Using a mixed methods approach, we explore a relationship between students' graph reasoning and graph selection via a fully online assessment. Our population includes 673 students enrolled in college algebra, an introductory undergraduate mathematics course, across four U.S. postsecondary institutions. The assessment is accessible on computers,…

This study explored a diagnostic assessment method that emphasized the cognitive process of algebra learning. The study utilized a design and a theory-driven model to examine the content knowledge. Using the theory driven model, the thinking skills of algebra learning was also examined. A Bayesian network model was applied to represent the theory…

Many educational data mining studies have explored methods for discovering cognitive models and have emphasized improving prediction accuracy. Too few studies have "closed the loop" by applying discovered models toward improving instruction and testing whether proposed improvements achieve higher student outcomes. We claim that such…

An important open issue in Functional Measurement is whether the three most important models of cognitive algebra are sufficient to describe the great majority of possible response behaviors. Generally speaking, the individual response "R" is a function of the subjective scale values "s[subscript k]" and can be imagined as a continuous manifold.…

Reducing problem complexity by isolating elements has been shown to be an effective instructional strategy. Novices, in particular, benefit from learning from worked examples that contain partially interacting elements rather than worked examples that provide full interacting elements. This study investigated whether the isolating-elements…

Newton's cradle, a device consisting of a chain of steel balls suspended in alignment, has been used extensively in physics teaching to demonstrate the principles of conservation of momentum and kinetic energy in elastic collisions. The apparent simplicity of the device allows one to test commonly hold views regarding the intuitive understanding…

This article reports on the results of research, the objective of which was to document and analyze the manner in which students relate different representations when solving problems. A total of 20 students attending their first year of university studies took part in the study. In order to design the problem, the underlying information in each…

This paper presents the results obtained from a study comparing the strategies used by 1,070 Chinese students and 1,002 Singaporean students from primary grade 6 to secondary year 2 in solving three fraction-related problems. It is part of the author's Ph.D. study, which involves more word problems on speed. The Chinese students performed better…

In the present study, I examine a modelling strategy as employed by a teacher in the context of an algebra lesson. The actions of this teacher suggest that a modelling approach will have a greater impact on enriching student learning if we do not lose sight of the need to manage associated cognitive loads that could either aid or hinder the…

The objective of the present investigation was to compare the adequacy of two cognitive models for predicting examinee performance on a sample of algebra I and II items from the March 2005 administration of the SAT[TM]. The two models included one generated from verbal reports provided by 21 examinees as they solved the SAT[TM] items, and the…

The purpose of the study is to present research focused on validating the four algebra cognitive models in Gierl, Wang, et al., using student response data collected with protocol analysis methods to evaluate the knowledge structures and processing skills used by a sample of SAT test takers.

This paper is structured in two sections. The first examines views of mathematical abstraction in two broad categories: empiricist and dialectical accounts. It documents the difficulties involved in and explores the potentialities of both accounts. Then it outlines a recent model which takes a dialectical materialist approach to abstraction in…

This article describes the Adaptive Control of Thought-Rational (ACT-R) cognitive architecture (Anderson et al., 2004; Anderson & Lebiere, 1998) and its detailed application to the learning of algebraic symbol manipulation. The theory is applied to modeling the data from a study by Qin, Anderson, Silk, Stenger, & Carter (2004) in which children…

The purpose of this paper is to provide researchers with a shared framework, terminology and tool to improve the coherence of research into learning mathematics with CAS and to assist its findings to accumulate into a significant body of knowledge. Experience with calculators in arithmetic led to a framework for number sense. There is an obvious…

This document presents a first approximation for a model for student performance in solving linear equations. First, characteristics of a model of student performance for solving linear equations is discussed. It is noted that there is a need for some theoretical structure to support and organize the bits and pieces of information provided by…