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Euclid once told the Hellenistic Pharaoh Ptolemy “There is no royal road to geometry”. Math is one concept built upon another concept built upon another concept, et cetera, et cetera. A student of mathematics must walk the path, learning each concept along the way. When one concept is misunderstood it becomes difficult to build upon. This is why it only takes one bad math teacher to end a good student’s career in math. Even a few sick days can impede someone’s progress.

Sitting one on one with a student, a good tutor can watch the mistakes made and recognize which concepts the student has developed a poor understanding. This is called “filling in the holes”. As the student/mentor relationship develops a good tutor will learn how an individual student prefers to approach a problem and present explanations tailored to their conceptions. The two most obvious are the algebraic, step by step, process approach and the geometric, diagrammed, conceptualized approach. Most people fall into one of these two mindsets and generally one will excel in algebra and hate geometry while the other will love geometry and loath working through the algebra. Lastly, a good tutor can watch the expressions of the student and recognize when the proverbial light bulb turns on before moving on to the next subject.

When I draw a “3” on the board and ask if it is a number, people give me funny looks. The answer is no, “3” is not a number. It is a symbol that represents a number. When we as English speakers see “3” we think of a pattern we know as three. But when we see γ or ۳ we do not recognize them as three. The first is Greek and the second is Arabic. All these are numerals or symbols that represent numbers. Once people are freed from the rigid numerals numbers are easier to manipulate.

Mathematics is the study and description of patterns. Many people believe that math is the study of numbers. Numbers are so rich in patterns that they are necessary to describe other patterns. The description of patterns must be rigorous to qualify as mathematics. However, descriptions in simple language are sufficient to practice the art of mathematics.

Mathematics is the only art in the school of Natural Sciences. All other arts fall within the school of Liberal Arts. People think of mathematics as a science because each well posed problem has only one solution. However, from each question in math there are many different paths to the correct answer. One should choose the path that is most enlightening to their own mindset because everyone frames their own perceptions of basic mathematical patterns differently.

Most people look at this diagram and notice that there are many paths from a math question to the correct answer. Some paths are shorter than others and some are long and convoluted. One should avoid the long path because it takes a long time, it can lead to mistakes, and the long path usually isn't very enlightening.

The most interesting thing about having different paths to the answer is that the answer can be perceived in many different ways. For example, if the answer is “five” one can approach the answer as: 2+3, 1+1+1+1+1, half of ten, or any number of different ways. This may seem trivial in many regards, but it is how one can know they have the correct answer. Someone can look at a pencil, pick it up and spin it around, and say “yes, this is a pencil”. If you can get an answer using one approach, stop and look from another approach and get the same answer, then the answer is probably correct.