Skill in calculating by hand.
Nowadays, is this skill so important?
Two categories of people
The answer to the question
Three useful remarks
2) When calculating by hand is not good
1. calculating in mind
2. avoiding algebra
3) Management aspect
The object of the article is to answer the question posed in the title of the article.
There may be two main opinions.
The first states that the skill is not important and is irrelevant.
The second states that the skill is important and is applicable.
The first opinion is not very popular today (especially among the schoolteachers) but supported by many outstanding authorities on mathematics.
The author of the article adheres to the second opinion, and gives several arguments for calculating by hand.
Here "by hand" means that one may type or write solutions using a computer but must not use math soft to solve a problem.
Note, the author of the article is in no way against the usage of mathematics software like Mathematica, Matlab, Mathcad. The author considers these systems as the most advanced and useful means in science, engineering and education.
Two categories of people
Before starting to put forward the arguments we should answer the following question.
Who need to have the skill in calculating by hand?
My answer is:
There are two categories of people.
The people who need to have the skill in calculating by hand belong to the first category of people.
Engineers, specialists in applied mathematics and physics need to have the skill. (Most of the programmers and a part of the economists may be called engineers.). So, they belong to the first category.
The people who need not to have the skill in calculating by hand belong to the second category of people.
The persons, whose profession is far from mathematics, need not to have the skill of course.
So, poets, musicians, dancers, hairdressers, manicurists, taxi drivers, managers, sportsmen, doctors, marketers and a lot of the others belong to the second category. Of course, these people are not a lower class or something like that. They just need not. Pushkin hated mathematics but he remains the sublime Pushkin. Moreover, these people need not to know any other subject of mathematics either.
The answer to the question
Now we come back to the first question that should be restated a bit.
Nowadays, is this skill so important for the first category of people?
I dare say, yes. More, I am sure, yes.
Here are my arguments for calculating by hand.
Look at the following formula.
Is there any doubt about what we see?
In the formula we can see that the functions x(t), y(t), z(t) must first be differentiated and only then raised to the square; then summed and the result of the summing is being multiplied by m/2.
But how do we know the order of the operations? I am sure we know it because we did it by hand many times in our life. This knowledge is not only a piece of knowledge but a skill.
Ok, math soft do it properly and quick. So we can skip tasks of calculating and teach math at once.
I tried it several times. (The first reason was that my pupils had little time before their entrance exams. The second solid reason was that the schools which they attended were concerned about upbringing (using psychology, pedagogy, etc) much more than about subjects. )
So I tried, but no results. Why?
Well, I had to come back to her highness Arithmetic every 5 minutes when tried to explain how to solve "big" math problems. Every time my pupil asked me: "Is it possible to do like this?"
Here are some equalities at which I had to stop after this question had arisen again.
So I had to stop my experiments and give Elementary Mathematics Task to my pupils so that not to get stuck on the next problems. After doing my task (it takes less than 1 month except "Algebra" section), my pupils went on very well. It was like a big train when it was trying to start moving. A train driver must give a jerk to let the wheels of the vans roll.
Another simple example.
Imagine a student is at a lecture or seminar. A professor explains Newton law of gravity.
But the student cannot see the law properly because he doubts whether coefficient G goes to the top of the fraction or to the bottom. Also he does not know that the more a denominator the less the corresponding fraction must be. But this fact is crucial to understand this law (Let alone raising r to the square). This kind of knowledge is not just a piece of knowledge. It's a skill. So the student has to get that skill somewhere. Ok, I would offer him my Elementary Mathematics Task. In this problem book most of rules and niceties of arithmetic are compressed to near 100 simple tasks.
Thus the problem "calculating by hand" is solved. We cannot rid of the calculating by hand but we can accelerate the process of studding it. To do it we should use computers, calculators, math soft. I use a calculator (It's effective for this purpose and cheap. But someone can use something else).
Please note again that all the arguments above have to do with only the first category of people.
Three useful remarks
Very often, especially in programming manuals, authors make the following example.
A driver needn't know how to fix his car or how the engine works. The driver needs only to know how to drive the car. The driver needn't know private properties and methods (that is all the complicated details and processes in the car) but needs only to know the interface (that is the steering wheel; the gas, clutch and brake pedals) in terms of object oriented programming. But the mechanical engineer who makes such a car or fixes it must know the interior of the car and how all that stuff works in detail.
By analogy with this, the second category of people may be called drivers, and the first category of people may be called mechanical engineers.
When calculating by hand is not good
1. Calculating in mind
Calculating by hand is not good when one tries to calculate mentally. The error probability is very high in this case. As we have computers, this way must die.
(There is one exception. We have to admit, there always are the people of great talent who can do long mathematical calculations, transformations, formula derivations plus can do complicated three-dimensional geometrical constructions in mind. Unfortunately it is a very rare case. But even they have to turn on a computer when the job is in question.)
Once I saw the accountant who first computed some numbers with Microsoft Office Excel; then she checked the result with a calculator, after that she did check it once and for all with an abacus. Actually, I have a great respect for such persons. Now I only regret that didn't say her that Excel was quite enough.
2. Avoiding algebra
Calculating by hand is not good when one tries to solve a problem using only arithmetic instead of algebra.
This approach is widely spread in the primary schools (6-11 years old).
Try to solve the following problem without algebra.
Initial positions of the 2 bodies: x1 = 0 m; x2 = 100 m;
Initial velocities of them: v1 = 10 m/s; v2 = 5m/s;
t – ? ; t – the moment of time when the first body catches the second body.
This problem is offered to children 8 years old. I have just written formulary the textual statement of the problem. The problem must be solved only by arithmetic means. No symbols (except 0, 1, ... , 9, +, -, *, /, = ) are available. The pure arithmetic solution is so artificial and long that I don't expose it here. It's long because the great efforts are made to explain how to get the relative velocity without giving the physical notion of it. While solving the problem one must not use characters to define variables but only the tongue and numbers. On the top of that there are not x1, x2, v1, v2, t, m/s notations in the text of the problem and must not be in the solution.
Also, I found the same way in "GMAT Math Workout" by Jack Schieffer. The whole chapter 8 of the book deals with this approach and called "Avoiding Algebra". But instead of simple solutions we see the solutions look like the craziest works of avant-garde artists.
The reason is obvious. In my humble opinion, calculating algebra is a result of civilization's efforts to simplify solving of math problems as well as their statements. So it’s not surprising when one cannot get a simple solution if algebra put aside.
My opponent might say: "The person who belongs to the second category may not understand great and awful algebra. So we must give him simple and kind arithmetic. Please don't injure the state of mind of the young."
My answer is:
If the person belongs to the second category, please let him (her) alone.
If the person belongs to the first category, give him (her) algebra when he (she) is even 6 years old.
I often think why many reasonable ideas remain in the publications and they are not being accepted by the modern schools.
In my humble opinion the reason is what the main purpose of the education is set in the school. Now the main purpose is upbringing. So most of the schools are full of psychiatrists, pedagogues and other specialists of liberal arts. Did you ever see an engineer in the school? I never did.
But someone may object: "The engineer must not be presented in the school because he (she) hasn't a corresponding pedagogical education." Well, that's right. Besides that, I think, no prosperous engineer would go to the school to earn money.
Nevertheless, in my humble opinion, the main purpose of a math teacher must be teaching math. The teacher must teach pupils math so that they could apply it practically. So this person should teach only the first category of people. As for the second category, let them alone. As for upbringing, leave it to parents and clergymen.
Author: Ferat Talat oglu.