Representing and Understanding Multiplication

Harries, Tony; Barmby, Patrick

doi:10.1080/14794800008520169

Cited by 18 publications

(21 citation statements)

References 10 publications

(9 reference statements)

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“…It may be assumed that by the end of primary school (age 11), all students in English mainstream education will have encountered rectangular arrays (both dots and grids), and that these representations will have been employed in the context of multiplication. This pedagogical practice is supported by recent findings that the array form has particularly good potential for supporting reasoning and developing understanding in multiplication, and is one of the best for demonstrating the commutative and distributive principles (Harries & Barmby, 2007). have identified various levels of sophistication in students' interaction with rectangular arrays of squares, and studies by Mitchelmore (2000, 2004) have linked increased regularity of structure observed in young children's 2D array drawings to their development of multiplicative strategies.…”

Section: D and 3d Arrays As Visuospatial Representations Of Multiplimentioning

confidence: 77%

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Between Counting and Multiplication: Low-Attaining Students’ Spatial Structuring, Enumeration and Errors in Concretely-Presented 3D Array Tasks

Finesilver

2017

Mathematical Thinking and Learning

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Section: D and 3d Arrays As Visuospatial Representations Of Multiplimentioning

confidence: 77%

“…Furthermore, the flexibility to switch pragmatically between different array structurings can support understanding of further properties of multiplication (e.g. associativity or commutativity, as recommended by Harries & Barmby (2007)).…”

Section: Relationship Between Spatial Structuring and Enumeration In mentioning

confidence: 99%

Between Counting and Multiplication: Low-Attaining Students’ Spatial Structuring, Enumeration and Errors in Concretely-Presented 3D Array Tasks

Finesilver

2017

Mathematical Thinking and Learning

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“…[28], "a combination of a progress bar and a leaderboard is likely to generate excitement, commitment, a will to finish a gamified activity in a successful manner, and even desire to repeat the experience" (p. 38). Additional MG features that promote motivation include providing help in the form of hints and displaying graphical explanatory feedback correctly depicting the wrongly answered multiplication fact for commonly observed mistakes [84]. Motivation is also enhanced by timing players that can be available only for the case of older or highly skilled pupils, in order to provoke their interest and maintain their engagement.…”

Section: The Multiplication Gamementioning

confidence: 99%

Techniques to Motivate Learner Improvement in Game-Based Assessment

2020

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Learner motivation to self-improve is a crucial effectiveness factor in all modes and settings of learning. Game-based learning was long used for attracting and maintaining students’ interest especially in small ages, deploying means such as scoring, timing, scores of peers (i.e., hall of fame), etc. These techniques can provide recognition for high-scoring players, while also developing a sense of safe “distance” in the impersonal electronic environment for low-scoring players. In addition, constructive feedback on mistakes a player makes can contribute to avoiding similar mistakes in the future, thus achieving better performance in the game, while constructing valuable new knowledge when a knowledge gap is detected. This paper investigates an integrated approach to designing, implementing, and using an adaptive game for assessing and gradually improving multiplication skills. Student motivation is fostered by incorporating the Open Learner Model approach, which exposes part of the underlying user model to the students in a graphically simplified manner that is easily perceivable and offers a clear picture of student performance. In addition, the Open Learner Model is expanded with visualizations of social comparison information, where students can access the progress of anonymous peers and summative class scores for improving self-reflection and fostering self-regulated learning. This paper also presents the feedback received by the preliminary testing of the game and discusses the effect of assessing multiplication skills of primary school pupils using the adaptive game-based approach on increasing pupil motivation to self-improve.

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“…In analysing the recordings obtained during the classroom sessions, we looked for ways in which the array supported or hindered children"s reasoning during multiplication calculations. In a previous paper (Harries & Barmby, 2007), we highlighted particular examples (by two pairs of children) of how the array might be used. In the following sections, we have looked in greater detail at the recordings and have categorised our observations, drawing on pupils" conversations as examples of their reasoning.…”

Section: Methodsmentioning

confidence: 99%

The array representation and primary children’s understanding and reasoning in multiplication

et al. 2008

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(2009) 'The array representation and primary children's understanding and reasoning in multiplication.', Educational studies in mathematics., 70 (3). pp. 217-241.Further information on publisher's website:http://dx.doi.org/10.1007/s10649-008-9145-1Publisher's copyright statement:The original publication is available at www.springerlink.com Additional information: Use policyThe full-text may be used and/or reproduced, and given to third parties in any format or medium, without prior permission or charge, for personal research or study, educational, or not-for-prot purposes provided that:• a full bibliographic reference is made to the original source • a link is made to the metadata record in DRO • the full-text is not changed in any way The full-text must not be sold in any format or medium without the formal permission of the copyright holders.Please consult the full DRO policy for further details. Abstract We examine whether the array representation can support children"s understanding and reasoning in multiplication. To begin, we define what we mean by understanding and reasoning. We adopt a "representational-reasoning" model of understanding, where understanding is seen as connections being made between mental representations of concepts, with reasoning linking together the different parts of the understanding. We examine in detail the implications of this model, drawing upon the wider literature on assessing understanding, multiple representations, self explanations and key developmental understandings. Having also established theoretically why the array representation might support children"s understanding and reasoning, we describe the results of a study which looked at children using the array for multiplication calculations. Children worked in pairs on laptop computers, using Flash Macromedia programs with the array representation to carry out multiplication calculations. In using this approach, we were able to record all the actions carried out by children on the computer, using a recording program called Camtasia. The analysis of the obtained audiovisual data identified ways in which the array representation helped children, and also problems that children had with using the array. Based on these results, implications for using the array in the classroom are considered.

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Representing and Understanding Multiplication

Cited by 18 publications

References 10 publications

Between Counting and Multiplication: Low-Attaining Students’ Spatial Structuring, Enumeration and Errors in Concretely-Presented 3D Array Tasks

Between Counting and Multiplication: Low-Attaining Students’ Spatial Structuring, Enumeration and Errors in Concretely-Presented 3D Array Tasks

Techniques to Motivate Learner Improvement in Game-Based Assessment

The array representation and primary children’s understanding and reasoning in multiplication

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