Volume 1, Home Education, pg 254-264
Of all the subjects a young child learns, the most important one might be arithmetic. It's not so much that he needs to be able to add that makes it important, but using the skills he needs to come up with the sum has a beneficial part in the rest of his education. This is so true that those who want math emphasized and those who want language emphasized have pretty much had total control over education until recently.
We don't need to say how arithmetic has practical value for everyone, no matter what their station in life. But arithmetic's practical value is the least of its benefits. The main value of arithmetic and higher math is the way it trains reasoning powers, habits of understanding, quickness, accuracy, and being truthful intellectually. No other single subject benefits as much from good teaching as arithmetic, and no other subject results in such damaging results if it's taught wrong. For instance, a child multiplies but doesn't get the right answer. So he tries division, but that doesn't work, either, so he tries to see if subtraction works. He doesn't see clearly how the problem needs one process and only using the correct process will get the right answer. A child who doesn't know when to add and when to divide with a simple problem, hasn't been taught properly from the beginning, even though he may be able to finish pages of multiplication problems or long division correctly.
How do we get the child to understand what kind of problem he's dealing with? Give him simple word problems he can understand from the beginning instead of lists of multiplication problems. Young, enthusiastic teachers love to assign complicated long division problems that fill the paper and keep the student busy for a good half-hour. When it's finished, the child is worn out and wearied with work that serves no practical purpose. And, on top of that, his answer isn't even right! The last two digits are wrong and the remainder is too much. But there's no time to do it over and the teacher doesn't want to discourage him after all that work, so she marks it 'almost right.' But there is no such thing as 'almost right' when it comes to arithmetic. Instead of assigning such a long, complicated task which offers no variety in exercising the brain, and which tends to make his mind wander, say,
'Mr. Jones sent 607 apples to school, and Mr. Stevens sent 819. The apples are to be divided between 27 boys on Monday. How many apples will each boy get?'
The student must ask himself some questions. 'How many apples are there altogether? How do I find out? And after I do that, I have to divide the apples into 27 piles to find out each boy's share.' In other words, the child figures out which processes he needs to use to get the required information. He is interested, the work is done eagerly and the answer is found in no time--and it's probably correct because his attention was focused on the work. Problems should be chosen carefully. They should be easy enough for him to do, but challenging enough to require a little mental effort.
The next thing is to demonstrate everything that can be demonstrated. A child can learn his multiplication tables and do a subtraction problem without ever understanding the reason for doing either one. He may even become good at figuring and applying the rules but never understand when or why to use them. Arithmetic becomes the first step in doing real math only when every process is clear in the child's mind. 2+2=4 is pretty obvious even without proving it. But 4x7=28 can be proved by demonstrating with manipulatives.
The child might have a bag of dried beans. He can place them in four rows of seven and then add the rows: 7 and 7 are 14, 7 more are 21, 7 more are 28. How many 7's are there in 28? 4. That's why we say 4x7=28. And the child sees for himself that multiplication is nothing more than a shorter way to do addition.
He should use a bag of beans, buttons or other counters in all his early arithmetic lessons. He should be able to manipulate them freely, and even to add, subtract, multiply, and divide in his head, before he's ever given a list of problems to figure on his paper.
He might arrange an addition chart like this with his beans--
0 0 0 = 3 beans
0
0 0
0 = 4 beans
0 0 0 0 0 = 5 beans
and be practiced until he can tell without counting, and without
looking at the beans, that 2+7=9, etc.
In this way, with 3, 4, 5, --all the way up to 9. As he learns each set of math facts, the 4's, for instance, he should practice with imaginary objects, such as 'four apples and nine apples,' or 'four nuts and six nuts.' Then, finally, he can work with abstract number symbols--6+5 or 6+8.
A subtraction chart can be worked on at the same time as addition. As he works out each line of addition facts, he can go over the same thing working backwards by taking away one bean or two beans instead of adding them, until he can answer readily, 'what is 2 from 7?' or 'How many is 2 from 5?' After working out each line of addition or subtraction facts, he may write that line on his paper with the proper symbols if he knows how to make them. It takes more mental effort to understand subtraction than addition. The teacher must be patient enough to go slowly--one finger from four fingers, or one nut from three nuts, etc., until he feels confident with it.
When the child can add and subtract freely up to 20, he can work out his multiplication and division tables with his beans until he gets to 6x12. At that point, he can break down the problem, such as 'two times six is twelve,' which he can see by laying down two rows of six beans.
When he's able to say quickly, without even glancing at his beans, that 2x8=16 or 2x7=14, then he can take 4 beans, 6, 8, 10, 12 and divide them in two piles. From that he can tell how many twos are in 10, 12, and 20, and then continue in the same way for each multiplication fact, working out division facts.
Now the child is ready for more challenging word problems, such as 'A boy has two baskets of ten apples. How many bags of four can he make?' He'll be able to work with a bigger variety of numbers, like 7+5-3. If he needs the beans, let him use them. But he should be encouraged to use imaginary beans as a way to get him closer to working with abstract numbers. Meticulously graduated teaching and some mental effort every day from the child from the very beginning might help him develop real ability in mathematics. And it will definitely help him develop habits of concentration and working the mental muscle.
When the child has no problem working with small numbers, he will face a challenge. How successfully he meets this challenge will determine whether he will appreciate mathematics as a science. On this rides his ability to learn from all the math problems he'll do from here on out. He must understand our system of notation [the written symbols we use to signify numbers and place value]. Here, just like before, it's best to begin with concrete, tangible objects. Let the child understand that ten single units is one group of ten, such as ten pennies in one dime.
Give him fifty-two pennies and point out how inconvenient it is to carry so many heavy coins around while shopping. So we use lighter money, such as dimes. How many pennies are in a dime? So then, how many dimes can he exchange for his fifty-two pennies? He divides his pennies into five piles with two left over and finds that fifty pennies are (or are worth) five dimes. If I buy two apples at twenty-one cents apiece, the clerk gives me a bill for 42 cents. Show the child how to put down the pennies, which are worth less, to the right, and the dimes, which are worth more, to the left.
When the child is able to work freely with dimes and pennies and he understands that the number two in the right hand column means two pennies and the number two in the left column means two dimes, introduce him to the concept of tens and units. Be patient and work slowly. Tell him about uncivilized peoples who can't count beyond five. When they want to express some immense number, they'll say, 'five-five beasts in the forest,' or 'five-five fish in the river.' But we can count as high as want, all day long for years on end without ever coming to the last number. That's because we only have a few numbers to count with and only a few symbols to express them with. We only have nine numerals and a zero. We can take the first numeral and the zero to express a new number: ten. After that, we have to begin again until we get two tens, then again until we reach three tens, and so on. We call two tens 'twenty' and three tens 'thirty' because 'ty' is from the old German word tig that means 'ten.' But if I see just a number, 4, how do I know if it means four tens or four ones? There's a simple solution. The tens have a place of their own. If you see the 4 in the ten's place, you know it means forty. The tens are always behind the units, at the left. When you see two numerals side by side such as 55, the left-hand numeral is the tens and the right hand number is the units.
Let the child work with tens and units until he has mastered the idea that the number on the left is ten times the number on the right. When he laughs at the idea of writing 7 single units in the tens column and making it look like 70, then he is ready to extend his understanding to hundreds. He will have no trouble with hundreds if he understands the principle clearly, that each place value to the left is ten times more. Meanwhile, don't give him lists of arithmetic problems to figure. Don't let him work with notation symbols larger than he's been taught. When he gets to the point of 'carrying' in addition or subtraction, make sure he says 'two tens' or 'three hundreds' and not just 'two' or 'three.'
If the child doesn't get a firm grasp at this stage, he'll never get beyond trying guess which rule to use. In the same way, he should learn about weights and measures first hand: by weighing and measuring real things. Let him use scales and sand or rice with paper bags. Let him put together perfectly measured bags of sand or rice in pounds and ounces. Although this exercise isn't arithmetic per se, it is very educational. It teaches the child to judge how much things weigh and it encourages neatness, skill in handling materials, and quickness. In the same way, let him work with a ruler and tape measure and draw up charts. Besides measuring the obvious things, let him try to estimate weights and measures. How many yards is the tablecloth? How many feet long and wide is the map, and the picture over the mantle? How much does he think this book would weigh if he wanted to mail it first-class? This kind of skill will serve him well in life and should be cultivated. While busy measuring and weighing, he will naturally come face to face with the concept of fractions, and 'half a pound,' and 'a quarter of an ounce,' etc.
Arithmetic is a great way to train children to be strictly accurate, but a bad teacher can encourage a disregard for truth. An inferior teacher allows copying, prompting, telling, helping over difficulties and working towards a solution when the answer is already known. Just as bad, she says that an answer is 'nearly' right, because just the last two digits are wrong, or whatever, and then she has the student work it over again. But a sum is either wrong or right--it can't be somewhere in between. And if it's wrong, it's wrong. The student shouldn't be allowed to think that what wasn't done properly the first time can just be fixed to make it right. There is no going back. But he can move forward. Maybe he'll get the next one right; a wise teacher will make sure that he does. She'll give him new hope. But the wrong sum needs to be left alone. Therefore, his progress should be carefully graduated. There is no subject like arithmetic where the teacher has a real sense of drawing out new power in a child from day to day. Don't offer him a crutch, he needs to be able to go in his own power. Give him short sums using words rather than figures. Excite him so that his enthusiasm prompts him to work more quickly and with greater focus. His mental growth will be as obvious as seedlings sprouting in springtime.
Instead of spending more time discussing elementary arithmetic, I'd like to recommend A B C Arithmetic by Sonnenschein & Nesbit. Their method is based on a passage from John Stuart Mill's Logic that says,
'The basic truths of the science of math rest on what we know with our senses. They are proved by seeing and touching objects, and figuring out naturally what numbers break down into. For instance, if you have ten balls, it's easy to see that they can be arranged in two groups of five, or six and four. All of the improved methods for teaching arithmetic work on that fact. Anyone who wants the child's mind to really understand when learning arithmetic, anyone who wants children to understand numbers and not just to work ciphers, is now teaching through the use of the senses [handling manipulatives].'
That's the only fault with this otherwise excellent book. It's true that the basic truths of numbers rest on the senses, but, after handling manipulatives for awhile, children do learn to associate numbers with objects so that they begin thinking in numbers instead of objects, which is the beginning of math. Therefore, I think that too many complicated manipulatives--an elaborate system of fancy cubes and props instead of simple tens, hundreds, and thousands, insults the child's intelligence by teaching more than is needed, and puts more emphasis on the manipulatives than on the numbers they're supposed to be illustrating.
But dominoes, beans, line graphs on the blackboard help children to grasp the concept of a large number by using a smaller number. Seeing a symbol of a large number is one thing. Working with that symbol is a different matter.
Except for that one minor flaw, which doesn't make the books any less effective, the books are delightful with their careful analysis of numbers and well-planned graduation of work so that only one difficulty is presented at a time. The examples and little word problems were written by someone who obviously knows and likes children. Anyone interested in teaching arithmetic should read Mr. Sonnenschein's paper on 'The Teaching of Arithmetic in Elementary Schools,' which is in a Board of Education publication.
In the 1840's and 50's, it was thought that continually being exposed to visible signs of geometrical forms would result in the inner mind developing mathematical genius, or at least developing an inclination towards math. But when educationalists of those days gave children boxes of geometric forms and taped cubes, hexagons, pentagons and other shapes on every inch of school wall space, they forgot one thing: we all tend to get bored, especially children. When something bores us, we feel repulsed by it. Dickens' Hard Times has an example of this in Mr. Gradgrind's schoolroom which included lots of outlined shapes. John Ruskin exposes the mistake in a more friendly way than Dickens did. He wrote that geometric shapes abound in nature, and children should experience them in the beauty of the living world. It's backwards to try to plant the image of a shape in a child's mind in artificial ways in the hopes that seeing the form of the shape will give him the idea of geometry. For a beginner, it's probably always the idea that begets the form, not the other way around. Only a trained mind could beget an idea just by looking at the form of a shape. I don't think children need any direct preparation to make them ready for math. If a child is allowed to think, and hasn't been pressured to cram for tests, he will be delighted with learning math when he's old enough. Mathematics are such a great subject because normal minds naturally love it and are able to study it. Too much elaboration, either by preparing or over-teaching, makes math less interesting.
Volume 3, School Education, pg 236
I don't need to discuss mathematics. It already receives enough attention, and is quickly becoming a subject that's taught with living methods.
As far as practical instruction in subjects like Science, Drawing, Manual and Physical Training, etc., I can't do any more than repeat our convictions again. The PNEU believes that children in all social classes have a right to be educated in all of these four subjects. For students under twelve, the same general curriculum should be fine for all of the children. I don't have anything to add to the way these subjects are taught, which is pretty widely accepted by everyone.
Volume 6, Philosophy of Education, pg 7
(i) Success in disciplinary subjects such as math and grammar depends on the ability of the teacher, although the students' habit of attention helps here, too.
Volume 6, Philosophy of Education, pg 51
I've said a lot about history and science, but math appeals directly to the mind and, although it's as challenging as scaling a mountain, it can be just as rewarding. Good math teachers know not to drown lessons in too many words.
Volume 6, Philosophy of Education, pg 110-111
M. Fouillee, who wrote Education From a National Standpoint, thought that the idea was everything in philosophy, and in education. But Fouillee barely touched on education's role in forming physical, intellectual and moral habits. Here's what he wrote:
'Descartes said that scientific truths are victories. If you tell students the key point in the victory, the most heroic battles in scientific discoveries, you'll get them interested in the end results of science. By getting them excited about the conquest of truth, you develop a scientific spirit in them. Imagine how fascinating math might be if we gave a short history of the major theorems of math. Imagine if the student felt like he'd witnessed the work of Pythagoras, or Plato or Euclid. Or imagine if he felt like he'd been there with modern intellects like Descartes, Pascal or Leibnitz. Great theories would no longer be lifeless, anonymous and abstract. They would become living truths, each with a thrilling history of its own, like a statue by Michelangelo, or a painting by Raphael.'
This is a way of applying Coleridge's 'captain idea' at the head of every train of thought. An idea shouldn't be some stark generalization that no child or adult could feed on. Ideas need to be clothed with both fact and story. That way, the mind can do its own work of selecting what it needs and initiating a new birth of ideas from a collection of colorful details.
Volume 6, Philosophy of Education, pg 151-153
Children can drive us crazy arguing a trivial point to death just because they enjoy using their reasoning power. Yet many dislike the very school subjects that seem like they'd give an outlet for their reasoning ability, and might even strengthen it. But very few children enjoy grammar, especially English grammar, which depends so little on inflection. Arithmetic and Math don't appeal to most children, either, no matter how intelligent. Most children are baffled by math, although they may love reasoning out questions of life in literature or history. Since so many dislike those subjects, maybe we should take that as a hint and stop putting so much pressure on those subjects. It would make sense to push grammar and math if children's reason was waiting for us to develop it. But when we see that they have plenty of ability to reason in other subjects, we have to face the fact that they have plenty of reason. They have as much ability to reason as they have ability to love. They don't need us to give them subjects to develop their reason. Our job is to give them lots of material for their reason to work on. If their reason gets sharper, it will be a side effect as they learn their other subjects. At the same time, we can't let them skip grammar and math. Some day they'll delight in language, and in the beauty of the most appropriate words to express a thought. They'll see that words are the vehicle of truth, and shouldn't be carelessly thrown around, or mutilated when written. We need to prepare them for that day. We should probably wait before we have them parse sentences until they're used to analyzing whether they make sense. We should let them play with figures of speech before making them try to break sentences down to small parts. We should keep proper grammatical terms to a minimum. The truth is, children can't really draw conclusions about abstract things. They're good at busily collecting particulars, but they don't commit themselves to deducing anything definite, and we shouldn't rush them. And if language has its own confounding rules, imagine how much more baffling it is for children to work with abstract lines and mathematical figures! We remember how John Ruskin amazed and taught us with his thesis that two and two make four, and the universe has no way of ever making two and two equal three or five. Children should approach math from the perspective of that unalterable law. They should understand how impressive it was when Euclid said that two and two equals three or five is an absurd possibility, as absurd as a man claiming that, on his tree, apples fell upwards. It's absurd to think that apples would break the law of gravity. Figures and abstract lines work just like an apple falling. They are confined to an unchangeable law. It's a great thing to understand the nature of these kinds of laws by experiencing them in their lowest application, gravity. A child who understands how immutable the laws of math are will never divide 15 pennies between five people and give them the wrong amount. He will understand that math answers aren't arbitrary, they're logical, and even a child can use reason to come to the right answer. Math can be enjoyable for a person who loves perceiving a law of nature and figuring out the law behind why things work the way they do. But not every child can be a star wrestler, and not every boy 'takes' to math. So perhaps teachers should make it their duty to expose the child to as many interests as possible. Math is just one subject in education, and it's one that not everyone excels at. So it shouldn't monopolize too much time in the school day. And youths shouldn't be denied good jobs because the subject they're the worst at is one that test examiners love. They probably love it because the answers are final and easy to grade. There are no essay questions to have to make subjective judgments about.
Volume 6, Philosophy of Education, pg 230-233
The subject of math is usually very important to educators. As long as the idea still prevailed that children's faculties needed to be developed, it seemed good to focus a lot of attention on a subject that could help develop the faculty of reasoning. But now we know that children come with reasoning powers already born in them, and those powers don't wait for training from us. They are there with or without us. So if we want to make math the main focus and priority of education, we'll have to find some other excuse to justify it. One strong case for giving math a central place in our curriculum is because of its truth and beauty. As John Ruskin points out, two and two always make four and couldn't possibly make five. That's a truth, an inevitable law. It's a great thing to come face to face with a law, with a whole natural law system that exists and is true whether we agree with it or not. Two straight lines can never enclose a space. That's a true fact that we can grasp, say, and act upon--but there's nothing we can do to change or alter it. This kind of truth helps children have a healthy sense of living with limitations, and inspires a reverent respect for natural law.
Being persons of integrity in all our dealings depends on Mr Micawber's [David Copperfield] golden rule about living within our limits ["Annual income twenty pounds, annual expenditure nineteen nineteen and six, result happiness. Annual income twenty pounds, annual expenditure twenty pounds ought and six, result misery."] But Harold Skimpole's [Bleak House, see this] disregard for these things is a moral offense against society. The mental challenge of math is good for us. Although it's true the body needs more than strenuous aerobic exercise, some exercise is invigorating and healthy. This is as true for the mind as it is for the body.
But education needs balance. No single subject should assume greatest importance at the expense of other subjects that a child needs to know about. Math is easy to test, and as long as education is ruled by test scores, we'll have teaching focused on training exactness and solving problems efficiently, instead of teaching to awaken a sense of awe in contemplating a field of knowledge where perfection lives with or without us.
Some will ask what's wrong with training exactness and problem-solving. But these qualities developed in math lessons don't carry over into other departments of our lives. If they did, then math would be the best way to get a total education. But that's not the case. The habits and powers trained for one specific educational subject will only work for that subject. The anecdote about Sir Isaac Newton making a large opening in his door for his large cat, and a small one for his small cat illustrates this. It isn't that his mind had a mental lapse, but his greatness was in a specialized area and didn't carry over to everything else he thought about. Specialized training only makes a person qualified to work in that specialized field. It's not uncommon to hear about a challenged student who takes to Bradshaw [railroad schedules and routes], or an accountant who's gifted at numbers but unable to function at anything else.
A boy can get straight A's in math, and yet not do well in history because the accuracy and problem-solving skills developed while doing his sums will only apply to working on his math worksheets. How valuable is math to everyday life? Those of us who never excelled in math will heartily agree with the respected military staff officer who said,
'I've never found that math, beyond simple addition, made any difference to my life except when taking my staff entrance exam. As far as the claim that math provides the challenge of mental exercise and training in accuracy, I don't agree that math is the most effective to develop that.'
Most of us have always believed that understanding the theory and practice of battle strategy depends on math. So it's worth considering the officer's words above. Our basic point is that math should be studied for its own sake, not for the purpose of making the mind smarter or quicker. Math is profoundly worthy of study for its own sake, and because it's connected to other equally noble subjects. We should strive for balance when putting together a curriculum, remembering that a brilliant mathematician who knows nothing about the history of his own country or any other isn't very well educated.
Yet we can't overlook the fact that genius has rights of its own. A mathematical genius should be allowed to pursue nothing but math, even if it means sacrificing other subjects that any person should learn. He will be naturally driven to solve math problems, and he should be indulged. He won't even need very much laborious teaching, a lot will come easily to him. But not very many students are math whizzes. Why should they be pressured to focus on math as if they were? And why should a person's success depend on his skill at one thing--the drudgery of math?
The tendency of our universities is to deny students entrance if they aren't strong in math., which means denying them the opportunity to get some jobs. So students who aren't gifted in math have to expend extra time and effort trying to excel at something they have no natural talent for--all the while neglecting the humanities that they're better equipped for. That hardly makes for the balanced, liberal education we hope for.
As a case in point, the bold claims of the London Matriculation exam [this must be like our SAT's] are acknowledged by many teachers to be out of step with the concept of a broad education.
Math, more than any other subject, depends on the teacher rather than on the textbook [because it's easier to grasp a math concept when you see it being explained and used, rather than trying to grasp it from reading a book?] Yet few subjects are taught worse, mostly because teachers don't usually have time to give the inspiring ideas that quicken the student's imagination. Coleridge calls those inspiring ideas the 'Captain' ideas.
Imagine how interesting and alive geometry would seem if students also knew about Euclid and his trials and challenges in discovering geometric principles!
To summarize, math is a necessary part of any education. It needs to be taught by someone who knows math. But math shouldn't take up so much time and attention that other subjects have to be squeezed out. Knowing about those other subjects is every student's natural right.
It's not necessary to exhibit any student math work, since it's the same kind of work that other schools are doing, and reaches the same standard. Having the habit of paying attention undoubtedly gives our PUS students an advantage.
Volume 6, Philosophy of Education, pg 334
It doesn't matter whether the knowledge is physics or literature. There seems to be some inborn quality in the mind that will only respond to a literary form and nothing else. I said 'in the beginning' because I think it's possible that once the mind is familiar with a certain type of knowledge, it unconsciously converts even the driest formula into living dialog. Maybe that has something to do with the reason why math seems to be the exception to the rule about knowledge being in literary form. Math, like music, is a language in itself. Its speech is logical without fail, and always clear. It meets the mind's requirements.
