Key Ideas about Algebraic Reasoning and Mathematics


Consider the following example of transforming ordinary arithmetic into an opportunity for algebraic reasoning. A teacher can extend a simple arithmetic computation (say, 5 + 7 = 12) into a class of problems (e.g., 3 + 7 = 10, 1 + 11 = 12, 5 + 9 = 14, etc.), from which, through student argumentation and teacher questioning, students might conjecture that ‘the sum of any two odd numbers is even’. In other words, a claim can be made about something that happens in general. Once such a conjecture is made, the teacher can extend this process by asking students to justify the conjecture, and to find more formal ways of writing it (perhaps by writing each odd number as a sum of an even number plus one). Once the conjecture has been established as true, it is called a generalization.

There are many other forms of algebraic reasoning. For example, students are reasoning algebraically when they are able to develop and express how many handshakes would be made if everyone in a group of arbitrary size shook hands once, or when they can interpret '=' as the expression of a relationship between quantities rather than a 'call for action', or when they can describe the amount of candies Mary has if she has 3 more than John and John has an unknown amount of candies.


As is now well established, reliable computational skill without understanding is impossible. On the other hand, computational skill requires practice. Our approach, as reflected in the activities included here, combines building mathematical understanding with skill development by providing opportunity for repeated computations in the context of finding and generalizing patterns and other mathematical relationships. Further, by embedding the arithmetic in interesting and challenging problem situations, students develop strong number sense because their calculations are purposeful, related to one another, and related to other knowledge that they are building.