Beyond linearity: Using IRT‐scaled level models to describe the relation between prior proportional reasoning skills and fraction learning outcomes

Abstract

Previous research on the role of prior skills like proportional reasoning skills for the development of mathematical concepts offers conclusions such as “more (prior skills) is better (for later learning).” Insights, which prior skill level goes along with which level of learning outcomes, may advance the understanding of the development of mathematical concepts. An exploratory approach is presented based on level models to describe the relation between symbolic proportional reasoning skills and fraction outcomes beyond linearity. Analyses draw on samples of German fourth to sixth graders from a scaling (2017, N = 325, 54.8% female) and longitudinal study (2018/2019, N = 436, 42.7% female). Particularly mastering natural and internal rational ratios in proportional reasoning seems relevant for successful fraction learning.

Mathematical knowledge and skills as longitudinal predictors of fraction learning among sixth-grade students.

“…It is noted that the data are further analyzed in other publications. These publications focus on item-response-theory -based level models of fraction knowledge (Schadl, 2020) and the relations between symbolic proportional reasoning skills and fraction knowledge beyond a more is better based on level models (Schadl & Ufer, 2023).…”

I have three more than you, you have three less than me? Levels of flexibility in dealing with additive situations

“…Level models can be created for one-dimensional skill constructs, or for single dimensions of multi-dimensional skill constructs. This has a long tradition in large scale assessments (Heine et al, 2013), but level models also exist for different mathematical knowledge areas such as broad arithmetic skills in primary school (Reiss and Obersteiner, 2019), proportional reasoning skills or fraction knowledge (Schadl and Ufer, 2023), and mathematical knowledge required for undergraduate mathematics learning at university (Rach and Ufer, 2020;Pustelnik et al, 2023). Often level models are seen as a preliminary step towards models that describe development such as learning trajectories (Simon, 1995) or learning progressions (Jin et al, 2019).…”

“…This is an interpretative process, that results in subsets of items that cluster along the difficulty scale alongside with verbal descriptions of the common item demands. Depending on the implementation of the method, one expert or a group of experts are involved in this interpretation (e.g., Reiss and Obersteiner, 2019;Rach and Ufer, 2020;10.3389/feduc.2024.1340322 Frontiers in Education 06 frontiersin.org Pustelnik et al, 2023;Schadl and Ufer, 2023). The item subsets from these analyses are interpreted as levels of demands regarding the skill construct.…”