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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.

Specialised Content Knowledge (SCK) is defined by Ball, Hoover-Thames, and Phelps (2008) as mathematical knowledge essential for effective teaching. It is knowledge of mathematics that is beyond knowledge which would be required outside of teaching; for instance, the capacity to determine what misconception(s) may lie behind an error in calculation. Such knowledge should be core business of teachers of mathematics, and any perceived shortfall in SCK viewed as problematic. The research reported on here is part of a large study about multiplicative thinking involving approximately two thousand children between nine and twelve years of age and their teachers. Data were generated from semi-structured interviews and a written diagnostic assessment quiz. As part of that large project, forty-four Australian and New Zealand primary and middle school teachers were asked to respond to student work related to multiplicative thinking, particularly to concepts of numbers of equal groups and commutativity. Participants' responses reflected confusion about a pivotal piece of SCK, the convention for naming arrays. As well, questionable assumptions about the children's work samples were made. Given that there is not unanimous agreement amongst mathematics educators about naming conventions, these observations may not be surprising. Due to the sample size, broad generalisations cannot be made, but the results suggest that many teachers may have limited SCK with regards to the important mathematical area of Multiplicative Thinking (MT).

Factors and multiples are important aspects of mathematical structure that support the understanding of a range of other ideas including multiplication and division, and later on, factorization. At primary school level, it is important that factors and multiples are taught as a connected enterprise and as vital parts of the multiplicative situation; that is multiplication and division. The primary objective of the study on which this paper is based was to determine the extent of children's understanding of factors and multiples. A written quiz containing questions about factors and multiples and asking for children to explain their responses, was administered. Results suggest that the language involved with factors and multiples may play a role in the extent to which children develop a conceptual understanding of them. Also, most children know some things about factors and multiples but struggled to connect and articulate ideas when factors and multiples were presented in a different context. In conclusion, the inconsistency of participant responses suggests that teaching about factors and multiples needs to emanate from a more conceptual and connected standpoint.

Multiplicative thinking is widely accepted as a critically important 'big idea' of mathematics that underpins much mathematical understanding beyond the primary years. It is therefore important to ensure not only that children can think multiplicatively, but that they can do so at a conceptual rather than procedural level. This paper reports on a large study of 530 primary school children in Australia, New Zealand and the United Kingdom. The research question was "To what extent do children of 10 and 11 years of age understand what happens to digit values when numbers are multiplied and divided by powers of ten?" A written multiplicative thinking quiz was administered and one section of four questions asked students to multiply and divide two digit whole and decimal numbers by a power of ten and then explain what happened to the numbers. Correct response rates for the four calculations ranged from 38.3% to 61.7%. Response rates for appropriate explanations about what happened to the numbers ranged from 2.6% to 5.5%. Most students who attempted to explain what happened did so at a 'pseudo-procedural' level with responses such as 'a zero is added' or 'we take off the zero'. The students who did explain it conceptually did so in terms of the digits moving a place for each power of ten. The implication is that teaching of multiplication and division needs to be done at a conceptual level, with attention paid to the underlying mathematical structure, rather than relying on 'pseudo-procedures' such as 'adding a zero' which are unsustainable and will likely lead to errors.

Multiplicative thinking is a "big idea" of mathematics that underpins much of the mathematics learned beyond the early primary school years. This article reports on a recent study that utilised an interview tool and a written quiz to gather data about children's multiplicative thinking. Our research has so far revealed that many primary aged children have a procedural view of multiplicative thinking which we believe inhibits their progress. There are two aspects to this article. First, we present some aspects of the interview tool and written quiz, along with some of findings, and we consider the implications of those findings. Second, we present a key teaching idea and an associated task that has been developed from our research. The main purpose of the article is to promote the development of conceptual understanding of the multiplicative situation as opposed to the teaching of procedures. In doing so, we encourage the explicit teaching of the many connections within the multiplicative situation and between it and other "big ideas" such as proportional reasoning and algebraic thinking.

Derek Hurrell shares a number of geometry tasks designed to tease out students' reasoning. Concrete materials are used to initiate discussions and develop a culture of reasoning in the mathematics classroom.

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. (Contains 4 figures and 1 table.)

The role of manipulative materials in mathematics can be a contentious one. It seems that in the mind of some teachers, manipulatives are fine in the early years but have no place in a rigorous maths classroom as the students get older. It is a topic worth our consideration.

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, the authors present a brief overview of research by mathematics educators and then provide a small selection of some of the many student work samples they have collected during their research into multiplicative thinking. They 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 article describes some of the essential mathematics that underpins the use of algorithms through a series of learning pathways. To begin, a graphic depicting the mathematical ideas and concepts that underpin the learning of algorithms for multiplication and division is provided. 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 article goes on to demonstrate how each element builds part of the underpinning structure needed to understand algorithms and to use them efficiently.

Improving learning through interdisciplinary STEM has come to the forefront of educational discourse, as schools attempt to attract more students to STEM study. However, little is known about how successful STEM curricula are established and sustained. This study details the establishment and sustaining of an exemplary STEM learning ecosystem in a girls' comprehensive secondary college. Results highlight exemplary STEM ecosystems demand multidimensional principal leadership, that over time, can develop effective generative proximal processes and levels of relational trust needed to implement the disruptive reforms associated with establishing interdisciplinary STEM curricula. Findings identify principals' ongoing engagement in the environment and dispositional and developmental assets are critical to successful school change supporting STEM innovation. [ABSTRACT FROM AUTHOR]

The characteristics that typify an effective teacher of mathematics and the environments that support effective teaching practices have been a long-term focus of educational research. In this article we report on an aspect of a larger study that investigated "best practice" in mathematics teaching and learning across all Australian states and territories. A case study from one Australian state was developed from data collected via classroom observations and semi-structured interviews with school leaders and teachers and analysed using Valsiner's zone theory. A finding of the study is that "successful" practice is strongly tied to school context and the cultural practices that have been developed by school leaders and teachers to optimise student learning opportunities. We illustrate such an alignment of school culture and practice through a vignette based on a case of one "successful" school.

A journey into multiplicative thinking by three teachers in a primary school is reported. A description of how the teachers learned to identify gaps in student knowledge is described along with how the teachers assisted students to connect multiplicative ideas in ways that make sense.

When teaching the measurement attribute of time, most teachers are aware that just getting students to read a clock is a task with its own challenges, but is developing this skill and understanding enough? What else do teachers need to make sure they cover, in order to give students a chance of developing this important component of being a numerate citizen? Siemon, Beswick, Brady, Clark, Faragher, and Warren (2015) write that there are three key aspects in developing an understanding of time. These are: (1) Understanding the concept of time duration; (2) Appreciating the passing and sequence of time; and (3) Determining a given point in time through the reading of a clock face (p. 471). In this article, Derek Hurrell discusses these aspects and provides concrete activities to support each one.

Multiplicative thinking is accepted as a "big idea" of mathematics that underpins important mathematical concepts such as fraction understanding, proportional reasoning, and algebraic thinking. It is characterised by understandings such as the multiplicative relationship between places in the number system, basic and extended number facts, and properties of operations and associated relationships. Using examples from a current Year 6 research project, this article highlights the importance of a combination of conceptual understanding and procedural fluency in developing multiplicative thinkers.

In providing a continued focus on tasks and activities that help to illustrate key ideas embedded in the "Australian Curriculum," this issue focuses on the Statistics and probability strand and the sub-strand of Chance. In the Australian Curriculum (ACARA, 2015), students are not asked to list outcomes of chance experiments and represent those probabilities until Year 5 (ACMSP116). Before this time, the focus was on identifying events that involve chance, and in developing everyday language to describe and order the events. Derek Hurrell notes, tt seems eminently sensible to involve students in activities which requires them to employ the language of chance, therefore Hurrell uses this article to introduce four activities that assist students in learning this concept.

In this article, the author looks at some key considerations which have proven to be very useful in the teaching of measurement in the primary classroom. Five ideas that can form the basis of focusing on measurement to access other strands of the mathematics curriculum are then examined.

Lorraine Day and Derek Hurrell provide a convincing argument for using arrays to promote students' understandings of mental computation strategies for multiplication. They also provide a range of different examples that illustrate the benefits of arrays in the primary classroom.

In providing a continued focus on tasks and activities that help to illustrate key ideas embedded in the new Australian Curriculum, this issue will focus on Number in the Number and Algebra strand. In this article Derek Hurrell provides a few tried and proven activities to develop place value understanding. These activities are provided for students who have moved through developing understanding through bundling and trading and are ready to look at numbers in their representational and abstract forms.

In providing a continued focus on tasks and activities that help to illustrate key ideas embedded in the new Australian Curriculum, the focus in this issue is on Measurement in the Measurement and Geometry strand. The small unit of work on measurement presented in this article has activities that can be modified to meet the requirements of particular year level descriptors in the aspects of the Measurement and Geometry strand. There is also a direct link to the Number and Algebra strand through the necessity to quantify.

As part of one of the undergraduate Bachelor of Education units offered as a specialisation unit in primary mathematics education at The University of Notre Dame Australia, the pre-service teachers were asked to complete an assignment. In this assignment they were given 12 typical algebra text book problems which they were asked to solve and then to take one of these problems and work extensively with it, to turn it into what was referred to as a "rich" learning experience. Working through the process of solving many problems provides experience to allow the identification of what constitutes a good task (Siemon, Beswick, Brady, Clark, Faragher & Warren, 2011). The main body of this article is the journey undertaken by one of the pre-service teachers, Courtney, in completing this assessment task and his reflections about the task. It should be noted that this assignment was preceded by much in the way of discussion and modelling of current best practice with an emphasis on developing sound pedagogy backed by current research. In answering the task requirements, Courtney drew the many strands of understanding together to create what was considered to be a cogent and thoughtful reflection which illustrates how a relatively trivial and context-free question can be teased out to provide not just an answer but a series of activities that can lead to developing some key understandings in the area of algebra. At the conclusion of this article is a small section which outlines some of the Australian Curriculum proficiencies and content links which have been addressed in the completion of this set of activities. (Contains 2 figures.)

Equivalence is mostly associated with the strand of Number and Algebra and is fundamental in both of the substrands. Equivalence is considered a key and foundational concept of algebra, and whereas an assumption might be made that equivalence is a relatively "easy" concept; studies have shown that when asked to balance an equation such as "8 + 4 = ___ + 5," students have substantial difficulties. This article provides activities which can be modified to meet the requirements of particular year level descriptors and although the descriptors articulated in this article from the Australian Curriculum start at Year 4, the work needs to, and can be developed, from a much younger age.

In providing a continued focus on tasks and activities that help to illustrate key ideas embedded in the new Australian Curriculum, this issue the authors focus, on Geometry in the Measurement and Geometry strand with strong links for an integrated focus on the Statistics and Probability strand. The small unit of work on the sorting and classifying of shapes presented in this article are activities that can be modified to meet the requirements of particular year level descriptors in the aspects of the Measurement and Geometry strand. There is also a direct link to the Statistics and Probability strand through the collection, the classification and the recording of the data. Where appropriate, elaborations are listed beneath each content descriptor.

The article discusses the efforts and studies of three primary school teachers within their professional learning community regarding developing multiplicative thinking in students. Topics discussed include the impact of mathematical concepts taught by the primary school, suggestions for teachers on how to develop an action plan, and the development of mental computation strategies.

The article discusses the significance of a combination of conceptual understanding and procedural fluency in developing multiplicative thinkers in the field of mathematics. Multiplicative thinking underpins important mathematical concepts and is characterized by an ability to recognise and solve a range of mathematical problems.

The potential of using Measurement as a way of "tuning students into mathematics" is demonstrated. Five ideas that can form the basis of focusing on measurement to access other strands of the mathematics curriculum are examined. [ABSTRACT FROM AUTHOR]

The article discusses the role of manipulative materials in the teaching of fractions. Topics covered include the positive effect of manipulatives on learning and how it can promote positive classroom behaviors, how the use of manipulatives can make a lesson more engaging, and the uniqueness of dynamic virtual manipulatives.

The article focuses on the significance of text books in mathematics education. It includes information on the role of textbooks in sequencing instructional activities, student engagement, and effective teaching. It also discusses topics including pedagogical approaches, contemporary learning theories, and symbolic representations.