How Mathematics Should Be Taught

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NEW SOLIDARITY April 29, 1980 Page 6

How Geometry Should Be Taught

A Scientist Looks at "New Math"by Dr. Steven Bardwell

The multitudes of American parents who have felt frustration and rage atwhat passes for mathematics in today's schools, the parents who have, in theend, resigned themselves to the fact that Johnny can't add, unconsciouslyadhere to a long, continuous line of mathematical thought-stretching fromthe mathematicians of Plato's Academy, Archimedes, through Nicholaus ofCusa and Leibniz to the great 19th-century school of German and French

mathematicians. This tradition is outstanding for two reasons: first, itsmembers are responsible for every essential mathematical discovery in thelast 2,000 years, and second, it has been pitted, since its inception, against acontrary tradition in mathematical thinking; today's parents are the front-line of that fight.

The New Math is not really new, any more than the inspiration for itsmethod is new. Lord Bertrand Russell and Swiss "child psychologist" JeanPiaget, are the modern progenitors of the development of the New Math'sideas. Both are quite explicit that their aim is to establish a non-Platonic

mathematics, based on the methods of Aristotle; both make unmistakablyclear that the fundamental issue is one concerning howmen think:

The "rational nature" of man is only a derivative. The subjectand object of knowledge are separate. . . On this point as onmany others' Aristotelian physics marks a return to ordinarythought rather than a continuation of the aspirations of Platonistmathematics.

--Jean Piaget: Mathematical Epistemology and Psychology

On the other side, perhaps the clearest statement of the Platonic view isgiven in a paper by the founder of the realtheory of sets, Georg Cantor:

We can speak of the reality or the existence of the wholenumbers, both the finite and the infinite ones in twosenses;however, these are the same two ways, to be sure in which any


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concepts or ideas can be considered. On the one hand we mayregard the whole numbers as real insofar as they take up a verydefinite place in our mind on the basis of definitions, becomeclearly differentiated from all the other components of ourthinking, stand in definite relations to them and thus modify thesubstance of mind in a definite way. Let me call this type ofreality of numbers their intra-subjective or immanent reality.Then again we can ascribe reality to numbers insofar as theymust be regarded as an expression or image of occurrences andrelationships in the external world confronting the intellect.This second type of reality I call the trans-subjective ortransient reality of the whole numbers. . .

There is no doubt in my mind that these two types of reality

will always be found together, in the sense that a concept to beregarded as existent in the first respect will always in certain,even in infinitely many ways, possess a transient reality as well.. .

This coherence of the two realities has its true foundation in theunity of the all, to which we ourselves belong as well.

This view of mathematics and science is what the New Math is designed todestroy. The Platonists have maintained that mathematics is an empirical

science whose subject (like that of any science) is what Plato called the"hypothesis of the higher hypothesis" and Cantor called the "Principle ofGeneration," both descriptions of the self-developing evolution of theUniverse. The Aristotelian opposition has counterposed the view thatmathematics (along with the other sciences) is a logical structure, lackingany essential connection to reality, and merely a product of the human mind,a mind which in their view has itself no essential connection to reality. (This

psychology is obviously self-validating, as the insanity of many of the mostillustrious of the latest generation of mathematicians testifies).

The fight between these two views in the 20th century has taken place overthe basic concepts of arithmetic numbers and arithmetic operations. The

biggest guns of the Aristotelian faction have, in fact, been aimed atoverturning the explicitly Platonic significance of the concept of numberdeveloped, as both sides recognize, by the discoverer of set theory GeorgCantor.


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Bertrand Russell spent ten years of his life producing a three volume book,Principia Mathematica which he hoped would show that mathematics could,through the use of set theory,be reduced to logic. He failed, but his book

became the model for three generations of formal logical mathematics to beused against Platonic methods in mathematics. On the pedagogical side,Jean Piaget took Russell's work and developed a theory of number and theconcept of number which he claims purifies Cantor of his Platonic excesses!

Fig. 1: Set Theory vs. "Set Theory,"

The new math is well-known for its love of set theory, as exemplified by the abovediagram from a first grade workbook illustrating the idea of number as a property of

sets of arbitrary objects. However, the set theory taught in the new math isdiametrically opposed to the concept as developed by its inventor, Georg Cantor. AsCantor makes clear over and over again, a set is notan arbitrary collection; it isdefined by the "rule" which determines membership in a set. There are real sets andcollections which are not sets. Cantor put into mathematical form, with hisdefinition of sets, the essentially Platonic idea of a universala "set" is a higher-order concept, not a simple aggregation of objects. The new math, based onRussell's bowdlerization of set theory, turns the whole concept into a nominalistgame. As the above picture shows, any collection can be a set even if the "rule" formembership is a totally arbitrary one.

The new math is the fruition of the Piaget-Russell attack on Platonicmathematics. Its incoherence, self-evident sterility, and destructive effect onchildren's minds are not accidentalthis is the essence of the Aristoteliantheory of mind.


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Two Examples

There have been many attacks on the New Math, but its epistemologicalsignificance remains largely unknown. The destructiveness of the NewMath is clear from two, examples taken from its curriculum which haveescaped the notice of critiques of New Math from conventional or practicalstandpoints; I want to concentrate on these here. The first is the concept ofan "algorithm" which is used as the basis for teaching arithmetic operations,and, second, the New Math concept of the structure of the number system.

1. Algorithms and Arithmetic

An algorithm is a set of rules, usually recursive, for performing some taskand for testing for the completion of the task. The concept of an algorithm

was a product of the development of machines which had to be "pro-grammed" with instructions for the actions required of the machine. Thepunched cards that controlled early spinning and knitting machines areclassic examples of an algorithmmove needle A to position 1, needle B to

position 2, move the red thread over needle 1, etc. Obviously, an algorithmis a powerful tool if certain conditions are satisfied:

1) The problem to be solved or task to be performed is completely-posedbeforehand;

2) The problem can be solved in a finite number of steps;

3) The quality of solution does not depend on factors known only after thealgorithm is begun (for example, singularities are excluded);

4) The rules for performing the algorithm are fixed or drawn from a fixedgroup. These assumptions are fine for a machine or a computer, but they areall violated by the simplest task required of human mentation! No algorithmcould be written for something as simple as getting out of bed (or gettingyour kids to school) in the morning.

In spite of this obvious fact, the algorithm has been taken as a prototype ofmathematical thinking by the Aristotelians and incorporated in the NewMath as the way of teaching arithmetic operations like addition andsubtraction. From a psychological and pedagogical standpoint this is absurd.Since people are not machines they perform tasks differently and they learnthem differently. In the same way, this method is absurd mathematically;


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Fig. 2: Division as Repeated Subtraction

The algorithmic approach to arithmetic shows its impracticality and inaccuracy inthe above figure taken from a fourth grade new math text. The diagram is anattempt to show how long division can be done by counting the number ofsubtractions of the divisor from the dividend.

arithmetic operations are only formally reducible to algorithmic techniques.They are actually synthetic concepts, higher, order concepts, and, when

reduced to their algorithmic counterpart, cease to be mathematics.

Long-division, long the terror of elementary children students, providedfertile ground for the New Math's algorithmic theory of arithmetic. Present-ed with the problem of dividing 90 by 8, the New Mathematician will tell usthe following (of course, he probably won't actually do the division this way

but this is what he says to the kids):

Step 1: Is 8 larger than 90? If yes, then quotient is 0; otherwise go to Step2.

Step 2: Subtract 8 from dividend. Add 1 to quotient.

Step 3: If 8 is larger than dividend then end; otherwise, go to Step 2.

The algorithm which he proposes counts the number of times that the divisor(8) can be subtracted from the dividend (90)this number of times is the


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quotient (11). This method is used, in actuality, only by the crudest ofmechanical calculatorseven computers have better ways of dividing!

Is this algorithm even division? Let's try it on the problem of 4 divided by12the answer, according to one student is -8. Certainly not. From amathematical standpoint, division is qualitatively different from subtraction

it is not compounded subtraction, unless, of course, you are a mechanicalcalculator. Subtraction of whole numbers, no matter how many times it is

performed, always produces whole numbers; but division, takes wholenumbers and produces a new kind of numbera rational number, orfraction. One can never get fractions from subtraction of whole numbers.

This reduction of division to an algorithm involving repeated subtraction isnot merely a mathematical travesty. The subject of mathematics, as all great

mathematicians have known, is not numbers and their manipulation; it is thehuman mind as a mirror of the Universe. Mathematics, as a product of thehuman mind, both reflects and modifies the structure and evolution of theUniverse. Cantor says that this connectionthe "unity of the all"ismathematics. Since neither the human mind nor the Universe satisfies anyof the four prerequisites for the applicability of an algorithm, to teachalgorithmic thinking as if it were mathematics is to systematically distort

both reality and human mentation. No wonder children hate the New Mathto understand it, they must deny the fundamental characteristic of theirability to think!

Let there be no mistake; the Aristotelian faction of mathematics agreesabout the implications of algorithmic thinking. They only disagree about theinapplicability of algorithmic methods to the mind and the Universe. Their

premise is that the laws governing both human thought and the Universe arefixed. Of course, they say, algorithms work precisely because human beingsand the Universe are machine-like.

2. The Structure of the Number System

The problem of long-division raises a more fundamental problem in arith-metic; the New Mathematician's reply to my objection that his algorithm forlong division could not generate fractions (because subtraction of wholenumbers can only generate whole numbers) would be the following: I can

provide you with an algorithm that is too simple, but just because subtrac-


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tion doesn't give you fractions, doesn't mean that there is no algorithm fordoing so.

The real argument here is not over an algorithm for long division, but rather,over the significance of these new numbers generated by division. Anyqualitative significance of division comes from its ability to generate thesenew numbers (fractions). The Platonic approach to mathematics hasmaintained, as Cantor and Dedekind were the first to show, that fractions(rational numbers) are a qualitatively different kind of number than wholenumbers. In addition, Cantor showed that the number system is, in fact, anested hierarchy of different kinds of numbers, each of which is generatedfrom the preceding by inherently non-algorithmic processes like limits ofinfinite series. To get irrational numbers from rational ones, for example,requires a complicated geometrical argument that demands new mathe-

matical rules for new numbers.

As Cantor points out, the significance of this hierarchical structure of thenumber system far transcends its mathematical applications. It is parallel to

a model ofthe similar nested, hierarchical structure of the physical

Fig. 3: The Number Line

Contrary to the implication of the new math, the number line (the continuum) has asubtle and important structure. All numbers on the number line are not the same,and, as Cantor stressed, the generation of one kind of number from a simpler one isa prototype not only for all mathematical reasoning but also for the evolution of theuniverse.


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Universe. Cantor showed, even more, that the fundamental feature of thishierarchy was not its structure at any one instant, but rather what he calledthe Principle of Generation which creates a new level of hierarchy out of its

predecessor. This transition from one level to the next (like from the wholenumbers to the rational numbers) is lawful but there is nothing in the lowerlevel that determinesbeforehand its successor. The Principle of Generationin mathematics has been called "negentropy" in physics but they are thesame.

By his algorithms, bastardized set-theory, and the like, the New Mathe-matician denies the qualitative structure of the number system. The crux ofthe Aristotelian approach is that the Platonic hierarchy does not exist.Russell's book was an attempt to prove the qualitative homogeneity ofmathematicsto prove that it was in totoreducible to a fixed set of logical

axioms. If he had been successful, it would have been possible to build acomputer which could prove every existing theorem in mathematics andevery theorem ever provable! He was not successful, but not because of anyshortcoming of his attempt; it is just that he and his New Math disciples arewrong about the nature of the human mind and physical universe.

Teaching Geometry by Plato's Method

by Laurence Hecht

You are standing at point A in the accompanying figure (fig. 4). Your task isto run to the wall marked "W," tag it and run back to point B in the least

possible time. The heavy line shows a perfectly logical, common sensesolution: run straight to the wall, because the perpendicular is the shortest

possible distance between a point and a line; then run straight from the wallto point B, because a straight line is the shortest distance between two

points.

Wrong! If this in any way approximates your line of reasoning about theproblem, you have not mastered the most fundamental principle of physicalscience known to Plato and his contemporaries 2,600 years agothePrinciple of Least Action.


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But don't feel discouraged. Though the concept was revived in the 18thcentury by Gottfried Leibniz, it has been lost to all but a few of today'sleading mathematicians and physicists as well. Reviving the study of

physics and mathematics from the standpoint of geometry as studied byPlato and the Neoplatonic scientists is the subject of an ongoing adulteducation class series in New York being led by Uwe Parpart and CarolWhite. Mr. Parpart and Mrs. White are both executive committee membersof the International Caucus of Labor Committees and have publishedextensively on mathematics and physics topics.

Geometry Is Not Euclid

For approximately the last year, Parpart has been developing a curriculumfor teaching geometry to children, using the method of Plato. "When the

word geometry is mentioned," Parpart told the 1979 conference of theInternational Caucus of Labor Committees, "what comes to almosteverybody's mind is the name of Euclid, because Euclid is virtuallysynonymous with the teaching and learning of geometry, especially in theEnglish-speaking countries. This is one of the biggest problems we have."Parpart instead has built his teaching method around the approach togeometry he associates with Plato, Archimedes, Leibniz and the 19thcentury German mathematician, Riemann. "The key to teaching geometry,"says Parpart, "is to find a way of looking at it not simply from the standpointof the specific method of generating one or another geometric object in the

classroom, but in a more profound sense, to identify the method ofgeneration in which real physical action and processes generate geometricalobjects."

Parpart's class series for adults began four weeks ago in New York. Itconsists of a weekly lecture session of approximately two hours durationsupplemented by a smaller-size weekly recitation session in which specific

problems are worked through. His method of presenting the solution to theproblem posed in the initial paragraph illustrates what he means by

incorporating geometry and physics.

The Ellipse Maker

To solve the problem we will first need to construct a simple device knownas an ellipse maker. This is how it was built in all of the recitation sectionsof the class series: Place two thumb tacks about three or four inches apart


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Uwe Parpart during a demonstration at the International Caucus of Labor

Committees 1979 conference

into a piece of paper. Now tie both ends of a piece of thread together so you

have a loop. This loop should be big enough to lasso around boththumbtacks with, a little slack left over.

Lasso the thumbtacks and stick a pencil point inside the loop. Now pull theloop taut with the pencil point and revolve the pencil around the twothumbtacks. You have drawn an ellipse.

The ellipse maker you have just created is the clue to solving the problem.Place a telephone book under figure 4 on this page and put 2 thumbtacks orsewing pins into the points A and B. Now tie your loop so that it is just longenough to make the ellipse drawn in fig. 4. If you have tied it right you willsee that this ellipse just touches the "wall" W. If the string is toolong, it will

pass behind the wall; if too short, it will never touch it. When you have thestring just right, draw in the ellipse, filling in the already printed line.


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Fig. 4: The Ellipse Maker

Now move your device back to the position where the threads exactly coverthe two dotted lines from A and B. Your thread is now taking the shortest

path from A to B that touches the line W. It issuggested that you test this

for yourself using a ruler, or better, a drawing compass and a measuring line.You will also see that the path makes the same angle going into the wall ascoming out of the wall, a point you can verify visually or precisely with a

protractor.

Points A and B are the focal points of the ellipse we have drawn. A veryspecial sort of geometric relation holds true within the space defined by theellipse. If we were to shape the ellipse out of a highly reflective materialsuch as mylar, and shine a light ray into it through one of the focal points,

the light ray would always be reflected off the inner surface back through thesecond focus. The dotted line in our figure 4 shows one case of what thiswould look like.


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The Principle of Least Action

In all such cases, the light would be taking the path of least action, that is thepath requiring the least expenditure of energy to get it from point A to pointB. Light and other forms of energy obey the law or Principle of LeastAction that we have just demonstrated in the special case of the ellipse. Thelaw of optics that the "angle of incidence equals the angle of reflection" isalso thus illustrated.

Why is this so? Why does the angle of incidence of light always equal theangle of reflection, whether on an ellipse, or on the surface of a plane mirroror any object? Various attempts have been made to explain this by reducingthe light ray to mechanical propertiesby saying that the light ray is a wave,or that it is a succession of tiny particles. Actually neither definition of light

is sufficient. It can readily be shown that light has both wave and particulateproperties.

The principle of least action, in fact, tells us more about the behavior of lightthan any attempt to reduce this complex natural phenomenon to the simpler

phenomenon of wave action or behavior of ballistic particles. We will fullyunderstand what light is when we understand what special geometry ofspace causes energy to behave in this way. We will never understand it byattempting to reduce it to simple mechanical properties.

NEW SOLIDARITY May 19, 1980 Page 6

LETTERS

Dear New Solidarity,

Laurence Hecht, in "Teaching Geometry by Plato's Method," finds theshortest path from a point to another point via a wall by constructing an

ellipse tangent to the wall, with vertices at the points in question. This isfine. However, in the tenth paragraph he writes: "You will also see that thepath makes the same angle going into the wall as coming out of the wall, apoint you can verify visually, or precisely with a protractor."

What? Empiricist mathematics? Actually, there is a very simple proof ofthis [shown in the accompanying figure].


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Construct B', the "mirror image" of B acrossthe wall (this is easily done with a compass.)Then let P be a point on the wall. By the

properties of right bisectors, PB = PB' (equal

distances). Likewise

CPB =

CPB'(where C P, but C is on W).

Thus the distance APB equals the distanceAPB'. Due to the Euclidean metric, theshortest path from A to B' through W is astraight line.

Thus APD = B'PC (since APB' and

DPC are both straight lines). And hence

APD = BPC.

Bruno Bauer Stanford, CA

P.S. Enclosed is a check for $25. Pleaserenew my subscription toNew Solidarity.

The editor replies:

Many thanks for your very useful addition to the Least Action Path problem.

Author Hecht points out that due to space limitation he was unable todiscuss the interesting historical aspects of the problem. It was well knownto the ancient Greeks, that when light is reflected from a mirror, the angle ofincidence is equal to the angle of reflection. But Heron, the 1st Century ADAlexandrian scientist, is generally credited with the discovery that this alsoimplies a least action path. This is, if we imagine the line W to be a mirror,and reflect light from point B to point A via the point P on W, this will bethe path of least distance that could connect A and B via W. Heron in facthad made both the ellipse construction that Hecht discussed in the referencedarticle, and your construction involving the virtual or "mirror" image of one

of the points.

Using both, Heron showed that if the point P were located anywhere elsethan where it is, a longer path from B to A would be defined. This discoveryformed the real germ of geometrical optics.

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How Mathematics Should Be Taught