Numbers are great for illustrating how things work. Add, subtract, multiply, divide, raise to a power, etc., you can’t fool anyone with numbers. If it can be done with numbers then it must be easy. On the other hand with numbers you can’t see structure. If the number 7 appears in some calculation you won’t necessarily know what it means. And is it the same “7” each time it appears? Maybe one is an interest rate and another is a maturity. To get to a deeper level of understanding you need symbols.
Symbols are great for showing structure, abstraction is always necessary if you are to go beyond mere arithmetic. The problem with symbols is that some people are frightened by them. And if you and I are used to using different types of symbols it may take some time before we fully understand each other. One could even be accidentally or deliberately confusing, throw in a symbol without a proper explanation and before you know it everyone is lost.
This is relevant to the teaching of mathematics in schools. People can become terrified of the subject at an early age if taught badly, with the result that they are probably forever lost. (Unless it’s possible to get therapy?) How often at dinner parties have we mathematicians heard the ever-so-original response to what we do for a living “I was terrible at maths at school, me!”? I read recently that the part of the brain that does maths is right next to the part that registers fear. I don’t know whether it’s true but it certainly makes sense.
I am forever hearing politicians wittering on about how maths education in schools needs to be made more fun, and more, what’s the word? Practical! Misguided fools! Not a single GCSE maths above grade D among them. The point of mathematics is that it is supposed to be abstract. If all your maths comes from counting apples then you are going to be stymied by the real thing. Mathematics is abstract, that is the beauty of it. And that’s what actually makes it fun. Teach mathematics properly, don’t terrify children by asking them how long it takes ten politicians to dig themselves into ten holes, explain to the young the beauty of the abstract.