People often talk about learning math, but what does it really mean to learn Math?

In today's world of performance indicators and metrics we tend to define our mastery of a subject by our ability to pass a test on it. A half a century ago a guy called Bloom from the University of Chicago established a taxonomy of learning objectives which the academic community has incorporated in the measurable SLO's (Student Learning Outcomes) that are the basis for every course we teach. Interestingly enough, "understanding" is not one of the action words we are allowed to use since there is no objective way to test it, even though many Math teachers feel that is exactly what we want to achieve.

I think there are two components to learning math.

- Learning the language of algebra. Algebra is a language with its own grammar and vocabulary. We test your mastery of this language the same way a Spanish teacher tests your mastery of Spanish. First we give you vocabulary words and simple sentences and have you translate them back and forth. For example, you would translate words like "To find the length of the border of a rectangle you add up the lengths of the four sides" into the formula "P=L+W+L+W"
Next you manipulate sentences like: "He goes to the store and she goes to the store" becomes "They go to the store."

In algebra we would say: P = L+W+L+W = 2(L+W)Ideally we want you to be thinking in Spanish or in algebra, but you can probably pass an objective test even if you just keep translating back and forth. In other words, to use Bloom's terms, we can't measure whether you actually understand Spanish or algebra, we can only measure your ability to translate it into another language. Incidentally, this can be a real bummer for foreign students who are trying to learn Spanish in a class with English speaking teachers. There are other challenges to learning a language, such as when a concept can not be translated. For example, when a Magyar speaking person, for whom all pronouns and adjectives are gender neutral, has to learn to use gender based pronouns or adjectives in English or Spanish.

- Once you have acquired a basic facility in a language then you want to use it. For example, in Spanish, you might try to read Don Quixote, by Miguel de Cervantes Saavedra. But what if you are a person who prefers prose to poetry. Or what if your objective is conversation with your Mexican friends. In other words Spanish can be used for many different purposes - each of which has its own vocabulary and grammar and rules.
Just so in mathematics, a treatise written for a person who is concerned about structural engineering may be totally unreadable to a person who lives in the world of algebraic topology. In fact, the word algebra in itself has a totally different meaning once you are at a level where elementary Algebra metamorphoses into abstract Algebra ( http://en.wikipedia.org/wiki/Algebra). The two are as different as a caterpillar is from a butterfly. There are other branches of mathematics, like the study of algorithms, category theory, and geometry, none of which really look like algebra either.

So, you need to decide, why are you studying Math? Do you just want to pass a test, or are you going to use it to communicate ideas?

You can find other perspectives about what it means to learn math here Lockhart's Lament

If you are interested in reading about advanced topics in Mathematics, here is a paper co-authored by the uncle of folk singer Joan Baez: categories