Are conceptual knowledge and procedural knowledge empirically separable? The case of fractions

Abstract

BackgroundConcerning students’ difficulties with fractions, many explanatory approaches are based on the distinction between conceptual knowledge and procedural knowledge. For further research in this field, it is thus crucial to make these constructs accessible to valid measurement.AimsIn this study, we aim at developing a test instrument that affords valid measurement of students’ conceptual and procedural fraction knowledge, including in particular empirical validation of this distinction.SampleThe data used in this study were from 8th‐ and 9th‐grade students (N = 235) in Germany.MethodsFacilitated by expert discussions, items from previous studies were developed further and assigned to either a conceptual scale or procedural scale. Confirmatory factor analysis was used to investigate the underlying structure of the data including model comparisons (1‐dimensional; conceptual–procedural, verbal–non‐verbal). Further analyses in terms of validation focused on reliability and on correlations of the knowledge types with general cognitive abilities.ResultsIt was found that the theoretically assumed 2‐dimensional model fitted the data best. Correlations of the two knowledge types with general cognitive abilities differed significantly. Furthermore, the latent constructs could be reliably estimated from its indicators.ConclusionsOur findings indicate that the empirical separation of conceptual and procedural fraction knowledge is possible: A theoretically grounded test instrument was developed that allows measuring the knowledge types with a sufficient degree of validity. These findings address a research gap that was pointed out repeatedly and gives rise to further research into reasons and remedies for students’ difficulties in dealing with fractions.

Individual Differences in Conceptual and Procedural Fraction Knowledge: What Makes the Difference and What Does it Look Like?

“…Residualized scores describe individual differences in terms of the discrepancy between expected and observed scores, which allows to eliminate shared variance between conceptual and procedural knowledge (Cohen, Cohen, West, & Aiken, 2013). This makes sense because conceptual and procedural knowledge are usually highly correlated (e.g., Hallett et al, 2010;Lenz et al, 2019a;Schneider & Stern, 2010). As residualized scores represent relative scores and not absolute scores, a more precise description of the identified clusters based on students' performance in the test is not possible.…”

“…To assess conceptual and procedural fraction knowledge, a test instrument was used which has been extensively validated (Lenz et al, 2019a). It comprises 37 items, 18 of which are on the procedural subscale (Cronbach's alpha α = .90) and 19 on the conceptual subscale (Cronbach's alpha α = .88) (see Table 1).…”

“…These strong positive correlations raise the question whether conceptual and procedural knowledge can be separated empirically at all. Consequently, previous studies have addressed the empirical separability, analyzing conceptual and procedural knowledge on a latent level (Hecht, Close, & Santisi, 2003;Lenz, Dreher, Holzäpfel, & Wittmann, 2019a). In particular, confirmatory factor analyses were conducted to determine whether conceptual knowledge and procedural knowledge represent two distinct dimensions by means of model comparisons.…”

Conceptual and Procedural Knowledge of Students of Nepal in Algebra: A Mixed Method Study

“…This debate leads to a question that which knowledge comes first. However, many research studies have done, such as Lenz et al (2020), Rittle-Johnson and Siegler (1998), Rittle-Johnson et al (2016), Ross (2010), Zuya (2017), etc., to visualize the relationship between procedural and conceptual knowledge so that we could teach better mathematics and students learn them meaningfully. As a mathematics teacher, most of the time, the researcher heard that for a mathematics knowledge acquisition, the rich concept should be established while teaching and procedures are also important in problem-solving.…”

“…But the teaching scenario changes when it comes to the classroom. The instructional practices emphasize more on students to memorize formulae, steps or procedures to solve the problems in mathematics than encourage them to be creative, ask questions, think critically, and play with the situation so that they use their highest potentiality to construct knowledge with the underlying concept which is rich in connection with deep meaning, a conceptual knowledge (Lenz et al, 2020;Rittle-Johnson, 2019;Star, 2005).…”