Dan Kennedy Baylor School Chattanooga, TN dkennedy@baylorschool

1. A 25th Anniversary Retrospective on American High School Mathematics Education: Change We Could Sometimes Believe In Dan KennedyBaylor SchoolChattanooga, TN dkennedy@baylorschool.org

2. Mathematics education in America began humbly. In the little red school house. Early technology. Early school. Before the 1800’s, not many American students studied any mathematics at all.

3. What about the famous three R’s? Reading ‘Riting Religion And, if you wanted to go to college, what you really needed was Latin. ‘Rithmetic didn’t join the party until people perceived that it was needed. This would take some time.

4. After all, people did not come to the New World to study mathematics. There were more important things to be done. At about the time that Napier was discovering logarithms… …colonists in Virginia were learning how to grow tobacco.

5. When Descartes published his famous Discours de la Méthode in 1637… …the first school in America (in New Amsterdam) was all of four years old.

6. When Newton published his famous Principia in 1687… …our ancestors were preparing to fight King William’s War.

7. While Leonhard Euler was changing the face of mathematics in the Old Country… In the New World, a country was being born. The Revolutionary War ended in 1783, the year that Euler died… while sipping tea and playing with his grandchildren.

8. So mathematics was alive and well, but America had basically been too busy to care. Schools, however, were gradually spreading, and many of them believed that teaching arithmetic was a good way to develop young minds. In 1745, Yale instituted an arithmetic requirement for admission. Hey, it was a step.

9. Phillips Exeter Academy was founded in 1781 by merchant John Phillips, funded largely by the Gilman family. The school has come a long way since then, but so have the United States of America. The Gilmans were involved in both stories. Nicholas Gilman signed the United States Constitution in 1787.

10. In 1802, the United States Military Academy opened at West Point. Harvard instituted algebra as an admission requirement in 1820. (Exeter, of course was on it.) In 1821, the English High School was founded in Boston. By this time, there was a serious debate brewing over why students needed to learn mathematics.

11. MATHEMATICS Technology Culture Mental Discipline Research(College Prep) QuantitativeLiteracy

12. By 1857 there were enough teachers to form an organization: the National Teachers Association. This group spawned the National Education Association in 1870. The college mathematicians, also feeling lonely, formed the American Mathematical Society in 1894. Almost immediately, both organizations began to look into the American mathematics curriculum.

13. There were two main issues that both groups felt had to be confronted, particularly in light of the diverse student population in America: 1) High school – college articulation; 2) What mathematics should be taught to whom, how and when.

14. The first group to tackle the curriculum was the Committee on Secondary School Studies, appointed by the NEA in 1892. They came to be known as the Committee of Ten. The chairman was Charles W. Eliot,the president of Harvard. They published reports in 1893and in 1894, recommending a curriculum focused on mentaldiscipline and college preparation. Much of it is still in place today, at least in mathematics.

15. In 1899 the NEA appointed the Committee on College Entrance Requirements, including members recommended by the AMS. They recommended less drilland more emphasis on logicalstructure, making connections,and solving problems. In 1915, college professors formed the Mathematical Association of America, which would concentrate more on teaching and less on research. They promptly formed a committee to study the American high school curriculum.

16. The MAA formed the National Committee on Mathematics Requirements in 1916. They published their report in 1923. This was to stand as the definitive study for more than three decades! Among other things, it gave us the unifying idea of functions.

17. It also came to the following conclusion about the mathematical needs of college-bound students and students headed straight to the workplace: “The separation of prospective college students from the others in the early years of the secondary school is neither feasible nor desirable…Fortunately, there appears to be no real conflict of interest between those students who ultimately go to college and those who do not, so far as mathematics is concerned.” Since 1923, that philosophy has prevailed in the mainstream of American education.

18. Another group that would extend the influence of the colleges on the high school curriculum came along in 1901: The College Entrance Examination Board. CEEB Originally, their only real objective was to validate, through impartial testing, a student’s ability to succeed in college.

19. The first CEEB tests were essay-type achievement tests in various subject areas, aligned with the 1923 NCMR report, like this 1928 exam in Elementary Algebra. The first Scholastic Aptitude Test was given in 1926. The SAT-V and SAT-M structure began in 1930.

20. By this time there was an organization for just about everyone interested in the high school mathematics curriculum…except for the high school mathematics teachers. There was an active groupin Chicago, the ChicagoMen’s Mathematics Club. In 1920 they became the first charter members of a new corporation: The National Council of Teachers of Mathematics.

21. Another group, the Association of Teachers of Mathematics in the Middle States and Maryland, had been publishing a journal called the Mathematics Teacher since 1908. NCTM took it over in 1921, and today it is one of the most powerful voices in education at any level.

22. Mathsucks. Math,yuck So, everyone was organized. Everyone was also worried about mathematics education, and almost everyone had written or read a report about it. Nonetheless, mathematics education was not going very well in the actual schools. This led everyone to complain about it. In other words, it was a lot like today.

23. The percentage of high school students taking algebra declined steadily from 56.9% in 1910 to 24.8% in 1955. In that same period, the percentage taking geometry declined from 30.9% to 11.4%. Many schools could not have taught more mathematics if they had wanted to. As late as 1954, only 26% of schools with a twelfth grade even offered trigonometry. College preparatory mathematics was hanging on in enough schools to keep the colleges fed, but it was available to a dwindling proportion of students.

24. Mathematical historian E.T. Bell wrote the following sober assessment in a 1935 article in the MAA’s American Mathematical Monthly: “It must now be obvious, even to a blind imbecile, that American mathematics and mathematicians are beginning to get their due share of those withering criticisms, motivated by a drastic revaluation of all our ideals and institutions, from the pursuit of truth for truth’s sake to democratic government, which are only the first, mild zephyrs of the storm that is about to overwhelm us all.”

25. Reform was badly needed, but the United States was, unfortunately, again too busy to deal with it. World War I Depression World War II While these events did delay education reform, they also served to convince many people that American mathematics education mattered to their welfare.

26. From the 1923 NCMR report until the end of World War II, the main evolutionary force in American mathematics was in the direction of making it more socially useful. Of course, there was still considerable confusion about how this was to be done. A new day, however, was about to dawn…

27. Things began to happen fast after the war. 1945: The Harvard ReportThis report emphasized college preparatory mathematics, although it was also big on its cultural value. Not much attention was paid to the non-college-bound. 1944-47: The Commission on Post-War Plans This NCTM report gave the mathematics education reaction to other reports. It was more specific about content and pedagogy, and it paid more attention to psychology and student development.

28. 1950: The National Science Foundation was established. Now there would be money to fund all this introspection. 1951: General Education in School and College This was an offspring of the Harvard Report that came from the faculties of Exeter, Andover, Lawrenceville, Harvard, Yale, and Princeton. It was notable for the following quote:

29. “No subject is more properly a major part of secondary education than mathematics. None has a more distinguished history or a finer tradition of teaching. Perhaps the very excellence of the topic has helped, in recent decades, to keep the content and order of its teaching largely unexamined. One of the most remarkable of our sessions was the one in which we consulted with a group of first-rate school and college teachers of mathematics and discovered, as the evening progressed, a very high degree of consensus on the view that school offerings in mathematics are ready for drastic alteration and improvement.”

30. 1951: The University of Illinois Committee on School Mathematics (UICSM) “The progenitor of all current curriculum projects in mathematics” was funded by the Carnegie Foundation, the NSF, and the USOE. It created curricula and materials, field-tested them, and refined them. It had great credibility among all the professional organizations, and it showed how change could actually be effected.

31. 1955: The Commission on Mathematics This group was formed by the CEEB to study “the mathematics needs of today’s American youth.” Its report did not come out until 1959, but its deliberations greatly influenced other committees along the way. This group specifically addressed the curriculum for college-bound secondary school students, deemed by the colleges to be the critical group most needy of educational reform.

32. 1958: The School Mathematics Study Group (SMSG) This group, the culmination of ten years of simmering reform, was formed by mathematicians. Every set of professional initials was in on it: AMS, MAA, NSF, NCTM, etc. They had the minds, and they had the money. Quite unexpectedly, they also had the full attention of the American people.

33. Although the reforms were well underway in mathematics education by October of 1957, they took on a new urgency in America when the Soviet Union launched Sputnik I into orbit. It didn’t take a rocket scientist to figure out what the government’s new priority would be: rocket scientists. And rocket scientists needed to know mathematics.

34. E. G. Begle of Yale directed the work of SMSG. He cited three goals: • Improve the school curriculum, preserving important skills and techniques while providing students with “a deeper understanding of the mathematics underlying these skills and techniques”; • Provide materials for the preparation of teachers, to enable them to teach the improved curriculum; • Make mathematics more interesting, to attract more students to the subject and retain them.

35. Dozens of mathematicians worked with SMSG through the 1960’s to write material. In time, the SMSG pilot textbooks were replaced by books from mainstream publishers, often from the same authors. There were other “reform projects” with similar goals and similar materials (not all of them in mathematics), but SMSG was certainly the biggest. The “New Math” had arrived!

36. Many here probably remember the New Math…

37. There were critics from the start. Morris Kline, a mathematician and author himself, called it “wholly misguided” and “sheer nonsense.” He felt that the reformers has replaced the “fruitful and rich essence of mathematics with sterile, peripheral, pedantic details.” • Other, less polemical critics concentrated on three shortcomings: • Disregard of the purposes of secondary education • Neglect of important concomitant outcomes (e.g., the ability to solve real-world problems) • Neglect of differential needs of various pupil groups

38. It also did not help that a great many people had no understanding or appreciation of the “new” parts of the New Math. Some authors tried to explain it to the masses, but their efforts were clearly doomed. Even before blogs and talk radio, the New Math became a hot-button topic.

39. Undaunted, the mathematicians continued to meet, and the NSF continued to pick up the tab. The Cambridge Conference in 1962 convened 25 mathematicians to discuss where the reforms would eventually lead. W. T. Martin (MIT) and Andrew Gleason (Harvard) chaired the committee. Their 1963 report, Goals for School Mathematics, tried to look ahead thirty years. Here is what they saw…

40. Dream on, math dudes! “A student who has worked through the full thirteen years of mathematics in grades K to 12 should have a level of training comparable to three years of top-level college training today; that is, we shall expect him to have the equivalent of two years of calculus, and one semester each of modern algebra and probability theory.”

41. There are many reasons why this did not happen. One of them began in 1954 with the report of the School and College Study of Admission with Advanced Standing. This was a task force, funded by the Ford Foundation, charged with coming up with an equitable way to award credit and/or advanced standing to students who had done college-level work in high school. Kenyon College

42. In 1955 this program was taken over by the Committee on Advanced Placement of the College Entrance Examination Board. It became, of course, the Advanced Placement program. Under the direction of Heinrich Brinkmann of Swarthmore College, the AP Mathematics Committee decided that the only mathematics course worth of the AP designation would be a full-year course in calculus.

43. In 1969, AP Calculus became two courses: AP Calculus AB and AP Calculus BC. The phenomenal growth of AP Calculus may have done more to affect the secondary mathematics curriculum than any of the previous reforms. Of course, there were other AP subjects as well, and their impact was also felt.

44. Unofficial 2009 point u u 2008276,004 exams 2003212,794 exams u 1993101,945 exams u 198651,273 exams u 196710,703 exams 1955285 exams u u

45. Once upon a time there were 11 AP courses. One of them was in mathematics. Today there are 37 AP exams in 20 subject areas. Three of them are in mathematics.

46. Number of AP Exams Taken Per Student in May, 2004

47. Cumulative AP Exams Per Student 2001-2004

48. Nobody at the Cambridge Conference in 1963 would have seen this coming. Our best students could not possibly accumulate as much mathematics as they were predicting. Instead, they would become AP scholars, taking AP courses in as many subjects as possible. It is how they would get into their colleges.

49. What effect is this AP scramble having on the students? On the one hand, they are condensing or skipping foundational courses, so they are less prepared for advanced courses. On the other hand, they are taking more advanced courses, assuring that their lack of preparation will be exposed!

50. “Currently, the greatest growth in the high school curriculum is in courses that have traditionally been taught in colleges. “The greatest growth in the college curriculum is in courses that have traditionally been taught in high schools. “It is not clear that either institution is serving its clients very well.” --Dr. Bernard Madison, Chair of the MAA Task Force on Articulation, 2002