Most people are familiar with the idea that Miss Mason advocates short lessons, which is true to a point. In all subjects, and perhaps especially math, Miss Mason advocates a brisk, short lesson, so as not to weary the young child. However, lesson length gradually increases, until by high school the students have 45 minute lessons and also homework.
Good teachers, says Miss Mason, will not drown out the teaching with much talk. Volume 6, page 51
Miss Mason advocates beginning with manipulatives (simple ones such as dominoes, beans, and figures drawn on the blackboard or whiteboard) and gradually moving over to abstract numbers. Those are the basics. But Miss Mason has much more of value to say about arithmetic.
"The practical value of arithmetic to persons in every class of life goes without remark. The chief value of arithmetic, like that of the higher mathematics, lies in the training it affords the reasoning powers, and in the habits of insight, readiness, accuracy, intellectual truthfulness it engenders." Volume One, Page 254 and following
"Of all his early studies, perhaps none is more important to the child as a means of education than that of arithmetic. That he should do sums is of comparatively small importance; but the use of those functions which 'summing' calls into play is a great part of education..."
More than any other subject, good teaching matters in math, while "slovenly teaching has more mischievous results."
It is a poor practice for a child to approach math as a random process- trying one thing after another to get the correct answer.
He must be able to look at a story problem and know which process to use to get the right answer, and the right answer is the only acceptable answer. Never, never tell a child that his answer was 'nearly right.' In math wrong is simply wrong.
Word Problems:
Give the children problems like this one:
"Mr. Jones sent six hundred and seven, and Mr. Stevens eight hundred and nineteen, apples to be divided amongst the twenty-seven boys at school on Monday. How many apples apiece did they get?"
Here he must ask himself certain questions. 'How many apples altogether? How shall I find out? Then I must divide the apples into twenty-seven heaps to find out each boy's share.' That is to say, the child perceives what rules he must apply to get the required information. He is interested; the work goes on briskly the sum is done in no time, and is probably right, because the attention of the child is concentrated on his work. Care must be taken to give the child such problems as he can work, but yet which are difficult enough to cause him some little mental effort."
Proceed from the concrete (using manipulatives) to the imagined concrete (suppose we had seven apples and Papa brought home five more) to the abstract (what is seven and five?)
"Demonstrate.––The next point is to demonstrate everything demonstrable. The child may learn the multiplication-table and do a subtraction sum without any insight into the rationale of either. He may even become a good arithmetician, applying rules aptly, without seeing the reason of them; but arithmetic becomes an elementary mathematical training only in so far as the reason why of every process is clear to the child. 2+2=4, is a self-evident fact, admitting of little demonstration; but 4x7=28 may be proved.
He has a bag of beans; places four rows with seven beans in a row; adds the rows thus: 7 and 7 are 14, and 7 are 21, and 7 are 28; how many sevens in 28? 4. Therefore it is right to say 4x7=28; and the child sees that multiplication is only a short way of doing addition.
A bag of beans, counters, or button should be used in all the early arithmetic lessons, and the child should be able to work with these freely, and even to add subtract, multiply, and divide mentally, without the aid of buttons or beans, before he is set to 'do sums' on his slate.
He may arrange an addition table with his beans, thus -
00 0 = 3 beans
00 00 = 4 "
00 000 = 5 "
and be exercised upon it until he can tell, first without counting, and then without looking at the beans, that 2+7=9, etc.
Thus with 3, 4, 5,––each of the digits: as he learns each line of his addition table he is exercised upon imaginary objects, '4 apples and 9 apples,' '4 nuts and 6 nuts' etc.; and lastly, with abstract numbers––6+5, 6+8.
A subtraction table is worked out simultaneously with the addition table. As he works out each line of additions, he goes over the same ground, only taking away one bean, or two beans, instead of adding, until he is able to answer quite readily, 2 from 7 ? 2 from 5?"
When the child has demonstrated a math fact using the beans (or some other manipulative; we use glass beads most often at our house), you can have him write out the math problem he has just shown on a white board or in a notebook, if he is old enough to write down his numbers. The beauty of the method of working with numbers is that a very young child can be working out a math problem with counters, even a child who is not yet able to do any writing.
Subtraction is usually harder for children to understand, and Miss Mason recommends that teachers move slowly, giving children time to grasp the material.
Once the children can add and subtract up to twenty with some ease, begin using the beans to work out multiplication tables.
"...the multiplication division tables may be worked out with beans, as far as 6x12; that is, 'twice six are 12' will be ascertained by means of two rows of beans, six beans in a row.
When the child can say readily, without even a glance at his beans, 2x8=16, 2x7=14, etc. , he will take 4, 6, 8, 10, 12 beans, and divide them into groups of two: then, how many twos in 10, in 12, in 20? And so on, with each line of the multiplication table that he works out.
Problems––Now he is ready for more ambitious problems: thus, 'A boy had twice ten apples; how many heaps of 4 could he make?' "
You can begin to work with adding and subtracting a series of numbers, such as 7 + 5 - 3. If the child still needs to use manipulatives, permit him, but also encourage him to work with imaginary numbers, as a step towards understanding abstract numbers.
"Carefully graduated teaching and daily mental effort on the child's part at this early stage may be the means of developing real mathematical power, and will certainly promote the habits of concentration and effort of mind."
Daily mental effort in arithmetic is important, but one needn't be seated at a desk to do this. While driving in your car give your child math problems, permitting him to use his fingers for manipulatives if need be. Sometimes I use license plate numbers as prompts- it's easier on my brain. I just pick numbers from license plates I see and use them in math problems- simple addition and subtraction at first.
You can also provide daily opportunities for mental effort in math while doing dishes together, reading stories together, making beds, switching out the laundry, folding clothes. Here are some examples:
While washing dishes, just ask your child questions such as how many forks are in the rinse. One he answers, ask him how many there would be if there were two more, or three more, or two fewer.
While folding clothes, set aside socks or washcloths and have the child add them and subtract them while you fold.
When reading a story, sometimes you can make up a few math problems when you come across a reference to a number (Jacob had twelve sons; five little ducks went out to play; three of us afloat in the meadow on the lea...). You don't want to do this all the time, as it would be distracting and irritating in every story, but you can often take a minute or two to do some math problems while reading. When you read about five ducks, for example, ask 'if there were two more ducks, how many would that be? If there were three more, how many would there be? What if three ran away, how many would be left? Can you use your fingers to show me two different ways to make five?' When looking at pictures in picture books, ask the children to count and add some of the things seen in the pictures.
At the grocery store, have the child help bag fruit, counting and adding it as he does. Or you can put three apples in a bag and tell him you want ten, asking him to help you add enough apples to make ten. Tell him you want three cans of beans and two of corn, and ask how many cans that makes all together.
Place Value is next
Miss Mason recommends that they learn this first through concrete objects. She suggests money, and fortunately, our money system is easily used to teach that ten pennies make ten cents, or one dime. She suggests giving a child a pile of pennies, perhaps 100, and pointing out how inconvenient it would be to have to carry around bags of pennies and count them out one by one to buy things.
Because of this, we use other coins- tell him how many pennies a dime is worth, and have him divide his pennies into piles to see how many dimes he could have for his pennies. After he has mastered dimes and pennies, move up to dollars.
To help him understand place value, when counting out his money, have him put the coins worth least to the right of coins worth more.
Teach him that every value has its own place or home- MUS is a good program for this.
Weighing and Measuring
Learn weights and lengths by meauring and weighing things. At the grocery store, have the child guess how much something is and then weight it to see how close he is. Ask him to fill up a bag with one pound of carrots, or 2 pounds of apples. If you have a good scale at home, give the child baggies and some items to weigh. Give him something like rice or sand and ask him to make a bag weighing one ounce, three ounces, six- the more he does this, the more capable he will be at estimating weight and being comfortable with terms like ounces and pounds (and grams, if you use a metric scale).
Do the same with measuring- give the child a ruler and ask him to find out how long his toes, his thumbs, his hands, his feet are- ask him to go find something six inches long, one foot long, 1/2 an inch long. Ask him to cut you out a piece of paper three inches long, or to draw a line four inches long.
Give him a yardstick and ask how long the table is, how wide the porch, how long the bed, how big the bathtub.
When you mail packages, ask him how much he guesses they weigh, and then find out together and see if he guessed too light or too heavy.
"Arithmetic is valuable as a means of training children in habits of strict accuracy," but we dilute that value, encourage lazy thinking, and even dishonesty when we permit children to copy the work of others, when we prompt them with the answers, when we tell them the answers, and especially when we tell them that an answer is 'nearly' right, or even worse, let them do a math problem over again.
MIss Mason says we must "Pronounce a sum wrong or right––it cannot be something between the two. That which is wrong must remain wrong: the child must not be let run away with the notion that wrong can be mended into right. The future is before him: he may get the next sum right, and the wise teacher will make it her business to see that he does, and that he starts with new hope. But the wrong sum must just be let alone."
If the child got the sum wrong because he didn't understand it, then go back to the concrete objects and teach him the material in a different way. If he got it wrong because he was lazy, either let him gt a bad grade, or set before him a new and different problem using the same concepts, and have him do the new math problems.
" Give him short sums, in words rather than in figures, and excite him in the enthusiasm which produces concentrated attention and rapid work. Let his arithmetic lesson be to the child a daily exercise in clear thinking and rapid, careful execution, and his mental growth will be as obvious as the sprouting of seedlings in the spring."
Math is more than puzzles and riddles- this is reducing it from what it is to something less than what it is.
Volume 4, pages 63 and following:
"Reason in Mathematics. -- Never are the operations of Reason more delightful and more perfect than in mathematics. Here men do not begin to reason with a notion which causes them to lean to this side or to that. By degrees, absolute truth unfolds itself. We are so made that truth, absolute and certain truth, is a perfect joy to us; and that is the joy that mathematics afford. Also, there is great joy in standing by, as it were, and watching our own thought work out an intricate problem..."
Volume 6, page 139 and following
Miss Mason expresses some really radical ideas about math here. She says that:
"Arithmetic, again, Mathematics, appeal only to a small percentage of a class or school, and, for the rest, however intelligent, its problems are baffling to the end... Perhaps we should accept this tacit vote of the majority and cease to put undue pressure" upon such studies, although, "we may not let children neglect either of these delightful studies."
MIss Mason says mathematics is valuable in teaching children some basic truths of the universe- the most basic being that there is absolute truth. Two and two "cannot by any possibility that the universe affords be made to make five or three."
The behavior of numbers and geometrical figures are fixed by unchangable law, and "it is a great thing to begin to see these laws even in their lowliest application." The proper approach to mathematics is that of discovering the different laws in effect, and he will see nonsense for what it is as he is able to " perceive that 'answers' are not purely arbitrary but are to be come at by a little boy's reason."
Miss Mason says that "Mathematics are delightful to the mind of man which revels in the perception of law, which may even go forth guessing at a new law until it discover that law; but not every boy can be a champion prize-fighter, nor can every boy 'stand up' to Mathematics. Therefore perhaps the business of teachers is to open as many doors as possible in the belief that Mathematics is one out of many studies which make for education, a study by no means accessible to everyone. Therefore it should not monopolise undue time..."
Volume 6, pages 230 and following
MATHEMATICS
We should communicate to our children the beauty and truth of mathematics. They should understand that it is a "great thing to be brought into the presence of a law, of a whole system of laws, that exist without our concurrence, -- that two straight lines cannot enclose a space is a fact which we can perceive, state, and act upon but cannot in any wise alter..." and this "should give to children the sense of limitation which is wholesome for all of us, and inspire that sursum corda which we should hear in all natural law.
What is this ‘sursum corda?’ Anglo-Catholics will know, but the rest of us (including myself) need some explanation.
In Episcopalian and other ‘high church’ services, there is something called a ‘versicle.’
This is, according to the 1913 Websters, “ a little verse; especially, a short verse or text said
or sung in public worship by the priest or minister, and followed by a response from the people.’
Sursum Coda is the name of a particular versicle in the service- it is the ‘lift up your hearts’ versicle.
So Sursum Coda refers to the portion of the service when the minister says to the church “Lift up your hearts”
and the congregation responds “We lift them to the Lord.”
It appears in the Book of Common Prayer and is used for thanksgiving over communion. This picture is inspiring.
Miss Mason is pointing out that all natural law is God’s law, and is part of God’s voice to us. Whenever we learn
of one of God’s natural laws, whether it be that two and two make four and never three or five, or that apples
fall down and not up, or that all things reproduce after their own kind, or that a blade of grass produces food
from sunlight in a process we now call photosynthesis- it should be to us as though that natural law were the
voice of God (which it is) saying to us “Life up your hearts,” and we should respond with awe and thanksgiving,
“We lift them up to the Lord.”
If we can communicate this beautiful idea to our children about their math, we can hardly be bored by the subject.
“Mathematics are to be studied for their own sake and not as they make for general intelligence and grasp of mind. But then how profoundly worthy are these subjects of study for their own sake, to say nothing of other great branches of knowledge to which they are ancillary! Lack of proportion should be our bête noire in drawing up a curriculum, remembering that the mathematician who knows little of the history of his own country or that of any other, is sparsely educated at the best. At the same time Genius has her own rights. The born mathematician must be allowed full scope even to the omission of much else that he should know. He soon asserts himself, sees into the intricacies of a problem with half an eye, and should have scope. He would prefer not to have much teaching. But why should the tortoise keep pace with the hare and why should a boy's success in life depend upon drudgery in Mathematics?” … To sum up, Mathematics are a necessary part of every man's education; they must be taught by those who know; but they may not engross the time and attention of the scholar in such wise as to shut out any of the score of 'subjects,' a knowledge of which is his natural right.”
Volume 3, page 230:
What a revolution should we have in our methods of education if we could once conceive that dry-as-dust subjects like grammar and arithmetic should come to children, living with the life of the Holy Spirit, who, we are told, 'shall teach you all things.' Nothing so Practical as Great Ideas. -- It may occur to some readers to consider that such lines of thought as I have suggested are perhaps interesting but not practical. Believe me, nothing is so practical as a great idea, because nothing produces such an abundant outcome of practical effort.
