This miniproject investigated, through direct experimental testing, the claims being made by several education researchers that asking learners to generate their own examples of mathematical concepts is a more effective teaching strategy than presenting worked examples of concepts to them directly.
First pilot study:
Once the instrument was finalised it was piloted with 10 clinical interviews with second- and third-year mathematics students at Loughborough University. In the clinical interviews, lasting each around one hour, the students were asked to work through the instrument at their own pace and ‘think aloud’ while solving the tasks. The students were asked to spend around 25 minutes on the first part (example generating tasks) of the instrument, and around 25 minutes on the second part (proof construction tasks). The data was collected in form of videos. The clinical interviews were then fully transcribed and analysed to monitor the fit of the instrument to the aims of the project. After careful scrutiny of the clinical interviews the decision was taken to change some of the tasks included in the instrument as they were deemed to be too complex.
Second pilot study:
Once the two instruments were ready they were piloted again with 10 PGCE students (secondary mathematics) at UEA. After analysis of the second pilot the two instruments were again slightly modified, in order to finalise versions to be used in the main study.
Data collection took place across three UK HEIs during March, April, and May 2009. The fact that the data collection was taking place so close to the exam period meant that few scripts were collected, despite having planned for several sessions of data collection, as detailed in the interim report. Further data collection took place in September (in one university in the US) giving a total of 53 participants. The questionnaire that was designed used the concept of fine function, adapted from Dahlberg & Housman [1].
The aim of this analysis, and the project, was to test the prediction that, on proof construction tasks, there would be an advantage for those participants who were asked to generate instances of a concept compared to those who passively read descriptions of a concept. The proof-tasks proved to be sufficiently discriminating, with an overall mean well away from any possible ceiling or floor effects.
To summarise, these findings offer no support to Watson & Mason’s [2] hypothesis (which predicted that the generating group would show an advantage over the reading group). Not only was the between-conditions difference non-significant, but the trend was in the opposite direction to that predicted. This is in some sense surprising giving the strong arguments made in the research literature.
[1] Dahlberg, R.P. and Housman, D.L. (1997) ‘Facilitating learning events through example generation’, Educational Studies in Mathematics, 33, 283-299.
[2] Watson, A. and Mason, J. (2005) ‘Mathematics as a Constructive Activity: learners generating examples’, Mahwah, NJ, USA: Erlbaum.