A Curriculum-Based Approach to Learning Trajectories in Middle School Algebra
Downloads
Abstract
Our aim is to contribute to the body of research on learning trajectories (LTs) in mathematics by making visible a process for articulating a hypothetical learning trajectory implicit in a widely adopted, reform-based, middle-grades mathematics curriculum. In doing so, we highlight considerations, decisions, and challenges we faced as part of this work. By describing our LT articulation process, our aim is to highlight ways in which curriculum-specific LTs can be articulated to serve as a more proximal and instrumental tool for teachers’ instructional practice. Furthermore, to illustrate we describe how the products of the work were used in practice-based professional learning experiences with middle-grades mathematics teachers.
Downloads
References
Barrett, J.E., Clements, D.H., Klanderman, D., Pennisi, S.J., & Polaki, M.V. (2006). Students' coordination of geometric reasoning and measuring strategies on a fixed perimeter task: Developing mathematical understanding of linear measurement. Journal for Research in Mathematics Education, 37(3), 187-221. https://doi.org/10.2307/30035058
Google Scholar CrossrefBerland, L.K., & McNeill, K.L. (2010). A learning progression for scientific argumentation: Understanding student work and designing supportive instructional contexts. Science Education, 94(5), 765-793. https://doi.org/10.1002/sce.20402
Google Scholar CrossrefBernbaum Wilmot, D., Schoenfeld, A. H., Wilson, M., Champney, D., & Zahner, W. (2011). Validating a learning progression in mathematical functions for college readiness. Mathematical Thinking & Learning, 13(4), 259-291. https://doi.org/10.1080/10986065.2011.608344
Google Scholar CrossrefBishop, Alan J., Lamb, L. L., Phillip, R. A. , Whitacre, I., & Schappelle, B. (2014). Using order to reason about negative numbers: The case of Violet. Educational Studies in Mathematics, 86(1), 39-59. https://doi.org/10.1007/s10649-013-9519-x
Google Scholar CrossrefBowers, Janet, Cobb, P., & McClain, K. (1999). The evolution of mathematical practices: A case study. Cognition and Instruction, 17(1), 25-66. https://doi.org/10.1207/s1532690xci1701_2
Google Scholar CrossrefBrown, Matthew W. (2009). The Teacher–Tool Relationship: Theorizing the Design and Use of Curriculum Materials. In J. Remillard, B. Herbel-Eisenmann & G. Lloyd (Eds.), Mathematics Teachers at Work. Routledge.
Google Scholar CrossrefClark, D. B. (2006). Longitudinal conceptual change in students' understanding of thermal equilibrium: An examination of the process of conceptual restructuring. Cognition and Instruction, 24(4), 467-563. https://doi.org/10.1207/s1532690xci2404_3
Google Scholar CrossrefClements, D.H., & Sarama, J. (2004). Learning Trajectories in Mathematics Education. Mathematical Thinking & Learning, 6(2), 81-89.
Google Scholar CrossrefClements, D.H., Wilson, D. C., & Sarama, J. (2004). Young children's composition of geometric figures: A learning trajectory. Mathematical Thinking & Learning, 6(2), 163-184.
Google Scholar CrossrefCobb, P. (1999). Individual and collective mathematical development: The case of statistical data analysis. Mathematical Thinking & Learning, 1(1), 5-43.
Google Scholar CrossrefCobb, P., & Whitenack, J. W. (1996). A method for conducting longitudinal analyses of classroom videorecordings and transcripts. Educational Studies in Mathematics, 30(3), 213-228. https://doi.org/10.1007/BF00304566
Google Scholar CrossrefCobb, P., & Yackel, E. (1996). Constructivist, emergent, and sociocultural perspectives in the context of developmental research. Educational Psychologist, 31(3/4), 175-190. https://doi.org/10.1080/00461520.1996.9653265
Google Scholar CrossrefCollopy, Rachel. (2003). Curriculum Materials as a Professional Development Tool: How a Mathematics Textbook Affected Two Teachers' Learning. The Elementary School Journal, 103(3), 287-311. https://doi.org/10.1086/499727
Google Scholar CrossrefCommon Core Standards Writing Team. (2013a). Grades 3-5, numbers and operations, fractions: A progression of the Common Core State Standards in Mathematics. Institute for Mathematics and Education, University of Arizona.
Google Scholar CrossrefCommon Core Standards Writing Team. (2013b). Preface for the draft progressions. Institute for Mathematics and Education, University of Arizona.
Google Scholar CrossrefConfrey, J., & Maloney, A. (2010). The construction, refinement, and early validation of the equipartitioning learning trajectory. Paper presented at the International Conference of the Learning Sciences, Chicago, US.
Google Scholar CrossrefConfrey, J., Maloney, A. P., & Corley, A. K. (2014). Learning Trajectories: a Framework for connecting standards with curriculum. ZDM-The International Journal on Mathematics Education, 46(5), 719-733. https://doi.org/10.1007/s11858-014-0598-7
Google Scholar CrossrefCorcoran, Tom, Mosher, Frederic A., Rogat, Aaron, & Consortium for Policy Research in, Education. (2009). Learning progressions in science: an evidence-based approach to reform. Consortium for Policy Research in Education.
Google Scholar CrossrefDaro, Phil, Mosher, Frederic A., Corcoran, Tom, Barrett, Jeffrey, & Consortium for Policy Research in, Education. (2011). Learning trajectories in mathematics: a foundation for standards, curriculum, assessment, and instruction. Consortium for Policy Research in Education.
Google Scholar CrossrefGravemeijer, K., Bowers, Janet, & Stephan, Michelle. (2003). A hypothetical learning trajectory on measurement and flexible arithmetic. In M. Stephan, J. Bowers, P. Cobb & K. Gravemeijer (Eds.), Supporting students' development of measurement conceptions: Analyzing students' learning in social context (Vol. 12, pp. 51-66). National Council of Teachers of Mathematics.
Google Scholar CrossrefGunckel, K.L., Mohan, L., Covitt, B. A., & Anderson, C. W. (2012). Addressing challenges in developing learning progressions for environmental science literacy. In A. C. Alonzo & A. W. Gotwals (Eds.), Learning progressions in science: Current challenges and future directions (pp. 39-75). Sense.
Google Scholar CrossrefHackenberg, A. J. (2014). Musings on three epistemic algebraic students. In K. C. Moore, L. P. Steffe & L. L. Hatfield (Eds.), Epistemic algebra students: Emerging models of students' algebraic knowing (Vol. 4, pp. 81-124). University of Wyoming.
Google Scholar CrossrefHackenberg, A. J., & Tillema, E.S. (2009). Students' whole number multiplicative concepts: A critical constructive resource for fraction composition schemes. The Journal of Mathematical Behavior, 28(1), 1-18. https://doi.org/10.1016/j.jmathb.2009.04.004
Google Scholar CrossrefJacobs, V., & Empson, S. (2015). Responding to children’s mathematical thinking in the moment: An emerging framework of teaching moves. ZDM: The International Journal on Mathematics Education, 48(1-2), 185-197. https://doi.org/10.1007/s11858-015-0717-0
Google Scholar CrossrefJin, H., & Anderson, C. W. (2012). A learning progression for energy in socio-ecological systems. Journal of Research in Science Teaching, 49(9), 1149-1180. https://doi.org/10.1002/tea.21051
Google Scholar CrossrefLobato, J., & Walters, C. D. (2017). A Taxonomy of Approaches to Learning Trajectories and Progressions. In J. Cai (Ed.), The Compendium for Research in Mathematics Education: National Council of Teachers of Mathematics.
Google Scholar CrossrefMason, J., Dreyfus, Tommy, Bills, L, Tsamir, P, Watson, A, & Zaslavsky, Orit. (2006). Exemplification in mathematics education. In J. Novotna, H. Moraova, M. Kratka & N. Stehlikova (Eds.), Proceedings of the 30th conference of the international group for the Psychology of Mathematics Education (Vol. 1, pp. 125-154). Prague, Czech Republic: PME.
Google Scholar CrossrefMeletiou-Mavrotheris, M, & Paparistodemou, E. (2015). Developing students' reasoning about samples and sampling in the context of informal inferences. Educational Studies in Mathematics, 88(3), 385-404. https://doi.org/10.1007/s10649-014-9551-5
Google Scholar CrossrefMitchelmore, Michael, & White, Paul. (2000). Development of angle concepts by progressive abstraction and generalisation. Educational Studies in Mathematics, 41(3), 209-238. https://doi.org/10.1023/A:1003927811079
Google Scholar CrossrefMoore, K. C. (2013). Making sense by measuring arcs: A teaching experiment in angle measure. Educational Studies in Mathematics, 83(2), 225-245. https://doi.org/10.1007/s10649-012-9450-6
Google Scholar CrossrefNabors, W.K. (2003). From fractions to proportional reasoning: A cognitive schemes of operation approach. The Journal of Mathematical Behavior, 22(2), 133-179. https://doi.org/10.1016/S0732-3123(03)00018-X
Google Scholar CrossrefPetty, O., & Jansson, L. (1987). Sequencing Examples and Nonexamples to Facilitate Concept Attainment. Journal for Research in Mathematics Education, 18(2), 112-125. https://doi.org/10.5951/jresematheduc.18.2.0112
Google Scholar CrossrefRasmussen, Chris, & Stephan, Michelle. (2008). A methodology for documenting collective activity. In A. E. Kelly, R. Lesh & J. Y. Baek (Eds.), Handbook of design research methods in education: Innovations in science, technology, engineering, and mathematics learning and teaching (pp. 195-215). Erlbaum.
Google Scholar CrossrefRemillard, Janine T. (2005). Examining Key Concepts in Research on Teachers’ Use of Mathematics Curricula. Review of Educational Research, 75(2), 211-246. https://doi.org/10.3102/00346543075002211
Google Scholar CrossrefSimon, M. (1995). Reconstructing Mathematics Pedagogy from a constructivist perspective. Journal for Research in Mathematics Education, 26, 114-145. https://doi.org/10.5951/jresematheduc.26.2.0114
Google Scholar CrossrefSteffe, L., & Thompson, P. (2000). Teaching Experiment Methodology: Underlying Principles and Essential Elements. In R. Lesh & A. E. Kelly (Eds.), Research Design in Mathematics and Science education. Erlbaum.
Google Scholar CrossrefStephan, Michelle, & Akyuz, D. (2012). A proposed instructional theory for integer addition and subtraction. Journal for Research in Mathematics Education, 43(4), 428-464. https://doi.org/10.5951/jresematheduc.43.4.0428
Google Scholar CrossrefSztajn, P., Confrey, J., Wilson, H., & Edgington, C. (2012). Learning trajectory based instruction: Toward a theory of teaching. Educational Researcher, 41(5), 147-156. https://doi.org/10.3102/0013189X12442801
Google Scholar CrossrefTillema, E.S. (2014). Students' coordination of lower and higher dimensional units in the context of constructing and evaluating sums of consecutive whole numbers. The Journal of Mathematical Behavior, 36, 51-72. https://doi.org/10.1016/j.jmathb.2014.07.005
Google Scholar CrossrefVermont Mathematics Partnership's Ongoing Assessment Project. (2013). OGAP proportional reasoning framework. Montpelier, VT.
Google Scholar CrossrefVermont Mathematics Partnership's Ongoing Assessment Project. (2014a). OGAP fraction framework. Montpelier, VT.
Google Scholar CrossrefVermont Mathematics Partnership's Ongoing Assessment Project. (2014b). OGAP multiplicative framework. Montpelier, VT.
Google Scholar CrossrefWeber, E., & Thompson, P.W. (2014). Students' images of two-variable functions and their graphs. Educational Studies in Mathematics, 87, 67-85.
Google Scholar CrossrefWilson, P. H., Sztajn, P., Edgington, C., & Confrey, J. (2013). Teachers' use of their mathematical knowledge for teaching in learning a mathematics learning trajectory. Journal of Mathematics Teacher Education, 17(2), 149–175. https://doi.org/10.1007/s10857-013-9256-1
Google Scholar CrossrefWilson, P., et al. (2013). Learning trajectories in teacher education: Supporting teachers’ understandings of students’ mathematical thinking. The Journal of Mathematical Behavior 32, 103–121. https://doi.org/10.1016/j.jmathb.2012.12.003
Google Scholar CrossrefDownloads
Published
Almetric
Dimensions
Issue
Section
License
Copyright (c) 2022 Mara V Martinez, Alison Castro-Superfine, Timothy Stoelinga
This work is licensed under a Creative Commons Attribution 4.0 International License.
Authors retain copyright and grant the journal the right of first publication but allow anyone to share: (unload, , reprint, distribute and/or copy) and adapt (remix, transform reuse, modify,) for any proposition, even commercial, always quoting the original source.
