Measurement Approach to Teaching Fractions: A Design Experiment in a Pre-service Course for Elementary Teachers


  • Georgeana Bobos Dawson College
  • Anna Sierpinska Concordia University


In this paper, we present a design experiment in a “Teaching Mathematics” course for prospective elementary teachers where we sought to develop a measurement approach to fractions.  We focus on the conceptualization of the mathematical content of the approach. We attribute our progress in the conceptualization to our efforts to overcome the challenges we had to face in bringing our students – prospective teachers – to thinking about fractions in a theoretical way. We describe some of these challenges in the paper. The approach was inspired by an approach under the same name, but addressed to children, developed by the psychologist V.V. Davydov and described in the paper by Davydov and Tsvetkovich (1991).  Davydov’s  measurement approach proposes that, in order to develop a concept of fraction with sources in reality, children’s attention should be directed to multiplicative relationships between quantities defined in terms of concrete units (such as kilos, inches or cups) rather than to indeterminate objects such as pizzas or cakes.  Our original contribution is a systemic study of these relationships and operations on them, as a theoretical system, using definitions derived from measurement situations, mathematical reasoning based on these definitions and generalizations of observed patterns. It is intended to support a gradual process of abstraction of the notion of fraction as an abstract number that represents a measure of the relationship between two quantities. Furthermore, our proposal for conceptualizing fractions in this way is addressed to teachers, not to children.  By its focus on relational reasoning about quantities and gradual construction of a theoretical system, the approach both requires and is expected to foster the development of quantitative reasoning and theoretical thinking.

Author Biographies

Georgeana Bobos, Dawson College

2015 PhD, Mathematics Education, Concordia University

Anna Sierpinska, Concordia University

1970 MSc Mathematics, Warsaw University (Poland)

1984 PhD Mathematics, specialization in Mathematics Education, Cracow Higher School of Pedagogy (presently Cracow University of Pedagogy) (Poland)

2006 Honorary doctorate, Lulea Tekniska Universiteit (Sweden)

1990- 1996 Associate Professor at Concordia University

1997 - present Full Professor at Concordia University 


Arcavi, A., & Schoenfeld, A. (1992). Mathematics tutoring through a constructivist lens: The challenges of sense-making. The Journal of Mathematical Behavior, 11, 321-36.

Artigue, M. (1992). Didactical engineering. In R. Douady, & A. Mercier, Research in Didactique of Mathematics. Selected papers (pp. 41-66). Grenoble: La Pensée Sauvage éditions.

Artigue, M. (2009). Didactical Design in Mathematics Education. In C. Winslow (Ed.), Nordic Research in Mathematics Education. Proceedings of NORMA08 (pp. 7-16). Rotterdam: Sense Publishers.

Artigue, M. (2014). Potentialities and limitations of the Theory of Didactic Situations for addressing the teaching and learning of mathematics at university level. Research in Mathematics Education, 16(2), 135-138.

Ball, L. D., & Bass, H. (2009). With an eye on the mathematical horizon: Knowing matematics for teaching to learners' mathematical futures. Based on keynote address at the 43rd Jahrestagung fuer Didaktik der Mathematik, Oldenburg, Germany, March 1-4, 2009. Retrieved from University of Dortmund - Mathematics:

Barbé, Q., Bosch, M., Espinoza, L., & Gascón, J. (2005). Didactic restrictions on the teacher's practice: The case of limits of functions. Educational Studies in Mathematics, 59, 235-268.

Behr, M., Harel, G., Post, T., & Lesh, R. (1993). Rational numbers: Toward a semantic analysis - emphasis on the operator construct. In T. Carpenter, E. Fennema, & T. A. Romberg, Rational numbers: An integration of research (pp. 13-47). Mahwah, NJ: Erlbaum.

Behr, M., Lesh, R., Post, T., & Silver, E. (1983). Rational number concepts. In R. Lesh, & M. Landau, Acquisition of mathematical concepts and processes (pp. 91-125). New York: Academic Press.

Bernstein, B. (1999). Vertical and horizontal discourse: an essay. British Journal of Sociology of Education, 20(2), 157-173.

Bobos-Kristof, G. (2015). Teaching fractions through a Measurement Approach to prospective elementary teachers: A design experiment in a Math Methods course. A PhD Thesis. Montreal : Concordia University.

Bright, G. W., Behr, M. J., Post, T. R., & Wachsmuth, I. (1988). Identifying fractions on number lines. Journal for Research in Mathematics Education, 19(3), 215-232.

Brousseau, G. (1997). Theory of didactical situations in mathematics. Dortrecht: Kluwer.

Brousseau, G., Brousseau, N., & Warfield, V. (2014). Teaching fractions through situations: A fundamental experiment. Springer.

Charalambous, Y., & Pitta-Pantazi, D. (2007). Drawing on a theoretical model to study students' understanding of fractions. Educational Studies in Mathematics, 64(3), 293-316.

Chevallard, Y. (1999). L'analyse des pratiques enseignantes en théorie anthropologique du didactique. Recherches en Didactique des Mathematiques, 19(2), 221-266.

Cobb, P., diSessa, A., Lehrer, R., & Schauble, L. (2003). Design experiments in educational research. Educational Researcher, 32(1), 9-13.
Cobb, P., McClain, K., & Gravemeijer, K. (2003). Learning about statistical covariation. Cognition and Instruction, 21(1), 1-78.

Cobb, P., Zhao, Q., & Dean, C. (2009). Conducting design experiments to support teachers' learning: A reflection from the field. Journal of the Learning Sciences, 18(2), 165-199. doi:10.1080/10508400902797933

Colburn, W. (1884). Warren Colburn's First Lessons. Intellectual Arithmetic upon the Inductive Method of instruction. Boston: Houghton, Mifflin and Co.

Confrey, J. (1990). What constructivism implies for teaching. In R. Davis, C. A. Maher, & N. Noddings, Constructivist views on the teaching and learning of mathematics (pp. 107-22). Reston, Virginia: NCTM.

Davydov, V. V. (1991a). A psychological analysis of the operation of multiplication. In V. Davydov, & L. P. Steffe, Soviet Studies in Mathematics Education, Volume 6. Psychological abilities of primary children in learning mathematics (pp. 9-85). Reston, VA: NCTM, Inc.

Davydov, V. V. (1991b). Preface to the Soviet edition. In V. Davydov, & L. P. Steffe, Soviet Studies in Mathematics Education. Volume 6. Psychological abilities of primary school chiildren in learning mathematics (pp. 1-8). Reston, Virginia: NCTM.

Davydov, V., & Tsvetkovich, Z. H. (1991). The object sources of the concept of fractions. In V. Davydov, & L. P. Steffe, Soviet studies in mathematics education. Volume 6. Psychological abilities of primary school children in learning mathematics (pp. 86-147). Reston, VA: National Council of Teachers of Mathematics.

Dorier, J.-L., Robert, A., Robinet, J., & Rogalski, M. (2000). The obstacle of formalism in linear algebra. In J.-L. Dorier, On the teaching of linear algebra (pp. 85-124). New York: Kluwer.

Eisenhart, M. (2009). Generalization from Qualitative Inquiry. In K. Ercikan, & W.-M. Roth, Generalizing from Educational Researcch: Beyond Qualitative and Quantitative Polarization (pp. 51-66). London: Routledge.

Erickson, H. (1986). Qualitative methods in research on teaching. In M. Wittrock (Ed.), Handbook of research on teaching (3rd ed., pp. 119-161). New York: Macmillan.

Fridman, L. (1991). Features of introducing the concept of concrete numbers in primary grades. In V. V. Davydov, & L. Steffe, Soviet studies in mathematics education. Volume 6. Psychological abilities of primary schoolchildren in learning mathematics (pp. 148-180). Reston, Virginia: NCTM.

Glaser, B., & Strauss, A. (1967). The discovery of grounded theory: Strategies for qualitative research. New York: Aldine.

Kieren, T. E. (1976). On the mathematical, cognitive, and instructional foundations of rational numbers. In R. Lesh, Number and measurement: Papers from a research workshop (pp. 101-144). Columbus, OH: ERIC/SMEAC.

Kieren, T. E. (1992). Rational and fractional numbers as mathematical and personal knowledge: Implications for curriculum and instruction. In G. Leinhardt, R. Putnam, & R. A. Hartrup, Analaysis of arithmetic for mathematics teaching (pp. 323-371). Hillsdale, New Jersey: Erlbaum.

Kieren, T. E. (1993). Rational and fractional numbers: From quotient fields to recursive understanding. In T. Carpenter, E. Fennema, & T. Romberg, Rational numbers: An integration of research (pp. 49-84). Mahwah, New Jersey: Erlbaum.

Kolmogorov, A. (1960). Introduction. In H. Lebesgue, Ob izmerenii velichin [On measuring magnitudes]. Second edition. Moscow: Uchpedgiz.

Lamon, S. J. (2001). Presenting and representing: From fractions to rational numbers. In A. Cuoco, & F. Curcio, The roles of representations in school mathematics - 2001 NCTM Yearbook (pp. 146-165). Reston, Virginia: NCTM.

Lamon, S. J. (2012). Teaching fractions and ratios for understanding. New York: Routledge.

Lerman, S. (2010). Theories in mathematics education: Is plurality a problem? . In B. Sriraman, & L. English, Theories of mathematics education. Seeking new frontiers (pp. 99-110). New York: Springer.

Lompscher, J. (1994). The sociohistorical school and the acquisition of mathematics. In R. Biehler, R. Scholtz, R. W. Straesser, & B. Winkelman, Didactics of mathematics as a scientific discipline (pp. 263-76). Dortrecht: Kluwer.

Ma, L. (1999). Knowing and teaching elementary mathematics. Teachers' understanding of fundamental mathematics in China and the United States. Mahwah, NJ: Erlbaum.

McLellan, J., & Dewey, J. (1900). The psychology of number. New York: Apple-Century-Crofts.

Niemi, D. (1996). Assessing Conceptual Understanding in Mathematics: Representations, Problem Solutions, Justifications, and Explanations. The Journal of Educational Research, 89(6), 351-363.

Noelting, G. (1978). The development of proportional reasoning in the child and adolescent through combination of logic and arithmetic. Proceedings of the Second PME Conference (pp. 242-277). Osnabrück, Germany: PME.

Ohlsson, S. (1988). Mathematical meaning and applicational meaning in the semantics of fractions and related concepts. In J. Hiebert, & M. Behr, Number concepts and operations in the middle grades (pp. 53-92). Reston, Virginia: NCTM.

Pantziara, M., & Philippou, G. (2012). Levels of students’ “conception” of fractions. Educational Studies in Mathematics, 79(1), 61-83.

Parker, T., & Baldridge, S. J. (2003). Elementary mathematics for teachers. Okemos, Michigan: Sefton-Ash Publishing.

Piaget, J. (1969/1990). Psychologie et pédagogie. Paris: Denoël/Folio.

Ratsimba-Rajohn, H. (1982). Éléments d'étude de deux méthodes de mesures rationnelles. Recherches en Didactique des Mathématiques, 3(1), 65-113.

Reys, R., Linquist, M. M., Lambdin, D. V., Smith, N. L., & Colgan, L. E. (2010). Helping children learn mathematics. Canadian edition. Mississauga, Ontario: John Wiley & Sons.

Schmittau, J. (2005). The development of algebraic thinking. A Vygotskian perspective. ZDM, 37(1), 16-22.

Schmittau, J., & Morris, A. (2004). The development of algebra in the elementary mathematics curriculum of V.V. Davydov. The Mathematics Educator, 8(1), 60-87.

Sierpinska, A. (2015). EDUC 387 Teaching Mathematics II - Course Notes Part II. Rational numbers. Ratio. Percent. (Revised Version). Retrieved from Academia paper sharing - Anna Sierpinska:

Sierpinska, A. (2016). Inquiry-based learning aproaches and the development of theoretical thinking in the mathematics education of future elementary school teachers. In B. Maj-Tatsis, M. Pytlak, & E. Swoboda, Inquiry-based mathematical education (pp. 23-57). Rzeszow: University of Rzeszow.

Sierpinska, A., & Bobos, G. (2015). EDUC 387 Teaching Mathematics II - Course Notes Part I - Fractions of quantities (Revised Version). Retrieved from Academia paper sharing - Anna Sierpinska:

Sierpinska, A., Bobos, G., & Pruncut, A. (2011). Teaching absolute value inequalities to mature students. Educational Studies in Mathematics, 78(3), 275-305.

Sowder, J., Sowder, L., & Nickerson, S. (2011). Reconceptualizing mathematics for elementary school teachers. Instructor's edition. New York: W.H. Freeman and Company.

Thompson, P. (1993). Quantitative reasoning, complexity and additive structures. Educational Studies in Mathematics, 25(3), 165-208.

Thompson, P. (1994). The development of the concept of speed and its relationship to the concepts of rate. In G. Harel, & J. Confrey, The development of multiplicative reasoning in the learning of mathematics (pp. 181-234). Albany, NY: SUNY Press.

Thompson, P., & Saldanha, L. (2003). Fractions and multiplicative reasoning. In J. Kilpatrick, W. G. Martin, & D. Schifter, A research companion to Principles and Standards for School Mathematics (pp. 95-113). Reston, VA: National Council of Teachers of Mathematics.

Van de Walle, J. A., & Lovin, L. H. (2006). Teaching student-centered mathematics: Grades 5-8. Pearson Ally & Bacon.

Vergnaud, G. (1994). Multiplicative conceptual field: What and why? In G. Harel, & J. Confrey, The development of multiplicative reasoning in the learning of mathematics (pp. 41-59). Albany, NY: SUNY Press.

Vygotsky, L. S. (1987). The collected works of L. S. Vygotsky. Volume I. Problems of general psychology. New York: Plenum Press.

Wittman, E. (1995). Mathematics education as a "design science". Educational Studies in Mathematics, 29(4), 355-374.

Wong, M., & Evans, D. (2008). Fractions as measure. In M. Goos, R. Brown, & K. Makar (Ed.), Proceedings of the 31st Annual Conference of the Mathematics Education Research Group of Australasia. MERGA Inc.

Wood, T. (1995). From alternative epistemologies to practice in education: Rethinking what it means to teach and learn. In L. P. Steffe, & J. Gale, Constructivism in education (pp. 331-40). Hillsdale, NJ: Erlbaum.

Yopp, D., Ely, R., & Johnson-Leung, J. (2015). Generic example proving criteria for all. For the Learning of Mathematics, 35(3), 8-13.