Flexible thinking and met-befores: Impact on learning mathematics

• The minus sign changes meaning subtly as the curriculum develops.
• Problematic aspects that arise from changes in meaning of the “−” symbol are identified.
• Cognitive demands on students are more complex than has generally been recognized.
• Use of a function machine helps students distinguish explicitly various meanings of the minus sign.
• Formative assessment increased teacher awareness of the consequences of prior instruction.

In this paper we study the difficulties resulting from changes in meaning of the minus sign, from an operation on numbers, to a sign designating a negative number, to the additive inverse of an algebraic symbol on students in two-year colleges and universities. Analysis of the development of algebra reveals that these successive meanings that the student has met before often become problematic, leading to a fragile knowledge structure that lacks flexibility and leads to confusion and long-term disaffection. The problematic aspects that arise from changes in meaning of the minus sign are identified and the iconic function machine is utilized as a supportive strategy, along with formative assessment to encourage teachers and learners to seek more flexible and effective ways of making sense of increasingly sophisticated mathematics.