A Curriculum-Based Approach to Learning Trajectories in Middle School Algebra

Authors

  • Mara V Martinez University of Illinois-Chicago
  • Alison Castro-Superfine University of Illinois - Chicago
  • Timothy Stoelinga Loyola University

https://doi.org/10.17583/redimat.5539

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

Download data is not yet available.

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 Crossref

Berland, 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 Crossref

Bernbaum 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 Crossref

Bishop, 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 Crossref

Bowers, 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 Crossref

Brown, 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 Crossref

Clark, 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 Crossref

Clements, D.H., & Sarama, J. (2004). Learning Trajectories in Mathematics Education. Mathematical Thinking & Learning, 6(2), 81-89.

Google Scholar Crossref

Clements, 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 Crossref

Cobb, P. (1999). Individual and collective mathematical development: The case of statistical data analysis. Mathematical Thinking & Learning, 1(1), 5-43.

Google Scholar Crossref

Cobb, 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 Crossref

Cobb, 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 Crossref

Collopy, 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 Crossref

Common 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 Crossref

Common Core Standards Writing Team. (2013b). Preface for the draft progressions. Institute for Mathematics and Education, University of Arizona.

Google Scholar Crossref

Confrey, 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 Crossref

Confrey, 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 Crossref

Corcoran, 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 Crossref

Daro, 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 Crossref

Gravemeijer, 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 Crossref

Gunckel, 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 Crossref

Hackenberg, 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 Crossref

Hackenberg, 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 Crossref

Jacobs, 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 Crossref

Jin, 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 Crossref

Lobato, 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 Crossref

Mason, 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 Crossref

Meletiou-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 Crossref

Mitchelmore, 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 Crossref

Moore, 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 Crossref

Nabors, 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 Crossref

Petty, 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 Crossref

Rasmussen, 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 Crossref

Remillard, 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 Crossref

Simon, 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 Crossref

Steffe, 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 Crossref

Stephan, 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 Crossref

Sztajn, 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 Crossref

Tillema, 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 Crossref

Vermont Mathematics Partnership's Ongoing Assessment Project. (2013). OGAP proportional reasoning framework. Montpelier, VT.

Google Scholar Crossref

Vermont Mathematics Partnership's Ongoing Assessment Project. (2014a). OGAP fraction framework. Montpelier, VT.

Google Scholar Crossref

Vermont Mathematics Partnership's Ongoing Assessment Project. (2014b). OGAP multiplicative framework. Montpelier, VT.

Google Scholar Crossref

Weber, E., & Thompson, P.W. (2014). Students' images of two-variable functions and their graphs. Educational Studies in Mathematics, 87, 67-85.

Google Scholar Crossref

Wilson, 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 Crossref

Wilson, 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 Crossref

Downloads

Published

2022-02-24

Almetric

Dimensions

Issue

Section

Articles

Most read articles by the same author(s)