Assessing covariation as a form of conceptual understanding through comparative judgement

Abstract

This paper focuses on the importance of covariational reasoning within the processes of mathematics teaching and learning. Despite the internationally recognized relevance of covariation, research shows that only a small percentage of students and teachers is able to adopt covariational reasoning and the majority of mathematics curricula do not contain explicit references to covariational skills. In particular, when covariational reasoning manifests as conceptual knowledge, it becomes challenging to assess, and the need for innovative methods of assessment emerges; there is a need for suitable assessment to highlight the characteristics of covariation and capture the various features that characterize conceptual understanding. Comparative judgement (CJ) is an innovative assessment method based on collective expert judgements of students’ work rather than requiring scoring rubrics. Due to its holistic approach, CJ is particularly suitable for assessing complex mathematical competencies, and, as we shall see in this study, it proved to be appropriate for the covariation’s assessment. In details, our study aims to investigate the perception and relevance attributed by mathematics teachers to covariation as a theoretical construct and the way CJ can help in the assessment of covariational reasoning skills underlying a less structured modelling task.

A pedagogical reflection on the interplay between variation and invariant: Variational thinking

“…This is about covariational reasoning and it emerges when one is able to observe "that there is an invariant relationship between their values that has the property that, in the person's conception, every value of one quantity determines exactly one value of the other" (Thompson & Carlson, 2017, p. 436), and "the operations that compose covariational reasoning are the very operations that enable one to see invariant relationships among quantities in dynamic situations" (Thompson, 2011, p. 46). Furthermore, covariation can be regarded as a form of conceptual understanding (Bagossi et al, 2022), and the observation of "dependency relation in which change in one variable causes a change in another, when one variation necessitates another variation" (Watson, 2017) These four strategic acts of variation are not in any hierarchical order, rather they interplay with each other in any discernment process. However, they can be arranged in a particular hierarchical order in a pedagogical design.…”

An exploratory study of spontaneous representations of covariational reasoning in middle school students

“…as dynamic objects that respond to algebraic tools and that are present in situations involving functional relationships. There are registers of the difficulties that arise in students when these reasoning processes do not reach the expected scope, as in the learning of functions (Bagossi et al, 2022;Rolfes et al, 2021;Thompson & Carlson, 2017). These difficulties, research is directed in principle to the study of covariation and the student's level of reasoning, in pre-class stages on the elaboration of planning or with the analysis of introductory tasks and instructions in basic level (primary and secondary), even with the instructional strategies that the teacher uses to facilitate reasoning, in post-class stages with the analysis of the construction of mental images obtained from dynamic situations and the ratio between covariate quantities (Castillo-Garsow, 2012;Johnson, 2015a;Miranda & Sánchez, 2019;Oehrtman et al, 2008;Panorkou & Maloney, 2016;Passaro, 2009;Swidan et al, 2022).…”

“…Other works analyze covariation with theories that delve into the type of problems or methodological frameworks that address covariational situations. Textbook exercises are analyzed, student mental processes when modeling situations, even connections between covariation and emotions of achievement when successfully understanding an activity are addressed (Bagossi et al, 2022;Carlson et al, 2002;Chen, 2023;Confrey & Smith, 1995;Nava-Guzmán et al, 2023;Syarifuddin et al, 2020;Tasova et al, 2020). Much research is qualitative in the analysis of covariational reasoning since each student conceives and represents dynamic events at different levels.…”

Efektivitas Model Pembelajaran Talking Stick Berbantuan Media Papan Statistika pada Materi Statistika

“…Pengetahuan hasil dari "pemberian" tidak akan bermakna. Adapun pengetahuan yang diperoleh melalui proses mengkonstruksi pengetahuan itu oleh setiap individu akan memberikan makna mendalam atau lebih dikuasai dan lebih lama tersimpan/diingat dalam setiap individu (Ayalon & Wilkie, 2020;Bagossi et al, 2022). Oleh karena itu, model pembelajaran yang didasarkan pada pandangan konstruktivisme adalah model pembalajaran kooperatif salah satunya yaitu model pembelajaran Talking Stick.…”