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\f0\b\fs32 \cf0 The Multi-Semiotic Nature of Mathematical Language and Its Secondary School Classroom Implications\
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\b0\fs28 \cf0 \ul \ulc0 Gregory D. Foley\ulnone \
Ohio University, USA\
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\cf0 \
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\i \cf0 This paper expands Schleppegrell's notion of the school mathematics register by integrating ideas from linguistics, mathematics, and mathematics education. It makes the case for literacy and fluency in this register as being critical components of mathematical proficiency and quantitative literacy, and for a language-rich, inquiry-based approach to secondary school mathematics teaching and learning as a way to achieve these ends.\
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\i0\b \cf0 THE MATHEMATICS REGISTER\
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\b0 \cf0 In simple terms, \expnd0\expndtw0\kerning0
the
\i mathematics register
\i0 is the formal academic approach to mathematical speaking and writing. Delving deeper, we see that \kerning1\expnd0\expndtw0 the language register used in secondary school mathematics is complicated and multi-faceted, involving several subfields of the mathematical sciences and the accompanying symbol systems and representational modes of these subfields. The school mathematics register not only includes numerical and algebraic notation, which is often cross-cultural, but also depends on specialized natural language (e.g., Arabic, Chinese, English, or Spanish), which varies across cultures. The mathematics register incorporates a multitude of representations (graphical, physical, pictorial, tabular) and combines natural language with specialized mathematical symbolism. Connecting and integrating these multiple representations is required for effective mathematical thinking and problem solving, and for communicating one's thinking and problem-solving methods to others!
in written or oral form. Moreover, comprehending the mathematical thinking and methods of others requires a degree of fluency in this multi-semiotic language we call mathematics. It is the integrated use of multiple modes of representation to communicate mathematical thinking that constitutes the
\b mathematics register
\b0 of a natural language. (See Table 1.)
\f1\fs20 \
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\f0\b\fs28 \cf0 Table 1.
\i\b0 Critical Features of the School Mathematics Register\
\itap1\trowd \taflags1 \trgaph108\trleft-108 \trbrdrt\brdrnil \trbrdrl\brdrnil \trbrdrr\brdrnil
\clvertalc \clshdrawnil \clwWidth5620\clftsWidth3 \clbrdrt\brdrs\brdrw20\brdrcf2 \clbrdrl\brdrs\brdrw20\brdrcf2 \clbrdrb\brdrs\brdrw20\brdrcf2 \clbrdrr\brdrs\brdrw20\brdrcf2 \clpadl100 \clpadr100 \gaph\cellx4320
\clvertalc \clshdrawnil \clwWidth3540\clftsWidth3 \clbrdrt\brdrs\brdrw20\brdrcf2 \clbrdrl\brdrs\brdrw20\brdrcf2 \clbrdrb\brdrs\brdrw20\brdrcf2 \clbrdrr\brdrs\brdrw20\brdrcf2 \clpadl100 \clpadr100 \gaph\cellx8640
\pard\intbl\itap1\tx5400\pardeftab720\ri5\sl-320\sa80\ql\qnatural
\i0\b\fs22 \cf0 Linked representational modes
\b0 (semiotic systems)
\i \cell
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\i0\b \cf0 Peculiar linguistic patterns
\i\b0\fs28 \cell \row
\itap1\trowd \taflags1 \trgaph108\trleft-108 \trbrdrl\brdrnil \trbrdrt\brdrnil \trbrdrr\brdrnil
\clvertalc \clshdrawnil \clwWidth5620\clftsWidth3 \clbrdrt\brdrs\brdrw20\brdrcf2 \clbrdrl\brdrs\brdrw20\brdrcf2 \clbrdrb\brdrs\brdrw20\brdrcf2 \clbrdrr\brdrs\brdrw20\brdrcf2 \clpadl100 \clpadr100 \gaph\cellx4320
\clvertalc \clshdrawnil \clwWidth3540\clftsWidth3 \clbrdrt\brdrs\brdrw20\brdrcf2 \clbrdrl\brdrs\brdrw20\brdrcf2 \clbrdrb\brdrs\brdrw20\brdrcf2 \clbrdrr\brdrs\brdrw20\brdrcf2 \clpadl100 \clpadr100 \gaph\cellx8640
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\i0\fs22 \cf0 Oral and written natural language\
Numerals and other mathematical symbols \
Tables of data, matrices, and other numerical representations\
Cartesian graphs and other graphical representations\
Diagrams and geometric figures\
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\cf0 Physical models and simulations
\i\fs28 \cell
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\i0\fs22 \cf0 Technical vocabulary\
Syntax of mathematical definitions \
Tacit and implicit logical relationships\
Uses of the verbs
\i to be
\i0 and
\i to have
\i0 \
Technical meanings of
\i and
\i0 and
\i or
\i0 \
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\cf0 Complicated and dense noun phrases
\i\fs28 \cell \lastrow\row
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\i0\fs20 \cf0 An adaptation and extension of a similar table by Schleppegrell (2007, p. 141)\
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\fs28 \cf0 In our classrooms, we tie mathematical concepts, processes, and practices to everyday, social, and scientific applications. Students are asked not only to manipulate numbers and other mathematical objects but also to solve in-context problems involving genuine data and to explain and justify their solution methods. The use of specialized logical reasoning in mathematics brings with it many linguistic wrinkles. As students continue their study of mathematics from lower to upper secondary school and then to tertiary education, the mathematics becomes increasingly based on reasoning and proof, which require sophisticated and technical uses of natural language. Recent research by applied linguists and mathematics educators into the mathematics register suggests that there are strong interactions between mathematics and language and that a student\'92s oral and written language skills are critical to the student\'92s success in mathematics (Lager, 2006; Schleppegre!
ll, 2007). \expnd0\expndtw0\kerning0
Because the mathematics register is so multi-faceted and involves so many linked symbol systems and representational modes, it is actually a
\b hyper-language,
\b0 analogous to the hyper-text used in computer-based media.\kerning1\expnd0\expndtw0 \
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\b \cf0 Mathematical thinking and language are inextricably linked.\
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\b0 \cf0 Thoughts are closely tied to words. Mathematical concepts are expressed in words and phrases, and their meanings develop in the students\'92 minds over time. Definitions, theorems, and other mathematical relationships and properties are expressed in natural-language sentences, often supported by mathematical symbolism, graphs, and other representations. \expnd0\expndtw0\kerning0
\'93In doing mathematics, it is not enough to be able to work with the language alone; mathematics draws on multiple semiotic (meaning-creating) systems to construct knowledge: symbols, oral language, written language, and visual representations such as graphs and diagrams\'94\kerning1\expnd0\expndtw0 (Schleppegrell, 2007, p. 141).\
Rotman (1988) analyzed the tight interplay between a mathematician\'92s thinking and the mathematician\'92s \'93scribbling\'94 in a mixture of words, symbols, and diagrams, thus making a strong case for the role of written mathematical language in mathematical thinking and sense-making. Vygotsky (1934/1987), on the other hand, focused attention on the importance of shared oral language for students to develop ideas, which are then internalized individually. Both written language and oral language are critical to the development of quantitative reasoning. The National Council of Teachers of Mathematics (NCTM, 2000) communication standard highlights this close relationship between mathematical language and mathematical thinking. \
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\b \cf0 MATHEMATICAL PROFICIENCY & QUANTITATIVE LITERACY\
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\b0 \cf0 In the National Research Council (NRC, 2001) report
\i Adding It Up
\i0 , Kilpatrick, Swafford, and Findell defined the construct of
\i mathematical proficiency
\i0 as the combination of five \'93interwoven and interdependent\'94 strands: conceptual understanding (comprehension), procedural fluency (computation), strategic competence (problem solving), adaptive reasoning (thinking and communication), and productive disposition (valuing mathematics, tenacity, and self-efficacy). To be precise, they defined
\i adaptive reasoning
\i0 as the "capacity for logical thought, reflection, explanation, and justification" (NRC, 2001, p. 5). In 2003 a blue-ribbon panel chaired by D. L. Ball, building on the 2001 NRC report, identified and elaborated three priorities for a strategic research and development program \'93to achieve mathematical proficiency for all students\'94; one of these priorities was \'93teaching and learning skills for mathematical thinking and problem solving\'94 (RAND Mathematics Study Panel, 2003, p. 7). The chapter in the RAND report that elaborated this priority focused on the notion of
\i mathematical practices
\i0 , examples of which include "mathematical representation, attentive use of mathematical language and definitions, articulated and reasoned claims, rationally negotiated disagreement, generalizing ideas, and recognizing patterns" (RAND Mathematics Study Panel, 2003, p. 32). So, mathematical thinking, mathematical proficiency, and mathematical practices are all closely linked to and dependent on language and communication.\
Steen (2004) noted that the literacy experts identify three aspects of literacy: prose literacy, document literacy (i.e., "reading charts and tables"), and quantitative literacy (i.e., "interpreting and reasoning with numbers") (p. xi). In view of our multi-semiotic definition of the mathematics register, we define
\b quantitative literacy
\b0 broadly to include all three of these aspects of literacy as well as other representational modes ignored in the literature on literacy. Usiskin's (1996) view that the increasing amount of quantitative information in students' everyday lives implies that "mathematics is becoming part of every student's native language repertoire" (p. 242) now appears overly optimistic. Secondary education must help students become mathematically proficient and quantitatively literate, and the mathematics education community must develop new curricula and instructional approaches that extend beyond the traditional bounds of school mathematics.\
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\b \cf0 STUDENT INQUIRY FOR PROFICIENCY & LITERACY\
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\cf0 Some history\
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\b0 \cf0 The focus on
\i mathematical practices
\i0 in the work of the RAND Mathematics Study Panel (2003) harkens back to the work of three mathematics education pioneers: H. P. Fawcett (1894\'961976), R. L. Moore (1882\'961974), and A. E. Ross (1906\'962002). All three of these pioneers used an
\i inquiry-based learning
\i0 approach to achieve dramatic levels of learning among their students via intense engagement in mathematical practices, especially the use of written and oral mathematical language. Fawcett (1938) documented his approach to secondary school geometry in which his students developed their own book, a practice repeated more recently by Healy (1993). Arnold Ross ran an inquiry- and writing-intensive summer mathematics program for 14\'9618 year olds from 1957 until 2000, which continues in his name. Year after year, Ross would get youngsters to construct and write their own proofs of substantial theorems in number theory. The Ross Program motto is to "think deeply of simple things." (See Jackson, 2001). Although Moore is best known for the 50 PhD mathematicians he supervised and their over 2000 mathematical descendants, he also taught inquiry-based beginning calculus at the University of Texas, in which students would go to the board to present the results of their work!
in written and oral form. The details are documented in the film
\i Challenge in the Classroom
\i0 (1966) and by Eyles (1998). Moore believed\'97"that student is
\i taught
\i0 best who is
\i told
\i0 the least." These three educational pioneers advocated a learner-centered communication-intensive approach to mathematical education, in which students learned mathematics via doing, writing, and speaking mathematics.\
It could be argued that Fawcett, Ross, and Moore worked with special students under special circumstances and that the RAND Mathematics Study Panel's intention of using a
\i mathematical practices
\i0 approach for the \'93equitable attainment\'94 of mathematical proficiency (2003, p. 29) is unattainable. In fact, Cain, Cary, and Lamb (1985) used their experience, which included evidence from aptitude-treatment interaction studies, to recommend a mathematics curriculum with a focus on mathematical thinking and problem solving only for the top 25% of secondary school students. Clearly, the inquiry-based methods of Fawcett, Moore, and Ross need to be adapted to reach a broad secondary school audience. \
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\b \cf0 A proposed course in quantitative literacy\
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\b0 \cf0 Foley and his colleagues (Foley, 2007; Foley, Butts, Casey, Moss, & Johanson, 2008; Foley & Jasper, in preparation) advocate for an inquiry-based communication-rich approach to learning in which secondary school students investigate and solve problems in context, carry out increasingly sophisticated projects, and present their results orally and in writing. Graphing calculators, interactive computer software, Web-based applets, and genuine data are used to support mathematics register development. Specifically, Foley et al. (2008) have proposed the development of an inquiry-based quantitative literacy course for 16\'9618 year old students with average prior mathematical achievement\'97a course with a communication-rich classroom environment that explicitly supports mathematics language development via exploration, investigation, and problem solving coupled with vocabulary instruction and classroom discourse, as well as frequent student presentations of ever more c!
hallenging activities and projects. (For details on vocabulary instruction and classroom discourse, see Foley & Crocker, 2004; Marzano & Pickering, 2005; Murray, 2004; NCTM, 2007; Winsor, 2007/2008.) Foley and his colleagues agree with the language-learner perspective of Bay-Williams and Herrara (2007) on several points:\
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\ls1\ilvl0\cf0 \'95 The mathematical practices approach needs to be used in concert with "intentional language instruction and support" (p. 46).\
\'95 "Literacy-rich classrooms foster . . . opportunities for simultaneous development of academic language proficiency and [mathematical] knowledge, skills, and capacities" (p. 45).\
\'95 Teachers should combine vocabulary development and classroom discourse, with an "eye on language and mathematics" (p. 48).\
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\cf0 What emerges is a method of teaching and learning that merges the mathematical-practices and language-learner approaches and that calls for students to learn by engaging in problems that cause them to think deeply and make connections, which they then articulate orally and in writing within a community of learners. \'93No choice needs to be made on the part of the teacher between the mastery of content and the development of practice\'94 (RAND Panel, 2003, p. 30).\
Through this process of inquiry, students develop the
\i cognitive tools
\i0 they need to become users and doers of mathematics. These cognitive tools include words and phrases (based on concept definitions and concept images: see Vinner, 1983) and the associated mathematical objects, algorithms, heuristics (P\'f3lya, 1957/1973), and relationships among concepts (axioms, definitions, theorems, and related images). Through this process, students develop proficiency in using these quantitative and linguistic cognitive tools and thereby develop quantitative literacy. Just as there is a close relationship between mathematical language and mathematical thinking, there is an equally tight connection between quantitative literacy and quantitative reasoning. The tools of language and literacy are needed to do mathematics and to solve mathematical problems. Proficiency in the mathematics register develops hand-in-glove with the development of mathematical proficiency.\
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\b \cf0 AN ILLUSTRATIVE EXAMPLE\
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\b0 \cf0 In order to explore the linguistic demands of secondary school mathematics for quantitative literacy, consider the following sample problem:\
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\b \cf0 Fun with matrix multiplication.
\b0 A sequence of matrices based on repeated multiplication by the same square matrix is a
\b Markov chain
\b0 . Named for Russian mathematician Andrey (Andrei) Markov (1856\'961922), Markov chains were first used in probability, but are now used widely in many fields. A Markov chain is essentially a geometric sequence of matrices. Using a graphing calculator for the computation, write out the first seven terms of the Markov chain with first term [1 1] that is produced by repeatedly multiplying by the transition matrix [[0 1][1 1]]. How is this sequence related to the Fibonacci sequence?
\fs26 \
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\fs28 \cf0 Now we analyze this example, sentence-by-sentence (cf. Table 1):\
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\ls2\ilvl0
\f1 \cf0 1.
\b \'93Fun with . . .\'94:
\b0 A casual, hopeful phrase but one that contains the mathe-matical term
\i matrix multiplication
\i0 , with which the student-reader may not be familiar. The student may know the terms
\b
\i\b0 matrix
\i0 and
\i multiplication
\i0\b
\b0 but may know little or nothing about
\b
\i\b0 matrix multiplication
\i0 . The student may not realize that a working knowledge of matrix multiplication is assumed by the author and needed by the student-reader to continue successfully.\
2. \'93A sequence of . . .\'94: An
\i informal
\i0 mathematical definition of the term
\i Markov chain
\i0 , indicated by the boldface type. A
\i formal definition
\i0 would be a full characterization of the concept, should be interpreted as an if-and-only-if statement, and could be used in mathematical proofs. Mathematical writing tends to be dense and is often replete with tacit assumptions.\
3. \'93Named for . . .\'94: An historical and biographical statement in academic English, with elements from both the mathematics register and the history register.
\i \
\ls2\ilvl0
\i0 4. \'93A Markov chain . . .\'94: The student reading this is expected to use this statement together with \'93repeated multiplication by the same square matrix\'94 to piece together a sufficient understanding of the term
\i Markov chain
\i0 to solve the problem that is about to be stated. Sentence (4) will be of little help if the student does not know the meaning of
\i geometric sequence
\i0 or does not call it to mind.\
5. \'93Using a graphing calculator . . .\'94: A complicated set of instructions to be carried out that draws on sentences (2) and (4), together with knowledge of sequence terminology and the use of a graphing calculator. \
6. \'93How is this sequence . . . ?\'94: A question calling for a comparative analysis of the results of the work just performed by the student with
\i the Fibonacci sequence
\i0 . This assumes the student has a working knowledge of the Fibonacci sequence concept. The question is open ended, and it may not be clear to the student what is expected.\
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\f0 \cf0 This example brings us face-to-face with P\'f3lya's (1957)
\i understand the problem
\i0 phase of problem solving, which as we have seen, is literacy intensive. \
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\cf0 All genuine mathematical practice is tied to the multi-semiotic language of mathematics. In order to achieve mathematical proficiency and quantitative literacy for the majority of secondary school students, we must use an approach that explicitly attends to both mathematics and language, and that engages students in the creative world of mathematical exploration and discovery.\
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\b \cf0 References\
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\b0\fs26 \cf0 Bay-Williams, J. M., & Herrara, S. (2007). Is \'93just good teaching\'94 enough to support the learning of English language learners? Insights from sociocultural learning theory. In W. G. Martin, M. E. Strutchens, & P. C. Elliott (Eds.),
\i The learning of mathematics: 69th yearbook
\i0 (pp. 43-63). Reston, VA: National Council of Teachers of Mathematics.\
Cain, R. W., Carry, L. R., & Lamb, C. E. (1985). Mathematics in secondary schools: Four points of view. In C. R. Hirsch & M. J. Zweng (Eds.),
\i The secondary school mathematics curriculum: 1985 yearbook
\i0 (pp. 22-28). Reston, VA: National Council of Teachers of Mathematics.\
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\i0 (1966). [A documentary film, now on DVD, about R. L. Moore and his teaching]. Washington, DC: Mathematical Association of America.\
Eyles, J. W. (1998). R. L. Moore's calculus course.
\f1 (Doctoral dissertation, The University of Texas at Austin, 1998).
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\f0 \
Fawcett, H. P. (1938).
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Foley, G. D. (2007, June).
\i M\'c1S\'97mathematics for
\b all
\b0 students: Communication is critical
\i0 . Talk presented at the southwest regional meeting of the American Mathematical Association of Two-Year Colleges, San Antonio, TX.\
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\i0 . A National Science Foundation Discovery Research K\'9612 research and development proposal.\
Foley, G. D., & Crocker, D. A. (2003). The words of mathematics.
\i The Centroid
\i0 ,
\i 29
\i0 (1), 8-9.\
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Healy, C. C. (1993).
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Murray, M. (2004).
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\i0 Kilpatrick, J., Swafford, J., & Findell, B. (Eds.). Mathematics Learning Study Committee, Center for Education, Division of Behavioral and Social Sciences and Education. Washington, DC: National Academy Press.\
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\cf0 Winsor, M. S. (2007/2008). Bridging the language barrier in mathematics.
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}