Search

The terms conceptual knowledge and procedural knowledge are often used by teachers and never more so than when discussing how teachers teach, and children learn mathematics. This paper will look at literature regarding conceptual and procedural knowledge and their place in the classroom, to offer teachers and teacher educators’ advice on some of the more pressing issues and understandings around them. A thorough synthesis of extant and seminal literature will provide advice to teachers and teacher educators on how a deeper insight into conceptual and procedural knowledge could improve the quality of mathematics teaching.

The terms conceptual knowledge and procedural knowledge are often used by teachers and never more so than when discussing how teachers teach, and children learn mathematics. This paper will look at literature regarding conceptual and procedural knowledge and their place in the classroom, to offer teachers and teacher educators’ advice on some of the more pressing issues and understandings around them. A thorough synthesis of extant and seminal literature will provide advice to teachers and teacher educators on how a deeper insight into conceptual and procedural knowledge could improve the quality of mathematics teaching.

This is the first of two articles on the use of a written multiplication algorithm and the mathematics that underpins it. In this first article, we present a brief overview of research by mathematics educators and will then provide a small selection of some of the many student work samples we have collected during our research into multiplicative thinking. We contend that many primary-aged children are taught algorithms for multiplication and division without an appropriate understanding of the mathematical structure and concepts that underpin those algorithms. This is not about demeaning the use of standard algorithms. They have stood the test of time and can be elegant ways of getting a solution. However, imagine the power we give to students if we underpin the strength of algorithms with understanding! In the second article, we elaborate on what we believe are the key mathematical underpinnings of algorithms. Algorithms are very useful methods for calculation when numbers are too large to mentally calculate quickly or accurately. For multiplication, this is generally when there is a need to multiply numbers of two digits or more by another number of a similar magnitude. For example, when attempting to multiply a single-digit number by a double-digit number, students should be considering other strategies, such as applying the distributive property, and exercising their understanding of place value (e.g., 17 x 6 is 10 x 6 which is 60 and 7 x 6 which is 42 so 17 x 6 is 60 + 42 = 102), which allows them to complete these calculations mentally. However, where algorithms are deemed as necessary it would be preferable if the user of the algorithm had an understanding of not only what they were doing, but also, why they are doing it.

Our previous article, Algorithms are great: Understanding them is even better, presented evidence from students and suggested that some were attempting to use written algorithms without having an adequate level of understanding of what they were doing and why. This article will describe some of the essential mathematics that underpins the use of algorithms through a series of learning pathways. The graphic below depicts the mathematical ideas and concepts that underpin the learning of algorithms for multiplication and division. The understanding and use of algorithms is informed by two important ideas—grid multiplication and extended multiplication facts. The graphic combines a number of learning pathways that lead to those two ideas. The discussion that follows shows how each element builds part of the underpinning structure needed to understand algorithms and to use them efficiently.

Since Shulman’s (1986) seminal work on Pedagogical Content Knowledge (PCK) was released, it has created opportunities for the creation of constructs to scaffold the knowledge and understandings that teachers need in order to be effective. Adapting this work from being a heuristic to an operational structure has seen the development of many models. One such model regarding Mathematical Knowledge for Teaching (MKT) (Hill et al., 2008) has enjoyed a good deal of attention. This paper aims to argue for a reconceptualisation of Hill et al.’s (2008) model to make it as informative as possible for teachers and teaching.

Since Shulman’s (1986) seminal work on Pedagogical Content Knowledge (PCK) was released, it has created opportunities for the creation of constructs to scaffold the knowledge and understandings that teachers need in order to be effective. Adapting this work from being a heuristic to an operational structure has seen the development of many models. One such model regarding Mathematical Knowledge for Teaching (MKT) (Hill et al., 2008) has enjoyed a good deal of attention. This paper aims to argue for a reconceptualisation of Hill et al.’s (2008) model to make it as informative as possible for teachers and teaching.

This study was motivated by the need to develop professional learning for primary school teachers that would support them to more effectively teach the mathematics topic of fractions. What seemed evident, was that previous professional learning attended by teachers had not adequately met their needs. The aim of this study was to investigate whether professional learning, with a focus on subject content knowledge, pedagogical knowledge and reflective practice could enhance primary school teachers’ PCK for teaching fractions and make them more confident teachers of fractions. Demonstrating this to be the case would have wide implications for the development of professional learning opportunities for in-service teachers and would also be highly beneficial in informing teacher education. This study brought together teachers from a variety of backgrounds and experiences. These experiences comprised not only what they had encountered in their teaching of mathematics, but also what they had encountered in their learning of mathematics. Therefore a study of the affective elements of attitudes, beliefs and self-efficacy were not only warranted, but pivotal. The professional learning was conducted over an extended period of time and the teachers were involved in workshops where clear links were explored between the required content and what the current research considered to be the most efficacious pedagogy. They were then required to take at least one of the activities from the workshops and use it in their classroom. After they had taught the lesson, they were asked to reflect upon the lesson and bring those reflections to the next session to share with the group. This cycle was repeated. This research showed that the professional learning amplified both Pedagogical Knowledge (PK) and Subject Matter Knowledge (SMK), which in turn provided pathways to increased PCK. The results also indicated that well-structured professional learning can have a positive effect on the belief