# Classroom engagement towards using definitions for developing mathematical objects: the case of function

- 611 Downloads
- 3 Citations

## Abstract

For mathematicians, definitions are the ultimate tool for reaching agreement about the nature and properties of mathematical objects. As research in school mathematics has revealed, however, mathematics learners are often reluctant, perhaps even unable, to help themselves with definitions while categorizing mathematical objects. In the research from which we take the data presented in this article, we have been following a group of prospective mathematics teachers studying functions. In the course of learning, the students gradually accepted the definition as the ultimate criterion to identify examples of function. And yet, this use was hindered by the difficulty the students experienced while trying to understand the logical structure of the definition. Our close analysis has shown that the determiners “for every” and “unique” constituted the main source of the difficulty. We propose that a brief introduction to logic and, in particular, to parsing complex mathematical sentences, may be a useful addition to mathematics curriculum.

## Keywords

Discourse analysis Sociocultural approach Mathematics definitions Learning processes Commognitive conflict## Notes

### Acknowledgments

This study is supported by the Israel Science Foundation, no. 446/10.

## References

- Aristotle (2015).
*Topics, VI, 4, (W. A. Pickard-Cambridge, Trans.) Logos virtual library*. Retrieved from http://www.logoslibrary.org/aristotle/topics/604.html. - Borasi, R. (1992).
*Learning mathematics through inquiry*. Portsmouth: Heinemann.Google Scholar - Breidenbach, D., Dubinsky, E., Hawks, J., & Nichols, D. (1992). Development of the process conception of function.
*Educational Studies in Mathematics, 23*(3), 247–285.CrossRefGoogle Scholar - Chapman, A. (1995). Intertextuality in school mathematics: The case of functions.
*Linguistics and Education, 7*(3), 243–362.Google Scholar - Chapman, A. P. (2003).
*Language practices in school mathematics. A social semiotic approach*. Lewiston: The Edwin Mellen Press.Google Scholar - Chiu, M. M., Kessel, C., Moschkovich, J. N., & Muñoz-Nuñez, A. (2001). Learning to grasp linear function: A case study of conceptual change.
*Cognition and Instruction, 19*(2), 215–252.Google Scholar - Copi, I. M. (1972).
*Introduction to logic*. New York: Macmillan Publishing Co., Inc.Google Scholar - De Villiers, M. D. (1994). The role and function of a hierarchical classification of quadrilaterals.
*For the Learning of Mathematics, 14*(1), 11–18.Google Scholar - Department for Education (DfE) (2009).
*Mathematics Curriculum for Middle Schools*. Retrieved from http://cms.education.gov.il/EducationCMS/Units/Tochniyot_Limudim/Math_Chatav/TachnitLimudim/. - Dubinsky, E., & Yiparaki, O. (2000). On student understanding of AE and EA quantification. In E. Dubinsky, A. H. Schoenfeld, & J. Kaput (Eds.),
*Research in collegiate mathematics education IV*(pp. 239–286). Providence: American Mathematical Society.Google Scholar - Edwards, B. S., & Ward, M. B. (2004). Surprises from mathematics education research: Student (mis)use of mathematical definitions.
*American Mathematical Monthly, 111*(5), 411–424.Google Scholar - Even, R. (1990). Subject matter knowledge for teaching and the case of functions.
*Educational Studies in Mathematics, 21*, 521–544.CrossRefGoogle Scholar - Fischbein, E. (1993). The theory of figural concepts.
*Educational Studies in Mathematics, 24*, 139–162.CrossRefGoogle Scholar - Fischbein, E. (1996). The psychological nature of concepts. In H. Mansfield, N. A. Pateman, & N. Bernardz (Eds.),
*Mathematics for young children. International perspectives on curriculum*. Dordrecht: Kluwer academic publishers.Google Scholar - Fischbein, E., & Nachlieli, T. (1998). Concepts and figures in geometrical reasoning.
*International Journal of Science Education, 20*(10), 1193–1211.CrossRefGoogle Scholar - Freudenthal, H. (1973).
*Mathematics as an educational task*. Springer Science & Bussiness Media.Google Scholar - Furinghetti, F. & Paola, D. (2000), Definition as a teaching object: A preliminary study. In T. Nakahara & M. Koyama (Eds.),
*Proceedings of the 24th International Conference on the Psychology of Mathematics Education*(vol 2 pp. 289–296). Japan: Hiroshima.Google Scholar - Halliday, M. A. K. (1978).
*Language as social semiotic: the social interpretation of language and meaning*. Edward Arnold.Google Scholar - Halliday, M. A. K. & Hasan, R. (1985). Language, context and text: aspects of language in a social semiotic perspective. Geelong, Vic: Deakin University.Google Scholar
- Heath, T. (1956).
*Euclid's elements*(Vol. 1). New York: Dover Publications.Google Scholar - Hershkowitz, R., & Vinner, S. (1983). The Role of Critical and Non Critical Attributes in the Concept Image of Geometrical Concepts. In R. Hershkowitz (Ed.),
*Proceedings of the 7th International Conference for the Psychology of Mathematical Education*(223–228). Rehovot, Israel: Weitzmann.Google Scholar - Hershkowitz, R., & Vinner, S. (1984). Children's concept in elementary geometry – a reflection of teacher's concepts? In B. Southwell (Ed).,
*Proceedings of the Eighth International Conference for the Psychology of Mathematical Education*(pp. 63–70). Sydney: International Group for the Psychology of Mathematics Education.Google Scholar - Keller, B. A., & Hirsch, C. R. (1998). Student preferences for representations of functions.
*International Journal of Mathematical Education in Science & Technology, 29*(1), 1–17.CrossRefGoogle Scholar - Kleiner, I. (1989). Evolution of the function concept: A brief survey.
*College Mathematics Journal, 20*, 282–300.Google Scholar - Leinhardt, G., Zaslavsky, O., & Stein, M. K. (1990). Functions, graphs, and graphing: Tasks, learning, and teaching.
*Review of Educational Research, 60*(1), 1–64.Google Scholar - Lemke, J. (2002). Mathematics in the middle: Measure, picture, gesture, sign, and word. In M. Anderson, A. Saenz-Ludlow, S. Zellweger, & V. Cifarelli (Eds.),
*Educational perspectives on mathematics as semiosis: From thinking to interpreting to knowing*(pp. 215–234). Ottawa: Legas Publishing.Google Scholar - Markovits, Z., Eylon, B. S., & Bruckheimer, M. (1986). Functions today and yesterday.
*For the learning of mathematics, 6*(2), 18–28.Google Scholar - Morgan, C. (2004). Word, definitions and concepts in discourses of mathematics, teaching and learning.
*Language and Education, 18*, 1–15.CrossRefGoogle Scholar - Moschkovich, J. N. (2004). Appropriating mathematical practices: A case study of learning to use and explore functions through interaction with a tutor.
*Educational Studies in Mathematics, 55*, 49–80.Google Scholar - Mountwitten, M. (1984).
*Processes of mathematical concept acquisition through definitions and through examples by elementary school students*. Unpublished doctoral dissertation, The Hebrew University, Jerusalem. (In Hebrew).Google Scholar - Nachlieli, T., & Tabach, M. (2012). Growing mathematical objects in the classroom—the case of function.
*International Journal of Educational Research, 51&52*, 10–27.CrossRefGoogle Scholar - O'Halloran, K. (2005).
*Mathematical discourse. Language, symbolism and visual images*. London: Continuum Press.Google Scholar - Ouvrier-Buffet, C. (2006). Exploring mathematical definition construction processes.
*Educational Studies in Mathematics, 63*(3), 259–282.CrossRefGoogle Scholar - Ouvrier-Buffet, C. (2011). A mathematical experience involving defining processes: in-action definitions and zero-definitions.
*Educational Studies in Mathematics, 76*(2), 165–182.Google Scholar - Piaget, J. (1957).
*Logic and psychology*. New York: Basic books.Google Scholar - Pimm, D. (1987).
*Communications in mathematics classrooms*. London: Routledge & Kegan Paul.Google Scholar - Schleppegrell, M. J. (2007). The linguistic challenges of mathematics teaching and learning: A research review.
*Reading & Writing Quarterly, 23*(2), 139–159.Google Scholar - Schleppegrell, M. J. (2010). Language in mathematics teaching and learning: A research review. In J. N. Moschkovich (Ed.),
*Language and mathematics education*(pp. 73–112). Charlotte: Information age publishing.Google Scholar - Schoenfeld, A. H., Smith, J. P., III, & Arcavi, A. (1993). Learning: The microgenetic analysis of one student's evolving understanding of a complex subject matter domain. In R. Glaser (Ed.),
*Advances in instructional psychology, 4*(pp. 55–175). Hillsdale: Erlbaum.Google Scholar - Schwarts, B. B., & Hershkowitz, R. (1999). Prototypes: Brakes or levers in learning the function concepts? The role of computer tools.
*Journal for Research in Mathematics Education, 30*(4), 362–389.CrossRefGoogle Scholar - Selden, J., & Selden, A. (1995). Unpacking the logic of mathematical statements.
*Educational Studies in Mathematics, 29*(2), 123–151.CrossRefGoogle Scholar - Sfard, A. (2007). When the rules of discourse change, but nobody tells you: Making sense of mathematics learning from a commognitive standpoint.
*Journal of Learning Sciences, 16*(4), 567–615.CrossRefGoogle Scholar - Sfard, A. (2008).
*Thinking as communicating: Human development, the growth of discourses, and mathematizing*. New York: Cambridge University Press.CrossRefGoogle Scholar - Shir, K. & Zaslavsky, O. (2002). Students’ Conceptions of an Acceptable Geometric Definition. In A. D. Cockburn & E. Nardi (Eds.),
*Proceedings of the 26th Conference of the International Group for the Psychology of Mathematics Education*(vol 4 pp. 201–208). Norwich: UK.Google Scholar - Sinclair, N., & Moss, J. (2012). The more it changes, the more it becomes the same: The development of the routine of shapes identification in dynamic geometry environment.
*International Journal of Educational Research, 51&52*(3), 28–44.CrossRefGoogle Scholar - Solomon, Y. (2006). Deficit or difference? The role of students' epistemologies of mathematics in their interactions with proof.
*Educational Studies in Mathematics, 61*, 373–393.CrossRefGoogle Scholar - Tall, D. (Ed.). (1991).
*Advanced mathematical thinking*. Dordrecht: Kluwer.Google Scholar - Tall, D., & Vinner, S. (1981). Concept image and concept definition in mathematics with particular reference to limits and continuity.
*Educational Studies in Mathematics, 12*, 151–169.CrossRefGoogle Scholar - Veel, R. (1999). Language, knowledge and authority in school mathematics. In F. Christie (Ed.),
*Pedagogy and the shaping of consciousness: Linguistic and social processes*(pp. 185–216). London: Continuum.Google Scholar - Vinner, S. (1982). Conflicts between definitions and intuitions – the case of a tangent.
*Proceedings of the Sixth International Conference for the Psychology of Mathematics Education*, 24–29.Google Scholar - Vinner, S. (1983). Concept definition, concept image and the notion of function.
*International Journal of Mathematical Education in Science and Technology, 14*(3), 293–305.Google Scholar - Vinner, S. (1990). Inconsistencies: Their causes and function in learning mathematics.
*Focus on Learning Problems in Mathematics, 12*(3&4), 85–98.Google Scholar - Vinner, S. (1991). The role of definitions in the teaching and learning of mathematics. In D. Tall (Ed.),
*Advanced mathematical thinking*(pp. 65–81). Dordrecht: Kluwer.Google Scholar - Vinner, S., & Dreyfus, T. (1989). Images and definitions for the concept of function.
*Journal for Research in Mathematics Education, 20*(4), 356–366.CrossRefGoogle Scholar - Vinner, S., & Hershkowitz, R. (1980). Concept Images and Common Cognitive Paths in the Development of Some Simple Geometrical Concepts. In R. Karplus (Ed.),
*Proceedings of the Fourth International Conference for the Psychology of Mathematical Education*(pp. 177–184). Berkeley: International Group for the Psychology of Mathematics Education.Google Scholar - Walter, J., & Gerson, H. (2000). Teachers personal agency: Making sense of slope through additive structures.
*Educational Studies in Mathematics, 65*(2), 203–233.CrossRefGoogle Scholar - Wawro, M., Sweeney, G., & Rabin, J. M. (2011). Subspace in linear algebra: Investigating students’ concept images and interactions with the formal definition.
*Educational Studies in Mathematics, 78*(1), 1–19. doi: 10.1007/s10649-011-9307-4.CrossRefGoogle Scholar - Wilson, P. S. (1990). Inconsistent ideas related to definitions and examples.
*Focus on Learning Problems in Mathematics, 12*(3&4), 31–47.Google Scholar - Yerushalmy, M. (2006). Slower algebra students meet faster tools: Solving algebra word problems with graphing software.
*Journal for Research in Mathematics Education, 37*(5), 356–387.Google Scholar - Zaslavski, O., & Shir, K. (2005). Students’ conceptions of a mathematical definition.
*Journal for Research in Mathematics Education, 36*(4), 317–346.Google Scholar - Zaslavsky, O., Sela, H., & Leron, U. (2002). Being sloppy about slope: The effect of changing the scale.
*Educational Studies in Mathematics, 49*, 119–140.Google Scholar