## Vojtěch Jarník on teaching

 The quotes below are taken from a number of sources, in particular: [1] V Jarník, Poznámky k otázkám vysokoškolské vyuky, Pokroky MFA 16 (1971), 5-8; [2] B Novak, In Memoriam Prof Vojtech Jarník, Pokroky Mat. Fyz. Astronom. 16 (1971), 1-5; [3] J Vesely, Pedagogical activities of Vojtěch Jarník, in B Novák (ed), Life and work of Vojtěch Jarník (Society of Czech Mathematicians and Physicists, Prague, 1999), 83-94. All the quotes are by Jarník himself except Quote 6:

Quote 1.

This quote is from [3] and was made by Jarník in 1952:

I am extremely fond of lecturing. Especially formerly, when I did not have so many offices and duties, I was a huge nuisance to the students. When one of my lectures was cancelled, whether it was for a holiday or for some other reason, I always tried hard to compensate by delivering it at some other time.

Quote 2.

This quote is from [3] and was made by Jarník in an unfinished text:

A mathematical lecture possesses one characteristic property: A correctly and purposefully performed chain of inferences leads with absolute reliability from the statement a theorem to its proof. But conversely, if we make a single mistake or if we do not maintain reasonable continuation of the chain of thoughts at any moment, the whole proof of a theorem or the solution of a problem collapses.

Quote 3.

This quote is from [1]:

To lecture from notes or without notes: for an introductory lecture I write down at most some points so that I do not forget anything, and also the data for the examples. For advanced lectures I have the text always with me, of course I "extemporize" but I check myself from time to time to make sure I haven't forgotten something I will need later. I also sometimes check the statements of theorems - for example, I formulate an auxiliary result including a complicated auxiliary formula which I will only prove later on. It is of course quite unnecessary to learn the formula by heart - apart from the possibility of a lapse of memory. Moreover, it would be incorrect also from a pedagogical point of view - I recommend to students that they should not memorize such things but first of all that they should understand the connections so that they know which results or arguments are to be used in a particular case, to be able to find it in the literature or, as the case may be, to be able to derive it independently by themselves. Trivial but lengthy transformations of complex expressions occur frequently, too. Such a routine procedure should be run through in the lecture as quickly as possible, and it is also important to check the result by comparing it with the prepared text in order not to be forced some moments later to look for an accidental mistake which occurred somewhere.

Quote 4.

This quote is from [1]:

I cannot help speaking when writing on the blackboard. Naturally, the sketches are rather primitive - I sketch a line and say ten words, I plot a point and say another ten or twenty words. I cannot imagine explaining, for example, the continuity of a function first and then to sketch a figure, or to sketch a figure first and then discuss it. Here it is important that the figure develops in accordance with my developing the thoughts related to it. Mathematics has the advantage that a figure only illustrates certain mathematical relations which could be explained without it - in this respect it is different from many other subjects. The advantage as compared with a book is precisely in the fact that the student sees the genesis of the figure (simultaneously with the genesis of notions or proofs) while the figure in a book is static and the reader must analyze it by themselves to find the procedure by which it has arisen.

Quote 5.

This quote is from [1]:

Introductory lectures on analysis have long since been accompanied by practical exercises. Formerly I led them myself when I had given the lecture. Now this is no longer possible because there have to be several parallel groups. I am rather sorry about this. Exercises classes seem more interesting and challenging for the lecturer than an elementary lecture. If a student suggests a different way of solving a given example to the way I had in mind, I must not "forbid him" to try it. On the contrary, I have to estimate whether the student's way leads to the goal, I must let them proceed along their way and I must be able to help them since sooner or later the student will very probably not know how to go on. Even if the student's method evidently cannot reach its aim, it is often better to let them try it in order to let them (as well as the others) find out where and why it fails. Of course, this requires considerable alertness from the teacher who meets here new problems directly in the teaching process. On the other hand the student penetrates from the very beginning of his studies into the spiritual workshop of his teacher: the latter must consider the problem, sometimes having to make several attempts before they finds the right way. And sometimes the teacher may fail together with their student, and some other time another student gets a lucky idea which the teacher had not found. There is no damage done (just the opposite), provided, of course, the students have recognized the high professional level of their teacher and the teacher does not try to look like an infallible oracle. The teacher can make a blunder even in a lecture (then it is of course completely their own fault). In such a case it is necessary to admit the mistake and not try to hush it up or to comfort the students by giving a plausible half-truth.

Quote 6.

This quote is from [2]:

One of the rare features of Professor Jarník: he was first and foremost a university teacher. He was equally excellent at lecturing both an introductory course of mathematical analysis as an advanced specialist course, and the latter not only from his own field. His lectures had an indescribable atmosphere of an intimate discourse between Mathematics, the reader and the students. Lot of papers and notes have been written about the way Jarník lectured. One essential feature should be emphasized once more. Although each lecture of his was prepared in every detail, it was never tedious or boring, it never reduced to a mere sequence of definitions, theorems and proofs, to the mere epsilon-delta symbolism. The truth is that his lectures always possessed all the necessary mathematical rigour, were delivered with extraordinary clarity, but the rigour in all details corresponded to the level of the students and to the subject. Jarník's art of lecturing consisted not so much in the accuracy of the exposition as in distinguishing the essential from the inessential, in emphasizing important and wider connections, in developing an informal idea of the topic being considered. This ability of his manifested itself especially in explaining complicated proofs which many teachers rather choose to omit. Jarník always first sketched roughly how the proof will proceed, explained in detail all possible problems, indicated the obstacles and frequently even substantiated the structure of the proof.

JOC/EFR November 2017

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