This dissertation case study examines how teachers' levels of units coordination afford their ability to reason multiplicatively and fractionally, as well as use their mathematics knowledge for teaching to analyze student multiplicative and fractional reasoning, when a two-year intervention is put into place to evoke new mathematical reasoning.…

As a component of a course on geometry for preservice elementary teachers (PSTs), we derive area formulas for a variety of polygons including triangles, quadrilaterals, and both regular and irregular shapes whose areas can be measured empirically using decomposition. Decomposing a circle to justify why its area can be measured using the standard…

Within teacher education, there is ongoing debate about the nature and extent of the propositional and conceptual knowledge that teachers need. In this paper we interrogate the learning tasks detailed in six learning modules offered in a formal qualification for South African Foundation Phase (grade R-3) teachers. Our purpose is to analyse to what…

In this article we present an integrative framework of knowledge for teaching the standard algorithms of the four basic arithmetic operations. The framework is based on a mathematical analysis of the algorithms, a connectionist perspective on teaching mathematics and an analogy with previous frameworks of knowledge for teaching arithmetic…

This study investigates practicing early-years teachers' concept images and concept definitions for triangles, circles, and cylinders. Teachers were requested to define each figure and then to identify various examples and non-examples of the figure. Teachers' use of correct and precise mathematical language and reference to critical and…

Supporting students to build robust fraction schemes and operations is an enduring challenge in mathematics education. Recent research has explored a developmental trajectory of fractions schemes and operations constructed by upper-elementary and middle school students in an effort to support student learning. This study broadened the existing…

The role of teacher education is to initiate purposeful conversation regarding specific content, to provide the scaffold for which preservice teachers can begin to understand the subject matter for the purpose of teaching, and to acknowledge gaps in their conceptual knowledge (Grossman and Shoenfeld with Lee 2005). Teacher candidates may enter…

The paper reports an investigation into how a group of elementary and middle school teachers collectively attempted to solve and understand a fraction division problem using an area model. Solving the word problem required that teachers determine how many two-thirds fit into three-fourths. The teachers struggled to conceptualise fraction division,…

This study reports teachers' insights and challenges after one year of adopting a curricular material designed to move students through carefully engineered, small steps and encourage learners through success and accessible challenges. The analysis of interviews showed that teachers 'followed' the material in different ways, not necessary in-line…

To be able to support meaningful mathematical experiences, preservice elementary school teachers (PSTs) must learn mathematics in deep and meaningful ways (Ma 1999). They need to experience investigating and making sense of the mathematics they will be called on to teach. To expand their own--often limited--views of what it means to teach and…

This study is the part of a larger study on investigating Hong Kong (HK) prospective teachers' (PTs) subject matter knowledge (SMK) and Pedagogical Content Knowledge (PCK). In this paper, five HK PTs' and PC on teaching one topic regarding square root were investigated. The results suggest that insufficient understanding on the concept of…

The need to design teacher preparation programs to ensure that pre-service teachers (PSTs) are prepared and equipped with knowledge, skills and practices to increase the chances that they will become effective novice mathematics teachers is of prime importance. Teacher educators are facing lingering challenges, since teacher education is…

Geometrical notions and properties occur in real-world problems, thus Geometry has an important place in school Mathematics curricula. Primary school curricula lays the foundation of Geometry knowledge, pupils learn Geometry notions and properties by exploring their environment. Thus it is very important that primary school teachers have a good…

Ambitious teaching is a form of teaching that requires a high level of teacher responsiveness to what students do as they actively engage with the subject matter. Thus, a teacher enacting ambitious teaching is often confronted with uncertainties about how to advance students' learning while also building on students' contributions. In…

Against the background of concerns around teaching and learning outcomes in primary school mathematics in South Africa, this article presents two studies conducted in American Samoa and seeks to draw implications for the teaching and learning of mathematics in South Africa. American Samoa has a very similar educational context to South Africa. The…