulatory talk for classroom management and contextual talk to bring in background information when students are solving word problems. (151)
  • Attention can be paid to unpacking the meanings in the dense noun phrases and clarifying relationships that are constructed in the verbs and conjunctions, as well as by making explicit what might have been left implicit in the formulation of the problem. Staub and Reusser (1995) also suggest a clearer consideration of the situations in the problems and their relevance to instructional goals. As Veel (1999) points out, most word problems are contrived by teachers or textbook writers to fit the particular calculation skills that are in focus. Although they purport to make mathematics real, or to connect with students’ actual lives, in fact, they often do not address the everyday experience of most students. (152)
  • Learning the mathematics register takes time, and teachers need to set goals that scaffold the development of precise ways of using language over lessons and units of study. Chapman (1995) shows how this can be done, tracking the understanding of a concept over several lessons and showing how it is built up in teacher talk, wholeclass teacher-student interaction, interaction between students, and reading of the textbook. The teacher focuses students on using appropriate language to construct the concept and contextualizes the textbook language through spoken language to help students understand the relationships in the textbook definitions. (154)
  • They changed the released math items from the National Assessment of Educational Progress (NAEP) 1992 in several ways, shortening nominal expressions, making conditional relationships more explicit, changing complex question phrases to simple question words and passive voice to active, and replacing less familiar or frequent nonmathematics vocabulary with more common terms.  Low-performing mathematics students benefited more from the revisions than those in higher mathematics and algebra, English Language Learners benefited more than proficient speakers of English, and students identified as having low socioeconomic status benefited more than others. This is promising work that can contribute to the development of materials that provide better support for students who are moving from the everyday language into the more technical. Other research has also demonstrated that the wording of math problems has a major influence on comprehension and children’s ability to solve them (Staub & Reusser, 1995). (156)
  • O’Halloran (2003) found that teachers of working class and female students used more informal and non-technical language in the oral discourse of the classroom, and suggests that there are social class and gender implications for access to knowledge when teachers use less technical language. (156)
  • Further research by applied linguists and mathematics educators can explore the linguistic challenges of mathematics learning in its multi-semiotic complexity to provide more support for teachers who want to engage struggling learners. (157)

Thoughts:

Doing this annotated bib on this particular article was really helpful.  In my own work, I need to use more Veel (1999), Chapman (1995).