Search

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 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.

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 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 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.

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.

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. [The title on the pdf is "Algorithms and Multiplicative Thinking: Are Children Prisoners of Process?"]

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.

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.

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.

It is important for students to make informed decisions about computation. This article highlights this importance and develops a framework which may assist teachers to help students to make effective computational choices.

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.

The recent review of Australian Curriculum represents an opportunity to significantly change the teaching of primary mathematics in Australia and perhaps elsewhere. The current curriculum document presents content in a linear and compartmentalized manner that dominates its structure and doesn't explicitly accentuate the links and connections found in the "big ideas" of mathematics. As well, it pays lip service to the "big process ideas"--the Proficiencies--which should be the vehicles for developing and making explicit links between and within the "big content ideas". The nature of key content needs to be reconsidered and organised on the basis of the "big ideas" of mathematics, and to emphasize the links and connections within and between them, as well as between them and real contexts. Such connections should be made explicit for children and hence teachers may need to hold their mathematical knowledge in different ways. The review document acknowledges the issue of the "crowded curriculum" and makes recommendations such as increasing the depth of content whilst reducing its breadth. Exactly how this will be achieved remains to be seen but it is strongly suggested in this position paper that there needs to be a reorganization of content on the basis of "big ideas of mathematics" or at least a substantial acknowledgement that this is important.

The relationships between three critical elements, and the associated mathematical language, to assist students to make the critical transition from additive to multiplicative thinking are examined in this article by Chris Hurst.

Most adults' attitudes to mathematics come from their experiences of mathematics in school when they were children. Children's mathematical worlds are complex places containing both cognitive and affective elements. One cannot ignore the affective domain if one wishes to understand children's mathematical learning. Teacher education students consistently use the following words in describing their memories of mathematics: repetition, monotony, boredom, not understanding, mystery, failure irrelevance, and humiliation. These memories group into feelings relating to the teaching of mathematics and their personal reaction to themselves as mathematics learners. In this article, the authors remind readers of the importance of helping students to develop positive attitudes to mathematics and provide practical suggestions as to how to engage students in a variety of stimulating activities. (Contains 4 figures.)