### Teaching Philosophy

Easy Simple Fun Math

All of these together?

You probably do not believe it.

Well, Math should be all those things but the way it is taught makes it not easy, not simple, and certainly not fun. After all, who enjoys a hard frustrating subject?

So how do I make Math easy, simple, fun, and at the same time allow students of all ages to learn by themselves?

To understand that, I have to explain the following:

- The problems that I have found students have with math.
- The nature of elementary, middle, and high school math.

How I combine points 1 and 2, and use my deep understanding of math and my experience of teaching at many levels to prevent the problems in #1. I sayprevent and not solve since with the books the problems do not arise, and therefore there is nothing to solve!

Problems that Students have with Math

The number one problem is the lack of a complete and concise set ofinstructions. Students are not provided with clear, step-by-step, procedures of how to solve a given type of problems. Often, the procedure is given only through an example, or is missing explicit explanation of all the steps. Therefore, students have to do with unclear and incomplete instructions. Unless the student can figure the missing or unclear part by herself, which only the few with a natural math intuition can, she is left puzzled and confused.

A related problem is that students are taught topics for which they lack all therequired skills and knowledge. Therefore, even if the instructions are clear and complete, those are not so from the student’s perspective. An example would be that addition and subtraction of fractions is taught before multiplication and division, although addition requires finding a common denominator, which requires multiplication (or division) of fractions.

The second problem is of generalization. All books, and teaching materials, make do with a limited set of examples that does not cover all the possible variations on a problem. However, only few students are able to intuitively grasp the underlying principles of math, and thus generalize the (often incomplete) procedure to the “new” type of problem. When faced with a similar but not the same type of exercise, they freeze, and do nothing. Often you would hear them complain that they have not learned that type of an exercise, while the teacher would claim they did. In fact, BOTH are right! The teacher did teach it, but the student did not learn it. This is a common problem and is not about being “smart” but about having mathematical intuition, which is not common.

Example: let us say that a student had learned how to multiply negative and positive integers (such as ), and learned how to multiply positive fractions (such as ). However, when faced with the following exercise, , the student would argue that s/he does not know how to do it and have not learned it. The student has not recognized the generalization of the two types of exercises already learned.

The third is the mental block. Students are not “born” with a mental block regarding math. The mental block is created when students are facing an incomplete and inconcise set of instructions and at the same time, a student who has intuitive understanding “jumps” with the solution. Even if the teacher does not create an explicit comparison, students themselves make that comparison. The conclusion is then “I don’t get math” or “I’m a math dummy”. After several such experiences students start believing that they are “dummies” or “just not good at math” and it becomes a self-fulfilling prophecy.

The fourth problem is practice. None of us would expect a basketball player to be any good without practice, and to be really good he would need a lot of practice. However, students do not like to practice math for a good reason. It is not fun to practice something for which you were not given all the instructions, and therefore is hard for you. To counter that, many approaches try to use games to make it fun. But it is telling that only good students seek those math games. The reason is that none of us enjoy playing a game we are not good at, even if it is “a game”. Only when the Math itself is easy and simple for us, we could consider it fun, regardless of whether it is in a form of a game or not.

Resulting from those problems is a lack of deep understanding. As had been found when TIMSS (see http://www.timss.org) results were analyzed, technical skills affect the ability to deeply understand math. The reason is simple. When students cannot see the path to the solution they cannot understand the more abstract points about the subject, since they are busy trying to figure out the way. Only when the “How” is known, the “Why” can be discussed. Both are important, but one is a necessary condition for the other; the “How” is necessary for the “Why”.

Lack of Motivation. Math is studied “as is”, usually without providing motivation to why a certain topic should be learned and what is it useful for. Given the difficulty arising from the above problems, few students have the motivation to study it. Although, some of the math studied in school is not really needed for anything most of us would encounter, but that is a much bigger and difficult issue to solve.

The Nature of Math taught in School

ALL the math taught in school, elementary to high, is not hard. By “not hard” I mean that it does not really require students to find NEW, unknown, methods. At least, it should not require it. It can be difficult, in the sense that following the instructions is not so easy and requires time and practice (much like shooting in basketball), but all the questions are answerable with the tools students studied.

One can compare school math to programming a DVD player to record a TV show. If you are given a complete and concise set of instructions, and if you do not have a mental block, you would be able to execute it successfully, and even easily. However, if the instructions are not complete, or not concise, or you lack some of the required knowledge, then the problem becomes Hard, not just difficult. Because, faced with this incompleteness, you do have to “figure it out” and in fact, invent solution methods. And after all, it took many mathematicians, who are people with intuitive sense of math, many centuries (millennia actually) to come up with the math knowledge and skills a high school students is expected to graduate with