Mathematics for Teaching Algebra When is it algebra and when is it arithmetic?

## When is it algebra and when is it arithmetic?

In the post Algebra vs Arithmetic, I distinguished between arithmetic and algebra by arguing that it has nothing to do with the use of letters. That algebra is about letters and arithmetic is about numbers is an oversimplified view of algebra and can create misconceptions. Here are more ways of characterizing algebra.

The following excerpts is from Paper 6 – Algebraic Reasoning of Nuffield Foundation Math Education Report. The review of math education studies in the project was written by Anne Watson. The following is her summary of about algebra and algebraic thinking. I hope you find it useful in your teaching. It’s good to be aware what we need to focus on when we are teaching algebra.

• The focus of algebra is on relations and not calculations; the relation a + b = c represents two unknown numbers in an additive relation, and while 3 + 5 = 8 is such a relation it is more usually seen as a representation of 8, so that 3 + 5 can be calculated whereas a + b cannot.
• Students have to understand inverses as well as operations, so that finding a hidden number can be done even if the answer is not obvious from knowing number bonds or multiplication facts; 7 + b = 4 can be done using knowledge of addition, but c + 63 = 197 is more easily done if subtraction is used as the inverse of addition. Some writers claim that understanding this structure is algebraic, while others say that doing arithmetic to find an unknown is arithmetical reasoning, not algebraic reasoning.
• Some situations have to be expressed algebraically in order to solve them, rather than starting a solution straight away. ‘I am 14 and my brother is 4 years older than me’ can be solved by addition, but ‘My brother is two years older than me, my sister is five years younger than me; she is 12, how old will my brother be in three years’ time?’ requires an analysis and representation of the relationships before solution. This could be with letters, so that the answer is obtained by finding k where k – 5 = 12 and substituting this value into (k + 2) + 3. Alternatively it could be done by mapping systems of points onto a numberline, or using other symbols for the unknowns. ‘Algebra’ in this situation means constructing a method for keeping track of the unknown as various operations act upon it.
• Letters and numbers are used together, so that numbers may have to be treated as symbols in a structure, and not evaluated. For example, the structure 2(a + b) is different from the structure of 2a + 2b although they are equivalent in computational terms.
• The equals sign has an expanded meaning; in arithmetic it often means ‘calculate’ but in algebra it more often means ‘is equal to’ or even ‘is equivalent to’.

If you ask your students to evaluate 3a+2b-c for values for example a = 3, b = -1 and c =5, do you think they are doing algebra or plain arithmetic?