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Translated and Republished in China by Renmin University Press!
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In the Equation for Excellence, renowned educational innovator Arvin Vohra (SAT Math Cognition, Vocabulary Synapse), teaches parents the fundamental principles and subtle techniques that they can use to make their children excel at math.
Learn how to use the Asian system for teaching math, how to improve your child’s self-perception, how to prepare your child for the SAT and SAT II, and how to effectively motivate your child to excel. If your child already excels at math, the Equation for Excellence will show you how to help him increase his lead. And if your child struggles with math, the Equation for Excellence will help you develop the permanent math abilities that will allow him to surpass his “gifted” peers.
Described as one of America’s most important educational innovators, Arvin Vohra is the author of SAT Math Cognition, the primary developer of Vocabulary Synapse, and the founder of Arvin Vohra Education (AVE). He holds degrees in mathematics and economics from Brown University in Providence, RI.
The Equation for Excellence Author Arvin Vohra on CBS
“The Equation for Excellence promises to be the defining educational work of the decade.”
– Dr. Ravi Ramachandran
President, American Alliance for Education
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)
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
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
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
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
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!
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
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)