Canadian Mathematical Society Winter Meeting 2016 – Niagara!

This weekend was really mathematical for me!

I went to Canadian Mathematical Society Annual Winter Meeting! This year it was held in Niagara falls. We didn’t get to see the falls and the fire alarm nearly chased us out of the conference halls, but overall the trip was incredibly productive and fun. I also met up with friends who I haven’t seen in such a long time!

Continue reading “Canadian Mathematical Society Winter Meeting 2016 – Niagara!”

Pedagogy -Balancing on the Border of the ‘Image’ and the ‘Body’ of Knowledge

apple, bag, client

I notice that I work better when I have very concrete goals in mind, or when I hold myself accountable to someone. I am in the process of creating a draft of my thesis and I will be sharing some excerpts with you. Most likely they will not be direct or exact excerpts, but rather adapted snippets.

Here is the first one in which I’ll tell you about the ‘image’ and the ‘body’ of mathematics, and about pedagogy that falls exactly into the intersection of these terms.

In summary, the body of knowledge encompasses the intellectual content that a certain scientific discipline is concerned with and the image of knowledge represents the attitudes, beliefs and concerns of the scientific community about the body of knowledge  Continue reading “Pedagogy -Balancing on the Border of the ‘Image’ and the ‘Body’ of Knowledge”

A ‘quirk’ that Dedekind, Mendeleev and Hilbert had in common :)

I always get an extra boost of creativity, motivation and productivity in my graduate research after periods of teaching.  I’ve been working at different math summer camps for the last few weeks, and the experience was tiring and challenging at times.  However, after the camps were finished I noticed that some ideas related to my research have just ‘appeared’ in my head – sounds great, right? …Except that when I share this experience with some of my friends in academia, they usually exclaim: “But teaching is a ‘distraction’ from your ‘real’ work, isn’t it!”

So I convinced myself that I’m just an ‘odd apple’ in academia with this strange quirk of not hating the process of teaching… Then I found out that I’m not the only one.  Apparently many works of Mendeleev, Dedekind and Hilbert were inspired, and driven by their teaching experiences, or dissatisfaction with existing teaching methods.

For instance, Dmitri Mendeleev’s table of elements was first published in [drum roll] a TEXTBOOK!  At the start of his career Mendeleev was one of the many professors with a high teaching load and not-so-high salary.  His famous table was a result of frustration with his students and the teaching materials that he was provided with.  Contrary to a relatively common belief, Mendeleev was NOT the first scientist to attempt organizing and classifying the elements.  Other classification systems were widely published in textbooks but Mendeleev’s class was unable to make sense out of those classification systems. Hence, Mendeleev set off to create his own way of organizing the elements and presenting them to his students.  Soon the famous periodic table appeared in an ordinary textbook for university level chemistry class.

Richard Dedekind was absolutely appalled by methods of teaching calculus at university level.  He was unhappy with all the gaps in students’ knowledge of math in general as well as in subject-specific areas.  This frustration with poor pedagogy of calculus has inspired his works on integers.  In fact, it inspired him to push the ‘boundaries’ of algebra so far that some mathematicians were doubting that his works should even be considered a part of the realm of algebra (for those who are specializing in history of sciences, I’m just trying to say that he has altered the ‘image’ of algebra).

David Hilbert always surrounded himself with as many talented students as he could, especially during the mature stages of his career.  He claimed that tuning into their ideas and bouncing his own ideas against them motivates him to expand his academic views and provides him with inspiration to keep learning.

There are countless examples of other scientists and mathematicians who were motivated by teaching such as Kolmogorov, Alexandrov, etc. (each of them deserves a BOOK  – not a paragraph).

Of course, there are equally as many successful and famous scientists who were sure that teaching is not their thing, which is fine, and deserves a discussion as well – but maybe in a different post 🙂

P.S.: mini-bibliography/inspiration sources:

A Well Ordered Thing: In the Shadow of the Periodic Table by Michael Gordin

History of Modern Algebra  by Leo Cory

Hilbert by Constance Reid

Congress of the Humanities and Social Sciences 2014


I was honored to be one of the speakers at the Annual Meeting of the Canadian Society for the History and Philosophy of Mathematics held with the Congress of the Humanities and Social Sciences. Once again, I was speaking of various regional differences in the Soviet mathematics education.  The comments from the attendees were extremely helpful!

The program and the list of abstracts – including mine – can be found here:


The congress took place close to the Niagara Falls at Brock University in the city of St. Catharines (May 25th – 27th).

After the Meeting was over I took a detour to see the Falls.

The first thing that I saw upon exiting the university campus was a demolition site


followed by a ‘haunted’ – or just abandoned – house


The flowers along the side of the Niagara River smelled wonderful before the rain has started


and Americans are building something on their side of the falls




Localities – Graduate Student Conference at York University

On May 3rd and 4th York University hosted the Localities conference. See the following link for the program that was offered

I was honored to be one of the speakers.  I was speaking of regional differences in mathematics and science education of the Soviet Students.  This was a new talk that I have recently prepared and I am happy that the audience was very attentive and responsive.

My presentation focused on discussion of various regional based differences in math and science education of soviet students based on primary sources such as laws and decrees issued between 1958 and 1980.  during the 1960`s the government was acknowledging the differences in educational needs of students in rural and urban areas.  By the 1970’s such acknowledgements became sparse.  The Communist ideology implied that all Soviet children were given exactly the same opportunities and were able to achieve high results regardless of their geographic location.  Although such statement could have ‘worked’ in the conditions of ‘perfect world’, in reality children from rural areas were often missing out on contacts with highly qualified scientists and artists whereas their urban peers benefited from such contacts.


I was pleasantly surprised and humbled by a tweet that was sent out regarding the illustrations that accompanied my presentation.

(<blockquote class=”twitter-tweet” lang=”en”>

Beautiful visuals of Soviet mathematics education in the 60s presented by @mariya_boyko12 at the @STS_YorkU grad conference #STSYU14

— Yana Boeva (@dropsmops) May 3, 2014



Marseilles, France: Academic Travel Edition of Collected Curiosities

This week I am participating in a “Sources in History of Mathematics” program in Marseilles, France

Here are some non-academic remarks:
—A bank representative asked me if Marseilles is in the USA…. not even sure how to comment on that.
—Airport shuttle bus signs in Marseilles duplicate all the information in Arabic and English
—I fear that if i did not wear a striped shirt on the day of my arrival to Marseilles, I would not be allowed to enter France.
—European students seem to be less obsessed with free wi-fi. For instance, not having wi-fi in a cafe does not seem to bother them too much.
—The soap bar in my dorm room is shaped like a fish. Another program participant remarked that it would be funny to have a soap bar that smells like fish
—Learning Chinese from scratch at university and becoming fluent in it is quite possible —  proven by a program participant

Academic remarks:
—NYU has a specific PhD program in history of mathematical education
— A certain ancient manuscript contained absurdities and mistakes on every page with strange regularity. Turned out that it wss not an original document.—  When a document that contained water damage at the bottom that went through about 50 pages was found, it became clear that it is the original. The stribes that copied the document simply could not read the damaged parts and simply made up the content.
— The content of mathematical texts and their meaning could be influenced by factors such as politics, social settings of the author and theirneconome status (obvious statement), as well as the physical objects or computational aids that were prevalent at that time (at least to me, that was not as obvious). For instance, descriptions of various ancient Chinese computations were composed in a way that would explicate the execution of these opeartions on an abacus.
— Education reforms in the 20th century France were motivated by bridging the gap between the contemporary math research and high school curriculum
— There are other connections between various historical texts that we can observe other than networks of citations. Various concepts could be mentioned in different texts. It is important to notice such references, especially if the authors refer to the same concept but use different terms to describe it.
— Chinese astronomy played a large role in politics. Various astronomical phenomena were often used as justifications of hiring or displacing politicians.

P.s. sorry if i made a ton of typos. My tablet refuses to do the spellcheck for me: (

The toughest thesis advisors

My mom’s thesis adviser wanted each student to have a research log book.  Weekly research goals needed to be designed and recorded by each student.  If the student failed to reach his own goals, the adviser would simply say: “I did not force you to put that goal down.  If you were not able to reach it, you should not have included it in your weekly goals.”  My mom became a great researcher, so whatever her adviser demanded, apparently, worked out very well.

When I entered my 6th week of the first year of graduate studies my mom was shocked when she found out that I do not have a research log book, moreover, no one demands that I do.  “You should record your research activities every day,” she said.   I, in turn, was shocked because it did not occur to me that I need one.  My first year was full of course work and other issues, so the idea of the research log slipped to the bottom of my priority list.

Now when I started my second year (and I am busier than ever with the remaining course work) and my first research work is starting, I recalled my mom’s and her thesis adviser’s words about the research log.  I will be honest – I cannot imagine jumping into this hard core weekly-research-goals routine, but I will attempt to start by recording interesting things (academic and not so academic) that I will be learning each day.  I will also be posting them (I will attempt to do it every week).

P.S. The fact that I impulsively bought a notebook another day has something to do with the above statement 20131009_224212

Here are the two things that I learned today:

Magnets would stop working after exposure to extremely high temperatures

Rene Descartes taught mathematics to women openly event though his contemporaries disagreed

Fields Medal Symposium in Toronto – done!

There is no Nobel Prize in mathematics—

[Many of you are now recalling the famous story about Alfred Nobel’s conflict with Gosta Mittag-Leffler over a woman. Sorry to disappoint you, but there is no historical evidence for that.  More details can be found on the website of the University of Waterloo]

— However, The Fields Medal is given to mathematicians under 40 years of age for their outstanding discoveries.   Professor of mathematics John Fields  established the award.  The first Fields Medal is awarded every four years since 1936 (except between 1939 and 1949).

The second Annual Fileds Medal Symposium – an event that promotes the recognition of the Fields Medal and honors the recent Files Medal recipients –  took place in Toronto this week.

The public opening on Monday night, the Student event on Tuesday night and numerous daytime lectures of the symposium scientific program were extremely intellectually stimulating and engaging.  I want to thank all of the participants for this celebration of mathematical knowledge.

If you want to know how exactly the Fileds Medal Symposium was going, see my live updates on the Fields Institute Twitter page at

Engineers on the pages of Russian socialist realist novels of 1910 – 1960


Russian society, before and after the Revolution of 1917, revered scientists and engineers whose expertise was utilized to increase the country’s industrial and military might.  The presence of numerous literary works influenced by the method of socialist realism, where engineers are represented as primary and secondary characters, illustrated society’s interest in the personal and professional lives of engineers.  The image of the engineer changed in the years 1910 to 1950 from a dictator-like figure to a romantic character and then to a fusion of the images of an engineer and a scientist.  Before the Russian Revolution most engineers had a high socio-economic status and came from upper class families. Workers were accustomed to the unquestioned authority of engineers.  Hence, prior to the emergence of socialist realism in the 1920s[1], engineers were portrayed as monarch-like figures with unlimited authority over the working class.  After the Russian Revolution, the communist government convinced workers that they must be the focus of all the social, cultural and political events.   Literature and science needed to cater to the tastes of the working class without lowering academic standards. Therefore, engineers needed to assimilate into the working class and to adapt science to their level.  Engineers were portrayed as ordinary human beings and as workaholics who led the workers by their own example.  As the method of socialist realism gained popularity in literature, the image of an engineer shifted towards a romantic figure with megalomaniac ideas, a builder of socialism and a carrier of progressive thought. This image was dictated by the political ideology of implementing the concepts of socialism in the entire universe.  The political situation drastically changed in the 1950s with the onset of the Cold War. Society and the government realized that advanced science would become a crucial factor in winning the ‘arms race’ and the ‘space race’. Thus, the traditions of socialist realism were challenged and the image of an engineer was fused with the image of a scientist.  This paper will discuss the political, and social factors that affected the changes in the image of the engineer in Russian literature between the 1910s and the 1950s.

Socialist realism is a method of artistic expression that is intended to highlight and glorify the idealistic goals and values of the socialist and communist regime. The term ‘socialist realism’ is also known as “Soviet culture’s literary system”[2] that monopolised the field of Russian creative writing at the turn of the twentieth century.  This method of literary expression allowed the authors to showcase the character traits that the citizens of socialist society must possess.   Socialist realism demanded a high degree of creativity from the authors.  The novelists needed to portray various social phenomena that have never occurred in Russia before.[3]  The working class was not politically active before the revolution.  In the 1920s, however, workers became very vocal in terms of the policy formations and social standards of life.  This rapid shift in social dynamics needed to be reflected in the literature.  The heroes of the socialist realist novels were workers who strived to meet the social and political standards of Communist society.  These heroes were often ‘deinvidualized’[4] because socialist society did not encourage individualistic self-expression.  The more selfless the character was and the more devoted he was to the achievement of the common goals of his community, the better he was perceived.  This image was perfect for portraying the workers, but it was challenging to represent engineers when the traditions of the socialist realism implied that the main character should meet the above requirements. According to the nature of their work, engineers were bound to be in leading positions.  Although their professional skills were valuable for the communist regime, their individualistic way of thinking was not accepted in communist society.  Engineers, in the novels and in reality, needed to adapt to the new social circumstances and to integrate the values of the working class into their practice. Socialist realism, however, did not come into being spontaneously and did not stay unchallenged.

The characteristics of socialist realism could be seen even in pre-revolution literature.  In his work of fiction Engineer Menni (1913) philosopher Alexander Bogdanov reflected the social and political values of the tsarist period and the gradual emersion of socialist ideas.  Bogdanov was a well-known utopian author.  His novel Red Star (1908), a prequel to Engineer Menni, was published in three editions in 1908, 1918, 1929[5] by a respected publishing house called the Comradeship of Publishing Artists.[6] Bogdanov’s philosophical views, disguised under the veil of fictional stories, drew Lenin’s attention and criticism.  Lenin dedicated three chapters of his book Materialism and Empirio-Criticism: Critical Comments On A Reactionary Philosophy (1908)[7]to the criticism of Bogdanov’s understanding of Marxism. Lenin’s disapproval was so harsh that the novel was not re-published after 1929. An abridged version of this novel was published only 60 years later, in 1963.[8] It is safe to assume that certain chapters of the book were omitted because of their politicized content. Nevertheless, the abridged version provided sufficient information regarding the transition that the image of an ideal engineer undergoes throughout the novel.

Engineer Menni started with a “Note from the Translator” who stated that Marsian society did not experience drastic political changes or revolutions.  By the year 1620, the last feudal state Tumasia surrendered and joined the common state that the Marsians had been forming since 1000. The Tumasian monarch died and his heir tried to resurrect the dynasty but failed and perished in battle.  His son Menni survived and became an engineer.  Menni asked the government to finance a project of building a series of giant waterways, some over 70 kilometers long, on the surface of Mars.  He demanded absolute control of the entire process, including handling of all the bureaucratic matters, technical issues and financial planning.  His project was accepted, and Menni was respected and liked by the workers at first.  Complications arose when one of the channels had to pass through a dangerous swamp area.  Many workers were dying.  Angry and scared, they formed unions and chose representatives to inform Menni about their complaints.  Menni, however, refused to accept the unions, saying that since not all the workers belong to their union, he cannot be certain that the representatives are not pursuing their own goals.  Soon Menni was betrayed by his colleague, who offered Menni to take part in an illegal financial transaction with the government.  Menni murdered his colleague on the spot and was sentenced to fifteen years of imprisonment.  Moreover, he was accused of wasting financial and human resources.  Menni was not aware of the existence of his son Netti, who was a worker at first and became an engineer later.  Netti wrote a book that unveiled the illegal transactions performed by the government and rehabilitated Menni’s reputation.  The new government asked Menni to resume the leadership over the great waterways project from his prison cell.  Netti became Menni’s assistant and tried to convince him that workers’ unions were essential. Menni did not agree with these ideas, but admitted that they were intelligent and logical.  After Menni’s death Netti soon retired from the waterways project and started educating the working class about science and technology.[9]

There were two distinct idealistic images of engineers in this novel represented by Menni and Netti.  Menni was a dictator at his workplace, a middleman between the men and the terrain, who felt his power not only over people, but also over nature itself.  He represented the ideal image of an engineer of the monarchic era. Menni however, was not presented as a distant leader capable of giving orders only.  He “took new measurements of the Livian canyon from north to south and from east to west himself with assistance of his helpers.”[10]  Moreover, he was willing to take care of all the bureaucratic aspects of the project, including the contract arrangements with workers.  Menni believed that “a worker who is not eating well or overworked does not possess the full working ability.  A worker who is not satisfied creates a threat of [unpleasant] surprises that will interrupt the course of action.”[11]  However, he had rather selfish motives for this statement, because then he added that he “does not need any [unpleasant] surprises”[12] on his worksite.  Moreover, later in the novel he revealed that to him, the labour of the human masses is just a mechanical force.[13] Although Menni considered human labour as a tool to implement his own ideas, he did not devise his waterways project simply to amuse himself.  He was aware of the geographic and economic needs of his country and tried to subdue nature to help humanity.  Menni was presented as a young over-ambitious man with megalomaniac ideas who, nevertheless, sincerely wanted the desert regions of Mars to become habitable.  Over the course of the novel, however, he took on the role of a dictator of the people as well as of nature.  Menni’s character may seem to be presented as an etalon of professional practice and honorable social behaviour.  However, the author did not allow the image of Menni to stay entirely flawless. Menni was portrayed as a blatant autocrat who could not be stopped by any laws or morals if they created obstacles on his way to success.  He abandoned Nella, a lady why loved him, because he wanted to be ‘utterly free’ and thought that only a loner could be completely invincible.[14]  Over twenty years Menni did not inquire about her fate and did not feel any remorse for his actions upon their next meeting.  Then, Menni killed his colleague right after blaming him for illegal cooperation with the government and did not regret his actions, telling the court that his deed was conscious and thoughtful.[15]

Half way through the novel, a new character Netti, Menni’s son and his polar opposite in terms of personal and professional goals, was introduced.  He was a democratic leader and a middleman between the working class and science, and a perfect engineer of the new socialist generation.  The author did not describe any situations where Netti’s behaviour could be considered even remotely immoral.  Although Netti cared about the future of the planet and humanity, and understood the importance of the great waterways project and his goals were more realistic.  He stated that the working class should take ownership of science, bring it down to the level of everyday ideas and relate it to labour.  Labour, according to Netti was the origin of science.[16] Therefore, Netti wanted all the workers to become educated in order to be able to have a voice in the decision making process regarding scientific matters related to the building of the channels.  His struggle with Menni regarding the purpose of the workers’ unions shows that the wellbeing of Marsians was a priority over the completion of the engineering plan.

Bogdanov’s novel had a high literary value because of the artistic language and the use of metaphors that related the fiction story to reality and the future, as it was seen at the beginning of the 1910s.  The author explained that, although there were wars on Mars, the Marsians were less violent and more environmentally conscious than the Earth dwellers.  Unlike on Earth, all of their political and social changes were gradual.[17] The author predicted the strive of the Soviet government to turn all the unhealthy lands into the fruitful ones by means of implementing the achievements of agricultural science.[18] It is particularly interesting that the author mentions that the “independent peasantry… has almost disappeared from the face of the planet [Mars]”[19] nearly five years prior to the revolution and the Soviet collectivization of private farms.  However, there were some very unrealistic assumptions in the novel.  For instance, a book published by a twenty four year old Netti created such a resonance that the entire government needed to retire.  Moreover, one of the members committed suicide.[20]

The image of an ideal engineer changed in the decade that immediately followed the Russian Revolution.  One of the representative authors of that time was Aleksei Tolstoy.  His futuristic science fiction novel Aelita, where the main character is an aero-space engineer, was written while he was abroad.   Tolstoy returned to Russia after the onset of the Soviet Regime and published his novel in two parts in a popular magazine called The Red Novelty[21] at first.  The magazine was published monthly and included popular science as well as philosophic articles.  Although authors such as Lenin, Bukharin and Frunze published their works in The Red Novelty, it could not be considered a fully proletarian magazine.  Nevertheless, the presence of popular philosophers give reasons to think that The Red Novelty was read widely enough, at least for the purposes of labelling Aelita a well-known novel.  It was re-published numerous times by various publishing houses, including the Children’s Publish House.[22]  Tolstoy’s work was acknowledged by three Stalin Prizes awarded after his death in 1941, 1943 and 1946.[23]Although by the 1940s the magazine became an object of criticism of the radically-oriented communist youth, it was widely read in the 1920s.[24]

Aelita’s main character was a lonely engineer Mstislav Los’ (which translates into Russian literally as a ‘moose’) who was building a spaceship intended for travel to Mars at the speed of light.  He was looking for a travel companion for his exploration trip.  An enthusiastic ex-soldier Gusev agreed to join Los’. They landed on Mars several days later and discovered an advanced civilization whose economic system was a very similar to extreme capitalism.  Gusev and Los’ were taken to a palace where princess Aelita, the daughter of the Marsian dictator Tascoob, taught Los’ and Gusev the Marsian language and educated them about the origins of the Marsian civilization. Marcians were decedents of Atlanteans, who left their continent after it was sunk.  Gusev and Los’ soon found out that the socio-economic status of the working class on Mars was extremely low.  The workers lived underground, close to the industrial machines. They also found out that the planet was approaching an environmental cataclysm because the ice caps stopped melting.  Gusev accidentally overheard the broadcast of Tascoob’s hostile speech in front of the assembly of the Marsian leaders.  He stated that Mars was declining and blowing up the city would be the only way for the upper class to face the fall of the planet with dignity.  Moreover, he ordered Aelita to poison Gusev and Los’.  Gusev became a leader of an uprising intended to overthrow the upper class leaders and to save the city. After the movement was crushed by the government and the city was destroyed, Los’   Gusev returned to Earth.  Aelita survived and sent signals to Los’ from Mars.[25]

The social and political goal that society was preoccupied with during the 1920s was the establishment of socialism in the entire universe.  Hence, the theme of space exploration and spreading of political ideas was reflected in popular literature.  The image of an engineer in Aelita was no longer dictator-like.  In contrast with Bogdanov’sMenni, Los’ did not exert his power over thousands of workers and did not even strive to gain such a power.  He built his spaceship with the help of several workers and did not express any desire to become rich or famous.  In fact, Los’ was a poetic dreamer who possessed the megalomaniac idea of travelling to Mars with the speed of light in only ten hours. People were avoiding him, as though he was mad.[26] Los’ was somewhat similar to Bogdanov’s Netti in his readiness to educate Gusev about space travel and the details of the spaceship’s construction.  At first, Tolstoy made the reader believe that scientists and engineers, represented in this novel by Los’, were the leading force of active space exploration.  However, it became evident over the course of the novel that the working class, represented by Gusev, was the cornerstone of all the social changes.  Illustration of Gusev’s socially active behaviour was a clear manifestation of the socialist realism expression method in this novel.  The working class provided the propelling power without which the science and technology would stay passive and useless, just as Los’, who spent most of his time idling and day dreaming while staying on Mars. Overall, the image of an ideal engineer shifted from the educator of the masses to the provider of scientific and technological services that were used by the working class for the dissemination of socialist ideas.

Aelita was full of unrealistic conversations.  Aelita often recited unreasonably lengthy monologues, such as her monologue about the Atlanteans.  It was difficult to determine the genre of the novel.  Tolstoy did not include enough psychological details about Los’ and Aelita’s relationship to call it a psychological drama.  On the other hand, the lack of victory over the Marsian upper class did not allow the novel to be labeled it as a utopia, even though it is conventionally labelled so.  Tolstoy included numerous descriptions that painted the images of the surrounding world in idealistic and realistic way at the same time.  For instance, the author said that a bird was singing with a voice that sounded ‘crystal from joy’.[27]

In order to illustrate further changes in the image of an ideal engineer, it is necessary to examine Valentin Kataev’s novel Time Forward! that was first published in 1932.  Kataev was a popular writer who published his works in literary magazines such as The Red Novelty, The New World and Russia.[28] He is known for producing numerous novels dedicated to the daily lives and ideals of the working class people as well as war veterans.  Kataev received the Stalin Prize in 1946 and the Hero of the Socialist Labor medal in 1974 for his accomplishments in literature.[29]

Time Forward! was a classic example of the socialist realism novel. Even though the main character was an engineer named David Margulies, the workers Konstantin Ishchenko and Mosya were the focus of the novel.  The novel took place at a construction site in the Ural Mountains. The entire story lasted for twenty four hours.  One morning Margulies found out that the world record of concrete-pouring was beat by someone in Kharkov.  At first, Margulies was not convinced that his workers should compete with Kharkov. He needed to check the scientific parameters of the cement and to verify that the quality will not suffer from the speedy production rate.  The enthusiastic worker Mosya was eager to beat the Kharkov’s record and did not understand why Margulies did not give the order to beat it right away.  In the afternoon Margulies’s sister called him and quoted an article which confirmed that the quality of the cement would not decrease if the production rate would increase.  Margulies immediately ordered the workers to try and break the record.  Another engineer Nalbandov was against this plan, but he preferred to let Margulies fail.  The workers faced numerous difficulties.  The warehouse refused to give the brigade extra cement and the water supply was cut off because the daily quotas of materials were used up. Moreover, the work was interrupted by a severe storm. Soon Margulies’ colleagues obtained two wagons of extra cement and restored the water supply.  By the end of the shift, the workers did not break Kharkov’s record, but Margulies reminded them that the time when the water supply was cut off needed to be compensated for.  There was just enough time to break the record. The brigade leader Ishchenko was accepted into the Communist party immediately and Mosya got his name published in a newspaper (it is implied from the context, although not stated explicitly).  The samples of solidified concrete passed the standard quality test.  Soon Margulies and his team found out that their new record was broken at another construction site.[30]

Time Forward! was written at the time of the peak popularity of Stalin’s Five Year Plans.  The workers were often rewarded for quick completion and over-production.  Hence, Kataev’s engineer was portrayed as an extremely practical and fast-thinking figure.  Despite his ‘crazyness’,[31] Margulies was shown as an extremely responsible professional who would never give orders that could lead to uncertain results.  Kataev’s ideal engineer was socially dynamic and eager to share his working enthusiasm.  The workers were portrayed as conscious citizens who were motivated to obtain their education by intrinsic factors such as attaining the right to be called an official member of the Communist Party, the pride of being praised in the local media or the possibility of professional promotion.  The engineer’s role of an active educator of the working class shifted towards a role of a mentor who needed to be on equal footing with the workers.

Kataev’s novel was filled with creative metaphors and humour.  Time was presented virtually as a distinct character in the novel.  In one instance Kataev compares time to a runner that constantly accompanied Margulies until “there was no substantial difference between him and time.”[32] Kataev used humour to highlight the everyday life situations that were imperfect or even unfortunate but their presence gave the novel a sense of connectedness with reality.  For instance, the scene where Margulies discovered that two boys were competing for the greatest number of painted words for posters and misspelled all the words[33], was presented as an unfortunate episode.  Nevertheless, it was described in a humorous tone.  On the other hand, Margulies as the main character, was portrayed as virtually flawless human being.  The only imperfection Margulies could be accused of was the fact that he was a workaholic.  This however was not considered a large character flaw because the population was preoccupied with completing the five-year plans, both in the novel and in reality.

Another author that depicted engineers using the socialist realism method of literary expression was Andrei Platonov.  He was a worker and an inventor.[34] Engineers, inventors and practical thinkers often became the main characters of his utopian socialist realism novels such as Chevengur and The Foundation Pit. Platonov always depicted the lives of workers in his novels and painted literary portraits of perfect socialist labourers.  He was a very productive writer and his works were published often.  Nevertheless, he was not always appraised by the soviet literary critics.  For instance, Platonov’s criticism of Soviet bureaucracy was labelled as an inaccurate understanding of the reality of communist society.[35]  Soviet critics would be very unlikely to criticize Platonov for his novel The Sea Of Youth (1931), where he portrayed an image of an engineer who strived to achieve ambitious professional goals in order to improve the well-being of the state farm he worked at.  Unlike Menni, who by 1931 could be branded as an engineer from the past generation, Platonov’s engineer placed his dedication to the well-being of the community above his ambitions.

The main character of Platonov’s The Sea of Youth was an electrical engineer named Nikolai Vermo, who was hired by a meat production state farm.  The leader of the farm Umrishchev suggested that Vermo should ‘mind his own business’ and avoid any confrontations or conflicts.  One of the secretaries named Nadezhda Bestaloeva, Vermo’s colleague, always thought that Umrishchev was too cynical.  The next day Vermo became a witness of the funeral of a milkmaid named Ayna.  She was a witness of the numerous crimes of Bozhev, who replaced the state farm cows with private farm cows.  Moreover, Ayna’s brother reported that Bozhev abused her physically and sexually.  Umrishchev was transferred to another state farm as a punishment for his inattentiveness to community life and Bozhev was executed shortly thereafter.  Bestaloeva became the leader of the meat production state farm and hired Vermo for the position of the chief engineer.  Vermo dreamt of the mechanization of labour and fantasised about futuristic cows that will have metal body parts that will allow them to eat faster and produce more milk.  He thought that the more machines will be created to replace the workers at their workplaces, the more time these workers will have for intellectual development and entertainment.  He conducted a series of technical reforms at the farm.  Vermo created a tower where the cows were killed by electric charges, installed a windmill and created a machine that was used to dig wells and to obtain the water from underground.  A delegation from Moscow praised Vermo’s work and sent him, along with Bestaloeva, to America for testing and the popularization of his well digging apparatus.[36]

Vermo possessed some of the qualities of ‘ideal’ engineers from previous decades.  Similarly to Kataev’s Margulies, Vermo dedicated his entire attention to the improvement of the community.  He did not possess any imperfect personal or professional traits, which is typical of pure socialist realism novels.  He was a romantic dreamer like Los’.  On the other hand, Vermo was never idle.  If he was not busy with his engineering projects, he was playing musical instruments.   Platonov portrayed an engineer as a well-rounded person with numerous professional and artistic interests.  Vermo completed his music education as a driver and a technician before he became an engineer.[37]  Platonov used a variety of writing styles in his novel.  In some parts he imitated simple language that could be used in remote villages.  Other parts of Platonov’s novel sound like poems, such as the opening part of the novel where Vermo’s travels are described.[38]

Despite the popularity of socialist realism, it did not go unchallenged for too long.  Daniil Granin, a famous author who is still living, was among the writers who chose not to follow all of the traditions of socialist realism.  Granin did not limit himself to portrayals of the lives of the manual labourers.  Scientists, professors, graduate students and engineers were often the main characters of his novels.  These characters were not always perfect in personal or professional ways.  Granin however, similarly to other socialist realist writers, portrayed his characters as honest and dedicated professionals who were willing to embrace their mistakes and to improve their skills.  His contributions to Soviet literature were rewarded with the Hero of Socialist Labour medal in 1989 and with numerous other prizes for his accomplishments in literature.[39]  Granin did not necessarily label his characters as ‘engineers’ or ‘scientists’.  He created the images of talented educated people who were ready to do any work that was necessary for the achievement of their professional goals.  Granin wrote Into the Storm in 1957, in the midst of the Cold War.  The government realized that scientific development would help the USSR to win the ‘arms race’ and the ‘space race’. As a result, “a pronounced atmosphere of respect for education and science had developed in the Soviet Union”.[40]  Granin conveyed this attitude very clearly.  He appropriated the image of a prefect engineer that was created by the writers of the previous generations and fused it with an image of a perfect scientist.  As a result, he produced a character who did not use science for immediate improvements only.  He was able to think and plan his research ahead of time and to inform the public about the expected results.

The main character of Into The Storm was Sergei Krylov.  He did not do very well at his first year of the physics program at university and initially received help from Oleg Tulin.  By the end of the third year of university Krylov proved to be an intelligent student.  Shortly thereafter, he was expelled due to his conflict with the dean.  He then decided to work at a plant. Krylov was passionate about science and sometimes appeared foolish to his coworkers when he was deeply immersed in thoughts.  The main engineer-constructor Gatenyan noticed Krylov’s scientific and engineering talent and hired him to work at his bureau.  Krylov published articles in technology-related journals and even spoke at a seminar at the institute of physics.  He left the plant and overcame numerous challenges before he could come back into academia.  Krylov started working with professor Dankevitch but found that the research was not moving forward fast enough.  He left the country on a geo-physics research ship and came back a year later to write a dissertation about his travels.  In the meantime, Dankevitch died but his ideas were proven to be true.  Krylov became famous because of his previous work with Dankevitch.  His fame however, resulted in conflicts with coworkers.  Krylov started working with Tulin who studied atmospheric electricity and designed various equipment and experiment procedures that would allow measuring the strength of the electric charges inside a storm cloud.  The experimental flight finished with the tragic death of a graduate student Richard who tried to save the disks with information obtained from the on-board measuring equipment.  The research project was closed, but Krylov kept working on it.  Once again, he was thought to be foolish but soon other researchers supported him and petitioned for the resumption of the experimental flights to study atmospheric electricity.[41]

Into the Storm had numerous story lines that were not mentioned in the above plot summary.  One of the Granin’s literary talents was to express himself laconically without losing the details of the story and the beauty of linguistic expression.  The author’s respect and reverence for scientists can be seen from the first lines of the novel, where he refers to Tulin as a ‘wizard’[42].  Granin illustrated, using Krylov as an example, that a scientist could not be considered fully educated unless he possessed practical knowledge.  Granin’s image of a scientist was inseparable from the image of an engineer.  The author kept a neutral position with respect to the decisions that the characters made.  For instance, Krylov’s decision to come back to academia was not immediate.  He discussed his plans with other characters and all of them offered different points of view that sounded equally logical and plausible.  None of Granin’’s characters were portrayed as perfect.  For instance, Krylov was socially awkward and Tulin was too cynical and even rude at times.  These imperfections would jeopardize the main characters of classic socialist realist novels, but Granin used those small character flaws to portray the scientists and engineers as ordinary people, who cannot be either absolutely evil or perfect.

The image of an engineer in Russian literature was influenced by political and social factors as well as by popular trends in literature.  The Socialist realism method of literary expression played a large role in the way that engineers were represented and how their interactions with the working class were portrayed.  The image of the engineer in pre-socialist  realist literature was monarch-like.  With the onset of the Russian revolution it shifted towards a servant of the working class who supplied the socialist state with the technological means of dispersing the popular political ideas.  In a few decades after the revolution the image of an engineer tended to become close to the image of an ordinary worker and a workaholic who dreamt of increasing the productivity rates by all possible means.  Finally, during the Cold War period, the image of an engineer and a scientist became inseparable due to the political turmoil that demanded immediate scientific advancements.  The popularity of engineers in Russian literature of the first half of the twentieth century reflect the society’s interest and awareness of the important, although always changing, role of engineer in social, political and professional matters.

[1]Timofeev L, Turaev. “Socialist Realism.”Russian Literature and Folklore (since 2002).Abridged Encyclopedia of Literature.Accessed April 17, 2012.

[2]Clark, Katerina. The Soviet Novel. Chicago. The University of Chicago Press, 1981, 9

[3]James, Vaughan. Soviet Socialist Realism.Origins and Theory. London: The Macmillan Press Ltd, 1973, 91.

[4]Clark. The Soviet Novel.


[7]Lenin, V. I. Materialism and Empirio-Criticism: Critical Comments on a Reactionary Philosophy. April, 2013. P. 147-155, 267 – 275, 389-400.

[8]Bogdanov, Alexander. Инженер Мєнни. (Engineer Menni). 2013. Accessed March, 2013. Last modified in 2010.

[9]Bogdanov.Инженер Мєнни. (Engineer Menni).

[10]Bogdanov.Инженер Мєнни. (Engineer Menni), 9.

FreetranslationfromRussian.Original: “Я сам со своими помощниками, – сказал он, – произвел новый промер Ливийской котловиныот юга к северу и от востока к западу”.

[11]Bogdanov.Инженер Мєнни. (EngineerMenni), 11.

Freetranslation.Original: “Рабочий, который плохо питается или переутомлен, не обладает полной рабочей силой. Рабочий, который недоволен, угрожает неожиданностями, нарушающими ход дела.”

[12]Bogdanov.Инженер Мєнни. (Engineer Menni), 11.

Freetranslation.Original:  “…мне не надо неожиданностей.”

[13]Bogdanov.Инженер Мєнни. (Engineer Menni), 49.

[14]Bogdanov.Инженер Мєнни. (Engineer Menni), 17.

[15]Bogdanov.Инженер Мєнни. (Engineer Menni), 31.

[16]Bogdanov.Инженер Мєнни. (Engineer Menni), 39.

[17]Bogdanov.Инженер Мєнни. (Engineer Menni), 4.

[18]Bogdanov.Инженер Мєнни. (Engineer Menni), 9.

[19]Bogdanov.Инженер Мєнни. (Engineer Menni), 19.

[20]Bogdanov.Инженер Мєнни. (Engineer Menni), 40.

[21] “Красная новь”

[22]Детское Государственное Издательство (ДЕТГИЗ)

[24]Краснаяновь (TheRedNovelty). Fundamental Digital Library of Russian Literature and Folklore.Accessed in April, 2013.

[25]Tolstoy, Aleksei. Аэлита (Aelita). 1923. Last modified 2009, Accessed in March, 2013.

[26]Tolstoy. Аэлита (Aelita). 5.

[27]Tolstoy. Аэлита (Aelita). 144.

[28]“КатаевВалентинПетрович” (“KataevValentinPetrovich”).Fundamental Digital Library of Russian Literature and Folclore.Accessed in April, 2013.

[29]“КатаевВалентинПетрович” (KataevValentinPetrovich). War Heroes database. Accessed in April, 2013.

[30]Kataev, Valentin. Time Forward!, 1932. Accesssed in March, 2013.

[31]Kataev, Valentin. Time Forward!53.

[32]Kataev, Valentin. Time Forward! 111.

[33]Kataev, Valentin. Time Forward! 9.

[34]“АндрейПлатоновичПлатонов” (Andrei PlatonovichPlatonov).Fundamental Encyclopedia of Russian Literature and Folclore.Accessed in April, 2013.

[35] АндрейПлатоновичПлатонов” (Andrei PlatonovichPlatonov).Fundamental Encyclopedia of Russian Literature and Folclore.Accessed in April, 2013.

[36]Platonov, Andrei. Ювенильноеморе (The Sea of Youth), 1931. Accessed in March, 2013.

[37]Platonov, Andrei. Ювенильноеморе (The Sea of Youth), 2.

[38]Platonov, Andrei. Ювенильноеморе (The Sea of Youth). 1.

[39]“ГранинДаниилАлександрович” (GraninDaniilAleksandrovich). War Heroes database. Accessed in April, 2013.

[40]Karp, Alexander, and Bruce R. Vogeli.Russuan Mathematics Education: History and World Significance.

New Jersey: World Scientific Publishing Co. Pte. Ltd., 2010. 93.

[41]Granin, Daniil. Idu Na Grozu.(Into The Storm).1957. Accessed in March 2013.

[42]Granin, Daniil. Idu Na Grozu.(Into The Storm). 1.

Work day of a mathematician

 “What does a usual work day of a mathematician look like?” (grade 8 student)

“Unlike writers and novelists, mathematicians do not even publish their work too often. What do they do all day?” (grade 9 student)

“What do professors do when they are not teaching undergraduate courses?” (grade 11 student)

“How do researchers manage their time when they do not have strict deadlines to follow?” (grade 11 student)

“How do co-authors work together when a lengthy research project needs to be completed?” (grade 9 student)

These are some of the questions questions that I often hear from high school students and even from junior undergraduates.  Most students do not observe mathematicians at work too often, so their questions are perfectly valid.  Every student knows what a shoe maker, a chef or a painter does because they see the direct products of these people’s work.  The situation is different with mathematician because often the outcomes of mathematician’s work cannot be immediately observed. Thinking about problems and experimenting with various solutions could take days, weeks or even months! Many attempts to solve a problem could fail.  Extreme persistence is needed to keep going forward and to avoid quitting. So, really, how do mathematicians keep themselves on track every day? How do they stay productive and motivated?  The answer is different for every mathematician. For example, here are some strategies for balancing academic work and hobbies that professor Andrei Kolmogorov and his co-author professor Pavel Alexandrov practiced:

First, they chose a pleasant setting to work in.  Both professors adored nature and often spent 3 to 4 days of the week outside Moscow in a cottage near a small river. Second, professors placed great value on physical activities to keep their minds fresh.  Their day usually started at 7am with various sports-related activities.  Both preferred taking walks and hikes every day after lunch (2pm) and short walks before bed time (10pm). Third, Kolmogorov and Alexandrov dedicated lengthy unbroken time periods to their research (usually from breakfast till lunch and from 3pm till dinnertime).  Fourth, both researchers did not neglect their shared hobby – music, and dedicated some time to listening and discussing various records every day. Fifth, professors aimed to get about 10 hours of sleep every day.  That often included short naps during the day.  Of course, when their research was going especially well, they altered their schedule and often spent entire days discussing the solutions to the posed problems.

In summary, in order to stay academically productive and motivated, try following these five simple suggestions:
1)      Find comfortable setting to work in
2)      Find common interests that you share with your co-author and dedicate time to pursue these interests
3)      Exercise regularly and get plenty of fresh air
4)      Get enough sleep!
5)      Organize your day in a way that will allow for lengthy unbroken work periods
6)      Be flexible in your planning and don’t hesitate to change your routine!
What do you think of these tips? Would you dare to try living several days by this demanding schedule?

Note: most of the information about professor Kolmogorov’s work habits was taken from the interview published in the “Quantum” (Квант) magazine in 1983.

The original version in Russian can be found here

The image is taken from


That day when I completed my coursework

20130225_104146 ed


This is my work desk. A usual desk of a grad school student. Do you think that in grad school we get paid to read interesting books? Sometimes we do. We like some books more than others, but the most important part of grad school is to convince yourself that something that you must read (say, it is required for your course work completion) IS really interesting and relevant for you.  It is one of the most difficult tasks to acquire.  At the end of the day, when all the confusing literature is read and set aside in a neat stack, it feels amazing to fill up a travel mug with some good tea and then open up all these files with biographies and memoirs of mysterious mathematicians.

First course in history of math and other exciting events of 2012

This is what seemed super-exciting to me nearly a year ago.  To be honest, I still think it is exciting (go ahead and call me a nerd :):) )

first posted at

My final year as an undergraduate at U of T was full of exciting academic events. I went to numerous Fields Undergraduate Network events and to a larger conference at Penn State.  One of the highlights of that academic year was the process of applying for graduate programs. It was a challenge to choose a sample of writing that was required for all the graduate applications. I took a course called Writing in Mathematics and wrote an essay on Mobius Transformations which later won the Dean’s Award for Writing.  Unfortunately, a purely mathematical essay would not satisfy the History and Philosophy of Sciences committee. A philosophical perspective was needed. I have never taken a course in philosophy (other than Modern Symbolic Logic described in one of the previous posts), so I was allowed to submit a paper I wrote in one of my pedagogy-related classes in addition to my math essay.  Several months later I was accepted to the History and Philosophy of Science and Technology Graduate program , History of Mathematics doctoral stream at U of T.

The most memorable course of my final year was called History of Mathematics after 1700. There I learned a lot of mathematics that was not taught in my previous math classes. For instance, the Vibrating String Controversy. I also got a chance to read some primary sources written by Euler and Bishop Berkley. An interesting theme that could be traced throughout the course was the absence of ‘revolutions’ in mathematics. A scientific revolution is a phenomenon when an old paradigm is abandoned and replaced by a new one, more suitable to solve problems posed in a certain science. For instance, once upon a time people thought that the Earth is the centre of the entire Universe. Then they learned that it is not true. That is a scientific revolution. We no longer use the knowledge from the previous paradigm and we know that it was wrong. In mathematics we do not have such drastic changes. For instance, we still use Pythagoras Theorem and the papers that were written thousands of years ago are still valid.

“You study math and what…???” – How math and history majors merged into one

As a student of the Concurrent Teacher Education Program I knew that eventually I will need to have two teachable subjects. Despite of all difficulties I really enjoyed my math classes and I did not even think of studying anything else. By the end of the second year I had a lot of math courses under my belt but my academic adviser kept reminding me that I need to choose the second teachable. Since I was interested in history before it seemed like a natural choice. I chose to complete a double major (Major of Mathematics and Major of History). I did not choose to complete the specialist because it contained many computer science courses and computer science was not my primary interest at that time.

Having such a combination of majors made me the “laugh of the town” among my friends for some time but I was confident in my choices. Turned out that this combination of academic interests was an asset for me while applying to the History of Science graduate program later.

first posted at

How I ended up in Grad School

This post was written a little over a year ago (in 2012) and was posted first at

This is how I ended up in a PhD program for History of Mathematics

My name is Mariya. I have just graduated from UofT majoring in Mathematics, History and        Eduation (Concurrent Teacher Education Program). I was born in Ukraine and came to Canada at the  lucky age of 13. The most important aspect of university life for me is balancing academic and social  activities. In my second year I got involved with the Mathematics and Computational Science Society   at UTM and stayed there as the VP of Advertising for three academic years. During that time I  learned   a lot about the academic culture of UofT. I met many academic celebrities and learned to communicate with a variety of people. Last summer I was a blog contributor for the Fields-MITACS program. Interviewing international undergraduate research assistants and writing about their achievements gave me an opportunity to learn culture-specific approaches to mathematics. I was always interested in the cultural aspects of sciences and scientific journalism but was never sure how to apply my interests to a possible career until I was told about the History of Science graduate program at UofT. Last year I took a history of mathematics course and finalized my decision about joining the History of Mathematics doctoral stream.

image from

The “New Math” Movement in the U.S. vs Kolmogorov’s Math Curriculum Reform in the U.S.S.R.

This is my first attempt to give an overview of math curriculum reforms in the US and the USSR during the Cold War period.

Andrey Kolmogorov’s Mathematical Education Reform in the USSR versus the “New Mathematics” Movement in the US during the 1950s, 1960s and beyond: The Analysis of the Legacies of the Two Reforms.

By Mariya Boyko

December 2012


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 curricula; 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]

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]

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.

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]

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 3×4 = 4×3 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 4×3 = 3×4 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]

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]

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]

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.

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.

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.

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]

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.


Bybee, Roger. “The Sputnik Era: Why Is This Educational Reform Different From All Other Reforms?” NationalAcademy of Science. Last modified 1997. Accessed December 18, 2012.

Hayden, Robert. “A History of the “New Math” Movement in the United States.” PhD diss., IowaStateUniversity, 1981.

Karp, Alexander, and Bruce R. Vogeli. Russuan Mathematics Education: History and World Significance. New Jersey: World Scientific Publishing Co. Pte. Ltd., 2010.

Kondratieva, Galina. “Why Our Children Do Not Know Mathematics?” (“Почему наши дети не знают математику?”). SchoolRelated Technologies (Школьные Технологии).  Research Institute for School-Related Technologies. (Научно-исслкдовательский институт школьных технологий). No 5. 2012.

Klein, David. “A Brief History of American K-12 Mathematics Education in the 20th Century”. Pre-preint. Final version published in Mathematical Cognition edited by James Royer, 2003. californoia State University North Ridge. Accessed on December 18, 2012

Kline, Morris. Why Johnny Can’t Add. New York: St. Martin’s Press. 1973.

Raimi, Ralf. “Ignorance and Innocence In Teaching of Mathematics”. University of Rochester. Published 2004. Accessed December 18, 2012.

Saunders, Debra. “New New Math”. National Reviews. 1995.

Schoenfeld, Allen. “The Math Wars” Educational Policy  vol. 18, no. 1. Corwin Press. 2004. 253 – 286.

Schubert, William. “Curriculum Reform”.

Specialized Education and Research Centre of the MoscowStateUniversity. Accessed on December 18, 2012.

Walmsley, Angela L. E. History of the “New Mathematics” Movement and Its Relationship With Current Mathematical Reform.Maryland: University Press of America Inc. 2003.

Wu, H. “The Mathematics Education Reform: What It Is and Why Should You Care?” University of California. Accessed December 18, 2012.

[1] Schubert, William. “Curriculum Reform”.

[2] Hayden, Robert. “A History of the “New Math” Movement in the United States.” PhD diss., IowaStateUniversity, 1981. 74 – 82.

[3] Walmsley, Angela L. E. History of the “New Mathematics” Movement and Its Relationship With Current Mathematical Reform.Maryland: University Press of America Inc. 2003. 13.

[4] Schoenfeld, Allen. “The Math Wars” Educational Policy  vol. 18, no. 1. Corwin Press. 2004. 256.

[5] Schubert, William. “Curriculum Reform”.

[6] Karp, Alexander, and Bruce R. Vogeli. Russuan Mathematics Education: History and World Significance. New Jersey: World Scientific Publishing Co. Pte. Ltd., 2010. 92, 87 – 141.

[7] Klein, David. “A Brief History of American K-12 Mathematics Education in the 20th Century”. Pre-preint. Final version published in Mathematical Cognition edited by James Royer, 2003. californoia State University North Ridge. Accessed on December 18, 2012

[8] Klein, David. “A Brief History of American K-12 Mathematics Education in the 20th Century”.

[9] Klein, David. “A Brief History of American K-12 Mathematics Education in the 20th Century

[10] Klein, David. “A Brief History of American K-12 Mathematics Education in the 20th Century”.

[11] Hayden, Robert. “A History of the “New Math” Movement in the United States.” PhD diss., IowaStateUniversity, 1981. 61.

[12] Klein, David. “A Brief History of American K-12 Mathematics Education in the 20th Century”.

[13] Klein, David. “A Brief History of American K-12 Mathematics Education in the 20th Century”. Pre-preint. Final version published in Mathematical Cognition edited by James Royer, 2003. californoia State University North Ridge. Accessed on December 18, 2012

[14] Kline, Morris. Why Johnny Can’t Add. New York: St. Martin’s Press. 1973. 17.

[15] Kondratieva, Galina. “Why Our Children Do Not Know Mathematics?” (“Почему наши дети не знают математику?”). School-Related Technologies (Школьные Технологии).  Research Institute for School-Related Technologies. (Научно-исслкдовательский институт школьных технологий). No 5. 2012.

[16] Karp, Alexander, and Bruce R. Vogeli. Russuan Mathematics Education: History and World Significance. New Jersey: World Scientific Publishing Co. Pte. Ltd., 2010. 33.

[17] Kondratieva, Galina. “Why Our Children Do Not Know Mathematics?” (“Почему наши дети не знают математику?”). School-Related Technologies (Школьные Технологии).

[18] Bybee, Roger. “The Sputnik Era: Why Is This Educational Reform Different From All Other Reforms?” NationalAcademy of Science. Last modified 1997. Accessed December 18, 2012.

[19] Hayden, Robert. “A History of the “New Math” Movement in the United States.” PhD diss., IowaStateUniversity, 1981. 82.

[20] Karp, Alexander, and Bruce R. Vogeli. Russuan Mathematics Education: History and World.  93.

[21] Karp, Alexander, and Bruce R. Vogeli. Russuan Mathematics Education: History and World.  95.

[22] Walmsley, Angela L. E. History of the “New Mathematics” Movement and Its Relationship With Current Mathematical Reform.Maryland: University Press of America Inc. 2003. 13.

[23] Walmsley, Angela L. E. History of the “New Mathematics” Movement and Its Relationship With Current Mathematical Reform.Maryland: University Press of America Inc. 2003. 13.

[24] Karp, Alexander, and Bruce R. Vogeli. Russuan Mathematics Education: History and World Significance. 95.

[25] Karp, Alexander, and Bruce R. Vogeli. Russuan Mathematics Education: History and World Significance.  93-97.

[26] Kline, Morris. Why Johnny Can’t Add. 24.

[27] Kline, Morris. Why Johnny Can’t Add. 41, 42.

[28] Kline, Morris. Why Johnny Can’t Add. 42, 50.

[29] Kline, Morris. Why Johnny Can’t Add. 50.

[30] Raimi, Ralf. “Ignorance and Innocence In Teaching of Mathematics”. University of Rochester. Published 2004. Accessed December 18, 2012.

[31] Karp, Alexander, and Bruce R. Vogeli. Russuan Mathematics Education: History and World Significance. New Jersey: World Scientific Publishing Co. Pte. Ltd., 2010. 113.

[32] Kline, Morris. Why Johnny Can’t Add. New York: St. Martin’s Press. 1973. 68 – 102.

[33] Karp, Alexander, and Bruce R. Vogeli. Russuan Mathematics Education: History and World Significance. 113 – 134.

[34] Kline, Morris. Why Johnny Can’t Add. 93.

[35] Kline, Morris. Why Johnny Can’t Add. 92.

[36] Karp, Alexander, and Bruce R. Vogeli. Russuan Mathematics Education: History and World Significance. 113 – 134.

[37] Kline, Morris. Why Johnny Can’t Add. 69.

[38] Karp, Alexander, and Bruce R. Vogeli. Russuan Mathematics Education: History and World Significance. 113 – 134.

[39] Kline, Morris. Why Johnny Can’t Add. 72.

[40] Karp, Alexander, and Bruce R. Vogeli. Russuan Mathematics Education: History and World Significance. 113 – 134.

[41] Karp, Alexander, and Bruce R. Vogeli. Russuan Mathematics Education: History and World Significance. 106.

[42] Walmsley, Angela L. E. History of the “New Mathematics” Movement and Its Relationship With Current Mathematical Reform.

[43] Karp, Alexander, and Bruce R. Vogeli. Russuan Mathematics Education: History and World Significance.

[44] Wu, H. “The Mathematics Education Reform: What It Is and Why Should You Care?” University of California. Accessed December 18, 2012.

[45] Walmsley, Angela L. E. History of the “New Mathematics” Movement and Its Relationship With Current Mathematical Reform.Maryland: University Press of America Inc. 2003.

[46] Karp, Alexander, and Bruce R. Vogeli. Russuan Mathematics Education: History and World Significance. New Jersey: World Scientific Publishing Co. Pte. Ltd., 2010. 119, 89 – 141.

[47] Walmsley, Angela L. E. History of the “New Mathematics” Movement and Its Relationship With Current Mathematical Reform.Maryland: University Press of America Inc. 2003. 85.

[48] Saunders, Debra. “New New Math”. National Reviews. 1995.

[49] Karp, Alexander, and Bruce R. Vogeli. Russuan Mathematics Education: History and World Significance. 120 – 124.

[50] Kolmogorov, Andrey. «Что такое функция?» (What is a function?). Квант (Qantum). 1970, no1.

[51] Квант (Qantum). 2012, no 4.

[52] Specialized Education and Research Centre of the MoscowStateUniversity. Accessed on December 18, 2012.