Modelling students’ thinking on a proportional reasoning task

Abstract

Production system models are presented of the thinking of a group of young adults on a proportional reasoning task in which the subjects were asked to compare the strengths of various mixtures of orange concentrate and water. The production system models were created in order to describe accurately learning gains made by the students as a result of a teaching experiment. A number of proportional strategies are identified: naive, addition, ‘doubling within’, ‘doubling between’, ‘simple multiplication within’, ‘simple multiplication between’, remainder, fractions, decimals. The addition strategy is modelled as a response to an impasse. A strategy based on the use of remainders is identified as a distinct strategy. Four groups of production rules are used in the models: comparison rules, equality rules, sequencing rules and encoding rules. Possible learning gains in each set of rules are identified. Comparison with student interviews before and after an experimental teaching episode showed that the models ca...