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Mental Maturity and Mathematics

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.
http://dl.dropbox.com/u/6378458/Column/Info/English/SpecialEvents.gifIn 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.


  1. Korgan
    January 5, 2012 at 8:44 pm

    I like this article. I was aware of my own transition between stages. I was an art student when I was 16-17 years old. No interest in math whatsoever, though I was interested in programming. When I reached 17-18 years old, I decided that to be a better programmer, I ought to learn math.
    So I learned from books at the library for about 2 years. I wanted to relearn everything I had been taught at school, because I had a feeling that it lacked. So I started from arithmetic and worked my way to calculus.
    After that time, I felt like a different person. My thought processes overall had changed. I felt like I had upgraded. My thinking process was more rigorous in all areas, relative to how I remember them to have been.
    It’s a sensation that I have found very difficult to explain to other people, but I suppose what happened was that I moved between stages of development.
    I think that I wouldn’t have ‘upgraded’ had I not studied math but, also, my brain was ready and hungry to use something as the vehicle for this development. If I didn’t have access and a hunger for math then, I’m not sure if I could have upgraded.
    So, my brain was hungry and primed to develop at a far greater rate than it had, and math books were there to facilitate that development.
    As a teacher, I think you are a feeder. You are providing fertile soil so that, if any of those seeds are ‘hungry’, ‘primed’ and ready to go, then they have the soil to climb through and they have the nutrition to make the journey. If they’re not ready, then they’re not ready. Can you make a ‘not ready’ brain into a ‘ready’ brain? I don’t know.
    At some point in a plant’s growth, it says “ok, I’m ready to grow outwards” and all of a sudden the plant that was growing upwards at some rate, suddenly, prolifically, starts growing outwards, gets bushy. I think our brains do the same thing.
    When I was 18, my brain got bushy and I’m glad of it. 🙂

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