# Professional Learning

# Learning progression &Trajectories

In many research documents and educational resources, the term learning trajectories and learning progressions are used interchangeably with the emphasis on the developmental progression of levels of thinking within a mathematical strand. Communicating research on learning trajectories to practitioners is critically important to teaching and learning (Clements & Sarama, 2004, 2009; Maloney, Confrey, & Nguyen, 2014; Simon, 1995).

Teaching that incorporates an understanding of learning trajectories has the potential to make teaching more efficient and help students to learn mathematics for understanding. Confrey (2009) described learning trajectory/progression as “a researcher‐conjectured, empirically‐supported description of the ordered network of constructs a student encounters through instruction (i.e. activities, tasks, tools, forms of interaction and methods of evaluation), in order to move from informal ideas, through successive refinements of representation, articulation, and reflection, towards increasingly complex concepts over time “(Confrey et al., 2009). According to Confrey (2012), there are five elements that can help unpack a learning trajectory. Teachers need to understand: 1) the conceptual principles and the development of the ideas underlying a concept; 2) strategies, representations, and conceptions that students bring; 3) meaningful distinctions, definitions and multiple models; 4) coherent structure – recognizing that there is a pattern in the development of mathematical ideas as a concept becomes more complex; and 5) bridging standards-understanding that there might be gaps between standards and knowing what underlying concepts are in between to bridge the gaps between the standards. Focusing on these important elements has potential for improved instructional planning, anticipation of student strategies, representations, and conceptions that can be attributed as students’ strengths and resources for building on their understanding.

The design of our modules is tightly coupled with hypotheses about students’ learning within a task (Barrett & Battista, 2014; Simon & Tzur, 2004), and represent one possible path of learning (Baroody et al., 2004; Stevens, Shin, & Krajcik, 2009). According to Blanton et al., (2015, p. 515), "the levels of thinking constituting the developmental progression of a learning trajectory are not intended to be interpreted as stages through which one must progress in a linear sequence but as states of mind with fluid boundaries through which students move bidirectionally as their learning progresses (Clements & Sarama, 2014). Students may skip levels altogether or may revert to previous levels when faced with changes in tasks (Clements & Sarama, 2014)"or changes in the learning environment or suffered stress/trauma (i.e. pandemic). Expanding the notion of competence is critically important in catalyzing change in the mathematics classroom and broadening the notion of doing mathematics (NCTM, 2020). Central to the first key recommendation in *Catalyzing Change in Early Childhood and Elementary School* (2020) is broadening the purposes of school math by “developing deep mathematical understanding as confident and capable learners” (p. 11). Positive and discourse-rich classrooms allow each student to have feelings of success and pride (NCTM, 2020). According to Gresalfi et al., (2008), what counts as “competent” gets constructed in particular classrooms through an interaction between the opportunities that a student has in a mathematics classroom, meaning that structures that promote equitable participation and interaction is the key. The instructional decision to formatively assess and highlight student thinking during discussions has important implications for assigning competence, as it suggested *what* students are accountable for and to *whom* they were responsible for sharing their thinking with (Gresalfi et al., 2009). We believe that finding strength in students’ multiple knowledge bases (Turner et al., 2016; Korbett & Karp, 2020), teachers are better able to assign competence in student thinking, while broadening the notion of what competence means and attend to building student agency and a positive sense of identity (Civil, 2007; Gonzalez, Moll, & Amanti, 2005; Aguirre et al., 2013; Cohen et al., 1999; Gresalfi et al., 2008).

In our professional development work (figure below), we focus on deepening our understanding of the learning trajectory so that we can situate the learning along a continuum and advance student thinking. We focus on five essential components, as : (a) learning goals, (b) a developmental progression that specifies increasingly sophisticated levels of thinking in which students might engage (Clements & Sarama , 2004), (c) formative assessment (d) instructional activities or an instructional sequence, and (e) evidence of students thinking to inform our next feedback loop. The formative assessment strategies are critical to mark students' strength and "finding the edge of students' learning" (Johnston, 2012) to advance their understanding. Designing purposeful tasks, routines and questions can help bridge and build strength for each and every student. The teachers role in facilitating classroom discourse to celebrate and honor students' thinking while highlighting multiple strategies helps affirm diverse mathematical competence within our students and engage some students who might not typically participate in the mathematics classroom.

Here are some resources to deepen your understanding of the learning trajectories. Enjoy!

**Learning Trajectory Resources**

Charles, R. (2005). Big ideas and understandings as the foundation for elementary and middle school mathematics. *Journal of Mathematics Education Leadership*, *7*(3), NCSM.

Clements, D. H., & Sarama, J. (2019). Learning and teaching with learning trajectories [LT]2. Marsico Institute, Morgridge College of Education, University of Denver. https://www.learningtrajectories.org/

Common Core Standards Writing Team. (2019). Progressions for the Common Core State Standards for Mathematics. Tucson, AZ: Institute for Mathematics and Education, University of Arizona.

Richardson, K. (2012). How Children Learn Number Concepts: A Guide to Critical Learning Phases. Bellingham: Math Perspectives Teacher Development Center.

Van De Walle, J., Karp, K. S., & Bay-Williams, J. M. (2018). *Elementary and Middle School Mathematics: Teaching Developmentally. *(10th edition) New York: Pearson (2019:9780134802084)

VDOE Curriculum Framework for All Grades - Standard of Learning Curriculum Framework (SOL)

Achieve the Core https://achievethecore.org/coherence-map/Learning Trajectories Levels for Primary Grade Mathematics (D. Clements): https://www.otffeo.on.ca/en/wp-content/uploads/sites/2/2017/05/Learning_trajectories_math.pdf

Math Mapper https://www.sudds.co Grades 6-9

Two minute teacher's guide -https://twominuteteachersguide.com/ Grades 6-12

https://earlymath.erikson.edu/why-early-math-everyday-math/big-ideas-learning-early-mathematics/https://earlymath.erikson.edu/diy-rekenrek-home-school/

**References cited in the Blog**

Baroody et al., 2004;

Barrett & Battista, 2014;

*Journal for Research in Mathematics Education,*

*46*(5), 511-558. doi:10.5951/jresematheduc.46.5.0511

Clements, D. H., & Sarama, J. (2004). Learning trajectories in mathematics education. Mathematical Thinking and Learning, 6, 81-89.

Confrey, J. (2012). Articulating a learning science foundation for learning trajectories in the CCSS-M. In Van Zoerst, L. R. Lo, J. J. & Kratky, J.L. (Eds.). Proceedings of the 34^{th} annual meeting of the North American Chapter of the International Group for the Psychology Mathematics Education (pp. 2-20). Kalamazoo, MI. Western Michigan University.

Confrey, J., & Maloney, A. (2010). The construction, refinement , and early validation of the equipartitioninglearning trajectory. In Gomez, K., Lyons, L., & Radinsky, J. (Eds.) Learning in the Disciplines: Proceedings of the 9th International Conference of the Learning Sciences (ICLS 2010) – Volume 1, Full Papers. International Society of the Learning Sciences: Chicago IL. Retrieved from http://dl.acm.org/citation.cfm?id=1854484.

Daro, P., Mosher, F. A. & Corcoran, T. (2011). Learning Trajectories in Mathematics: A Foundation for Standards, Curriculum, Assessment, and Instruction (Research Report No.68). Madison, WI: Consortium for POlicy Research in Education.

Gresalfi, Melissa & Martin, Taylor & Hand, Victoria & Greeno, James. (2009). Constructing competence: An analysis of student participation in the activity systems of mathematics classrooms. Educational Studies in Mathematics. 70. 49-70. 10.1007/s10649-008-9141-5.

Johnston, P. (2012). Opening minds: How classroom talk shapes children’s minds and their lives. Portland, ME: Stenhouse.

Korbett, B. M., & Karp, K.S. (2020). Strengths-Based Teaching and Learning in Mathematics: Five Teaching Turnarounds for Grades K–6. Corwin.

Lotan, R. (2003). Group-worthy tasks. Educational leadership: journal of the Department of Supervision and Curriculum Development, N.E.A. 60. 72-75.

Myers, M., Szajin, P., Wilson, P. H., & Edgington, C. (2015). From Implicit to Explicit: Articulating Equitable Learning Trajectories Based Instruction. Journal of Urban Mathematics Education, 8 (2), 11–22.

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

Simon, M. (1995). Reconstructing mathematics pedagogy from a constructivist perspective. Journal for Research in Mathematics Education, 26(2), 114-145.

Stein, M. K., & Smith, M. (2011). *5 practices for orchestrating productive mathematics discussions*. Reston, VA: National Council of Teachers of Mathematics.

Stevens, Shin, & Krajcik, 2009

**Suh, J. M., Birkhead, S., Frank, T., Baker, C., Galanti, T., & Seshaiyer, P. (2021). Developing an Asset-Based View of Students’ Mathematical Competencies Through Learning Trajectory-Based Lesson Study, ****Mathematics Teacher Educator****, ****9****(3), 229-245. Retrieved Aug 9, 2021, from ****https://pubs.nctm.org/view/journals/mte/9/3/article-p229_1.xml**

**Suh, J. M. & Seshaiyer, P. (2014). Examining teachers’ understanding of the mathematical learning progression through vertical articulation during Lesson Study. ****Journal of Mathematics Teacher Education****,**** 18****(3), 217-229. **

Sztajn, P., Confrey, J., Wilson, P. H., & Edgington, C. (2012). Learning trajectory based instruction: Toward a theory of teaching. Educational Researcher, 41(5), 147–156.

**Wills, T. (2019). Tips from Practitioners: Cultivating School Communities with Vertical Mathematics Tasks. Virginia Mathematics Teachers. VCTM. VA.**

Virginia LEARNS Resources to assist in the identification of content that can be connected when planning instruction and promoting deeper student understanding. Standards are considered a bridge when they: function as a bridge to which other content within the grade level/course is connected; serve as prerequisite knowledge for content to be addressed in future grade levels/courses; or possess endurance beyond a single unit of instruction within a grade level/course.