From my years as both a math professor and observer of people, I’ve come up with a list of hurdles for mathematical thinking. These are what I believe to be the essential set of skills a person must have if they want to understand and do mathematics. They don’t need to have all these skills to use mathematics but would need most of them if they want to progress far in mathematics. Identifying what sorts of conceptual barriers people may have could help in improving mathematics education.
I’ll first give the list and then explain what I mean by them.
1. Context dependent rules
2. Equivalence classes
3. Limits and infinitesimals
4. Formal logic
The first skill is the most important and is also necessary to be successful in life. In order to succeed, a person needs to understand rules and know when to apply them. Hence, to add fractions a person must first recognize a fraction and then remember or derive the rule to add them. Most people intuitively know that different social situations call for different codes of behavior but they don’t all extend that to mathematics. I think if people just realized that the key to understanding mathematics is simply the ability to follow instructions, less people would be math adverse.
The second skill is to understand the concept that sometimes things are the same even though they don’t look the same. A few years ago I was at the delicatessen counter at my local supermarket and I asked for point two pounds of turkey. The server told me that he could not give me point two points but he could give me point two zero zero pounds. I tried to inform him that 0.2 and 0.200 were the same thing but he looked at me as if I were a complete idiot. I don’t have the data but I would venture that a significant fraction of the population does not understand that 1/2 is the same as 2/4. It may seem completely intuitive to you but the abstract notion of an equivalence class is something that must be learned. It is also something that is absolutely necessary to understand mathematics.
I think mastering these first two skills will get a person through high school and much of university mathematics. However, if they want to study pure mathematics then they’ll need to understand the concept of the limit. While most people if they worked hard enough could probably acquire the first two skills, skill three may be the dividing line between whether or not they can ever truly understand mathematics. Many people may be able to take derivatives, compute limits and so forth but never really understand what they mean. It’s one thing to apply l’Hopital’s rule and quite another to understand the epsilon-delta definition of a limit. All of modern mathematics requires a gut level understanding of the limit properties of sequences and the difference between open and closed sets.
Although it would seem that skill 4 would be essential for life and doing mathematics, I think that you can actually get quite far in life without ever being completely logical. As I blogged before, most people use Bayesian plausible reasoning and not formal logic. I would venture that the majority of people do not know that the only conclusion you can draw from A implies B is that not B implies not A. Many people would make the mistake of thinking that not A implies not B. However, making this mistake is not all that costly. I also think that being completely logical is not even that necessary to do physics at a high level. As Feynman talked about in his “The Character of Physical Law” lectures, physics and math are very different. The ability to carry out calculations, which relies more on skills 1 and 2, is more important for physics then applying the rules of formal logic.
Finally, abstraction is important on two levels. The first is simply recognizing that things and concepts can be represented symbolically. I think most people get this since the use of metaphors seems to a major component of how we think. The second is the ability to crystallize a key component of a system and study that in isolation. This is a far more difficult skill to acquire and may be what separates strong mathematicians from average ones. It took mathematicians a long time to realize that arithmetic could be generalized to more abstract algebraic structures. Much of twentieth century mathematics is so abstract that it is very difficult to even explain the basic concepts to a layman and some have complained that abstraction is sometimes taken too far. The ability to abstract and generalize, while essential to create new mathematics, is probably not that important to understand or even teach mathematics.
So these are my five hurdles to mathematical thinking. I’m sure others will have a different opinion so I would like to hear them.