The work of Judita Cofman on didactics of mathematics

Резиме Јудита Цофман је студент прве генерације ђака уписаних на студије математике и физике на Филозофском факултету у Новом Саду, као и први доктор математиких наука на Универзитету у Новом Саду. Њена докторска дисертација, као и њен научни допринос до краја 70-их година прошлог века припадају области коначних пројективних и афиних равни и радови у оквиру ове теме објављени су у престижним међународним математичким часописима. Циљ овог рада јесте да скрене пажњу на допринос Јудите Цофман у области дидактике и наставе математике кроз рад са младим математичарима, чему је посветила други део свог живота и научног рада. Посебно су истакнута њена размишљања и ставови о важности наставе и научне области геометрије настали на основу њеног богатог искуства као математичарке и професорке у раду са средњошколцима. Abstract Judita Cofman was the first generation student of mathematics and physics at Faculty of Philosophy in Novi Sad, Serbia, and the first holder of doctoral degree in mathematics sciences at University of Novi Sad. Her PhD thesis as well as her scientific works till the end of 70's belongs to the field of finite projective and affine planes and the papers within this topic were published in prestigious international mathematical journals. The aim of this paper is to draw attention to Cofman’s contribution in didactics and teaching of mathematics through the activities with young mathematicians, to whom she devoted the second part of her life and scientific work. Her reflections on importance of geometry based on her experiences with high school students are specially pointed out. AMS Mathematics Subject Classification (1991): 01A60


Introduction
Mathematician Judita Cofman 2 was born in Vršac on 4th June 1936. She came from a well-known and formerly wealthy family of Zoffmanns whose arrival in Vršac is put at the time of the reign of Maria Theresa of Austria (1717-1780) where they came from a German region with a strong beer brewing tradition (Kuručev, D., 2007;157-163). Although the Zoffmanns were originally German, they gradually adopted Hungarian identity, so Judita declared herself as a Hungarian from Vojvodina. An environment of material and cultural wealth marked the life of Judita's father Ákos Zoffmann . Having received wide education in Germany, he became a great expert in beer brewing, and wine growing and storing industries. Judita's mother Lujza (1910Lujza ( -2000, born Kozics, comes from a Hungarian family of lawyers from her father's side, while her mother came from Vršac. Lujza's grandfather was a mathematics teacher at the Vršac Grammar School, and her uncle was the mayor of the town of Vršac. Despite the incertitude and horrors of World War II, Judita enjoyed a happy childhood at her family home.
She went to primary school in Hungarian and later to Serbian Grammar School in her hometown.
The family home, full of love and harmony, installed in her a great feeling that work, study, reading, as well as the knowledge of foreign languages are necessary preconditions for success in life. Besides being gifted for mathematics, Judita had a talent for languages, so, besides her mother tongue of Hungarian and the official Serbian language, as a child she learned German, Russian and, which was rare at that time, English. She later learned French and Italian. Judita Cofman's PhD thesis as well as her scientific work till the end of 70's belong to the theory of finite projective planes, Möbius planes and Sperner's spaces a very up-to-date and lively mathematical field, closely related to algebra and group theory. Her results within this topics were published in prestigious international mathematics journals (Mathematische Zeitschrift, Archives Mathematica, Canadian Journal of Mathematics, for example) and were presented at high ranking conferences devoted to these field of projective geometry. Her results complemented the results of many great geometricians of the early 20th century on the one hand, and on the other, the active follow-up and advancement of certain subfields of projective geometry rest upon her results (Nikolić, A., 2012 Prvanović (1956), and it was for the field of geometry. Judita Cofman was appointed as her Assistant in 1960. Judita had been the best student of mathematics for generations. Her younger colleagues, later professors at the University of Novi Sad, Irena Čomić and Danica Nikolić Despotović, remember that students had great respect for professors but also some kind of fear for them. Despite all efforts of professors to travel from Belgrade, they were not always accessible to their students. The professional literature in Serbian language was still insufficient at that time, and students could not use foreign titles because their knowledge of English, French, German or Russian was modest. The only person who was able to answer at any moment a variety of questions by curious students was Judita Cofman. As her knowledge of foreign languages was high, she was almost the only one among the students who could use German, English and Russian textbooks and widen her knowledge of mathematics, which she used to unselfishly share with her colleagues. They felt that she knew all there was to know about mathematics! As soon as she was made Assistant, in collaboration with students she She was the editor of journal Hypotenuse whose contents was in relation with her seminars and these camps. She also gave regular seminars in Germany for teachers of mathematics and students preparing to become teachers, as well as lectures within didactics  3. Mathematics is one of the earliest scientific disciplines, an important segment of human cultural heritage. This fact must be reflected on the teaching of mathematics: it is advisable to draw pupils attention, whenever an opportunity arises, to their historical background. The history of geometry is an important part of the history of mathematics, not only because geometry is one of the oldest branches of mathematics. The importance of geometry mostly lies in the fact that there were several major problems in this field, starting from the Ancient Greek age, which could finally be solved only in the 19th century. The solutions to these problems had been sought for for ages; the attempts led to a series of new discoveries and contributed to a further development of the entire science of mathematics. One of the famous problems of geometry was the so called Delian problem of doubling the cube. 4. Teaching of geometry can also play a useful role in illustrating the achievements in the most current fields of mathematics.
5. The knowledge gained in the study of geometry can contribute to a better understanding of the phenomena from different fields of natural sciences.
6. For teaching of geometrics to be successful, teaching personnel must have a solid knowledge of this subject. However, not only at schools, but also in university courses and other pedagogical institutions for training future mathematics teachers, there is a tendency of neglecting the study of geometry. This fact can lead to a drastic deterioration in the level of geometry teaching at schools. What is needed is an effort at elevating the respectability of geometry with the students of mathematics.

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In her pedagogical work, Judita Cofman had the ability to raise simple mathematical truths onto a higher level and turn the elementary into a science. She knew historical genesis of each problem, where it originated from and how it was solved throughout history. She deemed that an important reason for teaching mathematics in schools was to promote independent pupils' thinking processes and powers of observation. 8 Throughout their schooling, pupils should be made aware of the links between various phenomena and they should be given the opportunity to discover these links on their own whenever this is possible. Moreover, pupils should be motivated to search for interdependence between seemingly unrelated topics. How can this be achieved? The key answer to the above question and generally to teaching of mathematics she gave in several papers and five books dedicated to mathematics teaching methods, which represent an outstanding approach to solving non-standard mathematical problems. Her historical approach to science and mathematical problems was the focus of her books which feature problems based on famous topics from the history of mathematics and a selection of elementary problems treated by eminent twentieth-century mathematicians.