Fields Medal Symposium 2016

There’s no Nobel Prize in math but there’s a Fields Medal which is given to mathematicians under the age of 40 for their outstanding achievements. Every year Toronto’s Fields Institute hosts an annual Fields Medal Symposium to honor one of the medalists. This time we have Prof. Manjul Bhargava  – a number theorist born in Hamilton – is the honored guest.

Prof. Bhargava is the first Canadian born mathematician who received the Fields Medal! This makes this year’s symposium even more special. There were no empty seats in the Isabel Bader Theatre where the Public Opening of the Symposium was taking place yesterday evening. The audience was composed of professionals from a variety of fields and students. Prof. Bhargava’s talk was engaging and accessible for everyone in the audience.  Colorful diagrams and interesting facts about nature that he mentioned were especially appreciated.

We brought our Math Circles class from UTM to the public opening. This was their first time attending such a large event.  Luckily, they enjoyed the talk and the festive atmosphere around the Public Opening night.  Moreover, they even had a chance to take a picture with Prof. Bhargava!

Prof. Bhargava will be giving a special lecture for high school students this evening at the Fields Institute, but unfortunately I will not be able to attend it, cause I’ll be teaching my own students.

If you want to find out more information about the Fileds Medal Symposium, click here:

http://www.fields.utoronto.ca/activities/16-17/fieldsmedalsym

 

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

Teaching an online course: Online Math Kangaroo Enrichment Classes

This summer I have been teaching a Online Math Kangaroo Enrichment Classes for grade 4.

In each class we looked at several core concepts and then solved numerous problems that resembled problems from Math Kangaroo Contests.

Teaching an online class was a new experience for me and I am extremely happy that i had a chance to try it.

The trickiest part about teaching an online class was finding an effective way to communicate with the audience.  This challenge, however, could be resolved by asking the students to ‘raise their hand’ (by clicking an appropriate icon) and typing their questions in the chat window provided.

I found that giving the audience plenty of time to think about each posed question on their own.  Providing sufficient time allowed the students to understand posed problems and to formulate meaningful questions afterwards.

This summer the audience was quite active and well prepared for each class.  Although the group of students was quite diverse in terms of academic background and age, everyone caught up to the same speed and academic level very fast.

I am quite proud of my students and I wish them the best of luck in their next academic year and the upcoming Math Kangaroo Contests!

Although the summer course is over, please see the related information at https://kangaroo.math.ca/index.php?kn_mod=news  and stay tuned for information on further classes

 

Classroom Adventures in Mathematics Summer Institute 2014

Last week Department of Mathematics of UofT welcomed school teachers and teachers in training for a week of professional enrichment – Classroom Adventures in Mathematics Summer Institute 2014.  UofT faculty and graduate students spoke to the teachers about enrichment of traditional curriculum for students of different ages and academic backgrounds as well as highlighting the wide possibilities of interdisciplinary studies of STEM subjects.

Please see http://mathplus.math.utoronto.ca/home/caimsi to read more about the covered topics and stay tuned for the videos of selected lectures that will be posted soon!

Meeting experienced teachers from local school boards was an invaluable learning opportunity and I am very glad that I could be a part of the program!

 

 

Canada Math Camp 2014

DSC_4700

Last week I had a pleasure of being a camp coordinator of Canada Math Camp! Working at a camp is always intense but very exciting.  Although it can be tiring at times, I can’t wait to do it all over again already!

Canada Math Camp is for students who achieved high scores at the Canada Open Math Challenge (COMC) or for those who were recommended by their mathematics teachers or other members of the mathematics community.

Check out our daily updates here: http://mathplus.math.utoronto.ca/home/b_cmc2014

General information about the camp can be found here: http://mathplus.math.toronto.edu/home/cmc

If you re feeling nostalgic for the last year’s CMC camp, CMC 2013 info and daily updates can be found here: https://cmc.math.ca/home/blog/category/daily-updates/

 

(image from https://cmc.math.ca/home/wp-content/uploads/2013/07/DSC_4700.jpg)

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 http://yorkustsgradconference2014.wordpress.com/program/

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

)

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Kangaroos are coming!!!

Last weekend all of the University of Toronto campuses hosted the last preparation session before the Mathematics Kangaroo Contest of 2014! The students and the instructors had a great time. The kids couldn’t wait to write the contest!

The contest will take place tomorrow! Math students all over Canada will be writing it. If you are planning to participate, you still have the entire Saturday night to try some of the contest problems from last year.

You can find them here https://kangaroo.math.ca/index.php?kn_mod=samples&year=NO

Good luck for tomorrow’s contestants! Have some good night sleep and we will see you on Sunday morning!

Math Kangaroo March Break Camp 2014 has started!

LogoCanada (image from https://kangaroo.math.ca/)

I am so excited to be working at the Math Kangaroo March Break Camp and Math Academy this week! The camp started out on a great note. For instance, grades 4 and 5 started their day by looking at crocodiles from a mathematical point of view! Picture can be found here: https://twitter.com/mariya_boyko12/status/443053861318955008/photo/1

Fields Institute wins!

Last week, and the week before, we were visited by a group of students from St. Andrews Junior High School. They chose to come to the Fields Institute even though they could go to the Playdium or any other place in Toronto for their activity day!

I gave them a mini lecture on history of calculation devices, and then they created abaci from scratch. Here is a link to their photo:

Biography of one typewriter (that puts biographies of our smartphones to shame)

It was a Remington typewriter, probably like the one in the photo below. It had a sturdy case. The owner used it heavily and took it everywhere he went. He was a frequent traveler

From http://mytypewriter.com/remingtonportableof1920s.aspx

At the end of the 1920’s the typewriter was lucky enough to go on a lengthy boat trip on the Volga river. The owner and two other men started their journey in Yaroslavl and sailed to Kazan’ city, where one of them got off the boat. The owner of the typewriter and his new friend continued sailing toward Samara city. Sometimes they used a primitive home-made sail. But most of the time the typewriter’s owner was the main rower, and his friend was the navigator.  At that time they did not yet know that they will remain best friends for the rest of their lives. They were enjoying the weather, the nature, and conversations about math.  The typewriter was sitting in the boat next to a box of biscuits.  The history omits the circumstances under which both aforementioned items ended up in the Volga river. We also do not know anything about the fact of the biscuits (there are good reasons to expect that they have stopped existing in their biscuit-like form at that moment. The fauna of the river probably had a feast though). The typewriter, however, was rescued successfully.  As soon as it dried up it resumed working just as it has worked before.

The typewriter continued following its owner around the world. It has witnessed all of his mathematical discoveries, professional crises, conversations with his students and colleagues. This typewriter was meant to live a dangerous life, just like its owner.  On October 15th, 1945 it followed its owner along the streets of Moscow that was immersing itself into the dusk of a gloomy and sad evening. The tanks were rolling along the Gorky street toward the centre of Moscow.  It was a miracle that the typewriter along with its owner did not get hit by one of the tanks while crossing the street. Suddenly, the typewriter was violently shaken inside its case. The owner heard a sharp metallic sound. The typewriter’s case hit the light pole. A scratch remained on it forever, but the typewriter was still working as though nothing has ever happened. This fact, however, was discovered only after its owner’s arrival to Kazan – the place where he needed to be evacuated along with his life long friend.

After 1945 the typewriter did not experience serious injuries and has never even been to a repair shop! Now, all of us would love to be THAT healthy!

During the late 1970‘s, this typewriter still belonged to its original owner – Professor of mathematics Pavel Alexandrov. His life long friend was Professor of mathematics Andrei Kolmogorov.

P. S.: compare this biography with a biography of any smartphone… it usually ends with a tragic drowning in a kitchen sink, a coffee mug or a toilet; or with a fatal fall from a pocket onto the pavement. “What a tragic and ungraceful way to go” – Alexandrov’s typewriter would think

information was taken from Pavel Alexandrov’s memoirs. You can read them in original language (in Russian) here http://www.rulit.net/books/stranicy-avtobiografii-read-227532-1.html

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: (

Collected Curiosities: Darwin, Banksy, Tstar Nicholas II, etc

 Darwin said that men endowed with mathematical talent “seem to have an extra sense” (Lecture at the Fields Institute)
Banksy trouble:
According to a group of my friends I was the last person (if not on the entire Earth, than in North America:)) who did not know the name of a graffiti artist Banksy. When they told me that he is famous for hiding his identity I asked why he does that and if anyone asks him to reveal his identity.  Asking that was a big mistake.  Not only my friends got upset, but I also did not get the answer to my question.  If there is anyone else who did not know about Banksy, here is his website http://www.banksyny.com/
Tango was brought to Paris in the 1910’s by rich Argentinian young men who toured Europe and actively participated in Paris’ night life. Soon tango became so popular that there were tango lectures, tea parties, exhibitions, etc. (Tango! : the dance, the song, the story by Collier)
Stravinsky wrote the first act of “Petrushka” ballet in an attic of a small house near a hospital where his wife was placed before the birth of their son. (Stravinsky by B. Yarustovsky)
Russian tsar Nicolas II likely was the first monarch who saw tango being danced. He even liked it! (Tango! : the dance, the song, the story by Collier)
Apparently soap is the most recommended souvenir to buy in Marselles, France. (internet)
Robert Hooke convinced his contemporaries that ‘minute bodies’ (very small organisms and very small things in general) are capable of having complex structure.  In his book Micrographia (1665) he illustrated that a flies have 360 degree field of vision as well as many other intricacies of small organisms and plants. His work was especially impressive because before the seventeenth century smallness was considered an obstacle to having a complex structure. (Micrographia by Robert Hooke)

Revelations of a teacher-in-training

 

Since I was in the Concurrent Teacher Education Program (CTEP) the second semester of my fourth year was a professional semester. We had various courses related to pedagogy in January and then a 7 week teaching practicum at a local school. The Curriculum, Instruction and Assessment course was designed to prepare us for teaching mathematics. We learned many valuable teaching strategies and were exposed to cutting edge pedagogical research on teaching math. For instance, a guest speaker taught us to deal with students suffering from dyscalculia – a learning disability characterized by difficulties of learning and comprehending arithmetic. Some people even refer to it as ‘dyslexia but in terms of numbers’. One of the useful tips to fight both problems was to implement a buddy system in the classroom. Students of similar academic levels can be paired up (or put into groups of three) and asked to solve a problem. Students with a low level of confidence in their mathematical abilities would benefit from such a practice because they will see that their classmates struggle as well, but it is quite possible to find a solution if everyone thinks together. Psychological Foundations of Learning was an interesting course as well. We learned about various theories of human psychological and intellectual development. The professor always used examples from his own teaching practice to illustrate the concepts we covered in class. For instance, he told us about a student who was sent to a special education class and mistakenly diagnosed with Attention Deficit Disorder, when in fact she was just a kinaesthetic learner (a student who retains the information best when a physical activity is involved).

In my practicum I got to teach math in a grade 9 gifted and in a grade 10 applied class. Certain advice that the pedagogy professor gave us were extremely useful but some advice turned out to be inapplicable to the specific school or a specific classroom. After the first two weeks of teaching I realized that no matter how much time we spend discussing pedagogy in a university classroom, the main part of learning will come from practice. Every group of students is unique and it is not possible to describe the perfect ‘recipe’ for classroom management. I also realized that teaching and acting can mix at times. In order to explain the transformations of parabolas to my applied math class I needed to choreograph a Parabola Dance for them (students were asked to imitate the transformations of a parabola using their hands). We also had to sing a Quadratic Formula Song at times. Working with gifted students was also very pleasant. I especially liked when some of them came up to me before class and told me about their math, computer science and physics related ideas.

The main thing I understood during my practicum is the fact that the students do not understand where the mathematical concepts are coming from and why there was a need to develop them. For instance, many students asked “Solving a cubic equation is complicated. Why can’t we just have a formula like for a quadratic?” or “Why would someone even think of creating something like a parabola?” The students lack the knowledge of the historical background of mathematics and there is not enough age-appropriate historical literature for them. Even if such literature was present, the curriculum obliges the teachers to go through the concepts very fast and to omit some useful historical information. Shortly after completing my practicum I decided to join the graduate program for History of Mathematics and I hope that in the future I will be able to create some supplementary materials that would help teachers to integrate the historical background of mathematics in the fast paced curriculum effectively.

posted at https://math.escalator.utoronto.ca/home/blog/mariya-mathematics-history-and-eduation-week-9/

image www.educationworld.com

Freshmen, this one is dedicated to you :)

If you happen to be going into your first year of university, I have some pointers to give you here (https://math.escalator.utoronto.ca/home/blog/mariya-mathematics-history-and-eduation-week-3/)

first-week-of-the-first-year-of-university over here (https://math.escalator.utoronto.ca/home/blog/mariya-week4/)

more on specific math classes here (https://math.escalator.utoronto.ca/home/blog/mariya-mathematics-history-and-eduation-week-6/)

Let me know if that was helpful :):):)

How I got into a Teacher Education program

As a child I always had many hobbies at the same time. I played piano, read historical novels, studied English and was enrolled in a dance school. As a result my career preferences tended to change every several years. I remember wanting to be a teacher, a paramedic, a geneticist, a historian, a psychiatrist…

In elementary school I often struggled with mathematics but I always enjoyed studying it. I never considered it as a possible career choice or a university program possibility until my senior years of high school. I realized that mathematics can be applied to many situations in the real world and that attracted me to the discipline even more. I always looked at math from the point of view of an observer. The historical aspects of it fascinated me. To me, mathematical progress reflects the evolution of human thinking and of our vision of the world.  During my high school years I was also heavily involved with the math club, the peer tutoring club and the early child education centre, so applying my love for mathematics to teaching seemed like a logical choice. I joined the Concurrent Teacher Education Program to get a degree in math simultaneously with a teaching degree.

image from sticaqui.wikispaces.com

first published at https://math.escalator.utoronto.ca/home/blog/mariya-week/

How I ended up in Grad School

This post was written a little over a year ago (in 2012) and was posted first at https://math.escalator.utoronto.ca/home/blog/mariya-week1/

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 http://columbia-advising.blogspot.ca/2009_02_01_archive.html

Wikipedia: teacher`s enemy or a friend?

Much of Wikipedia's appeal lies in the way it creates a community.

One of my high school teachers used to say that “Wikipedia is an enemy of research”. I have heard similar phrases throughout my university career, at OISE and at my teaching practicums. All those teachers were making valid points. First of all, Wikipedia is too easily editable. Many articles (especially historical or political ones) are biased or inaccurate. Moreover, we never know who the author of the article is and who the editors are, hence, we cannot be sure that they are professionals. The ready availability of Wikipedia articles might give the students a false idea of what research is, leading them to think that in order to understand a subject one just needs to click on several links. Over-relying on Wikipedia promotes poor time management skills because many students use it to cram before their tests, so it is understandable why many teachers would encourage their students to opt out of using Wikipedia altogether. Such an aggressive reaction always surprised me. I can certainly see how the overuse of Wikipedia can negatively impact students’ research skills and how inaccurate information may lead them astray but as teachers we need to face reality. As much as we want our students to consult reliable sources, they tend to consult Wikipedia when faced with anything unfamiliar.

I was reviewing the topics that would be presented at the CMS camp and I was shocked by their level of difficulty. I have taught a gifted math class before but the enhanced curriculum never covered game theory, advanced combinatorics or Diophantine equations.  I thought it would be interesting to see what Wikipedia had to say about them (since the students probably looked all of them up already). I was pleasantly surprised. Most topics were covered in great detail and I did not spot any inaccuracies. Moreover, the descriptions contained links to interesting mathematical facts and theorems that could lead the students into interdisciplinary ways of thinking. For instance, after typing the word “Triangle” in Wikipedia search box I was expecting to see a one-screen description with a picture of a triangle. Wikipedia went above and beyond my expectations listing all interesting facts about triangles, related theorems and formulas, historical notes and more. There were at least 10 different formulas for finding the area of a triangle! (As a teacher I was lucky to meet students who knew one formula (A = ½(bh) ) and I was extra lucky if that student could explain why the formula is working.) There were other links to applications of mathematical concepts and numerous highlighted keywords linked to the relevant articles. Each page contained detailed diagrams, sometimes even the interactive ones. Many links made me regret that I did not think of consulting Wikipedia before preparing some of my lesson plans.

For me it is always important to motivate students to explore math in their own way and at their own pace. Wikipedia is a useful tool to get the student started on a topic. The first paragraph of a typical Wikipedia page is a concise summary that allows the student to quickly see if they are interested in the presented material. For instance, if a student heard an unfamiliar term he or she can quickly look it up and get a rough idea of what branch of math it belongs to, etc. Of course as teachers we need to be able to explain to the students which sources are suitable for effective research and which are not, but this part comes later, when the initial interest in a topic is present already. Wikipedia is just one of the numerous objects in our lives which have their designated uses but can be dangerous when misused. For instance, we never hear anyone calling a kitchen knife an enemy of human fingers. Wikipedia can be turned into teachers’ helper as well if the teachers will clearly define its role for the students. After all the phrase “Do NOT use Wikipedia” may cause more students to use it simply to ‘rebel’.

original at https://math.escalator.utoronto.ca/home/blog/category/points-of-note/

image from http://www.theage.com.au/news/web/fast-facts-found-online/2007/02/21/1171733770530.html

Interview with Professor of Mathematics Stephen Smale

This is an interview with Professor of Mathematics Stephen Smale, the Fileds Medalist of 1966.  The interview was filmed at the Fields Institute for Research in Mathematical Science, Toronto, in 2011.

 

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.

BIBLIOGRAPHY

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. http://www.nas.edu/sputnik/bybee4.htm

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 http://www.csun.edu/~vcmth00m/AHistory.html

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. http://www.math.rochester.edu/people/faculty/rarm/igno.html

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. http://internat.msu.ru/

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. http://math.berkeley.edu/~wu/reform3.pdf


[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 http://www.csun.edu/~vcmth00m/AHistory.html

[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 http://www.csun.edu/~vcmth00m/AHistory.html

[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. http://www.nas.edu/sputnik/bybee4.htm

[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. http://www.math.rochester.edu/people/faculty/rarm/igno.html

[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. http://math.berkeley.edu/~wu/reform3.pdf

[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.

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[52] Specialized Education and Research Centre of the MoscowStateUniversity. Accessed on December 18, 2012. http://internat.msu.ru/