What the “New Math” Teaches Your Child
PARENTS frequently find the “new math” perplexing, especially when their child gets 100 percent for writing 1 + 1 = 10 or 8 + 6 = 2. It is no wonder that a mother was heard to exclaim, “It’s beyond me!” when she saw her fifth grader’s homework.
Not just a few parents in the United States have been deeply disturbed to find their children using mathematical systems completely foreign to their experience. “In the old days,” wrote one concerned father, “a kid could bring home his homework and his parents would go over it with him, making corrections or giving encouragement. But today the homework is so complicated that neither the kid nor his parents know what is going on.”
Even teachers have had to be reeducated. And in some places night courses in the “new math” have been established for parents. But not all like the idea of going to them. One mother, who had received university training for two years, refused to go. “Do you know what it’s like at my age to feel inadequate with second-grade subjects?” she asked. Another parent complained: “This stuff is ruining my relationship with my kids. They think I’m stupid!”
What is the “new math”? Why is it being taught? Is it really better than the old way of teaching mathematics?
Why the “New Math”
The average person no doubt thinks of mathematics as a static subject, but it is far from that. It is estimated that more new mathematics has been created in the past sixty years or so than in all previous centuries combined. Yet the content of math courses changed little in three hundred years. One authority observed that a seventeenth-century teacher could walk into a math class a few years ago and start teaching without difficulty. But a teacher of history, science or language could not do this, since the content of these courses had changed radically. So educators had long felt the need to update mathematics courses.
In the United States public support for such changes was provided when Russia successfully launched its Sputnik in 1957. After that startling space achievement an urgent need was felt for more and better scientists, and, since science rests on mathematics, better math courses. The reformation in math teaching had already begun to a limited extent in higher schools of learning. Now it gained momentum, moving down through elementary schools too.
The purpose of the “new math” courses is to give children a confident understanding of the structure and relation of numbers to one another. They aim to help students to understand the way numeral systems are built up and the laws that govern their behavior. So, instead of simply prescribing rules and emphasizing drill in applying them, “new math” endeavors to go back to the sources of the rules to show that they are valid.
“New math” also introduces children early to advanced mathematical concepts. It shows the interrelation of the various branches of mathematics, such as algebra and geometry, rather than considering them as separate topics.
The “new math” might be compared to a cooking course in which effort is made, not only to provide practice in following the prescribed steps of a recipe, but also to help the student to understand the properties of the various ingredients and their effect when in combination with other ingredients. So the student not only learns how to prepare a particular dish, but also learns why the end product comes out the way it does. Thus the student is helped to obtain a better overall picture of cooking, and so, hopefully, will be a better cook.
Similarly, by helping young math students see the reason for rules and introducing them early to advanced concepts, it is hoped that they will be better equipped to work out solutions to problems and to pursue courses in higher mathematics.
Putting Numerals Together
Not all “new math” courses are alike. There may be considerable variety in them from one school to the next. But as a rule the courses try to teach children why numerals are put together in the way they are. This may seem simple enough, but really it is an artful development of centuries.
For instance, if you could ask someone unacquainted with our present-day numeral system what would be left if 5 was taken from 155, he would probably say 15. Do not be surprised, or think him ignorant. For consider: Does it not really seem that if you take 5 away from 155 it leaves only 15?
Do you say the answer should be 150? But where did you get the 0? Why did you turn one of the 5’s into 0? Could 15 really be the right answer? The “new math” endeavors to answer such basic questions so that children really understand, and do not simply give answers according to the dictates of rules.
If an ancient Egyptian were here he would probably give the answer 15 to the above problem. And he no doubt would assert strongly that he was correct. Do you know why? It is because Egyptians and other ancient peoples used a different numeral system. If they took one numeral (that is, a symbol representing a number) away from a series of numerals, the new sum would simply be the total of the remaining numerals. The sum did not depend upon the order in which the numerals were placed; the numerals kept their respective value whatever their position.
But this is not true today, is it? For 155 is not the same as 551. Why do the 5’s have a different value depending upon their position? It is because we today have a numeral system different from those of the ancient Egyptians, Greeks and other peoples. It is a system created long ago in which the numerals have different values, depending upon their position. The “new math” impresses upon children how this place-value system works.
Decimal Numeral System
Today the decimal numeral system is used in most parts of the world. It is a system that employs ten numerals, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9. In this system each position has a value ten times greater than the position to its right. The numeral in the first position represents a number equal to itself. So the numeral 5 represents the number 5. But if the 5 is one place to the left of the first position, it represents 5 tens, if two positions to the left, 5 hundreds, three positions to the left, 5 thousands, and so on.
The “new math” courses endeavor to demonstrate to children the value of numerals according to their position. Thus students may be taught to add like this:
5,555 = 5,000 + 500 + 50 + 5
2,222 = 2,000 + 200 + 20 + 2
7,000 + 700 + 70 + 7 = 7,777
And they may learn to subtract something like this:
346 = 300 + 40 + 6 = 300 + 30 + 16
239 = 200 + 30 + 9 = 200 + 30 + 9
100 + 00 + 7 = 107
Different Numeral Systems
The decimal system is called a base-ten numeral system. But any other numeral base can be used. Babylonians used a complex base-sixty system and the Mayas of Yucatán calculated in base twenty. Today computers use the base-two system. “New math” courses familiarize young children with different numeral systems. The purpose of this is to help them gain a better understanding of the familiar decimal system, and of arithmetic in general.
The base-five system is perhaps the easiest one to learn, and it may be taught to your fourth or fifth grader. In this system, which uses only the numerals 0, 1, 2, 3, 4, each position has a value five times greater than the position to the right. Thus in the number 324, the first numeral represents itself, or 4. The second numeral, instead of representing 2 tens as in the decimal system, represents 2 fives. And the third numeral, instead of representing 3 hundreds, represents 3 twenty-fives. So 324 in the base-five numeral system is really 89 in the base-ten system!
This same pattern is followed in every numeral system. Thus in the base-six system each position has a value six times greater than the position to the right. And in the base-eight system each position has a value eight times greater than the position to the right. Note the value of the number 324 in the numeral systems below as compared with its value in the decimal system.
324 in base five = 75 + 10 + 4 or 89
324 in base six = 108 + 12 + 4 or 124
324 in base eight = 192 + 16 + 4 or 212
Now do you see why your child may get 100 percent for writing 1 + 1 = 10? In the base-two system the result of 1 + 1 can be written as 10, because the 0 equals nothing and the 1 that is one position to the left of the 0 represents, not ten as it would in the decimal system, but only two! The base-two system uses only the two numerals, 0 and 1. And each position has a value two times greater than the position to the right. So do you see why 111 in the base-two numeral system is equal to 7 in the decimal system? And why 1111 is equal to 15? Can you figure out what 1010 in the base-two system is equal to in the decimal system?
“But how does 8 + 6 = 2?” you may ask. “How can a child be correct in giving this answer?” It is the right answer in the modulo-twelve system.
Modular arithmetic is used to describe events that occur in regular cycles. A common cycle that occurs twice a day, in millions of homes is the passage of the hands of a clock past the numerals standing for the hours of the day. A typical “new math” problem for perhaps fifth or sixth graders is: “If it’s eight o’clock now, what time will it be in six hours?” The answer, disregarding a.m. and p.m., is two o’clock. So 8 + 6 does equal 2!
“New math” students are thus introduced to concepts that may be met in their greater complexities later on. Modular arithmetic, for instance, is used to describe the functioning of electrical generators and gasoline engines in mathematical terms. Its mastery is essential to the work of some persons.
Set Concept
At the core of many “new math” courses is the set concept, which is taught at every grade level. It is a concept so all-prevailing that it permeates the advanced writings of mathematicians, and yet it can be used to teach arithmetic principles to the youngest children.
For instance, a kindergarten child may be shown a picture that has sets of 3 birds, 2 balloons, 3 apples, 2 boys, 3 bicycles and 4 lollipops, and be asked to circle each set that has 3 objects in it. In this way the child learns the idea of a number as the common property of these sets. Then the child can move on to grasp the idea of numbers expressed as numerals.
By gaining familiarity with the way sets operate, children learn elements common to arithmetic, algebra and geometry. It is hoped that this will prepare them to handle more advanced mathematics later on.
Appraisal of “New Math”
Many educators are enthusiastic about the “new math” programs. They feel that students learn much faster. Professor David A. Page, who edited a new elementary math program, asserted: “I can now teach third- or fourth-graders more about mathematical functions in one hour than I used to be able to teach college freshmen in two weeks.”
But this enthusiasm for “new math” is by no means shared by all. In addition to the resounding complaints heard from confused parents, many teachers are also perplexed. Reported Professor Robert Wirtz after visiting more than a hundred elementary schools in the United States: “The teachers I found are frightened. They don’t understand the new math or why they are supposed to teach it.”
Many mathematicians, too, are far from satisfied, including persons who worked on new programs. They feel that some of the programs are too sophisticated, too abstract, and that they fail to place enough emphasis on application to everyday life. One of the leading pioneers of the reform, Max Beberman, expressed fears that modern math may be “raising a generation of kids who can’t do computational arithmetic.”
So “new math” programs do have their shortcomings. Perhaps it was the feeling of urgency to keep abreast with Soviet space achievements that resulted in pitching many programs to the level of students with mathematical adeptness, neglecting the educational needs of others. Also, the lack of teachers who grasp the new concepts sufficiently well to teach them has been another shortcoming. And not to be minimized is the way “new math” has contributed to the generation gap in many homes, serving to alienate parents and children. So, whereas improvements were obviously needed in former math programs, it is questionable whether all the changes made have been the best ones.