A Preliminary Validity Evaluation of a Learning Progression for the Concept of Function

Abstract

Learning progressions (LPs) describe the development of domain‐specific knowledge, skills, and understanding. Each level of an LP characterizes a phase of student thinking en route to a target performance. The rationale behind LP development is to provide road maps that can be used to guide student thinking from one level to the next. The validity of an LP cannot be taken for granted, however. LPs evolve from a synthesis of multiple research studies, subject‐matter expertise, and standards documents. They are working models of student development that may require revision in light of critique and empirical evidence. The formulation of an LP is an iterative process in which expert feedback is elicited, data are collected, and the LP is revised accordingly. We developed an LP for the concept of function both because the concept of function is challenging to attain and because it is central to the study of algebra and higher mathematics. We report early findings with respect to the validity of the concept of function LP, based on small‐scale cognitive interviews and expert reviews.

Validation of cognitive models for subtraction of time involving years and centuries

“…Thus, the cognitive models constructed are regarded as a new theory of learning and cognition (Nichols et al, 2017) which has never been demonstrated to be valid (Nichols, 1994). In other words, the attributes associated with students' cognitive processes in solving the related tasks are only hypothesized by the experts (Graf et al, 2019). Thus, empirical evidence needs to be accumulated to verify the validity of the cognitive models constructed (Graf et al, 2019).…”

“…In other words, the attributes associated with students' cognitive processes in solving the related tasks are only hypothesized by the experts (Graf et al, 2019). Thus, empirical evidence needs to be accumulated to verify the validity of the cognitive models constructed (Graf et al, 2019).…”

“…The empirical evidence can be collected from various sources based on the nature of constructs and tasks. Most of the studies (i.e., Akbay et al, 2018;Chen et al, 2017;Graf et al, 2019;Langenfeld et al, 2020;Sia et al, 2019) employed cognitive-psychometric modelling to validate the cognitive models because the fit of cognitive models to the data can be computed by using the mathematics formula which take into consideration of the constraints imposed in the cognitive models (Keehner et al, 2017). In order to triangulate the data, Graf et al (2019), Langenfeld et al (2020) and Sia et al (2019) further analysed the protocols collected by using the think-aloud method and conducting cognitive labs, respectively.…”

A Learning Progression for Variability

“…(ETS, n.d., List Item 2)An LP identifies what Clements and Sarama (2004) have called “developmental progression[s] of levels of thinking” (p. 83). Each level of an LP “characterizes a phase of student thinking en route to target performance” (Graf et al, 2019, p. 1). The rationale behind LP development is to provide a map that can guide assessment design, instructional practice, and student learning (Graf et al, 2019).…”

“…Each level of an LP “characterizes a phase of student thinking en route to target performance” (Graf et al, 2019, p. 1). The rationale behind LP development is to provide a map that can guide assessment design, instructional practice, and student learning (Graf et al, 2019). While LPs in related domains will have points of connection between them, thus creating networks of learning paths, the developmental levels are best understood by recognizing individual LPs belonging to individual domains (Daro et al, 2011).…”