(Courtesy of the Lilly Library, Indiana University. Title page of the 1684 volume of Acta Eruditorum, the journal Leibniz created, published, and made famous with his first work on calculus, which appeared in this volume. Through formal and informal tutoring, by way of letters and casual discussion, in the absence of input from universities, they encouraged, challenged, instructed, refuted, and inspired each other.įigure 1. In the exciting days of the late seventeenth and early eighteenth centuries, after Gottfried Wilhelm Leibniz had opened the floodgates with a short, difficult article on calculus in his own journal, Acta Eruditorum, two generations of Bernoullis and two generations of Eulers were discovering, and teaching each other, the mathematics that made the modern age possible. This was just as true when modern analysis was getting its start.
And the best way to teach it is to explore with students how the original innovators learned and taught mathematics by investigating its rich history. Today, the best way to learn mathematics is to teach it. Another reason is the need to find ways to recast material to make it accessible to learners – to meet them where they are in the process of discovery. A common refrain among teachers of mathematics is, “I never really understood this material until I taught it.” Part of the reason for this, of course, is that the repetition and the need to know the material well enough to answer questions solidifies the subject in the mind of the teacher. Teaching and learning are part of the same process. True, he was very busy and so refused flatly to give me private lessons but he gave me much more valuable advice to start reading more difficult mathematical books on my own and to study them as diligently as I could if I came across some obstacle or difficulty, I was given permission to visit him freely every Sunday afternoon and he kindly explained to me everything I could not understand … and this, undoubtedly, is the best method to succeed in mathematical subjects. I soon found an opportunity to be introduced to a famous professor Johann Bernoulli.
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