Archive for the ‘Intellectual Development’ Category

Mental Maturity and Mathematics

January 4, 2012 1 comment

Mental Maturity and Mathematics

By James H. Choi
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I like watching nature programs on TV. Just as the rain is best appreciated when you are dry, lions’ claws are most spectacular when they are safely contained inside the cathode ray tube of my television set. Thanks to TV programs, I get to think of all the animals in there, running for their lives, and the not-so-fast ones being eaten alive, whenever I see a forest, an ocean, mountains, or any form of wilderness. In the wilderness, something somewhere is being eaten alive.

Although helpless human infants grow to sing “Born to Run” in later years — probably out of an infant inferiority complex — wild animals quietly practice it out of necessity. The very first thing many wild animals do after being born, even before getting their mothers’ milk, is run. Just think about that. Without having had any time to learn anything, a newborn deer knows Newton’s law of mechanics (gravity, acceleration, velocity and distance), how to interpret stereo vision, who his family is, who his predators are and how to run to save his life. This pre-wiring of the food chain diagram, an accurate “you are here” marker in the deer’s map, is remarkable, especially in contrast to the general helplessness of a human infant.

In fact, human infants are the least prepared of all animals for this world. Human babies are born unfinished, so that their birth is possible. In other words, a fully functioning brain is too big for birth, thus we were forced out of our mothers into life unfinished. We finished developing outside. Jean Piaget (1896-1980), a Swiss psychologist and a student of Maria Montessori, presented an influential theory on the human development stages.

Piaget observed the development of human babies and classified what he saw into four stages (each with several sub-stages) as these:

  1. Sensory Motor Stage (0 to 2 years old)
  2. Preoperational Stage (2 to 7)
  3. Concrete Operations Stage (7 to 11)
  4. Formal Operations Stage (11 to 15)

As a math teacher, I find the transition from one stage to another very remarkable because one’s stage directly affects a student’s ability to understand mathematical concepts. I will skip the Sensory Motor Stage and discuss only the stages that I come into contact with in my classes. the preoperational stage (typically age 2 to 7), children believe an amount of water changes simply after being poured into a different container that makes it look different. For example, if water in a cup is poured into a dish (without spilling), these students think there is less water because of the new shallow depth. When the water in the dish is poured back into the cup, they have no trouble accepting that the water volume has increased back to its original size. In other words, these students do not grasp the concept of “conservation of volume.” They also do not grasp the concept of “conservation of mass” in this stage. If you have two blocks of clay, for example, but stretch one of them so it is long, children in this stage will believe the stretched-out block has more clay inside. They will also think the clay has gotten heavier.

How can you teach children in the preoperational stage that the volume of clay is conserved? You can teach them this by repeating the experiment and pointing out that the volume didn’t change. Although the water seems to get bigger and smaller as it is poured into different containers, for instance, the volume didn’t really change, and this is what you must keep repeating to them.

If repeated enough times, they will learn to say, “The water volume didn`t change” when you ask them. Using this same method, you can also teach them to say “1 + 1 = 3” with equal conviction. Basically, you can turn them into a mocking bird or Pavlov’s dog. At this age, it would be impossible to teach physics, let alone algebra. However, children can handle such simple operations as counting numbers.

But once they enter the concrete operations stage (age 7 to 11), the children suddenly understand “conservation of volume” — even without any education. In fact, they find the concept to be so obvious that they wonder why you are asking them such a dumb question. In this stage, their understanding is still limited to concrete objects that can be either visualized or touched, however. In terms of mathematics, their understanding is limited to concrete concepts such as addition, subtraction, multiplication and division. Students at this age will have difficulty understanding concepts such as imaginary numbers or fractional exponents. Many will also have difficulty with the concept of using “x” as a variable.

Only when children enter the formal operations stage (age 11 to 15), are they able to think in the abstract domain. Invisible objects such as force fields, impossible-to-experience quantities such as infinity and intangible concepts such as justice can be understood only at this stage. Learning infinite series, probability and calculus is possible only after the student attains this power of abstraction. Only now, at this stage, is the student able to manipulate logic.

The transition from the concrete operations stage to the formal operations stage is often seen in math classes that I teach. I sometimes teach a 5th or 6th grade student who just won’t understand the concept of algebra in spite of a sizeable struggle to do so. The possible reasons for this inability to attain knowledge are many, including weak arithmetic foundation and a lack of interest. However, just as often, the cause is the student’s developmental stage. Fifth and sixth grades are when the students get out of the concrete operations stage and enter their formal operations stage. A failing student in algebra can suddenly become a smart student in a matter of a few months.

In addition to feeling relief and pride in my student’s newly found excellence, I also wonder something: Was it worth going through all those sweat and tears? Wouldn’t it have been wiser just to wait until he naturally reached the formal operations stage?

Maybe. But regardless of their stage, young students are able to learn algebra — even as early as fourth grade. They are able to learn the concepts, solve equations and even give correct answers. But I do wonder if they really understand the concepts, or if instead they are just repeating “The water volume didn’t change” in another form, trying to please me in a more elaborate way. The widespread difficulties they have in solving word problems indicate to me that they are merely repeating, at least to a certain extent.

At times I wonder about the causal relationship of this repetition they learn. In other words, what is the cause and what is the consequence of all that struggling to accelerate the student into the next stage? If the concept of “conservation of volume” comes without any education, shouldn’t the ability of mathematical abstraction come just as naturally? Did my lessons actually accelerate the student into the next stage? Or should his education wait until the student naturally reaches the new stage?

The question gets even more fundamental. What exactly is education anyway? Is it a way to move the children to the next stage? Or is it a way to squeeze out their full potential within their current development stage? When the students are taught high-level math too early, are we pulling out trees, or are we providing nutrients in the name of making the trees grow faster?

I do not know the answers. I know only that some students at a particular age often hit brick walls when trying to have an early start in algebra, only to understand everything after some time, suddenly and as if magically.

Although I would like to claim full credit in each student’s progress, the fact is I am not even sure if my teaching has anything to do with their sudden enlightenment. The only thing I can tell myself for sure is that I didn’t slow their intellectual progress. I imagine a news headline that reads like this: Teacher Didn’t Slow Down Student’s Progress!

After all, what more could we ask for from a teacher?

Such is the frustration and reward of teaching algebra to 5th and 6th graders. If your 5th grade child used to do well in arithmetic and then started struggling with algebra, perhaps you should not worry too much. As long as your child’s arithmetic is solid and sound, then he or she will enter the formal operations stage in due time, maybe as little as a few months. Then, your child will suddenly be able to understand the whole concept. While waiting for this to happen, it’s you let your child focus on the arithmetic and word-problem solving.

Six Reasons for Sudden Drop in Math Grades

January 4, 2012 8 comments

 All good math students are alike;
Six Reasons for a Sudden Drop in Math Grades

By James H. Choi
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“All happy families are all alike;
every unhappy family is unhappy in its own way”
–Leo Tolstoy, the first line of Anna Karenina–

“All good math students are alike;
every bad math student becomes bad in one of 6 ways.”
— James H. Choi–

When you open a newspaper, it seems there are only troubles and accidents in this world. When you listen to a math teacher, it would seem all students are falling into dangerous mathematical traps. After all, no respectable newspaper would report that “absolutely nothing happened to the vast majority of people today,” nor would a competent math teacher get a call from a stranger proclaiming, “My daughter is doing very well in math and we don’t need you!”


As a math teacher, most calls I get go this way:

“My son’s math grade is going down! Do you know why?”
“It depends.”
“Depends on what?”
“On many things.”
“Many things, you say?”
“Yes. For example, are all of his grades going down or just his math grade?”

1. Biological stage:

“Blood is thicker than water,
and hormones even thicker.”
–James H. Choi–

If all of his grades were going down, then it is probably not a math problem. The hormone factory runs in full gear at the high school age. As the students’ heights, tastes, hobbies, interests and even values go through changes, it would be strange indeed if only their interest in mathematics remained intact. The interest in mathematics may share the same fate of childhood toys and dolls.

This is a highly insecure age and, perhaps as a result, an age of revolt. They will fight to win their freedom from parents’ authority, only so that they can submit themselves to their peers’ authority. At this age, the definition of truth is “All my friends do it, too.”

I am not sure of what can be done to fight this awesome force of nature, if it has to be fought at all. Mathematical ability doesn’t seem to be very important in reaching the ultimate goal of life, which is supposed to be happiness. Besides, who you know is supposed to be more important than what you know. Perhaps socializing and befriending are indeed more important than anything else. After all, the world is full of successful and happy people who proudly claim their mathematical incompetence. The question is more philosophical than mathematical. I shall leave the solution to the philosophers and poets.


Only the math grades are going down; he is a smart student. I am not sure why he is suddenly struggling,” the mother said.
“Did he ever say that he hates the class or the teacher?” I asked.

2. Social Situation:

“Don`t let school interfere with your education.”
–Mark Twain–

Two important social factors affecting students’ math grades are their teachers and friends.

The effect of bad teachers needs no mention. It is hard enough to study with the finest teachers. Having a bad teacher makes every step an uphill battle. A transition from a good teacher to a bad one is jarring psychologically and creates a noticeable change on a report card.

Bad teachers come in three varieties: ignorant, indifferent and unlikable.

When teacher’s knowledge is substandard, then it is no surprise the students end up ignorant as well. Worse, the students reach a state of ignorance through a path of endless confusion and frustration, muddying the water, making future learning more difficult.

Other teachers know their subjects but present them in such mind-numbing manners that the lesson disengages the students. With such indifferent teachers, students are left on their own to learn the material.

A teacher’s likeability is just as important as competence or enthusiasm. Likeability is usually linked to the scholastic competence and pedagogical passion the teacher displays, but it could also be linked to some other force of nature. When a student likes his teacher, his grade generally improves. The reverse is also true. Thus, math grades might suffer from the student’s purely personal dislike for his teacher., of course, there are unique teachers that embody ignorance, indifference and dislikeability all in one, making them singularly effective in lowering students’ math grades.

There is not much I can advise or suggest here because, by and large, public school teacher selection is done by that overpowering force in our life: luck. And also, if this problem could be fixed easily, the whole test-preparation and tutoring industry would never have existed in the first place. But for those who want to take action, I would like to recommend the following book: Bad Teachers by Guy Trickland for advice on how to deal with the school’s bureaucracy.

I’d also like to wish you what matters the most: good luck.


“No, he likes his math teacher,” the mother replied.
“What is he learning in math now?” I asked.
“I am not sure, I would have to ask.”
“Did he just start geometry?”

3. Content Change:

“Change alone is unchanging.”
–Heraclitus(c.540-c.475 BC) Greek philosopher–

Mathematics is a collection of many different disciplines. Although they all use logical reasoning and build upon one another, some use abstract, spatial, or arithmetic skills. Throughout high school, students go through several transitions within math that can be jarringly disruptive.

The transition into geometry is particularly difficult for some. In the United States, the typical high school curriculum suspends algebra after one year to teach geometry, then resume algebra again, calling it algebra 2. Thus geometry starts and ends abruptly. In my opinion, this is not the best arrangement, but that’s what the higher educational power of the United States decided, and I only work here.

One might think that geometry should run in our blood by now because it didn’t change much for 2,400 years. Traveling back in time for 2,400 years, one would pass Jesus’ time and reach all the way into Buddha’s time. That’s how ancient of an art geometry is. The usual complaint (“The world is changing so quickly, I cannot keep up!”) does not apply to geometry at all.

Even so, many students find it difficult to deal with spatial relationship and the rigor of logical proof. What is frustrating is that some understand it without any difficulty. Geometry is also very difficult to teach in a classroom setting because of the natural divergence of student levels. I will write about geometry — how to prepare for it and how to study for it — in another section.

Another notable disruptive transition occurs while beginning to study probability. In the discipline of probability, you cannot test your answer by plugging it in. In fact, you cannot test the answer even if you possess all the time in the world. Probability is constructed by pure logic and can be validated or disproved only by superior logic — not by an experiment or demonstration.

If the grade drop happened during these subject transitions, then the drop is not likely to be a reflection of a student’s change, but rather of the change in use of their brain’s area. In other words, the student didn’t change, but the math world changed on him even though it is still called “mathematics.” The good news is that the student’s performance will go back to normal once he has finished these new topics. The bad news is that some subject last for a year, and others never end.

The only right answer is to admit that some topics are more difficult for some people, and after admitting this, compensate for one’s uneven gift with more effort — however unfair the concept of more effort might seem. It is worth remembering that in the end, only results matter. The effort you had to put into the subject, however inordinate, won’t dilute your achievement.


“I don’t think so,” she said. “He learned geometry already.”
“I see,” I said. “How good is he with mental calculation? Does he depend on the calculator for everything?”

4. Arithmetic Weakness:

“I cannot even balance my own checkbook! Haha!”

Arithmetic and mathematics are used interchangeably in everyday language, but they are different. Arithmetic is about addition, subtraction, multiplication and division. Mathematics is about logic and solving problems. The timeless rhetoric about the inability to balance the checkbook is not about mathematics at all; it is about arithmetic. The current high school generation was raised with calculators. Perhaps as a result, their arithmetic incompetence is staggering.

Although mathematics is more than just arithmetic, arithmetic is absolutely necessary for understanding mathematics.

The current generation of high school students grew up in a world flooded with calculators, and as a result they are showing an alarmingly low level of competence with arithmetic.

Calculators by themselves are not the culprit. If one searches for studies on the use of calculators in math education, one can find many arguments in favor of the device. Indeed, calculators can be used to enhance learning. I myself use the most powerful calculator in the world in my own classes: Wolfram Research’s Mathematica software. We cannot live without calculators.

The problem is the way the calculators are used. The calculator usage habits of high school students are shocking. Many reach out for the calculator when they are faced with “12 x 2.” They do not know if 0.2 x 0.2 would be 0.4 or 0.04 without working it out, and they have difficulty telling if a division by ½ would increase or decrease the number.

The inability to perform arithmetic is a crippling deficiency, but the even bigger problem is the loss of the notion of numbers themselves. These calculator-dependent students grow so numb toward numbers they can no longer compare the relative sizes of complex numbers. This weakness will eventually bring down the student’s math grade, typically sometime between algebra 1 and algebra 2.

What is really surprising about this weak arithmetic skill is the current collective neglect in fixing it. From the teacher’s side, nothing is done to address this problem. There is no high school course on “How to compute quickly with accuracy.” In fact, arithmetic incompetence is not even recognized as a problem because students can “just use calculators.” From the students’ side, they all wish the problem away by saying, “Oh, that was just a stupid mistake.” Meanwhile, mathematics seems to get harder and harder for them, as those “stupid mistakes” become more frequent.

Arithmetic weakness is a very difficult problem to solve. It takes time, patience and determination to overcome. Yet without addressing this problem first, learning higher math would be like pouring water into a leaky jar.

I would like to be more optimistic, but typically this problem doesn’t go away without professional help. It will continue to limit the student’s mathematical performance until his last math class.


“No, he did Kumon when he was young,” the mother said. “He handles numbers easily.”
“You said his math grade is falling now,” I said. “How well did he do in previous years? Did he get perfect scores? Did he sometimes get a B?”

5. Mathematics Foundation:

“Success depends upon previous preparation,
and without such preparation there is sure to be failure.”

More than any other discipline, mathematics is built upon a clear sequence, with well established dependencies among its concepts. For example, solving second-degree equations is impossible without first understanding first-degree equations. Solving a probability problem is impossible without first knowing how permutations work. There are some quasi-independent branches like geometry, but for the most part, even when a new topic deals with a radically new concept, it still utilizes many of the previous concepts and builds upon them.

This is why it is common to see students getting worse grades in mathematics, yet it is rare to see a student getting better. Once the math foundation is weak, anything built upon it is shaky. When a tough test like the SAT hits, delicate houses of cards will collapse.

Although a “B” is supposed to be “Good,” these days a “B” stands for “Bad.” School grades are so inflated that I find many holes in the understandings of even “A” students. Getting an “A” is no longer a sign of competence, whereas getting a “B” is a sure alarm that the student is losing grip.

Thus, getting a grade of “B” or worse in math could indicate difficult times as little as a year later. Sooner or later, something else will be built upon the concept a student is currently getting a “B” in, and the student’s understanding is likely to collapse due to the foundation’s weakness. Typically, a student tries to overcome a problem with current math curriculum by studying only the new material, but the real problem lies in what this student learned a year ago. It takes a professional math teacher to analyze a problem of this nature and direct the student to the right material to review. Without this teacher’s guidance, the student is likely to repeatedly crash into that brick wall and lose both confidence and interest after putting in so much time, which proved to be only misguided effort.


“He never got a B in math before,” the mother said. “He was not the top, but he didn’t have difficulty with math before.”
“I see. Do you know if he started precalculus or calculus?” I asked.

6. Abstraction Faculty:

“Imagination is more important than knowledge.”
–Albert Einstein–

By definition, infinity cannot be grasped, and adding an infinitely many numbers cannot be performed even by the most advanced computers in the world. Yet students are asked to do in five minutes what super computers cannot do in 1,000 years. For example, what would be the answer to the following infinite addition that will go on until the end of the world?

or how about this one that will also keep adding until the end of the universe?

The first sum adds up to 1 only if you add until the end of the universe. If you stop in the middle, the sum will be less than 1. The second one’s answer is infinity, but only if you add all the numbers. If you stop anywhere, the answer will be very big but not infinity.

How do we know this? Did anyone actually do it once to find out? These are addition problems, but adding will never lead to the answer because you cannot finish the addition in this universe’s lifetime even on a supercomputer. Only software like Mathematica can solve it, but Mathematica doesn’t find the answer by adding: It solves the problem by applying abstraction.

So how can the tender brains of high school students be expected to perform this level of computation?

Abstraction is the answer: The brain can imagine infinity and beyond and solve this kind of problem through abstraction, or by “thinking.” However, the brain used in this type of “thinking” is very different from the one used to add numbers. Mathematics is a discipline of abstraction, not of computation. The more advanced mathematics becomes, the fewer numbers it uses. Indeed, balancing a checkbook is only arithmetic; it has nothing to do with mathematics of abstraction.

Precalculus and calculus are concepts based on abstraction. Calculators become useless at this stage. The only way to master these topics is by “getting it.”

This is the point at which studying by memorization fails. Those who used to get an “A” by memorizing formulae will begin to fail at this point. By now, there are too many formulae and concepts to be juggled. Without a coherent unifying abstraction, all these formulae, theorems, equations and topics seem to fight with each other, causing conflict and confusion. Only by mastering the abstraction can a student fit all of them will into one coherent picture, in which one concept complements another. Without reaching this enlightened state, students will continue to view mathematics as a walk in the dark.


“He didn’t start calculus yet. He said he would learn it next year,” she said.
“Then it must be precalculus,” I said.
“Do you know why his math grade is falling?”
“I think so, but I would have to talk to him to know for sure.”
“What is the problem?” she asked.
“I think it is his transition into abstraction. That could be difficult for some.” I said.
“Can he overcome it?”
“Any high school math can be taught to any motivated student.”

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