Seeing Spots and Developing Multiplicative Sense Making

Matney, Gabriel T; Daugherty, Brooke N.

doi:10.5951/mathteacmiddscho.19.3.0148

Cited by 6 publications

(4 citation statements)

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“…There appear to be some clear links between the use of the array, an understanding of place value partitioning (Jazby & Pearn, 2015), and the distributive property (Matney & Daugherty, 2013;Kinzer & Stafford, 2015). Similarly, it is reasonable to suggest that the base ten property of place value underpins the use of extended number facts (Ross, 2002).…”

Section: Theoretical Frameworkmentioning

confidence: 99%

“…It was noted earlier that the proportion of Year 6 students who drew an array was less than for the Year 5s and it was suggested that this might be due to a progressive lessening in the use of the array with older students. However, as discussed earlier (Benson et al, 2013;Matney & Daugherty, 2013), the array underpins the distributive property which in turn underpins the grid method and written multiplication algorithm. It is possible that these students have been taught to use the algorithm without being exposed to its progressive development from the array to the grid method.…”

Section: Use Of the Array To Show 4 ×mentioning

confidence: 99%

“…By following a developmental progression from arrays through the grid method to the formal algorithm, students understand better the purpose of the algorithm by understanding the component sub-structures on which it is built. Matney and Daugherty (2013) also suggested that the array is a useful pre-requisite to ideas of commutativity and distributivity, and that manipulation of arrays offers the valuable experience of recognizing efficient ways of grouping numbers and links to the partial products needed for the algorithm. By offering students dot arrays for grouping, they are encouraged to seek efficient and fluent methods for counting the dots.…”

Section: Introductionmentioning

confidence: 99%

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Distributivity, partitioning, and the multiplication algorithm

Hurst

Huntley²

2020

J. Res. Adv. Math. Educ

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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.

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Section: Theoretical Frameworkmentioning

confidence: 99%

Section: Use Of the Array To Show 4 ×mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Distributivity, partitioning, and the multiplication algorithm

Hurst

Huntley²

2020

J. Res. Adv. Math. Educ

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“…For this to occur, teachers need to understand how and why the array is important. Matney and Daugherty (2013) are very clear about the power of the array to help students visualize mathematical structures like commutativity and distributivity, adding that students' computational errors may be compounded by their lack of understanding of what they are doing. They note that students are better able to understand commutativity when they rotate the array through ninety degrees.…”

Section: Arrays and Equal Groupsmentioning

confidence: 99%

Children Have the Capacity to Think Multiplicatively, as long as …

Hurst

2017

EJSTEME

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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.

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Teaching the partial products algorithm with the concrete representational abstract sequence and the strategic instruction model

Flores

Moore

Meyer

2020

Psychology in the Schools

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Psychol Schs. 2020;57:946-958. wileyonlinelibrary.com/journal/pits 946 | Abstract Elementary standards include multiplication of single-digit numbers and students advance to solve complex problems and demonstrate procedural fluency in algorithms. The ability to illustrate procedural fluency in algorithms is dependent on the development of understanding and reasoning in multiplication. Development of multiplicative reasoning provides the foundation for advanced mathematics and algebraic reasoning. For students who struggle in mathematics, instruction in multiplication algorithms should ensure conceptual understanding so that students have a foundation for success in advanced mathematics. The concrete representational abstract (CRA) sequence addresses conceptual understanding and the strategic instruction model (SIM) supports procedural knowledge.The current pilot study combined these methods to teach elementary students the partial products algorithm. Twelve students in grades four and five participated in the study, receiving instruction from teachers in their school during an intervention period. Within a pre-experimental design, using pre-and postintervention data, students showed a significant change in performance. The article will describe and show how teachers implemented the CRA-SIM interventions and discuss implications for practice. K E Y W O R D S concrete representational abstract, mathematics, partial products

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Seeing Spots and Developing Multiplicative Sense Making

Abstract: Seeing Spots Cans on a grocery store shelf and Hirst's Capric Acid Amide can illustrate dot arrays, thus helping students understand the distributive property, partial products, and the standard algorithm for multiplication.

Cited by 6 publications

References 1 publication

Distributivity, partitioning, and the multiplication algorithm

Distributivity, partitioning, and the multiplication algorithm

Children Have the Capacity to Think Multiplicatively, as long as …

Teaching the partial products algorithm with the concrete representational abstract sequence and the strategic instruction model

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