Progressions in young learners’ understandings of parity arguments

References

• Arcavi, A., Drijvers, P., & Stacey, K. (2017). The learning and teaching of algebra: Ideas, insights, and activities. Routledge.
   Google Scholar
• Balacheff, N. (1988). Aspects of proof in pupils’ practice of school mathematics. In D. Pimm (Ed.), Mathematics, teachers and children (pp. 216–235). Hodder & Stoughton.
   Google Scholar
• Ball, D. L., & Bass, H. (2003). Making mathematics reasonable in school. In J. Kilpatrick, W. G. Martin, & D. Schifter (Eds.), A research companion to principles and standards for school mathematics (pp. 27–44). National Council of Teachers of Mathematics.
   Google Scholar
• Ball, D. L., Hoyles, C., Jahnke, H. N., & Movshovitz-Hadar, N. (2002). The teaching of proof. In L. I. Tatsien (Ed.), Proceedings of the international congress of mathematicians (Vol. III, pp. 907–920). Beijing, China: Higher Education Press.
   Google Scholar
• Baroody, A. J., Cibulskis, M., Lai, M.-L., & Li, X. (2004). Comments on the use of learning trajectories in curriculum development and research. Mathematical Thinking and Learning, 6(2), 227–260. https://doi.org/10.1207/s15327833mtl0602_8
   Google Scholar
• Barrett, J., & Battista, M. (2014). Two approaches to describing the development of students’ reasoning about length: A case study for coordinated related trajectories. In A. Maloney, J. Confrey, & K. Nguyen (Eds.), Learning over time: Learning trajectories in mathematics education (pp. 97–124). Information Age.
   Google Scholar
• Battista, M. (2004). Applying cognition-based assessment to elementary school students’ development of understanding of area and volume measurement. Mathematical Thinking and Learning, 6(2), 185–204. https://doi.org/10.1207/s15327833mtl0602_6
   Google Scholar
• Bieda, K. N. (2010). Enacting proof-related tasks in middle school mathematics: Challenges and opportunities. Journal for Research in Mathematics Education, 41(1), 351–382. https://doi.org/10.5951/jresematheduc.41.4.0351
   Google Scholar
• Bieda, K. N., Ji, X., Drwencke, J., & Picard, A. (2014). Reasoning-and-proving opportunities in elementary mathematics textbooks. International Journal of Educational Research, 64, 71–80. https://doi.org/10.1016/j.ijer.2013.06.005
   Web of Science ®Google Scholar
• Blanton, M., Brizuela, B., Gardiner, A., & Sawrey, K. (2017). A progression in first-grade children’s thinking about variable and variable notation in functional relationships. Educational Studies in Mathematics, 95(2), 181–202. https://doi.org/10.1007/s10649-016-9745-0
   Web of Science ®Google Scholar
• Blanton, M., Brizuela, B., Gardiner, A., Sawrey, K., & Newman-Owens, A. (2015). A learning trajectory in 6-year-olds’ thinking about generalizing functional relationships. Journal for Research in Mathematics Education, 46(5), 511–558. https://doi.org/10.5951/jresematheduc.46.5.0511
   Web of Science ®Google Scholar
• Blanton, M., Brizuela, B., Stephens, A., Knuth, E., Isler, I., Gardiner, A., Stroud, R., Fonger, N., & Stylianou, D. (2018a). Implementing a framework for early algebra. In C. Kieran (Ed.), Teaching and learning algebraic thinking with 5- to 12-year-olds: The global evolution of an emerging field of research and practice (pp. 27–49). Springer International Publishing.
   Google Scholar
• Blanton, M., Otalora Sevilla, Y., Brizuela, B., Gardiner, A., Sawrey, K., & Gibbons, A. (2018b). Exploring kindergarten students’ early understandings of the equal sign. Mathematical Thinking and Learning, 20(3), 167–201. https://doi.org/10.1080/10986065.2018.1474534
   Web of Science ®Google Scholar
• Blanton, M., Stroud, R., Stephens, A., Gardiner, A., Stylianou, D., Knuth, E., Isler, I., & Strachota, S. (2019). Does early algebra matter? The effectiveness of an early algebra intervention in grades 3–5. American Educational Research Journal, 56(5), 1930–1972. https://doi.org/10.3102/0002831219832301
   Web of Science ®Google Scholar
• Blanton, M., & Stylianou, D. (2014). Understanding the role of transactive reasoning in classroom discourse as students learn to construct proofs. Journal of Mathematical Behavior, 34, 76–98. https://doi.org/10.1016/j.jmathb.2014.02.001
   Google Scholar
• Boaler, J., Chen, L., Williams, C., & Cordero, M. (2016). Seeing as understanding: The importance of visual mathematics for our brain and learning. Journal of Applied & Computational Mathematics, 5(5 1–6). https://doi.org/10.4172/2168-9679.1000325
   Google Scholar
• Brizuela, B. M., & Earnest, D. (2008). Multiple notational systems and algebraic understandings: The case of the “best deal” problem. In J. Kaput, D. Carraher, & M. Blanton (Eds.), Algebra in the early grades (pp. 273–301). Erlbaum.
   Google Scholar
• Campbell, T. G., Boyle, J. D., & King, S. (2020). Proof and argumentation in K-12 mathematics: A review of conceptions, content, and support. International Journal of Mathematical Education in Science and Technology, 51(5), 754–774. https://doi.org/10.1080/0020739X.2019.1626503
   Web of Science ®Google Scholar
• Carpenter, T. P., Franke, M. L., & Levi, L. (2003). Thinking mathematically: Integrating arithmetic and algebra in elementary school. Heinemann.
   Google Scholar
• Clements, D. H., & Sarama, J. (2004). Learning trajectories in mathematics education. Mathematical Thinking and Learning, 6(2), 81–89. https://doi.org/10.1207/s15327833mtl0602_1
   Google Scholar
• Clements, D. H., & Sarama, J. (2014). Learning trajectories: Foundations for effective, research-based education. In A. P. Maloney, J. Confrey, & K. H. Nguyen (Eds.), Learning over time: Learning trajectories in mathematics education (pp. 1–30). Information Age Publishing.
   Google Scholar
• Clements, D. H., Sarama, J., & Barrett, J. E. (2007). A longitudinal examination of children’s developing knowledge of measurement. National Science Foundation.
   Google Scholar
• Cobb, P. (2000). Conducting teaching experiments in collaboration with teachers. In A. E. Kelly & R. A. Lesh (Eds.), Handbook of research design in mathematics and science education (pp. 307–333). NJ.
   Google Scholar
• Coe, R., & Ruthven, K. (1994). Proof practices and constructs of advanced mathematics students. Educational Research Journal, 20 (1) , 41–53 http://www.jstor.org/stable/1500862.
   Google Scholar
• Cooper, T. J., & Warren, E. (2011). Years 2 to 6 students’ ability to generalise: Models, representations and theory for teaching and learning. In J. Cai & E. Knuth (Eds.), Early algebraization: A global dialogue from multiple perspectives (pp. 187–214). Springer. https://doi.org/10.1007/978-3-642-17735-4_12
   Google Scholar
• Duschl, R. A., Schweingruber, H. A., & Shouse, A. W. (Eds.). (2007). Taking science to school: Learning and teaching science in grades K–8. National Academies Press.
   Google Scholar
• Falmagne, R. J. (1980). The development of logical competence: A psycholinguistic perspective. In R. Kluwe & H. Spada (Eds.), Developmental models of thinking (pp. 171–197). Academic Press.
   Google Scholar
• Fisher, A. V. (2011). Processing of perceptual information is more robust than processing of conceptual information in preschool-age children: Evidence from costs of switching. Cognition, 119(2), 253–264. https://doi.org/10.1016/j.cognition.2011.01.015
   PubMed Web of Science ®Google Scholar
• Fonger, N. L., Stephens, A., Blanton, M., Isler, I., Knuth, E., & Gardiner, A. (2018). Developing a learning progression for curriculum, instruction, and student learning: An example from mathematics education. Cognition and Instruction, 36(1), 30–55. https://doi.org/10.1080/07370008.2017.1392965
   Web of Science ®Google Scholar
• Glaser, B. G. (1998). Doing grounded theory: Issues and discussions. Sociology Press.
   Google Scholar
• Gravemeijer, K. (2004). Local instruction theories as means of support for teachers in reform mathematics education. Mathematical Thinking and Learning, 6(2), 105–128. https://doi.org/10.1207/s15327833mtl0602_3
   Google Scholar
• Harel, G., & Soto, O. (2017). Structural reasoning. International Journal for Research in Undergraduate Mathematics Education, 3(1), 225–242. https://doi.org/10.1007/s40753-016-0041-2
   Google Scholar
• Harel, G., & Sowder, L. (1998). Students‘ proof schemes: Results from exploratory studies. In A. H. Schoenfeld, J. Kaput, & E. Dubinsky (Eds.), Research in collegiate mathematics education III. American Mathematical Society, 234–283.
   Google Scholar
• Harel, G., & Sowder, L. (2007). Toward a comprehensive perspective on proof. In F. Lester (Ed.), Second handbook of research on mathematics teaching and learning. (pp. 805–842). Information Age Publishing.
   Google Scholar
• Healy, L., & Hoyles, C. (2000). A study of proof conceptions in algebra. Journal for Research in Mathematics Education, 31(4), 396–428. https://doi.org/10.2307/749651
   Web of Science ®Google Scholar
• Kaput, J. (2008). What is algebra? What is algebraic reasoning? In J. J. Kaput, D. W. Carraher, & M. L. Blanton (Eds.), Algebra in the early grades (pp. 5–17). Erlbaum.
   Google Scholar
• Kieran, C. (2007). Learning and teaching of algebra at the middle school through college levels: Building meaning for symbols and their manipulation. In F. K. Lester Jr. (Ed.), Second handbook of research on mathematics teaching and learning (Vol. 2, pp. 707–762). Information Age.
   Google Scholar
• Knuth, E., Choppin, J., & Bieda, K. (2009). Middle school students’ production of mathematical justifications. In D. Stylianou, M. Blanton, & E. Knuth (Eds.), Teaching and learning proof across the grades: A K–16 perspective (pp. 153–170). Routledge.
   Google Scholar
• Knuth, E. J., Choppin, J., Slaughter, M., & Sutherland, J. (2002). Mapping the conceptual terrain of middle school students’ competencies in justifying and proving. In D. S. Mewborn, P. Sztajn, D. Y. White, H. G. Weigel, R. L. Bryant, & K. Nooney (Eds.), Proceedings of the 24th Annual Meeting of the North American chapter of the international group for the psychology of mathematics education (Vol. 4, pp. 1693). Athens, GA: Clearinghouse for Science, Mathematics, and Environmental Education.
   Google Scholar
• Knuth, E. J., Stephens, A. C., McNeil, N. M., & Alibali, M. W. (2006). Does understanding the equal sign matter? Evidence from solving equations. Journal for Research in Mathematics Education, 37(4), 297–312 http://www.jstor.org/stable/30034852.
   Web of Science ®Google Scholar
• Knuth, E. J., Zaslavsky, O., & Ellis, A. (2019). The role and use of examples in learning to prove. Journal of Mathematical Behavior, 53, 256–262. https://doi.org/10.1016/j.jmathb.2017.06.002
   Web of Science ®Google Scholar
• Lampert, M. (1990). When the problem is not the question and the solution is not the answer: Mathematical knowing and teaching. American Educational Research Journal, 27(1), 29–63. https://doi.org/10.3102/00028312027001029
   Web of Science ®Google Scholar
• Lampert, M. (1992). Practices and problems in teaching authentic mathematics. In F. K. Oser, A. Dick, & J. Patry (Eds.), Effective and responsible teaching: The new synthesis (pp. 295–314). Jossey-Bass Publishers.
   Google Scholar
• Lesh, R., & Lehrer, R. (2000). Iterative refinement cycles for videotape analyses of conceptual change. In A. Kelly & R. Lesh (Eds.), Handbook of research in mathematics education (pp. 665–708). Erlbaum.
   Google Scholar
• Linchevski, L., & Livneh, D. (1999). Structure sense: The relationship between algebraic and numerical contexts. Educational Studies in Mathematics, 40(2), 173–196. https://doi.org/10.1023/A:1003606308064
   Google Scholar
• Maher, C. A., & Martino, A. M. (1996). The development of the idea of a mathematical proof: A 5-year case study. Journal for Research in Mathematics Education, 27(2), 194–214. https://doi.org/10.2307/749600
   Web of Science ®Google Scholar
• Maloney, A., Confrey, J., & Nguyen, K. (2014). Learning over time: Learning trajectories in mathematics education. Information Age.
   Google Scholar
• Mason, J., & Pimm, D. (1984). Generic examples: Seeing the general in the particular. Educational Studies in Mathematics 15, 15(3), 277–289. https://doi.org/10.1007/BF00312078
   Google Scholar
• Matthews, P., Rittle-Johnson, B., McEldoon, K., & Taylor, R. (2012). Measure for measure: What combining diverse measures reveals about children’s understanding of the equal sign as an indicator of mathematical equality. Journal for Research in Mathematics Education, 43(3), 316–350. https://doi.org/10.5951/jresematheduc.43.3.0316
   Web of Science ®Google Scholar
• Mulligan, J., & Mitchelmore, M. (2009). Awareness of pattern and structure in early mathematical development. Mathematics Education Research Journal, 21(2), 33–49. https://doi.org/10.1007/BF03217544
   Google Scholar
• National Council of Teachers of Mathematics. (2000). Principle and standards for school mathematics. Author.
   Google Scholar
• National Governors Association Center for Best Practices & Council of Chief State School Officers. (2010). Common core state standards for mathematics. http://www.corestandards.org/assets/CCSSI_Math%20Standards.pdf
   Google Scholar
• Pimm, D. (1995). Symbols and meanings in school mathematics. Routledge. https://doi.org/10.4324/9780203428610
   Google Scholar
• Reid, D. A. (2002). Conjectures and refutations in grade 5 mathematics. Journal for Research in Mathematics Education, 33(1), 5–29. https://doi.org/10.2307/749867
   Web of Science ®Google Scholar
• Rittle-Johnson, B., Matthews, P. G., Taylor, R. S., & McEldoon, K. L. (2011). Assessing knowledge of mathematical equivalence: A construct-modeling approach. Journal of Educational Psychology, 103(1), 85–104. https://doi.org/10.1037/a0021334
   Web of Science ®Google Scholar
• Schifter, D. (2009). Representation-based proof in the elementary grades. In D. Stylianou, M. Blanton, & E. Knuth (Eds.), Teaching and learning proof across grades: A K-16 Perspective (pp. 71–86). Routledge.
   Google Scholar
• Schoenfeld, A. (2009). The soul of mathematics. In D. Stylianou, M. Blanton, & E. J. Knuth (Eds.), Teaching and learning proof across the grades: A K – 16 perspective (pp. xii–xvi). Routledge/Taylor & Francis Group.
   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. doi:10.1007/BF00302715
   Google Scholar
• Shin, N., Stevens, S. Y., Short, H., & Krajcik, J. S. (2009). Learning progressions to support coherence curricula in instructional material, instruction, and assessment design. Paper presented at the Learning Progressions in Science (LeaPS) Conference, Iowa City, IA.
   Google Scholar
• Simon, M. (1995). Reconstructing mathematics pedagogy from a constructivist perspective. Journal for Research in Mathematics Education 26(2), 114–145.
   Web of Science ®Google Scholar
• Simon, M. A., & Tzur, R. (2004). Explicating the role of mathematical tasks in conceptual learning: An elaboration of the hypothetical learning trajectory. Mathematical Thinking and Learning, 6(2), 91–104. https://doi.org/10.1207/s15327833mtl0602_2
   Google Scholar
• Staples, M. E., Bartlo, J., & Thanheiser, E. (2012). Justification as a teaching and learning practice: Its (potential) multifaceted role in middle grades mathematics classrooms. The Journal of Mathematical Behavior, 31(4), 447–462. https://doi.org/10.1016/j.jmathb.2012.07.001
   Google Scholar
• Stephens, A. C., Fonger, N. L., Strachota, S., Isler, I., Blanton, M., Knuth, E., & Gardiner, A. (2017). A learning progression for elementary students’ functional thinking. Mathematical Thinking and Learning, 19(3), 143–166. https://doi.org/10.1080/10986065.2017.1328636
   Web of Science ®Google Scholar
• Stevens, S. Y., Shin, N., & Krajcik, J. S. (2009). Towards a model for the development of an empirically tested learning progression. Paper presented at the learning progressions in science (LeaPS) conference, Iowa City, IA. http://www.education.msu.edu/projects/leaps/proceedings/Stevens.pdf
   Google Scholar
• Strachota, S., Morton, K., Veltri-Torres, R., Stephens, A., Blanton, M., Gardiner, A., Sung, Y., Stroud, R., & Knuth, E. (in preparation). The role of tools in supporting students’ generalizing about even and odd numbers. .
   Google Scholar
• Strauss, A. L., & Corbin, J. (1990). Basics of qualitative research: grounded theory procedures and techniques. Sage.
   Google Scholar
• Stylianides, A. J. (2007a). Proof and proving in school mathematics. Journal for Research in Mathematics Education, 38(3), 289–321. doi:10.2307/30034869.
   Web of Science ®Google Scholar
• Stylianides, A. J. (2007b). The notion of proof in the context of elementary school mathematics. Educational Studies in Mathematics, 65(1), 1–20. https://doi.org/10.1007/s10649-006-9038-0
   Google Scholar
• Stylianides, A. J. (2016). Proving in the elementary mathematics classroom. Oxford University Press.
   Google Scholar
• Stylianides, A. J., & Ball, D. L. (2008). Understanding an describing mathematical knowledge for teaching: Knowledge about proof for engaging students in the activity of proving. Journal of Mathematics Teacher Education, 11(4), 307–332. https://doi.org/10.1007/s10857-008-9077-9
   Google Scholar
• Stylianides, G. J., & Stylianides, A. J. (2008). Proof in school mathematics: Insights from psychological research into students’ ability for deductive reasoning. Mathematical Thinking and Learning, 10(2), 103–133. https://doi.org/10.1080/10986060701854425
   Google Scholar
• Stylianides, G. J., Stylianides, A. J., & Shilling-Traina, L. N. (2013). Prospective teachers’ challenges in teaching reasoning-and-proving. International Journal of Science & Mathematics Education, 11(6), 1463–1490. https://doi.org/10.1007/s10763-013-9409-9
   Web of Science ®Google Scholar
• Stylianides, G. J., Stylianides, A. J., & Weber, K. (2019). Research on the teaching and learning of proof: Taking stock and moving forward. In J. Cai (Ed.), Compendium for research in mathematics education (pp. 237–266). National Council of Teachers of Mathematics.
   Google Scholar
• Stylianou, D., Blanton, M., & Rotou, O. (2015). Undergraduate students’ proof conceptions: Relationships between understanding, beliefs, and classroom experiences with learning proof. International Journal for Research in Undergraduate Mathematics Education, 1(1), 91–134. https://doi.org/10.1007/s40753-015-0003-0
   Google Scholar
• Van Ness, C. K. (2017). Students’ Argumentation about the Density of Fractions. In C. A. Maher & D. Yankelewitz (Eds.), Children’s reasoning while building fraction ideas. mathematics teaching and learning. Sense Publishers,163–181. https://doi.org/10.1007/978-94-6351-008-0_16
   Google Scholar
• Van Ness, C. K., & Maher, C. A. (2019). Analysis of the argumentation of nine-year-olds engaged in discourse about comparing fraction models. Journal of Mathematical Behavior, 53, 13–41. https://doi.org/10.1016/j.jmathb.2018.04.004
   Web of Science ®Google Scholar
• Ventura, A. C., Brizuela, B. M., Blanton, M., Sawrey, K., Gardiner, A., & Newman-Owens, A. (2021). A learning trajectory in Kindergarten and first grade students’ thinking of variable and use of variable notation to represent indeterminate quantities. Journal of Mathematical Behavior, 62, 100866. https://doi.org/10.1016/j.jmathb.2021.100866
   Web of Science ®Google Scholar
• Verhoeven, H. (2013). Flexible categorization in preschool-age children: Comparing color to taxonomic matching strategies. http://writing.rochester.edu/celebrating/2013/Verhoeven.pdf
   Google Scholar
• Wagner, P. A., Smith, R. C., Conner, A., Singletary, L. M., & Francisco, R. T. (2014). Using Toulmin’s model to develop prospective secondary mathematics teachers’ conceptions of collective argumentation. Mathematics Teacher Educator, 3(1), 8–26. https://doi.org/10.5951/mathteaceduc.3.1.0008
   Google Scholar
• Witzel, B. S. (2005). Using CRA to teach algebra to students with math difficulties in inclusive settings. Learning Disabilities—A Contemporary Journal, 32(2), 49–60.
   Google Scholar
• Witzel, B. S., & Little, M. E. (2016). Teaching elementary mathematics to struggling learners. The Guilford Press.
   Google Scholar
• Yackel, E. (2002). What we can learn from analyzing the teacher’s role in collective argumentation. Journal of Mathematical Behavior, 21(4), 423–440. https://doi.org/10.1016/S0732-3123(02)00143-8
   Google Scholar