The new Insight
Abstract: At school, the debate is about how to teach children arithmetic. Schools more and more try to explain maths, but children can
better develop insights in underlying concepts and processes by
staying home, playing with Lego or helping mum with cooking.
At school, a debate is raging about the way children are taught arithmetic. Traditionally, children were first taught to recite times-tables. Then, they were taught to draw and recognise individual numbers and symbols such as +, /, -, =, ! and %. They then had to recite examples of manipulations of numbers using these symbols.
Children spent years on the same maths exercises and tests; mental arithmetic, rules and formulas were drilled into children as if it was a grave sin to make errors against the 'laws of maths and physics'. Maths was presented to children as the framework of nature, the truth, the law of God. Formulas and rules were memorized by sheer endless repetition. If any comprehension followed, that was a bonus, but not the aim. It didn't
matter whether children understood what they were doing, as long as their answers were correct. The teacher was there to discipline, not to explain.
Children who had problems with that were called 'lazy' and were punished simply for 'being stupid'. In the old days, school was a place of discipline and even the most obedient child was familiar with the strap or
cane. Children left school at an early age, ready to take up a trade. Mental arithmetic was seen as essential in many jobs, e.g. in counting money, cutting pieces of fabric to size, weighing groceries and laying bricks.
Today's new Situation
Today, many say that there is not as much discipline at school as there used to be. At 'lower' (i.e. younger age) levels, children do not sit in rows any more, all facing the teacher; instead, children often have
joined their tables together and 'work in groups'. Especially in new subjects such as dance and drama, children are talking to each other, moving, playing and even laughing in the classroom.
Today, children stay longer at school; often they are not interested in the academic side of school, they readily admit that they do not learn very much. Their skills and progress are not tested anyway, children move up one grade according to age, not according to competence. Further schooling does not make them much brighter, but it looks better on their record than being unemployed.
In maths, the perception that children needed to learn the basic rules first, from which comprehension then might follow, was replaced by the idea that mere exposure to situations, problems and solutions would give children insight. The'New Maths' method became popular, presenting children with problems in the context of 'real life' situations, described in words, rather than with numbers, symbols and formulas. 'New Maths'-advocates argue that in the traditional method of teaching arithmetic, meaning is detached from exercises. They argue that children should be confronted with `real' situations, rather than boring exercises. They argue that, in these days of calculators, electronic cash registers and computers, mental arithmetic is out of date. They argue that it is more important that children understand what they are doing, than to complete exercises without the slightest error.
A Return to the Basics?
And already, before it is even being widely implemented, the New Maths method has come under increased criticism from conservative forces that like to see a return to the traditional way arithmetic was taught at school in the old days. 'Back to Basics' is their slogan and they make a lot of fuzz about recent studies indicating that many students leave school unable to perform the most basic calculations.
Conservative teachers complain that children will not only be unable to get a job without the 'basic skills', but that they will be mentally crippled, unable to reason logically and unable to function as a 'complete human being'. They want a return to the discipline of the past. New subjects such as dance and drama are regarded as undermining discipline at
school. They blame videogames, TV and popular music for a perceived lack of respect for teachers.
Conservative teachers regard the lack of testing and grading of children as a socialist plot to hide the fact that not all children are equal. They regard old-fashioned arithmetic as the best instrument to find out who is working hard and who isn't.
Insight instead of Reason
New Maths-advocates try to explain maths by giving examples of real life situations. They may describe how six donkeys can equally share eighteen apples. The children pay attention to the teacher; they like listening to stories about donkeys and apples. But they do not make the step: 18 : 6 = 3.
The problem is that such descriptions are speech. Like song lyrics and conversation, speech is basically audio and as such it is first processed in the audio-part of the brain. Understanding of the meaning of words develops by means of association. Speech can put emphasis on words by intonation, speaking louder and making some words longer, but this does not explain the meaning of those words. By using speech, teachers may think they are teaching logic and reasoning; but even if they are, their logic and reason are part of verbal language; they do not give children new insights in visual processes, new experiences in movement, weight or volume.
In class, children may learn how to pronounce numbers, but that is not the same as counting. Children count with their fingers, they use touch, pressure, vision (which finger is lifted); they only use words as teachers
want to hear the answers.
Children use all kinds of senses when they are busy with concepts such as weight, size, length and shape, and in processes such as movement, growth and acceleration. Children are born with an interest for patterns, circles, symmetry, etc. In most cases, use of the eyes is essential; children will develop 'visual logic' or insight, before they can express this in words.
In the old days, many people used scales and balances: The balance is like a mathematical equation. What is on the right hand scale equals what is on the left hand scale. Selling apples, flour, salt or gold meant using weights and balances. To make a cake from a recipe, one had to measure salt, flour, sugar, etc. Sewing meant measuring fabrics and cutting them at length. Making a house meant looking at a map or plan, measuring lengths of wood and counting bricks. Such processes are not heard but observed, usually in silence for more concentration.
From childhood, people became familiar with such activities taking place in and around the house, on the market, in daily life. To start gaining insight in these activities, all they had to do was watch. Children developed many insights simply by experimenting, by observing and imitating others. Children gained an insight in the meaning of numbers, lines, arrows and other symbols on recipes, maps, receipts and other records of measurements, not by being lectured, but by watching examples and trying it out for themselves. If anything, such insights should be regarded as the 'basic skills'.
Homeschooling is better
School prevents children from gaining such skills today as school denies children exposure to real-life examples. Children are now locked up in school, they cannot watch their parents at work in and around the house. At school, they are expected to learn basic skills from the teacher, who only uses speech and the blackboard. On the blackboard, the teacher starts using symbols assuming that it is sufficient to verbally explain the meaning of such symbols.
But children have not had the opportunity to relate either these symbols, or the way they are pronounced, to anything else; children have had too little practical experience to relate to. For them, what the teacher does in front of the classroom, is like watching a cartoon on TV. All they see is someone writing funny symbols on a blackboard.
No matter how many examples the teacher uses, how often explanations are repeated or how many drawings are made on the blackboard, the children will not understand much of it, because they have not been exposed to the real-life situations to get a clear vision of the basic underlying processes.
Homeschooling is much better in this regard. The real-life situations are often very stimulating for the children. They are naturally interested in food and how it is made. Children can learn more 'maths' by playing with Lego at home, than by going to school.
At school, there is no reward for those who are sincerely interested. At school, children are denied the pleasure of practical experience and accomplishment. They are totally isolated from reality.
The Power of Computers
When maths is stripped from its fancy words and symbols, it quickly loses its magic. Maths is something that has been put together by people,just like a magician's show. What you see looks terribly difficult, it seems almost impossible to figure out how to do it. Many children fall for this trick and regard maths as a subject that is out of their reach. But once you understand how it works, the magic disappears. Maths becomes a cheap trick when used by school to discipline children, to give them feelings of inferiority and to make them look up at school as a place of wisdom.
The funny thing is that computers are much better in such tricks than teachers. If a teacher derives respect from such maths tricks, the computer should deserve an even bigger respect. If one wants to learn maths, one can better spend one's time with a computer than goig to school.
The Power of the Media
But who wants to learn maths? Why should anyone learn maths? It is the education system that elevates maths to a high status, but outside education, few people use maths. If necessary, people use calculators, cash registers and computers; even accountants are hesitant to use mental arithmetic when preparing tax forms for clients.
Computers, VCRs and TV can be tremendous educational tools, complementing real-life experiences with an abundance of further examples. History, geography, language, maths and other typical school subjects can all be learned so well from the screen, that it makes school a joke. Even for physical exercise, it is more convenient to do exercises with a video, than to go all the way to the sport field. That does not mean that we should watch the screen all day, it is in our own hands to use these media as appropriate.
Multimedia technology will change the way many people look at TV, books, computers and other media. For many, TV is moving pictures, often drenched with commercials, passive entertainment; the telephone is more active, it is used for casual as well as business conversations; computers are regarded by many as sophisticated calculators or typewriters; children are often told to spend less time with TV, video and computergames and to read more books.
Multimedia technology blurs the distinctions between all such media and thus also such prejudices. Today's equipment combines elements of many media that were previously regarded as separate and independent. CD-ROMs can now contain movies, music, text and data. Databases can transmit films, news, books and photos. People can communicate verbally or by exchanging images.
The debate about arithmetic is a debate about the world of the past. The media have already decided their future shape and school is a joke, if not a horrific waste of talent of children. New Maths is a failure as it is part of school and it tries to explain verbal concepts instead of giving insights in real life. School messes up children's brains as it tries to explain things that become clear by watching, imitating and experimenting. Visual logic is fundamentally different from verbal rhetoric. The new insight is that sound can be sound, video can be video and communication can be direct.
Appendix A: Questionable Virtues of the Three 'R's
Reading, writing and arithmetic were long regarded as cornerstones of education, often referred to as the three R's. Within compulsory education, they are the most compulsory of all. But the more vigorously they are defended by school, the more their virtues should be questioned. When
children are made to do their exercises, it is not all numeracy and literacy they are being taught. The fact that they have to submit to instructions from a teacher, to blindly follow orders, is a hidden message, subconsciously accepted and never challenged.
The insidious way in which children's brains are manipulated makes it hard to shed off this servile cloak when reality calls for action in later life. Maths exercises try to make children believe that, for any problem in life, there is only one right answer. Maths tries to make children believe that if they cannot find that answer, they are stupid. Maths tries to make children believe that if things are complicated, there is always one person with all the answers: the teacher.
School, and in particular maths exercises, try to prepare children to fit into society as the Government has shaped it; they should obey orders, listen to authority, accept that they are often wrong, but they should feel secure in the knowledge that there is always one person who has the answer: the Judge!
(Above text extracted from: The Four 'R's, Optionality Magazine, September 1993)
Appendix B: The Math Myth
School claims that maths has its roots in physics, that it reflects the Law of Nature, the Truth. But the fact is that school has stripped math exercises from everything associated with reality.
Take the equation A + B = C as an example. Maths argues that if there is a certain situation (say: A) and a certain element is added (say: B), then the result is that a different situation will eventuate (say: C). What maths then wants us to believe is that if that element B is again taken away, the situation will revert to the original situation A. But this is simply incorrect. The situation will never be the same again, it has been exposed twice to a manipulation and will have reacted to this exposure. In reality, any action has an impact and in some cases that impact may not be revertible at all. Even if the two situations are similar, they will
never be exactly the same. As a minimum, they are separated in time. Maths' presumptions defy both physical evidence and logic. It presumes that every time a manipulation is repeated, it will have the same result, while physicians are resorting to relativity, quantum mechanics, even Chaos-theory, in efforts to explain why it does not.
The point is not that the difference may be so small that it is negligible. The point is that, in reality, two situations are never the same. This whole system of maths, this giant structure of formulas, manipulations and equations, is a house of cards, built upon something that is not real and cannot be extrapolated beyond a narrow window of applications. The myth is that somewhere there is an a-priori set of numbers, including things such as pi, that times tables existed before mankind, that somewhere in heaven there is a place where maths is floating around, amidst all kinds of scientific laws. But in reality, maths was dreamed up by people and its early manifesta uons wereexpressions of language, rather than inspirations of statutory logic.
It is important to distinguish maths from logic. Logic (as well as experience) tells us there always are multiple possibilities, different circumstances and other approaches. Logic didn't put maths together in the way it is being taught at schools. It was lust for power that did that. And abuse of power has led to this arrogant portrayal of one system, one method as the only true one. In that sense, maths has become a religion which tries to explain not so much the spiritual world, but which claims to rule the natural world. There is no logic behind the myth, if anything,
maths defies logic; it only makes sense to expose this evil plot, notjust because logic commands it, but because defabricating this image of maths leads to a better society in a moral and economic sense.
(Extract from: The Math Myth, Optionality Magazine, September 1993)
Appendix C: The Evil of the Metric System exposed
The Metric System is a compulsory standard, apparently unable to earn a place next to other systems on its merits.
Calculations that use fingers are only mechanical manipulataons, used for adding and subtracting; such manipulations are fabrications of accountants involved in activities in which numbers are represented as fixed points on a line. They do not feature prominently in nature, as in reality a standard continuum built up from equally distanced units does not exist. More fundamentally, the combination of a sequential numbering system with a rational system is a questionable practice. Nowhere in nature does a straight line exist; nor does an infinite number of units each of equal length occur anywhere in nature, as the Metric System is trying to make us believe. Most importantly, the concept of zero does not exist in nature, let alone manipulations in which something is divided by zero.
Rather than standardizing any numbering system, multiple systems should co-exist, such that users continuously have choices in appreciation of potential differences.
(Free from: Standards of Measurement, Optionality Magazine, January 1994)