Comparing example generation with classification in the learning of new mathematics concepts

ABSTRACT

Prompting learners to generate examples has been proposed as an effective way of developing understanding of a new concept. However, empirical support for this approach is lacking. This article presents two empirical studies on the use of example-generation tasks in an online course in introductory university mathematics. The first study compares the effectiveness of a task prompting learners to generate examples of increasing and decreasing sequences, with a task inviting them to classify given examples; it also investigates the effectiveness of different sequences of generation and classification tasks. The second study replicates the investigation of interactions between generation and classification tasks. The findings suggest that there is little difference between the two types of task, in terms of students' ability to answer later questions about the concept.

KEYWORDS:

• Learner-generated examples
• e-assessment
• undergraduate mathematics

Disclosure statement

No potential conflict of interest was reported by the author.

Data availability statement

All data and analysis code is available at https://doi.org/10.17605/osf.io/gry6v.

Notes

1 There were 53 students in Study 1 of Iannone et al. (2011) and 132 in the experiment of Alcock and Simpson (2017).

2 Students who declined to participate (N = 38) were given access to a version of the course materials that was unchanged from the previous year.

3 The definition given was of strictly increasing/decreasing: “The sequence $u_1,u_2,u_3,\dots$ is said to be increasing if $u_{n+1}>u_n$ for all n, and decreasing if $u_{n+1}<u_n$ for all n”.

4 The sequence of terms beginning $10\frac{1}{2},10\frac{3}{4},\dots$ was changed to $\frac{1}{2},\frac{3}{4},\dots$ since it is not straightforward to represent mixed numbers in the online system. Similarly, the sequence defined by $3+(-1{)}^n$ was instead presented as $(-1{)}^n+3$. Finally, for the graphical representation of an increasing sequence, the vertical axis was dilated to accentuate the pattern.

5 These priors are weakly informative normal distributions, with the scale determined by the observed standard deviation; see http://mc-stan.org/rstanarm/articles/priors.html.