Sputnik in Context

The anniversary of Sputnik’s launch is a good time to put it and its effects in perspective. It is often seen as the watershed event in propelling math and science education forward. However, this is not the case. Since my expertise is in mathematics teaching, I concentrate on why that curriculum looks like it does (why is Geometry taught between Algebra I and Algebra II?) , and how the history of our country and the history of math teaching have developed. My bias toward implementing research-based ideas is clear.

 

The First National Curriculum. Educational policy is state-specific. Historically, citizens of the United States have been suspicious of power vested in a central, federal government. Challenges to central government power started with our country’s founding fathers and their conception of our nation. These founders were so unwilling to create a single voice for the thirteen colonies that they only bound themselves together with Articles of Confederation (indeed, the word confederation itself resounds with anti-federal sentiments).These articles were only renounced when they realized that some central power is necessary for survival — a policy reaffirmed a century later that resulted in our Civil War. This philosophy forced John Jay, James Madison, and Thomas Jefferson to extraordinary efforts to convince the new country’s population to ratify the Constitution through their anonymous Federalist Papers . Our constitution would not have been ratified without its specific caveat that any power not specifically granted to the federal government would be reserved “to the states, or to the people.”

The regulation of education is a perfect example of this constitutional right. Each state adopts its own standards, which can be further refined by county and city boards of education. Furthermore, individual people can tailor-make refinements by home-schooling their children. It is considered perfectly “American”, therefore, to have different standards for every family in the United States.

Thankfully, this variance is not reality. It never has been. Even in the post-revolution society of early America, there were strong forces that dictated national standards and practices of education: textbooks. Webster’s Blue Backed Speller set national standards for spelling and pronunciation. A British textbook, Daball’s Schoolmaster’s Assistant spread standard arithmetic skills to most of the academies and schools until 1850.

Davies University Arithmetic

From Davies, a page showing conversion of shillings to the various US State currencies

Even with the standardization provided by textbooks, early American schools were far from uniform, and this wide diversity would fuel the largest revolution in the history of mathematics education.[†]

Fuel for a Revolution. The primary purpose for the private academies in colonial times was to prepare a student for admission to a university. Students who gained admission to one of the early American universities, such as Harvard or Yale, would then study the same content that had been studied for hundreds of years in Europe. “Educated” men studied the medieval trivium [‡](grammar, rhetoric and logic) and quadrivium (arithmetic, astronomy, geometry, and music) to prepare themselves for society or professional school. Because of this historical tie, Greek and Latin were core subjects in the pre-collegiate academies.

Each university set its own admission requirements. Although there were relatively few universities in existence, their entrance requirements varied widely. Yale, for example, required an elementary knowledge of arithmetic for admission in 1745, whereas most colleges (including Harvard and Princeton) did not. They would not require it until 1807. Algebra was not set as a requirement until the mid-1800s, and geometry would not be required until well after the Civil War. Ever wonder why high school math has a traditional sequence Algebra I, Geometry, Algebra II? Because that’s the order that they were pushed out of the Universities, into the high schools.

The situation in the other subject areas was no different. The state of college entrance requirements at this time is described as “comic, had it not been so preposterous.”

Psychological theories were quite different as well. The “mental faculty” approach to education was the order of the day. The mind was a muscle, and it was therefore strengthened through exercise. Mental calisthenics consisted of readings from Cicero, Virgil, and Caesar, along with a healthy dose of arithmetic. Mathematics was taught as much for “mental training” as it was for anything else. The question of relevance to students’ lives or future needs never arose. These subjects had always been studied, and were required for college admission%mdash;justification enough.

The Morrill Land-Grant Act of the mid-1800s that created many of our existing State universities furthered this confusion. Educators considered mechanical, technical and agricultural needs for the first time, so some secondary school—not all—supplemented their curricula with practical” math. (Civil) war, industrialization, and western expansion all contributed to the feeling that practicality should be a guiding force in content for schools. Curriculum materials failed to deliver this practical math. The aforementioned Schoolmaster’s Assistant, for example, demonstrated its financial mathematics through examples using shillings and pounds for seventy years after the formation of our national currency.

Imagine, then, the predicament of schools before the dawn of the 20th century. Colleges all set their own standards for admission. Society increasingly called for pragmatic content because of its ever-increasing industrial needs, for the first time admitted that farmers needed different math than lawyers. Mathematics courses varied from one school to another, both in name and in content. Our nation, far from an educated populace, only had about 7% of its school-aged children in school. All of this in a school system that could only graduate half its students from secondary (high) school, with only twenty percent prepared for a college curriculum [§]. It is not surprising that the call for reform came from these high schools.

The NEA Committee on Secondary School Studies. The first revolution — regarded by many to be the beginning of our modern educational system%mdash;emerged in the 1890s from the newly formed National Education Association. Commonly known as the Committee of Ten, the report of the subcommittee on mathematics revolutionized the way that the United States educated its young by doing the impossible—standardizing college entrance requirements. The College Entrance Examination Board (CEEB), a major player in a future revolution, emerged from this group.The Committee viewed the situation of mathematics education at the time of their report as dismal. The writers of the report detailed the shocking statistic that there were thirty-six different subjects taught to high school students, six of which were mathematics. [**] To cure this malady, the committee organized mathematics under the courses and names that we still use today—courses like Algebra I, Algebra II, and Geometry.

The “Committee of Ten” report

They were also the first group to suggest that the curriculum contained extraneous math that the educational system was reluctant to drop. Historically, as public opinion and college admissions changed, the curriculum grew. Nothing was removed. The Committee of Ten proposed the then-novel idea that topics irrelevant to students could be omitted.Indeed, they made specific recommendations of irrelevant topics to be eliminated, such as

• Compound proportion
• Manual extraction of cube roots
• Abstract mensuration
• Fractional periods of compound interest

Their radical ideas did not stop there. Our current system of units (as in “three math units required for graduation”), the idea of tracking, and the allowance for students to choose their own course of study (although only among four choices) can be directly traced to the work of these men. The appeal of varied teaching methods and the condemnation of “dry and lifeless system of instruction by text-book?” is found in this report.It is interesting to note that one of their ideas that drew the most criticism was the omission of Greek and Latin as college requirements. For the first time, the trivium was not driving the secondary schools’ curriculum, although college preparation still was.

The committee, for all of its work, grounded itself in no research base. Many reasons could account for this, including the fact that mathematics education, as a separate psychological discipline, was decades away. The first sentence of their report echoes a sentiment that draws criticism in today’s reform. “The Conference was, from the beginning of its deliberations unanimously of the opinion (italics added) that a radical change in the teaching of arithmetic was necessary.” They were correct—a radical change was necessary, and the call for reform existed.There was not, however, research that called for a change in teaching. Nobody looked at the curriculum, adjusted it, and studied the change to see that teaching could be improved. For all the precedents that the Committee of Ten created, this idea—the idea that reform in education could be effected without a call (or guidance) from researchers—might have been the most dangerous.

Until World War II. Technology, social order, and the need for mathematicians would evolve rapidly during the time period after the issuance of the Committee of Ten’s report. In the two decades after the release of the report, two major disciplines, mathematics education and psychology, were founded. The First World War shocked twentieth century culture into the realization that technology could shape the future of the world, and that mathematics was the underpinning for these applied scientific ideas.The school population was changing. World War I and a following depression caused school enrollment (and therefore enrollment in mathematics classes) to decline. The need for practical math was as evident as ever. For all the good that the Committee of Ten’s report did, its lack of attention to applied mathematics and the needs of the non-college bound student started to show through.

 

The 1923 Report

  In 1918, just after the end of World War I, a group was organized to address these concerns. In their own words, they were working “in response to an insistent demand that national expression be given to various movements looking toward reform in the teaching of mathematics.” Liberal education was devalued, except where it addressed a need of society. Traditional liberal arts courses were combined or given less emphasis. For example, history and geography could not be justified as separate subjects, so they were merged into the courses we have today, and given the name Social Studies.

Entitled The Reorganization of Mathematics in Secondary Education, the mathematics subcommittee’s report echoed the mood of the times, not the findings of educational researchers. Its 1923 release date earned it the moniker “The 1923 Report”, easily the most influential report until after World War II. The 1923 report mirrored society’s need for practicality, and many of its calls sound as if they are straight from today’s headlines. The committee recommended, for example, a reduction in the amount of manipulation in algebra and a reduction in the memorization of proofs. For the first time, the concept of function was given a central role in mathematics courses. The junior high school (part of the “6-3-3″ system) and the “general math” course for the non-college bound student have their roots in this report. Similarly, the requirements that all students stay in school until the age of 16, and that all students should receive the same mathematical instruction until 8^th grade, are founded in this work.For the first time, a curriculum reform group specifically looked at the needs of the “average student”. Where the Committee of Ten gave us our college-bound track, the 1923 report gave us our vocational track.

All of these ideas succeeded, but, like their predecessors, they are ideas that were not drawn from verified, researched ideas. Nobody studied the 6-3-3 system before it was instituted. No studies tested the efficacy of vocational mathematics before the courses were designed. The report was the production of professional, well-meaning people, but it was not the end product of educational research.Educational research did exist at this time, but not in regards to direct curricular reform. It manifested itself as new ideas of national standardized testing.The hallmarks of standardized tests—AP tests and the SAT—gained prominence in this period, and this popularity required close scrutiny and loads of statistics. However, although these tests are often spoken of in the same breath, they ended up affecting the educational system from opposite ends. Where the APtests provided published content standards for students to meet, the SAT tested broad, non-specific skills. In effect, the AP tests promoted content standards, and the SAT undermined them. These same effects are easily seen today.

Progressive Education.  The movement toward applied mathematics had its critics.Critics wrote of a clear tendency to adjust all courses down to the lowest ability level, so that even the “dull-normal” pupil could be promoted, while others refer to the impact of the progressive movement as a dumbed-down version of John Dewey’s ideas. There was an ineffective treatment of arithmetic in elementary school, and a postponement of college-preparatory mathematics until the later years of high school. The creation of the general mathematics course caused a drop in the enrollments of algebra classes from 45 percent to 30 percent.

During this time period, reform sprouted in other facets of American life. Politics had Theodore Roosevelt, law had Oliver Wendell Holmes, and social work had Jane Addams, all pushing the “progressive” movement. Progressive educators thought that the school could be an instrument of this reform, but the schools themselves needed reconstruction for this to happen.They attacked the school’s tradition of passive, teacher-centered learning, and sought to educate the child through student interests.

When looking at reformations, it is unusual to get one major reform report per decade. In 1943, the math education world produced two—one from the Progressive Education Association (PEA), and one from a joint committee of the Mathematical Association of America and the NCTM.

 The PEA Report.

 

The Joint Commission Report

Although approaching it from different philosophies, both sought a broader base for the study of mathematics, perhaps because of the changing times of the era. The progressive movement was affecting most other social areas of life, so change in the school was inevitable and dictated by changing societal norms. These reports concerned themselves with how the change would look, and how it would be accomplished.

Residuals from both of these groups survive. From the Joint Commission’s report, we have the birth of the junior college system and the first attempt to change mathematics instruction through teacher education. The PEA, although not as widely read or accepted, called for ideas that seem right out of today’s standards movement: that the teacher is a guide, that interest is the best motivator for work, and that students should not be passive learners that simply memorized facts. Both reports continued the tradition of the 1923 report’s “function as a thread” idea by extending it to other topics in mathematics.

However, neither of these reports made sweeping changes in the educational system or mathematics teaching in the field. Perhaps, this could be a result of historical events. The first half of the 1940s saw the entire world sink into another world war, and attention toward education waned.

World War II and Data for Change. Not only did World War II mark out exit from the Great Depression and destroy gender roles in the workplace, but it also gave the country its first widespread study of mathematical abilities: from the military. Since the military’s draft board was pulling in men from all over the country and from different walks of life, the data from soldiers in World War II provided a huge, quantifiable base from which to draw conclusions of our public school failings.

The situation was not good. Reports from The Committee on Post-War Plans, especially their third report in 1947, addressed concerns that secondary school mathematics did not adequately prepare students for a life that would increasingly involve mathematics.

The war saw huge improvements in technology and applied mathematics. Mathematicians and scientists that had formerly existed only in ivory-tower think tanks had applied their knowledge toward a common goal: the allied victory. After the war was over, they returned to their university with new ideas on mathematics’ uses, and with a mountain of data (from the army’s huge database) that showed inadequate preparation in mathematics.The trends from the 1923 report still ruled the day. Math was practical, technical, and applied for most students. Most of the attention of the mathematics curriculum focused on skills acquisition. Textbooks reflected this factory-worker attitude by displaying huge numbers of problems at the end of each section. Topics were not integrated, and, according to the mathematicians, students were not doing “real” math.

A flurry of post-war reports followed. In addition to the three reports from the Committee on Post-War plans, groups such as the University of Illinois Commission on School Mathematics, also known as the UICSM (1951) and Ball State Teacher’s College (1955) closely examined the existing math curriculum. As professional mathematicians, they had a grasp of axiomatic structure of mathematics and proposed changes to the secondary school curriculum so that it would reflect this “modern” view, the view that the professional mathematicians brought back from the war: A mathematician of the time stated that “During the war, we were aware of exciting new applications of math. Returning to the classroom, we saw a stale curriculum”. This new modern attitude said that students well prepared in mathematical theory should be able to apply mathematics to completely novel situations. First theory, then practice.

 

Cambridge Report, one of the many post-war reforms

The UICSM project described this attitude in detail. Like its recent predecessors, it did not receive much attention, because America was economically and militarily on top. Although we had a new enemy in the USSR, new technologies were not seen as vital, because we had just demonstrated out superiority by winning a war on two fronts. Our former allies in the Soviet Union would have to go a long way to surpass us. As it turns out, that’s exactly what they did.

Sputnik Leads the Way. The Soviet Union launched Sputnik in 1957, shocking the United States. Somehow, the Russians had moved ahead in the space race, suddenly prodding mathematics and science education to the front of national interests.Congress needed a new plan— a modern plan—to revamp mathematics education. It looked, and found our group of mathematics educators with just such a solution.

The College Entrance Examination Board (remember them from the Committee of Ten?) began refining the work of the 1951 UICSM materials in 1955, two years before Sputnik. Like the Illinois report, it showed a novel direction to push mathematics education. Like the Illinois report, it increased rigor. Like the Illinois report, in brought the mathematics curriculum more in line with the work of real mathematicians. Unlike the Illinois report, though, it received attention, and, more importantly, money. Sputnik’s launch opened the purse strings of Congress so that this report could be implemented. Remember—the ideas of this group date from 1951, and the group was appointed in 1955. They only received attention (in a rather dramatic fashion) because of the newsworthiness of Sputnik. Sputnik did not create this group or its report, it simply made people read it.

The Old “New Math”.  The CEEB report spurned what is known today as the “New Math” movement. A misnomer, certainly, because the mathematics wasn’t new—in fact, it was quite old. New math was old math in a new program that emphasized mathematics as a way of thinking, rather than a set of arbitrary rules.

The effectiveness of the “new math” movement came from speed, totally ruling out a time for careful, deliberate research. Recognizing the important methodological influence of the first national curriculum, the textbook, the New Math proponents immediately started spending money to develop materials. The writing arm of the new math, the School Mathematics Study Group (SMSG), created materials from scratch in less than a year. This paper reform solidified its position with widespread, easily available teacher training. It could be considered one of the fastest revolutions in mathematics education.

Algebra was no longer generalized arithmetic. Geometry was now based on Hilbert’s postulates, not Euclid’s. Substantial units of logic (especially in geometry) appeared. Concepts such as set theory, algebraic properties, cardinal numbers arrived in early childhood curricula. Mathematics previously reserved for undergraduate study was now liberally sprinkled in the secondary school, partially because mathematicians—not classroom teachers—were the main sources for the curriculum’s content.

This change was, of course, a magnet to criticism. Parents, teachers, and mathematics education professionals all raised concerns that the new math was certainly new, but it wasn’t math. The speed of development didn’t allow for extensive research to support the program—writing textbooks was more important. Discussing the modern math movement, B. F. Skinner said

The most widely publicized efforts to improve education show an extraordinary lack of method. Learning and teaching are not analyzed, and almost no effort is made to improve teaching as such. 

This very well could have been a purposeful move. Although psychologists had made great strides in understanding learning, content resources were more readily available. There were plenty of professional mathematicians around to assist secondary teachers in the content shifts, but comparable assistance from psychologists and learning theory experts was much more difficult to find. The state of the psychologist’s art was not at a high enough level to provide this assistance. Even proponents of the new math conceded that the problem of finding a level of abstraction appropriate for a diverse student population had not been found.

The most famous criticism came from Morris Kline, a leading mathematics historian, in the now-infamous book Why Johnny Can’t Add . Kline called for a return to the basics of mathematics. He said, for example, that addition, subtraction, multiplication, and division had to be mastered before any of the higher topics could be considered. Working with “properties” and “sets” should come later.

 

Also, the new math didn’t raise everyone’s standards, only those with identified talent. For the lower and middle track, math classes had lower expectations and less homework than before. After several years, the new math’s popularity started to erode.

The New Math Fades. The battle over the “new math”

movement was the first one to take place in the media. Previous struggles were mostly contained within the mathematics education community, where this one appeared in popular books and newspapers. Because of this, the new math created something else new—a public anti-standards movement.

Many felt that expectations were lowered during the reign of the new math. SAT scores declined from 1966 through 1980. Parents complained that they could no longer help their children with their math homework. Social change and civil rights—not satellites—became the issue-of-the-day. Overall, the U. S. was becoming less convinced that the new math (and its crowning achievement, the moon landing) could solve our social and economic problems.

Basic facts again came in vogue. Educational researchers, like E. D. Hirsch in his book Cultural Literacy, called for a back-to-basics movement where facts were important.“New math” soon became a joke, scoffed at in professional circles, and used as a basis for popular humorists of the day [††] . Students slowly (but never completely) returned to learning “the three R’s”

 as before.

 

Ed Hisrsch’s Cultural Literacy

Again, the “new math” movement illustrates a common theme in education reform. Research in teaching did not initiate the reform, dissatisfaction in the existing curriculum and test scores did. Similarly, the new math fell out of favor because of dissatisfaction with the curriculum. No one carried a standard saying “Teaching is better done this way—

and I can prove it.”

Standards Arise Again. The abandoning of the “new math” left a void in the mathematics education curriculum. Generally labeled the “back to basics” movement, no major curriculum status report emerged until 1980.

In 1980, the U.S. elected Ronald Reagan as President after one of the first campaigns where national education was an issue.Statistics have been quoted of this period that show SAT scores had declined for the past eleven years, that American students lagged behind students of other countries in mathematical abilities, and that colleges reported that freshmen came to college unprepared.

President Reagan fulfilled his campaign promise to the NEA when he (and his Secretary of Education Terrel Bell) established the National Commission on Excellence in Education in 1981. Composed of members from the education and corporate worlds, the committee issued its report in April of 1993, entitled A Nation at Risk: The Imperative for Educational Reform.

A Nation at Risk is a very small document. It contains broad-stroke, non-curriculum-specific recommendations, such as

• increased graduation requirements, including 3 years of mathematics
• lengthening of the school day and year
• raising college admission requirements
• raising the reading level of textbooks
• institute career ladders and merit-pay programs for teachers

These requirements were, for the most part, at least attempted. Some (such as the increased mathematics requirement) succeeded, while others (merit pay, or lengthening of the school year) were not universally instituted.A Nation at Risk contained an unusual amount of research. Forty-four papers were commissioned to study a wide variety of educational issues, although none of them compared amongst specific teaching methods. Really, though, it doesn’t matter, because A Nation at Risk provided another, more needed service for the math education community—it re-warmed the climate for reform that the “new math” had extinguished, and provided a precedent for effecting school improvement through curricular standards. Once again, educators saw a need for reform, and the time was right to institute it. Enter the most recent player in the field, the National Council of Teachers of Mathematics.The Modern Standards Movement.  The NCTM’s 1989 Curriculum and Evaluation Standards were not the beginning of the modern standards movement. Indeed, they could be seen as the last piece in a puzzle that started nearly a decade earlier.In 1980, the NCTM set a goal to present ideas for curriculum reform for the 1980s. Reacting to reports of the Mathematical Association of America (the PRIME report of 1980) and data from the Second National Assessment of Educational Progress (NAEP), the NCTM worked with Ohio State University to begin the Priorities in School Mathematics project. The recommendations of this group resound with a familiar ring. They recommended emphasis on

• Problem Solving
• Basic Skills
• Computers and Calculators
• Standards
• Evaluation issues
• Greater requirements and options for math classes
• Teacher Professionalism
• Public support

 

The PRISM Report

In the above list, the fourth entry is quite interesting. In thePRISM report, an NCTM document, there was a call for standards in school mathematics. Although other reforms had done it before, the NCTM finessed a bold move with the PRISM report. They called for research, then answered their own call. By polling existing mathematics teachers (including samples taken from the subscription rolls of NCTM journals), the NCTM partially answered its own challenge with An Agenda for Action: Recommendations for School Mathematics of the 1980s . Although the NCTM billed this document as representative of mathematics educators from many facets of the profession, it mostly came from the surveys and minds of those who served on the committee. The in-the-field mathematics teachers returned surprisingly few responses because they were unsure what the questions were asking.

 

Although not reported as a “recommendation,” the Agenda produced another precedent with its publication. It established the NCTM as a leader in curricular reform. Having called for reform (through the PRISM project) and establishing itself as a leader in this reform (through the Agenda ), the NCTM was ready to produce its most extensive work.

The Curriculum and Evaluation Standards. The Standards were initiated, according to the NCTM, because of calls for reform from several reports. They cited other reports (many authored by NCTM policy makers) that agreed with their objectives, such as 1985′s Mathematics: Options for the 1990′s, 1987′s Workforce 2000, and 1988′s The Mathematics Report Card.

Just as planned, the standards were groundbreaking.They set the tone for curriculum reform in other subject areas, particularly science and social studies.Indeed, some of the mistakes and missions of the Curriculum and Evaluation Standards (many of which are corrected in the Professional and Assessment standards issued years later) gave the other disciplines a “leg-up” during their own development phases. The NSF’s science standards do not sound like the NCTM standards by accident.

 

The NCTM unified mathematics curricula across grade and ability levels with their four themes of mathematics as problem solving, communication, reasoning, and mathematical connections. The concept of mathematical power—a term much broader and more powerful than the nebulous number sense — showed the standards were interested in fostering mathematical abilities of all students.

The NCTM surveyed teachers and other education professionals for several years during the draft stages of the Standards publication, but did not take these recommendations to a classroom before publishing them. There is a research base for the standards, in that they grew from previous movements, but not one from testing the standards in a classroom setting before releasing them in a final form. Of course, they gleaned some “experiential” research information because many of their “innovative” ideas were not new, but were remains of the other reforms discussed here. Some examples might be the following:

• The idea of an editable curriculum, and a college track, came from the Committee of Ten;
• The idea of function , and the concept of reduced manipulation in algebra came from the 1923 report;
• Professional standards for teachers were originated in the Joint Commission report;
• The role of teacher as less authoritarian and more guide-like came from the Progressive Education Association report
• Logic, rigor, and proof are largely residuals of the “new math” movement;
• More mathematics for every student came from a Nation at Risk ; and
• Computer and calculator integration, as well as mathematics laboratories, emerged from the Agenda for Action .

Have these standards been successful? Perhaps. It depends on how one defines success. The NCTM standards, like the “new math”, have had a huge impact on textbooks. Like many other reports, the Standards are being revised, not discarded, so some of their original content must have been valid. The Standards have generated volumes of writing, research, and testimonials. However, data from recent international studies show mathematics achievement, especially in the higher grades, to lag far behind international comparisons. Some organizations, like California’s Mathematically Correct, are voicing strong opposition to the Standards , and are calling for another back-to-basics movement. Common sense notions, like deciding what every child should learn, seem to elude both camps.

---

 

[†] Today, the situation is no different. Textbooks, although in a tug-of-war between dictating and mirroring content fads, are arguably the main factor that decide a course’

s content. Unfortunately, they have as much to do with profit as they do with pedagogy.[‡]Note that our modern English word trivia—unimportant details—came from this word. 

[§] For comparison, today’s school system graduates 83% of its school-aged students, 62% of whom go on to post-secondary education.

 

[**]Imagine their opinion on today’s system of education. American high schools today offer more than 2100 distinct course titles to students.

 

[††] Following is some of the text of a routine by Tom Lehrer, a Ph.D. mathematician who rose to fame in the ‘60s because of his irreverent songs about current events.

“Some of you who have small children may have perhaps been put in the embarrassing position of being unable to do your child’s arithmetic homework because of the current revolution in mathematics teaching known as the New Math. So as a public service here tonight I thought I would offer a brief lesson in the New Math. Tonight we’re going to cover subtraction. This is the first room I’ve worked for a while that didn’t have a blackboard so we will have to make due with more primitive visual aids, as they say in the “ad biz.” Consider the following subtraction problem, which I will put up here: 342–173. Now remember how we used to do that. 3 from 2 is 9 carried to 1 and if you’re under 35 or went to a private school you say 7 from 3 is 6, but if you’re over 35 and went to a public school you say 8 from 4 is 6, carried to 1 so we have 169, but in the new approach, as you know, the important thing is to understand what you’re doing rather than to get the right answer.

Now actually, that is not the answer that I had in mind, because the book that I got this problem out of wants you to do it in base eight. But don’t panic. Base eight is just like base ten really&emdash;if you’re missing two fingers. Shall we have a go at it?”