Academic journal article Educational Technology & Society

Classroom-Based Cognitive Diagnostic Model for a Teacher-Made Fraction-Decimal Test

Academic journal article Educational Technology & Society

Classroom-Based Cognitive Diagnostic Model for a Teacher-Made Fraction-Decimal Test

Article excerpt

Introduction

Teacher-made tests have recently gained attention because of their functions of formative assessment in students' learning (Dekker, 2007; Frey, Petersen, Edwards, Pedrotti, & Peyton, 2005; Frey & Schmitt, 2010; Holler, Gareis, Martin, Clouser, & Miller, 2008; Nodoushan, 2011; Wiliam, Lee, Harrison, & Black, 2004). To facilitate students' learning effectively, teachers need the competencies of test construction and learning diagnoses in class. Although several pedagogical disciplines have taught teachers how to construct items (Frey et al., 2005; Hopkins, 1998; Kubiszyn & Borich, 2007), how well tests have been conducted was frequently based on subjectively judgmental criteria. Items can be objectively qualified only after a large number of samples had participated on the tests. This, however, seems impractical in classroom setting because no large samples can be provided in a traditionally sized class and even if this is the case, the qualified items might lose their original assessment purposes.

As known, teacher-made tests can fulfill the diagnostic function of understanding students' misconceptions on a certain subject unit. A misconception or an error occurs when expected knowledge by experts or teachers is structurally inconsistent with a student's or novice's actual knowledge (Hartnett & Gelman, 1998). Analyzing students' learning errors to enhance their conceptual understandings has been highly valued and recognized in many fields (e.g., Borasi, 1994; Hartman, 2001). In mathematics, students often find it difficult to learn fractions because they are quite different from whole numbers and may involve many complex quantities (such as decimals) and even language (Bezuk & Bieck, 1993; Mack, 1995; Ni, 2001; Ni & Zhou, 2005). A common misunderstanding related to fractions on multiplication and division is that "multiplication makes bigger" and "division makes smaller." In other words, students expect a greater product and a smaller quotient than both whole numbers themselves (Greer, 1988; Sowder, 1988). To understand students' conceptual misunderstandings, procedural mistakes, or even psychological response aberrances (e.g., guessing and carelessness) in fractions and decimals, teachers undoubtedly need some immediately detailed feedbacks from students' responses on the teacher-made tests that they will use in the classroom.

To analyze what students had mastered or misunderstood, many cognitive diagnosis models (CDMs) were developed to provide informative profiles about students' learning and progresses. Unlike the item response theory (IRT) models only providing item characteristics estimated (e.g., difficulty, discrimination, and guessing parameters), the CDMs make it possible to investigate underlying content knowledge in items (McGlohen & Chang, 2008). However, as McGlohen and Chang indicated, most CDMs can estimate only students' knowledge states but not their ability levels. In addition, most CDMs are still novel and limited because of a few readily available computer programs (de la Torre, 2009). Moreover, desirable parameters in CDMs are usually estimated through large sample sizes. Consequently, being lack of convenient tools for estimation and big sample size requirements, most teachers in classroom are not acquainted with and even fear using modern CDMs. Fortunately, aberrance indices can be deemed as an adequate option for CDMs because of their in-depth reflections on students' misconceptions without through large samples or complex estimation requirements (Liu, Douglas & Henson, 2009; Liu & Yu, 2011; Seol, 1999).

Typically, aberrance indices may be classified into two areas: IRT-based and group-based categories (Meijer & Sijtsma, 1999). The IRT-based indices usually use a specific IRT model as a standard norm so that any response pattern departing from the model will be classified as aberrant. They may include the residual-based approaches (e. …

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