Continuing our critique of the classical derivation of the formula for the area of a disk, we focus on the limiting processes in geometry. Evidence suggests that intuitive approaches in arguing about infinity, when geometric configurations are involved, are inadequate, and could easily lead to erroneous conclusions. We expose weaknesses and…

Gap reasoning is an inappropriate strategy for comparing fractions. In this article, Patrick Sullivan and Joann Barnett look at the persistence of this misconception amongst students and the insights teachers can draw about students' reasoning.

In this article we present the results of a study focused on engaging students in argumentation to support their growth as mathematical learners, which in turn strengthens their science learning experiences. We identify five argumentation categories that promote the learning of argumentation skills and enrich mathematical reasoning at the…

Mathematical reasoning is one of the four proficiencies in the Australian Curriculum: Mathematics (AC:M) where it is described as: "[the] capacity for logical thought and actions, such as analysing, proving, evaluating, explaining, inferring, justifying and generalising" (Australian Curriculum, Assessment and Reporting Authority [ACARA],…

"Modeling Mathematical Ideas" combining current research and practical strategies to build teachers and students strategic competence in problem solving.This must-have book supports teachers in understanding learning progressions that addresses conceptual guiding posts as well as students' common misconceptions in investigating and…

Misconceptions in geometry are an essential problem in the understanding of geometric terms by primary and pre-primary aged children. Present research shows some misconceptions in geometry demonstrated in the understanding of circles, squares, triangles and oblongs for children in the last year of kindergarten and pupils in the last year of…

For students to be successful in algebra, they must have a truly conceptual understanding of key algebraic features as well as the procedural skills to complete a problem. One strategy to correct students' misconceptions combines the use of worked example problems in the classroom with student self-explanation. "Self-explanation" is the…

In the authors' examination of various instructional programs, they observed that most provide all the necessary data to solve proportion problems, employ compatible numbers that are usually unrealistic, present numbers (data) in the order in which they are to be manipulated, discuss contexts that cannot be easily replicated, and present…

Adult developmental mathematics students often work under great pressure to complete the mathematics sequences designed to help them achieve success (Bryk & Treisman, 2010). Results of a teaching experiment demonstrate how the ability to reason can be impeded by flaws in students' mental representations of mathematics. The earnestness of the…

Discusses Vygotsky's theories about concept formation, his distinctions between everyday and theoretical concepts, and how empirical generalizations can lead to misconceptions. Examines the implications of these theories for mathematics instruction and its relationship to the current mathematics reform. (34 references) (MDH)

Discusses geometrical reasoning in the framework of the theory of Figural Concepts to highlight the interaction between the figural and conceptual components of geometrical concepts. Examples of students' difficulties and errors in geometrical reasoning are interpreted according to the internal tension that appears in figural concepts resulting…

Examined inconsistencies in secondary school students' reasoning about the probability concept of equally likely events. Results of two studies suggest that the number of students who understand the concept of independence is much lower than the latest National Assessment of Educational Progress results indicate. (Contains 22 references.) (MDH)

Pragmatic and semantic problem solving are examined as processes that enhance acquisition of mathematical knowledge. It is suggested that development of mathematical cognition involves restructuring and that math teachers can help restructure children's knowledge systems by providing them with situations in which semantic and pragmatic problem…

Proposes analogy as the central mechanism of knowledge acquisition in formal domains. Discusses experimental data on preschoolers' knowledge of one-to-one correspondence and college students' understanding of force decomposition. Suggests that a knowledge base domain is a thematically organized knowledge structure and that thematic relations in a…