Science and Technology
Document
Education
Series No. 35
Mathematics,
Education, and Society
Reports and papers presented in the Fifth Day Special Programme on “Mathematics, Education, and Society” at the 6th International Congress on Mathematical Education Budapest, 27 July - 3 August, 1988
edited by Christine Keitel (Chief editor) and Peter Damerow, Alan Bishop, Paulus Gerdes (Chief organizers)
Division of Science Technical and Environmental Education ED-89/WS/94
UNESCO
Paris, 1989
for all of us, one in which our belief in the value of discussion and group work for learning mathematics has been reinforced. References Association of Teachers of Mathematics (1986). These have workedfor us at A-level. A.T.M. Burton, L. (Ed.) (1986). Girls into maths can go. London: Holt, Rinehart and Winston. Buxton, L. (1981). Do you panic about maths? Heinemann Education. Harvey, R., Kerslake, D., Shuard, H., & Torbe, M. (1982). Language teaching ana’ learning: 6. Mathematics. Ward Loek Educational. Mason, J., with Burton, L., & Stacey, K. (1982). Thinking mathematically. Reading, MA: AddisonWesley.
Why ICMI? The 19th century saw the birth of modem educational systems in Europe including elementary education. The movement of building such educational systems increased in the last three decades of this century. Until the beginning of the 20th century, the teaching of mathematics in the elementary schools was related to the well-known three Rs of the year 1828, where one “r” refers to arithmetic. The teaching of mathematics, especially in the elementary schools, had put emphasis on the achievement of mechanical skills. ICMI was established to support better education of mathematics at all levels and to secure public appreciation of its importance. This means that the mathematicians, who met in Rome in 1908, were of the opinion that the development of mathematics depends on the development of mathematics education. ICMI and Mathematics Education Until the Year 1957
ICMI and the Crisis of Mathematics Education: What Kind of Reform is Needed? George Malaty Mathematics Cultivation and the Establishing of ICMI The cultivation of mathematics in the European societies has been affected by different factors. One of these factors was the invention of printing. Another important factor was the progress of transportation, which facilitated the moving of mathematicians and their product from one society to another. The conclusive factor, which was also affected by the abovementioned two factors, was the education of mathematics. The result of mathematics cultivation in the European societies, was the birth of local mathematical societies to facilitate communication between the increasing number of mathematicians in these societies. The spread of mathematical centers crossed the Atlantic to establish the American Mathematical Society in 1888 after the Hamburg society in 1690, the Amsterdam society in 1778, the Moscow society in 1864, the London society in 1865, the society of France in 1873, the Edinburgh society in 1883, among others. To facilitate the direct communication between mathematicians all over the world, in 1897 the First International Congress of Mathematicians was held in Zurich with 204 participants. The Fourth International Congress of Mathematicians was held in Rome in 1908 with 535 participants. By a resolution of this congress, a central committee of ICMI was established with members from Germany, England, and Switzerland.
In the beginning of this century, ICMI was able to organize four international congresses in 1910, 1911, 1912, 1914 and to reestablish a central committee in 1928. Until the year 1957 the most formal activities of ICMI can be seen in the short time of the years 1910 to 1914. The work of ICMI at that time was oriented intensively towards the emphasis of understanding of mathematics instead of the mechanical skills. This active work of ICMI received an echo in different places all over the world. For instance, in Finland in 1913 the mathematician and distinguished pedagogue L. Neovius-Nevanlinna wrote his book Havainto-oppi or Perception, in which he mentioned the new orientation in teaching mathematics. Despite the fact that the work of ICMI was interrupted by the First World War, the work towards the understanding of mathematics continued to form a general tendency in mathematics education in the 30s. Different specialists in different countries have done research, which showed that the understanding of mathematics leads to better attainment. They had gotten support from the gestaltists, especially after W. Kohler’s work The mentality of apes in 1924. One of the factors, which decreased the importance of drills and gave a space for emphasizing the meaning, was the revision of E. L. Thomdike in 1932 to his law of exercise to make it unimportant in the learning process. In other words, finally the theory based on the behavior of cats made the place to a theory based on the behavior of more intelligent animals, the apes. The work of the specialists in methods of teaching mathematics continued in the same direction of emphasizing the understanding and found support from other educators. In the fall of 1956, B. Bloom and
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others wrote the result of their classification of educational objectives in the cognitive domain, which had been appreciated by the specialists in teaching mathematics. ICMI and Mathematics Education Since 1957 Between the world wars, mathematicians were able to hold 5 international congresses. The last was their 10th congress, in Oslo in 1936 with 487 participants. In Cambridge, Mass., USA in 1950, the first postwar International Congress of Mathematicians was held with 1,917 participants. In 1952, the newly formed IMU reestablished the ICMI. Not until 1969 did ICMI start the new intemational ICME congresses after the “new math” movement took place in different countries. The first three congresses mainly dealt with the new curricula and the ongoing projects. At the time of the “new math,” the main problem, which specialists tried to face was the public opinion. The explanations of the specialists and the advertisements and publicity of the publishers were not able to face the anger of the parents towards the “new math” textbooks. The introduction of sets and Venn-diagrams in elementary schools caused the public press on specialists to choose the slogan “back-to-basics” in 1978. The declaration of this slogan was the declaration of the crisis in mathematics education. Up to the present, the crisis has continued for ten years. This period is not long in the history of mathematics education, but how long will it continue? It is difficult to give an answer to this question. ICMI-The Crisis and the Reform
The Interests of the 80s and the new Slogans In the 8Os, most of the specialists have been interested in one or more of the following fields: computer education, computer as a tool, problem solving, games, modeling and visual geometry. These interests are not new: new is the attempt to have one of them as a leading tendency, which can get the support of the media and the public. These activities can only be positive if they can bring understanding of mathematics as a starting point to fulfil1 the other direct and indirect objectives of teaching mathematics. In the case of ICME 6, the most expected slogans are “Mathematics for all” and “Everyday life mathematics.” These slogans can finally get the support of the public and the media, but they lead to no mathematics. Mathematics by its nature is not that of the socalled “Everyday life.” Mathematics can be for all, mathematics can be a simple and lovely subject when the students can understand it. The Kind of Work Needed Since 1957, with the increase of the role of the media, many slogans have been invented. Slogans cannot belong to science. One of the main properties of science is continuity, therefore we have to study the work of the specialists up until 1957, the work which had been interrupted by the “new math” and “back-to-basics” movements. The back-to-basics movement did not take us back to that time, but to the time before the establishment of ICMI in 1908. While the specialists in methods of teaching mathematics tried to place emphasis on understanding until 1957, the dissertation of L. Saad in 1957 showed that the teaching of mathematics was far from fulfilling this objective. The number of mathematics educators now is much higher than in the year 1957. Their work can help in fulfilling this objective. ICMI and the Other School of Education The work in mathematics education in the USSR presents a special case, from which we have to learn. Mathematics education in this society has been a continuous work since Leonard Euler came to St. Petersburg in 1730 to work at the Em2eror’s Academy and teach mathematics in the Academy’s school (gimnazii). One of the specific characteristics of mathematics education there lies in the structure of the curriculum. This curriculum is organized to introduce gradually to all students the abstract, deductive nature of mathematics from the 1st grade on and the
Since 1980, the participants of ICME attend the congress to see if there will be a new slogan or not. The use of slogans to describe and show a new tendency, became the major expected outcome of ICME’s work. On the one hand, this reflects the need of security. On the other hand, this shows how we become users of an unscientific tool which belongs to the media and marketing activities, the slogans. The slogans’ system is wrong. The use of the slogans “new math,” “the revolution of mathematics,” “radical changes,” etc., affected not only the general opinion to support. the changes in the 6Os, but also affected the work of mathematical educators and teachers of mathematics all over the world. The main challenge for ICMI is how it can lead a reform of our concepts of the work in the field of mathematics education, a reform which can enable us to work again, without noise, towards better education in mathematics and thus find a way out of the crisis.
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axiomatic structure from the 6th grade on. Gifted students receive special education from the 7th grade on, mathematical clubs and competitions take place from the 4th grade on. As a result, mathematics is the most popular subject. To show how successful they are in teaching mathematics there, let us take a look at the following example: The 4th grade student of ten years of age is asked in one of the textbook exercises to prove that the vertically opposite angles are equal. To reach such a level, a gradual preparation is begun in the 1st grade. The specialized methodologists of the first three grades of the Academy of Pedagogical Sciences of the USSR, M. I. Moro, M. A, Bantova, G. V. Bel’tyukova, A. S. Ptshyolko, and A. M. Pyshkalo wrote the textbooks of these grades about 20 years ago, but they are continuously developing these books on the basis of continuous experimental work. On the other hand, their textbooks are further developed versions of the previous textbooks. In 1967, a reform committee under the leadership of Kolmogorov, refused the kind of reform taking place in the Western countries. For instance, the reform which the committee accepted included the use of the concepts of Sets from the 4th grade, but not the writing of a special topic in Sets. They accepted the use of more of the transformations approach in geometry, but within a more rigorous axiomatic structure of the Euclidean geometry from the 6th grade. The committee did not accept the unification of geometry and algebra in one textbook, to keep to geometry its lessons and importance. The continuous improvement of mathematics education under the supervision of the most well-known mathematicians all over the country is supported by public appreciation of mathematics and its structure. The appreciation of the structure of mathematics and its beauty is one of the main factors which lead to the resolution of 1985 to separate the computing education in the 9th and 10th grade from the teaching of mathematics to keep to mathematics its importance. The appreciation of mathematics and its structure is a result of mathematics cultivation, which has been affected by the reform of education of the common
people at the beginning of this century. The teaching of mathematics to these common people was based on the work of the distinguished mathematician and pedagogue A. P. Kiselev (1852-1940). Kiselev’s phenomenon is a result of the long cultivation of mathematics in this society, his textbooks of the end of the 19th century have been used until the 60s of the 20th century. Kiselev had continuously revised his textbooks to some of about 40 editions. After his death, his books were revised again. Until the present, these books form the basic reference material for building the curricula and writing the textbooks. On the other hand, these textbooks have relations with the previous textbooks. In conclusion, the education in mathematics is a continuous work, which has not been influenced by any slogan or mode. The question now to ICMI and also to ICME is: Can we learn something from the other main school of education, which represents the USSR school?
References Kolmogorov, A. N. (1986). 0 skaliarnykh velitshinakh. Matematika v shkole, 3, 32-33. Mat Neille, H. (1970). Mathematical societies and publications. Encyclopedia Americana, 18, 428430b. Malaty, G. (1988). What is wrong with the “back-tobasics” movement and what was wrong with the “new math”movement? International Journal of Mathematical Education in Science and Technology, 1, 5765. Neovius-Nevanlinna, L. (1913). Matemaattinen havainto-oppi. Jyvaskyla: Syd&r-Suomen kirjakauppa. Saad, L. (1957). Understanding in mathematics. Unpublished Ph.D. Dissertation, University of Birmingham. Shtokalo, I. Z. i dr. (1975). Istoriya matematitsheskovo obrazovanijya v SSSR (red.). Kiev: Naukova dun&a.
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