This paper introduces two theoretical constructs, open-loop covariation and closed-loop covariation, that combine covariational reasoning and causality to characterize the way that three preservice mathematics teachers conceptualize a feedback loop relationship in a mathematical task related to climate change. The study's results suggest that the…

We report on findings from two one-on-one teaching experiments with prospective middle school teachers (PTs). The focus of each teaching experiment was on identifying and explicating the mental processes and types of intermediate, supporting reasoning that each PT used in their development of combinatorial reasoning. The teaching experiments were…

Diagrams are powerful opportunities for grappling with and learning abstract relationships, for example learning the relations between elements in an ecosystem rather than simply memorizing the objects within the system. Further, what is crucial from any diagrammatic learning opportunity is the ability to use this relational knowledge in a new…

The purpose of this paper is twofold: On the one hand, this work frames a variety of considerations on cognitive processes underlying mathematical concept construction in two research strands, namely an actions-first strand and an objects-first strand, that mainly shapes past and current approaches on abstraction in learning mathematics. This…

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…

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…

Advocates and teachers of critical thinking tend to deny that intuition and justification are logical, even though they assume that both processes are rational. However, it can be demonstrated that the relation between intuition and inference, between justification and explanation, is dialectical and complementary, so that there is no mystery as…

Contemporary dispute among teachers of rhetoric between those who prefer the classical tradition of rhetoric and those who champion an epistemic view of rhetoric has antecedents among the disputes of the ancient Greek scholars. Some of the vital themes of epistemology can be traced back to Protagoras of Abdera, one of the two great leaders of the…

This paper presents two puzzles that have emerged in a series of completed and ongoing studies at Queen's University, Ontario. These puzzles concern teachers' professional knowledge, a term used to refer to the non-propositional forms of knowledge assumed to be of importance to professional action. This research assumed that the professional…

A theory of adult cognitive development which includes two post-formal operational stages is described. The paper is divided into three sections. The first section provides background for discussion of the theory. A case study in which various employees in a hypothetical organization react to a problem is provided. Examples of pre-logical,…

Preference scores were determined for several different groups of students ranging in age from seven to adult. In addition, instruments were administered to determine the cognitive level at which the students were capable of functioning. Comparisons between the students' cognitive level and preferences were made. The results indicated that the…

This paper argues that if conclusions about children's grasp of logical concepts are to be reached and acceptable lines of research followed, then more precise definitions of the concept "logical necessity" must be formulated. The paper defines logical necessity as "the unconditional guarantee of truth that accompanies valid…

Despite the current emphasis on thinking skills and the resulting concentration on lists and taxonomies that do not succeed beyond research contexts, all reflective people know that critical thought relies not on applying mental steps but on simply trying to figure out what might be right or wrong. This depends on one basic cognitive act,…

A long-standing issue in cognitive psychology is the question of content effects upon problem-solving skills, that is, do students' problem-solving abilities generalize across specific subject matter domains? Although Piaget argued that formal reasoning strategies are independent of content, the well-known decalogs of Piaget can be interpreted as…