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Easy. Simple. Fun.
Math.
Can all of these exist together?
You probably don't 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, and fun, and at the same time allow students of all ages to learn by themselves?
First, let me tell you about the following topics:
The basic problems that students have with math.
The nature of elementary, middle, and high school math education.
How I combine points 1 and 2, and use my understanding of math and my experience of teaching to
prevent the problems in #1. I say prevent and
not solve, since with my books, the problems do not arise, and therefore there is nothing to solve!
The problems that Students have with Math
1. The number one problem students have with math is the lack of a complete and concise set of instructions. Students are not provided with clear, step-by-step procedures of how to solve a given type of problem. Unless the student can figure the missing or unclear part by herself, which only a few with natural mathematical intuition can, she is left puzzled and confused.
- A related problem is that students are taught topics for which they lack all the required skills and knowledge for that topic. Therefore, even if the instructions are clear and complete, they will not help the students. An example is teaching the addition and subtraction of fractions before teaching multiplication and division. Adding fractions requires finding a common denominator, which requires the multiplication (or division) of fractions.
2. The second problem is of generalization. All books and teaching materials make do with a limited set of examples that do not cover all the possible variations of a problem. However, only a few students are able to intuitively grasp the underlying principles of mathematics, and thus generalize the (often incomplete) procedure to the "new" type of problem. When faced with a similar but not the same exercise, most students freeze, and do nothing. They might complain that they haven’t learned how to do this type of exercise, and the teacher might claim that they do. In fact, BOTH are right! The teacher did teach it, but the students did not learn it. This is a common problem. It is not about being "smart" but about having mathematical intuition, which is not unusual.
3. The third problem is the mental block. Students are not "born" with a mental block regarding math. The mental block is created when some students face an incomplete and inconcise set of instructions, and get stuck, while other students with an intuitive understanding of math find the solution. Even if the teacher does not make an explicit comparison, the students themselves will. The conclusion reached is, "I don't get math" or "I'm a math dummy". After several such experiences, students start believing they are "dummies" or "just not good at math", which becomes a self-fulfilling prophecy.
4. 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. Many approaches try to use games to make learning math fun, but it is revealing that only good math students seek out these games. The same reasoning applies: none of us enjoys playing games we are not good at, even if it is "just a game". Only when mathematics itself is easy and simple we could consider it fun, regardless of whether it is in the form of a game or not.
5. These problems lead to a lack of deep understanding. As had been found when TIMSS results were analyzed (see www.timss.org), technical skills affect the ability to deeply understand mathematics. The reason is simple. When students cannot see the path to the solution, they cannot understand the more abstract points about the subject. Only when the "How" is known, can the "Why" be discussed. Both are important, but one is a necessary condition for the other; the "How" is necessary for the "Why".
6. Lack of Motivation. Traditionally, mathematics is studied without motivation as to why we should learn it, and what it is useful for. Given the difficulty arising from the above problems, few students have the motivation to study mathematics.
The Nature of Math taught in School
None of the math taught in school, elementary to high school, is hard. By "not hard", I mean that students do not need to find NEW, unknown methods. Solving math problems can be difficult, in the sense that following the instructions is not that easy and requires time and practice (much like shooting in basketball). However, all questions should be answerable using the tools that the students have 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 should be able to program your DVD player to record a show 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. You might even have to invent a solution method. It took many mathematicians, who had an intuitive sense of mathematics, many centuries, millennia even, to come up with the mathematical knowledge and skills that a high school student is expected to graduate with.
My Approach
My approach uses simple principles to prevent the problems described above and to help make math easy (not hard), simple (not complex), and fun to practice (not frustrating). My principles are the following:
Be complete and concise. I provide a complete set of instructions at the beginning of each topic. Everything is clearly outlined and explained in simple language using simple sentences and step-by-step procedures that are easy to follow.
The next smallest step. Using my deep understanding of math, I break down each topic to its ingredients and create a skill map. This skill map outlines the progression of skills and requirements for different topics. When I present something new, I advance only by the smallest step possible. If the new step requires a modification of the original solution method, I explicitly present the entire modified method, not just the modification.
This practice ensures that all the necessary skills are being studied. If a skill is outside the scope of the book I say so.
Provide all possible examples. To prevent the generalization problem, I provide examples for all possible variations. This way, when students reach the practice questions, they can draw on an example to apply the procedure they have learned. There is no need to guess how the "generalized" procedure works in the specific exercise since the example will show it explicitly.
Be spare. I do not put too much information on any given page. A page has at most one new method and a few examples. Students are not flooded with information and therefore can manage the information flow, and the relationship between the examples and the methods taught.
Focus on each topic. Each book is focused on a specific subject or topic.
Explain the how and why. For each topic I first explain the method. Then I explain the mathematical principle that makes this method work.
Motivation. For each subject, I provide motivation: the reasons why this subject is important and how that knowledge can be used for practical problems we face in our life. I try to make the problems applicable to kids…
The result is a thorough, complete, simple, and easy to follow book.
These qualities make Math fun, since we have fun when something is easy.
Especially if at first, we thought it was not!
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