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,…

Past studies have documented some pre-service teachers' (PSTs) difficulties in reasoning about sampling variability. This study adds to the body of literature by investigating the ideas that PSTs employ in reasoning about sampling variability, and by conjecturing what is behind the difficulties especially during the contextuality episodes. This…

Considers the cognitive mechanisms available to individuals which enable them to operate successfully in different parts of the mathematics curriculum, such as children's arithmetic shows divergence in performance. Explains how students cope with the transition to advanced mathematical thinking in different ways, leading once more to a diverging…

Proposes that children progressively recognize deeper and deeper similarities between their physical angle experiences, and classify them firstly into specific situations, then into more general contexts, and finally into abstract domains. Indicates that the standard angle concept first develops in situations where both arms of the angle are…

The framework for this paper is a recently developed theory of abstraction in context. The paper reports on data collected from one student working on tasks concerned with absolute value functions. It examines the relationship between mathematical constructions and abstractions. It argues that an abstraction is a consolidated construction that can…

An experimental study of children's confusion between if" and if and only if" statements when phrased in different language and used in different contexts (grades 4, 6, 8, and 10). (MM)

Six fifth-grade children participated in a four-day teaching experiment on complex additively structured problems to investigate difficulties in concurrently holding in mind situations in which items play multiple roles. Difficulties included distinguishing take-away difference from comparison difference and indirect evaluation of an additive…

Some potential difficulties involved in using "real-life situations" in teaching problem solving are illustrated with a racing car problem. The importance of developing mental images and the ability to abstract in the solution process is indicated. (MP)

Beginning geometry students misunderstand the requirements of formal proof because of confusion between deductive reasoning and argumentation. Presented is a cognitive analysis of deductive organization versus argumentative organization of reasoning and the applications of this analysis to learning. Implications of a study analyzing students'…