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" 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.