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On Numbers: Concepts, Operations, and Structure

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The Second Handbook of Research on the Psychology of Mathematics Education

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Abstract

Mathematics as well as mathematics education research has long progressed beyond the study of number. Nevertheless, numbers and understanding numbers by learners, continue to fascinate researchers and bring new insights about these fundamental notions of mathematics.

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References

  • Alatorre, S., & Saiz, M. (2008). Mexican primary school teachers’ misconceptions on decimal numbers. Proceedings of PME 32, 2, 25–32.

    Google Scholar 

  • Alatorre, S., Mendiola, E., Moreno, F., Sáiz, M., & Torres, R. (2011). How teachers confront fractions. Proceedings of PME 35, 2, 17–24.

    Google Scholar 

  • Amato, S. A. (2006). Improving student teachers’ understanding of fractions. Proceedings of PME 30, 2, 41–48.

    Google Scholar 

  • Amato, S. A. (2009). Improving elementary student teachers’ knowledge of multiplication of rational numbers. Proceedings of PME 33, 2, 41–48.

    Google Scholar 

  • Amato, S. A. (2011). Ratio: A neglected division at schools. Proceedings of PME 35, 2, 25–32.

    Google Scholar 

  • Askew, M., Abdulhamid, L., & Mathews, C. (2014). Embodiment in teaching and learning early counting: Grounding metaphors. Proceedings of PME 38, 2, 65–72.

    Google Scholar 

  • Bakker, M., van den Heuvel-Panhuizen, M., & Robitzsch, A. (2013). What children know about multiplicative reasoning before being taught. Proceedings of PME 37, 2, 49–56.

    Google Scholar 

  • Barkai, R., Tabach, M., Tirosh, D., Tsamir, P., & Dreyfus, T. (2009). Highschool teachers’ knowledge about elementary number proofs constructed by students. Proceedings of PME 33, 2, 113–120.

    Google Scholar 

  • Battista, M. T. (1990). Spatial visualization and gender differences in high school geometry. Journal for Research in Mathematics Education, 21(1), 47–60.

    Article  Google Scholar 

  • Bofferding, L., & Hoffman, A. (2014). Learning negative integer concepts: Benefits of playing linear board games. Proceedings of PME 38, 2, 169–176.

    Google Scholar 

  • Bolite Frant, J., Quintaneiro, W., & Powell, A. B. (2014). Embodiment and argumentation theories: Time axis in periodic phenomena. Proceedings of PME 38, 6, 290.

    Google Scholar 

  • Bonotto, C. (2006). Extending students’ understanding of decimal numbers via realistic mathematical modeling and problem posing. Proceedings of PME 30, 2, 193–200.

    Google Scholar 

  • Bruno, A., & Cabrera, N. (2006). Types of representations of the number line in textbooks. Proceedings of PME 30, 2, 249–256.

    Google Scholar 

  • Callejo, M. L., Fernandez, C., & Marquez, M. (2013). Pre-service primary teachers’ knowledge for teaching of quotitive division word problems. Proceedings of PME 37, 2, 145–152.

    Google Scholar 

  • Cayton, G. A., & Brizuela, B. M. (2008). Relationships between children’s external representations of number. Proceedings of PME 32, 2, 265–272.

    Google Scholar 

  • Charalambous, C. Y. (2007). Developing and testing a scale for measuring students’ understanding of fractions. Proceedings of PME 31, 2, 105–112.

    Google Scholar 

  • Charalambous, C. Y., & Pitta-Pantazi, D. (2005). Revisiting a theoretical model on fractions: Implications for testing and research. Proceedings of PME 29, 2, 233–240.

    Google Scholar 

  • Chick, H., Baker, M., Pham, T., & Cheng, H. (2006). Aspects of teachers’ pedagogical content knowledge for decimals. Proceedings of PME 30, 2, 297–304.

    Google Scholar 

  • Chrysostomou, M., & Mousoulides, N. (2010). Pre-service teachers’ knowledge of negative numbers. Proceedings of PME 34, 2, 265–272.

    Google Scholar 

  • Cimen, O. A., & Campbell, S. R. (2012). Studying, self-reporting, and restudying basic concepts of elementary number theory. Proceedings of PME 36, 2, 163–170.

    Google Scholar 

  • Clarke, B., Clarke, D. M., & Horne, M. (2006). A longitudinal study of children’s mental computation strategies. Proceedings of PME 30, 2, 329–336.

    Google Scholar 

  • Clarke, D. M., Sukenik, M., Roche, A., & Mitchell, A. (2006). Assessing fraction understanding using task-based interviews. Proceedings of PME 30, 2, 337–344.

    Google Scholar 

  • Coles, A. (2014). Ordinality, neuroscience, and the early learning of number. Proceedings of PME 38, 2, 329–336.

    Google Scholar 

  • Cooper, J. (2014). Mathematical discourse for teaching: A discursive framework for analyzing professional development. Proceedings of PME 38, 2, 337–344.

    Google Scholar 

  • Cortina, J. L., & Zúñiga, C. (2008). Ratio-like comparisons as an alternative to equal-partitioning in supporting initial learning of fractions. Proceedings of PME 32, 2, 385–392.

    Google Scholar 

  • Csikos, C. (2012). Success and strategies in 10 year old students’ mental three-digit addition. Proceedings of PME 36, 2, 179–186.

    Google Scholar 

  • Deliyianni, E., Panaoura, A., Elia, I., & Gagatsis, A. (2008). A structural model for fraction understanding related to representations and problem solving. Proceedings of PME 32, 2, 399–406.

    Google Scholar 

  • Deliyianni, E., Elia, I., Panaoura, A., & Gagatsis, A. (2009). A structural model for the understanding of decimal numbers in primary and secondary education. Proceedings of PME 33, 2, 393–400.

    Google Scholar 

  • Department for Education and Skills (DfES). (2006). Primary national strategy: Primary framework for literacy and mathematics. Norwich: DfES Publications.

    Google Scholar 

  • Diezmann, C., & Lowrie, T. (2007). The development of primary students’ knowledge of the structured number line. Proceedings of PME 31, 2, 201–208.

    Google Scholar 

  • Doritou, M., & Gray, E. (2009). The number line and its demonstration for arithmetic operations. Proceedings of PME 33, 2, 457–464.

    Google Scholar 

  • Dreher, A., Kuntze, S., & Winkel, K. (2014). Empirical study of a competence structure model regarding conversions of representations – the case of fractions. Proceedings of PME 38, 2, 425–432.

    Google Scholar 

  • Dubinsky, E., Weller, K., McDonald, M. A., & Brown, A. (2005a). Some historical issues and paradoxes regarding the concept of infinity: An APOS-based analysis: Part 1. Educational Studies in Mathematics, 58, 335–359.

    Article  Google Scholar 

  • Dubinsky, E., Weller, K., McDonald, M. A., & Brown, A. (2005b). Some historical issues and paradoxes regarding the concept of infinity: An APOS-based analysis: Part 2. Educational Studies in Mathematics, 60, 253–266.

    Article  Google Scholar 

  • Dubinsky, E., Weller, K., Stenger, C., & Vidakovic, D. (2008). Infinite iterative processes: The tennis ball problem. European Journal of Pure and Applied Mathematics, 1(1), 99–121.

    Google Scholar 

  • Ekol, G. (2010). Operations with negative integers in a dynamic geometry environment. Proceedings of PME 34, 2, 337–344.

    Google Scholar 

  • Ellemor-Collins, D., & Wright, R. J. (2008). From counting by ones to facile higher decade edition: The case of Robyn. Proceedings of PME 32, 2, 439–446.

    Google Scholar 

  • Ernest, P. (1985). The number line as a teaching aid. Educational Studies in Mathematics, 16, 411–424.

    Article  Google Scholar 

  • Freudenthal, H. (1983). Didactical phenomenology of mathematical structures. Dordrecht, Holland: Kluwer.

    Google Scholar 

  • Fuson, K. C., Wearne, D., Hiebert, J. C., Murray, H. G., Human, P. G., Olivier, A. I., Carpenter, T. P., & Fennema, E. (1997). Children’s conceptual structures for multidigit numbers and methods of multidigit addition and subtraction. Journal for Research in Mathematics Education, 28, 130–162.

    Article  Google Scholar 

  • Gabel, M., & Dreyfus, T. (2013). The flow of a proof – the example of the Euclidean algorithm. Proceedings of PME 37, 2, 321–328.

    Google Scholar 

  • Gallardo, A., & Hernandez, A. (2005). The duality of zero in the transition from arithmetic to algebra. Proceedings of PME 29, 3, 17–24.

    Google Scholar 

  • Gallardo, A., & Hernandez, A. (2006). The zero and negativity among secondary school students. Proceedings of PME 30, 3, 153–160.

    Google Scholar 

  • Geiger, V., Dole, S., & Goos, M. (2011). The role of digital technologies in numeracy. Proceedings of PME 35, 2, 385–392.

    Google Scholar 

  • Gervasoni, A. (2006). Insights about the addition strategies used by grade 1 and grade 2 children who are vulnerable in number learning. Proceedings of PME 30, 3, 177–184.

    Google Scholar 

  • Gervasoni, A., Brandenburg, R., Turkenburg, K., & Hadden, T. (2009). Caught in the middle: Tensions rise when students and teachers relinquish algorithms. Proceedings of PME 33, 3, 57–64.

    Google Scholar 

  • Gervasoni, A., Parish, L., Bevan, K., Croswell, M., Hadden, T., Livesey, C., & Turkenburg, K. (2011). Exploring the mystery of children who read, write, and order 2-digit numbers, but cannot locate 50 on a number line. Proceedings of PME 35, 2, 401–408.

    Google Scholar 

  • Gilmore, C., & Inglis, M. (2008). Process- and object-based thinking in arithmetic. Proceedings of PME 32, 3, 73–80.

    Google Scholar 

  • Gomez, D. M., Jimenez, A., Bobadilla, R., Reyes, C., & Dartnell, P. (2014). Exploring fraction comparison in school children. Proceedings of PME 38, 3, 185–192.

    Google Scholar 

  • González-Martín, A. S., Giraldo, V., & Machado Souto, A. (2011). Representations and tasks involving real numbers in school textbooks. Proceedings of PME 35, 2, 449–456.

    Google Scholar 

  • Gray, E., & Doritou, M. (2008). The number line: Ambiguity and interpretation. Proceedings of PME 32, 3, 97–104.

    Google Scholar 

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

    Article  Google Scholar 

  • Gunnarsson, R., Hernell, B., & Sonnerhed, W. W. (2012). Useless brackets in arithmetic expressions with mixed operations. Proceedings of PME 36, 2, 275–282.

    Google Scholar 

  • Hannah, J., Stewart, S., & Thomas, M. (2014). Teaching linear algebra in the embodied, symbolic, and formal worlds of mathematical thinking: Is there a preferred order? Proceedings of PME 38, 3, 241–248.

    Google Scholar 

  • Heemsoth, T., & Heinze, A. (2013). Learning fractions from errors. Proceedings of PME 37, 3, 25–32.

    Google Scholar 

  • Heemsoth, T., & Heinze, A. (2014). How should students reflect upon their own errors with respect to fraction problems? Proceedings of PME 38, 3, 265–272.

    Google Scholar 

  • Heinze, A., Marschick, F., & Lipowsky, W. (2009). Addition and subtraction of three-digit numbers: Adaptive strategy use and the influence of instruction in German third grade. ZDM, 41, 591–604.

    Article  Google Scholar 

  • Heirdsfield, A., & Lamb, J. (2006). Teacher actions: Enhancing the learning of mental computation in year 2. Proceedings of PME 30, 3, 281–288.

    Google Scholar 

  • Ho, S. Y., & Lai, M. Y. (2012). Pre-service teachers’ specialized content knowledge on multiplication of fractions. Proceedings of PME 36, 2, 291–298.

    Google Scholar 

  • Hodgen, J., Kuchemann, D., Brown, M., & Coe, R. (2010). Multiplicative reasoning, ratio and decimals: A 30-year comparison of lower secondary students understanding. Proceedings of PME 34, 3, 89–96.

    Google Scholar 

  • Iannece, D., Mellone, M., & Tortora, R. (2009). Counting vs. measuring: Reflections on number roots between epistemology and neuroscience. Proceedings of PME 33, 3, 209–216.

    Google Scholar 

  • Izsak, A. (2006). Knowledge for teaching fraction arithmetic: Partitioning drawn representations. Proceedings of PME 30, 3, 345–352.

    Google Scholar 

  • Jones, I., & Inglis, M. (2015). The problem of assessing problem solving: Can comparative judgement help? Educational Studies in Mathematics, 89(3), 337–355.

    Article  Google Scholar 

  • Jones, I., Inglis, M., Gilmore, C., & Hodgen, J. (2013). Measuring conceptual understanding: The case of fractions. Proceedings of PME 37, 3, 113–120.

    Google Scholar 

  • Kalogirou, P., Gagatsis, A., Michael, P., & Deliyianni, E. (2010). An attempt to overcome the epistemological obstacle in the case of fraction division. Proceedings of PME 34, 3, 153–160.

    Google Scholar 

  • Kempen, L., & Biehler, R. (2014). The quality of argumentations of first-year pre-service teachers. Proceedings of PME 38, 3, 425–432.

    Google Scholar 

  • Kieren, T. (1980). The rational number construct–Its elements and mechanisms. In T. Kieren (Ed.), Recent research on number learning (pp. 125–149). Columbus, OH: ERIC/SMEAC.

    Google Scholar 

  • Kieren, T. (1988). Personal knowledge of rational numbers: Its intuitive and formal development. In J. Hiebert & M. Behr (Eds.), Research agenda for mathematics education: Number concepts and operations in the middle grades (Vol. 2, pp. 162–181). Mahwah, NJ: Lawrence Erlbaum.

    Google Scholar 

  • Kilhamn, C. (2009). The notion of number sense in relation to negative numbers. Proceedings of PME 33, 3, 329–336.

    Google Scholar 

  • Kim, D., Sfard, A., & Ferrini-Mundy, J. (2005). Students’ colloquial and mathematical discourses on infinity and limit. Proceedings of PME 29, 3, 201–208.

    Google Scholar 

  • Kim, D., Ferrini-Mundy, J., & Sfard, A. (2012). How does language impact the learning of mathematics? Comparison of English and Korean speaking university students’ discourses on infinity. International Journal of Educational Research, 51–52, 86–108.

    Article  Google Scholar 

  • Koichu, B. (2008). If not, what yes? International Journal of Mathematical Education in Science and Technology, 39(4), 443–454.

    Article  Google Scholar 

  • Kortenkamp, U., & Ladel, S. (2013). Designing a technology based learning environment for place value using artifact-centric activity theory. Proceedings of PME 37, 1, 188–192.

    Google Scholar 

  • Kortenkamp, U., & Ladel, S. (2014). Flexible use and understanding of place value via traditional and digital tools. Proceedings of PME 38, 4, 33–40.

    Google Scholar 

  • Koukkoufis, A., & Williams, J. (2006). Integer instruction: A semiotic analysis of the “compensation strategy”. Proceedings of PME 30, 3, 473–480.

    Google Scholar 

  • Kullberg, A., Watson, A., & Mason, J. (2009). Variation within, and covariation between, representations. Proceedings of PME 33, 3, 433–440.

    Google Scholar 

  • Kyriakides, A. O. (2006). Modelling fractions with area: The salience of vertical partitioning. Proceedings of PME 30, 4, 17–24.

    Google Scholar 

  • Ladel, S., & Kortenkamp, U. (2013). An activity-theoretic approach to multi-touch tools in early maths learning. The International Journal for Technology in Mathematics Education, 20(1), 3–8.

    Google Scholar 

  • Lampert, M., & Tzur, R. (2009). Participatory stages toward counting-on: A conceptual cause for ‘regress’ to counting-all. Proceedings of PME 33, 3, 473–479.

    Google Scholar 

  • Leinonen, J., & Pehkonen, E. (2011). Teacher students’ improvements in calculation and understanding in the case of division. Proceedings of PME 35, 3, 129–136.

    Google Scholar 

  • Lesh, R., Post, T., & Behr, M. (1987). Representations and translations among representations in mathematics learning and problem solving. In C. Janvier (Ed.), Problems of representation in the teaching and learning of mathematics (pp. 33–40). Hillsdale, NJ: Lawrence Erlbaum.

    Google Scholar 

  • Levenson, E. S. (2014). Exploring the relationship between explanations and examples: Parity and equivalent fractions. Proceedings of PME 38, 4, 105–112.

    Google Scholar 

  • Li, Y., & Smith, D. (2007). Prospective middle school teachers’ knowledge in mathematics and pedagogy for teaching – the case of fraction division. Proceedings of PME 31, 3, 185–192.

    Google Scholar 

  • Linchevski, L., & Livneh, D. (1999). Structure sense: The relationship between algebraic and numerical contexts. Educational studies in Mathematics, 40(2), 173–196.

    Article  Google Scholar 

  • Ling, L. M., & Runesson, U. (2007). Teachers’ learning from learning studies: An example of teaching and learning fractions in primary four. Proceedings of PME 31, 1, 157–161.

    Google Scholar 

  • Lo, J., & Grant, T. (2012). Prospective elementary teachers’ conception of fractional units. Proceedings of PME 36, 3, 169–176.

    Google Scholar 

  • Maeda, Y., & Yoon, S. Y. (2013). A meta-analysis on gender differences in mental rotation ability measured by the purdue spatial visualization tests: Visualization of rotations. Educational Psychology Review 25, 69–94.

    Article  Google Scholar 

  • Maes, R., Cornet, E., Verhoef, N., & Hendrikse, P. (2011). High school students’ problems with infinity. Proceedings of PME 35, 3, 169–176.

    Google Scholar 

  • Mamede, E., & Cardoso, P. (2010). Insights on students (mis)understanding of fractions. Proceedings of PME 34, 3, 257–264.

    Google Scholar 

  • Mamede, E., & Nunes, T. (2008). Building on children’s informal knowledge in the teaching of fractions. Proceedings of PME 32, 3, 345–352.

    Google Scholar 

  • Mamolo, A. (2007). Infinite magnitude vs infinite representation: The story of π. Proceedings of PME 31, 3, 233–240.

    Google Scholar 

  • Mamolo, A. (2014a). Cardinality and cardinal number of an infinite set: A nuanced relationship. Proceedings of PME 38, 4, 169–176.

    Google Scholar 

  • Mamolo, A. (2014b). How to act? A question of encapsulating infinity. Canadian Journal of Science, Mathematics, and Technology Education, 14(1), 1–22.

    Article  Google Scholar 

  • Mamolo, A., & Zazkis, R. (2008a). Paradoxes as a lens for exploring notions of infinity. Proceedings of PME 32, 3, 353–360.

    Google Scholar 

  • Mamolo, A., & Zazkis, R. (2008b). Paradoxes as a window to infinity. Research in Mathematics Education, 10(2), 167–182.

    Article  Google Scholar 

  • Marchini, C., & Papadopoulos, I. (2011). Are useless brackets useful tools for teaching? Proceedings of PME 35, 3, 185–192.

    Google Scholar 

  • Markovits, Z., & Pang, J. (2007). The ability of sixth grade students in Korea and Israel to cope with number sense tasks. Proceedings of PME 31, 3, 241–248.

    Google Scholar 

  • Mason, R., Taylor, P., Simmt, E., & Gourdeau, F. (2015). Should we continue to teach fractions in school? Panel discussion presented at the 39th Annual Meeting of the Canadian Mathematics Education Study Group, Moncton, NB, Canada.

    Google Scholar 

  • Merenluoto, K., & Lehtinen, E. (2006). Conceptual change in the number concept: Dealing with continuity and limit. Proceedings of PME 30, 1, 163–164.

    Google Scholar 

  • Ministry of Education and Culture, Finland. (2015). Basic education of the future – Let’s turn the trend! Retrieved May 25, 2015, from www.minedu.fi/OPM/Verkkouutiset/2015/03/tomorrows_school.html?lang=en

  • Montes, M., Carrillo, J., & Ribeiro, C. M. (2014). Teachers knowledge of infinity, and its role in classroom practice. Proceedings of PME 38, 4, 233–230.

    Google Scholar 

  • Morselli, F. (2006). Use of examples in conjecturing and proving: An exploratory study. Proceedings of PME 30, 4, 185–192.

    Google Scholar 

  • Murphy, C. (2006). Embodiment and reasoning in children’s invented calculation strategies. Proceedings of PME 30, 4, 217–224.

    Google Scholar 

  • Murphy, C. (2008). The use of the empty number line in England and the Netherlands. Proceedings of PME 32, 4, 9–16.

    Google Scholar 

  • Naik, S., & Subramaniam, K. (2008). Integrating the measure and quotient interpretation of fractions. Proceedings of PME 32, 4, 17–24.

    Google Scholar 

  • Narli, S., Delice, A., & Narli, O. (2009). Secondary school students’ concept of infinity: Primary and secondary intuitions. Proceedings of PME 33, 4, 209–216.

    Google Scholar 

  • Narode, R., Board, J., & Davenport, L. (1993). Algorithms supplant understanding: Case studies of primary students’ strategies for double-digit addition and subtraction. Proceedings PME-NA 15, 1, 254–260.

    Google Scholar 

  • Nikolaou, A. A., & Pitta-Pantazi, D. (2013). Hierarchical levels of fraction understanding at the elementary school. Proceedings of PME 37, 3, 377–384.

    Google Scholar 

  • Obersteiner, A., Moll, G., Beitlich, J. T., Cui, C., Schmidt, M., Khmelivska, T., & Reiss, K. (2014). Expert mathematicians’ strategies for comparing the numerical values of fractions – evidence from eye movements. Proceedings of PME 38, 4, 337–344.

    Google Scholar 

  • Obersteiner, A., Van Hoof, J., & Verschaffel, L. (2013). Expert mathematicians’ natural number bias in fraction comparison. Proceedings of PME 37, 3, 393–400.

    Google Scholar 

  • Olive, J. (2011). Fractions on a dynamic number line. Proceedings of PME 35, 3, 289–296.

    Google Scholar 

  • Pehkonen, E., & Kaasila, R. (2009). Understanding and reasoning in a non-standard division task. Proceedings of PME 33, 4, 345–352.

    Google Scholar 

  • Pehkonen, E., Hannula, M., Maujala, H., & Soro, R. (2006). Infinity of numbers: How students understand it. Proceedings of PME 30, 4, 345–352.

    Google Scholar 

  • Peled, I., Meron, R., & Rota, S. (2007). Using a multiplicative approach to construct decimal structure. Proceedings PME 31, 4, 65–72.

    Google Scholar 

  • Pittalis, M., Pitta-Pantazi, D., & Christou, C. (2013). A longitudinal study tracing the development of number sense components. Proceedings of PME 37, 4, 41–48.

    Google Scholar 

  • Prediger, S. (2006). Continuities and discontinuities for fractions: A proposal for analysing in different levels. Proceedings of PME 30, 4, 377–384.

    Google Scholar 

  • Prediger, S., & Schink, A. (2009). “Three eighths of which whole?” – dealing with changing referent wholes as a key to the part-of-part-model for the multiplication of fractions. Proceedings of PME 33, 4, 409–416.

    Google Scholar 

  • Radford, L. (2003). Gestures, speech, and the sprouting of signs. Mathematical Thinking and Learning, 5(1), 37–70.

    Article  Google Scholar 

  • Radu, I., & Weber, K. (2011). Refinements in mathematics undergraduate students’ reasoning on completed infinite iterative processes. Educational Studies in Mathematics, 78, 165–180.

    Article  Google Scholar 

  • Ribeiro, C. M., Mellone, M., & Jakobsen, A. (2013). Characterizing prospective teachers’ knowledge in/for interpreting students’ solutions. Proceedings of PME 37, 4, 89–96.

    Google Scholar 

  • Roche, A., & Clarke, D. M. (2006). When successful comparison of decimals doesn’t tell the full story. Proceedings of PME 30, 4, 425–432.

    Google Scholar 

  • Selter, C. (2001). Addition and subtraction of three-digit numbers: German elementary children’s success, methods and strategies. Educational Studies in Mathematics, 47, 145–173.

    Article  Google Scholar 

  • Sfard, A. (2004). What could be more practical than good research? On mutual relations between research and practice of mathematics education. Plenary address at ICME-10. Proceedings of the 10th International Congress for Mathematics Education (pp. 76–92).

    Google Scholar 

  • Sfard, A. (2014). Mathematics learning: Does language make a difference? Plenary address at the 17th SIGMAA on RUME Conference, Denver, CO.

    Google Scholar 

  • Shinno, Y. (2007). On the teaching situation of conceptual change: Epistemological considerations of irrational numbers. Proceedings of PME 31, 4, 185–192.

    Google Scholar 

  • Shinno, Y. (2013). Semiotic chaining and reification in learning of square root numbers: On the development of mathematical discourse. Proceedings of PME 37, 4, 209–216.

    Google Scholar 

  • Shinno, Y., & Iwasaki, H. (2009). An analysis of process of conceptual change in mathematics lessons: In the case of irrational numbers. Proceedings of PME 33, 5, 81–88.

    Google Scholar 

  • Siegler, R. S., & Booth, J. L. (2004). Development of numerical estimation in young children. Child Development, 75(2), 428–444.

    Article  Google Scholar 

  • Singer, F. M., & Voica, C. (2009). When the infinite sets uncover structures: An analysis of students’ reasoning on infinity. Proceedings of PME 33, 5, 121–128.

    Google Scholar 

  • Spinillo, A. G., & Lautret, S. L. (2006). Exploring the role played by the remainder in the solution of division problems. Proceedings of PME 30, 5, 153–160.

    Google Scholar 

  • Steinle, V., & Pierce, R. (2006). Incomplete or incorrect understanding of decimals: An important deficit for student nurses. Proceedings of PME 30, 5, 161–168.

    Google Scholar 

  • Suh, J. M., & Moyer-Packenman, P. S. (2007). The application of dual coding theory in multi-representational virtual mathematics environments. Proceedings of PME 31, 4, 209–216.

    Google Scholar 

  • Suh, J. M., & Moyer-Packenman, P. S. (2008). Scaffolding special needs students’ learning of fraction equivalence using virtual manipulatives. Proceedings of PME 32, 4, 297–304.

    Google Scholar 

  • Swidan, O., & Yerushalmy, M. (2013). Embodying the convergence of the Riemann accumulation function in a technology environment. Proceedings of PME 37, 4, 257–264.

    Google Scholar 

  • Tall, D. (1980). The notion of infinite measuring numbers and its relevance in the intuition of infinity. Educational Studies in Mathematics, 11, 271–284.

    Article  Google Scholar 

  • Thompson, P. W., & Saldanha, L. A. (2003). Fractions and multiplicative reasoning. In J. Kilpatrick, G. Martin, & D. Schifter (Eds.), Research companion to the principles and standards for school mathematics (pp. 95–113). Reston, VA: National Council of Teachers of Mathematics.

    Google Scholar 

  • Tjoe, H. (2014). When understanding evokes appreciation: The effect of mathematics content knowledge on aesthetic predisposition. Proceedings of PME 38, 5, 249–256.

    Google Scholar 

  • Toh, P. C., Leong, Y. H., Toh, T. L., & Ho, F. H. (2014). Designing tasks for conjecturing and proving in number theory. Proceedings of PME 38, 5, 257–265.

    Google Scholar 

  • Torbeyns, J., Vanderveken, L., Verschaffel, L., & Ghesquière, P. (2006). Adaptive expertise in the number domain 20–100. Proceedings of PME 30, 5, 289–296.

    Google Scholar 

  • Tsamir, P., & Tirosh, D. (2006). PME 1 to 30 – summing up and looking ahead: A personal perspective on infinite sets. Proceedings of PME 30, 1, 49–66.

    Google Scholar 

  • Tsamir, P., Tirosh, D., Dreyfus, T., Barkai, R., & Tabach, M. (2008). Inservice teachers’ judgement of proofs in ENT. Proceedings of PME 32, 4, 345–352.

    Google Scholar 

  • Tzur, R., & Depue, B. E. (2014). Conceptual and brain processing of unit fraction comparisons: A cogneuro-mathed study. Proceedings of PME 38, 5, 297–304.

    Google Scholar 

  • Tzur, R., Xin, Y. P., Si, L., Woodward, J., Jin, X., et al. (2009). Promoting transition from participatory to anticipatory stage: Chad’s case of multiplicative mixed-unit coordination (mmuc). Proceedings of PME 33, 5, 249–256.

    Google Scholar 

  • Tzur, R., Johnson, H., McClintock, E., Xin, Y. P., Si, L., Kenney, R., Woodward, J., Hord, C., & Jin, X. (2012). Children’s development of multiplicative reasoning: A schemes and tasks framework. Proceedings of PME 36, 4, 155–162.

    Google Scholar 

  • Vamvakoussi, X., & Vosniadou, S. (2006). Aspects of students’ understanding of rational numbers. Proceedings of PME 30, 1, 161–162.

    Google Scholar 

  • Vamvakoussi, X., Christou, K. P., & Van Dooren, W. (2010). Greek and Flemish students’ understanding of the density of rational numbers: more similar than different. Proceedings of PME 34, 4, 249–256.

    Google Scholar 

  • Van Dooren, W., Van Hoof, J., Lijnen, T., & Verschaffel, L. (2012). How students understand aspects of linearity: Searching for obstacles in representational flexibility. Proceedings of PME 36, 4, 187–194.

    Google Scholar 

  • Van Hoof, J., Vandewalle, J., & Van Dooren, W. (2013). In search for the natural number bias in secondary school students when solving algebraic expressions. Proceedings of PME 37, 4, 329–336.

    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 

  • Weller, K., Arnon, I., & Dubinsky, E. (2011). Preservice teachers’ understanding of the relation between a fraction or integer and its decimal expansion: Strength and stability of belief. Canadian Journal of Science, Mathematics, and Technology Education, 11(2), 129–159.

    Article  Google Scholar 

  • Weller, K., Arnon, I., & Dubinsky, E. (2013). Preservice teachers’ understanding of the relation between a fraction or integer and its decimal expansion: The case of 0.999… and 1. Canadian Journal of Science, Mathematics, and Technology Education, 13(3), 232–258.

    Article  Google Scholar 

  • Widjaja, W., & Stacey, K. (2006). Promoting pre-service teachers’ understanding of decimal notation and its teaching. Proceedings of PME 30, 5, 385–392.

    Google Scholar 

  • Williamson, J. (2013). Young children’s cognitive representations of number and their number line estimations. Proceedings of PME 37, 4, 401–408.

    Google Scholar 

  • Woodward, J., Kenney, R., Zhang, D., Guebert, A., Cetintas, S., Tzur, R., & Xin, Y. P. (2009). Conceptually based task design: Megan’s progress to the anticipatory stage of multiplicative double counting (mDC). Proceedings of PME-NA 31, 5, 1378–1385.

    Google Scholar 

  • Zazkis, R. (2009). Number theory in mathematics education: Queen and servant. Mediterranean Journal of Mathematics Education, 8(1), 73–88.

    Google Scholar 

  • Zazkis, R., & Chernoff, E. (2006). Cognitive conflict and its resolution via pivotal/bridging example. Proceedings of PME 30, 5, 465–472.

    Google Scholar 

  • Zazkis, R., & Zazkis, D. (2013). Exploring mathematics via imagined role-playing. Proceedings of PME 37, 4, 433–440.

    Google Scholar 

  • Zazkis, R., & Zazkis, D. (2014). Script writing in the mathematics classroom: Imaginary conversations on the structure of numbers. Research in Mathematics Education, 16(1), 54–70.

    Article  Google Scholar 

  • Zoitsakos, S., Zachariades, T., & Sakonidis, C. (2013). Secondary mathematics teachers’ understanding of the infinite decimal expansion of rational numbers. Proceedings of PME 37, 4, 441–450.

    Google Scholar 

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Authors and Affiliations

  1. Simon Fraser University, Vancouver, Canada

    Rina Zazkis

  2. University of Ontario Institute of Technology, Toronto, Canada

    Ami Mamolo

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  1. Rina Zazkis
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  2. Ami Mamolo
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  1. Universidad de Valencia, Spain

    Ángel Gutiérrez

  2. Monash University, Australia

    Gilah C. Leder

  3. Università di Genova, Italy

    Paolo Boero

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Zazkis, R., Mamolo, A. (2016). On Numbers: Concepts, Operations, and Structure. In: Gutiérrez, Á., Leder, G.C., Boero, P. (eds) The Second Handbook of Research on the Psychology of Mathematics Education. SensePublishers, Rotterdam. https://doi.org/10.1007/978-94-6300-561-6_2

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  • DOI: https://doi.org/10.1007/978-94-6300-561-6_2

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