# Prospective teachers’ understanding of the multiplicative part-whole relationship of fraction

- 722 Downloads
- 2 Citations

## Abstract

The part-whole multiplicative relationship, as a topic that gives rise to the concept of fraction, is fundamental in education at the primary school level, and must therefore be included in training courses for prospective primary school teachers (PSTs). In this paper, we introduce a first study of a larger project, which aims to understand the usefulness of a semantic triangle in studying prospective teachers’ knowledge of various mathematical topics. In particular, we present the results of a study focused in the starting level of the fraction concept, based on the multiplicative part-whole relationship. We carried out this study with PSTs by means of a questionnaire. We analyzed the collected responses using a framework of three components that form a semantic triangle, in terms of their conceptual structure, system of representations, and contexts and modes of use of the part-whole relationship. The results show different typologies of meaning expressed by the participants in terms of the semantic triangle. Each typology emphasizes some aspects of the meaning of the part-whole relationship such as the equality of parts, the model of area as representation and the context of division.

## Keywords

Fractions Part-whole relationship Pre-service teacher education Semantic triangle## Notes

### Acknowledgment

This study was supported from project “Procesos de aprendizaje del profesor de matemáticas en formación” EDU2012-33030 (MICINN) and by the Group FQM-193 of the 3rd Andalusian Research Plan (PAIDI).

## References

- Ball, D. (1990). Preservice elementary and secondary teachers’ understanding of division.
*Journal for Research in Mathematics Education, 21*(2), 132–144.CrossRefGoogle Scholar - Behr, M., Khoury, H., Harel, G., Post, T., & Lesh, R. (1997). Conceptual units analysis of preservice elementary school teachers’ strategies on a rational number as operator task.
*Journal for Research in Mathematics Education, 28*(1), 48–69.CrossRefGoogle Scholar - Behr, M. J., Lesh, R., Post, T. R., & Silver, E. A. (1983). Rational number concept. In R. Lesh & M. Landau (Eds.),
*Acquisitions of mathematics concepts and processes*(pp. 91–126). New York: Academy Press.Google Scholar - Carpenter, T. P., Fennema, E., & Romberg, T. A. (1993).
*Rational numbers: An integration of research*. Hillsdale: LEA.Google Scholar - Castro-Rodríguez, E., Rico, L., & Gómez, P. (2013, July).
*Meanings of fractions as demonstrated by future primary teachers in the initial phase of teacher education*. Paper presented at the 12th ICME, Seoul, Korea.Google Scholar - Charalambous, C. Y., & Pitta-Pantazi, D. (2007). Drawing on theoretical model to study students’ understanding of fractions.
*Educational Studies in Mathematics, 64*(3), 293–316.CrossRefGoogle Scholar - Cluff, J. J. (2005).
*Fraction multiplication and division image change in pre-service elementary teachers*(Unpublished doctoral dissertation). Brigham Young University, USA.Google Scholar - Cramer, K., & Lesh, R. (1988). Rational number knowledge of preservice elementary education teachers. In M. Behr (Ed.),
*Proceedings of the 10th annual meeting of the North American chapter of the international group for psychology of mathematics education*(pp. 425–431). DeKalb: PME.Google Scholar - Domoney, B. (2001). Student teachers’ understanding of rational numbers. In J. Winter (Ed.),
*Proceedings of the British society for research into learning mathematics*(Vol. 21, pp. 13–18). Southampton: BSRLM.Google Scholar - Figueras, O. (1988).
*Dificultades de aprendizaje en dos modelos de enseñanza de los racionales*(Unpublished doctoral dissertation). Cinvestav, Mexico.Google Scholar - Freudenthal, H. (1983).
*Didactical phenomenology of mathematical structures*. Dordrecht: Reidel.Google Scholar - Hiebert, J., & Carpenter, T. P. (1992). Learning and teaching with understanding. In D. Grouws (Ed.),
*Handbook of research on mathematics learning and teaching*(pp. 65–97). New York: Macmillan Publishing Company.Google Scholar - Hill, H. C., Ball, D. L., & Schilling, S. G. (2008). Unpacking pedagogical content knowledge: Conceptualizing and measuring teachers’ topic-specific knowledge of students.
*Journal for Research in Mathematics Education, 39*(4), 372–400.Google Scholar - Kaput, J. (1987). Towards a theory of symbol use in mathematics. In C. Janvier (Ed.),
*Problems of representation in the teaching and learning of mathematics*(pp. 159–195). Hillsdale: LEA.Google Scholar - Kieren, T. E. (1976). On the mathematical, cognitive, and instructional foundations of rational numbers. In R. Lesh (Ed.),
*Number and measurement*(pp. 101–144). Columbus: ERIC-SMEAC.Google Scholar - Kieren, T. E. (1993).
*The learning of fractions: maturing in a fraction*. Paper presented at the conference fraction learning and instruction, GA, Athens.Google Scholar - Kurt, G., & Cakiroglu, E. (2009). Middle grade students’ performances in translating among representations of fractions: A Turkish perspective.
*Learning and Individual Differences, 19*, 404–410.CrossRefGoogle Scholar - Lo, J., & Grant, T. (2012). Preservice elementary teachers’ conceptions of fractional units. In T. Y. Tso (Ed.),
*Proceedings of the 36th PME*(Vol. 3, pp. 169–176). Taipei: PME.Google Scholar - McMillan, J. H., & Schumacher, S. (2006).
*Research in education*. New York: Longman.Google Scholar - Morgan, C., & Kynigos, C. (2014). Digital artefacts as representations: Forging connections between a constructionist and a social semiotic perspective.
*Educational Studies in Mathematics, 85*(3), 357–379.CrossRefGoogle Scholar - Newton, K. J. (2008). An extensive analysis of preservice elementary teachers’ knowledge of fractions.
*American Educational Research Journal, 45*(4), 1080–1110.CrossRefGoogle Scholar - Park, J., Güçler, B., & McCrory, R. (2012). Teaching preservice teachers about fractions: Historical and pedagogical perspectives.
*Educational Studies in Mathematics, 82*(3), 455–479.CrossRefGoogle Scholar - Philippou, G., & Christou, C. (1994). Preservice elementary teachers’ conceptual and procedural knowledge of fractions. In J. P. Ponte & J. F. Matos (Eds.),
*Proceedings of the 18*^{th}*conference of the international group for the psychology of mathematics education*(Vol. 4, pp. 33–40). Lisbon: PME.Google Scholar - Piaget, J., Inhelder, B., & Szeminska, A. (1960).
*The Child’s conception of geometry*. New York, NY: Basic Books.Google Scholar - Pinto, M., & Tall, D. (1996). Student teachers’ conceptions of the rational number. In L. Puig & A. Gutierrez (Eds.),
*Proceedings of the 20th conference of the international group for the psychology of mathematics education*(Vol. 4, pp. 139–146). Valencia: PME.Google Scholar - Pitkethly, A., & Hunting, R. (1996). A review of recent research in the area of initial fraction concepts.
*Educational Studies in Mathematics, 30*(1), 5–38.CrossRefGoogle Scholar - Ponte, J. P., & Chapman, O. (2008). Preservice mathematics teachers’ knowledge and development. In L. D. English (Ed.),
*Handbook of international research in mathematics education*(pp. 223–261). New York: Routledge.Google Scholar - Post, T., Harel, G., Behr, M., & Lesh, R. (1988). Intermediate teachers knowledge of rational number concepts. In E. Fennema, T. P. Carpenter, & S. J. Lamon (Eds.),
*Papers from first Wisconsin symposium for research on teaching and learning mathematics*(pp. 194–219). Madison: WCER.Google Scholar - Radford, L. (2003). Gestures, speech, and the sprouting of signs: A semiotic-cultural approach to learners’ types of generalization.
*Mathematical Thinking and Learning, 5*(1), 37–70.CrossRefGoogle Scholar - Rico, L. (2009). Sobre las nociones de representación y comprensión en la investigación en educación matemática.
*PNA, 4*(1), 1–14.Google Scholar - Sáenz-Ludlow, A. (2006). Learning mathematics: Increasing the value of initial mathematical wealth.
*Revista Latinoamericana de Investigación en Matemática Educativa*[Special issue], 225–245.Google Scholar - Steffe, L. P., & Olive, J. (1990). Constructing fractions in computer microworlds. In G. Booker, P. Cobb, & T. N. de Mendicuti (Eds.),
*Proceedings of the 14th conference of the international group for the psychology of mathematics education*(Vol. 3, pp. 59–66). Mexico: CONACYT.Google Scholar - Steffe, L. P., & Olive, J. (1993).
*Children’s construction of the rational numbers of arithmetic*. Paper presented at the international study group on the rational numbers of arithmetic, GA, Athens.Google Scholar - Steinbring, H. (1989). Routine and meaning in the mathematics classroom.
*For the Learning of Mathematics, 9*(1), 24–33.Google Scholar - Steinbring, H. (2006). What makes a sign a mathematical sign? An epistemological perspective on mathematical interaction.
*Educational Studies in Mathematics, 61*, 133–162.CrossRefGoogle Scholar - Streefland, L. (1991).
*Fractions in realistic mathematics education: a paradigm of developmental research*. Dordrecht: Kluwer Academic Publishers.CrossRefGoogle Scholar - Tatto, M. T., Schwille, J., Senk, S. L., Bankov, K., Rodriguez, M., Reckase, M., . . . Peck, R. (2012).
*The Teacher Education Study in Mathematics (TEDS-M): Policy, practice, and readiness to teach primary and secondary mathematics. Findings from the IEA Study of the mathematics preparation of future teachers*. Amsterdam, the Netherlands: IEA.Google Scholar - Tichá, M., & Hŏspesová, A. (2013). Developing teachers’ subject didactic competence through problem posing.
*Educational Studies in Mathematics, 83*(1), 133–143.CrossRefGoogle Scholar - Toluk-Uçar, Z. (2009). Developing pre-service teachers’ understanding of fractions through problem posing.
*Teaching and Teacher Education, 25*(1), 166–175.CrossRefGoogle Scholar - Van den Heuvel-Panhuizen, M. (2014). Didactical phenomenology (Freudenthal). In S. Lerman (Ed.),
*Encyclopedia of mathematics education*(pp. 174–176). Heidelberg: Springer.Google Scholar - Wright, K. B. (2008).
*Assessing ec-4 preservice teachers’ mathematics knowledge for teaching fractions concepts*. Unpublished dissertation, University of Texas, USA.Google Scholar