### Why use algebra when you can use arithmetic?

This post was jointly written with my colleague, Steven Francis. It follows from my earlier post, Welcome to Maths class: Leave your common sense at the door. In this post, we provide examples that highlight the importance of allowing students to solve problems from first principle. At times, insisting on specific formal methods can diminish students' ability to solve problems.

In the video above, the renowned physicist Richard Feynman relates how he could solve linear equations early but was told "You did it with arithmetic. You have to do it with algebra."
Feynman then reflects with visible annoyance:
There's no such a thing as you don't do it by arithmetic, you do it by algebra. It's a false thing that they had invented in school so that the children who have to study algebra can all pass it. They had invented a set of rules which you follow them (sic) without thinking to produce the answer.

As teachers who do instruct students in these methods, our observation has been that Feynman was right! We often see some students' ability to solve problems diminish as they learn more and more formal curriculum Mathematics.
The following problem comes from a year 9 book. Year 7s find it much easier to solve, especially before they have been taught algebra!
In 1974, a mother is 6 times as old as her daughter. The mother turned 50 in the year 2000. In what year was the mother double her daughter's age?
Given a few minutes to think about it, year 7s will reason in the following way:
• In the year 2000, the mother was 50 => she was born in 1950
• In the year 1974, the mother must have been 24 => her daughter was 4 years old
• The mother is 20 years older than her daughter
• When the daughter was 20 years old, the mother was 40 (1950 + 40 = 1990)
• Answer: The mother was double her daughter's age in 1990.
Year 9 students, on the other hand, tend to reason as follows:
• This question is an application of algebra => I need to define pronumerals (The word used in Australia for a variable that represents a specific unknown number)
Let x = the daughter's ageLet y = the mother's ageIn 1974, y = 6xIn 2000, y = 50 but we don't know what x is then!Okay, I am stuck!
We are not suggesting that students should not be taught any formal methods. We are suggesting that they should be encouraged to explore a variety of ways to represent and solve problems. Algebra is the science of abstraction and, before working in the abstract, students need to explore concrete ways of thinking. Future posts will change tone and present concrete activities that can help achieve this goal.
If you have personal experiences or teaching ideas that can help students solve problems from first principles, please share them here.