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Motor performance is multifaceted and refers to several different dimensions, including force production, precision control, movement speed, resistance to fatigue, motor adaptation, and motor skill learning. The investigation of the placebo effect on different motor functions represents a useful approach to extend our knowledge about the pervasive nature of this effect and build new models for the study of its neural underpinnings. To date, while many behavioral studies have demonstrated the efficacy of placebo and nocebo procedures in influencing some aspects of motor performance, others remain unexplored. In this chapter, two categories of motor functions will be considered: those performed with the upper limb and those requiring the whole body. Within the first category, evidence using a fundamental function through which repeated practice forms isolated movements into well-performed skills will be presented on force production, goal-directed movements, and motor sequence learning. Within the second category, evidence will be presented on balance control, a fundamental motor function, which allows to maintain a stable stance and to prevent falls, as well as on leg-extension performance.

This small study sought to determine students’ knowledge of multiplication and division and whether they are able to use sets of bundling sticks to demonstrate their knowledge. Manipulatives are widely used in primary and some middle school classrooms, and can assist children to connect multiplicative concepts to physical representations. Qualitative data were generated from semi-structured interviews with 32 primary and middle school children aged nine to eleven years. Participants were asked to work out the answer to multiplication and division examples and explain their thinking using bundling sticks. Results suggest that the majority of participant students may have a limited knowledge of aspects of the multiplication process and even less knowledge of the division process. The study also identified that many of the students appeared uncomfortable and/or unfamiliar with using bundling sticks and a number of them had difficulty in using bundling sticks to explain the multiplication and division processes. We conclude that manipulatives such as bundling sticks do not magically lead children to mathematical learning but are sufficiently powerful to warrant teachers familiarising themselves with how manipulatives can be used to develop conceptual understanding.

Multiplicative thinking underpins much of the mathematics learned beyond the middle primary years. As such, it needs to be understood conceptually to highlight the connections between its many aspects. This paper focuses on one such connection; that is how the array, place value partitioning and the distributive property of multiplication are related. It is important that students understand how the property informs the written multiplication algorithm. Another component of successful use of the standard multiplication algorithm is extended number facts and the paper also explores students’ ability to understand and generate them. One purpose of the study was to investigate the extent to which students used the standard multiplication algorithm and if their use of it is supported by an understanding of the underpinning components of the array, partitioning, the distributive property, and extended number facts. That is, we seek to learn if students have a conceptual understanding of the multiplication algorithm and its underpinning mathematics that would enable them to transfer their knowledge to a range of contexts, or if they have procedurally learned mathematics. In this qualitative study, data were generated from the administration of a Multiplicative Thinking Quiz with a sample of 36 primary aged students. Data were analyzed manually and reported using descriptive statistics. The main implications of the study are that the connections among the multiplicative array, place value partitioning, base ten property of place value, distributive property of multiplication, and extended number facts need to be made explicit for children in terms of how they inform the use of the written algorithm for multiplication.

Multiplicative thinking is a critical component of mathematics which largely determines the extent to which people develop mathematical understanding beyond middle primary years. We contend that there are several major issues, one being that much teaching about multiplicative ideas is focussed on algorithms and procedures. An associated issue is the extent to which algorithms are taught without the necessary explicit connections to key mathematical ideas. This article explores the extent to which some primary students use the algorithm as a preferred choice of method and whether they can recognise and use alternative ways of calculating answers. We also consider the extent to which the students understand ideas that underpin algorithms. Our findings suggest that most students in the sample are ‘prisoners to procedures and processes’ irrespective of whether or not they understand the mathematics behind the algorithms.

Big idea thinking provides an opportunity to re-conceptualize how we view and teach primary mathematics. Contemporary curricula continue to be organized in a linear fashion with content allocated to year levels, which encourages a narrow view of what needs to be taught. Big idea thinking has the capacity to change that. The real value of big ideas lies in interpreting the mathematics within them. Big ideas are those which connect mathematical understandings into a coherent whole, and are central to the learning of mathematics. Big ideas comprise a network of ‘little ideas’ or ‘micro-content’ and teachers who think in terms of them are able to look forwards and backwards from their own year level to identify specific content that a student may not know, and to lay the foundations for what the student needs to know next. Big idea teachers are not limited in their thinking by curriculum boundaries. Most importantly, big idea thinking encourages teachers to deconstruct and reconstruct their knowledge. Teachers can actively engage in this by beginning with a mathematical idea such as place value and building a concept map showing the various pieces of ‘micro-content’ that contribute to the development of the concept and considering how the content is connected.

Multiplicative thinking is a key aspect of primary and middle school mathematics and is considered to be a predictor of students’ capacity to progress beyond basic mathematical learning. It is characterised by a complex set of connecting ideas about which teachers need to have a broad and deep understanding. The study on which this symposium is based began in 2014 in Western Australia. It has involved over 1900 primary school students of ages 9 to 11 years, approximately 120 teachers, and 16 schools. This symposium presents an overview of the project and then focuses on the New Zealand phase of the project. Assessment of students’ multiplicative thinking in the form of a written quiz and semi-structured interviews enabled teachers and researchers to identify students’ knowledge and understanding of multiplicative concepts and led to the structuring of a targeted teaching program over several months. Parallel pre and post quizzes were used to investigate the extent of student learning that occurred. A highly significant increase in student attainment was noted. The use of manipulative materials to identify the extent of students’ multiplicative thinking was also investigated through semi-structured interviews. Teachers’ content knowledge was explored with particular emphasis on the use of student tasks targeting specific aspects of multiplicative thinking. It was found that teachers became more confident in teaching multiplicative concepts, showed a greater awareness of connections between ideas, and demonstrated a growing awareness of the importance of explicit mathematical language.

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.

Evidence suggests that some students have learned procedures with little or no underpinning understanding while others have a much more connected and conceptual levels of understanding. In this article, the work of four primary students is discussed in terms of their contextual understanding of multiplicative concepts. The difference between teaching for understanding and procedural teaching is highlighted. An analogy is drawn with Charles Dickens' character, Mr Thomas Gradgrind, who would have endorsed procedural teaching and abhorred teaching that encouraged understanding. The work of four primary students is discussed in terms of their contextual understanding of multiplicative concepts. The difference between teaching for understanding and procedural teaching is highlighted.

Multiplicative thinking has been widely accepted as a critically important ‘big idea’ of mathematics and one which underpins much mathematical understanding beyond the primary years of schooling. It is therefore of importance to consider the capacity of children to think multiplicatively but also to consider the capacity of their teachers to teach multiplicative thinking in a conceptual manner. This article focusses specifically on the conceptual links between the multiplicative array, the notion of numbers of equal groups in the multiplicative situation, factors and multiples, the commutative property of multiplication, and the inverse relationship between multiplication and division. A study involving a large sample of primary school students found that whilst most students demonstrated an understanding of some of the aforementioned elements, hardly any of the students were able to connect the ideas or to explain them in terms of each other. As a consequence of the findings, the impact of teacher knowledge on children’s capacity to think multiplicatively was considered.

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.

Multiplicative thinking is a critical stage of mathematical understanding upon which many mathematical ideas are built. The myriad aspects of multiplicative thinking and the connections between them need to be explicitly developed. One such connection is the relationship between place value partitioning and the distributive property of multiplication. In this paper we explore the extent to which students understand partitioning and relate it to the distributive property and whether they understand how the property is used in the standard multiplication algorithm.

Background: Teacher knowledge continues to be a topic of debate in Australasia and in other parts of the world. There have been many attempts by mathematics educators and researchers to define the knowledge needed by teachers to teach mathematics effectively. A plethora of terms, such as mathematical content knowledge, pedagogical content knowledge, horizon content knowledge and specialised content knowledge, have been used to describe aspects of such knowledge. Purpose: This paper proposes a model for teacher knowledge in mathematics that embraces and develops aspects of earlier models. It focuses on the notions of contingent knowledge and the connectedness of ‘big ideas’ of mathematics to enact what is described as ‘powerful teaching’. It involves the teacher’s ability to set up and provoke contingent moments to extend children’s mathematical horizons. The model proposed here considers the various cognitive and affective components and domains that teachers may require to enact ‘powerful teaching’. The intention is to validate the proposed model empirically during a future stage of research. Sources of evidence: Contingency is described in Rowland’s Knowledge Quartet as the ability to respond to children’s questions, misconceptions and actions and to be able to deviate from a teaching plan as needed. The notion of ‘horizon content knowledge’ (Ball et al.) is a key aspect of the proposed model and has provoked a discussion in this article about students’ mathematical horizons and what these might comprise. Together with a deep mathematical content knowledge and a sensibility for students and their mathematical horizons, these ideas form the foundations of the proposed model. Main argument: It follows that a deeper level of knowledge might enable a teacher to respond better and to plan and anticipate contingent moments. By taking this further and considering teacher knowledge as ‘dynamic’, this paper suggests that instead of responding to contingent events, ‘powerfu