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Scottish Extensions


5-14 Mathematics in Scotland: The Relevance of Intensive Quantities (2003 - 2005)

This project has now completed. Major outputs include:


Professor Christine Howe (Strathclyde University)
Professor Terezinha Nunes (Oxford Brookes University)
Professor Peter Bryant (University of Oxford)


Initial Project Summary

This Scottish Extension was linked to the Phase II project, 'The Role of Awareness in the Teaching and Learning of Literacy and Numeracy in Key Stage 2'.

In mathematics and science education, there is an important distinction to be made between two types of quantity, extensive (e.g., height, weight, volume) and intensive (e.g., speed, density, temperature). This distinction is recognised elsewhere in the world, but it has been neglected throughout the UK, where teaching concentrates on extensive quantities. This seems likely to have serious implications for pupil attainment, as there is evidence that in the absence of teaching, intensive quantities are treated as if they were extensive.

Until recently, there has been little psychological analysis of the contrasts between intensive and extensive quantities, of what pupils should learn in order to avoid confusion, and of how they should be taught. However, previous TLRP work in England by two of the investigators demonstrates difficulties that vary with the logical moves required by the problem, the language used in problem description, and the specific type of intensive quantity. These difficulties cannot be remedied by the teaching of extensive quantities alone. An equivalent situation can be anticipated in Scotland, although differences stemming from the Scottish 5-14 Programme indicate that their form may vary.

In view of the above, this two-year Scottish extension aimed:

  1. to document the ways in which primary school pupils in Scotland learn about intensive quantities, and the difficulties that they have with this learning;
  2. to identify modes of teaching that are both effective and sensitive to the Scottish context;
  3. to ensure that the results have maximal impact upon policy and practice.

The specific objectives were:

  • To provide researchers, policy makers, teachers and trainers of teachers with reliable information about the extent to which intensive quantities are currently grasped by primary school pupils in Scotland, and about areas of particular difficulty;
  • To develop a whole-class plus small-group approach to the teaching of intensive quantities, and to make controlled comparisons with alternative methods;
  • To contrast the efficacy of ratio and fractional approaches to the teaching of intensive quantities;
  • To explore how base-line knowledge of intensive quantities, and responses to the teaching intervention, vary with pupil age, sex and mathematical ability;
  • To develop an impact strategy, and to implement this with policy makers, trainee teachers and relevant academic staff.

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