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Productive ambiguity in unconventional representations: “what the fraction is going on?”

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Abstract

We explore the responses of 26 prospective elementary-school teachers to the claim “1/6.5 is not a fraction” asserted by a hypothetical classroom student. The data comprise scripted dialogues that depict how the participants envisioned a classroom discussion of this claim to evolve, as well as an accompanying commentary where they described their personal understanding of the notion of a fraction. The analysis is presented from the perspective of productive ambiguity, where different types of ambiguity highlight the prospective teachers’ mathematical interpretations and pedagogical choices. In particular, we focus on the ambiguity inherent in the aforementioned unconventional representation and how the teachers reconciled it by invoking various models and interpretations of a fraction. We conclude with a description of how the perspective of productive ambiguity can enrich teacher education and classroom discourse.

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Fig. 1

taken from Lee & Lee, 2021)

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Notes

  1. https://dictionary.cambridge.org/dictionary/english/technically.

  2. Some text emphasized here, not in original assignment.

  3. While the PT’s drawing involved rectangles rather than squares to denote the “parts”, we believe the intention in the text is nonetheless clear.

References

  • Ball, D. L. (1993). Halves, pieces, and twoths: Constructing and using representational contexts in teaching fractions. In T. P. Carpenter, E. Fennema, & T. A. Romberg (Eds.), Rational numbers: An integration of research (pp. 157–195). Mahwah.

    Google Scholar 

  • Ball, D. L., Thames, M. H., & Phelps, G. (2008). Content knowledge for teaching: What makes it special? Journal of Teacher Education, 59(5), 389–407.

    Article  Google Scholar 

  • Barwell, R. (2005). Ambiguity in the mathematics classroom. Language and Education, 19(2), 117–125.

    Article  Google Scholar 

  • Buchbinder, O., & Cook, A. (2018). Examining the mathematical knowledge for teaching of proving in scenarios written by pre-service teachers. In O. Buchbinder & S. Kuntze (Eds.), Mathematics teachers engaging with representations of practice (pp. 131–154). Springer.

    Chapter  Google Scholar 

  • Byers, W. (2007). How mathematicians think: Using ambiguity, contradiction, and paradox to create mathematics. Princeton University Press.

    Google Scholar 

  • Čadež, T. H., & Kolar, V. M. (2018). How fifth-grade pupils reason about fractions: A reliance on part-whole subconstructs. Educational Studies in Mathematics, 99(3), 335–357.

    Article  Google Scholar 

  • Castro-Rodríguez, E., Pitta-Pantazi, D., Rico, L., & Gómez, P. (2016). Prospective teachers’ understanding of the multiplicative part-whole relationship of fraction. Educational Studies in Mathematics, 92, 129–146.

    Article  Google Scholar 

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

    Article  Google Scholar 

  • Ciosek, M., & Samborska, M. (2016). A false belief about fractions: What is its source? Journal of Mathematical Behavior, 42, 20–32.

    Article  Google Scholar 

  • Depaepe, F., Van Roy, P., Torbeyns, J., Kleickmann, T., Van Dooren, W., & Verschaffel, L. (2018). Stimulating pre-service teachers’ content and pedagogical content knowledge on rational numbers. Educational Studies in Mathematics, 99, 197–216.

    Article  Google Scholar 

  • Domoney, B. (2002). Student teachers’ understanding of rational number: Part-whole and numerical constructs. Research in Mathematics Education, 4(1), 53–67.

    Article  Google Scholar 

  • Duval, R. (2017). Understanding the mathematical way of thinking: The registers of semiotic representations. (T. M. M. Campos, Ed.). Cham: Springer.

  • Duval, R. (2006). A cognitive analysis of problems of comprehension in a learning of mathematics. Educational Studies in Mathematics, 61(1–2), 103–131.

    Article  Google Scholar 

  • Even, R. (1990). Subject matter knowledge for teaching and the case of functions. Educational Studies in Mathematics, 21(6), 521–544.

    Article  Google Scholar 

  • Foster, C. (2011). Productive ambiguity in the learning of mathematics. For the Learning of Mathematics, 31(2), 3–7.

    Google Scholar 

  • Gray, E. M., & Tall, D. (1994). Duality, ambiguity, and flexibility: A “proceptual” view of simple arithmetic. Journal for Research in Mathematics Education, 25(2), 116–140.

    Google Scholar 

  • Grosholz, E. R. (2007). Representation and productive ambiguity in mathematics and the sciences. Oxford University Press.

    Google Scholar 

  • Grossman, P., Hammerness, K., & McDonald, M. (2009). Redefining teaching, re-imagining teacher education. Teachers and Teaching, 15(2), 273–289.

    Article  Google Scholar 

  • Hazewinkel, M. (Ed.). (1989). Encyclopaedia of mathematics: An updated and annotated translation of the Soviet “Mathematical Encyclopaedia” (Vol. 4). Kluwer Academic Publishers.

    Google Scholar 

  • Herbst, P., Chazan, D., Chen, C.-L., Chieu, V.-M., & Weiss, M. (2011). Using comics-based representations of teaching, and technology, to bring practice to teacher education courses. ZDM Mathematics Education, 43(1), 91–103.

    Article  Google Scholar 

  • Hewitt, D. (1999). Arbitrary and necessary part 1: A way of viewing the mathematics curriculum. For the Learning of Mathematics, 19(3), 2–9.

    Google Scholar 

  • Jayakody, G., & Zazkis, R. (2015). Continuous problem of function continuity. For the Learning of Mathematics, 35(1), 8–14.

    Google Scholar 

  • Joutsenlahti, J., & Perkkilä, P. (2019). Sustainability development in mathematics education: A case study of what kind of meanings do prospective class teachers find for the mathematical symbol “2/3?” Sustainability, 11(2), 1–15.

    Article  Google Scholar 

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

  • Koestler, A. (1964). The act of creation. Picador Pan Books.

    Google Scholar 

  • Kontorovich, I. (2018). Roots in real and complex numbers: A case of unacceptable discrepancy. For the Learning of Mathematics, 38(1), 17–25.

    Google Scholar 

  • Krippendorff, K. (2004). Content analysis: An introduction to its methodology (2nd ed.). Sage.

    Google Scholar 

  • Lee, M. Y., & Lee, J.-E. (2021). Pre-service teachers’ selection, interpretation, and sequence of fraction examples. International Journal of Science and Mathematics Education, 19, 539–558.

    Article  Google Scholar 

  • Mamolo, A. (2010). Polysemy of symbols: Signs of ambiguity. The Mathematics Enthusiast, 7(2), 247–262.

    Article  Google Scholar 

  • Mamolo, A. (2018). Eyes, ears, and expectations: Scripting as a multi-lens tool. In R. Zazkis & P. Herbst (Eds.), Scripting approaches in mathematics education: Mathematical dialogues in research and practice (pp. 229–248). Springer.

    Chapter  Google Scholar 

  • Marmur, O., Yan, X., & Zazkis, R. (2020). Fraction images: The case of six and a half. Research in Mathematics Education, 22(1), 22–47.

    Article  Google Scholar 

  • Marmur, O., & Zazkis, R. (2018). Space of fuzziness: Avoidance of deterministic decisions in the case of the inverse function. Educational Studies in Mathematics, 99(3), 261–275.

    Article  Google Scholar 

  • Mason, J., & Pimm, D. (1984). Generic examples: Seeing the general in the particular. Educational Studies in Mathematics, 15(3), 277–289.

    Article  Google Scholar 

  • Moschkovich, J. N. (2018). Recommendations for research on language and learning mathematics. In: J. N. Moschkovich, D. Wagner, A. Bose, J. Rodrigues Mendes, & M. Schütte (Eds.), Language and communication in mathematics education: International perspectives (pp. 37–47). Cham, Switzerland: Springer.

  • Murray, E., & Baldinger, E. E. (2018). Impact of abstract algebra on teachers’ understanding of and approaches to instruction in solving equations. In N. H. Wasserman (Ed.), Connecting abstract algebra to secondary mathematics, for secondary mathematics teachers (pp. 403–429). Springer.

    Chapter  Google Scholar 

  • Ni, Y., & Zhou, Y.-D. (2005). Teaching and learning fraction and rational numbers: The origins and implications of whole number bias. Educational Psychologist, 40(1), 27–52.

    Article  Google Scholar 

  • Osana, H. P., & Royea, D. A. (2011). Obstacles and challenges in preservice teachers’ explorations with fractions: A view from a small-scale intervention study. Journal of Mathematical Behavior, 30(4), 333–352.

    Article  Google 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.

    Article  Google Scholar 

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

    Article  Google Scholar 

  • Petit, M. M., Laird, R. E., & Marsden, E. L. (2010). A focus on fractions: Bringing research to the classroom. Routledge.

    Book  Google Scholar 

  • Pimm, D. (1987). Speaking mathematically: Communication in mathematics classrooms. Routledge.

    Google Scholar 

  • Priestley, W. M. (2013). Wandering about: Analogy, ambiguity and humanistic mathematics. Journal of Humanistic Mathematics, 3(1), 115–135.

    Article  Google Scholar 

  • Rathouz, M. (2010). Ambiguity in units and their referents: Teaching and learning about rational number operations. For the Learning of Mathematics, 30(1), 44–52.

    Google Scholar 

  • Sahin, N., Gault, R., Tapp, L., & Dixon, J. K. (2020). Pre-service teachers making sense of fraction division with remainders. International Electronic Journal of Mathematics Education, 15(1), 1–13.

    Google Scholar 

  • Schleppegrell, M. J. (2010). Language in mathematics teaching and learning: A research review. In J. N. Moschkovich (Ed.), Language and mathematics education: Multiple perspectives and directions for research (pp. 73–112). Information Age Publishing.

    Google Scholar 

  • Sfard, A. (1991). On the dual nature of mathematical conceptions: Reflections on processes and objects as different sides of the same coin. Educational Studies in Mathematics, 22(1), 1–36.

    Article  Google Scholar 

  • Shultz, T. R., & Pilon, R. (1973). Development of the ability to detect linguistic ambiguity. Child Development, 44(4), 728–733.

    Article  Google Scholar 

  • Simon, M. A., Placa, N., Avitzur, A., & Kara, M. (2018). Promoting a concept of fraction-as-measure: A study of the Learning Through Activity research program. Journal of Mathematical Behavior, 52, 122–133.

    Article  Google Scholar 

  • Sinclair, N., & Robutti, O. (2012). Technology and the role of proof: The case of dynamic geometry. In: M. A. Clements, A. J. Bishop, C. Keitel, J. Kilpatrick, & F. K. S. Leung (Eds.), Third international handbook of mathematics education (pp. 571–596). New York, NY: Springer New York.

  • Sinclair, N., & Moss, J. (2012). The more it changes, the more it becomes the same: The development of the routine of shape identification in dynamic geometry environment. International Journal of Educational Research, 51–52, 28–44.

    Article  Google Scholar 

  • Stevenson, A., & Lindberg, C. A. (Eds.). (2010). New Oxford American Dictionary (3rd ed.). Oxford University Press.

    Google Scholar 

  • Strauss, A. L., & Corbin, J. M. (1998). Basics of qualitative research: Techniques and procedures for developing grounded theory (2nd ed.). Sage.

    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.

    Article  Google Scholar 

  • Tanton, J. (2005). Encyclopedia of mathematics. New York, NY: Facts On File.

    Google Scholar 

  • Tirosh, D., & Even, R. (1997). To define or not to define: The case of (-8)1/3. Educational Studies in Mathematics, 33(3), 321–330.

    Article  Google Scholar 

  • Tobias, J. M. (2013). Prospective elementary teachers’ development of fraction language for defining the whole. Journal of Mathematics Teacher Education, 16(2), 85–103.

    Article  Google Scholar 

  • Vamvakoussi, X., & Vosniadou, S. (2004). Understanding the structure of the set of rational numbers: A conceptual change approach. Learning and Instruction, 14(5), 453–467.

    Article  Google Scholar 

  • Van de Walle, J. A., Karp, K. S., Bay-Williams, J. M., & McGarvey, L. M. (2018). Elementary and middle school mathematics: Teaching developmentally (Fifth Canadian edition). Pearson.

    Google Scholar 

  • Weller, K., Arnon, I., & Dubinsky, E. (2009). Preservice teachers’ understanding of the relation between a fraction or integer and its decimal expansion. Canadian Journal of Science, Mathematics and Technology Education, 9(1), 5–28.

    Article  Google Scholar 

  • Whitacre, I., Atabaş, Ş, & Findley, K. (2019). Exploring unfamiliar paths through familiar mathematical territory: Constraints and affordances in a preservice teacher’s reasoning about fraction comparisons. Journal of Mathematical Behavior, 53, 148–163.

    Article  Google Scholar 

  • White, M. D., & Marsh, E. E. (2006). Content analysis: A flexible methodology. Library Trends, 55(1), 22–45.

    Article  Google Scholar 

  • Zakaryan, D., & Ribeiro, M. (2019). Mathematics teachers’ specialized knowledge: A secondary teacher’s knowledge of rational numbers. Research in Mathematics Education, 21(1), 25–42.

    Article  Google Scholar 

  • Zazkis, R. (2000). Using code-switching as a tool for learning mathematical language. For the Learning of Mathematics, 20(3), 38–43.

    Google Scholar 

  • Zazkis, R., & Herbst, P. (Eds.). (2018). Scripting approaches in mathematics education: Mathematical dialogues in research and practice. Springer.

    Google Scholar 

  • Zazkis, R., & Kontorovich, I. (2016). A curious case of superscript (−1): Prospective secondary mathematics teachers explain. Journal of Mathematical Behavior, 43, 98–110.

    Article  Google Scholar 

  • Zazkis, R., & Leikin, R. (2008). Exemplifying definitions: A case of a square. Educational Studies in Mathematics, 69(2), 131–148.

    Article  Google Scholar 

  • Zazkis, R., & Marmur, O. (2018). Scripting tasks as a springboard for extending prospective teachers’ example spaces: A case of generating functions. Canadian Journal of Science, Mathematics and Technology Education, 18(4), 291–312.

    Article  Google Scholar 

  • Zazkis, R., & Marmur, O. (2021). Pedagogical tasks toward extending mathematical knowledge: Notes on the work of teacher educators. In M. Goos & K. Beswick (Eds.), The learning and development of mathematics teacher educators: International perspectives and challenges (pp. 83–108). Springer.

    Chapter  Google Scholar 

  • Zazkis, R., Sinclair, N., & Liljedahl, P. G. (2013). Lesson play in mathematics education: A tool for research and professional development. Springer.

    Book  Google Scholar 

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Author information

Authors and Affiliations

  1. Mathematics Education Unit, Department of Mathematics, Faculty of Science, The University of Auckland, 38 Princes St., Auckland Central, Auckland, 1010, New Zealand

    Ofer Marmur

  2. Faculty of Education, Simon Fraser University, 8888 University Drive, Burnaby, BC, V5A 1S6, Canada

    Rina Zazkis

Authors
  1. Ofer Marmur
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  2. Rina Zazkis
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Correspondence to Ofer Marmur.

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“What the fraction is going on?” is a title that one of the participants gave to the task used in this study.

Appendix: The scripting task

Appendix: The scripting task

Classroom scenario

Teacher: I would like you to work in pairs and find as many fractions as possible between \(\frac{1}{6}\) and \(\frac{1}{7}\).

[The students work in pairs as instructed and after a while the teacher hears the following conversation between two of the students in class]

Natalie: I think \(\frac{1}{6.5}\) is between \(\frac{1}{6}\) and \(\frac{1}{7}\).

Emma: How did you get to that?! That sounds weird!

Natalie: It’s not weird! 6.5 is between 6 and 7, so \(\frac{1}{6.5}\) has to be between \(\frac{1}{6}\) and \(\frac{1}{7}\).

Emma: But \(\frac{1}{6.5}\) is not a fraction!

…

Assignment description

Read the above classroom scenario and respond to the following:

(A) Continuation of the dialogue

Imagine yourself in the role of the teacher and write a dialogue that continues the interaction above as you imagine it continuing in a real classroom situation (for example, you may decide to have the teacher engage in dialogue with Natalie and Emma from the get-go, or to have the teacher first listen to the students’ interaction and only later begin engaging in dialogue with Natalie and Emma; another option to consider is whether the teacher converses only with Natalie and Emma or opens the discussion to the entire classroom; these are only several examples to consider and the decision on how to continue is yours to make). The continuation of the discussion is the main part of the assignment.

(B) Additional questions

Please also respond to the following questions. It is recommended to already start thinking about these questions before beginning writing the dialogue continuation, as these questions could help guide you in imagining how the classroom situation may play out.

  1. 1.

    Examine the task given by the teacher (finding as many fractions as possible between \(\frac{1}{6}\) and \(\frac{1}{7}\)): Which difficulties do you expect students to experience during this activity? What are potential student errors and/or misconceptions in your opinion?

  2. 2.

    Examine the beginning of the dialogue presented above: Were you surprised by Natalie’s suggestion? Were you surprised by Emma’s response? Why yes or why not? Any other thoughts that went through your mind as you were reading the dialogue?

  3. 3.

    Explain your choice/s of the pedagogical approach and actions that are demonstrated in your continuation of the dialogue.

  4. 4.

    Address whether a discussion with a teacher colleague about the mathematics involved in the task would be different from the conversation in your imagined continuation of the dialogue (for example, a conversation with a colleague might include mathematics that is beyond students’ mathematical knowledge). If so, in what way?

  5. 5.

    [OPTIONAL QUESTION] Reflect on your engagement with the assignment: How have you dealt with it? Did you have any difficulties and/or dilemmas while working on the assignment? What have you learned through the process?

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Marmur, O., Zazkis, R. Productive ambiguity in unconventional representations: “what the fraction is going on?”. J Math Teacher Educ 25, 637–665 (2022). https://doi.org/10.1007/s10857-021-09510-7

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