# Calculation without Understanding

Calculation without Understanding

Early in our institutional education system we learn arithmetic. We learn to add, subtract, multiply, and divide. We learn to calculate without understanding.

This mode of education follows us throughout our formal education system. We learn to develop answers devoid of understanding. We do this because, in a society focused upon maximizing production and consumption, most citizens need only sufficient education to perform mechanical type operations; that is perhaps why our electronic gadgets fit so well within our culture.

If we think about this situation we might well say that this form of education best serves our needs. It is efficient and quick. However, beyond the process of maximizing production and consumption we are ill prepared to deal with many of life’s problems because we have learned only how to develop answers that are “algorithmically friendly”.

In grade school we are taught to manipulate numerals (symbols) not numbers (concepts). We are taught in grade school not ideas about numbers but automatic algorithmic processes that give consistent and stable results when dealing with symbols. With such capability we do not learn meaningful content about the nature of numbers but we do get results useful for a culture of production and consumption.

We have a common metaphor Numbers are Things in the World, which has deep consequences. “The first is the wide spread view of mathematical Platonism…[it] leads to the metaphorical conclusion that numbers have an objective existence as real entities out there as a part of the universe…Given this metaphorical inference, other equally metaphorical inferences follow, shaping the intuitive core of the philosophy of mathematical Platonism.”

Quotes from Where Mathematics Comes From by Lakoff and Nunez

Okay coberst, this concise topic is more cereberally palatable. Education in a public system is like herding cattle through the stock yards. Having teachers who know how to teach down to adolescent minds is where the problem lies. Teaching abstract ideas in mathematics comes easy for some teachers and students. Some children’s brains are wired to deal with numbers, so they can grasp the concept easier. Having teachers who recognize the children that don’t and can think down to their reasoning levels is the key.

If children could have real world applications to help associate the need for math, then their interest levels would increase. One problem teachers have is time constraints and another is progression planning that add to the learning conumdrums students face. If the learning procedures could adjust to those factors, then it is quite possible young people could advance easier and sooner.

I can safely say that I was more intelligent than all my mathematics teachers from kindergarten to college.

I was held back and eventually-lost interest in math as a study. I went to Philosophy instead. I would not be here right now, if…

RU, there seems to be a pattern developing concerning you being held back. There are some correlations between you and my son.

I assure you that it is no coincidence.

To be more literal; I mean: ‘gravity’ is the main problem.

Your teachers and parents never had number lines, blocks, clocks, or pennies?

When children are young, good parents will ask them, “Good job! Now how many blocks are in your stack?” Parents will have their kids watch Sesame Street or other educational programs where they pose the same types of questions and scenarios: “Two bats! Ah-ah-ah!” While it is now natural, or ‘done without understanding,’ for you to count now, it wasn’t when you were three.

Then when they get counting and pattern recognition down, they start learning addition and subtraction, usually done with visual aides so they can rely on their previous, unthinking or naturalized skill of counting.

Each skill builds on the next, not just for efficiency, but out of necessity. I am an engineer not because I think about addition when I do calculus; I’ve already crossed that bridge of understanding. Once you understand it, you move onto the next thing. Algebra used to be hard, now it ‘just makes sense’, and rough calculus is what I consider ‘hard.’ If algebra didn’t come so easy to me, I wouldn’t have been able to learn calculus.

Granted, some children will not be able to grasp the next level of mathematical development and will just memorize 3+5=8 for a quick-and-dirty solution to the expectations of the educational system. Once a kid realizes he can’t get through algebra, though, he chalks it up to his nature and moves onto business school. Or philosophy. Or art. It’s a pejorative joke put out there by engineers and their ilk, but also the way of the world and there’s really nothing wrong with it. “We need someone to talk to and someone to sweep the floors.”

It’s also not a terrible thing to force children to learn that way. Maybe there’s a better way for some kids, but if they can’t get it despite a teacher’s best efforts, or even if the teacher doesn’t care about the student, then they still need to know 3+5=8 or they’re going to have a rough time getting through life.

Those who truly ‘get it’ as far as math goes usually come to realize math is just another language, another way to communicate and think about the world to supplement other thought processes. Those who don’t ‘get it’ are probably not that smart anyway, or are smart enough to realize math and such mechanical operations of the mind aren’t their thing and will develop themselves in another area.

Maybe I’m just biased because I went through public school, had good teachers and bad, and I think I’m doing alright. Then again, maybe I still can’t grasp the relationship between symbols and the real world because my clock says 2:03 AM and I probably should’ve been in bed about an hour ago.

Repost.

Just to interrupt… I hated High school. One of the problems I found is that people assume the “level” of others intelligence. All of my high school career I was put into a hierarchical system; lowest “level” being foundations then extensions and the highest honors then for the ‘gifted children’ advanced placement. I was held back and was very inconsistent in the public ‘learning’ system. That is if you consider learning memorization then regurgitation, repeat…Not to mention the hormonally driven cruelty.

You get out what you put in.

Tell that to a 4-year-old in preschool. I am sure it will do wonders for his/her “education”.

…or is it, “institutionalization”?

I would not entirely blame myself nor would i entirely blame the system. Attitude will reflect leadership as will leadership reflect attitude. Often it is the environment that the system attempts to control which is the problem.

I was referring to Double 'O’s time in high school. By that time, you’re old enough to make that choice.

Why?

I only mentioned highschool but the worst of it was in grade school. I was put in readiness for reasons I’m still not entirely aware of; lacking social skills perhaps. But instead of investigation of the psychological cause(s) this condition was immediately equated with an intelligence deficiency. My teachers would talk down to me, and use social status blackmail to enforce there codes–“Now all your friends can blame you for making them miss recess time”

A bad social rep in school will only lead to more problems and it will follow you…

I second that

Because, I’m guessing your stubborn countenance would demand such respect by high school age.

That, and I fully recognized when I was not making the most of my education by the time I was twelve. I was OK with being lazy and just relaxing my way through school with A’s and B’s to forgo some amount of learning.

Does a person demand respect, command respect, or earn it?

I was very lazy in high school and I got a 4.0 GPA. The teachers loved me. I didn’t even have to think at all.

You earn respect, but someone with your attitude seems to demand it. Expect it. That dosen’t mean you’ll get it, but your contrary, argumentative attitude makes me feel that you think that way.

Besides, I meant a teenager wants his decisions to be respected and acknowledged. Unless you’re not saying you’re capable of making decisions…

I got a 4.23 (or something like that), actually, because we had a weighted grading scale for AP, gifted, and honors classes, despite the fact that I made it a point for a lot of teachers to hate me. GPA doesn’t mean squat, though. Some of the stupidest people I know got four-Os; and by that I mean a 4.5 something (which is what our valedictorian got).

My attitude revolves around my expectations and disappointments in other people. That is it. I don’t care much for where other people respect me. They either do or don’t. I do not make it easy for people to earn my respect, anymore than I give it freely, because nothing in life is free. If you want to be “smart”, “intelligent”, and/or “reasonable”, then you have to at least act like it or pretend like you know what you’re talking about. Either way, if you enter into dialogue with me, then I can quickly find out where people are lying.

I wouldn’t trust a high school student to any task except mowing my lawn (if I ever have one).

[b]especially[/b] raising kids for that matter. I am just making a point here; I digress.

What does that tell you when a person can “ace” the institution but fail outside of it???