9

“The Speedier Expedition of Their Learning”:
Thoughts on Teaching and Learning Mathematics

“ARITHMETIC IS A VERY DULL STUDY TO CHILDREN, AND IF THE ROD AND THE slate hang side by side, it cannot fail to be a disagreeable one,” wrote Mrs. Lovechild in 1785. This chapter showcases a few of the many ways that mathematics teaching has been thought about and practiced over the years, and some of the ways different writers have tried to avoid making mathematics “dull and disagreeable.”

We will see both examples of practice and more reflective discussions: Humfrey Baker’s list of the mathematical subjects he taught to the “servants and children” at his London school, or the agonies of just how to present Euclid for the modern learner (with algebra? without axioms?) The dialogue format we have met in Chapters 2 and 4 receives more than one airing, continuing into the twentieth century.

We also pick up the story of arithmetic teaching, which in Chapter 2 we took no further than the early twentieth century. From the 1960s this subject became a divisive one, and the “New Math” appears here as, perhaps, the natural development of earlier concerns to teach methods rather than rote-learned rules. Chapter 8 has shown some examples of how that tension was played out in the mathematical workplace.

The most attractive theme of this chapter, though, and perhaps the most important, is that mathematics can and should be fun. The “mathematical toys” of the eighteenth century and the “game of logic” of the nineteenth receive a wonderful modern echo in one of my favorite extracts, the last in this chapter, on the mathematics of “turtle fun.”

“To Have Their Children or Servants Instructed”

Humfrey Baker, 1590

We met Humfrey Baker in Chapter 1, as the author of the arithmetic textbook The Welspring of Sciences. Here we see a single-page advertisement for his mathematical school in London, showing in detail the “arts and faculties” which were taught there. Few hints are given of his teaching methods, but he has in common with later writers on mathematics pedagogy that he begins from a detailed understanding of how mathematics can be conceived and subdivided.

Humfrey Baker (fl. 1557–1574), Such as are desirous ([London], c. 1590), single page.

Such as are desirous, either themselves to learn, or to have their children or servants instructed in any of these Arts and Faculties hereunder named, it may please them to repair unto the house of Humfrey Baker, dwelling on the North side of the Royal Exchange, next adjoining to the sign of the ship, where they shall find the professors of the said Arts, etc., ready to do their diligent endeavours for a reasonable consideration. Also, if any be minded to have their children boarded at the said house, for the speedier expedition of their learning, they shall be well and resonably used, to their contentation.

The Arts and Faculties to be taught, are these.

Arithmetic vulgar, namely Numeration, Addition, Subtraction, Multiplication, Division, Progression, in Whole numbers, Fractions both Arithmetical and Astronomical. The application of in operation, that is to say: The rule of 4 numbers proportional; The rule of 5 numbers or more; The rule of gain and loss with time; The rules of barter° both simple and compound; The rules of fellowship without time, and with time; The rules of interest both simple and compound; The rule of Alligation; The Virgins’ rule; The rule of suppositions, commonly called false positions. in Whole numbers, Fractions. With the extraction of all kinds of roots of numbers, Numeration, Addition, Subtraction, Multiplication and Division of Proportions. The like in Surd and Cossic numbers,° with the application of them in the rule of Equation, for Equations Arithmetical and Geometrical, and also the second quantities° of Algebra.

How to measure Lands, Woods, and all other platforms whatsoever, and to reduce the same platforms into greater or lesser shape, at your own pleasure; Timber, Stone, and all kind of Solid bodies whatsoever.

The principles of Geometry, to be applied to the aid of all Mechanical workmen.

The use of the Quadrant, Geometrical square, and Baculus Jacob.°

The composition of the Astronomer staff, Astrolabe, and Ruler of Ptolemy,° with their uses.

The calculation of the Tables of Sines, and Chords, with their proper use.

Also the perfect order how to keep any accounts by Debitor and Creditor, after a more plain manner than hath heretofore been usually taught by any man within this City. Likewise how to rectifie and make perfect any difficult or intricate account, depending in variance between two or more partners, and thereby to show which of them shall be indebted the one to the other.

Notes

rules of barter: rules that determine how much of one kind of goods should fairly be exchanged for a given amount of another if the price of each is known; the rules of fellowship determine how the profits should be shared from an enterprise where several people have contributed different amounts initially; the rule of alligation determines the price of a mixture or “mixed lot” of goods if the prices of the ingredients or components are known; the Virgin’s rule is a mystery; while the rule of false position is in effect a strategy for solving linear equations by applying a correction to an initial guess.

Cossic numbers: unknown quantities or the multiples, squares, cubes, etc., of them, expressed using the sixteenth-century system of “cossic” notation, in effect an early form of algebraic notation.

second quantities: Baker may well mean areas, “first quantities” being lengths.

Quadrant, Geometrical square, Baculus Jacob: instruments used in surveying.

Astronomer’s staff, Astrolabe, and Ruler of Ptolemy: astronomical instruments

Euclid with Algebra

Isaac Barrow, 1660

Isaac Barrow was Isaac Newton’s predecessor as Lucasian Professor of Mathematics at Cambridge; later he left mathematics to concentrate on theology, and for many years he was remembered chiefly for his sermons. Also, for a time, a professor of Greek at Cambridge, he produced important editions of Euclid in both Latin and English. Here he introduces his Euclid and explains his aims in using the new algebraic notation wherever possible as a help to “the studious.”

Isaac Barrow (1630–1677), Euclide’s Elements; The whole Fifteen Books compendiously Demonstrated By Mr Isaac Barrow Fellow of Trinity Colledge in Cambridge. And Translated out of the Latin. (London, 1660), 2^r–3^v.

In order to the Reader’s satisfaction concerning the Book put into his hand, I am to advertise him of some few things, and that according to the nature of the Work, briefly, as followeth. My Undertaking aimed principally at two Ends: the first of which was to conjoin the greatest Compendiousness of Demonstration with as much Perspicuity as the quality of the subject would admit, that so the Volume might bear no bigger bulk than would render it conveniently portable. Which I have so far attained, that though possibly some other person might with greater curiosity, yet (I presume) none could with more conciseness have demonstrated most propositions, especially since I have altered nothing in the number and order of the Propositions, nor taken the liberty to leave out any one of Euclid’s as less necessary, or to reduce certain of the easiest into the Class of Axioms.

But I had a different Purpose from the beginning; not to compose Elements of Geometry any-wise at my discretion, but to demonstrate Euclid himself, and all of him, and that with all possible brevity. For as for Four of his Books, the Seventh, Eighth, Ninth and Tenth, although they do not so nearly pertain to the Elements of Plane and Solid geometry, as the Six First and the Two subsequent, yet no man that has arrived to any measure of skill in Geometry is ignorant how exceedingly useful they are in Geometrical matters, as well in regard of the very near alliance between Arithmetic and geometry, as for the knowledge of Commensurable and Incommensurable Magnitudes which is highly important to the understanding both of Plane and Solid Figures. And the noble Theory of the Five Regular Bodies, contained in the Three Last Books, could not be omitted without prejudice and injury; since our Author of these Elements, being a follower of Plato’s School, is reported to have compiled the whole System only in reference to that Contemplation .

Moreover, I was easily induced to believe, that it would be acceptable to all Lovers of these Sciences to have the Entire work of Euclid by them, as it is usually cited and recommended by all men. Wherefore I determined to leave out no Book or Proposition . Upon the same account also I purposed to use generally no other than Euclid’s own Demonstrations, contracted into a more succinct form, saving perchance in the Second and Thirteenth, and sparingly in the Seventh, Eighth, and Ninth Books, where it seemed convenient to vary something from him. So that it may be reasonably hoped that in this Particular our own Design and the Wishes of the Studious are in some manner satisfied.

The other End aimed at was in favour of Their desires who more affect Symbolical then Verbal Demonstrations. In which kind, seeing most of our own Nation are accustomed to the Notes of Mr. Oughtred,° I esteemed it more convenient to make use of them principally throughout. For no man hitherto that I know of, saving only Peter Herigon,° has attempted to set forth and interpret Euclid according to this way. The Method of which most learned Person, though in many other respects very excellent, and exactly accommodated to his peculiar purpose, seemed to me notwithstanding doubly defective. First, in that, whereas of several Propositions brought to the proving of some one Theorem or Problem the Latter does not always depend on the Former, yet when they do cohere one with another, and when not, cannot readily enough be known, either from their order or any other way; whence it not seldom comes to pass, that through the want of Conjunctions and Adjectives, Ergo, rursus,° etc., there arises difficulty and occasion of doubting, especially to such as are but little versed therein.

And in the next place, it oftentimes falls out that the said Method cannot avoid superfluous repetitions, whereby the Demonstrations become sometimes prolix, and sometimes perplexed and intricate. All which Inconveniences are easily remedied in our Way by the intermingling of Words and Signs at discretion. And thus much may suffice to be premised concerning the Intent and Method of this Compendium. I shall not allege in favour of myself the scantness of time allotted to this Work, nor the avocations of affairs, nor the scarcity of Helps to this sort of Studies amongst us (as I might not untruly) out of fear lest my Performance should not throughly please every body. But I wholly submit to the fair censure and Judgement of the Ingenuous Reader, what I have undertaken for the advantage of his Studies; to be approved, if he find it serviceable thereunto; or, if otherwise, rejected.

Notes

Mr. Oughtred: William Oughtred (1574–1660), an important English writer on algebra.

Peter Herigon: Pierre Hérigone, pseudonym of Baron Clément Cyriaque de Mangin (1580–1643). His translation of Euclid appeared posthumously in 1644.

Ergo, rursus: therefore, on the other hand.

The Idea of Velocity

Leonhard Euler, 1760

Leonhard Euler wrote his wildly popular Letters to a German Princess to the “young and sensible female,” the princess of Anhalt-Dessau, at a rate of one or two each week, between 1760 and 1762. They ranged across the physical sciences and philosophy (including the intriguing “Electrification of men and animals”) and covered a great deal of applied mathematics, always expressed using geometry rather than algebra. Euler’s lucid but unpatronizing exposition makes the collected letters still one of the best introductions to the science of their period. Their English translator enthused about their potential to contribute to female education:

The time, I trust, is at hand, when the Letters of Euler, or some such book, will be daily on the breakfasting table, in the parlour of every female academy in the kingdom; and when a young woman, while learning the useful arts of pastry and plain-work [sewing], may likewise be acquainting herself with the phases of the moon, and the flux and reflux of the tides.

Here we see Euler, near the beginning of the series of letters, introducing one of the fundamental concepts of applied mathematics: velocity.

Leonhard Euler (1707–1783), trans. Henry Hunter, Letters of Euler to a German Princess, on different subjects in physics and philosophy (London, 1795), vol. 1, pp. 6–9 (letter 2, dated 22 April 1760).

I proceed to unfold the idea of velocity, which is a particular species of extension, and susceptible of increase and of diminution. When any substance is transported, that is, when it passes from one place to another, we ascribe to it a velocity. Let two persons, the one on horseback, the other on foot, proceed from Berlin to Magdeburg; we have, in both cases, the idea of a certain velocity, but it will be immediately affirmed that the velocity of the former exceeds that of the latter. The question, then, is: Wherein consists the difference which we observe between these several degrees of velocity? The road is the same to him who rides, and to him who walks: but the difference evidently lies in the time which each employs in performing the same course. The velocity of the horseman is the greater of the two, as he employs less time on the road from Berlin to Magdeburg; and the velocity of the other is less, because he employs more time in travelling the same distance. Hence it is clear, that in order to form an accurate idea of velocity, we must attend at once to two kinds of quantity: namely to the length of the road, and to the time employed.

A body, therefore, which, in the same time, passes through double the space which another body does, has double its velocity; if, in the same time, it passes through thrice the distance, it is said to have thrice the velocity, and so on. We shall comprehend, then, the velocity of a body, when we are informed of the space through which it passes in a certain quantity of time. In order to know the velocity of my pace, when I walk to Lytzow,° I have observed that I make 120 steps in a minute, and one of my steps is equal to two feet and a half. My velocity, then, is such as to carry me 300 feet in a minute, and a space sixty times greater, or 18,000 feet, in an hour, which, however does not amount to a mile, for this, being 24,000 feet, would require an hour and 20 minutes. Were I, therefore, to walk from hence to Magdeburg, it would take exactly 24 hours. This conveys an accurate idea of the velocity with which I am able to walk.

Now it is easy to comprehend what is meant by a greater or less velocity. For if a courier were to go from hence to Magdeburg in 12 hours, his velocity would be the double of mine; if he went in eight hours, his velocity would be triple. We remark a very great difference in the degrees of velocity. The tortoise furnishes an example of a velocity extremely small. If she advances only one foot in a minute, her velocity is 300 times less than mine, for I advance 300 feet in the same time. We are likewise acquainted with velocities much greater. That of the wind admits of great variation. A moderate wind goes at the rate of 10 feet in a second, or 600 feet in a minute; its velocity therefore is the double of mine. A wind that runs 20 feet in a second is extremely violent, though its velocity is only 10 times greater than mine, and would take two hours and twenty-four minutes to blow from hence to Magdeburg.

The velocity of sound comes next, which moves 1000 feet° in a second, and 60,000 in a minute. This velocity, therefore, is 200 times greater than that of my pace; and were a cannon to be fired at Magdeburg, if the report could be heard at Berlin, it would arrive there in seven minutes. A cannon ball moves with nearly the same velocity; but when the piece is loaded to the utmost, the ball is supposed capable of flying 2,000 feet in a second, or 120,000 in a minute. This velocity appears prodigious, though it is only 400 times greater than that of my pace in walking to Lytzow; it is, at the same time, the greatest velocity known upon earth.

But there are in the heavens velocities far greater, though their motion appears to be extremely deliberate. You know that the earth turns round on its axis in 24 hours: every point of its surface, then, under the equator, moves , in 24 hours, while I am able to get though only miles. Its velocity is, accordingly, 300 times greater than mine, and less, notwithstanding, than the greatest possible velocity of a cannon ball. The earth performs its revolution round the sun in the space of a year, proceeding at a rate of in 24 hours. Its velocity, therefore, is 18 times more rapid than that of a cannon ball. The greatest velocity of which we have any knowledge is, undoubtedly, that of light, which moves of miles every minute, and exceeds the velocity of a cannon ball, 400,000 times.

Notes

Lytzow: A village about a league from Berlin. (Translator’s note.)

1000 feet: The velocity of sound is generally computed at 1,142 feet each second but varies with the elasticity and density of the air. The earth travels in her orbit 1,612,000 miles in the space of 24 hours and, therefore, with a velocity more than 50 times greater than that of a cannon ball. Light moves about 13 millions of miles every minute. (Translator’s note.)

Mathematical Toys

“Mrs Lovechild,” 1785

This unfortunately-named pedagogue produced a whole series of books aimed to help mothers teach their children; the emphasis was on making learning fun, and “Lovechild” described games and tricks as well as more orthodox teaching aids. The selections given below show something of her philosophy of teaching as well as some of the mathematical games she considered suitable to engage the curiosity of small children.

“Mrs Lovechild” (Eleanor Fenn, 1743–1813), The Art of Teaching in Sport; Designed as a Prelude to a set of Toys, for enabling Ladies to Instill the Rudiments of Spelling, Reading, Grammar, and Arithmetic, under the Idea of Amusement (London, 1785), pp. 51–54, 60–62.

Arithmetic is a very dull study to children, and if the rod and the slate hang side by side, it cannot fail to be a disagreeable one.

There are amusing books, calculated to excite application in children in learning to read; but for figures, what near prospect of pleasure appears as an incentive?

A boy is required to learn accounts; he drudges in obedience to his parents, gets with difficulty through the first rules of Arithmetic, and contracts an aversion to figures for life.

Authority may place a child in the path of learning, but pleasure only can entice him on; let us therefore endeavour to strew the entrance with flowers, which may induce him to proceed with alacrity.

The Box must be held sacred: the little people must not be allowed to touch it, nor to look in the book which contains the arcana.

The Box is to be produced occasionally, as a favour, and some of the sports indulged to the children, according to their progress. Ladies would do well to procure great abundance and variety of cuts, selected with care. The present set could then be distributed gradually, and replaced; thus the charm of novelty would long remain, and occasions of much instruction be introduced at a small expense.

Tricks with Cards.

I.

Turn the cards. For instance, a five. Then an eight. A young child adds the numbers: 8 and 5 are 13. Or an older one subtracts the smaller number: 5 from 8, and there remain 3. Or a still older multiplies the two numbers together: five times 8 are 40.

II.

Take nine cards, viz. Ace, , etc. Place them so as to make 15 eight ways.

A lady may not choose to take the trouble of discovering how they are to be placed—though she will contentedly drudge at 2 and 2, for the benefit of her children—so the order is shown. A child must be pretty well versed in figures before he will be able to discover how they are to be placed.

III.

The pack of cards the whole amount of the pips of a pack (not counting the tenth cards) is 180. One player takes a card. The other is to discover what that card is, by missing the number of it. Thus: The player is requested to take a card below ten; suppose takes a four. I then miss four from 180, and have only to run the pack over again, to see which four is absent. Or, if you count the cards as they pass in review before you, casting out , till you see what is wanted of the last ten. This requires a readiness in addition to do it well. For a young child, take cards whose pips amount to 100. For a very young one, 20 only, and that number composed of small cards.

The Merchant, or Commerce.

A merchant sold beans; he was of so suspicious a disposition that he apprehended every person meant to impose on him; he was never satisfied with telling his money or beans once or twice, but counted them several times, and in every possible manner. If he had twenty beans he would first count them thus: two and two are four, four and two are six, and so on by two at a time to twenty. Then he counted 3 and 3 are 6, and 3 are 9, and 3 are 12, and 3 are 15, and 3 are 18, and 2 are 20. Then 4 and 4 are 8, and so on to twenty. Then 5 and 5 are 10, and so on to twenty. Then 6 and 6, etc.

This sport should be enlivened by secreting a counter or bean (occasionally), the child to miss it.

A Mother Explains Comets

Catherine Vale Whitwell, 1823

Comparable in some ways to Newton for the Ladies (Chapter 5), this account of astronomical theories in dialogue form illustrates one author’s concern that mathematical subjects be made accessible even to the very young, and to girls. Catherine Vale Whitwell, also the author of a “scheme of simplicity and economy” addressed to mothers, is ambitious in the complexity of the astronomical information she relates (much of it taken from Charles Hutton’s mathematical dictionary of 1795–1796), although she stops short of giving a quantitative or geometric account of comets’ orbital motions.

Catherine Vale Whitwell, An Astronomical Catechism; or, Dialogues between a mother and her daughter (London, 1823), pp. 124–129.

Are not comets important bodies belonging to our system? Will you kindly inform me, how I may distinguish them?

Comets are solid opaque bodies, of different magnitudes, like the planets, but are distinguished by fiery tails, or long beards of transparent hair.

From which side does the tail issue?

From that side of them which is furthest from the sun.

Has the tail the same degree of lustre throughout?

No: the lustre is the most considerable near the body of the comet, and becomes less and less, the further it is removed thence.

Is there not a popular division of comets into three kinds?

Yes; there is such a division, but it rather relates to the several circumstances of the same comet, than to the phenomena of several: the divisions are the bearded, the tailed, and the hairy comet.

Will you explain this remark by an illustration?

When a comet is eastward of the sun, and moves from it, it is said to be bearded, because the light precedes it, in the manner of a beard.

What is it termed when the comet is westward of the sun, and sets after it?

It is said to be tailed, because the transparent hairlike appendage follows it in the manner of a tail.

When the comet and the sun are diametrically opposite, the earth being between them, what is the term employed?

Then the train is hid behind the body of the comet, except a little which appears round it, in the form of a border or tuft of hair, or coma; hence it is called hairy, and from hence the name comet is derived.

Is the body as well as the tail of the comet subject to apparent changes?

Yes; and these Sir Isaac Newton ascribes to changes in the atmosphere of the comet. His opinion was confirmed by observations on the comet of 1774.

At what periods have very interesting comets made their appearance?

In Dr. Rees’ Cyclopedia you will find the elements of ninety-seven comets, and the names of the authors who have calculated their orbits. But a few words respecting the comets of 1680, 1661, and 1664, will answer my purpose.

As the elements of so large a body of comets are given, they have, I suppose, in common with other heavenly bodies, claimed the attention of astronomers in every age?

Not so; and nothing is more to be regretted by posterity, than that the ancients considered them as portentous signals of approaching calamity, instead of regarding them in a philosophical point of view.

What, then, were they altogether neglected by ancient astronomers?

They were, at most, considered only as meteors, or sublunary vapors floating in the atmosphere.

Whose attention did they first excite, and by whom were they first regarded as bodies of more importance?

Seneca, who was born six years before Christ, paid considerable attention to them, in consequence of the appearance of two in his own time. But his observations remain unrecorded. Who then was the first, who described with any degree of accuracy the path of a comet?

Nicephorus Gregorius,° an astronomer and historian of Constantinople. Deeply impressed with the neglect of his predecessors, he composed that stupendous effort of human industry, “The Table of the Elements of Comets.”

O mamma, with what multiplied specimens of energy, and with what vast achievements of industry, do you furnish me! Surely I ought to catch one spark of ambition from the fire burning on that altar. With a reputation ever gathering, shall the name of such a man descend to posterity. But may I ask, what was the appearance of the comet of 1680?

Sturmius° tells us that when he viewed it with a telescope, it appeared like a coal dimly glowing, or a rude mass of matter, illuminated with a dusky fumed light, less sensible at the extremes than in the centre.

What was the appearance of the comet of 1661?

Hevelius observed, that its body was of a yellowish colour, very bright and conspicuous, but without glittering light. In the middle was a dense ruddy nucleus, almost equal to Jupiter, encompassed with a much fainter, thinner matter.

But do these instances evince that the body of the comet is subject to apparent changes?

No, they do not; therefore I will enter a little more into the detail. On the 5th of February, the head of the last-mentioned comet was somewhat bigger, and brighter, than it had before been; it was of a gold colour, but its light was more dusky than the rest of the stars, whilst the nucleus appeared divided into several parts. February the 6th, the disk was lessened, the nuclei still existed, though less than before; February the 8th, its body was round, and represented a very lucid little star; the nuclei were still encompassed with another kind of matter. February the 10th, the head was somewhat more obscure, and the nuclei more confused, but brighter at the top than at the bottom. February the 13th, the head was much diminished, both in magnitude and brightness. March the 2nd, its roundness was a little impaired, its edges lacerated, etc. March the 28th, it was very pale and exceedingly thin; its matter was much dispersed, and no distinct nucleus appeared.

What is the description given of the comet?

Weigelius° says that he saw the comet, the moon, and a little cloud illumined by the sun at the same time; and he observed that the moon, through a telescope, appeared of a continued luminous surface, but the comet very different, being exactly like a little cloud in the horizon illumined by the sun.

I have heard various conjectures respecting the cause of the tails of the comets; will you have the goodness to mention to me the most plausible opinion?

It is that formed by Hevelius,° who supposes that the thinnest parts of the atmosphere of a comet are rarefied by the force of the heat, and driven from the forepart and from each side of the comet, towards the parts turned from the sun.

What is the greatest distance from the sun at which a comet can be situated, to be visible to us on the earth?

About three times the distance of the sun from the earth.

At what distance from the sun was the comet of 1680, at the period of its greatest heat?

About four-sevenths of the earth’s distance from the sun, or forty-six millions, two hundred and eighty-five thousand miles, nearly.

When a comet has been once seen, can the period of its return be foretold?

Not with any degree of certainty, though it is possible that the periods of three of them may have been ascertained. The first of these appeared in the years 1531, 1607, and 1682, and is expected to return every 75th year: there is also another, which by its return in 1758 gratified astronomers, as its appearance corresponded with the prediction of Dr. Halley.°

When did the second appear?

In the years 1522, and 1662; and it was expected that it would again make its appearance in 1789, but in this the astronomers of the present day have been disappointed.

Notes

Nicephorus Gregorius: or Gregoras (c. 1295–1360), remembered for his historical writings.

Sturmius: Johann Christophorus Sturm (1635–1703), German astronomer and mathematician.

Weigelius: Erhard Weigel (1625–1699), German mathematician, astronomer, and philosopher.

Hevelius: Johannes Hevelius (1611–1687), politician and astronomer, of Danzig.

Dr. Halley: Edmond Halley (1656–1742), who had indeed predicted that the comet now named after him would reappear, although the details were worked out by others.

“Geometry without Axioms”

Thomas Perronet Thompson, 1833

Something of an oddity among nineteenth-century Euclids, of which there were not surprisingly several, Thompson’s attempted, as the title page explained, “to get rid of Axioms and Postulates; and particluarly to establish the Theory of Parallel Lines without recourse to any principle not grounded on previous demonstration.” It seems to have been motived at least in part by unease about the status of the “parallel postulate,” but one cannot help feeling that in rewriting Euclid without axioms a baby has been lost as well as some bathwater. By the time we reach proposition I.47, however, things look more or less as in most Euclids.

Thomas Perronet Thompson (1783–1869) is mainly remembered as an army officer and politician; he had been governor of Sierra Leone by the time his Euclid appeared and would later distinguish himself as a political writer and radical MP.

Thomas Perronet Thompson, Geometry without Axioms, or The First Book of Euclid’s Elements. With Alterations and Familiar Notes; and an Intercalary Book in which the straight line and plane are derived from properties of the Sphere. (London, 4th edition, 1833), pp. v–ix, 1–2, 119–120.

In the preceding Editions endeavour had been made to get rid of Axioms, and particularly to establish the Theory of Parallel Lines without recourse to any principle not grounded on previous demonstration.

On showing the results to some of the leading mathematicians at Cambridge, they replied by the remark that they had always felt something to be more urgently wanted for the emendation of Geometry—which was, information on the nature and construction of the straight line and plane.

It had been stated, about the time when the circumstances were engrossing the attention of the public, that Napoleon on his voyage from Egypt amused himself and staff with circular geometry. What circular geometry might be, could only be collected from the tradition that the problem given by the future Emperor was “to divide the circumference of a circle into four equal parts by means of circles only.” But this sufficed to indicate that the idea which had passed through the mind of that eminent practical geometer was that in the properties of the circle, or still more probably of the sphere, might be discovered the elements of geometrical organization.

The author had in consequence been led at different times to attempt the collecting of the conditions, under which figures of various kinds may be turned about certain points and be what may be termed introgyrant, or turn upon their own ground without change of place. And on receipt of the remark mentioned above, this track was pursued with renewed vigour, and the results are presented in the sequel.

These results, together with the proofs offered of what have usually been termed the Axioms and Postulates, have been formed into an Intercalary Book, with a view to facilitate their postponement in the case of students beginning geometry for the first time. And for the further convenience of this class, a Recapitulation of the principal contents of the Intercalary Book has been given , thereby placing the beginner in the same situation as by the ordinary proceeding. The best time for commencing the Intercalary Book would probably be after having gone once through as much of the Elements of Euclid as is usually read; not, of course, that the contents are in any degree dependent upon what follows, but to take advantage of the habits of reasoning that may be thus acquired, before attempting what must be characterized as, in some parts, at least equal in complexity to anything in the succeeding Books.

If this process is objected to as irregular, it may be a great irregularity that nature should not have framed the elements of geometry so as to present a whole with the easiest parts always foremost and the Planes in the Eleventh Book. But if she has not, or till somebody can establish that she has, there seems to be no cause why bad reasoning should be admitted for the sake of a conventional arrangement. If the sphere is the simplest of figures and the properties of all others are derivable from it, it is more reasonable to be thankful for the knowledge than to quarrel with the dispensation.

In labouring to get rid of Axioms, the object has been to assail the belief in the existence of such things as self-evident truths. Nothing is self-evident, except perhaps an identical proposition. There may be things of which the evidence is continually before the senses; but these are not self-evident, but proved by the continual evidence of the senses. There may be things whose connexion with other things is so constantly impressed upon us by experience that few people ever think of inquiring into the cause, but for that very reason there is often considerable difficulty in clearly explaining the cause, and among this class of things the admirers of axioms have found their greatest crop of self-evident truths. In arguments on the general affairs of life, the place where every man is most to be suspected is in what he starts from as “what nobody can deny.” It was therefore of evil example that science of any kind should be supposed to be founded on axioms, and it is no answer to say that in a particular case they were true. The Second Book of Euclid would be true, if the First existed only in the shape of the heads of the Propositions under the title of Axioms, but this would make a most lame and imperfect specimen of reasoning.

The Game of Logic

Lewis Carroll, 1887

It would be a strange anthology of popular mathematics that passed by the nineteenth century without mentioning Lewis Carroll, as Charles Lutwidge Dodgson is better known. Few readers will need to be introduced to Alice’s Adventures in Wonderland or The Hunting of the Snark, but Dodgson wrote much else, including this attractive book, The Game of Logic, which I hope may be less familiar.

The book is introduced by the following splendid line: “There foam’d rebellious Logic, gagg’d and bound”—which turns out (Carroll does not tell us) to be from Alexander Pope’s mock epic The Dunciad (1728/1741); logic, there, was “gagg’d and bound,” alongside science, wit, and rhetoric, beneath the throne of “dullness” or stupidity.

Lewis Carroll, The Game of Logic (London, 1887), pp. 1–6.

Preface

This Game requires nine Counters—four of one colour and five of another, say four red and five grey.

Besides the nine Counters, it also requires one Player, AT LEAST. I am not aware of any Game that can be played with LESS than this number, while there are several that require MORE: take Cricket, for instance, which requires twenty-two. How much easier it is, when you want to play a Game, to find ONE Player than twenty-two. At the same time, though one Player is enough, a good deal more amusement may be got by two working at it together, and correcting each other’s mistakes.

A second advantage possessed by this Game is that, besides being an endless source of amusement (the number of arguments that may be worked by it being infinite), it will give the Players a little instruction as well. But is there any great harm in THAT, so long as you get plenty of amusement?

Propositions.

“Some new Cakes are nice.”

“No new Cakes are nice.”

“All new cakes are nice.”

There are three “PROPOSITIONS” for you—the only three kinds we are going to use in this Game; and the first thing to be done is to learn how to express them on the Board.

Let us begin with

“Some new Cakes are nice.”

But before doing so, a remark has to be made—one that is rather important, and by no means easy to understand all in a moment; so please to read this VERY carefully.

The world contains many THINGS (such as “Buns,” “Babies,” “Beetles,” “Battledores,” etc.), and these Things possess many ATTRIBUTES (such as “baked,” “beautiful,” “black,” “broken,” etc.: in fact, whatever can be “attributed to,” that is “said to belong to,” any Thing, is an Attribute). Whenever we wish to mention a Thing, we use a SUBSTANTIVE; when we wish to mention an Attribute, we use an ADJECTIVE. People have asked the question “Can a Thing exist without any Attributes belonging to it?” It is a very puzzling question, and I’m not going to try to answer it; let us turn up our noses, and treat it with contemptuous silence, as if it really wasn’t worth noticing. But, if they put it the other way, and ask “Can an Attribute exist without any Thing for it to belong to?”, we may say at once “No: no more than a Baby could go a railway-journey with no one to take care of it!” You never saw “beautiful” floating about in the air, or littered about on the floor, without any Thing to BE beautiful, now did you?

And now what am I driving at, in all this long rigmarole? It is this. You may put “is” or “are” between names of two THINGS (for example, “some Pigs are fat Animals”), or between the names of two ATTRIBUTES (for example, “pink is light-red”), and in each case it will make good sense. But, if you put “is” or “are” between the name of a THING and the name of an ATTRIBUTE (for example, “some Pigs are pink”), you do NOT make good sense (for how can a Thing BE an Attribute?) unless you have an understanding with the person to whom you are speaking. And the simplest understanding would, I think, be this—that the Substantive shall be supposed to be repeated at the end of the sentence, so that the sentence, if written out in full, would be “some Pigs are pink (Pigs).” And now the word “are” makes quite good sense.

Thus, in order to make good sense of the Proposition “some new Cakes are nice,” we must suppose it to be written out in full, in the form “some new Cakes are nice (Cakes).” Now this contains two “TERMS”—“new Cakes” being one of them, and “nice (Cakes)” the other. “New Cakes,” being the one we are talking about, is called the “SUBJECT” of the Proposition, and “nice (Cakes)” the “PREDICATE.” Also this Proposition is said to be a “PARTICULAR” one, since it does not speak of the WHOLE of its Subject, but only of a PART of it. The other two kinds are said to be “UNIVERSAL,” because they speak of the WHOLE of their Subjects—the one denying niceness, and the other asserting it, of the WHOLE class of “new Cakes.” Lastly, if you would like to have a definition of the word “PROPOSITION” itself, you may take this: “a sentence stating that some, or none, or all, of the Things belonging to a certain class, called its ‘Subject’, are also Things belonging to a certain other class, called its ‘Predicate’.”

You will find these seven words—PROPOSITION, ATTRIBUTE, TERM, SUBJECT, PREDICATE, PARTICULAR, UNIVERSAL—charmingly useful, if any friend should happen to ask if you have ever studied Logic. Mind you bring all seven words into your answer, and you friend will go away deeply impressed—“a sadder and a wiser man.”

Now please to look at the smaller Diagram of the Board (Figure 9.1), and suppose it to be a cupboard, intended for all the Cakes in the world (it would have to be a good large one, of course). And let us suppose all the new ones to be put into the upper half (marked “x”), and all the rest (that is, the NOT-new ones) into the lower half (marked “x′ ”). Thus the lower half would contain ELDERLY Cakes, AGED Cakes, ANTE-DILUVIAN Cakes—if there are any: I haven’t seen many, myself—and so on. Let us also suppose all the nice Cakes to be put into the left-hand half (marked “y”), and all the rest (that is, the not-nice ones) into the right-hand half (marked “y′ ”). At present, then, we must understand x to mean “new,” x′ “not-new,” y “nice,” and y′ “not-nice.”

Figure 9.1. The playing board for the Game of Logic. ©Princeton University Library. Rare Books Division, Department of Rare Books and Special Collections, BC135 .D6.

And now what kind of Cakes would you expect to find in compartment No. 5?

It is part of the upper half, you see, so that, if it has any Cakes in it, they must be NEW; and it is part of the left-hand half, so that they must be NICE. Hence if there are any Cakes in this compartment, they must have the double “ATTRIBUTE” “new and nice;” or, if we use letters, they must be “x y.”

Observe that the letters x, y are written on two of the edges of this compartment. This you will find a very convenient rule for knowing what Attributes belong to the Things in any compartment. Take No. 7, for instance. If there are any Cakes there, they must be “x′ y,” that is, they must be “not-new and nice.”

Figure 9.2. Fixing our attention on this upper half, suppose we found it marked like this. (Carroll, p. 6. ©Princeton University Library. Rare Books Division, Department of Rare Books and Special Collections, BC135 .D6.)

Now let us make another agreement—that a red counter in a compartment shall mean that it is “OCCUPIED,” that is, that there are SOME Cakes in it. (The word “some,” in Logic, means “one or more” so that a single Cake in a compartment would be quite enough reason for saying “there are SOME Cakes here”). Also let us agree that a grey counter in a compartment shall mean that it is “EMPTY,” that is that there are NO Cakes in it. In the following Diagrams, I shall put “1” (meaning “one or more”) where you are to put a RED counter, and “0” (meaning “none”) where you are to put a GREY one.

As the Subject of our Proposition is to be “new Cakes,” we are only concerned, at present, with the UPPER half of the cupboard, where all the Cakes have the attribute x, that is, “new.”

Now, fixing our attention on this upper half, suppose we found it marked like this (see Figure 9.2), that is, with a red counter in No. 5. What would this tell us, with regard to the class of “new Cakes”?

Would it not tell us that there are SOME of them in the x y-compartment? That is, that some of them (besides having the Attribute x, which belongs to both compartments) have the Attribute y (that is, “nice”). This we might express by saying “some x-Cakes are y-(Cakes),” or, putting words instead of letters,

“Some new Cakes are nice (Cakes),”

or, in a shorter form,

“Some new Cakes are nice.”

At last we have found out how to represent the first Proposition of this Section. If you have not CLEARLY understood all I have said, go no further, but read it over and over again, till you DO understand it. After that is once mastered, you will find all the rest quite easy.

Higher Mathematics for Women

Mrs. Henry Sidgwick, 1912

In 1911–1912 the British Board of Education assembled a volume of papers reporting on the teaching of mathematics for presentation to the International Congress of Mathematicians in Cambridge in 1912. The eighteenth paper in the series discussed “mathematics in the education of girls and women,” and the three authors displayed a range of views on the subject. Eleanor Mildred Sidgwick (her husband was the philosopher Henry Sidgwick), in her section titled “Higher Mathematics for Women,” was easily the most optimistic, in line with her role as principal of the all-female Newnham College, Cambridge, and a promoter of university education for women. Her warning against teaching and learning mathematics only “because it is useful for something else” retains its force in the twenty-first century.

Mrs. Henry Sidgwick (1845–1936), “Higher Mathematics for Women,” in E. R. Gwatkin, Sara A. Burstall, and Mrs. Henry Sidgwick, “Mathematics in the Education of Girls and Women”; The Board of Education, Special Reports on Educational Subjects: The Teaching of Mathematics in the United Kingdom, no. 18 (London, 1912), pp. 26–27, 31.

I have been asked, in connection with the work of the International Commission on the Teaching of Mathematics, to write a paper on the above subject, and my attention has been specially invited to the question whether, for instance, the study of Mathematics by women should have a special bearing on the subsequent study of Economics, or of statistical inquiries into sociological questions, and therefore perhaps differ in kind from the Mathematics required by, e.g., a future engineer. I understand the expression Higher Mathematics to be used, not in the sense that mathematicians might give it, but merely to mean Mathematics as studied at the Universities by those who are at least to some extent specialising in the subject, as distinct from the Mathematics which form a part of general education at secondary schools.

It will perhaps conduce to clearness if I state at once that, for reasons set forth below, it does not appear to me necessary or advisable to have separate programmes of mathematical study for men and for women at the Universities. If an inherent difference in mathematical ability between the two sexes were established it might seem at first sight to furnish a reason for a difference of programme. But experience has shown that some women have sufficient ability and sufficient liking for Mathematics to justify their making it their principal subject of study at the University, and this being so, the inquiry how their powers compare with those of men, either on the average, or as regards the highest degrees of mathematical ability, becomes irrelevant to the question before us. For the great differences in ability among the men who specialise in Mathematics afford a range amply wide enough to include all the women who could on any hypothesis be reasonably advised to do so.

In considering the nature of the demand for mathematical education on the part of women, I may take as fairly typical the present (1911–12) students of Newnham College at Cambridge. This is convenient, both because information about them is easily accessible to me, and because the extent to which specialisation is carried at Cambridge, and the consequent sharp differentiation between the different Triposes, makes classification comparatively easy. I should explain that, with the exception of a few graduates from other Universities, and a few other students taking special courses, the women students at Cambridge, whether at Girton or Newnham,° are reading for Honours in one or other of the Tripos examinations, i.e., examinations for degrees in Honours. Out of 213 students at Newnham in November 1911 were 30 reading Mathematics, 2 Engineering, 35 Classics, 3 Moral Sciences, 1 Law, 33 Natural Sciences (including Geography), 39 History, 61 Medieval and Modern Languages, 9 Economics. Over 14 per cent of the students therefore were working at Mathematics, and this proportion is somewhat less than has been usual in other years. Of the 30 reading Mathematics, 9 were in their first year, 13 in their second year, and 8 in their third year at the University. A certain number will not continue to work at Mathematics through their whole course, but after obtaining Honours in Part I of the Mathematical Tripos at the end of their first or second year will turn their attention to some other subject, e.g., Science, especially Practical Physics for the first or second part of the Natural Science Tripos, or Geography for the Diploma in Geography, or possibly Economics for the Economics Tripos. They might even take some subject unconnected with Mathematics, such as History or Languages. The proportion of students of Mathematics is usually much the same at Girton as at Newnham, but probably the proportion at Cambridge is in excess of that among women students at the Universities generally, as the prestige of Cambridge in Mathematics no doubt tends to attract mathematical students there.

The next question to be considered is what leads these women to make a special study of mathematics. It is almost always, I think, that they have liked the subject and succeeded fairly well in it during their school education, and have consequently been advised by their school teachers to go on with it at the University as a subject they are likely to do well in. If, as is frequently the case, they intend to become teachers, they think it is the subject they will like best to teach, and which they are most likely to teach well and to find profitable employment in. The estimate thus formed of their powers and tastes, either by the students themselves or their teachers, does not always prove correct. It happens occasionally that facility in manipulating algebraical formulae has been mistaken for mathematical grasp. It happens, occasionally, that taste for the subject diminishes as the difficulty increases. It happens, also, sometimes that the amount of previous preparation required for success in the Cambridge course, by any but persons of unusual ability, is under-estimated, and that a student who has come to the University with the intention of studying Mathematics, but insufficiently prepared, has to be advised to change her plans. It does not often happen that, on the other hand, a student who comes up intending to devote herself to some other subject, changes to Mathematics.

If I am right in the above diagnosis of the motives that lead women to the study of Mathematics at Cambridge, the subject is in almost all cases studied mainly for its own sake—not because it is useful as an adjunct or stepping-stone to something else. And educationally it is very important that this should be so. A subject which is studied, not for its own sake, but because it is useful for something else, is almost always degraded in the process, and loses much of its educational value, whether the ultimate object be merely to pass an examination or to acquire the minimum knowledge necessary for dealing with some other and different subject of study. It is, no doubt, quite possible to learn Mathematics after a fashion, and up to a certain point, purely as an instrument for some ulterior purpose. But probably for everyone, and certainly for anyone with any mathematical ability, the loss from such a method of study would be great. There would be loss in mental training, loss in knowledge, and loss in interest. A recent writer in the Engineering Supplement of “The Times” advocating a continued attention to Mathematics for its own sake on the part of engineers, said, “Its importance in education is to form deliberate judgment, to assign a measure to results, to disclose fallacies, to discourage narrowness, to reveal the unity of whatsoever things are true, and generally to exalt the mind.” I should be inclined to add, to stimulate the imagination. The study of mathematics reveals to the eyes that can see whole vistas of knowledge, whole aspects of the universe, unsuspected by those without mathematical perception.

I would not, therefore, limit the opportunities of pursuing the study of Mathematics for its own sake in any way, either for women or for men. It must not of course, be forgotten that among those who study Mathematics at the Universities a few—one here and there—may have sufficient power and originality to carry the subject in some direction or other beyond the present limits of our knowledge. But quite apart from this, it is important to keep before everyone, and before women especially, the value of knowledge for its own sake. I say before women especially, because I think the education of women has suffered more than that of men from what I may call, for shortness, the commercial or utilitarian point of view. The old bad education so prevalent in the girls’ schools and the private schoolrooms of 50 or 60 years ago—drawing-room accomplishments, needlework, a little arithmetic and something of general information—was a kind of degenerate technical education for domestic life. The standard was low because neither thoroughness nor a high standard were needed for the purposes aimed at, and the standard tended to get worse rather than better because the teachers, being taught in the same schools, had no better opportunities of learning than their pupils. As a general result extremely little intellectual training or cultivation was, as a rule, attained, and very little interest was developed in anything beyond trivialities. I fear a return towards a similar state of things when I hear it suggested that Science should be taught to girls and women as the handmaid of Cookery and Hygiene, mathematics as the handmaid of Economics, and Economics again studied because a knowledge of Economics may be useful in social work.

To sum up, my conclusion is that in the first place it is highly desirable that women with mathematical ability, even if it be not of a very high order, should be encouraged to study the subject for its own sake, and not with limits prescribed by the utility of Mathematics for something else, for it is thus that they will get the utmost value out of it both in pleasure and in mental training. And I conclude in the second place that so far as it may be necessary to make arrangements for the study of mathematics merely as a stepping stone to other studies, there is no need to consider the case of women separately from that of men.

Note

Girton or Newnham: At this time these were the only two Cambridge colleges which admitted women, and they both admitted only women.

A New Aspect of Mathematical Method

George Pólya, 1945

George Pólya, a Hungarian emigré to the United States, worked in real and complex analysis, as well as other areas of mathematics, and also studied mathematical pedagogy. How to Solve It, his book on mathematical heuristics, strategies of problem solving, and how to teach and learn them, is said to have sold over a million copies in seventeen languages and is one of the classics of modern popular mathematics.

While the dialogue form is one we have met before, in Catherine Whitwell’s Astronomical Catechism and in Newton for the Ladies in Chapter 5, what Pólya intends to convey by it, concerning the learning of mathematics as a method of thought, has a distinctly different flavor from anything else in this book.

G. Pólya (1887–1985), How to Solve It: A New Aspect of Mathematical Method (Princeton, 1945, 1957; London, 1990), pp. 25–29. © 1945 Princeton University Press, 1973 renewed by Princeton University Press, 2004 Princeton University Press, Expanded Princeton Science Library. Reprinted by permission of Princeton University Press.

A problem to prove.

Two angles are in different planes but each side of one is parallel to the corresponding side of the other, and has also the same direction. Prove that such angles are equal.

Figure 9.3. Two angles are in different planes, but each side of one is parallel to the corresponding side of the other and has also the same direction.

What we have to prove is a fundamental theorem of solid geometry. The problem may be proposed to students who are familiar with plane geometry and acquainted with those few facts of solid geometry which prepare the present theorem in Euclid’s Elements. (The theorem that we have stated and are going to prove is the proposition 10 of Book XI of Euclid.)

“What is the hypothesis?”

“Two angles are in different planes. Each side of one is parallel to the corresponding side of the other, and has also the same direction.”

“What is the conclusion?”

“The angles are equal.”

“Draw a figure. Introduce suitable notation.”

The student draws the lines of Figure 9.3 and chooses, helped more or less by the teacher, the letters as in the figure.

“What is the hypothesis? Say it, please, using your notation.”

“A, B, C are not in the same plane as A′, B′, C′. And AB || A′B′, AC || A′C′. Also AB has the same direction as A′B′, and AC the same as A′C′.”

“What is the conclusion?”

“BAC = B′A′C′.”

“Look at the conclusion! And try to think of a familiar theorem having the same or a similar conclusion.”

“If two triangles are congruent, the corresponding angles are equal.”

“Very good! Now here is a theorem related to yours and proved before. Could you use it?”

“I think so but I do not see yet quite how.”

“Should you introduce some auxiliary element in order to make its use possible?”

“Well, the theorem which you quoted so well is about triangles, about a pair of congruent triangles. Have you any triangles in your figure?”

“No. But I could introduce some. Let me join B to C, and B′ to C′. Then there are two triangles, ΔABC, ΔA′B′C′.”

“Well done. But what are these triangles good for?”

“To prove the conclusion, BAC = B′A′C′.”

“Good! If you wish to prove this, what kind of triangles do you need?”

“Congruent triangles. Yes, of course, I may choose B, C, B′, C′ so that

AB = A′B′, AC = A′C′.”

“Very good! Now, what do you wish to prove?”

“I wish to prove that the triangles are congruent,

ΔABC = ΔA′B′C′.

If I could prove this, the conclusion BAC = B′A′C′ would follow immediately.”

“Fine! You have a new aim, you aim at a new conclusion. Look at the conclusion! And try to think of a familiar theorem having the same or a similar conclusion.”

“Two triangles are congruent if—if the three sides of the one are equal respectively to the three sides of the other.”

“Well done. You could have chosen a worse one. Now here is a theorem related to yours and proved before. Could you use it?”

“I could use it if I knew that BC = B′C′.”

“That is right! Thus, what is your aim?”

“To prove that BC = B′C′.”

“Try to think of a familiar theorem having the same or a similar conclusion.”

“Yes, I know a theorem finishing: ‘. . . then the two lines are equal.’ But it does not fit in.”

“Should you introduce some auxiliary element in order to make its use possible?”

“You see, how could you prove BC = B′C′ when there is no connection in the figure between BC and B′C′?”

. . .

“Did you use the hypothesis? What is the hypothesis?”

“We suppose that AB || A′B′, AC || A′C′. Yes, of course, I must use that.”

“Did you use the whole hypothesis? You say that AB || A′B′. Is that all that you know about these lines?”

“No; AB is also equal to A′B′, by construction. They are parallel and equal to each other. And so are AC and A′C′.”

“Two parallel lines of equal length—it is an interesting configuration. Have you seen it before?”

“Of course! Yes! Parallelogram! Let me join A to A′, B to B′, and C to C′.”

“The idea is not so bad. How many parallelograms have you now in your figure?”

“Two. No, three. No, two. I mean, there are two of which you can prove immediately that they are parallelograms. There is a third which seems to be a parallelogram; I hope I can prove that it is one. And then the proof will be finished!”

We could have gathered from his foregoing answers that the student is intelligent. But after this last remark of his, there is no doubt.

This student is able to guess a mathematical result and to distinguish clearly between proof and guess. He knows also that guesses can be more or less plausible. Really, he did profit something from his mathematics classes; he has some real experience in solving problems, he can conceive and exploit a good idea.

New Math for Parents

Evelyn Sharp, 1966

Evelyn Sharp’s book aimed “to explain to parents something of the tremendous changes taking place in . . . school mathematics” during the 1960s. She admitted that “there must be many mothers and fathers who are bewildered by the new approach to the subject” and gave a readable and comprehensible approach to mathematics along New Math lines. Here we have her outline of the motivations for the new methods and her discussion of the basic laws of arithmetic, forming a remarkable contrast with the older discussions of arithmetic in Chapter 2 of this book.

Evelyn Sharp, edited by L. C. Pascoe, A Parent’s Guide to New Mathematics (London, 1966 (original U.S. edition 1964)), pp. 1–2, 47–49, 71–72.

The Mathematics Revolution

If you have a child in school you may have begun to feel the impact of a change in mathematics—a change so far-reaching that it may well revolutionize the teaching of the subject at all levels and in many countries in the Western world, at least. Some time in the middle 1950s it became abundantly clear to the U.S.A. that drastic modification of a limited and old-fashioned (non-progressive) syllabus in the schools was urgently needed. With determination and enthusiasm, research was undertaken and experiments were started. By 1965, great progress had been made; very large numbers of American schools had been able to introduce courses of study, even at an early age, aimed to develop a logical understanding of mathematics and the capacity to carry out thought processes of deductive reasoning. The aim was to avoid a purely automatic use of manipulative processes and learning by rote, with no comprehension of the principles involved. Naturally enough, word of the schemes crossed the Atlantic and, although the needs in Britain were not the same, experiments began in various parts of the country, sometimes sponsored by university research workers and sometimes by schools. Needless to say, the teams soon linked up. It is, after all, difficult to introduce new approaches to teaching unless one has children able and willing to participate. At this stage there seemed to be two points of view which could be taken: on the one hand, it could be argued that there was already a comprehensive cross-section of mathematics in schools; on the other, that much of the existing curriculum of the average good school was unimaginative, stereotyped and archaic. One thing is, however, clear—that there was not such an urgent need for violent modification. The process has already started but it is, at present at any rate, a cautious and unhurried transformation.

Properties of Number Systems: The ACD (associative, commutative, distributive) laws and arithmetic

The simplest of these laws, stated in mathematical language, is the commutative property for addition: a + b = b + a. For example, 5 + 1 = 1 + 5. The commutative property for multiplication is ab = ba, eg 7 × 3 = 3 × 7.

The law of association declares an equally obvious fact: that 2 + (3 + 4) = (2 + 3) + 4. (Do the part in parentheses first and then add the other number.) Formally stated, the associative property for addition is: a + (b + c) = (a + b) + c. It is necessary because addition, like marriage, is a binary operation—only two can take part in it at a time. To add three numbers, you first add any two of them, then add that answer to the third one. The law of association merely states what you already know—that it does not matter which two you add first.

This property plays a vital role in the way children are now taught to add, as you can see in the following example:

They start with sets, regroup them by using the law of association, always building sets of ten, and then add as the union of sets. “We join sets, we add numbers,” they say.

A great deal hinges on the child’s ability to split the second number into two parts, one of which will combine with the first number to make ten. There are pages of practice on this.

Often the way has been prepared by using, in kindergarten, the Cuisenaire rods. The system consists of a set of wooden pieces varying from one unit to ten units in length. Cuisenaire, a Belgian, used one centimetre as his unit. The system introduces the child to numbers through measuring, rather than by counting, the traditional gateway to arithmetic. If Billy wants to find 3 + 2, instead of counting 3 clowns here and 2 clowns there, he places the 3-block end-to-end with the 2-block and then finds the block whose length measures the same.

A characteristic colour is used for each number; for example, the Cuisenaire rod for 10 is orange. The children learn to identify the block both by length and colour. If the blocks are piled up so that only part of the orange block is showing, Susie recognizes it as the 10-block by its colour, although she cannot see the full length.

The use of these rods may be carried on for some time in the child’s school life. This accords with current pedagogical procedure, which is to present number concepts first through physical operations with three-dimensional objects, follow this with two-dimensional marks on paper, and then proceed to traditional Hindu–Arabic numerals.

Soon the children are handling sets of more than one ten—twenty, thirty, etc. (The objects in their sets have now lost some of their barnyard flavour—not so many baby ducks and frolicking lambs.) 22 + 4 becomes (20 + 2) + 4, then 20 + (2 + 4) by the law of association, then 20 + 6, and finally 26.

More complicated examples require the laws of both commutation and association.

As progress is made the problems get harder:

True, nobody is going to go through life putting all this down whenever he wants to add. But this is a sound basic process on which to build. It is in fact easier, as skill develops, to add the tens, thus:

and later still

Eventually an automatic mental process is evolved: 65, 85, 92.

“Merely a Formal Statement of the Way We Think”

Robert E. Eicholz and Phares G. O’Daffer, 1964

Dating, like the previous extract, from the beginning of the New Math era, Eicholz and O’Daffer’s book, with set notation on page 1 and the whole numbers defined through set cardinality a few pages later, exemplifies the characteristics (and problems) of what was intended as a thoroughly systematic approach to learning mathematics. It aimed to give teachers a thorough understanding of the structure of mathematics underlying the New Math pedagogy. We have come a long way from Humfrey Baker’s list of the parts of mathematics he proposed to teach, or indeed from the number games of Mrs. Lovechild’s Art of Teaching.

Figure 9.4. Sets S and T are disjoint; sets Q and R are not. (Eicholz et al., p. 6.)

Robert E. Eicholz and Phares G. O’Daffer, A First Look at Modern Mathematics (Palo Alto, 1964), pp. 6–7, 12–14.

Cardinal Numbers

If two sets S and T have the property that the elements of S can be paired with the elements of T so that each element in S is paired with exactly one element in T, and vice versa, then the sets S and T are said to be equivalent to each other. The relation shown by the pairing of elements is called a one-to-one correspondence between the sets.

Examples of pairs of equivalent sets are shown in Figure 9.4. Note that the sets S and T are disjoint with respect to each other, and that the sets Q and R are not disjoint. In the case of Q and R, we think of the square which is common to both sets as being paired with itself. Expanding upon this idea of pairing an object with itself, we point out that each set is equivalent to itself; that is, each element can be paired with itself.

With these observations as background, we can now make some statements about cardinal numbers. We shall not attempt a precise definition of the set of all cardinal numbers, but will confine ourselves to a simple description and some examples. Let us say that a cardinal number is a class of equivalent sets. For example, cardinal number 2 is the class of all sets which are equivalent to the set in Figure 9.5; cardinal number 3 is the class of all sets equivalent to the set {a, b, c}; etc. These statements are depicted in Figure 9.6. Cardinal number zero is the number of the empty set.

Since we think of each cardinal number as a class of sets, we can speak of choosing a set A from a cardinal number. If we choose a set from cardinal number 2, we have a set of two elements.

Figure 9.5. Cardinal number 2 is the class of all sets which are equivalent to this one. (Eicholz et al., p. 6.)

Figure 9.6. Cardinal numbers 2 and 3 as classes of equivalent sets. (Eicholz et al., p. 6.)

We have presented the idea of a class of equivalent sets in an abstract form. To explain this concept of number to a young child, we simply point repeatedly at representative sets and say the proper words. This is exactly how children are exposed to their first number concepts, and an understanding of the cardinal-number concept is well under way for most children by the time they enter school. For example, a child has heard the word “two” many times with respect to many different sets (two cows, two lollipops, two brothers) and has thus gained a feeling for the idea of “twoness.”

Order for the set of cardinal numbers is easier to understand intuitively than it is to state formally. We state below precisely what it means to say that cardinal number a is greater than cardinal number b.

Consider two cardinal numbers, a and b, and sets from these cardinal numbers, A and B respectively. If the set A contains a proper subset which is equivalent to B, then we say that cardinal number a is greater than cardinal number b, and that cardinal number b is less than cardinal number a. We write a > b (a is greater than b) and b < a (b is less than a).

Careful examination of this definition will reveal that it is merely a formal statement of the way we think when we compare cardinal numbers.

Addition of Cardinal Numbers

We are now in a position to say exactly what is meant by the sum of two cardinal numbers.

Consider two cardinal numbers a and b and sets A and B from these cardinal numbers such that A B is empty. (A and B are disjoint.) The cardinal number of the set A B is the sum of cardinal numbers a and b (written a + b). The numbers a and b are called addends of the sum a + b.

An example illustrates that this definition is little more than a formal statement of how we introduce children to the concept of addition. Suppose we wish to show a child that 2 + 3 = 5. We would probably begin by displaying a set of 2 and a set of 3 and having the child identify the numbers of these sets. (Select a set from cardinal 2 and a set from cardinal 3.) Now we would have the child push the two sets together. (Form the union.) We would ask, “How many in all?” (The sum is the cardinal number of the union.)

Observe the importance of the word disjoint in the definition of the sum of two cardinal numbers. Suppose we wish to demonstrate the sum of 4 and 5 and fail to select disjoint sets.

The cardinal number of the union is 7, and this is not the number that we wish to call the sum of 4 and 5. Obviously, in order to obtain the desired result, we must select disjoint sets.

Some special remarks will be helpful.

(1) We add numbers, not sets.

(2) We form the union of sets, not of numbers.

(3) If A and B are sets, A = B means that A and B are the same set; that is A and B are different names for the same set. If A and B are equivalent but different sets, we do not write A = B.

(4) If a and b are cardinal numbers, and we write a = b, we mean that a and b are the same cardinal number; that is, a and b are different names for the same cardinal number.

(5) Greater than and less than are relations defined for numbers, not for sets. We do not write A > B or B < A for sets A and B.

1. From the definition of union of sets, we can see that A B is the same set as B A. What does this fact tell you about the cardinal numbers a + b and b + a?

2. Suppose that a, b, and c are cardinal numbers such that a + b = c. What can you conclude if a = c? if c > a?

3. Show a set from cardinal 4 and a set from cardinal 5 such that the cardinal number of the union of these two sets is 5. Make choices so that the cardinal number of the union is 6; so that it is 9. Can sets be chosen from 4 and 5 so the cardinal number of the union is greater than 9? less than 5?

4. If the cardinal number of the union of two sets is less than the sum of the cardinal numbers of the sets, what can you say about intersection of the two sets?

5. Consider three cardinal numbers a, b, and c such that a + b = c and a + c = b. What can you say about cardinal number a?

6. At a school picnic, 16 children drank cherry soda and 15 had lemon soda. There were only 25 children at the picnic. Explain in set language why one cannot argue that there were 31 children at the picnic.

7. List the six possible pairs of addends of 7.

Turtle Fun

Serafim Gascoigne, 1985

Many readers will remember the LOGO turtle, but it comes as something of a surprise to recall that it was described in the 1980s as representing an entirely new form of geometry. As we saw in Chapter 6, the late twentieth century saw the rise of computers and calculators as the new “mathematical instruments,” and children received instruction in their use just as adults did.

Nothing illustrates this better than these passages about the turtle and its antics. The same book also has a wonderful passage about another piece of microchip wildlife, the “mouse”: “The busy executive can sit back in his leather swivel chair and call all sorts of information on to the screen simply by moving and squeezing the mouse!”

You can try out the instructions given here using one of the many turtle programs on the web, such as www.mathsnet.net/logo/turtlelogo.

Serafim Gascoigne (dates unknown), Turtle Fun: LOGO for the Spectrum 48K (London, 1985), pp. vii, 6–8, 34–41, reproduced with permission of Palgrave Macmillan.

The world of microcomputers is forever changing. There is always something new to learn and find out. How can you keep up? Will you be able to program the computers of the future? And what about “intelligent” machines and robots, especially the LOGO turtle?

This book will help you to keep up with events by introducing you to a very powerful control language of the future, called LOGO. LOGO has in fact been developed by computer scientists working in a new science called Artificial Intelligence. LOGO has been written for young programmers. It comes from another language called LISP, which is one of the main languages used to control “intelligent” computers and robots. Using such commands as STARTROBOT or REPEAT [FORWARD 50 LEFT 120] for example, you can either drive a small mechanical device called the floor turtle (see Figure 9.7) or control the screen turtle, a movable cursor on your TV screen. You can also teach the turtle your own commands, called “procedures,” which can include graphics, text and sound.

Figure 9.7. A small mechanical device called the floor turtle. (Gascoigne, p. 6.) Reproduced with permission of Palgrave Macmillan.

Learning LOGO will not only teach you another computer language but it will also help you to organise your thinking and ideas in a clear and structured way. Apart from the scientific value of learning LOGO, it is fun! The world of the LOGO turtle is an exciting and intriguing world that ranges from simple screen doodling to the study of turtle geometry—a new geometry, totally different from school geometry but surprisingly familiar!

LOGO allows you to invent your own language, to be creative and above all, to make your programming a personal venture.

How to move the turtle

Before you begin to explore the ideas in this book, you will need to know how to use some basic commands. If you already know how to move the turtle, you may skip this section.

To call up the turtle, you type SHOWTURTLE (ST). You will now see a turtle or cursor-shape at the centre of the screen. This is the LOGO screen turtle.

To move the turtle you use the following commands

These commands can be shortened to FD (FORWARD), BK (BACK), RT (RIGHT) and LT (LEFT). Most LOGO commands that you will meet in this book can in fact be shortened.

Typing FORWARD 10 or FD 10 (and pressing ENTER) will move the turtle forward ten steps on the screen, and FORWARD 20 or FD 20 (press ENTER) will move the turtle forward twenty steps.

Typing BACK 10 or BK 10 (press ENTER) will move the turtle backwards ten steps, and so on.

Try changing the number of steps after the commands, FORWARD and BACK.

To turn the turtle to the left you type: LEFT followed by a number. This is the number of degrees. For example LEFT 90 or LT 90 (press ENTER) will turn the turtle to the left 90 degrees. (If you are not sure what a degree is, it is a unit of measurement like miles or litres. Degrees are used to measure angles just as miles are used to measure distance or litres are used to measure quantities of liquid.)

RIGHT 90 or RT 90 (press ENTER) will turn the turtle to the right 90 degrees. LEFT 1 or LT 1 will turn the turtle to the left 1 degree.

RIGHT 45 or RT 45 will turn the turtle to the right 45 degrees.

Try changing the number of degrees after the commands, LEFT and RIGHT. Don’t forget to press ENTER whenever you want the computer to carry out your commands.

As the turtle moves, you will have noticed that it draws a line. If you wish, you can move the turtle without drawing a line by typing PENUP or PU. When you wish to resume drawing, type PENDOWN or PD.

You can also hide the turtle as it draws by typing HIDETURTLE. To make the turtle reappear simply type SHOWTURTLE.

The basic commands so far are

One of the important things to do with LOGO is to experiment! Try out these commands and see what happens. Some important ideas have been discovered by people doodling on the screen. From the very moment that you start moving the turtle, you are ready to discover new ideas. See what you can find.

Pretty poly

You may have enjoyed drawing polygons at school using a ruler and pencil. Even if you did not enjoy drawing them, using the turtle to draw polygons can be an exciting experience. The power of the turtle allows you to produce some spectacular effects at the press of a button. Figures that would require hours of precision drawing with a pencil can be produced on the screen instantly.

Here is a simple procedure to get you started. You can draw polygons of all shapes and sizes. The procedure, which I have called POLY, is as follows

TO POLY :SIZE :ANGLE

FD :SIZE

LT :ANGLE

POLY :SIZE :ANGLE

END

By typing POLY, followed by an input for the sides and an input for the angle, you can produce some amazing figures on the scceen. The last line in the procedure is recursive. It tells the turtle to start again.

You might like to experiment with the ANGLE input. Can you make any predictions as to what kind of shape the turtle will produce? How many sides will it have? How many points or vertices? Will the lines cross each other?

POLY with subprocedures

You can also use subprocedures inside the POLY procedure. Instead of FD, for example, use the name of a procedure such as TRIANGLE or SQUARE.

To demonstrate this, let’s write a new version of POLY and call it POLY1.

TO POLY1 :SIZE :ANGLE

TRIANGLE :SIZE

RT :ANGLE

POLY1 :SIZE :ANGLE

END

If you have not already got a procedure called TRIANGLE, here is one for you to experiment with.

TO TRIANGLE :SIZE

REPEAT 3 [FD :SIZE RT 120]

END

You can of course invent your own program to draw a triangle. Try

POLY1 60 144

Polyspirals

From the simple POLY procedure you can go on to produce what are called polyspirals. For this let’s create a new procedure and call it POLYSPI.

TO POLYSPI :SIZE :ANGLE

FD :SIZE

LT :ANGLE

POLYSPI :SIZE + 3 :ANGLE

END

This new procedure is really the POLY procedure again. However the last line

POLYSPI :SIZE + 3 :ANGLE

tells the turtle to increase the size of the figure by 3 each time. The increase in size is written in the last line, which also uses recursion to repeat POLYSPI.

You can go a stage further and make the increase in size (called an increment) a variable as well. Let’s call it INC.

TO POLYSPI1 :SIZE :ANGLE :INC

FD :SIZE

LT :ANGLE

POLYSPI1 :SIZE + :INC :ANGLE :INC

END

This procedure allows you to chooose the increment or increase in step each time. However the increment remains the same throughout the procedure. It is the :SIZE that gets bigger. Note that in the last line of this procedure :INC has to be written twice! POLYSPI1 needs three inputs. The combination :SIZE + :INC is regarded as one input for :SIZE. Therefore :INC is written again in the same line.

The next procedure, called POLYSPI2, makes the increment grow this time in proportion to :SIZE.

TO POLYSPI2 :SIZE :ANGLE

FD :SIZE

LT :ANGLE

POLYSPI2 :SIZE + :SIZE/10 :ANGLE

END

Here you can see that the increment is not written as a variable but is included in the variable :SIZE. In the recursion line I have written

:SIZE + :SIZE/10

This simply means, add the value of :SIZE to :SIZE divided by 10.

You can carry out any arithmetical operations you like on input. For example, you can use +, −, and /. (* is used for multiply and / is used for divide.) Have a go at using different operators inside your procedures. Here are just a few examples.

FD :SIZE * 4 (multiplication)

FD :SIZE / 2 (division)

FD :SIZE + 10 (addition)

You can even use operators with the inputs to your procedures. You can type, for example

POLYSPI2 3 360/7

where you are asking the turtle to divide the angle of 360 degrees by 7.

Here is another program that uses BK as well as FD and uses + and / as operators.

TO POLYSPI3 :SIZE :ANGLE :INC

WINDOW SETBG 1 SETPC 3

FD :SIZE SETPC 7 FD 20 + :SIZE/2

BK :SIZE/2 RT :ANGLE

POLYSPI3 :SIZE + :INC :ANGLE :INC

END

Figure 9.8. Turtle frenzy: spirals and polyspirals. (Gascoigne, p. 38.) Reproduced with permission of Palgrave Macmillan.

I have used some cosmetics here! You can set your own background and pen colour. As you can see, the size inputs are divided by 2, except in the last line.

Here (see Figure 9.8) is a list of inputs that you might like to try.

Stopping your procedures

You will have noticed that POLY and POLYSPI do not stop. This is because we have used recursion. To stop a procedure that is recursive you need a conditional statement.

IF :SIZE > 100 [STOP]

This tells the turtle to stop if the size of the figure is greater than 100. This conditional statement using IF can be used with any procedure in which you are using recursion.

Using RANDOM

Here is another poly program that uses RANDOM.

TO RANDP :SIZE :ANGLE :INC

MAKE “ANS RANDOM 10

IF :ANS = 0 [PD][PU]

FD :SIZE LT :ANGLE

RANDP :SIZE + :INC :ANGLE :INC

END

The conditional statement in this procedure tells the turtle to pendown if the random number is 0. If not, then penup.