In his paper The Design of Multiple Representation Tasks To Foster Conceptual Development, Professor Malcolm Swan of University of Nottingham presented five types of tasks that foster conceptual understanding of mathematical concepts. This was developed through their work with teachers. This classification of tasks is a very good framework to use in designing instruction. I have used this framework in one of our lesson study projects. Five tasks ‘types’ that encourage conceptual understanding.
- Classifying mathematical objects
Students devise their own classifications for mathematical objects, and/or apply classification devised by others. In doing this, they learn to discriminate carefully and recognize the properties of objects. They also develop mathematical language and definitions. The objects might be anything from shapes to quadratic equations.
- Interpreting multiple representations
Students work together matching cards that show different representations of the same mathematical idea. They draw links between representations and develop new mental images for concepts.
- Evaluating mathematical statements
Students decide whether given statements are always, sometimes or never true. They are encouraged to develop mathematical arguments and justifications, and devise examples and counterexamples to defend their reasoning.
- Creating problems
Students are asked to create problems for other students to solve. When the solver become stuck, the problem ‘creators’ take on the role of teacher and explainer. In these activities, the ‘doing’ and ‘undoing’ processes of mathematics are exemplified.
- Analyzing reasoning solutions
Students compare different methods for doing a problem, organize solutions and/or diagnose the causes of errors in solutions. They begin to recognize that there are alternative pathways through a problem, and develop their own chains of reasoning.
Professor Malcolm Swan is also the author of the books Collaborative Learning in Mathematics: A Challenge to Our Beliefs and Practices and The Language of Functions and Graphs An Examination Module for Secondary Schools.