2 Finding the general in the particular
Early algebra involves moving between the particular and the general.
When we talk about moving to the general in mathematics (also called generalising), we are concerned with finding features of a situation, thinking about how they may change and extending our reasoning to cover those changes. This has many similarities with our aims for learning – we want our learners to generalise beyond today’s lesson.
You have seen that algebraic thinking involves working with unknown quantities, so you can think of this movement in two ways.
First, teachers and learners can choose different representations for the unknown quantities. Some representations are closely linked to their own particular classroom experiences, while others draw on generally accepted conventions for symbolic notation.
Second, the indeterminate quantity being represented can be thought of as having a specific (but unknown) value or it can be thought of as a variable, capable of taking a range of values. Mathematical statements about a specific unknown are likely to lead to finding its value; whereas statements about a variable say something more general, since they must hold for any of its possible values.
This section looks at these two movements, starting with representations.