Jere Confrey*, Distinguished University Professor of Mathematics Education, Emerita, North Carolina State University; President, The Math Door*

Alan Maloney*, Vice-President, The Math Door*

The idea of a “hypothetical learning trajectory” proposed by Simon (1995) was a description of a tentative plan to help move students from naive to more sophisticated thinking about a target concept. It was proposed to provide guidance for teachers as they observed students’ responses to tasks that invited and challenged them to engage with increasingly complex ideas. The assumption in that work was that a learning trajectory (LT) was valuable–though always tentative–for imagining and planning instruction in advance. It helped teachers to hypothesize how students might encounter and solve increasingly difficult aspects of a complex idea. Key to that progression was an understanding of what propels conceptual depth—an idea Piaget described as “genetic epistemology” (Piaget, 1970). Genetic epistemology implies that the meaning of an idea unfolds and becomes enriched as one sees various aspects of it and specifically recognizes how it can be refined to make sense of broader related tasks and deeper connections. It recognizes that the constructive process evolves gradually. Fundamentally, it acknowledges that early reasoning is almost always partial–solving *part* of the problem space–and may misrepresent or ignore other parts, and thus requires further accommodations, revisions, or distinctions to become complete. The field has now progressed to develop and synthesize research-based learning trajectories that describe patterns in students’ responses to solving challenging tasks as they advance from naive to sophisticated reasoning about target concepts.

We view LTs using the metaphor of a climbing wall (Confrey, Shah, & Maloney, 2022). This vision is in marked contrast to the metaphor of a ladder or to Piaget’s stage theories which postulate a single path with prerequisite steps (Confrey, 2019). Climbing walls, comprising a set of handholds, footholds, and obstacles, can be accessed from various positions and the paths themselves can vary. Some paths prove more likely, hence the possible paths on the wall are probabilistic, not deterministic. Different positions are located at different heights from the ground/floor, capturing the recognition (based on research data) that some aspects of an idea are more sophisticated than others. This approach recognizes that in LTs “one size does not fit all”, and it allows for students to enter the space with varied experience and to utilize diverse fonts of knowledge. LTs provide a means to inform practitioners and learners about important findings from the learning sciences concerning specific mathematical ideas.

A further point about LTs is that they can be usefully shared with students to support student agency. One application of LTs is to create psychometrically valid measures of LTs. The Math Door’s diagnostic assessment tool, Math-Mapper 6-9 (Confrey, 2015), establishes LTs for all of typical middle grades mathematics, and then documents student progress along LTs; these are *formative* assessments which are returned to both teachers *and* students as feedback on and for instructional actions. Students are provided opportunities to review the data with others, to revise and resubmit their responses, to practice, and to retake psychometrically equivalent assessments as desired. In classrooms, we have observed students spontaneously adopting the language of the LT levels to focus their attention on topics in which they are personally weaker, as well as to assist peers on LTs or levels.

To promote equitable outcomes using LTs coupled with such an assessment tool, the following stipulations are key:

- The original and ongoing studies must involve students from diverse cultural and economic backgrounds.
- The levels should be treated as probabilistic, not deterministic.
- The focus should be
*movement**along*LTs; “positions” in any LT should be regarded as transient, and should*not*be a means to categorize learners into static and restrictive ability profiles. - The levels are
*not*propositions to be taught but rather scaffolds, conditions, and guides on how to use interesting and compelling tasks to provoke further evolution of thought. - The focus is on students’ ideas, not their deficits, and on supporting the learner's agency.

A critical point to achieving a strong equity perspective is that LTs can change. Researchers and teachers should always be listening for something new to emerge. That emergence may emanate from various sources—differences in a problem’s context, language, representation, purpose, and motivation. Our research team conducted validation annually of the measurement of the LTs in Math-Mapper (Confrey, Toutkoushian & Shah, 2019). It was based on a model of linear regression between item difficulty and level. We investigated two types of variation: inter-level variation and intra-level variation, and sought to eliminate construct-irrelevant variation. Factors influencing unexpected item behavior included student group composition and instructional assignment, as well as consideration of the task’s numeric values, phrasing, and complexity. In addition to changes in the task itself, we examined whether levels needed clarification and whether an entire trajectory needed reworking or recontextualizing. Approximately 92% of the 44 LTs with sufficient data showed moderate or strong correlation between item difficulty and level, with major shifts from moderate to strong correlation for most of the LTs between the first and second validation cycles (Confrey, Toutkoushian, & Shah, 2020). These results emphasize a critical need for such diagnostic assessments to be undertaken at scale and for the development of further psychometric approaches that lead to systematic progress towards more scientifically secure, valid, and reliable results. Further, the fact that LTs can be tentative, changeable, and responsive suggests that their development should reside in a trading zone (Lehrer & Schauble, 2015; Confrey, Shah, & Toutkoushian, 2021) with participants, including practitioners, learning scientists (in math), psychometricians, and even perhaps with roles for students and community members.

Another critical equity target is providing more consistent, persistent, and high-quality professional development on LTs. Any school, but particularly schools with high teacher turnover rates and/or fewer math-certified teachers, should commit to long-term (multi-year) efforts to incorporate LTs into instruction gradually. Teachers need to learn to use and trust LTs. One model for successful implementation includes conducting professional learning communities (PLCs). In our studies (Confrey, Maloney, Belcher, McGowan, Hennessey, & Shah, 2018), teachers shared topic-specific data during monthly grade-level PLCs and diagnosed patterns of weak and strong student performance. Teachers with stronger outcomes in a topic shared instructional strategies and students’ reactions, thus increasing instructional capacity in the participating schools. Long-term success required setting and maintaining administrative supports and incentives towards gradual implementation and improvement and continued outreach to new teachers.

We have directly experienced and observed challenges to the value of LTs in talks, publications, and reviews and discussions of research proposals. These have come primarily from two directions, both with strong equity connections and with some degree of mischaracterizations of LTs. Challenges are typically grounded in historical precedents that had detrimental effects on minority children. But are these critiques overly global?—Do these critiques actually apply to LTs and if so, to what degree, and can they be resolved?

The first is opposition to most forms of measurement of LTs as a direct consequence of the negative effects of two decades of high-stakes testing. This position represents a justified response to overly coarse high-stakes tests with formulaic items and restricted forms of response. High-stakes tests provided minimal feedback, often too generalized to serve any real instructional purposes. Though they shed light on disparities in achievement, they resulted in a narrowed curriculum and a tendency to describe minority learners in a deficit mode. This has led many to reject any and all development of measures of learning trajectory levels. This in turn risks denying students precise and timely feedback that effectively allows them to know what they know and what they need to learn. In our recent work with the Young People’s Project in the Algebra Grand Challenge (Bill and Melinda Gates Foundation), Math-Mapper was used collaboratively by Math Literacy Workers to strengthen their own understanding of key concepts (and to build a group commitment to mastery by everyone), in preparation for working peer-to-peer with middle school students. Furthermore, our research reveals that LT measures can also show the degree to which* all* students are missing out on learning the most sophisticated levels of the LTs (often associated with widespread, overly procedural instruction).

A second source of direct challenge to the value of LTs is related to the influence of sociopolitical theory. In particular, a post-structural analysis (Gutierrez, 2013) can lead researchers to claim that any form of structured and sequenced description of levels restricts students’ opportunity to express their own choices about content development and identity and limits their agency. For instance, Guitierrez questioned the implementation of “reform math” programs, suggesting they can reinforce a “static classification system'' that is “complicit in the practice of constructing brown and black bodies in a deficit and overly simplistic manner” (ibid, p. 45.) As pointed out previously (#3 above), we emphasize the formative use of LTs in order to guard against using them for “static classification”. However, using the ”static classification” argument to reject LTs wholesale seems truly ironic because it ignores the fact that the levels* come from* studies of students’ own inventions, and that the emphasis (particularly via the diagnostic assessment data) is on student movement and growth, not stasis and classification. In fact, research on LTs *increases* the focus on learners as legitimate and central contributors to mathematics classroom practices. In making such learning sciences research widely available to teachers, LTs support them in scaffolding students to learn specific concepts deeply. This represents a "both-and" approach to addressing content dimensions *and* equity, as Confrey has advocated for previously (Confrey, 2010).

LTs primary roots derive from the learning sciences, and some aspects, such as the emphasis on student voice, bring in sociocultural and sociopolitical considerations. This positioning makes a discussion of equity dimensions essential, and in this blog post, we identify fundamental elements of the research that are essential to support equity, and others that can benefit from more attention. Further, we respond to some critiques of LTs by clarifying certain dimensions and advocating for further work in others. For instance, we advocate that, to strengthen equity in mathematics education, LT researchers lean into that work in order to build and implement comprehensive, flexible diagnostic systems for documenting learners’ progress along LTs (Confrey, 2023).

**REFERENCES**

Confrey, J. (2010). “Both And”—Equity *and* mathematics: A response to Martin, Gholson, and Leonard. *Journal of Urban Mathematics Education 3*(2), 25-33.

Confrey, J. (2015). Math-Mapper 6-9. Accessed at www.sudds.co. Raleigh, NC.

Confrey, J. (2019). *A Synthesis of Research on Learning Trajectories/Progressions in Mathematics*. Commissioned for the OECD 2030 Learning Framework, by OECD Mathematics Curriculum Document Analysis Project Workshop. Access: http://www.oecd.org/education/2030-project/about/documents/A_Synthesis_of_Research_on_Learning_Trajectories_Progressions_in_Mathematics.pdf

Confrey J. (2023). Strengthening the Instructional Core with Low-Stakes, Formative Diagnostic Assessment based on Mathematics Learning Trajectories. 2023 IES Mathematics Summit; September 19; Washington, DC, United States.

Confrey, J., Maloney, A. P., Belcher, M., McGowan, W., Hennessey, M., Shah, M. (2018). The concept of an agile curriculum as applied to a middle school mathematics digital learning system (DLS). *International Journal of Educational Research, 92*, 158-172.

Confrey, J., Shah, M., & Maloney, A. (2022). Learning trajectories for vertical coherence. *Mathematics Teacher: Learning and Teaching PK-12*, *115*(2), 90-103.

Confrey J., Shah, M. and Toutkoushian, E. (2021). Validation of a learning trajectory-based diagnostic mathematics assessment system as a trading zone. *Frontiers in Education, 6*, 654353.

Confrey, J., Toutkoushian, E., Shah, M. (2020). Working at scale to initiate ongoing validation of learning trajectory-based classroom assessments for middle grade mathematics. *Journal of Mathematical Behavior*, *60*, 100818.

Confrey, J., Toutkoushian, E. P., Shah, M. P. (2019). A validation argument from soup to nuts: Assessing progress on learning trajectories for middle school mathematics. *Applied Measurement in Education,* *32*(1), 23-42.

Lehrer, R., & Schauble, L. (2015). Learning progressions: The whole world is NOT a stage. *Science Education*, *99*(3), 432-437.

Piaget, J. (1970). *Genetic epistemology*. Columbia University Press.

Simon, M. A. (1995). Reconstructing Mathematics Pedagogy from a Constructivist Perspective. *Journal for Research in Mathematics Education*, *26*(2), 114-145.