Necessity of algebraic symbolism

Question

We solve different problems algebraically .For example,if we add $20$ with a number and the sum is $42$.What is the value of the number.To solve we denote the number as $x$ and write like this

$x+20=42$ or $x=42-20$ or $x=22$

My question is that why we use symbol and use algebraic method to solve this kind problems instead of manual method? A student of 6th grade solve the problem in the following way(manually) before learning algebraic rules.

He firstly subtract $20$ from $42$ and got the desired result i.e. he followed the same procedure as I did algebraically. So why do I use this symbol and how do I make him understand that my algebraic method is better than his method for these kind of problems?

Answer 1

Interesting question. Well, you can solve a problem either way. It is always good to find an answer just by looking at the problem.

What you do with your method is trying to denote what is going on in the world by making a model of the world. You make some simple world in which x + 20 is always equal to 42. In that world, you can only find the answer 22. If you have a more general world, let us say: x + y = 30, you can have a lot of solutions. You can take 10 and 20, 15 and 15, 29 and 1, ... .

What you always want to do when solving a problem is making a mathematical description for it, to solve the problem. For new fresh math learners it might be hard to grasp those systems, since the true insight into mathematical systems only really becomes apparant when you learn to solve harder systems such as in linear algebra courses. There the student really begins to understand the notion of a model an really knows that what he is doing is actually making an abstraction of the world by describing his problem in symbols.

Your way is a standardized way to solve problems. You should show him why your method makes really sense. Show him that to solve x + 20 = 42, you actually subtract 20 from both sides to find the solution, so you do, x + 20 - 20 = 42 - 20, which gives the solution at the end, namely x = 22. Show him that this rule makes sense. 6 = 6, is the same as 6 - 3 = 6 - 3, etc. Then he knows that what you are doing is a valid rule, and that you can use also symbols to solve a problem. In the simple case, I would prefer his method of finding a solution, however, when it becomes more hard, it can be better to use your method, thus a systematic real algebraic method to solve the problem. Since methods as guessing or seeing it immediatly will become harder the more advanced problems will become. But: always show why the rules you give them, make sense, and then he will accept to follow those rules, I am sure about that. :)

Answer 2

I think you should ask René Descartes.

Just a short background

In the age of the Pythagoreans it was discovered that the diagonal of a square is incommensurable with its side. This means that the ratio between them (known as $\sqrt 2$) can never be expressed as a ratio of "numbers" which back then only included whole numbers. After that point the world of geometry and the world of numbers were regarded to be separate fields of study that only occasionally coincided, with the world of geometry being richer than that of numbers.

In 1637 Descartes set out a course eventually leading to a reunion of the two subjects. He introduced arithmetic manipulations using algebra to deal more freely with geometry, instead of being limited to tedious geometric constructions. He introduced the convention of using $x,y,z$ as unknown and $a,b,c$ as constants etc. Today we merely regard $x,x^2$, and $x^3$ as powers of numbers that can be plotted geometrically as graphs of functions, but in the $16^{th}$ century those were regarded as a line, a square, and a cube.