Open Access Te Herenga Waka-Victoria University of Wellington
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Do structured manipulatives aid understanding of additive reasoning, and addition and subtraction fluency in a sample of Year 7 and 8 students?

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Version 2 2023-09-26, 01:37
Version 1 2021-12-08, 23:38
posted on 2023-09-26, 01:37 authored by Green, Daniel

Mathematical achievement may impact on outcomes in later life; thus, identifying and improving key mathematical skills is a focus of a large body of educational research. Both additive reasoning, and knowledge of addition and subtraction facts, appear to predict later mathematical achievement. The current study explores the impact of a short intervention with a small group of year 7 and 8 students working at lower than expected academic levels. The current study is based on Cognitive Load Theory and research suggesting that counting strategies overload working memory. A mixed-methods approach was used to identify whether structured manipulatives improved the additive reasoning and, addition and subtraction fluency in a sample of ten participants. Participants attended after-school intervention sessions of 45 minutes for seven weeks. The intervention focused on teaching additive reasoning and fluency using structured manipulatives. Inferential statistical analysis showed a statistically significant mean improvement in participants’ ability to answer simple addition and subtraction questions. Tests constructed to operationalise additive reasoning also showed statistically significant mean improvement. Participants answered diagnostic questions operationalising various aspects of additive reasoning. Individual differences in understanding of additive reasoning were observed, and the inverse relationship between addition and subtraction proved to be a challenging concept. Semi-structured interviews provided themes of valuing the intervention and the manipulatives used. Due to the size and design of this study, it is not possible to extrapolate findings to other learners. However, the study may provide directions for future research. Structured manipulatives may have a role to play in enabling learners to begin to learn additive relationships and further securing recall of addition and subtraction facts. Students at years 7 and 8 may still need considerable exposure to additive concepts; moreover, returning to manipulatives may develop this knowledge. Finally, the findings from the diagnostic questions help show the complexity of additive reasoning. Classroom practitioners may need to further develop their knowledge of additive reasoning, its importance, and the individual differences and misconceptions that learners hold in order to provide considered learning experiences.


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Date of Award



Te Herenga Waka—Victoria University of Wellington

Rights License


Degree Discipline


Degree Grantor

Te Herenga Waka—Victoria University of Wellington

Degree Level


Degree Name

Master of Education

ANZSRC Type Of Activity code


Victoria University of Wellington Item Type

Awarded Research Masters Thesis



Victoria University of Wellington School

School of Education


Averill, Robin