Andrey Kolmogorov’s Mathematical Education Reform in the USSR versus the “New Mathematics” Movement
Mathematics should be studied, at the very least, because it brings order to the mind.
(M. B. Lomonosov)
(Математику уже затем учить следует, что она ум в порядок приводит.)
(М. В. Ломоносов)
A university graduate meets his professor 15 years after graduation.
Professor: I am so glad to see you! You were my best mathematics student! Please tell me if you had a chance to use any of the math skills I have taught you in your everyday life?
Student: Yes, professor! Indeed! There was a situation when I used my knowledge of advanced mathematics. Once I was walking down a street and the wind took my hat and landed it into a giant puddle. It was rather an expensive hat and I wanted to get it back, but the puddle was too large and deep. Then I saw a piece of wire nearby. I bent it in the shape of the integral symbol and used it to pick my hat up from the puddle.
(Common joke)
Education often becomes the topic of public discussion. It is not surprising because of its tremendous importance for society’s future. Some topics in education are rarely questioned. Nobody questions the importance of basic literacy skills and the niche they occupy in elementary and high school curricula. Adults rarely go a day without having to read or write. The lack of literacy skills considerably impedes a person’s ability to get around a town, place an order in a café and even to utilize the advantages of technology. The situation with mathematics is very different. Even though mathematics is used in everyday life, there is a common misconception that modern technology allows for the avoidance of the use of mathematics beyond the basic arithmetic operations
Most of the basic mathematics used daily is learned in elementary school and high school. The intellectual skills learned early in life influence choice of occupation and level of contribution to society. Consequently, important political and social events provoke the need for adjustments of the curriculum in order to stimulate the active development of certain areas of the industrial sector that promise to maximize society’s productivity. The ‘space race’ and the ‘arms race’ during the Cold War created a constantly growing demand for new and advanced technologies [1]. The new generation of scientists and future citizens had to be provided with a quality mathematics education to be able to invent and make use of these technologies. [2] Both the US and the USSR spared no expense [3] to fund mathematics education reforms in the hope of creating the best curriculum suited for the purpose of winning the Cold War. [4] The “New Math” reform in the US and Andrey Kolmogorov’s Reform in the USSR both took place in the 1950s and 1960s. Despite of the similarities in their content and execution they had very different effects on the further development of math education in both societies. “Reform” will be defined as reshaping, reconfiguration or alternation, but not necessarily absolute improvement. “Curriculum” will be defined as a set of implemented courses and “whatever is advocated for teaching and learning” including “both school and non-school environments; both overt and hidden curriculums; and broad as well as narrow notions of content - its development, acquisition, and consequences.” [5]
The reform which took place in the Soviet Union beginning in the 1950s later became known as Kolmogorov’s reform because Kolmogorov was one of the most enthusiastic promoters of the improvement of math education. He was also an active reviewer and co-author of numerous textbooks that were implemented as a part of the reform. Officially Kolmogorov’s reform is accepted to have begun in 1970 because Kolmogorov was appointed head of the math committee of the Scientific Methodological Council in this year. This is however an arbitrary choice of dates because there is no clear beginning and end dates for these reforms. In fact, the Soviet government initiated a larger education reform, also referred to as “Khrushchev’s education reform”, as early as the 1950s. The reform was supposed to encompass changes in math education. Kolmogorov soon became an active participant of discussions of this reform and remained an active promoter of math education and later reforms in the 1960s and 1970s. Moreover, Kolmogorov drafted his first ideas regarding the changes of the math curriculum back in the 1940s. [6] At same time as these reforms were happening in the USSR, a similar math reform called the “New Math” movement was also occurring in the US.
The goal of this essay is to outline the academic, political and social similarities and differences of the “New Math” and Kolmogorov’s reform, to examine the legacies that they left behind, as well as to illustrate that despite of the numerous shortcomings and criticisms, Kolmogorov’s reform left a longer lasting and more productive legacy for the future development of mathematics education in the USSR than the “New Math” did in the US. The historical origins of the US and the USSR mathematics curricula prior to the 1950’s will be discussed to highlight the prominent changes that were brought about by the reforms. The implementations of set theory and the deductive logical approach to the study of mathematics along with their criticisms will be examined as examples of the similarities of the two reforms. Then the legacies of the reforms will be inspected and the evidence of their longevity and productivity will be presented.
William Kilpatrick was a professor at the Teachers College of Columbia University and an influential education leader of the beginning of the twentieth century in the US. His advisor John Dewey asserted that "In the best sense of the words, progressive education and the work of Dr. Kilpatrick are virtually synonymous.” [7] Sharing the mainstream views of progressivism in education, Kilpatrick strongly believed that studying mathematics in elementary school and high school is not beneficial for the development of mental discipline in students. He stated that the math curriculum should be restricted to learning of utilitarian skills because, according to him, mathematics was "harmful rather than helpful to the kind of thinking necessary for ordinary living." [8] He advocated for the student-centered discovery learning methods of teaching even though such methods slowed down the pace of learning. Kilpatrick also proposed to exclude algebra and geometry from the high school curriculum, labelling them as an “intellectual luxury” [9] and pointing out that “We have in the past taught algebra and geometry to too many, not too few.” [10] Kilpatrick based his opinions on psychological research by Edward Thorndike and R. S. Woodworth who discredited the importance of learning mathematics. According to Thorndike and Woodworth the skills gained while studying math were not transferable and therefore could not contribute to the general reasoning ability of the students. [11] Kilpatrick’s ideas inspired the Activity Movement in the 1930s. The movement’s main goal was to "teach children, not subject matter" and some of its proponents did not even regard the learning of multiplication tables or reading as legitimate activities. [12]
By the 1940s it was clear that the youth educated in progressivism lacked even basic mathematical skills. This was most apparent in army recruits who were unable to execute bookkeeping and gunnery. Despite of these unsatisfactory results, the Life Adjustment Movement emerged in the mid 1940s with a bold statement that “secondary schools were ‘too devoted to an academic curriculum’ ”. The education leaders behind this movement stated that over 60% of students do not possess the intellectual skills that would enable them to go to college or to hold a position requiring specific intellectual skills. Therefore, new courses with focus on purely practical applications of knowledge, including mathematics programs, should be introduced for those students. Home economics, insurance and taxation were favoured. Algebra, geometry and trigonometry were to be excluded. Most educators supported this movement and even demanded that it must be available for all the students but parents and journalists often resisted and criticized the movement for dramatically reducing the academic content of the mathematics curriculum. [13]
In the meantime, educators attempted to determine what caused the youth’s mathematical abilities to decline. They concluded that “mathematical education had failed because the traditional curriculum offered antiquated mathematics, by which they meant mathematics created before 1700.” [14] These educators assumed that the students were aware of the ‘antique’ nature of school mathematics and refused to learn it for that reason. They did not seem to account for the fact that mathematics is a cumulative discipline and that cutting edge modern research cannot be learned unless the ‘older’ concepts are mastered first.
The mathematics curriculum in the USSR before the 1950s took a drastically different course. The historical origins of it trace back to the late nineteenth and early twentieth century pre-Russian Revolution period when the main primary and secondary educational institutions were classical academic gymnasiums and ‘real schools’. Gymnasiums prepared students for entering universities and later becoming teachers, lawyers or politicians. The grades for final exams that the students completed before graduating from gymnasiums were used to grant acceptance to universities [15] similarly to the modern Canadian system. The ‘real school’ graduates were not given permission to enter universities [16] but were prepared to start a career in banking or technical engineering. Algebra, geometry and trigonometry were taught in both types of schools. Classical gymnasiums, however, focused on theoretical knowledge whereas ‘real schools’ emphasised the practical applications of acquired theoretical concepts. Even though the students were expected to be academically prepared before entering gymnasiums and ‘real schools’, these institutions were only available for upper-middle class students with an above average socio-economic status. After the revolution the curriculum needed to be adapted in accordance with Soviet values of equality. Therefore, in the 1930s the math curriculum of elite gymnasiums was adapted to be available for a wider audience of students, including the ones with lower socio-economic statuses, but academic expectations were not lowered. This curriculum proved to be so effective that there were no drastic revisions until the 1950s. [17]
nbsp; The ‘space race’ and the ‘arms race’ emerged against the backdrop of the Cold War and the first artificial Earth satellite Sputnik was launched in 1957. [18] At that moment the Soviet Union realized that it took a leading position in the ‘space race’ and in order to stay in this leading position, more highly-qualified scientists, mathematicians and engineers were needed. Moreover, both opposing superpowers were aware of the role of the education of the new generation in their prospects of winning the Cold War and ‘outdoing’ each other in military and scientific fields. American Admiral H. G Rickover assured that “trained manpower has become a weapon in [the] cold war” and made the nation aware that the shortage of scientists and engineers could leave the country defenceless. [19]
nbsp; By the 1950s “a pronounced atmosphere of respect for education and science had developed in the Soviet Union.” [20] The general public grew to understand the significance of education and people of various ages and backgrounds strived to complete their high school and elementary school education that was either interrupted by WWII or not obtained before the war. A wide network of evening schools for adults was set up across the country. This interest in education and especially in the exact sciences was largely inspired by the technological advancements that became available in the USSR as well as by national pride. At the same time the Soviet leader Nikita Khrushchev introduced the set of education reforms that included alternations of the math curriculum and the one year increase of the mandatory minimum education from seven to eight years of school. [21] The curriculum and teaching methods that were created in the 1930s were quite robust. Nevertheless, the newly formed attitudes in the society and Khruschev’s increase of the education minimum demanded changes in the mathematics curriculum not only to produce more specialists that would be able to ‘fight’ in the Cold War, but also to ensure that the society that just realized the importance of education remains interested in the exact sciences on a voluntary basis
nbsp; The launch of Sputnik caused so much panic among the American population and brought about such reactionary government actions that the director of The National Science Foundation Waterman described the situation as a “scientific Pearl Harbor.” [22] The National Defence Education Act (NDEA) allocated a billion dollars that were to be spent on the promotion of math, science and foreign languages over the next four years. The Act, however “did not address quality of education but instead was an anxious move by Congress following Sputnik to improve college-level education - particularly in applied science and engineering.” [23] Therefore it is not surprising that some decisions regarding the “New Math” movement appear rushed in retrospect. The situation in the Soviet Union was similar. The government felt the pressure to keep up in the competition with the US. The first attempts of the math curriculum change in the framework of Khrushchev’s reform were prepared in such haste that they did not bring about any useful changes. New textbooks for grades six to eight in geometry by I.N. Nikitin, algebra by A. N. Barsukov and trigonometry by Novoselov, which at that time was a new subject, were introduced to the curriculum in 1956. Despite of the authors’ attempts to create a new representation of the familiar content, the textbooks differed very little from their predecessors in terms of the concepts included and the methods of their presentation. All the texts were heavily criticized [24] by the education community and did not survive in schools for more than several years. A competition for new textbooks was held in 1962 with the participation of eighty six groups of authors but the texts they produced were again short-lived. By 1964 it became clear that in order to execute the math curriculum reform in particular, and the education reform in general, a more systematic approach was needed. For this purpose the vice president of the Academy of Pedagogical Science and a well known mathematician A. I. Markushevich was chosen as a chair of the Central Committee for Developing the Content of School Education in 1965. It is not surprising that Kolmogorov and Markushevich soon started active cooperation regarding the creation of the altered math curriculum because they “were linked by long-standing relations of mutual respect”. Moreover, at that time Kolmogorov already made a firm decision regarding his active involvement in primary and secondary math education, engaged in lively discussions of possible curriculum reforms and even established a mathematics-physics oriented boarding school in 1963. [25]
nbsp; One of the prominent features of mathematics curriculum reforms in the US and the USSR was the introduction of the deductive approach to mathematics into the curriculum. The deductive logical approach requires the learner to start with “definitions and axioms and [to prove] conclusions, called theorems, deductively.” [26] This approach was previously used in geometry but in the framework of the “New Math” and Kolmogorov’s reform it was being applied to arithmetics, algebra and trigonometry. One of the arguments against the logical approach is that in the middle of the nineteenth century mathematicians used logic to justify the properties of various types of numbers they discovered rather than to determine these properties. The created theorems were largely artificial in nature and served to “satisfy the needs of professional mathematicians” only. The approach was never intended to become a pedagogical tool. [27] The logical approach must follow from the utility of a concept or from the experience that the students have with certain mathematical concepts. Students understand intuitively that 3x4 = 4x3 because it follows from their experience and they can observe that three groupings of four objects and four groupings of three objects add up to the same value. Therefore, “the commutative axiom is correct because 4x3 = 3x4 and not the other way around”. The majority of students can mimic the usage of the term ‘commutative’ without understanding it, illustrating Pascal’s statement in his Provincial Letters, “fix this term in his memory because it means nothing to his intelligence”. He also condemned reason as “a slow and tortuous method.” [28]
nbsp; Another criticism of the deductive logical approach was that it was misleading and created an impression in students that new results in mathematics are produced only by mystical geniuses who start with a set of simplistic axioms and work their way to advanced theorems using strict rules of reason. Mathematician Felix Klein stated that mathematicians’ work is similar to the work of an investigator. An imagination and experiment based approach rather than a deductive one must be used in order to prove new results as well as to learn older ones. Kline labels the deductive approach as an intellectually dishonest pedagogical method as well as “one of the most devitalizing influences in the teaching of school mathematics.” [29] According to Kline, the deductive approach could pose practical complications. It will take the students twice as long to label each step that they take (such as ‘commutative law’, ‘associative law’, etc) while simplifying an expression. These mathematical properties must be grasped so well that the students do not have to think about using them too much. As a result of introducing numerous axioms [30] that the students were to memorize, some of the new textbooks contained up to eighty axioms. Therefore Learning was largely based on rote memorization, which is explicitly what the “New Math” reformers tried to avoid. Even Henri Lebesgue stated that “no discovery has been made in mathematics… by an effort of deductive logic” which is a plausible argument because day-to-day and academic decision making requires judgement and not just pure facts. The deductive logical approach was criticised by historical figures such as Rene Decarte and Roger Bacon as well as by more modern figures such as Bertrand Russell. Their opinions should have been taken into account while implementing education reforms in both the US and the USSR. Soviet educators later criticised the deductive exposition of mathematical ideas for its shock factor for the students who have not seen it before. The students were under the impression that the goal of mathematics is to prove obvious concepts using other obvious concepts and did not understand the goal of such mind exercises. [31]
nbsp; Another addition to the math curriculum that was common to the US and the USSR was the heavy emphasis on set theory. Textbooks written with Kolmogorov’s co-authorship and the “New Math” textbooks sought to define as many mathematical concepts in terms of sets as possible. The solution to algebraic equations were supposed to be presented as sets of values [32] and most geometric figures were presented as sets of points. One of the most active critics of the excessive implementation of set theory into the USSR math curriculum was a mathematician named L.S. Pontryagin who later labelled such pedagogical presentation as unsatisfactory. [33] Richard Feynman, a Nobel Prize winner of 1965 asserted that new textbooks that emphasise set theory suffer from presenting a small number of concepts in an excessive number of words that are not absolutely necessary for understanding the mathematical concepts. He also stated that that the “material about sets is never used - nor is any explanation given as to why the concept is of any particular interest or utility.” Moreover, the students understand the basic notion of a set as of a collection of objects on an intuitive level and this is sufficient for understanding elementary school and even high school mathematical concepts. [34] According to Kline, set theory plays an important role in advanced mathematics, “but in elementary mathematics it plays none.” [35] Russian critics of Kolmogorov’s reform agreed and even stated that the introduction of such topics and their representation was killing the students’ interest not only towards mathematics but also towards other exact sciences. [36] Soviet and American textbooks suffered from the excessive use of new terms and symbols that were often unnecessary. As a result, the meaning of the concept itself was lost among the unfamiliar terms. The term ‘binary operation’ was introduced to replace the usual ‘addition’ or ‘multiplication’ operations. Feynman stated that “often the total number of facts that are learned is quite small, while the total number of words is very great”. [37] Pontryagin criticised Kolmogorov’s textbooks and pedagogical method for a similar reason stating that they diffuse the core mathematical concepts among the less important details . [38]
nbsp; Kolmogorov’s attitude towards the concept of rigour and detailed definitions of various mathematical concepts differed from the attitude of the proponents of the “New Math”. The goal of American authors was to be as precise as possible in all of the definitions and the rigour was used for its own sake rather than for the sake of clarity. In his article “New Textbook for the New Mathematics” Feynman wrote that language in textbooks was “claimed to be precise, but precise for what purpose?” [39] In contrast, Kolmogorov had a very clear idea of what rigour should be used for in the school curriculum. In his interview for the newspaper Izvestiya (The News) he stated that he wants to “eliminate the distinction between the ‘rigorous’ methods of pure mathematicians and the ‘non-rigorous’ methods of pure reason employed by applied mathematicians, physicists, engineers.” [40] Kolmogorov wanted his students to become so familiar with advanced mathematical concepts that they would use them as freely as they use other daily common sense notions
nbsp; Kolmogorov’s reform differed from the “New Math” movement in terms of the newly implemented topics and their emphasis. The “New Math” proponents introduced set theory, bases of number systems, congruence, symbolic logic, introductory notions of abstract algebra and groups and fields into the curriculum with heavy emphasis on set theory and logic. In contrast, Kolmogorov emphasised elements of introductory calculus, vector algebra, analytic geometry and geometric transformations far more than the notion of set theory. Moreover, the Soviet educators set an ambitious goal to restructure the entire school curriculum so that mathematics would be coherently integrated with other subjects. Kolmogorov was the first soviet educator who introduced the idea of elective courses in the framework of the reform. He believed that subjects like radio technology, foundations of evolution, foreign languages, arts and physical education deserved extra time to be allocated to them. He also emphasised that mathematics and other math-related courses should be studied throughout the entire school year. [41] This is a very effective approach because it eliminates the possibility of forgetting the material over extensive breaks and creates continuity in terms of presenting material.
nbsp; Many aspects of both reforms failed to turn out as they were originally planned. The late 1960s showed that the students who were educated by the “New Math” curriculum were unable to do well on standardized tests. [42] Soviet universities were also puzzled by the task of composing entrance exams for the students who knew a lot of theoretical material, yet missed out on many technical details and calculation methods. [43] Moreover, very little has changed in terms of concepts covered by the curriculum. The main culprits of these failures were insufficient preparation of school teachers for the new curriculum and the lack of an adequate time frame to prepare the new curriculum documents and to adjust the pedagogical methods. [44] The authors of the reforms in both countries had different academic backgrounds than the teachers who were expected to execute the reforms. In the US the writers of the new curriculum were teachers with very high mathematical abilities who worked mostly with gifted students. Therefore a curriculum that was designed for gifted students could not be implemented for all students without excessive training of teachers and modifications of pedagogical techniques. Unfortunately, the time frame of the reform did not allow for that re-training period. Even though additional courses were offered to teachers, educators were so busy that they simply did not have time to complete these new courses. [45] Kolmogorov and other authors of the Soviet reform were all professors of mathematics who lacked professional training in pedagogy. Certain concepts like set theory seemed extremely important to them because they were accustomed to the advanced mathematics where set theory played a large role. Mathematicians failed to adapt their understanding of mathematics to the level of an ordinary student. Why did university professors think that they are suitable for the authorship of the school curriculum reform? The answer can be found in a common phrase used in the USSR that has become a cliché and a common joke but reflected the Soviet values well. Lenin was misquoted as having said that “even the lowliest worker can run the state”. With this mentality, professors did not hesitate to try their hand in a different academic realm. The textbooks were written in a hurry and many of them contained numerous mathematical inaccuracies that were the direct result of unrealistic time frame. Kolmogorov needed to co-author many of the textbooks of the reform period precisely because the authors had difficulties completing their work in allocated time frames of less than a year or two. [46] American authors went even further completing seven years worth of curriculum in only one year. [47] With such unrealistic time frames it is not surprising that the reform builders were unable to predict all the difficulties that were about to arise with the new curriculum. Teachers could not re-train overnight and students had difficulties adjusting to the new representation of mathematical ideas.
nbsp; Despite of all the failures and numerous criticisms, the effects of Kolmogorov’s reform proved to be longer lasting than the effects of the “New Math”. The “New Math” faded away by the 1970s when the progressivism movement in education that it was a part of faded away. The “New Math” did not disappear entirely though. The new education movement called the “new New Math” emerged. It encouraged students to write essays about the meaning of mathematics in their lives and the educators suggested skipping counting in elementary school in order to move right into multiplication tables. [48] Kolmogorov’s reform and his other curriculum-related ideas, however, stayed in the USSR for a long time. The textbook that Kolmogorov co-authored with a group of mathematicians and methodologists and included newly implemented aspects of calculus, algebra and geometry is still in use today in Russian schools and colleges. [49] Moreover, Kolmogorov was involved in the production of the science-math focused magazine for youth called Kvant (Quantum) that first appeared in the USSR in 1970. Kolmogorov was the head of the mathematics division and published articles for students frequently up until his death in 1987. His articles were clear and concise, although it is evident that he preferred to write for audiences of mathematically gifted students. [50] Kvant is still being produced today bi-monthly. The articles published in this magazine now reflects modern aspects of students’ lives, such as usage of internet, importance of cryptography in online banking, etc. Nevertheless, the administration of the magazine is striving to keep up the high standards that Kolmogorov set up in the 1970s. [51] Kolmogorov’s experimental boarding school now called the Specialized Education and Research Centre of the Moscow State University or simply A. N. Kolmogorov School is still operating with the protection of the Moscow State University. [52]
nbsp; In retrospect the mathematics education reforms in the US and the USSR cannot be considered absolutely successful. Even the ideas that seemed effective at the beginning, such as inclusion of rigorous definitions, implementing calculus and introducing set theory were not executed effectively because of the extremely restricted time frame for composing the new curriculum and the lack of adjustment time provided for ordinary teachers. Such rushed decisions were caused by the constant threat of nuclear war and the desire to win the Cold War as soon as possible. Therefore it is not surprising that the governments of the US and the USSR were willing to do anything they could in order to ‘harvest’ more scientists and engineers who could help them win the war. Both systems were criticised for overuse of technical terms and overloading the texts with unnecessary symbols that diverged the students’ attention from the main concepts. Nevertheless, both reforms provided a useful lesson for both countries. The legacies of the reforms still lingered after the official reforms were over. The “New Math” provoked the emergence of the “New New Math” and Kolmogorov’s Reform left behind textbooks, journals and general academic traditions that are alive to this day.