Despite statements about algebra in national curricula and textbooks, algebra begins with the baby in the cot. When a children are experimenting with sounds and with actions, when they are investigating the dynamics of their mind (of course not intentionally or consciously, but ‘embedded in’ their experimenting) (Gattegno 1973).
When children burst into speech, they are generalizing, because to use a word correctly requires an implicit sense of when it is appropriate and what it means, independent of the specific situation.
As soon as learners are expected to develop a ‘method’ for doing arithmetical tasks, they are expected to generalise. Thus you cannot learn arithmetic without generalising, which means without beginning to think algebraically even if you don’t use symbols to express those generalities [see
Generalisation,
Mathematical powers].
School algebra is about developing a manipulable language for expressing generalities about numbers and their properties, making it much easier to
conjecture & convince that it would be using only words and pictures.
The manipulation of algebraic symbols is sometimes referred to as ‘arithmetic with letters’ and as ‘generalized arithmetic’ [see
what is a number?]). Traditionally algebra has been introduced as ‘manipulating letters as if they were numbers’, which lacks motivation [see
purpose & utility] and efficacy and suggests that there are a lot of rules to be learned in order to ‘do algebra’. Another approach is to prompt learners to express generalities as early as possible, and gradually to move to more and more succinct expressions, often in situations in which there are
multiple expressions for the same thing. Learners then develop the wish to work out how to transform one expression into another, and they can draw upon their knowledge of arithmetic in doing this. Thus algebra emerges as a language which can be used to express and to manipulate generalisations.