Notes on the History


of Math Teaching and Math Books

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NOTE: This article is about the English math books in America. The earliest math books published in the Americas were in Spanish. You can find more about these early non-English math books at my link on Arithmetic

Prior to 1800

The first English Language arithmetic book was probably the 1537 edition of An Introduction for to : Lerne to Reckon with the Pen and with the Counters after the True Cast of Arsmetyke or Awgrym. The last word seems to be one of the many intermediate steps along the line to our present word, algorithm.. This was an English translation of the 1494 Latin Arithmetic of Luca Pacioli, Summa de Arithmetica, Geometrica, Proportioni et Proportionalita (Everything about Arithmetic, Geometry, and Proportion). You can see a some pages from the book, and a famous picture of Friar Luca here

The first really popular arithmetic in English was the 1542 Arithmetick: or, The Grounde of Arts written by Robert Recorde, author of The Whetstone of Whet. Note in the image at right that the "afterward" has a contribution by Englands famous alchemist and early mathematician, John Dee.

According to David Eugene Smith's "History of Mathematics" Vol.2, "...our earliest native American arithmetic, the Greenwood book of 1729,..." and "...the first in what is now the United States was a reprint of Hodder's English arithmetic, Boston, 1719." The full title of Hodder's book was Arithmetick: or, That Necessary Art Made Most Easy. The American Printing was a copy of the 25th edition in England.

Benjamin Franklin' autobiography makes mention that he studied another common English translation, Cocker's Arithmetic, after he moved from his home to Pennsylvania, " And now it was that, being on some occasion made asham'd of my ignorance in figures, which I had twice failed in learning when at school, I took Cocker's book of Arithmetick, and went through the whole by myself with great ease. I also read Seller's and Shermy's books of Navigation, and became acquainted with the little geometry they contain; but never proceeded far in that science. And I read about this time Locke On Human Understanding, and the Art of Thinking, by Messrs. du Port Royal. "

Here is a link to an image of the covers of three early arithmetics that were used in America from the digital library at Pitt.

Perhaps the most popular English copy printed in America was Thomas Dilworth's, The Schoolmaster's Assitant, Being a Compendium of Arithmetic both Practical and Theoretical. The book reached from the introductory topics to the advanced in five sections:
Section I, Whole Numbers included the basis of the four operations and proceeded to topics on interest, rebates, partnership, weights and measures, the double rule of three, alligation, mediation and permutations.
Section II delt with common fractions.
Section III delt with decimal fraction operations and included roots up to the fourth power, and work on annuitites and pensions.
Section IV was a collection of 104 problems to be solved. As was common in many older texts, the questions were sometimes stated in rhyme. Lessons for students were for memorization and recitation.

Q. How many sorts of Addition are there?
A. Two, viz. simple and compound

Section V was on Duodecimals, working with fractions in which the only denominators were twelfths. These types of problems continue in textbooks and appear in my 1870 edition of White's Complete Arithmetic in the appendix. The definition states, "A Duodecimal is a denominate number in which twelve units of any denomination make a unit of the next higher denomination. Duodecimals are used by artificers in measuring surfaces and solids. The foot is divided into primes marked '; the primes into seconds marked ''; the seconds into thirds marked ''', etc.
Here is the table of contents from the Dilworth book.
Other popular English arithmetics printed in America included Cocker's Arithmetick and John Bonneycastle's The Scholar's Guide to Arithmetic.

In 1729 the first arithmetic published by a Native of Colonial America was published by Isaac Greenwood, a professor at Harvard. The title was Arithmetic, Vulgar (common) and decimal.(see earlier image for cover) Greenwood seemed to have a short and somewhat checkered life (1702-1745) He graduated from Harvard in 1722 and went on to become the Hollis professor of Mathematics, but was censured in 1737 for drunkenness and dismissed on August 30, 1738. He was replaced as Hollis professor, I believe, by John Winthrop of the Massachusetts Bay Winthrops.

Only one year after Greenwood's book the second book by a colonial was published. This one, Arithmetica of Cyfer Konst by Pieter Venema, was in Dutch. Venema was a teacher, and probably a preacher in the Pennsylvania Dutch Community, who had been a well respected mathematics teacher and textbook author in Groeningen, Holland. His problems with the authority of the church may have prompted his departure to America. Interestingly, this book may well have been the first book published in the future United States that had a serious algebra content, and over sixty years would pass before there was another. Of additional interest, the book's title page indicate that it was printed on the presses of Peter Zenger.

Neither these, or several other arithmetics published in the next fifty years were very popular, and none of them approached the English translations mentioned before. The first really popular English language arithmetic by an American born author was in 1788 when Nicholas Pike published A New and Complete System of Arithmetic, which he said was "Composed for the use of the citizens of the United States". Well, patriotism probably won't hurt sells in a new country. Pikes book carried endorsements from several noted persons, including the governor of Massachusetts, James Bowdoin, and Yale President Ezra Stiles. The book even included a copy of the Act of Congress of 1786 which created the U. S. Federal Money System with denominations of mills (1/1000 of a dollar), cents, dimes, dollars, and Eagles (ten dollars). With all this emphasis on the new USA, it seems strange that none of the problems in the book involved the new American money, but instead were based on the English system. [The 2nd edition, in 1797, includes in the (very long) title; "adapted to the Federal Currency by Nathaniel Lord, A.M.;Boston]" An image of the cover can be seen in this text from the digital library at Pitt. Units to be memorized in the book included measures for cloth, wine, and beer, and both Troy and Apothacary weight. Just so you know, beer measures consisted of pint, quart, gallon, firkin, kilderkin, barrel, hogshead, puncheon, and butt. One other book that is worth mentioning before 1800, although it was not very popular, is Chancey Lee's 1797 publication of American Accomptant which is ofted credited with being the first book to use the $ symbol in print, although he used something very similar to the dollar symbol for both the dime and the dollar. .

From The History of Mathematics at the Univ of Georgia. "In an attempt to recreate a plausible description of the mathematics syllabus one may assume that it consisted in a sequential coverage of a portion of one of the standard texts available at the time. An arithmetic text by Nicholas Pike, with the extended title 'A New and Complete System of Arithmetic composed for the Use of the Citizens of the United States', was published in Worcester, Massachusetts in 1788[this location seems to be an error as shown in an image of the cover from the digital library at Pitt., and also as quoted in a "Bibliography of Early American Textbooks on Algebra..." by Lao Genevra Simons in 1936 gives Newbury-Port; Printed and sold by John Mycall.... the Algebra in it is mostly, by admission of the author in the preface, directly from Bonnycastle's ]. This text would be the second math book in the future U.S. area which containing algebra. This book was the first arithmetic text to be written in America which received wide-spread distribution. (An earlier book, published in 1729, was used only in a few classes at Harvard.) The preface to Pike’s Arithmetic contains testimonials by the Presidents of Dartmouth, Harvard and Yale. The book is approximately 500 pages long, with some 200 sections. It begins with the rules for the elementary operations for integers, together with many examples worked out in detail. These are followed by sections about vulgar fractions, decimal fractions, rules for exchanging currency, tricks for rapid computation, extraction of square roots, computation of interest, commissions, annuities, the volumes of particular solids, and topics from elementary mechanics. The book may be summarized as a compendium of useful techniques and formulas, with examples completely worked out, in a wide diversity of practical applications. There are very few proofs. The formulas from the slightly more advanced topics which are presented are given with very little detail. The first three quarters of the book is an excellent collection of common calculations, of the sort we hope are mastered by students by the time they reach high school."

Prior to 1800 American schools were mostly one-room affairs with limited resources. Here is a note from a 1961 U.S. Department of Health, Education, and Welfare, Office of Education publication called "Technology in American education, 1650-1900."

The colonial schools had no blackboards, slates, or maps, although some flourishing schools could boast of owning a glove. Almost all of the school supplies for pupils were homemade. The pens were goose-quills. In fact, a teacher was sometimes hired more for his ability to cut and mend quills than for his ability to teach. If the schoolmaster was an expert penmaker, a great deal of his time would be consumed in that activity if he had a large school.

Each family supplied their children with homemade ink, usually by dissolving ink powder in water. Many of the country fold gathered the bark of swamp-maple and boiled it down for ink. These homemade inks were often weak and pallid and sometimes dried up.

The paper ordinarily bought for school purposes was rough and dark. Its high cost led the scholars to use it sparingly and in the new and poorer communities children frequently had to write on birch bark. The paper came in foolscap size (approximately 13" x 17") and was unruled. The pupils would fold the paper and make separate pages out of it, cover the pages with a course brown wrapping paper or wallpaper, and then carefully sew it into a "copy-book" or "sum-book."

Once the copy book was made, the children had to rule the paper in preparation for writing. This was done with little strips of sheet lead or "leaden plummets" as they were called. Regardless of the primitive equipment, however, the handwriting of the colonial children seemed to suffer no visible damage.


1800-1850

In 1776 there were seven colleges in America; by 1800 there were nineteen. The typical college consisted of one or two buildings which housed approximately 100 young men. The programs were mostly modeled after the one at Yale; those at the other colleges, Harvard, William and Mary, Princeton, Dartmouth, Rutgers, Middlebury, … were much the same at the time, At the University of Georgia, then newly founded (and called Franklin College??? check this), "The course of studies at the Franklin College in a typical term in the early years was as follows: Freshmen studied arithmetic from about 5 o’clock until breakfast, Latin from 9 until 12 and Greek from 2 until 4. Sophomores advanced to algebra and geometry; with more Latin and Greek. The list of topics studied during the last two years included trigonometry, astronomy, natural philosophy (i.e. physics and chemistry), logic, history, composition and forensic disputation, and Latin and Greek. All students in each of the Freshman, Sophomore, Junior and Senior classes had the same courses. Most of the time in class was spent on recitation."

History and Use of the Blackboard
It appears that the blackboard first came into American education around 1800. They seem to have very quickly become an essential part of daily school life. [From a web page of Prof. Rickey]

Perhaps no one method has so influenced the quality of the instruction of the cadets as the blackboard recitations. Major Thayer insisted on this form, although old records show that it was introduced at West Point by Mr. George Baron, a civilian teacher, who in the autumn of 1801 gave to Cadet Swift "a specimen of his mode of teaching at the blackboard." Today it is the prominent feature in Academic instruction. [Quoted from Richardson 1917, p. 25] There is indication that the blackboard was used in a few schools in the US before it was used at USMA. See Charnel Anderson, Technology in American Education, 1650-1900, published by the US Dept of Health, Education, and Welfare 1961 [Kent State: FS 5.234: 34018. Rickey has photocopy of pp. vi +16-25, 32-35, 40-47, 50-53; the last few pages are the bibliography of this small booklet.]
The Publication on Technology in American Education mentioned above gives a different date for the introduction of blackboards to West Point, "The blackboard was introduced to West Point Military Academy in 1817 by Claude Crozet, who had been an ex-officer under Napoleon, then appointed to the Academy as professor of engineering. When he arrived at the school, he found to his amazement that there were no textbooks in English. Thus he was faced with the problem of teaching a new science without a textbook, and he himself unfamiliar with the English language. Crozet solved the problem with the aid of a carpenter and painter who turned out the first blackboard to be used in that part of the country. (AJE, 1867-1869, pp. 31-38)"

The blackboard seems to have made a strong and immediate impact on teaching "technology". Previous practice methods, such as the horn book abrutly disappear as indicated in this quote from OED about Hornbooks, "a1842 HONE in A. W. Tuer Hist. Horn-Bk. I. i. 7 A large wholesale dealer in..school requisites recollects that the last order he received for Horn-books came from the country, about the year 1799. From that time the demand wholly ceased..In the course of sixty years, he and his predecessors in business had executed orders for several millions of Horn-books". The US Government Publication, Technology in American Education, 1650-1900, has the following:

In the 1830's educators stopped regarding the blackboard as a curious innovation and began to look upon it as essential to teaching. A lecturer in 1830 listed it as one of four essential apparatuses every school should have.

One or more of these should be found in every school . . . This piece of school-furniture is almost invaluable. In some schools it has been deemed so important as to form part of the WALL, all around the room. (Adams, 1830, pp. 345-346)

The Connecticut Common School Journal of February 15, 1839, advised its schools: "In all the operations performed by the pupils . . . blackboards should be used for demonstrations and illustrations." (CCSJ, 1841,p. 92) In the same journal a letter from a teacher ventured that "the most useful piece of school apparatus, may be simply a black board painted or stained black, attached to the wall or to a movable stand.... It is employed in teaching scholars of every stage of advancement." (CCSJ, 1841, p. 48-49)

By the 1840's the blackboard was firmly entrenched in the school systems. Texts for teachers on the use of the blackboard began to be written.

A few quotes from one of these (pp. VII and VIII) shows us, possibly with some exaggeration, that teachers were convinced that the blackboard was here to stay.

I should feel in the schoolroom, without the blackboard, as though the LAST PLANK had been taken from under me!

And again:

The inventor or introducer of the blackboard system deserves to be ranked among the best contributors to learning and science, if not among the greatest benefactors of mankind.

In 1842, the Connecticut Common School Journal dedicated five entire issues including illustrations to "Slate and Black Board Exercises for Common Schools."

Early blackboards were usually made of wood, (but some may have been made of paper mache') and painted with many coats as true slate boards were very expensive. Schools purchased large pots of "slate paint" for regular repainting of the boards. The Earliest quotes from the OED date to 1823. .

1823 PILLANS Contrib. Cause Educ. (1856) 378 A large black board served my purpose. On it I wrote in chalk. 1835 Musical Libr. Supp., Aug. 77 The assistant wrote down the words..on a blackboard. 1846 Rep. Inspect. Schools I. 147 The uses of the black board are not yet fully developed.

However under "slates" I found the earlier quotation, "1698 FRYER Acc. E. India & P. 112 A Board plastered over, which with Cotton they wipe out, when full, as we do from Slates or Table-Books" which indicates that boards covered with Plaster or other materials were used to write upon much earlier than the earliest use of "blackboard". The frequently mentioned publication on technology in education comments on the history of slates, "The origin of the slate reaches far back into antiquity. Its predecessors were the wax tablets used by the Geeks and Romans and the black wax tablet used by the Hindus around 1000 A.D. The modern slate seems to have been in common use in Europe by the 1400's. (Chaucer makes the first allusion to it in English in his Astrolabe II (p. 44), "enter hit into thy slate." (Karpinski, 1863, p. 341) [For
more on blackboards and slates] Slates were slowly introduced into American schools. Although there were some allusions to them in the colonial period, it was not until the 1800's that they were commonly used. In William B. Fowle's report to the trustees of the Boston Monitorial School in 1825, he says: "Every child in school is furnished with a slate and pencil, which are considered a part of the furniture of the school." ".

In 1841, a Professor Haskell listing apparatuses in use in New York schools gave a list of 32 items. They included the following of interest to the math/science teacher. (some of these are not known to me, and I would love to have input on any of them, pictures, descriptions, etc)

3. The arithmetical cards of thirteen folio pages.
4. The arithmetical card with movable counters.[Perhaps the Arithmeticon shown below]
Several small globes were listed, and then

#13. The cylindrical revolving Macerator's chart. [This seems to be a Mercator projection laid onto a cylinder for dispalay so that it could be rotated about the polar axis like a globe. In chapter three (pg 109) of The Autobiography and Other Memorials of Mrs Gilbert, Formerly Ann Taylor by Ann Taylor, edited by Josiah Gilbert. London: Henry S. King & Co., 1874; she writes "In order to facilitate the reading of voyages and travels he had strained a large Mercator's chart, round a revolving cylinder, upon which, with pith-headed pins, representing the Pacific voyager or African traveller, his pupils could follow the wanderings of each ]

14. The card of geometrical figures.
15. The protractor.
16. The machine to illustrate angles, chords, sines, tangents, etc.
17. Geometrical solids.
18. The planetarium.
19. The machine showing the inclination of the orbits of the planets
20. The Tellurian.
[one I know, these were also sometimes called a tellurium (and may still be by some) until Martin Klaproth named a newly discovered metal by that name.) The image at right shows an example from the Glossary of Globe Terminology by George Glazer.]

21. The tide globe.[I wonder how many schools have one of these today?, a web search turned up no hits]

22. The machine showing the earth as an oblate spheroid was simply a globe; (but to make it more expensive, it would flatten at the poles when whirled with the finger.)

23. The machine explaining the moon's roads. (????)
24. The celestial globe.
25. Diagram of the solar system.
26. The frame of pulleys and levers.
27. The machine for showing action and reaction.
28. The intermittent fountain ( to demonstrate the phenomena of siphoning.)
29. The machine to show the resistance of the air.(????)
30. The prism.
31. An electrical air-pump and chemical apparatus.
32. A box of family apparatus.(???????)
"

In 1851, Practical illustrations of the principles of school architecture By Henry Barnard(1811-1900) listed among the desirable apparatus, "The Gonigraph is a small instrument composed of a number of flat rods connected by pivots, which can be put into all possible geometrical figures that consist of straight lines and angles, as triangles, squares, pentagons, hexagons, octagons, &c." They also recommended, and described the use of the "Arithmeticon" as shown below:

The book also describes the construction of desks, blackboards, etc. The book is online at the "Making of America" Database at the University of Michigan; http://www.hti.umich.edu/m/moagrp/

While visiting Brugge I came across an old school room set up in the Folk Museum there [Museum voor Volkskunde]. The classroom had an incredible array of mathematical and scientific tools, as shown in the photos, and one wonders if every school could have been so well equipped. Note the large number of models behind the "teacher". Many of these were sliced to show cross sections. In the display case were a number of counting type tools, and a classic cube set for showing the Pythagorean theorem for a 3-4-5 triangle.












1850-1900

1854 Joseph Ray establishes a "Mathematical Department" in The Ohio Journal of Education, devoted to the solution and discussion of mathematical problems. Many prominent Ohio mathematics teachers will contribute problems and solutions over the next six years. By the 1850s, books had become much cheaper and texts more uniform. This cover is from a book in a series of math texts by Joseph Ray (1807-1855), a Cincinnati teacher. Ray’s first arithmetic text appeared in 1834, and his books were widely used as late as 1913.

In 1893 "An international Congress of Mathematicians is held at the World's Columbian Exposition in Chicago, August 21-26. Felix Klein and E.H. Moore occupy center stage. The Committee of Ten on Secondary School Studies recommends a year of algebra, followed by two years of plane and solid geometry to be taught side by side with more algebra. The first year's course in algebra is recommended for all students."(from Milestones in (Ohio)Mathematics, by David E. Kullman)

Compare these recommendations with this description from the History of the Math Department at the University of Nebraska in Lincoln, which opened in September of 1871. Especially note the entrance requirements for the present:

The initial program of courses in mathematics was extremely modest. Geometry, Plane and Spherical Trigonometry, Analytic Geometry and Calculus were the only subjects offered. It was early accepted that mathematics, in particular calculus, would not be a required study for all students. In the first University Catalog it is noted that "Surgery and Calculus are not required of students in the Classical course." Exemptions from other mathematics were soon included. The University Register and Catalog for the Fourth Session, 1874-1875, explained that (except in the Scientific course) General Geometry and Calculus were optional courses "from the fact that those students who, through lack of taste or ability, are not likely to gain a clear understanding of the principles will derive more benefit from other studies than from the memorizing of their rules and formulae". Such rationale, in forms modified to fit the situations, has persisted through the years. The pattern of minimizing mathematical exposure established then has continued to exist within the College to this day. However, every college at UNL now requires students to take at least one course in the Math/Stat area, which includes Math 104, 106, 107, 203, and Stat 180 as well as some quantitative courses that are in other departments. In addition, UNL now has an entrance requirement of four years of high school mathematics which must include 2 years of algebra, 1 year of geometry, and one year that builds on algebra and geometry. (It use to be 2 years of math with only the stipulation that it include one year of algebra.)

Here is a listing of the collection of Early American Mathematical books at the University of California at Berkley.


1900-1950

Graph paper, a math class staple, was developed between 1890 and 1910. During this period the number of high school students in the U.S. quadrupled, and by 1920, according to E.L Thorndike, one of every three teenagers in America "enters High School", compared to one in ten in 1890. The population of "high school age" people had also grown so that the total number of people entering HS was six times as great as only three decades before. Research mathematicians and educators took an active interest in improving high school education. E. H. Moore, a distinguished mathematician at the University of Chicago, served on mathematics education panels and wrote at length on the advantages of teaching students to graph curves using paper with “squared lines.” When the University of Chicago opened in 1892 E.H. Moore was the acting head of the mathematics department. "Moore was born in Marietta, Ohio, in 1862, and graduated from Woodward High School in Cincinnati. "(from Milestones in (Ohio)Mathematics, by David E. Kullman) Moore was President of the American Mathematical Society in 1902. The Fourth Yearbook of the NCTM, Significant Changes and Trends in the Teaching of Mathematics Throughout the World Since 1910, published in 1929, has on page 159, "The graph, of great and growing importance, began to receive the attention of mathematics teachers during the first decade of the present century (20th)" . Later on page 160 they continue, "The graph appeared somewhat prior to 108, and although used to excess for a time, has held its position about as long and as successfully as any proposed reform. Owing to the prominence of the statistical graph, and the increased interest in educational statistics, graphic work is assured a permanent place in our courses in mathematics." [emphasis added]

Hall and Stevens "A school Arithmetic", printed in 1919, has a chapter on graphing on "squared paper". Some more notes on Graph paper can be found here.

The idea of a number line seems to have made its way into instruction much later than might have been expected. On Jeff Miller's web page on the earliest use of some terms in mathematics we find:

An earlier term was scale of numbers. In 1912, First Year Algebra by Webster Wells and Walter W. Hart has: "Thus, the positive and negative numbers together form a complete scale extending in both directions from zero." Lawrence M. Henderson, "An Alternative Technique for Teaching Subtraction of Signed Numbers," The Mathematics Teacher, Nov. 1945 has: "In teaching subtraction of signed numbers, I first draw a number scale." Number line is found in January 1956 in "An exploratory approach to solving equations" by Max Beberman and Bruce E. Meserve in The Mathematics Teacher: "In an earlier paper we described a procedure by which students could 'solve' equations and inequalities using a number line. The set of points on a number line is in one-to-one correspondence with the set of real numbers."
Max Beberman (1925-1971), a professor of mathematics education at the University of Illinois, is generally regarded as the father of the New Math.

The idea of a number scale had been used earlier, as in the image below from Wentworth's Elements of Algebra (1893).


The term "number line" and even "number scale" did not seem to develop quickly. Dave L. Renfro found a journal referrence that predates the 1955 date given above, "The phrase "Number Line" (capitalized in this way) is used several times in:
Herbert Ellsworth Slaught, "The evolution of numbers -- An historical drama in two acts", American Mathematical Monthly 35 #3 (March 1928), 146-154." . Yet the Wells and Hart mentioned above by Miller did not use it as late as their 1934 Progressive Algebra in which they display what we would today call a number line and comment that, "We naturally represent these numbers on a straight line, as below". In Rosenbach & Whitman, College Algebra (1939) they continue to refer to the "numbered scale". The 1942 edition of Ginn and Company's Useful Mathematics in the "supplementary work" at the end of the book(pg 416) they use, "First we make a number scale..." and include a number line illustration. Nadyne Lapi sent me a note that in ALGEBRA by Welcons and Krickenberger (Book One) 1953," 'Number Scales' is on page 77 and only has integers for scale markings."

In a note from Dave Mason of South Taho High School he included a note that, "The number line on page 55 uses decimals, and shows where the square root of 2 would be.." The image shows a section of a line but doesn't use the term "number line", and also doesn't indicate in any way that the line continues forever, in fact the language of the text refers to "the interval from one to two." [See the page here.] The first image I have in my personal collection that includes a general number line with integers and some type of fractional representation is in Merrill's Modern Algebra (1962) where the image below appears on page 65 (another is on page 63, but only the positive ray).


Sam Koski from Miami Springs Senior High sent me an image of an earlier use in the 1960 edition of Algebra, Its Big Ideas and Basic Skills(McGraw-Hill) which I have added here.


I have not, at this point, seen a book (or article) prior to 1960 which illustrates a general number line with anything other than integers. (now extended below). OK, and now I have seen then much earlier. In the summer of 2009 I received some documents from Dave Renfro, and amongst them found a number line in Simon Newcomb's Algebra for Schools and Colleges in 1888, which describes a line going "indefinitly in both directions" with positive and negative integres In the same bundle was a page from Henry Burchard Fine's A College Algebra, 1904 which showed an "indefinite straight line" with integers and rational fractions. And a third document had a page from the 1907, First Course in Algebra by Albert Harry Wheeler, Teacher of Mathematics in the Enlish High School at Worcester, Massachusetts. Again, none of these used the direct term "number line". By 1962, however, Exploring Arithmetic by Webster Publishing had "number line" in the index with references to, "addition on, decimals on, division on, fractions on..." etc.

It is worth noting here that the earliest demonstration of a number line is usually credited to John Wallis, for a presentation in his "Treatise on Algebra", in 1685. Wallis' intention in the chapter was to develop a geometric algebra to deal with "negative squares and imaginary roots" and to lay the foundation he described how negative numbers could have meaning in representing a different "sense" than a positive number. Here are his words, as gratiously explained to me by Professor Phillip Beeley, who works at The Wallis Project at the University of Oxford Centre for Linguistics and Philology. (That meets my standard for someone in a postion to know) .

"As for instance: Supposing a man to have advanced or moved forward, (from A to B,) 5 yards; and then to retreat (from B to C) 2 yards: If it be asked, how much he had advanced (upon the whole march) when at C? Or how many yards he is now forwarder than when he was at A? I find (by subducting 2 from 5,) that he is advanced 3 yards. (Because +5 -2 = +3.)

D A C B
|...|...|...|---|---|---|---|---|

But if, having advanced 5 yards to B, he thence retreat 8 yards to D; and it be then asked, How much is he advanced when at D, or how much forwarder than when he was at A: I say -3 yards. (Because +5 -8 = -3.) That is to say, he is advanced 3 yards less than nothing."
He was also kind enough to send an image of the page. I have captured the relevant section below.

Wallis then draws a similar analogy from the line to the plane, and begins his assault on the complex numbers. For those who wish to see the full page and preceeding page, which Professor Beeley copied for me, the pdf file is here .


The Slide Rule, which had been invented in the 1600's seems to have made its way slowly into the educational scene if the US Naval academy was typical; "In 1929 the slide rule became a formal part of the mathematics program and served as the midshipmen's primary calculator until 1976.” [From “A Brief History of the Department of Mathematics” by Professor T. J. Benac http://www.usna.edu/MathDept/website/mathdept_history.pdf]

My personal memories of attending HS in the US in the Early sixties included a large class demonstration model like the one at right in my advanced classes. In Cajori's HISTORY OF THE LOGARITHMIC SLIDE RULE, (1909) we find several notes about the use of slide rules that seem to support a late development in education:

In the United States, as previously pointed out, brief instructions for the use of the slide rule were printed in a few arithmetics of the latter part of the eighteenth century. An edition of Thomas Dilworth's Schoolmaster.s Assistant (an English book) was brought out in Philadelphia in 1805 by Robert Patterson, professor of mathematics in the University of Pennsylvania. It devotes half a dozen pages to the use of the slide rule in gauging. Another English work, Honey's Complete Measurer (1st ed., London, 1717) was printed in Baltimore in 1813. It describes the Carpenter's Rule as well as Verie's Slide Rule. for gauging. Of American works, the Arithmetic of Nicolas Pike (1788) gives brief directions for the use of slide rules, while Bowditch's Navigator, 1802, gives one page to the explanation of the slide rule, but in working examples, Gunter's line alone is used. From these data it is difficult to draw reliable conclusions as to the extent to which the slide rule was then actually used in the United States. We surmise that it was practically unknown to the engineering profession as a whole. There were, no doubt, isolated instances of its use. The Swiss-American geodesist, F. R. Hassler, who was the first superintendent of the U. S. Coast and Geodetic Survey, possessed a slide rule.we shall describe it later. Mr. C. H. Progler, of Ripley, W. Va., informs me that in 1848-50 he was in the employ of Hassler's & son in U. S. geodetic work in North Carolina, that they had a slide rule in camp, but did not use it in connection with the survey. I have seen a reference to the slide rule in a book issued about 1838 by a professor of the Rensselaer Polytechnic Institute.

it appears that, before 1880 or 1885, the slide rule was little known and very little used in the United States. References to it are seldom found in engineering literature. Appleton.s Dictionary of Engineering, Vol. I, New York, 1868, does not name the slide rule, but devotes three or four lines to a description of it under the head of Gunter's line.. Knight's American Mechanical Dictionary, Vol. II, 1881, describes Gunter's line; then, without naming the slide rule, describes it in four lines and ends with the statement to the effect that these "instruments" are now common among intelligent workmen..

Wider interest in the slide rule was awakened about 1881. It was in that year that Edwin Thacher, a graduate of the Rensselaer Polytechnic Institute, now a bridge engineer, patented his well-known Cylindrical Slide Rule (Fig. 9). It was in 1881 that Robert Riddle published in Philadelphia his booklet on The Slide Rule Simplified, in which he describes a rule of Coggeshall's type.

Since 1881 the interest in the slide rule has been steadily increasing in America. Professor Calvin M. Woodwind of Washington University writes that he does not remember having seen a slide rule while an under-graduate student, that his attention was first called to it by Professor C. A. Smith in the (18)70's and more strongly by Professor J. B. Johnson in the (18)80's; Johnson required every student in engineering to use it. Its popularity began with the introduction of the Mannheim type about the year 1890. At that time William Cox began his propaganda in the Engineering News. In recent years many of the rules for special purposes have originated in the United States. An inquiry instituted by C. A. Holder showed that in about half of the engineering schools of the United States attention is given to the use of the slide rule. (Holder's article is in Engineering News, Vol. 45, 1901, p. 405.)
The slide rule appears in the 1924 College Algebra by Walter Burton Ford of the University of Michigan in a short note in which he describes them as "coming into general use, especially by engineers' and points out that they are "inexpensive and may ordinarily be secured from booksellers."

The 1920 edition of MacMillian's Logarithmic and Trigonometric Tables has directions and illustrations for a student to make their own sliderule with which to practice

Some snips from A personal quote from Bill Robinson to the Yahoo elist on slide rules indicates that they became popular in many American high schools at or about the same time they were introduced at the Naval Academy,

I can only relate this from my personal experience, but from talking to others over the years I believe my experiences were similar to those of others in many cities across our country during the late 1930's to the late 1940's. I went to Seattle public elementary schools in the 1930's; public high school starting in 1940; and the University of Washington after returning from WW II service. Slide rules were very common when I was in school. In the late 1930's you could buy a good workable one at the five-and-dime for 50 cents. And for a few dollars more at a Blueprint store that carried engineering supplies, or any upscale Stationary store you could buy a beauty. .... Some books of my day had short sections on the slide rule, but most did not. None of my high school math, physics, or chemistry texts had sections on the slide rule. The reason for these omissions is simple. It is that every slide rule came with a manual of instructions. There was no need for a text book explanation of something that was explained in the manual you received when you purchased your slide rule. And was further covered in your classes.

When I signed up for my Freshman engineering classes in 1946, all students were required to purchase a designated slide rule as well as the assigned texts. All Freshman were expected to be slide rule knowledgeable, but the first classes involved what seemed like weeks of endless slide rule drills to bring all students up to speed. Except for my assigned copy of Eshbach, " Handbook of Engineering Fundamentals" that devoted two whole pages (out of a few thousand) to the slide rule, none of my college text books had sections on the slide rule.

From the same e-list it seems that they were used much later in the United Kingdom, at least according to the recollections of Cyril Catt,

Although slide rules appear to have been common in certain classes in USA high schools by the 40s, in my UK high schools in the late 40s and early 50s they were hardly used at all. Only about a quarter of my final year science classmates had a slide rule. Log tables were the ubiquitous tool for calculations. Similarly in my B.Sc class at university in the UK in the late 50s and early 60s.

However, when searching bric a brac and collectibles shops and boot sales, and reading school related publications, in the UK and Australia, I get the impression that the slide rule did get more exposure in high schools in the 60s, because a higher proportion of the cohort of early baby boomers appear to know about them than in other age groups.

The Junior High School was another innovation that led to changes in the teaching of mathematics in the early years of the 20th century. The NCTM Fourth Yearbook from 1929, mentioned earlier states, "The junior high school movement began to be an important factor in American education about 1915. .... Here was a chance for progress, and we find a considerable change made in the curriculum in many schools. Many textbook writers on junior high school mathematics organized the work in units and introduced considerable intuitive geometry and some algebra in the seventh and eighth grades." (page 146) Several notes in the book caught my eye and reminded me of the teaching changes NCTM pressed for in the last two decades of the 20th century. On page 161 the book lists in two adjacent columns the syllabus of 1910 from NY state, and the "new syllabus, which will go into effect next year"(1930?). In the left column we find "9. quadratic equations: solutions of pure quadratics and complete quadratics by factoring, by completing the square, and by formula." On the opposite side is, "9. Study of Quadratics Optional". Earlier in item three the "new syllabus" suggested students should be responsible for "taking out a common monomial factor and factoring the difference of two squares. Factoring trinomials optional." [See the entire list here There were also frequent calls for less "traditional geometry" in favor of a more "intuitive geometry." Despite this new direction by NCTM, they list in recent "Fads in Teaching", "There was the laboratory method... especially in Geometry, where the pupil discovered for himself the truths to be learned." (page 175)


During the period from 1900 to 1950 the "progressive" movement in education led to a reduction of the level of mathematical rigor, a reduction of the number of topics, and an altered focus for math education. One consequence was a reduction in both the percentage, and actual number of students taking advanced math classes. The following is from a pre-print of an article by David Klein called "A Brief History of American K-12 Mathematics Education in the 20th Century ". The complete document can be found at http://www.csun.edu/~vcmth00m/AHistory.html
During the previous half century, enrollment in advanced high school mathematics courses, and other academic subjects, had steadily decreased, thanks at least in part to progressive education. From 1933 to 1954 not only did the percentage of students taking high school geometry decrease, even the actual numbers of students decreased in spite of soaring enrollments. The following table gives percentages of high school students enrolled in high school math courses.30

Percentages of U.S. High School Students Enrolled in Various Courses

School Year
Algebra
Geometry
Trigonometry
1909 to 1910
56.9%
30.9%
1.9%
1914 to 1915
48.8%
26.5%
1.5%
1921 to 1922
40.2%
22.7%
1.5%
1927 to 1928
35.2%
19.8%
1.3%
1933 to 1934
30.4%
17.1%
1.3%
1948 to 1949
26.8%
12.8%
2.0%
1952 to 1953
24.6%
11.6%
1.7%
1954 to 1955
24.8%
11.4%
2.6%

 
 

After Sputnik went up in 1957 a "golden age of science education" began with lots of money provided for training of teachers and students. Compare the results above with the report on math course completion given by the US Dept of Education

The percentage of 17-year-olds who completed higher-level math courses?algebra II and precalculus or calculus?rose from 1978 to 1996. In 1996, 50 percent of all students completed algebra II, an increase of 13 percentage points from 1978. In 1996, 13 percent of students completed precalculus or calculus, nearly double the percentage in 1978. In 1996, more students completed their high school math program at the more advanced levels of geometry, algebra II, precalculus or calculus, than those who completed work only through the prealgebra/general or algebra I levels.

The proportion of both males and females taking higher-level math courses increased over the 18-year span. The percentage of females completing algebra II increased by 16 points (37 to 53 percent of students), and the percentage of males completing algebra II increased by 9 points (38 to 47 percent).

The proportion of minority students taking higher-level math courses also rose from 1978 to 1996. The percentage of Hispanic students completing algebra II increased by 18 points, from 23 to 41 percent. The percentage of black students completing algebra II rose 17 points, from 28 to 45 percent.

The percentage of students NOT passing Alg I in 1996 was less than 10% for both males and females. (SOURCE: U.S. Department of Education, National Center for Education Statistics, National Assessment of Educational Progress (NAEP), 1996 Trends in Academic Progress, 1997.)


after 1950

A major change in mathematics classroom education in the last half of the 20th century came in the increase of non-textbook materials used. In particular the use of manipulative materials started to flourish in the sixties and seventies. One such device that became very popular was the geoboard, created by Caleb Gattegno, a mathematics professor at London University. Notes on the Geoboard, with additional links is here. Gattegno was also involved in the development and popularization of Cuisenaire rods, which had been invented by Georges Cuisenaire, a Belgian classroom and music teacher. Cuisenaire had created the rods using the idea of a musical keyboard as his inspiration. He wondered that his music students could understand fractional notes so easily, but not understand the same ideas in math. In 1953, Gattegno met with Cuisenaire and the meeting led to a joint agreement to create and sell the manipulatives. (notes on the two men from the corporate web page). Gattegno began writing and publishing materials for classroom use of the rods and by 1958 they had met with Fritz Kunz in New York, who acquired the US rights for the colored rods and other products. Perhaps the biggest boost to the success of the rods in the US came at the 1983 NCTM convention in Detroit when the Companies directors met with the head of the Texas State materials adoption, Alice Kidd. By the next year, Cuisenaire Company had formed an agreement with three textbook companies to include sets of manipulatives to accompany their texts. Only those three textbooks were adopted that year in Texas.


Notes to sort out later the minimum age of entrance to the University at that time was thirteen. This may be interpreted as a scrap of evidence to support the "age and stage" theory — the natural mathematical stage of youngsters 185 years ago at age 13 was not essentially different from that of our own young people. Link to history of curriculum at Colorado College where Cajori taught http://www.cc.colorado.edu/Dept/Ma/History/Topics/EvolCurr.html Books in Jefferson's Library on math and science http://www.math.virginia.edu/Jefferson/image3.htm History of mathematics at Rutgers http://www.math.rutgers.edu/docs/history.html From the above,