Algebra vs Arithmetic Thinking

Algebra had always been associated with high school mathematics while arithmetic, the study of numbers, is associated with elementary school mathematics. One of the solutions to help students understand algebra in high school is to start the study of algebra earlier hence the elementary school curriculum incorporated some content topics traditionally studied in high school. However, I believe that more than knowledge of additional content, pupils can best be prepared for further mathematics work by engaging them in activities in deeper and more challenging ways using the traditional content of elementary school mathematics. I believe that children who become familiar with algebraic thinking from an early age and in meaningful contexts will do better in mathematics.

There is this study which I read in this paper titled A cognitive gap between arithmetic and algebra. This study distinguished algebra and arithmetic in terms of the type of equation tasks. According to them if the equation only involves one unknown then that is an arithmetic task. If the equation involves two unknowns then it is an algebra task. For example,

(1) 15 + ____ = 40 is an arithmetic task while

(2) ____ = 4 + _____ is an algebra task.

This distinction, in a way, makes sense. To answer Equation (1), a child only need to ask: What number should I put in the blank so that when I add it to 15, it gives 40? Equation (2) involves the concept of a variable. There are infinite values that you can  put in the two blanks. It also involves  the concept of function, the relationships between two numbers. The two numbers in the blanks cannot be just any number. The two numbers must differ by 4 and the number in the first blank should always be the greater number. This relationship is a “very algebraic” concept. But, even then, I’m still not very excited about this distinction between algebra and arithmetic!

I believe that one is engaged in algebra when one thinks relationally. Equation (1) for example is not necessarily an arithmetic task. If a student solves it by reasoning “because they are equal, even if I subtract 15 from both side of the equality sign then I still maintain the equality then he is doing algebra.  Another solution to Equation (1) is to express 40 as 15 plus another number, i.e., 15 + ___ = 15 + 25. This may be a simple solution but it involves another very important principle: Since the quantities on both sides of the equal sign are equal and 15 is equal to 15, then the blank must be equal to 25! This is algebraic reasoning! As for Equation (2) even if students can generate hundreds of correct pairs of values, if they cannot see the relationship between the two numbers that goes to the blanks then they are not yet engaging in algebra. So it is not so much the task or the problem but the solutions we use to solve it that could tell whether we are doing algebra or not.

What is algebraic thinking?

Algebraic thinking in working with numbers as described by Kieran is characterized by a focus on relation between numbers and not merely on the calculation; a focus on operations and their inverses and on the related idea of doing/undoing; a focus on both representing and solving problems rather than on merely solving it; a focus on the meaning of equal sign not as a signal to perform operation but as denoting equivalence.  Algebraic thinking involves deliberate generalization, active exploration and conjecture (Kaput, NCTM, 1993) and reasoning in terms of relationships and structure.

I suggest you also read Prof Keith Devlin What is Algebra?

More activities about algebraic thinking: