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The United States of America was honored to be invited to make one of the National Presentations at the 12th International Congress on Mathematical Education in Seoul, Korea.

It is a great pleasure for me to have the opportunity to address you, during this opening ceremony for the 12th International Congress on Mathematical Education, in my capacity as President of the International Mathematical Union, or IMU.

I am delighted to be here, to open the 12th International Congress of Mathematics Education—to be honest, it is a moment I have been looking forward to for more than 4 years. Our community is very fortunate to have attracted a conference bid from Korea, and our Korean friends are already proving to us that we made a very good decision to accept their bid.

I would like to express my utmost gratitude to His Excellency Lee Myung-bak, the President of the Republic of Korea for preparing a welcoming message for us despite his busy schedule.

First of all, congratulations on the opening of the 12th International Congress on Mathematical Education. I am glad that this important math event is being held in Korea this year. Also, it is a great pleasure to welcome math education researchers and math teachers from more than 100 countries. With the aim of transforming Korea into a nation of great science and technology capacity, and a nation of outstanding human talent, the Ministry of Education, Science and Technology of Korea is focusing on three important points in designing and implementing its policies. The three points are “creativity”, “convergence”, and “human talent”. Creativity enables us to think outside the box, convergence allows us to go beyond the traditional boundaries between disciplines, and finally human talent builds the very foundation that make all these possible.

A wonderful part of the opening session of the ICME congresses is the ICMI Awards ceremony. The 2012 ceremony, which was presided over by Prof. Carolyn Kieran, the chair of the ICMI Awards Committee, was no exception. Congress participants shared in congratulating the recipients of the 2009 and 2011 competitions for the Klein and Freudenthal awards. The Korean Minister of Education, Science, and Technology, the Honorable Mr. Ju-Ho Lee, did us the honor of presenting each award.

It is very unusual for a mathematical idea to disseminate into the society at large. An interesting example is chaos theory, popularized by Lorenz’s butterfly effect: “does the flap of a butterfly’s wings in Brazil set off a tornado in Texas?” A tiny cause can generate big consequences! Can one adequately summarize chaos theory in such a simple minded way? Are mathematicians responsible for the inadequate transmission of their theories outside of their own community? What is the precise message that Lorenz wanted to convey? Some of the main characters of the history of chaos were indeed concerned with the problem of communicating their ideas to other scientists or non-scientists. I’ll try to discuss their successes and failures. The education of future mathematicians should include specific training to teach them how to explain mathematics outside their community. This is more and more necessary due to the increasing complexity of mathematics. A necessity and a challenge!

I wish in this lecture to reflect on the links between mathematics and didactics of mathematics, each being considered as a scientific discipline in its own right. Such a discussion extends quite naturally to the professional communities connected to these domains, mathematicians in the first instance and mathematics educators (didacticians) and teachers in the other. The framework I mainly use to support my reflections is that offered by the International Commission on Mathematical Instruction (ICMI), a body established more than a century ago and which has played, and still plays, a crucial role at the interface between mathematics and didactics of mathematics. I also stress the specificity and complementarity of the roles incumbent upon mathematicians and upon didacticians, and discuss possible ways of fostering their collaboration and making it more productive.

Mathematics has been recognized and justified to be placed in the prime core of the formal curriculum for general education. In this paper, however, some reflections are made on the national curriculum together with mathematics education in accordance with the tradition of “liberal education.” Liberal education is education for liberal men. The basic education of liberal human being is the discipline of his rational powers and the cultivation of his intellect. It has sustained its meaning and value to be different from the vocational training for the purpose of earning one’s living. But John Dewey differently contends that the vocational training may claim a pertinent candidate to the position playing a role in cultivating the human mind, the intellect (or intelligence). For Dewey, important is not the content of teaching but rather the intelligence in its operation.Intelligence is “equipped” with some properties that are functionally related to the properties of the problematic situation, which they take on the character of “method.” A kind of mental process, “a methodic process,” connecting problematic situations and resolved consequences is what Dewey qualified to be “reflective thinking,” where the intelligence keeps itself alive and activating for its full operation. Then, we would have two different, but closely related tasks. One is (i) the self-habituation of methodic activity; and the other is (ii) the nurturing of children in methods. The curricular device is bound to gratify a variety of different needs and motives. No matter how worth studying mathematics may be, it can never be learnt unless the body of learning materials are so organized that students may cope with its degree of difficulty settled for the teaching purpose. Then contents must be appropriately selected and efficiently programmed on the part of learners. Learnability is prior to the academic loftiness at least in educational situations.

The topic of this paper is mathematical modelling or—as it is often, more broadly, called—applications and modelling. This has been an important topic in mathematics education during the last few decades, beginning with Pollak’s survey lecture (New Trends in Mathematics Teaching IV, Paris, pp. 232–248, 1979) at ICME-3, Karlsruhe 1976. By using the term “applications and modelling”, both the products and the processes in the interplay between the real world and mathematics are addressed. In this paper, I will try to summarize some important aspects, in particular, concerning the teaching of applications and modelling.