Persevering in problem solving and constructing and critiquing mathematical arguments are some of the mathematical practices included in the Common Core State Standards for Mathematics (CCSSI 2010). To solve unfamiliar problems, students must make sense of the situation and apply current knowledge. Teachers can present such opportunities by…

This study examined 361 Chinese and 345 Singaporean sixth-grade students' performance and problem-solving strategies for solving 14 problems about speed. By focusing on students from two distinct high-performing countries in East Asia, we provide a useful perspective on the differences that exist in the preparation and problem-solving strategies…

The aim of this study is to reveal teacher candidates' preference regarding uses of verbal, symbolic, number line, and/or model representations of fraction divisions, and to investigate their skill of transferring from one representation type to the others. Case study was used as the research method in this study. The case that is examined within…

Pattern blocks are inexpensive wooden, foam, or plastic manipulatives developed in the 1960s to help students build an understanding of shapes, proportions, equivalence, and fractions (EDC 1968). The colorful collection of basic shapes in classic pattern block kits affords opportunities for amazing puzzle-like problem-solving tasks and for…

This article describes how a sixth-grade teacher's "missing word" task uncovered higher-level thinking and engaged her students in the Standards for Mathematical Practice. The role and selection of the task promotes higher-level thinking and connects to the Common Core State Standards for Mathematics. The task required students to apply…

In this study, we examined how curricular resources supported three expert teachers in their enactment of progressions. Using a video-stimulated interview process, we documented the multiple types of progressions identified, described, and enacted by the teachers. Results indicate that the teachers used four different types of…

Pirie and Kieren (1989 "For the learning of mathematics", 9(3)7-11, 1992 "Journal of Mathematical Behavior", 11, 243-257, 1994a "Educational Studies in Mathematics", 26, 61-86, 1994b "For the Learning of Mathematics":, 14(1)39-43) created a model (P-K) that describes a dynamic and recursive process by which…

This paper reports a study on retention of core concepts in differential and integral calculus by examining the knowledge of two pre-service mathematics students. The study is conducted using a mixed method approach and the obtained data were analyzed using theory of three worlds of mathematics. The results showed that having good understanding of…

Generalizing--along with conjecturing, representing, justifying, and refuting--are forms of mathematical reasoning important in all branches of mathematics (Lannin, Ellis, and Elliott 2011). Increasingly, however, generalizing is recognized as the essence of thinking in algebra (Mason, Graham, and Johnston-Wilder 2010; Kaput, Carraher, and Blanton…

This paper provides a method for analysing undergraduate teaching of proof-based courses using Toulmin's model (1969) of argumentation. It presents a case study of one instructor's presentation of proofs. The analysis shows that the instructor presents different levels of detail in different proofs; thus, the students have an inconsistent set of…

In this article we explore approaches to curriculum in the primary school in order to map and manage the omissions implicit in the current unfolding of the Curriculum and Assessment Policy Statement for mathematics. The focus of school-based research has been on curriculum coverage and cognitive depth. To address the challenges of teaching…

Teachers' knowledge of the early algebra content that is to be taught is crucial for effective pedagogy and ensuring that the students' understanding of early algebra is not flawed. This article reports the findings of two of the activities that a group of in-service teachers participated in during a professional learning intervention program that…

This paper reports from a broad investigation of mathematics knowledge as dependent on interconnected concepts. The paper focuses specifically on illustrating how network analysis may be used in examining spatiotemporal relationships between learned mathematics concepts, or curriculum outcomes, and concepts inherent in assessment items.…

This paper addresses the role of learning progressions in informing many international standards documents, discussing the affordances and limitations of building standards and curricula from a learning progression model. An alternate model, the hypothetical learning trajectory, is introduced and contrasted with learning progressions. Using the…

An historical case is presented in which extra-mathematical certainties lead to invalid mathematics reasonings, and this is compared to a similar case that arose in the area of virtual education. A theoretical-methodological instrument is proposed for analysis of certainties. The article suggests the need for teachers to be aware that certainties…