A Validation Approach to Middle-Grades Learning Trajectories Within a Digital Learning System Applied to the “Measuring Characteristics of Circles”

Validation of a Learning Trajectory-Based Diagnostic Mathematics Assessment System as a Trading Zone

“…With respect to MM6-8, Confrey et al (2020) summarized the results of the first round of validation studies across all 60 constructs in the learning map. Two other articles describe our validation framework and report on validation studies focusing on the first four claims for two clusters (Measuring Characteristics of Circles and Finding Key Percent Relationships) (Confrey et al, 2017;Confrey and Toutkoushian, 2019). Confrey et al (2019a) and Confrey and Shah (2021) report on patterns of how teachers use MM6-8 in practice, both individually and collectively in PLCs.…”

“…Students' movement towards more difficult tasks requires them to engage in a process Piaget called "reflective abstraction", critical to the development of mathematical reasoning. LTs, for our team, are likened to pathways up a climbing wall with differing points of entry, but with predictable handholds, footholds, and obstacles (Confrey and Toutkoushian, 2019). Levels can be viewed as likely landing places that indicate progress towards the target construct.…”

“…Firstly, in mathematics, cases associated with a specific level can legitimately vary in difficulty based on many factors, and secondly, students' performance on these items can be influenced by instructional factors, including opportunity to learn. These considerations led us to distinguish three categories of variability that affect the difficulty of items: intra-level variation ("Intra-LV"), inter-level variation ("Inter-LV"), and construct-irrelevant variation ("Irrel-V") (Confrey and Toutkoushian, 2019). If we encountered a situation where on average, items are more difficult for an earlier level than a later level, and we have ruled out Irrel-V, then we would aggressively seek an explanation based on the mathematical choices affecting the complexity of the problem (as related to the meaning of the level) or instructional practices shared across lots of teachers.…”

“…We adapted and elaborated on their framework to create a validation framework tailored to MM6-8. The modifications of this approach for MM6-8, and applications of a principled assessment design perspective (Nichols et al, 2016) are described in further detail in Confrey and Toutkoushian (2019) and Confrey et al (2019b).…”

“…To illustrate the basis of claims 4 and 5 in the validation argument, data from one authentic heatmap will be analyzed to illustrate the intended use of the LT-DDDM content-and learner-centered elements. Claim 6 has been discussed elsewhere (Toutkoushian et al, 2019); the two aforementioned analyses illustrate the importance of building, validating, using, and refining LTs as a robust collaborative activity conducted in trading zones.…”