Math and Abstraction

Because I am considering a career in education, and because my university background is in math, I spend a lot of time reading and thinking about teaching math. One of the common themes I read about on the web regarding math is students' (and even teachers') confusion about what math is, and what it's for. This is especially true for more advanced math, such as algebra. Even in daily conversations about education, I often here people say things like, kids don't need to study math past arithmetic because they'll never use it, or, they will just use calculators when they're older. Statements like these show a fundamental misunderstanding of what math is.

Math, as a subject, is the study of patterns and relationships. The processes of math, such as multiplication, algebra, and geometry, are actually the tools of math. We use the tools to help us better understand the real subject matter of math, and it is important to understand that these tools are abstract--they deal with patterns and relationships only, not with the things exhibiting these properties.

Consider the following two problems from multiplication.

1. For a birthday party, you need 3 chocolates for each guest, and you have invited 5 people. How many chocolates do you need to buy?

2. Your bedroom is 3 meters by 5 meters. How much carpet do you need to buy to re-carpet your floor?

Both problems deal with the numbers three and five. However, to know that we are dealing with multiplication in both situations, we have to understand the underlying patterns. In the first case, we have 5 sets of 3 chocolates. In the second case, we have 5 sets of 3 square meters (this case also requires a conceptual understanding of measurement of area to understand what is really happening). The key in both cases is that we have a fixed number of units, which we deal with as a set, and make a bunch of copies of the set, and then count up how many units are in all the sets combined. Although these situations are quite different (one deals with physical objects, the other with arbitrary divisions of space), the relationship between the quantities is the same. By focusing on the relationship, rather than the entities, we can deal with many different problems in a single, unified manner. Multiplication is the process we can apply to find the total quantity of unites when we want to copy sets.

It might be argued that children do not need to understand such abstractions. Regarding this objection, when children are first learning multiplication, not only do children not need to understand the underlying abstraction, they should not be taught it in any way. They should always start with a specific, concrete situation which they can preferably work with using physical media. This is vital for them to get an idea of what is really going on. Once they have firmly grasped how multiplication works in one concrete situation, they can be introduced to another situation where multiplication applies. If they have truly understood the first situation, they will mostly be able to understand the second quickly, and I would argue that most children will automatically make at least a part of the connection at some unconscious or conscious level. However quickly or easily children learn the second situation, once they have mastered it they will be ready to learn the underlying abstraction. They must wait until this time because you can't make an abstraction from one thing. An abstraction, by definition, combines the essential characteristics of two or more entities or concepts. However, this step is also essential. If students do not make this integration, then in order to be able to handle just these two situations, they will need to memorize multiplication twice (or those parts which they weren't able to integrate themselves). Furthermore, in order to apply multiplication to new situations, they will effectively need to learn multiplication again. This is especially true when they start applying multiplication beyond the whole numbers. However, if they learn the abstraction, and learn to apply it (the next step in their development in multiplication), they will be able to apply multiplication in any situation with very little new learning being needed. Even subjects in multiplication which are traditionally very difficult, such as multiplying fractions, become simple once the underlying structure can be connected to the abstraction the children already hold. Learning to multiply fractions becomes learning what a fraction is and what it means, a task they will have to do anyway.

As with this example from multiplication, all of math is the study of relationships and the processes that can be used to manipulate and gain insight into those relationships. Math is essentially a subject of abstraction. It focuses on the underlying, fundamental characteristics and does not deal at all with the superficial features of its subject matter. Until teachers and students alike can grasp this key point, math will continue to be considered a difficult subject and those who manage to grasp the truth despite the confused teaching they received will continue to be considered gifted in math. The mathematically gifted of today are gifted only in the sense that a man with two legs is a gifted walker when compared to a man with no legs--he is an ordinary man among cripples.