Developing mathematical fluency: comparing exercises and rich tasks
Achieving fluency in important mathematical procedures is fundamental to students’ mathematical development. The usual way to develop procedural fluency is to practise repetitive exercises, but is this the only effective way?This paper reports three quasi experimental studies carried out in a total...
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| Format: | Article |
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Springer
2017
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| Online Access: | https://eprints.nottingham.ac.uk/46887/ |
| _version_ | 1848797419319328768 |
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| author | Foster, Colin |
| author_facet | Foster, Colin |
| author_sort | Foster, Colin |
| building | Nottingham Research Data Repository |
| collection | Online Access |
| description | Achieving fluency in important mathematical procedures is fundamental to students’ mathematical development. The usual way to develop procedural fluency is to practise repetitive exercises, but is this the only effective way?This paper reports three quasi experimental studies carried out in a total of 11 secondary schools involving altogether 528 students aged 12–15. In each study, parallel classes were taught the same mathematical procedure before one class undertook traditional exercises while the other worked on a "mathematical etude" (Foster International Journal of Mathematical Education in Science and Technology, 44(5), 765–774, 2013b), designed to be a richer task involving extensive opportunities for practice of the relevant procedure. Bayesian t tests on the gain scores between pre- and post-tests in each study provided evidence of no difference between the two conditions. A Bayesian meta-analysis of the three studies gave a combined Bayes factor of 5.83, constituting Bsubstantial^ evidence (Jeffreys, 1961) in favour of the null hypothesis that etudes and exercises were equally effective, relative to the alternative hypothesis that they were not. These data support the conclusion that the mathematical etudes trialled are comparable to traditional exercises in their effects on procedural fluency. This could make etudes a viable alternative to exercises, since they offer the possibility of richer, more creative problem-solving activity, with comparable effectiveness in developing procedural fluency. |
| first_indexed | 2025-11-14T20:03:34Z |
| format | Article |
| id | nottingham-46887 |
| institution | University of Nottingham Malaysia Campus |
| institution_category | Local University |
| last_indexed | 2025-11-14T20:03:34Z |
| publishDate | 2017 |
| publisher | Springer |
| recordtype | eprints |
| repository_type | Digital Repository |
| spelling | nottingham-468872020-05-04T19:08:44Z https://eprints.nottingham.ac.uk/46887/ Developing mathematical fluency: comparing exercises and rich tasks Foster, Colin Achieving fluency in important mathematical procedures is fundamental to students’ mathematical development. The usual way to develop procedural fluency is to practise repetitive exercises, but is this the only effective way?This paper reports three quasi experimental studies carried out in a total of 11 secondary schools involving altogether 528 students aged 12–15. In each study, parallel classes were taught the same mathematical procedure before one class undertook traditional exercises while the other worked on a "mathematical etude" (Foster International Journal of Mathematical Education in Science and Technology, 44(5), 765–774, 2013b), designed to be a richer task involving extensive opportunities for practice of the relevant procedure. Bayesian t tests on the gain scores between pre- and post-tests in each study provided evidence of no difference between the two conditions. A Bayesian meta-analysis of the three studies gave a combined Bayes factor of 5.83, constituting Bsubstantial^ evidence (Jeffreys, 1961) in favour of the null hypothesis that etudes and exercises were equally effective, relative to the alternative hypothesis that they were not. These data support the conclusion that the mathematical etudes trialled are comparable to traditional exercises in their effects on procedural fluency. This could make etudes a viable alternative to exercises, since they offer the possibility of richer, more creative problem-solving activity, with comparable effectiveness in developing procedural fluency. Springer 2017-09-26 Article PeerReviewed Foster, Colin (2017) Developing mathematical fluency: comparing exercises and rich tasks. Educational Studies in Mathematics . pp. 1-21. ISSN 1573-0816 Bayesian hypothesis testing Enlargements Linear equations Fluency Practice Mastery Mathematics education Procedures Rich tasks https://link.springer.com/article/10.1007%2Fs10649-017-9788-x doi:10.1007/s10649-017-9788-x doi:10.1007/s10649-017-9788-x |
| spellingShingle | Bayesian hypothesis testing Enlargements Linear equations Fluency Practice Mastery Mathematics education Procedures Rich tasks Foster, Colin Developing mathematical fluency: comparing exercises and rich tasks |
| title | Developing mathematical fluency: comparing exercises and rich tasks |
| title_full | Developing mathematical fluency: comparing exercises and rich tasks |
| title_fullStr | Developing mathematical fluency: comparing exercises and rich tasks |
| title_full_unstemmed | Developing mathematical fluency: comparing exercises and rich tasks |
| title_short | Developing mathematical fluency: comparing exercises and rich tasks |
| title_sort | developing mathematical fluency: comparing exercises and rich tasks |
| topic | Bayesian hypothesis testing Enlargements Linear equations Fluency Practice Mastery Mathematics education Procedures Rich tasks |
| url | https://eprints.nottingham.ac.uk/46887/ https://eprints.nottingham.ac.uk/46887/ https://eprints.nottingham.ac.uk/46887/ |