A fierce war rages over control of your child's mathematics education.
It is a cruel war, fought mostly on the sly, in which the only casualties
are children. The war is between those who want your child to understand
mathematics and those who just want her to learn it. (For convenience,
I'm going to call "your child" Dana from now on). The stakes for
Dana are enormous. Leaving aside personal benefits, which are substantial,
a good math education is a vital key to entry to a good, or even not so
good, university. Once there, it is a key to entry into the school
of her choice within that university. And a good education is the
key to a comfortable and productive life. Income distribution now
is based largely on the level and quality of education, with highly educated
people moving swiftly into well paying jobs and becoming part of the "cognitive
elite." Others toil for a pittance and suffer the bumps and jolts
that come with life in the slow lane.
Why is proficiency in mathematics required for college work? Certainly
it is required if Dana wants to be an engineer or physicist or mathematician,
but why is it required for a career in literature or philosophy or poetry?
The answer is that proficiency in math is used as a filter to reduce the
number of qualified applicants to many schools. State universities,
for example, are required to accept all qualified resident applicants.
In today's climate, where the importance of a college degree is clear to
all, applications flood those universities. The response is to
shrink the pool of qualified applicants by raising the entrance requirements. Math
proficiency makes a great barrier,
especially in view of the terrible job in math education being done in
so many high schools and grade schools. And what if Dana has great
mechanical aptitude and would make a fine engineer if she chose to become
one, except for a distaste for math acquired in grade school? She
can probably grit her teeth and make it into the university system, but
she is likely to end up with a degree in history and a job in human resources,
not engineering or science. In brief, she will find it harder to
approach her potential in life without a good background in math.
The message of this booklet is that a young child cannot understand
math, not even arithmetic, because her brain is not yet wired for the task.
She can learn to perform arithmetic, all of it, but she can't understand
it. Virtually no one in her age group can, not anywhere in the world.
But there are those who believe she
can understand arithmetic, that
she can explore the subject and discover the underlying principles,
and that she can come to
understand it through that process.
That is the reason for the war, and that is the reason Dana can be harmed
by a system that commands her to do something she cannot do.
What is it about grade school arithmetic that is impossible for children
to understand? Our base 10, Arabic numeral system of working with
numbers certainly seems simple enough. It's not rocket science, after
all. Or is it? We will see that the system is a very sophisticated
method of handling numbers. It is the end product of thousands of
years of thought by some very, very smart people. We also will see
that a child's brain is psychologically immature. It is not wired
to duplicate (rediscover) the work of mature scholars from the past.
Someday, Dana may stand on the shoulders of those scholars and make contributions
of her own, but first she must sit at their feet and become expert in performing
their method. When all is said and done, it is impossible to measure
a schoolchild's understanding of mathematics. But it is quite possible
to measure performance, and that is exactly what college entrance boards
do.
Before we visit the battlefields of this war for control of Dana's mathematical
education, let's take a look at the complexity of the subject matter and
the mental equipment she brings to the classroom.
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-
Levels of mathematical sophistication
–– pre-modern
A. Hunter-Gatherers
There are a few hunter-gatherer societies still in existence, and we
may assume that their level of mathematical development has remained unchanged
throughout the millennia. Many of these cultures have no words for
numbers higher than "two" or "three." In these societies, numbers
sometimes are attached to specific objects. Thus, there may be no
generic term for, much less abstract concept of, the number "two."
The "two" used to quantify one type of animal may differ from the "two"
used to quantify another. There is a faint echo of this in modern
English when we refer to a brace of pheasants or a pair of
twins. The people are primitive (by our standards), but their brain
structures are not. For example, there are no primitive languages.
The languages of the most primitive societies are as grammatically complex
and rhetorically supple as those of the most modern societies. It
is not that primitive people lack the brain power to comprehend more sophisticated
mathematical structures, but rather that they lack the need for a higher
concept of mathematics. In other words, they don't have much to count.
Australian Aborigine children who are exposed to "modern" educational structures
perform as well, on average, as their nonnative counterparts.
B. Herding Societies
Herdsmen have numbers high enough to count the animals in their herds.
There is no concept of numbers as abstract entities or of formal arithmetic
operations. If four animals are to be sold to one buyer and a price
per animal is agreed upon, each animal must be sold in a separate transaction
to avoid confusion and mistrust. As with the hunter-gatherers, they
have developed mathematical structures sufficient to meet their needs,
and no more.
C. Ancient Civilizations
These societies had words and symbols sufficient to generate very large
numbers, and, though they needed mechanical aids to perform calculations,
they could do the arithmetic for any activity from state bookkeeping to
huge construction projects to planning grain production for whole populations.
The Roman Empire was one such civilization, and its method of using Roman
numerals to record numbers, together with counting boards to perform the
arithmetic, is the one that eventually was replaced by Arabic numerals
and algorithms.
The Roman method, which did not use place value, expressed each number
as the addition or subtraction of certain symbols, as with MM DC LXXX VIII,
which we express as 2,688. It is called an additive system
of notation as opposed to our positional system of notation.
Like symbols were added to obtain a subtotal, and the subtotals
were added to produce a total. I, X, C, and M were symbols for ascending
powers of 10, with values of 1, 10, 100, and 1000, respectively, while
V, L, and D were symbols for the intermediate values 5, 50, and 500.
Larger numbers were formed by adding bars to the tops of letters.
A single bar over "V," for example, indicated a multiple of 1,000, for
a final value of 5,000.
Additive systems, by their nature, do not accommodate paper and pencil
work, and so counting boards and counters were required to perform arithmetic.
An addition problem was expressed in Roman numerals, the operation was
done on a counting board, and the result was written in Roman numerals.
The counting board was a very simple adding machine that functioned as
a conversion device between additive Roman numerals and positional
place value, with numerals replaced by counters and the value of the counters
determined by their place (the column they occupied) on the board.
The figure below
illustrates the process of the Roman Counting Board. It is not a
true reproduction.
The counters actually produced two values, additive and positional.
Three individual counters within a place on the counting board were
added to produce an additive value of 3. The particular place those
three counters occupied determined whether they would be multiplied by
1, 10, 100, 1,000, or a higher power of ten. Thus, the three counters
could have a positional value of 3, 30, 300, or 3,000, in the same way that the value
of our "3" is determined by its place within a number. The difference
is that our "3" is a symbol in which the addition (1+1+1) already has been
done. It is a critical difference, as we shall see.
Although the use of Roman numerals and the counting board was a workable
method of performing arithmetic and recording numbers, it still was primitive
by comparison to our place value method of manipulating
numbers
instead of objects. Our modern system can deal with any number,
no matter how large, using only ten symbols (0 - 9), but the Roman system,
where ten "I"s made one "X," ten "X"s made one "C," and ten "C"s made one
"M," required an infinity of symbols to handle an infinity of numbers.
That said, the Roman numeral – counting board system did indeed work.
It met the needs of the Roman Empire, and no more. Almost anyone
could learn to use it, and there was an added benefit in that it was difficult
to alter Roman numerals on commercial documents. (It is comparatively
easy to alter "333" by changing a "3" to an "8," by adding a "0" at the
end, or by adding a "9" at the beginning; that's why we express numbers
on bank checks in both Arabic numerals and words). The system functioned
well enough to remain in place until the greatly expanded commercial activities
of the Industrial Revolution required a faster and more efficient method
of dealing with numbers.
D. Babylonian base 60
Mesopotamia, one of the great centers of civilization of the ancient
world, developed a unique base 60 numbering system using place value.
(Base 60 means dealing with groups of 60 instead of groups of 10).
A remnant of that system divides our modern days into seconds, minutes,
and hours, with 60 seconds in a minute and 60 minutes, or 3600 (60 x 60)
seconds, in an hour. For example, 11:22:33 A.M. means 11 hours, 22
minutes, and 33 seconds after midnight, a simplified expression of (11
x 3600) + (22 x 60) + (33 x 1) seconds after midnight (the total is 40,953,
illustrated below). We know from archeological evidence that the
system was employed in the days of King Hammurabi of Babylonia (c. 1700 BC),and
the sophistication of the mathematical calculations indicates
it was in use long before then. Base 60 used only two numerals (as
opposed to our ten Arabic numerals or the Roman infinity of numerals),
a vertical wedge shape that designated "1" and a horizontal wedge shape
that designated "10." When combined, those symbols had both additive
and positional values. Two horizontal wedges and two vertical wedges
(10+10+1+1), for instance, had an additive value of 22. The
positional value of 22 was determined by its place within a number.
The figure below shows how the Babylonians would have recorded 11:22:33
(A.M.).
Babylonian place
value notation with cuneiform numerals - these are not counters!
Base 60 had several advantages over base 10. Very large numbers could
be generated and handled quite easily. The repetitive nature of the
numbering system eliminated the need to assign differing symbols for the
ascending powers of 60 (as opposed to the Roman method of using letters
of the alphabet for the ascending powers of 10). Fractions were easier
to handle because 60 has more factors than 10 (1, 2, 3, 4 ,5 ,6, 10, 15,
20, 30, and 60 all divide evenly into it). Perhaps the most important
advantage was that base 60 dovetailed nicely with their 360 day calendar
and 360 degree circle. The scientific communities of Greece, Rome,
and eventually all Medieval Europe, used base 60 as a sophisticated alternative
to the more common Roman (or Greek) numerals until it, too, was replaced
by Arabic numerals and algorithms.
The Babylonian base 60 numbering system was the first recorded example
of written place value and held splendid promise, but there were
drawbacks. First, there was no zero. An empty place
was held by an empty space. In the example "11 22 33," given
above, if the middle place were empty (11 0 33), the number would
be recorded "11 33." Carelessness
with spacing caused uncertainty about the value of the number.
Second, it required additive notation within each place, so that
"59" would be written as 5 horizontal wedges and 9 vertical wedges, a clumsy
system. (If Arabic numerals were restricted to two symbols, "5" and
"1," we would write "873" as "5111 511 111"). Therefore,
evolution into a place value system similar to our own would have required
60 individual numerals (including zero), a disabling obstacle.
Babylonian astronomers and scholars who worked with base 60 were, to say the very least, extraordinary
mathematicians who handled very complex problems with very simple tools. Counting boards, which were useful for base 10
arithmetic but not base 60, were not in play. Beyond that, high-end Babylonian math was far, far more complicated
than simple arithmetic (although it's likely Babylonian merchants handled simple arithmetic in much the same way as
the Romans). What the mathematicians did was solve complex problems once, then record the event on a clay tablet. Over
time, the tablets evolved into formal tables that were stored in libraries and used by future generations to
solve other, more complex problems. No one is sure how the original problems were solved.
Why were Babylonians forced to rely on large libraries of mathematical tables (inscribed on clay tablets) instead
of working individual problems out
using the equivalent of paper and pencil? One reason is that was no papyrus, much less paper. Clay was the available
medium. Papyrus had been invented in Egypt, but didn't appear in Mesopotamia until much later. A more
fundamental reason is that manipulating numbers on paper is not possible unless one already has memorized the
requisite addition, subtraction, multiplication, and division facts. Excluding the zero, there are 9 working numbers
in base 10. That means there are a total of 81 (9 x 9) addition facts, and 81 subtraction facts. Had base 60 been
equipped with a full array of numerals (instead of only two), it would have been necessary to memorize 3,481 (59 x 59) addition facts, and
an equal number of subtraction, multiplication, and division facts, for a grand total of 13,924 facts, an extremely
daunting task, even for those mathematical heroes.
In any event, base 60 met many needs, and so it lived on and spread to the learned
classes in Greece, then Rome, then, eventually, all of Europe. It also spread to the Indian subcontinent, and there
it became involved in one of the most momentous events in the history of mankind.
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The development of our modern
numbering system
A. India and the birth of "Arabic" numerals
Like Greece and Rome, India retained the use of base 10 (with letters
of the alphabet representing the ascending powers of 10) for trading purposes
and for basic arithmetic. Both systems required the use of a counting
board. Around 500 AD, a truly intelligent person got to thinking
about how base 60 numeration worked so well using a limited number of numerals
(two only, the vertical and horizontal wedges), and how convenient it was
to have the value of these numerals determined by their position in a number,
as the value of counters was determined by their position on a board.
He must have pondered how nice it would be to have that same simplicity
of repeating numerals in base 10.
Whatever his thought processes,
this unique individual combined the principles of the counting board and
base 60, and developed a system of repeating numerals for base 10.
There were 9 numerals involved (1 - 9, zero had been developed by the Persians
during their tenure in Mesopotamia around 300 BC, but the concept had
not spread). Astoundingly, he devised a set of numerals in which
the additive nature of the existing system was bypassed because the numerals
were symbols for addition that had already been done! This
insight occurred only once in the history of mankind, and so it seems a
shame that the singular accomplishment of this unknown Indian genius has
been given the misleading popular name
Arabic numerals (the formal
name is Hindu Arabic numerals, which seems small compensation).
Roughly two hundred years later, Hindus incorporated zero into the system
and almost all the components of our modern base 10, positional notation
system of numbers were in place.
B. Islam constructs a cultural highway from India to Europe
In 711 AD, the recently established and rapidly expanding Islamic Empire launched successful
military campaigns against both Europe and India, thereby connecting Europe
to all the intellectual riches from Alexandrian Egypt to the Indus Plain.
The cultural flow was one way, since what had been the Western Roman Empire
was by then ruled by the children of barbarians. Islamic scholars
acquired the Hindu base 10 place value system and, in ninth century Baghdad,
a Persian mathematician named al-Khowarazmi published a book in Arabic
about the new numbers and the rules (algorithms) for dealing with them.
(Algorithm is a corruption of his name; the rules he passed
along remain unchanged to this day).
Hindu Arabic numerals were not on the fast track. Al-Khowarazmi
wrote his book in the ninth century, but the news didn't reach Moslem Spain
until the eleventh century. Christian scholars learned the "new"
method from the Moslems and brought it to the rest of Europe, where it
languished for hundreds of years. Throughout the remainder of the Middle
Ages, the Renaissance, the Reconquista of Spain, the conquest and colonization of the New World,
the Reformation and the Counter-Reformation and the countless wars, most
Europeans, excluding the great Italian banking families, clung to their
tried-and-true Roman numerals and counting boards. Why was this so?
One reason is that Arabic numerals, which initially require more discipline
and work (to learn the arithmetic facts and the algorithms), exceeded the
needs of the Europeans. Roman numerals met their needs exactly, no
more, no less. When Roman numerals and counting boards failed to
meet the needs of the Industrial Revolution, Arabic numerals and algorithms
were embraced, at last.
Scholars and scientists also refused to adopt Arabic numerals.
They were content with the ancient Babylonian base 60 method, and for another
very good reason. Base 60 could deal with decimal fractions;
Arabic numerals couldn't (yet). Moving from right to left in base
60, the number in the first (units) place is multiplied by 1, the number
in the second place is multiplied by 60, the number in the third place
is multiplied by 3,600, and so on. But three thousand years earlier
the Mesopotamians had figured out that numbers also could move to the right
of the units place. Thus, the first number to the right was multiplied
by 1/60, the second number was multiplied by 1/3600, and so on. The
powers of 60 could be negative as well as positive.
This is how they
would have expressed the powers of 60, had they the use of Arabic numerals
and exponents. Powers of 60 could be negative.
|
602
|
601
|
600
|
60-1
|
60-2
|
|
3600
|
60
|
1
|
1/60
|
1/3600
|
It wasn't until the fifteenth century that a Moslem scholar perceived
that "10" also could have negative powers.
Powers of 10 can
also be negative.
|
102
|
101
|
100
|
10-1
|
10-2
|
|
100
|
10
|
1
|
1/10
|
1/100
|
Again, the new knowledge spread to Europe, and in the seventeenth century
the Europeans made one of their few contributions to the Hindu Arabic base
10 place value system, and a splendid one it was: the decimal point.
Arabic numerals were ready and waiting for the Industrial Revolution, which
arrived one hundred and fifty years later.
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The six elements of our modern
numbering system
1. Arabic numerals (0, 1, 2, 3, 4, 5, 6, 7, 8, 9)
All of the numbering systems used before the arrival of Arabic numerals
were additive. For example, the Romans had a word for "8," octo,
but they needed 4 symbols to write it, "VIII." It was necessary to
add the values of those symbols to arrive at the intended quantity, 8.
In contrast, the quantity represented by the Arabic numeral "8" needs but
a single symbol to represent it because the addition already has been done.
2. Ten, the fundamental number, is invisible.
Ten is the fundamental number of our base 10, place value system,
but we never see it because, unlike the Romans, we have no symbol for it;
we never see the fundamental number of any place value system. In
base 10, if we start from zero and add single units (0+1=1, 1+1=2, 2+1=3...),
we will soon arrive at the fundamental number, ten. Just as
with an old-fashioned adding machine, the ten is represented by
a "1" which has been moved to another place (the adjoining column
to the left), where it now represents 1 group of ten "1"s,
and a zero, which occupies the first column.
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If, for some reason, we were to convert our place value system
from base 10 to base 8, an interesting thing would happen.
We would still have a word for 8, eight, but we no longer would
have use of the numeral. Starting from zero and adding single units,
as above, we would arrive at the new fundamental number, eight,
at which point we would write a "1," which now would mean 1 group
of eight 1's, in another place and append a zero to the right as
place holder. If Dana had eight dollars, we would say, "Dana
has eight dollars," but write, "Dana has 10 dollars." Strictly
speaking, the written term "base 10" can refer to any base whatsoever.
"There are 10 types of people: those who understand binary and those who
don't" is a written witticism that affords merriment to those who
understand that "10" here expresses the base 2 value of "two."
Continuing on in base 10, if we keep adding single units we soon will accumulate
2 groups of ten units, then 3, 4, and so on until we have 10 groups of
10 units. We can't keep a ten in any column (because ten,
as our fundamental number, has no symbol), so the new ten is rolled
over into the next (third) column and represented by a "1," which now means
1 group of 10 x 10 units. Another way to write 10 x 10
is 102, where the "2" is called an exponent
(an exponent indicates how many times a number is multiplied by itself).
We write it as "102," but we say "10 squared"
or "10 to the power of 2."
3. Place value
The place in place value refers to the position of the various
Arabic numerals (0 - 9) in any number. Place determines the
final value of each number by attaching a hidden multiplier (the
fundamental ten raised to some power) to that number. To
the left of the decimal point, the number in the first place is
multiplied by 1, the number in the second place is multiplied by
10, the number in the third place is multiplied by 100, etc.
Moving to the right of the decimal point, the number in the first place
is
multiplied by 1/10, the number in the second place is multiplied
by 1/100, the number in the third place is multiplied by 1/1000,
etc. The figures below illustrate the meaning of 345.34.
The meaning of 345.34.
|
102=100
|
101=10
|
100=1
|
10-1=1/10
|
10-2=1/100
|
|
x 3
|
x 4
|
x 5
|
x 3
|
x 4
|
|
345.34 =
|
300 +
|
40 +
|
5 +
|
3/10 +
|
4/100
|
4. Shorthand notation
"345" is shorthand for 300 + 40 + 5, which is shorthand for (3 x 100)
+ (4 x 10) + (5 x 1), which is shorthand for (3 x 10 x 10) + (4 x 10) +
(5 x 10/10). That is a lot of shorthand. The genius of Arabic
numerals is that we can manipulate them, using the standard algorithms,
without giving thought to the shorthand involved.
-
This amazing simplicity of operations is why children
can learn to do arithmetic without understanding it.
And the same simplicity is what misleads so many educators. They expect your child to
be capable of understanding something that appears to be so simple. But arithmetic is not
simple at all, and understanding it is not easy. It is beyond the reach of small children.
5. Zero
The Roman system, where 202 was expressed as CCII, did not require a
zero. Our place value system does. Zero has two functions:
as a place holder (in 202, for example, where we could as easily write
2 - 2, or 2 2), and as a working number (5 + 0 = 5).
In effect, zero has replaced ten.
6. Algorithms
Every numbering system needs an operating system. The Romans used
their counting boards; we use algorithms, which are step-by-step procedures
for solving certain categories of mathematics problems. Taken together,
they form the operating system for Arabic numerals. Each type of
arithmetic problem has its own algorithm. For example, here is the
algorithm for addition of whole numbers, without carry-over.
Adding 123 and 456
The addition algorithm seems such a simple and straightforward way to
combine two quantities that there is a temptation to view it as easily
discoverable. It is important, however, to realize that this algorithm
is the end product of over 5,000 years of human thought. The first
agricultural settlements appeared in Mesopotamia between 5,000 and 6,000
BC The Sumerians, who established the basic features of Mesopotamian
civilization, arrived about 3,500 BC Four thousand years passed
before the unsung Hindu genius invented "Arabic" numerals, and it took
another 1,200 years or so before the numerals and operating system were
refined to their present form. In all the years since the algorithm
was published by al-Khowarizmi , no better method of dealing with "123
+ 456" has been devised.
As mentioned above, learning Arabic numerals and how to operate them
requires discipline and work. The discipline must be imposed and
the work must be directed. In other words, it must be taught.
We will review the history of how this was done and why it is being done
the way it is today, but first let's take a look at the tools Dana will
bring to the educational process.
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Psychological development of
children
A. Infancy
Dana was born into a personal paradise.
Call it her Garden of Eden, if you wish. There were three huge differences
between her world and ours. First, since her brain could not as yet
comprehend the relationship between (apparent) size and distance, she perceived
solid objects as being elastic. Her mother's face would grow large,
then small, then large again, as if by magic. Second, being physically
helpless, Dana did not work. Everything was brought to her; everything
was done for her. If things weren't exactly to her liking, she used
her inborn instincts to promote change. An involuntary smile or cry
made her mother's face grow large. Warmth and comfort and nourishment
followed. She maintained eye contact with her mother while nursing,
which prolonged the encounter and secured a full meal, and she did those
things without volition. Third, she was literally a learning machine
and learned without effort. Among her most important powers was an
inborn ability to acquire language without formal instruction of any kind.
Born with a portion of her brain already hardwired for the task, that
area was fully engaged even as her family prattled "meaningless" words
to her.
Her horizon expanded as she learned to crawl, then walk. She relentlessly
investigated every single aspect of her kingdom and continued to learn
at a prodigious rate, effortlessly. By age four, she had acquired
at least one complete language, and her language skills could outperform
the world's largest computer. She even displayed a glimmering of
formal logic when she ran into the house screaming, "Johnny hurted me!"
because she was applying previous examples of past tense verb formation
to current events. And she was absolutely superb at absorbing facts,
although she couldn't form a logical connection between chasing a ball
into the street and the very grave danger of being hit by a passing car.
The power of formal reasoning lay years in the future.
B. Pre-operational stage of development
Dana's magical kingdom began to dissolve as her brain adjusted for the
puzzling changes in the size and shape of things. But newer concepts,
such as "how much" and "how many," were quite hazy. She would choose
orange juice in a tall, narrow glass over an equal volume of orange juice
in a short, wide glass because there appeared to be more juice in the tall
glass, and she would stay with her choice even when the juice was poured
back and forth from one glass to the other as proof that the volume of
juice was the same in either glass. She would choose a widely spaced
group of 5 jelly beans over a tightly spaced group of 5 jelly beans for
the same reason, in spite of having solemnly counted out "5" in each group.
Counting out loud ("one, two, three...") was not much more than a vocabulary
building exercise. Although her parents were thrilled that she was
"learning her numbers," it had nothing to do with learning math.
As her fine motor skills improved and she learned to write symbols for
those words (1, 2, 3...), it still had nothing to do with learning math.
A loosely spaced "5" was considered to be greater than a tightly spaced
"5." Dana was at the "pre-operational" level of development where
numbers existed but had no real meaning.
And then one day something happened. Dana could no longer be fooled
by the jelly bean trick. She had achieved conservation of number,
whereby the space occupied by the jelly beans no longer distorted
the
quantity. It was not something she was taught; it can't
be taught. It was a direct consequence of the ongoing development
of her brain into a complex, bilateral structure. Dana was slowly
moving from her own magical kingdom into the world of reason.
Unfortunately for Dana, there is a tradeoff attached to the growing
sophistication of her brain. As her power to reason increases, her
ability to learn without effort decreases. Everything previously
done so effortlessly was done because of her instinctive tendency to do
those things. She does not, however, have an instinctive tendency
to learn the addition and subtraction facts. That will require work,
a concept with which Dana will become familiar. Hopefully, she will
learn that work can be rewarding because tears and smiles, while still
somewhat effective with Mom and Dad, are proving much less effective with
siblings and playmates and teachers.
C. Concrete operational stage of development – attaching numbers
to objects
Dana has entered what many psychologists call the concrete operational
stage of development. Space and motion no longer distort quantity.
She conserves number, and so is ready to deal with concrete manipulatives,
such as counters or beans, and learn a little math. At this level,
number
is regarded as an attribute of objects, much as color or size or shape
or texture. This level of understanding can be built upon and carry
her through grade school arithmetic and beyond. The ancient world
operated at this level. Armies were paid, grain production for entire
empires was analyzed, the Egyptian pyramids and the Hanging Gardens of
Babylon were constructed. But there is a higher level of understanding
numbers, a level that views numbers as independent, abstract entities.
This level will be beyond Dana until she reaches the final stage of psychological
development.
D. Formal operational stage of development – power to reason
with formal logic
There was a dramatic change in Dana's mental apparatus when she attained
conservation of number and entered the concrete operational stage of development.
She won't experience another such change until she is somewhere between
11 and 15 years old, at which time she will enter the final stage of psychological
development, called formal operational by some psychologists, and
become equipped to deal with formal logic and abstract concepts.
Just as there was no way to hurry the change from pre-operational to concrete
operational, there is no way to hurry the change to formal operational.
Her brain will complete its bilateral hardwiring at its own pace.
Dana's brain can be nourished and exercised, but it can't be hurried.
This is not to say that one day Dana won't be able to employ formal reasoning
and the next day she will. Rather, it means that there will come
a time when she is ready for an introduction to ideas employing formal
logic, but that time is not now.
This final stage of psychological development is marked by the potential
to view numbers in an entirely different way. If this potential is
reached, Dana someday will view numbers, not as attributes of concrete
objects, but as independent and abstract entities with infinities of forms.
It is at this level that the difference between numbers and numerals becomes
clear. When five, for example, is considered in the abstract,
it can be represented by many different forms, an infinity of forms, in
fact, much as the abstract concept of beauty can have an infinity of forms.
Five
can be represented symbolically as 5, 8-3, y+2,
the
square root of 25, the 3rd prime number, and so on. Numbers
at this level can be viewed as anti-words because numbers have only
one meaning, but many forms, whereas words have many meanings,
but only one form. (Logicians have developed a synthesis called Symbolic
Logic, where the "symbols" have one form only and one meaning only,
but that way lies madness). Please keep these thoughts in mind when
we review the history of New Math and the ruinous effects of introducing
abstract concepts to first and second graders.
Now that we know something of the hidden complexity and sophistication
of grade school arithmetic and the developmental stage of Dana's brain,
it is time to visit the war between those who want Dana to
understand
math and those who just want her to learn it.
-
-
A brief history of European
and American mathematics instruction
A. From the Middle Ages to Sputnik
In Medieval Europe, warfare was the business of the rulers and the burden
of the ruled. Many male children judged unsuitable for war duty were
encouraged to join the Church, and there they became the clerics who kept
the books for the various kingdoms, bishoprics, and fiefdoms. They
performed their calculations on counting boards and wrote the answers in
Roman numerals. The Industrial Revolution changed that system.
Since there were not enough clerics to staff the proliferating counting
rooms of Europe, training for mathematics was separated from training for
the religious life. Arabic numerals were taught, and the products
of this teaching were called clerks (a modification of cleric).
The new students tended to be more unruly than their predecessors, and
so mind numbing, unending calculations formed the backbone of the curriculum,
both as a teaching method and as a disciplinary device. No effort
was made to encourage understanding of the material. This system
of training students in mathematics was transferred to the New World and
remained intact through World War II. But many educators were unhappy
with the system and sought a better teaching method, a way to encourage
understanding
of the material, thereby eliminating the need for the constant, boring
drills that turned so many students away from math. Though money
was scarce, several universities experimented with new teaching methods.
These new methods were similar and later were given the collective name
of New Math. Experimentation was measured, results were analyzed,
and progress, though slow, was steady.
B. From New Math to Newer Math
The successful flight of Russia's Sputnik in 1957 changed things
forever. The federal government, already worried about potential
shortages of qualified engineers and scientists to staff our increasingly
technical industrial base, became terrified that we were falling behind
the Russians. Purse strings were opened, and harmful effects of the
new money soon appeared. The measured pace of research was replaced
by a frantic scramble for market share in textbooks and teaching programs.
It quickly became a gold rush in which everybody had a license to "improve"
the teaching of mathematics. Set theory became the grand vehicle
that would lead to understanding. Never mind that set theory, until
then, had been the domain of professional mathematicians and graduate students.
Children who lacked the ability to deal with abstract concepts because
of the immaturity of their brain structures were informed of the difference
between numbers and numerals, that "5," when written, was a numeral,
a symbolic representation of the set of all mathematical operations from
which an abstract, unwritten number "5" could be derived.
The properties of numbers (not numerals) were introduced: associative
[(5+6)+7 = 5+(6+7)], commutative (5+6 = 6+5), and distributive [5x(6+7)
= 5x6 + 5x7]. These were the tools that would lead to discovery,
then understanding. Children were reminded that as soon as they recorded
a certain number in writing, they were using a numeral to
do so. There is more. The children were expected, to a great
extent, to explore these concepts on their own!
Those of you reading the handsome, cyber-bound edition of
this booklet on the web may be interested to know of another consequence
of Sputnik. A federal bureaucracy called ARPA (Advanced Research
Projects Agency) was established to oversee the research activities of
increasingly worried military services and increasingly busy defense
contractors. Around 1970, a system was devised that would allow various
components of ARPA to transfer data via computers in the event of
nuclear war. ARPAnet was born. From a union of ARPA and the
National Science Foundation came NSFnet. From NSFnet came the Internet,
which, as time went by, developed an internal organ called the World Wide
Web. And so the beneficial apparatus derived from one consequence
of Sputnik enables us to communicate about the harmful effects of another.
New Math was a catastrophe. Teachers were untrained (and untrainable),
parents were ignored, school officials were assailed, but only the children
were harmed. So great was the damage from New Math that a counterrevolution,
called Back to Basics, was formed. Back to Basics was gaining
momentum when the well funded and well connected (remember the chase for
market share) New Math camp counterattacked by calling up painful images
of the way math was taught in the bad old days. This is the Space
Age, they argued, we must do more than teach future farmers how to plan
for next year's crop. New Math had its shortcomings, they admitted,
but they had reformed the method. In fact, they called it
Reform Math. Reform was an exquisitely clever choice of words
because reform, after all, was what the Back to Basics people wanted.
Many parents mistakenly believed that Reform Math and Back to Basics were
the same.
Parents were pushed out of the loop from the first days of New Math.
In earlier days, a child who needed help with math stood a good chance
of finding it at home, from a parent or an older sibling. But in
a system that even competent and dedicated teachers couldn't comprehend,
parents, including those who enjoyed math and were good at it, were helpless
because the textbooks seemed to make no sense at all. Parents tilted
at windmills as teachers floundered, school district officials shook hands
with the textbook dealers, and the various state governors politely yawned
and stood above it all. New teaching programs still are introduced
with great fanfare, but the programs do not work. Pattern recognition
is now the rage. It is declared that pattern recognition will lead
to mathpower. Underlying everything is exploration
and group work, calculators and games.
Many school administrators now pin their hopes on computers, although there
probably is not one truly effective math teaching software program available.
Talk about a race for market share. (The Math Path computer program
teaches facts, not mathematics. The Math Path system teaches
mathematics).
In effect, students are expected to teach themselves. Any way
to arrive at a particular answer, any way at all, is fine, for it demonstrates
that the student has gained an insight into the underlying process.
Beyond that, the worst of the schools don't demand a correct answer.
The effort expended, and the improved self image from the praise that effort
elicits from the teacher, are quite sufficient. Mathpower is sure
to follow. Sadly, many teachers never even pose specific arithmetic
problems. Instead, they assign groups to discover for themselves,
for example, how a small, urban business can turn a profit (so much for
the Space Age; at least we've moved off the farm). Group work, of
course, provides the perfect cover for shy or lazy children.
There are those who want Dana to understand math and those who
want her to learn it. For the first group, we can lump together their
many programs and call them Newer Math (my term, others use the pejorative
"Whole Math," in reference to the equally disastrous "Whole Word" approach
to reading, or "Feel Good Math," from the constant praise heaped on flailing
students). Parents and some concerned teachers who form the bulk
of the second group might call their various programs "Basic Math."
The pejorative for this is "rote learning." Newer Math students are
encouraged to discover their own algorithms and gain understanding through
that process. We have seen that the standard algorithms are the end
product of over 5,000 years of human thought on numbers and how to manipulate
them, so it's not likely that Dana will duplicate this work in her first
few years of grade school. Basic Math students are taught
the standard algorithms, but there is indifference (duly noted and advertised
by the Newer Math crowd) as to the amount of understanding acquired in
the process. Here is a summary of the two philosophies:
Newer Math –
teaches for
understanding. |
Basic Math –
teaches for
performance. |
| **pejorative: "feel good"
math |
**pejorative: "rote learning"
math |
| **algorithms self-taught |
**algorithms taught |
**algorithms expected to be
understood |
**algorithms not expected
to be understood |
**arithmetic operations
de-emphasized |
**arithmetic operations
emphasized |
| **calculator use encouraged |
**calculator use forbidden |
**student performance is hard
to measure |
**student performance is easy
to measure |
| **teachers need special skills |
**teachers need no special
skills |
**students become comfortable
with themselves |
**students become comfortable
with numbers |
| **curriculum is wide and shallow |
**curriculum is narrow and
deep |
Since there is only so much time and energy available for the acquisition
of math skills, common sense tells us that a compromise should be possible
between the competing philosophies, but there is no common ground.
From the viewpoint of those in the Newer Math group, teaching for understanding
and teaching the formal algorithms are mutually exclusive. Teaching
a specific algorithm forecloses the exploration process that is the basis
of their philosophy. If someone tells Dana how to do it, how
is she going to discover how to do it? Huh?
Consider this. A system of dealing with numbers as sophisticated,
though seemingly simple, as our own was developed only once throughout
the entire course of human history. Why then, should we believe that
the Newer Math crowd is capable of inducing Dana, a pre-adolescent years
away from entering the formal operational level of psychological development,
to "discover" this system. The answer is that we shouldn't believe
it. Dana is capable of learning the algorithms, and of understanding
the mechanics involved, but she is not capable of discovering anything
of mathematical value on her own. Perhaps someday, in her mature
years, Dana will emerge as a giant and invent a better method of dealing
with numbers. If that happens, people surely will remark how simple
and straightforward her system is, how painfully obvious.
What about the Basic Math crowd? They are not opposed, in principle,
to children understanding math. But they feel that the present pursuit
of understanding reduces the likelihood of their children doing well enough
at math to get in and out of a good college and get a good job afterward.
Math power may foster Dana's self esteem, but comfort with numbers and
expertise with the algorithms will enable her to take her math education
as far as her talents will allow. She might even become a graduate
student in the mathematics department of a major university. That
will be the time to seek true understanding of mathematics, perhaps by
investigating set theory or pattern analysis.
-
-
The truth about "true" understanding
There is a world of difference between true understanding of math (or
even numbers) and reaching a level of competence where things seem to make
sense. Bertrand Russell, a very gifted man who sought a true understanding
of mathematics, became confused while pursuing the logical basis of 5 plus
3 equals 8. A group of prominent French mathematicians, the Bourbaki
Group, produced a paper dealing with the fundamental characteristics of
numbers. The entry for "1" ran to over 200 pages.
The plain truth is that Dana probably never will achieve a true understanding
of even one segment of math. If she is lucky enough not to become
the victim of a "feel good" math program or of math phobia, the math she
is learning now and the math she has yet to learn will start to make sense
in high school, after her brain has formed the hardwiring necessary to
make the logical connections. Well before then, however, she can
become expert in performing grade school arithmetic, and even some algebra.
By "expert," I mean that she should be able to perform rule based calculations
without consciously thinking of the rules. Using that definition,
most of us are expert drivers. We jump in the car, drive to the store,
and jump out without having given one thought to the rules of driving.
Yet very few of us could be said to truly understand cars. Certainly,
when we first learned to drive, no one gave us the keys to a car and told
us to drive off and discover the truth about automotive transportation.
We were thoroughly grounded in the rules of operating cars, and now we
are expert at the operating system and hardly ever think about the rules
we so nervously mastered. Our status as expert drivers gives us an
independence and freedom without which we would be substantially handicapped
in life.
Continuing the automobile analogy, "Why do I have to learn to drive
a car?" is a complaint none of us have ever heard a teenager make.
The benefits of learning to drive are so obvious and so immediate that
failure in that area is not an option. It is, in fact, unthinkable.
Whatever actions are necessary to obtain a driver's license will be undertaken.
Yet "Why do I have to learn math?" is a constant refrain. The child
complains because neither the benefits of success nor the consequences
of failure are immediately apparent. But if Dana fails to become
expert, first at arithmetic and then at the higher levels of math required
for college work, her options in life will be substantially narrowed.
Therefore, it is the parents' job to keep in mind that failing to learn
math is not an option for Dana. It is unthinkable.
-
-
What's a poor Parent to do?
Many parents have discovered that the ruinous effects of the "whole
word" reading system can be overcome by a comparatively brief period of
tutoring, using any of the various phonics systems that are commercially
available. Those parents who have taken advantage of the phonics
programs have, in effect, wrested from the school the responsibility for
their child's progress in reading. In like manner, the harmful effects
of "whole word" math programs can be overcome when parents assume responsibility
for their child's progress in basic arithmetic.
Monitoring progress in math is more difficult than in reading.
Parents can monitor a child's progress in reading simply by asking her
to read for them. Either she has trouble reading or she doesn't.
But they can't monitor her progress in Newer Math because they have been
cut out of the loop. They don't know what the programs are supposed
to do nor how to measure progress.
Tutoring also is more difficult in math because
the critical learning period is much longer. Everything in math is
built upon, and requires mastery of, things that went before. Memorizing
the addition facts requires knowledge of numbers. Executing the addition
algorithm requires mastery of the addition facts. The multiplication
algorithm requires mastery of multiplication facts, addition facts, and
the addition algorithm. Building upon what went before continues
through advanced calculus. Mathematics is structured like an upside
down pyramid, and basic arithmetic forms the small base. If the base
is weak, nothing of mathematical consequence can ever be built upon it.
You must monitor your child's progress in the fundamentals of basic
grade school arithmetic. If the school can't or won't teach basic
arithmetic, then you must arrange for it to be done elsewhere. You
can do it yourself or pay an outsider to do it. Your child must become
expert in addition, subtraction, multiplication and division because those
areas, together with fractions, form the foundation of higher mathematics.
A child who isn't comfortable with the addition facts will experience discomfort
with every addition problem. This discomfort will increase as higher
levels of arithmetic are built upon an increasingly unstable foundation.
Your child is too young to have acquired math phobia or an attitude of
"I can't do math," but that time may come if care isn't taken. Start
now. Trying to play catch-up by reviewing fundamentals with a recalcitrant
sixth grader is nearly impossible.
Your child is at an age where she will cooperate in acquiring math skills.
She will proudly display them. But her cooperation will vanish if she develops
a distaste for math. She is not old enough to contemplate what contempt
for math will mean to her future prospects. Contemplating her future
is your job.
Summary
1. Inability to understand math is not a bar to achieving comfort
with the mechanical operations and performing them expertly.
2. Arithmetic must be taught. It cannot be discovered.
3. Most adults (and most teenagers) can teach basic
arithmetic. The operations are simple and mastery of them is easily measured.
Don't complain about the school system to your child. Just do the job that
must be done.
4. It is the expectation of success that fuels achievement
in mathematics. Schoolchildren can't be fooled by a "feel good" environment.
They know they are not really learning math. Accordingly, they learn
to anticipate failure instead of success. Learning, really learning,
numbers, facts, and operations will foster an expectation of success and
help Dana to persevere and succeed on her own in algebra and beyond.
You have it in your power to make this happen.
