When children ask why they need to study math, the answer usually has something to do with either daily life or applications to science and technology. The problem with the first motivation is that it is an obvious and transparent lie. The second type of “motivation” tends to have the opposite of the intended effect.

The “daily life” explanation tells students that they will need math for their daily activities. For example, they will need to calculate the tip in a restaurant, or determine how much they should pay for their groceries. Most students are quick to point out that this problem can be solved by carrying around a calculator. And anyone who is worried about running out of batteries can carry around a spare set of batteries, or even two calculators. Even cell phones have built in calculators.

The arguments against the “daily life” explanation continue. In daily life, you never need to do more than add, subtract, multiply, or divide. Why learn trigonometry? Why study calculus? Why do anything beyond arithmetic? Even math-oriented jobs rarely require any really advanced math. When I worked as an actuary, the only math I used on the job was multiplication and the occasional exponent (the actuarial profession is one of the most math-oriented professions in the world.)

The other rationale for studying math focuses on science and technology. We need math to design space shuttles and satellites, to work in laboratories, and to build the newest computers. In one way this argument makes sense. Much of that work requires intensive use of advanced math. But very few people work in those areas. Those that work in those areas usually do so because of an internal passion, not because of any external motivation.

In fact, from the perspective of most students, there is very little external motivation to be a scientist. The strongest external motivators for most teenagers are money, fame, power, popularity, and attraction to the opposite sex. None of these powerfully motivate students to pursue careers in science. For every million dollars a scientist makes, the businessmen for whom he works make a billion. For every famous scientist, there are a thousand famous musicians and actors. The scientists who made the nuclear bomb were not the ones to use it; that power belonged to politicians. And in American culture, scientists have no more popularity or sex appeal than anyone else.

Thus, this argument not only fails to motivate students; it actually does the reverse. A student with no interest in being a scientist who hears the technology argument now thinks that advanced math is useful only for scientists. Thus, he does not need to learn it. If his goal is personal gain, his time is better spent doing almost anything else – studying politics, learning to play the guitar, working out, or thinking of ways to make himself rich. Math becomes just an annoying requirement.

So then why should a student learn math at all? (close)

The belief that Asians are good at math is held with good reason: even in America, students with Asian parents tend to significantly outperform every other ethnic group. For example, 2005 math proficiency testing showed that Asian students had higher math proficiency scores than White, Black, and Hispanic students at all age levels (Source: Child Trends Databank).

This section examines the techniques used by Asian parents and educators. Of course, there are variations depending on the country of origin and the individual, but there are techniques and principles that are almost universal among Asian parents and educators.

The Asian system is built on memorization. At an early age, children are taught to memorize multiplication tables and the like. As they get older, they memorize formulas, and even memorize step by step ways to solve specific problem types.

The Asian system is radically different from current American methods, which emphasize understanding over memorization. Where American parents and math teachers focus on explaining why a technique works, the Asian educators simply require that the student memorize the technique, and be ready to use it.

One might expect that such a technique would create students who simply have formulas memorized and are unable to understand what they are doing. But the reality is just the opposite. Once students have the information memorized, the understanding seems to come naturally. On the other hand, systems that drop memorization and focus on only understanding seem to have the reverse effect. Students often end up confused – unable to understand the problem, or to solve it.

This is one of the strangest paradoxes in math education, one that I wrestled with extensively at the beginning of my career as an educator. Why does memorization work in math? Why does focusing exclusively on understanding fail? Isn’t math about understanding? Shouldn’t memorization be saved for history?

To unravel this mystery, we will undertake a journey that will help us understand some of the most important cognitive principles involved in math education. (close)

Situation 1:

You just started seventh grade, and it is the first math class of the year. The teacher explains something, and you notice that others in the class seem to understand the concepts faster than you do. You are able to solve the problems, but it takes you a bit longer than the rest. By the end of the class, you are starting to think that you just might be bad at math.

The next day confirms it, as the other students seem to race ahead of you. In reality, they are at best 5% faster, but from your perspective it seems like they are at least ten times better at math than you are. At this point (two days into the year), you are already pretty sure that you are bad at math. By the end of the week, you are absolutely certain. In fact, being bad at math soon becomes part of your identity. When the teacher gives a challenging problem, you do not really try that hard to solve it. That is just not who you are. You are the kid who struggles with math; what chance do you have of solving a hard problem?

You do your homework, but if you cannot get a problem, you are not too concerned. Athletes play sports. Rockstars sing. Your role in life is to get math problems wrong.

Every so often, you get some evidence that does not fit your theory. You get a 9/10 on a quiz that everyone else in the class fails. But you and the rest of the class are now so sure that you are bad at math that you decide that the quiz is flawed. Everyone laughs about it. How did the bad math student do better than everyone else on the quiz? Even your parents think it is kind of funny.

From time to time you get a math teacher that tells you that you can be an excellent math student, and that there is nothing wrong with your abilities. Sure, he is something of an authority. But you have years of experience, the opinions of other teachers, and the opinions of your friends on the opposite side. Obviously, you assume that the heretical math teacher is wrong and that every other person on Earth is right. (close)

...For the mind to have the incentive to develop, two things are necessary. First, it must encounter a problem that it is unable to do; the process of figuring out how to solve this initially unsolvable problem causes the mind to develop. If a student is only given problems at his current ability level, what incentive does the mind have to improve? Just as lifting a half-pound weight will not make a person physically stronger, doing an easy math problem will not make a person mentally stronger.

Parents and teachers of gifted students often overlook this, and just allow them to work at a comfortable pace. The result is that the gifted students never get the opportunity to realize their full potential. Like natural athletes who never train hard, they end up squandering their innate talents. (close)

If you ask the parents of an outstanding math student what kind of rewards they use to motivate their child to do his homework, for a split second they will look at you as if you asked what kind of reward their child gets for not wearing diapers anymore. They will then quickly recover and come up with some polite answer that sounds good. But that initial look tells you everything you need to know. Parents of outstanding math students only use incentives when the child is very young, and then only for a short period of time. The approach is almost like toilet training. When a child is first toilet trained, parents may give great approval, and even some small reward, to the child when he correctly uses the toilet. But very few parents cheer a 15 year old of normal intelligence when he manages to use a toilet correctly. By that age it is just taken for granted. Similarly, no approval is given when a child does his homework every day. It is just taken for granted.

So how do you get there? How do you make homework a permanent part of a child’s daily life? (close)

We learned in the last chapter that motivating a child to do his daily homework has more to do with self-perception than with incentives. However, to really bring out the best in a child, you will need to provide the right type of long-term incentives, in addition to helping him develop the right self-perception.

This is especially important for brilliant but lazy children who think outside the box. Because these students tend to look at situations objectively, self-perception does not have the same momentum with them as it does as with others. Thus, these students need the right objective, long-term incentives to bring out their best.

Why might a student not do his homework? Sometimes students look at the short-term gains from not doing their homework (immediate fun) and ignore the long-term benefits of strong cognitive skills. For other students, particularly for students with learning differences such as dyslexia and students with delayed intellectual development, the homework is such a painful process that they avoid doing it at all costs.

But in one of the most common types of situations, neither applies. The main details are usually the same. The child is raised in an upper middle class family, in a wealthy neighborhood. His parents are successful professionals, often doctors or lawyers, and he has seen the positive results of an excellent education. The large house in which he lives, the car in which he rides, the luxuries he enjoys resulted from educational success. If anyone on Earth should be convinced that education is important, it is this child.

As one looks more closely at the situation, it makes even less sense. The child is unusually bright, and his memory and processing speed are excellent. Even his attention span may be above average. This is not a case in which the child is overwhelmed by the homework, or unable to do it. Nor is the child too young to understand the consequences; typically the child is between 13 and 15. It seems as if the child just does not want to do any homework, so he does not do it. The parents (and teachers) are often tearing their hair in frustration, as increasingly elaborate threats and bribes fail to motivate the student.

Why would an intelligent, able student who obviously knows the value of an education not do homework? Why would he not bother to study for a test? Is he just short sighted? Does he just not see the big picture?

Surprisingly enough, the answer is often just the opposite! The child is not really thinking only about the short-term; in his own way, he is thinking about the long-term. (close)

This is a section about stereotypes, and as with all stereotypes, there will be many exceptions. However, understanding the nature of these differences will help you teach math to both girls and boys. You might find that “girl” techniques are appropriate for your son, or that “boy” techniques work well for your daughter. Or you may find that neither are completely appropriate for your child. However, there is a good chance that the gender specific techniques in this chapter will match exactly.

Take a boy and a girl in the same school, with the same grades, the same level of ability, and the same level of understanding. It would be no surprise to discover that the boy thinks of himself as excellent at math, and the girl thinks of herself as horrendous at math.

There are a variety of reasons for this discrepancy. Girls tend to attribute failures to permanent, internal qualities. For example, a girl who fails a test might think, “I failed this test because I am not good at math.”

This is an internal attribution. The girl blames herself (instead of some external factor, like a teacher or textbook) for the math failure. The girl may also think of the attribute as permanent. She believes that she is not good at math in the same way she might believe her blood type is B: it is a permanent and unchangeable feature.

The boy who fails the same test will be more likely to think, “I failed the test because the test was unfair/stupid/ridiculous. The next one will be better.”

The boy attributes the failure to the test itself, rather than to his own lack of ability. This is an external attribution. He also views the problem as temporary: “The next test will be better.” Thus, a failing grade does not affect his confidence as severely. (close)

Here is what you need to know about innate math ability:
1. People have varying natural math abilities. Some people naturally find math easy to learn. Others naturally find math harder to learn.
2. So what?

If you have a slow metabolism, you can either give up on being fit, or you can accept the fact that you have a slow metabolism and train harder than everyone else. If your child struggles with math, you can make excuses for him, and hold him to a lower standard. Or you can accept the fact that your child will have to work twice as hard as his peers, and that if he does so, he will be at the top of his class.

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What Does the sat test? To most people, the SAT Math section is a mystery. It only covers algebra and geometry; however students who have taken calculus often get several problems wrong. Students who get A’s in honors math classes sometimes do worse on the math section of the SAT than do students who struggle in regular math classes. In addition, the SAT math section does not cover any advanced topics; a student who knows basic algebra and geometry knows every concept and formula needed to solve any problem on the SAT. These basic concepts include the formula to find the area of a circle, the Pythagorean theorem, and how to factor a trinomial. There are no logarithms, no esoteric geometry theorems, and no trigonometry. In fact, most of the necessary formulas are actually given at the beginning of each SAT math section! (close)

Often, a child will do extremely well on math quizzes, but horrendously on exams. The student’s quiz grades may be the highest in the class, and yet he may be getting D’s on his exams. This is not due to a lack of studying – the student studies for days for each math exam, putting in far more effort than his peers. In fact, many of his peers do not study at all for math exams, and still do much better than he does. It seems that no matter how much he studies for an exam, he does horrendously. (close)

The most significant change to American math education during the last fifteen years has been the incorporation of calculators. As early as seventh grade, students are using graphing calculators as part of their math education.

You already know that for most students, math is valuable not for its potential applications, but for the cognitive abilities and reasoning skills it develops. For example, a student should study geometry so that he can develop his logical and spatial reasoning skills, not because he will one day need to decide whether or not two triangles are congruent. A student should learn to factor polynomials because of the cognitive skills that the task develops, not because he is likely to use this skill in any part of his adult life.

However, many teachers and administrators do not fully appreciate this fact. Instead, they believe that the goal of math education is to give students the ability to somehow get the answer to a large number of math problem types. Because calculators can often allow students to get the answer more quickly, many teachers teach students to solve problems using calculators. For example, rather than teaching analytical techniques for graphing, teachers just teach students how to graph using a calculator. In the absolute worst cases, teachers actually teach basic math concepts, such as adding and multiplying fractions, with calculators.

Of course, this approach completely misses the point of math. The answers to the problems themselves do not matter; only the cognitive skills the problems develop matter. (close)

A child cannot control the pace of a school math class. If a student does not learn how to do a problem type, he just falls behind. He will later have to struggle to catch up, or just end up getting low grades in the course. The course moves at a fixed pace, and one student’s performance will not speed up or slow down the class.

However, when a child is working with a parent, his performance does control the pace of the training. If he moves slowly and is unable to figure out how to solve problems, his parent may move slower and give the child easier problems. The child will end up actually doing less total work.

On the other hand, if the child works quickly and uses his full ability, the parent may increase the number of problems and the level of difficulty. By working at his peak, the student has actually increased his total workload.

Thus, a student has a strong incentive to drag his feet and use only a fraction of his full abilities when learning math from his parents. At some level, he realizes that the worse he does, the less total work he will end up doing. Thus, he might take ten minutes to solve a problem that he could do in one minute, or even give up on problems that he would normally be able to solve. Rather than developing his cognitive abilities, the child ends up wasting his time.

Thus, to ensure that the child gains the benefits of parental math training, the parent must give the child an incentive to push his mind to the limit. Historically, parents have used physical and verbal punishments to create incentives. But what if there was a more elegant and effective system? What if any “punishment” actually benefited the child? (close)

The Asian method can be used by all parents and teachers, regardless of their math ability. The Ladder, the Rule of Two, and the Hydra can be used by most parents. In this section we will examine the Microchallenge method, which can be used by par- ents and educators with excellent math skills. Since developing the Microchallenge method, I have used it daily with excellent results, even when teaching students with major deficits in their math skills. While the Microchallenge method can be difficult to master, those who can master it will find it extremely effective, especially when combined with the Hydra. It builds a thorough understand- ing of mathematical concepts, while simultaneously developing many necessary cognitive skills. This method, combined with the Hydra, will allow you to turn students with almost no math abil- ity into strong math students, and to supercharge the abilities of strong math students.

The Microchallenge method requires both patience and a strong understanding of the subject. You should only use the Microchallenge method with material with which you are comfortable. For example, if you are not comfortable with trigonometry, you probably will not be able to use the Microchallenge method to teach trigonometry; instead you might use the Asian System to teach trigonometry. However, you will still be able to use the Microchallenge method to teach the other parts of math.

The Microchallenge method involves helping the student understand how to solve a difficult problem by leading him through a series of “microchallenges” that run parallel to the required cognitive steps of the problem. This ensures that the student develops the underlying understanding and cognitive abilities necessary to do the problem, and does not instead just memorize a few steps. (close)