Mathematical skills > Example generation

Question 53
How can e-assessments be designed to expand and enrich students' example spaces?

Learning about concepts by developing a rich example space has been suggested as educationally valuable (e.g., Watson & Mason, 2006), with e-assessment proposed as a particularly suitable mechanism for gathering and checking examples. However, there are currently few examples of this being done in practice, and little in the way of guidance about designing effective tasks.

What motivates this question?

Watson and Mason (2006) build on Michener’s notion of ‘example spaces’ and highlight students’ mathematical development in terms of (i) expanding the range of examples students are aware of within a particular mathematical area, and (ii) enriching the knowledge of those examples. This is often done by presenting students with new examples that they need to be aware of, such as classic examples (e.g. modulus function as a continuous but not differentiable function) but could also be done by asking students to play around and come up with examples that satisfy certain constraints. In some cases it may be the process of finding these examples that is of more value than the specific examples found.

One difficulty with larger classes is checking that students’ examples are actually mathematically correct, and do not in fact indicate mistakes and misunderstandings. CAA such as STACK would be ideal for checking that constraints on examples are satisfied, and, if not, perhaps saying “One or more constraints are not satisfied” so as to avoid making it too easy for the student and forcing them to go back and check carefully.

In school-level mathematics, there are many example-generation task types used, many of which might be considerably enhanced with CAA. For example, Venn diagrams are a convenient way to set up problems of separate and overlapping constraints, where the student is required (if possible) to provide an example for each region, or (if not) explain why it is impossible. Could this be set up in CAA?

What might an answer look like?

The core of the question here is about how to design tasks to achieve a particular effect, so the answer could take the form of (i) a collection of particular examples drawn from a variety of topics (e.g. Sangwin, 2004, describes a task sequence about polynomials), or (ii) some general principles for the design of tasks that prompt students to develop their example spaces, e.g. building on Watson & Mason (2006, Chapter 6). Evidence of effectiveness might be provided by pre- and post-tests of ‘richness of example space’, as compared with paper and pencil tasks in a comparison condition.

Choices would have to be made about the kind of feedback provided. Individualised, immediate feedback could be beneficial, or perhaps not. In even a moderately-sized lecture, the lecturer will not have time to go round and evaluate each student’s examples carefully, whereas CAA could, so there is a potential route to greater effectiveness here. At the first stage, it might be more promising to see whether a motivational and guiding feedback is capable of encouraging students to generate a particular type of examples, for instance those that are more sophisticated or more rare.

References

Sangwin, C. J. (2004). On building polynomials. The Mathematical Gazette, 89(516), 441–450. https://doi.org/10.1017/S0025557200178295

Watson, A., & Mason, J. (2006). Mathematics as a constructive activity: Learners generating examples. Routledge.