Abstract
How does teacher-guided whole-class interaction contribute to expanding students’ potential for making and exchanging mathematical meanings? We address this question through an interpretative analysis of a whole-group conversation in a sixth grade class taught by an experienced teacher in a school in Southern Argentina. The extended interaction (circa 160 speaker turns) concerned a problem involving remainders that calls for building a direct formula and expressing it in symbolic algebraic language. Our functional-grammatical analysis of the selected conversation revealed that, in synch with the deployment and use of the function table, the manner in which the teacher worded her questions was instrumental to the simultaneous accomplishment of three goals. First, her use of the personal pronouns “I” and “we” with generalized referents helped shape a collective zone of proximal development for the class to think aloud together about the algebraizing task at hand. Second, her verb choices supported students in moving among material, mental, operative, and relational processes on the road toward the targeted formula. Finally, by means of repetition, conjunction, demonstrative pronouns, and pointing gestures, the teacher molded the conversation as a coherent and cohesive multi-semiotic text.
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Notes
Examples of texts are everyday conversations, email exchanges, recipes, instructional manuals, geometric proofs, lab reports, novels, poems, political speeches, etc.
“It appears that we can recognize a generalized interpersonal gateway whereby new meanings are first construed in interpersonal contexts and only later transferred to ideational ones, experiential and/or logical” (Halliday 1993, p. 103).
Worth highlighting here is, also, the shift in verb tense from “I” + Past to “You” + Present.
Declaratives are clauses with subject-verb order, imperatives are clauses with no subject, and interrogatives are those in which the auxiliary verb comes before the subject.
Our analysis attended also to Theme (i.e. what the clause is about and what the speaker chooses as the point of departure for the message) and Rheme (the part of the clause which develops the Theme). We do not include that analysis in here given that, in Spanish, Theme marked-ness functions differently than in English (Taboada 2004) thus requiring a lengthy discussion of the translation effect.
Given the non-obligatory status of personal pronouns in Spanish, we identified instances where the teacher did use personal pronouns. We view those instances as further evidence that the teacher’s use of first person singular and plural supported students in appropriating a generalized mathematizing other.
Nominalizing, turning processes and actions (verbs) into objects (nouns), allows mathematical objects to appear as subjects in the clause thereby suggesting they carry causality and agency (cf. Halliday and Martin 1993).
We noted two reference shifts in the teacher person choices: (1) from “you” referring to and addressing either an individual student or a group—Some of you seem to have some trouble with…; Joaquin, how did you understand….—to “we” referring to and addressing the whole class herself included—How can we understand….; (2) from “they” and” them” referring to the ants, to “you” referring to the students: when/if they march, if they go…, if it were you, 25 kids, …., if they go by 2 s, how many rows can I make?, If the 25 ants, I make them march--a personification aimed at helping students place themselves in the situation.
Implicit in Teo's reasoning is that, since the length of the repeating pattern in the last two digits is 5 (25, 85, 45, 05, 65), the 10th term ends in the same two digits as the 5th one. Requesting him to make this reasoning explicit was perhaps a missed ‘teachable moment’.
References
Arendt, H. (1978). The life of the mind. New York: Harcourt Brace Jovanovich.
Arzarello, F. (2006). Semiosis as a multi-modal process. Revista Latinoamericana de Investigación en Matemática Educativa, 267–299.
Bakker, A., & Hoffmann, M. (2005). Diagrammatic reasoning as the basis for developing concepts. Educational Studies in Mathematics, 60, 333–358.
Christie, F. (2002). Classroom discourse analysis: a functional perspective. London & New York: Continuum.
Dewey, J. (1938). Logic: The theory of inquiry. New York: Henry Holt.
Dörfler, W. (2008). En route from patterns to algebra: comments and reflections. ZDM—The International Journal on Mathematics Education, 40(1), 143–160.
Driscoll, M. (1999). Fostering algebraic thinking: a guide for teachers, grades 6–10. Portsmouth: Heinemann.
Driscoll, M., et al. (2007). Fostering geometric thinking: a guide for teachers, grades 5–10. Portsmouth: Heinemann.
Fortanet, I. (2004). The use of ‘we’ in university lectures: reference and function. English for Specific Purposes, 23, 45–66.
Freudenthal, H. (1991). Revisiting mathematics education: China Lectures. Dordrecht: Kluwer.
Halliday, M. A. K. (1993). Towards a language-based theory of learning. Linguistics and Education, 5, 93–116.
Halliday, M. A. K. (1994). Introduction to functional grammar (2nd ed.). London: Edward Arnold.
Hasan, R. (1996). Ways of saying: Ways of meaning. London & New York: Cassell.
Herbel-Eisenmann, B., & Wagner, D. (2010). Appraising lexical bundles in mathematics classroom discourse: obligation and choice. Educational Studies in Mathematics, 75(1), 43–63.
Ju, M. K., & Kwon, O. N. (2007). Ways of talking and ways of positioning: students’ beliefs in an inquiry-oriented differential equations class. Journal of Mathematical Behavior, 26, 267–280.
Lampert, M. (1990). When the problem is not the question and the solution is not the answer. American Education Research Journal, 27(1), 29–63.
Lotman, Y. M. (1988). Text within a text. Soviet Psychology, 26, 32–51.
Mead, G. H. (1964a). In Strauss (Ed.), George Herbert Mead on social psychology. Selected Papers. Chicago and London: University of Chicago Press.
Mead, G.H. (1964b). In A. J. Reck (Ed.), Selected writings. Chicago: University Chicago Press.
Mercer, N. (2000). Words and minds: How we use language to think together. London: Routledge.
Nathan, M. J., & Kim, S. (2009). Regulation of teacher elicitations in the mathematics classroom. Cognition and Instruction, 27(2), 91–120.
O’Halloran, K. (2000). Classroom discourse in mathematics: a multi-semiotic analysis. Linguistics and Education, 10(3), 359–388.
O’Halloran, K. (2005). Mathematical discourse: Language, symbolism, and visual images. London & New York: Continuum.
Pimm, D. (1987). Speaking mathematically: Communication in mathematics classrooms. London: Routledge and Kegan Paul.
Radford, L. (2000). Signs and meanings in students’ emergent algebraic thinking: a semiotic analysis. Educational Studies in Mathematics, 42(3), 237–268.
Radford, L. (2008). Iconicity and contraction: a semiotic investigation of forms of algebraic generalizations of patterns in different contexts. ZDM—The International Journal on Mathematics Education, 40, 83–96.
Radford, L. (2014). Towards an embodied, cultural, and material conception of mathematics cognition. ZDM—The International Journal on Mathematics Education, 46, 349–361. doi:10.1007/s1858-014-0591-1.
Rounds, P. (1987a). Multifunctional personal pronoun use in an educational setting. English for Specific Purposes, 6(1), 13–29.
Rounds, P. (1987b). Characterizing successful classroom discourse for NNS teaching assistant training. TESOL Quarterly, 21(4), 643–671.
Rowland, T. (1992). Pointing with pronouns. For the Learning of Mathematics, 12(2), 44–48.
Rowland, T. (1999). Pronouns in mathematical talk: power, vagueness, and generalization. For the Learning of Mathematics, 19(2), 19–26.
Rowland, T. (2000). The pragmatics of mathematics education: Vagueness in mathematical discourse. New York: Falmer Press.
Schleppegrell, M. (2007). The linguistic challenges of mathematics teaching and learning: a research review. Reading and Writing Quarterly, 23, 139–159.
Shreyar, S., Zolkower, B., & Pérez, S. (2010). Thinking aloud together: a teacher’s semiotic mediation of a whole-class conversation about percents. Educational Studies in Mathematics, 73, 21–53.
Singer, M. (1989). Pronouns, persons, and the semiotic self. In B. Lee & G. Urban (Eds.), Semiotics, self, and society. Berlin and NY: Mouton de Gruyter.
Smit, J., & van Eerde, D. (2013). What counts as evidence for the long-term realisation of whole-class scaffolding? Learning, Culture, and Social Interaction, 2, 22–31.
Staats, S., & Batteen, C. (2010). Linguistic indexicality in algebra discussions. Journal of Mathematical Behavior, 29, 41–56.
Taboada, M. T. (2004). Building coherence and cohesion: Task-oriented dialogue in English and Spanish. John Benjamins.
Veel, R. (1999). Language, knowledge, and authority in school mathematics. In F. Christie (Ed.), Pedagogy and the shaping of consciousness (pp. 185–215). London & New York: Cassell.
Vygotsky, L. S. (1978). Mind in society: The development of higher psychological functions. Cambridge: Harvard University Press.
Vygotsky, L. S. (1986). Thought and language. Cambridge: MIT Press.
Zolkower, B., & Shreyar, S. (2007). A teacher’s mediation of a thinking aloud discussion in a 6th grade mathematics classroom. Educational Studies in Mathematics, 65, 177–202.
Acknowledgments
The authors are grateful to Mary Schleppegrell as well as to the three anonymous reviewers for their comments and suggestions. An earlier version of this paper was presented at the 2014 Annual Meetings of the American Education Research Association (Philadelphia).
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Zolkower, B., Shreyar, S. & Pérez, S. Teacher guidance of algebraic formula building: functional grammatical analysis of a whole-class conversation. ZDM Mathematics Education 47, 1323–1336 (2015). https://doi.org/10.1007/s11858-015-0701-8
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DOI: https://doi.org/10.1007/s11858-015-0701-8