Three Mathematical Cultures: What Can We Learn?

Mark Saul

July 2023

Everyone needs mathematics. It is the heavy industry of scientific development, the unseen basis on which the more spectacular advances in science, in technology, and in medicine are often built. And mathematics is cheap. We rarely need fancy equipment to pursue our research. A pad and paper, sometimes a computer, and sometimes even less to start the process.

In some sense there is only one mathematics. Mathematicians everywhere tend to agree on the nature of mathematical truth or whether a particular proof is valid or not. In that sense, mathematics transcends culture. But in another sense it does not. The attitudes toward the subject, the place it holds in the canon of scholarship, the value it takes for average citizens or even professional academics–all these may indeed vary from culture to culture, and across time and place. And this variation affects the nature of mathematics education.

In my visits abroad, working with mathematicians and educators, I have had to both adapt to and learn from these different mathematical cultures. Here I write about three of them, from which I have learned the most. These three cultures are:

>East Asia, with its strong ability to bring large numbers of students up to a high level of mathematical competence

>The US, where students are urged to develop and use their creativity;

>East Europe, where the mathematical community inspires passion for the subject in its students.

Disclaimer: one of these is my own culture, of the United States of America. I include it because I find it to be a strong influence on others. The reader can decide whether this portion of my account is sufficiently objective.

The cultures of East Asia are well-known for providing a solid mathematical foundation for masses of people. Countries such as Taiwan, Singapore, Korea, and Japan shine in results reported by TIMSS and PISA (Mullis et al., 2020, 7ff, 147ff). These results have been well studied (see for example Leung, 2020) and the cultures that produced them analyzed. In a nutshell, these largely Confucian cultures revere learning. The Chinese imperial examinations sometimes offered a path through the feudal economy from a rural village to the bureaucracy and splendor of the capital. There are, to this day, local deities who receive offerings at their temples for success in examinations. And the Chinese Empire, like the Roman Empire in the West, had a tremendous influence on nearby cultures. These attitudes, and even some of these practices, persist.

The profession of teaching is revered in those cultures. The teacher is respected as a source of knowledge. In Taiwan, I saw students bow *en masse* to the teacher at the start of the lesson, and again at the end, thanking him or her . Classes of 50 or 60 are not unusual on the high school level. “Classroom management” is rarely an issue. The student automatically respects and trusts the teacher, and the respect and trust is generally returned. I have returned to Taiwan often over the years, and have observed as pedagogical methods there have opened up to include work by groups of students and project-based learning. While the methods of teaching have changed and developed, the respect between the student and the teacher abides.

The public education outcomes of these societies are the envy of the world. But in the countries themselves, something is often perceived to be lacking. The Japanese call it ‘zest for life ‘ (see for example Mori, or Ministry of Culture, both 2023) and have been struggling to restore to students their childhood. This quest is on an emotional level, but of course it has its intellectual ramifications. The Japanese seek to balance their children’s intellectual achievement with an emotional goal.

Other cultures have goals that are more directly intellectual. Often one of them is a quest for creativity. In many countries of this region, learning is seen as mastering a body of knowledge which has been well worked out and enshrined in the literature. Teachers and educators—and even politicians—in those countries complain that there is little room for students to create their own knowledge or to invent new ways to reorganize the knowledge.

And in fact they often look to the US for models of ‘creativity’. This kept coming up, for example, on a tour of China by a group of educators in 2010. In a series of fascinating meetings, the question aroseup: “How do you encourage creativity in your students?” It turned out that the Ministry of Education had disseminated a memorandum about creativity. Unfortunately, what I often found was that people asking that question would expect an algorithm for creativity: institute practices A, B, and C, and you will get creative thinking.

The question stymied me. It doesn’t work that way. How could I even begin to answer it? My own view, from inside the culture, is clouded.

One of the strengths of Confucian societies is the great respect for teachers and mentors. Many have pointed out that the TIMSS and PISA data are a reflection of this deep and strong tradition. But this very strength, which allows these cultures success in mass education, at the same time hamstrings them in getting students to think for themselves. How can the student know something the teacher doesn’t? This question did not come up from people working in, say, China. But many Chinese scientists who work in other countries have pointed out that this is the crux of the problem.

It is difficult for us to see this from within America, but we are very good at just this goal. It is difficult partly because ‘thinking for oneself’ is hard to measure. Only careful, sensitive, and intensive observation, considering the student’s immediate history and relationship to the teacher, can uncover the creative act.

But when it happens in the classroom, the moment is magical. That look on a student’s face, that turn of phrase that you as a teacher have never heard, but which matches exactly the necessary intuitions—including the teacher’s pedagogical intuitions—are hard to forget.

So what is it that unleashes the minds of US students? As with most cultural phenomena, this question is difficult to answer from within the culture. The fundamental levels of culture, of ethnicity, are unconscious, and it is hard for someone participating in them to dig them out. Too, the wellsprings of creativity are also in the unconscious mind, which adds another level of mystery to the question.

So as an American educator I am uniquely unqualified to say more about this phenomenon. I know about it only because my hosts describe it when I travel abroad. But it is true that Americans value creativity in ways that other cultures often overlook. A recent Gallup poll (Gallup, 2019; Saad, 2019) found that teachers who promote creativity see results in achievement. I don’t think this finding is particularly shocking. The point is that an American institution (Gallup, funded partly by Apple Computers) thought to ask the question and seek the answer. This happens less often in other cultures.

Credit in this development must be given to the European legacy in America. British, Dutch, and Scandinavian schools have valued and encouraged independent thinking for a while. Perhaps this is another example of the historical commonplace that America has played the role for European culture that Rome played for Greece. In each case, the one extended and spread the innovations of the other. Maybe.

Credit must also be given to more recent borrowings from Eastern Europe. An extraordinary mathematical culture has developed in these countries, whose strength is not simply creativity, but passion.

The end of the Cold War gave Americans greater access to these countries, and more inter-cultural exchange. A form of mathematical engagement new to Americans, the math circle, was largely a borrowing from these cultures.

The cultures of Eastern Europe have been strong in mathematics since their industrial revolution opened up education to their middle classes. I have elsewhere (Saul, 1992,2003, 2017, 2022) chronicled these developments in Bulgaria, Romania, and Russia (or the USSR). Tibor Frank (2007) has written about his native country, Hungary, which followed a slightly different path.

In brief, there are two forces at work here, neither of which apply directly to the American situation. One is that of a small nation struggling to assert itself and consolidate its own progress. This is a large part of the Hungarian experience. Hungary underwent an industrialization in the late 1800s as part of the Hapsburg Empire. There was a national resolve to make the most of the nation’s young minds, a tradition which persists to the present. High school teachers knew, and still know, members of the mathematical research community, and would inform them of students with noticeable talent in mathematics. A group of research mathematicians and physicists began the journal KöMaL (see MATFUND, n.d.) which to this day binds together the mathematical community. The result is that this tiny nation of less than 10 million has become famous for producing mathematicians and scientists.

The other force, best exemplified by the USSR, is totalitarianism, which tended to push active minds into mathematics as a refuge. In brief, Soviet totalitarianism pervaded (by definition) its citizens’ entire lives. Thinking was dangerous. Artists and writers worked under strict controls. Biologists were not allowed to study evolution. Psychologists and educators could not mention Piaget. Researchers in the physical sciences, dependent on laboratory resources, generally had to work for the military. Computer science was neglected, even after the Second World War.

But mathematicians were relatively free. They needed little or no equipment, and the applications of their work were sufficiently distant from the work itself to obviate any political ‘channeling’.

So active minds, people who might otherwise have become chemists, economists, even poets, gravitated to mathematics. Mathematics departments and classrooms became centers for a silent rejection of totalitarian values. The social lives of mathematicians, math educators, and math students, were also their professional lives, a subculture within the Soviet system. And like any subculture, it strove to reproduce itself. Research mathematicians and graduate students found common cause with teacher and pre-college students, in math circles, summer camps, and competitions. A rich literature of advanced mathematics from an elementary standpoint developed, authored by some of the country’s most prominent researchers. The Anneli Lax New Mathematical Library is an example of American borrowings from this culture, as is the more recent flourishing of math circles in the US. This passion for mathematics, a love of thinking about the subject, is a strong aspect of East European culture.

I have left out here many more mathematical cultures. The venerable mathematical traditions of South India, the current stirrings in Latin America, the vibrant mathematical culture of Western Europe, all deserve more note than this margin will allow. And the continent of Africa is a giant just beginning to feel its strength. In my work there, I find active young minds, and programs just starting to develop them. This second most populous continent has only begun to offer the benefits to mankind that it surely will in future.

This description of ‘three cultures’ can be read as a set of stereotypes. Like all stereotypes, they contain some truth. But like all stereotypes, they cannot be applied to individual cases. There are certainly creative Asians, passionate Americans, and schools in Eastern Europe that reach all their students with deep and important mathematics. The point is that each of these stereotypes describes a strength of the culture, an area from with others can learn. I hope we continue to learn.

WORKS CITED

Frank, Tibor. (2007). *The social construction of Hungarian genius (1867-1930)*. Background paper for the conference on John von Neumann, organized by the Princeton Institute for International & Regional Studies and the John Templeton Foundation, The Witherspoon Institute, Princeton, NJ. October 5-6, 2007.

https://www.researchgate.net/publication/242572746_The_Social_Construction_of_Hungarian_Genius

Gallup (2019) *Creativity in Learning. *https://www.gallup.com/education/267449/creativity-learning-transformative-technology-gallup-report-2019.aspx

#### Leung, Frederick K.S. (2017) “Making Sense of Mathematics Achievement in East Asia: Does Culture Really Matter?”, in Gabriele Kaiser (ed.), Proceedings of the 13th International Congress on Mathematical Education, ICME-13, 201-218. https://link.springer.com/book/10.1007/978-3-319-62597-3#author-1-0

MATFUND Foundation (n.d.)* What is K**ö**MaL? * http://www.komal.hu/info/miazakomal.e.shtml

#### Ministry of Education, Culture, Sports, Science and Technology (2023) “Encouraging Zest for Living”, in Priorities and Prospects for a Lifelong Learning Society: Increasing Diversification and Sophistication, Chapter 3, Section 3, Notice from the Ministry of Education, Culture, Sports, Science and Technology, May 25, 2023 (in Japanese). https://www.mext.go.jp/b_menu/hakusho/html/hpae199601/hpae199601_2_042.html

Mori, Lynsey Helen (2023) “Having a Zest for Life: SEL in Japan”, in *Exploring Social Emotional Learning in Diverse Academic Settings*, Regina Rahimi, Delores Liston, Eds., 102-125. https://www.igi-global.com/chapter/having-a-zest-for-life/321384

Mullis, Ina V. S., Martin, Michael O., Foy, Pierre, Kelly, Dana L., Fishbein, Bethany. (2020*) Timss 2019 International Results in Mathematics And Science*, Lynch School of Eucation and Human Deelopmnet, Boston College and International Assiciation for the Evaluation of Educational Achievement, Boston, chrome-extension://efaidnbmnnnibpcajpcglclefindmkaj/https://www.iea.nl/sites/default/files/2020-12/TIMSS-2019-International-Results-in-Mathematics-and-Science_0.pdf

Saad, Lydia (2019). “Teachers Who Promote Creativity See Educational Results”, in Gallup *Blog*, October 28, 2019. https://news.gallup.com/opinion/gallup/245600/teachers-promote-creativity-educational-results.aspx

Saul, Mark (1992) “Love Among the Ruins: The Education of High-Ability Mathematics Students in the USSR,” in *Focus*, Vol. 12, No. 1, February, 1992.

_______ (2003) “Mathematics in a Small Place: Notes on the Mathematics of Romania and Bulgaria”, in AMS *Notices, *May, 2003, 561-565;

_______ and Fomin, Dmitri (2017) “Russian Traditions in Mathematics Education and Russian Mathematical Contests” (with D. Fomin), in *Russian Mathematics Education: History and World Significance.* Edited by Alexander Karp and Bruce R Vogeli. Singapore: World Scientific, 2010. Translated into Russian as российские традиции в математическом образовании и российские математичес кие соревновании, in российское математическое образование, Москва, мгпу, 2017, p. 209;

_______ (2022) “Russian influences on American mathematics education after 1991: historical roots of changes in extracurricular programs” in *Zeitschrift fur Didaktik in Mathematics*, 53(7), 1605-1616, DOI 10.1007/s11858-021-01293-8;