This case study examines the salient features of two individuals' reasoning when confronted with a task concerning the cardinality and associated cardinal number of equinumerous infinite sets. The APOS Theory was used as a framework to interpret their efforts to resolve the "infinite balls paradox" and one of its variants. These cases…

Proportional reasoning is important for informed decisions in proportional problem situations. This paper reports on mathematical content knowledge related to proportional reasoning of second-year, pre-service teachers. Responses to two ratio items provide insights into their correct method of solutions and common misconceptions. Anchor Points…

This study investigates students' individual patterns of adherence to a norm for creating and assessing definitions in an undergraduate real analysis classroom. Our findings indicate that, though all students understood the expectation to define, students differed in their individual adherence to the norm depending upon their perception of the…

The following is a report of an exploration of what mathematical reasoning might look like in classrooms. Focusing on just one lesson in one classroom, data are presented that indicate that upper primary students are willing and able to reason for themselves, especially in classrooms in which the culture for such reasoning has been established. It…

This paper reports on an analyis of data from a novel "Peer Instruction" application, named DALITE. The Peer Instruction paradigm is well suited to take advantage of peer-input in web-based learning environments. DALITE implements an asynchronous instantiation of peer instruction: after submitting their answer to a multiple-choice…

This research investigates the question of what growth, if any, is shown by teachers in identifying the components of children's reasoning using an upper and lower bounds argument for a fraction task. Specifically, it reports on assessment outcomes from design-based research in teacher education that measures teachers' identification of children's…

This research report examines how two groups of bilingual algebra students made connections among representations while solving a non-routine generalization problem. Using a socio-cultural orientation to mathematics learning, together with a semiotic lens on students' joint mathematical activity, this report details the type of connections among…

The routine jobs of yesterday are being replaced by technology and/or shipped off-shore. In their place, job categories that require knowledge management, abstract reasoning, and personal services seem to be growing. The modern workplace requires workers to have broad cognitive and affective skills. Often referred to as "21st century…

It is commonly assumed that psychologists studying human reasoning may safely ignore the normative issue of rationality, and that philosophers may analyze rationality without reference to empirical research on actual human reasoning. Presenting five arguments against such a mutually exclusive dichotomy, this paper concludes that although empirical…

The study was designed to test the hierarchy of difficulty in the acquisition of principles of class and conditional logic for 120 children of normal intelligence, and to compare the relative rates of logic development for those children of normal intelligence with the rate of logic development found in a previous study involving 120 gifted…

Research both in cognitive psychology and psychobiology suggests that political behavior is often less rational than individuals believe it to be. Information processing, memory, and decision making are interlinked processes. Studies in cognitive psychology reveal that even though decision making requires rationality, individuals often adopt…

A common approach used for introducing algebra to young adolescents is an exploration of visual patterns and expressing these patterns as functions and algebraic expressions. Past research has indicated that many adolescents experience difficulties with this approach. This paper explores teaching actions and thinking that begins to bridge many of…

It is claimed that, since mathematics is essentially a self-contained system, mathematical objects may best be described as "abstract-apart." On the other hand, fundamental mathematical ideas are closely related to the real world and their learning involves empirical concepts. These concepts may be called "abstract-general" because they embody…

What is involved in consolidating a new mathematical abstraction? This paper examines the work of one student who was working on a task designed to consolidate two recently constructed absolute function abstractions. The study adopts an activity theoretic model of abstraction in context. Selected protocol data are presented. The initial state of…

A study explored the degree to which people can capture the meaning of logical terms, using mental imagery. It was then hypothesized that: (1) subjects generally would fail to express logical relationships accurately, and (2) abstract images would be more difficult to capture than would concrete ones. Subjects, 25 college students, were asked to…