A essay written for my Preliminary Examination for doctoral candidacy in Curriculum & Instruction at Texas A&M University, Fall 2009.
One of the recurring themes in the history of education is the decision by families and societies of who may or must learn mathematics. This issue is very closely related to the funding of mathematics education.
Schooling, be it public or private, requires significant financial resources, and the USA is very generous in its public support. According to an OECD (2009)1 report published in 2005 the USA and Switzerland spend more than $11,000 per student per year in the public school system. This amount is the highest among OECD countries.
Today there is the USA no controversy of who should learn mathematics. The answer is everyone and it has been so for about one hundred years. The controversy in mathematics education has moved to other areas, the curriculum and the instructional methods. This controversy is so intense that it has been called the “Math Wars” (Loveless, 2001; Latterell, 2005).
Our present situation in the USA is the result of a long historical process where the questions of whether one should learn mathematics in the first place and if so, who should learn it, have had various answers.
Historically, state support of education in general and of mathematics specifically is a relatively recent phenomenon. This concept arose in Europe during the 17-18th centuries. Previously, families or religious institutions would financially support the education of its members.
The first time in known history that an author wrote on education was by Plato in his two political treatises, The Republic (380 BCE) and Laws (Harris, 1991, p. 100). The first book (Adam, 1902) is much better known than the second (Plato, 1967/1968), however, his recommendation for universal education is actually in his last work.
[7.804c] …we have described buildings for the public gymnasia as well as schools in three divisions within the city, and also in three divisions round about the city training-grounds and race-courses for horses, arranged for archery and other long-distance shooting, and for the teaching and practicing of the youth: if, however, our previous description of these was inadequate, let them now be described and legally regulated. In all these establishments there should reside teachers [7.804d] attracted by pay from abroad for each several subject, to instruct the pupils in all matters relating to was and to music; and not father shall either send his son as a pupil or keep him away from the training-school at his own sweet will, but every “man jack” of them all (as the saying goes) must, so far as possible, be compelled to be educated, inasmuch as they are children of the State even more than children of their parents. For females, too, my law will lay down the same regulations as for men, and training of an identical kind.
In the above passage we should notice the mentioning of the concept of competitive pay for teachers. In the same dialogue we also have injunctions for the implementation of a form of school administration
[6.764c] …It will be proper next to appoint officials for music and gymnastics, two grades for each department, the one for education, the other for managing competitions. By education-officers the law means supervisors of gymnasia and schools, both in respect of their discipline [6.764d] and teaching and of the control of the attendances and accommodations both for girls and boys.
Plato directly mentions mathematics
[7.809c] …first, literature, next lyre-playing; also arithmetic, of which I said there ought to be as much as everyone needs to learn for purposes of war, house-management and civic administration; together with what it is useful for these same purposes to learn about the courses of the heavenly bodies – stars and sun and moon – in so far as every State [7.809d] is obliged to take them into account.
In The Republic the passages relevant to education are presented as a fictitious conversation between Glaucon and Socrates (as a stand-in for Plato himself). They discuss, among others, the concepts of curriculum, motivations for learning mathematics, support by the state, and economic, cultural and military aspects of mathematics knowledge. Here Glaucon starts
[7.522b] “Surely; and yet what other study is left apart from music, gymnastics and the arts?” “Come,” said I, “if we are unable to discover anything outside of these, let us take [7.522c] something that applies to all alike.” “What?” “Why, for example, this common thing that all arts and forms of thought and all sciences employ, and which is among the first things that everybody must learn.” “What?” he said. “This trifling matter, “ I said, “of distinguishing on and two and three. I mean, in sum, number and calculation. Is it not true of them that every art and science must necessarily partake of them?” “Indeed it is,” he said. “The art of war too?” said I. “Most necessarily,” he said.
[7.525b] …“And the qualities of number appear to lead to the apprehension of truth.” “Beyond anything,” he said. “Then, as it seems, these would be among the studies we are seeking. For a soldier must learn them in order to marshal his troops, and a philosopher, because he must rise out of the region of generation and lay hold on essence or he can never become a true reckoner.” “It is so,” he said. “And our guardian is soldier and philosopher in one.” “Of course.” “It is befitting, then, Glaucon, that this branch of learning should be prescribed by our law and that we should induce those who are to share the highest function of state [7.525] to enter upon that study of calculation and take hold of it, not as amateurs, but to follow it up until they attain to the contemplation of the nature of number, by pure thought, not for the purpose of buying and selling, as if they were preparing to be merchants or hucksters, but for the uses of war and for facilitating the conversion of the soul itself from the world of generation to essence and truth.”
[7.527c] “Then nothing is surer,” said I, “than that we must require that the men of your Fair City shall never neglect geometry, for even the by-products of such study are not slight.” “What are they?” said he. “What you mentioned,” said I, “its uses in war, and also we are aware that for the better reception of all studies there will be immeasurable difference between the student who has been imbued with geometry and the one who has not.” “Immense indeed, by Zeus,” he said. “Shall we, then, lay this down as a second branch of study for our lads?” “Let us do so,” he said. …[7.522e] “Shall we not then,” I said, “set down as a study requisite for a soldier the ability to reckon and number?” “Most certainly, if he is to know anything whatever of the ordering of his troops – or rather if he is to be a man at all.”(cf. Laws 819d)
Plato realizes that the study of mathematics is difficult, but has many positive aspects to it, and should certainly be pursued
[7.526b] …“Again, have you ever noticed this, that natural reckoners are by nature quick in virtually all their studies? And the slow, if they are trained and drilled in this, even if no other benefit results, all improve and become quicker than they were?” “It is so,” he said [7.526c] “And, further, as I believe, studies that demand more toil in the learning and practice than this we shall not discover easily nor find many of them.” “You will not in fact.” “Then, for all these reasons, we must not neglect this study, but must use it in the education of the best endowed natures.” “I agree,” he said …
Plato also mentions the importance of the use of public funds for mathematics research and even its organization
[7.528b] …“but this subject, Socrates, does not appear to have been investigated yet.” “There are two causes of that,” said I: “first, inasmuch as no city holds them in honor, these inquiries are languidly pursued owing to their difficulty. And secondly, the investigators need a director, who is indispensable for success and who, to begin with, is not easy to find, and then, if he could be found, as things are now, seekers in this field would be too arrogant. [7.528c] to submit to his guidance. But if the state as a whole should join in superintending these studies and honor them, these specialists would accept advice, and continuous and strenuous investigation would bring out the truth. Since even now, lightly esteemed as they are by the multitude and hampered by the ignorance of their students as to the true reasons for pursuing them, they nevertheless in the face of all these obstacles force their way by their inherent charm [7.528d] and if would not surprise us if the truth about them were made apparent.” …[7.528e] “assuming that this science” …“is available, provided that is, that the state pursues it.”
Some elements that would seem very modern to us
[7.536d] …“Now, all this study of reckoning (logismon) and geometry and all the preliminary studies that are indispensable for dialectic must be presented to them while still young, not in the form of compulsory instruction.” “Why so?” “Because,” said I, [7.536e] “a free soul ought not to pursue and study slavishly; for while the bodily labors performed under constraint do not harm the body, nothing that is learned under compulsion stays with the mind.” …
The other great philosopher of that time, Aristotle (1944), discussed education in the context of a political treatise called Politics, just as Plato did. However, his approach was much more practical and much less Utopian. He did write about the importance of the involvement of the state in education and some form of universal education
[8.1337a] Now nobody would dispute that the education of the young requires the special attention of the lawgiver. Indeed the neglect of this in states is injurious to their constitutions; for education ought to be adapted to the particular form of constitution, …And as much as the end for the whole state is one, it is manifest that education also must necessarily be one and the same for all and that the superintendence of this must be public, and not on private lines, in the way in which at present each man superintends the education of his own children, teaching them privately, and whatever special branch of knowledge he thinks fit. But matters of pubic interest ought to be under public supervision; at the same time we ought not to think that any of the citizens belongs to himself, but that all belong to the state, for each is a part of the state, and it is natural for the superintendence of the several parts to have regard to the superintendence of the whole. And one might praise the Spartans in respect of this, for they pay the greatest attention to the training of their children, and conduct it on a public system.
Then Aristotle discussed about the legislation of education and mentions conflicting ideas about the curriculum in his time. He also tells us about the curriculum that “There are perhaps four customary subjects of education, reading and writing, gymnastics, music, and fourth, with some people, drawing; …” (8.1337b)
Aristotle makes reference to an, otherwise unknown, Phaleas of Chalcedon, who considered “it fundamentally necessary for states to have equality in these two things, property and education.” Then Aristotle discusses the connection between education and property and their inequalities among the people (1266b).
He also seems to have supported the teaching of both sexes, again for political reasons
[1260b] …For since every household is part of a state …it is necessary that the education both of the children and of the women should be carried on with a regard to the form of constitution, …
It is interesting to note that Aristotle considers education important for women because they “are a half of the free population,” (1260b). It has been noted that today’s patriarchal societies waste half of their human resources and how in a globally competing world this is hurting the economy of those countries.
After reading these quotes on education written about 2,400 years ago, one may think that almost everything there is to say about education was written then. We have discussions about curricula, school ages, school support, obligatory education for all,and the need for state regulation. With special reference for mathematics, Plato proposes state support for research, and stresses the great relevance of education for the state of morality, intellectual development, and the economic and military vitality of the state. He also cautions the reader acknowledging that it is the most difficult subject, and should not be taught against the wishes of the student.
Reality was much different. Xenophon (1979) wrote in his Memorabilia (371 BCE) that in his time education was responsibility of the parents, who may send their children to a teacher “at a cost, and strive their utmost that the children may turn out as well as possible.” (2.2.6). There was then no universal education and there will not be for two millennia. It took only slightly less time to have state support of education.
The Greek civilization became the Hellenistic civilization and then the Roman/Hellenistic civilization. Afterwards civilization in Europe would be mostly confined to Moorish Iberia, Sicily and to Cyprus. There were schools in Europe, of course, but instruction did usually not include mathematics. Usually the curriculum consisted only of the Latin language and some Latin literature (Black, 2001, p. 1). The first stirrings of change in the condition of mathematics education occurred in Italy during the thirteenth century. Commerce brought Italian merchants in contact with the much more civilized Near East and North Africa. Leonardo of Pisa published in 1202 his Liber Abaci that popularized the Indian system of numeration in Italy and then the rest of Europe. The new number system was soon adopted by the Italian merchants (Smith & Kapinski, 1911, pp. 128–151). At the same time we have the development of the modern banking system starting with the “Monte dei Paschi” of Siena.2 Finally, the modern double-entry accounting system came into general use in Italy (e.g. Smith, 1951).
Private schools were created that were devoted to the teaching of mathematics for the sons of merchants and bankers. They were called the “abbaco,” the abacus school. The curriculum today would be described as ‘business mathematics.’ The main textbook was the Trattato d’Abbaco by Piero della Francesca.3 Not only business people were trained in these schools, but also architects, painters, and artisans. For instance Machiavelli and Leonardo da Vinci went to the abacus school (Gamba & Montebelli, 1989, p. 19). Nicolo’ Tartaglia was a teacher at an abacus school in Verona.4 There was no public support for these schools and neither was it universal education, but a whole new social class was sending all their sons to these schools with the result that mathematics knowledge increased greatly in society.
The important factor, in my opinion, is that from here on the ruling class slowly becomes aware of the importance of mathematics for the wealth and defense of the country. In Italy it happened quickly because most governments were directly controlled by the business class. The governments in the rest of Europe were still in the hands of the nobles who worried about the business class mostly when they had to borrow money to finance their wars or raise taxes.
John Amos Comenius (1592–1670) lived during the Thirty Years’ War (1618–1648). This was one of the most savage and destructive wars in Europe. He was an educator and a rare voice for tolerance, equality, and reason. Like the ancient philosophers he advocated universal education. In his Magna Didactica he wrote
…sed omnes pariter, nobiles at ignobiles, divites et pauperes pueros et puellas per omnes urbes et oppida pagosque et villas scholis esse adhibendos sequentia evincunt. (9.1)
That is, all alike, noble and ignoble, rich and poor boys and girls …should be sent to school (Comenius, 1894, p. 60).
Comenius does not single out mathematics from the other school subjects. He mentions it in chapter 30, “Scholae Latinae delineatio” where delineates his curriculum with a list of 13 items. Item four is arithmetic and five is geometry. They are listed immediately after the language items.
Comenius realized that his education reform program would face many obstacles. It was after all a reform of profound political implications. In the last chapter of his book (33) he appeals to the rulers of Europe (dominatores populorum at magistratus politici) to erect schools (aedificaveritis synagogas) for the children (parvulos) (p. 238). Comenius tries to convince them that the cost of funding schools is justified because education is more important than military training (militarem scientiam instrui) and infrastructure, because a “vir enmim bonus et sapiens pretiosissimum est totius rei-publicae cimelion,” (p. 239). That is, a good and wise man is the most precious treasure of the state. He continues to exhort the political class using Biblical quotations and concludes the book with a prayer.
Unfortunately, his appeal to noble ideals would not convince the rulers to fund schools where mathematics could be taught. Mathematician teachers and professors found a more convincing argument that actually was the opposite of what Comenius wrote. That is, military training was more valuable than goodness and wisdom.
The introduction of gun powder in Europe changed warfare from a “sport between nobles” to a “chess game between generals.” Guns and artillery improved and became more and more important on the battle field. The mathematicians of the Renaissance in Italy hoped that they could solicit state support by showing the ruling class how mathematics could help calculate trajectories, arrange armies, and build defenses. Nicolo’ Tartaglia published a book titled La Nova Scientia in 1537 and dedicated it to the Duke of Urbino. It contains, among other subjects, methods to calculate the range of cannon balls. Galileo Galilei published in 1606 a manual with military applications titled Le operazioni del compasso geometrico et militare and dedicated it to the Prince of Tuscany. He was the first mathematician to discover how to calculate the maximum range of cannon balls. The Renaissance military architect De’ Zanchi wrote that military engineers needed to know geometry and arithmetic (Gamba & Montebelli, 1989, p. 36).
These efforts did not have much success, most probably because ballistics was more an art than a science. The mathematical calculations did not take into account air drag and other factors that determined the trajectory of a cannon ball. The calculations of Tartaglia and Galileo would work only for short trajectories where air drag would still be negligible (Steele, 1994).
The people who would make ballistics a real science were the English Benjamin Robins (1707–1751) and the German Leonhard Euler (1707–1783). Robins published in 1742 the New Principles of Gunnery. In 1745, at request of the King of Prussia Frederick the Great,5 Euler translated into German Robins’ work and added new material that Euler himself had developed. At that time Euler was employed by the King of Prussia and Frederick wanted his artillery officers to have the best ballistics book available at that time (Steele, 1994)
Soon afterward the Seven Years’ War started (1754–1763) where the new and improved ballistics would be applied. England and Prussia were victorious and France and its allies lost. The terms of peace were dire for France, which lost Canada, India and other oversee territories to the English. The French government realized that one of the causes for the defeat was the technological inferiority of the artillery. Prussian artillery pieces and their officers were superior to their French equivalents.
On 23 August 1774 the French minister of marine wrote to King Louis XVI
Le clèbre Léonard Euler, un des plus grands mathématiciens de l’Europe, a composé deux ouvrages qui pourraient être très-utiles pour les écoles de la Marine et de l’Artillerie. L’un est un Traité de la Contruction et de la Manoeuvre des vaisseaux; l’autre est un commentaire sur les principles d’artillerie de Robins, traduit en français. Je propose à Votre Majesté d’en ordonner l’impression …(Henry, 1883, p. 180)
He proposed that the two books written by Euler be translated and used by the navy and artillery schools.
By this time the main governments in Europe had established military schools and the U.S.A. would not be far behind. These schools were funded by the state treasury and one of the most important subjects in the military schools was mathematics. The famous mathematician Pierre-Simon Laplace taught at the École Militaire in Paris. This institution was founded by King Louis XV in 1752. Its most famous graduate was Napoléon Bonaparte, who was good at mathematics, and completed the school in one year instead of two. His examiner was the same Laplace. Napoléon became an artillery officer and used the, by now, superior French artillery to become the great general. We could say that it was the “power of mathematics.”
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