Euler and the Bernoullis: Learning by Teaching - Teaching the Way Euler and the Bernoullis Learned

The foregoing lengthy example of teaching and learning is one the author has used with much success in both precalculus and calculus classes.  First, it is necessary to decide how deeply into the subject to go.  For a precalculus class, obviously the differentiation arguments advanced by Bernoulli would not be appropriate.  But the symmetry arguments are exactly right for such a class (providing that the differential equations can be eliminated or explained in some other manner).  If the historical context, with intriguing asides on personalities and some historical context are first provided, the topic comes to life.  The immediacy and fascination which we wish to inspire in students can be achieved more easily with an honest revelation of the uncertainties, disputes, and corrections of the innovators of the field. Indeed, it seems unlikely that students will ever be inspired to become innovators who think for themselves if mathematics is always presented to them as a brilliant and elegant fait accompli.

Consider how the questions asked by students today recapitulate the history of mathematical discovery far better than the textbooks generally do.  We do a disservice to students when we fail to inform them that the very same questions troubled the great minds who discovered the answers.  For example, when first presented with the concept of negative quantities, students often respond with incredulity or skepticism:  “How can a number be negative?  What can this mean?”  Though Girolamo Cardano (1501–1576) in 1545 laid out the rules for negative numbers, it would be more than a century before coefficients in equations were explicitly permitted to take on negative values, in the work of Jan Hudde (1628–1704) in 1713, in De reduction aequationem. [10]  (Bernoulli met Hudde in 1681; see Note 10.)  Finding ways to answer these questions is just as important to the development of the mathematical intuition of the teacher as it is to the growth of the learner. 

The questions faced by Leibniz, by Jakob and Johann Bernoulli, and by Leonhard Euler were the same basic questions students face today: 

• What are these objects we are investigating?  
• What rules do they follow, and how general are those rules?  
• What are the underlying principles that unite these investigations?  

They found answers through discussion in journals and letters, and through personal, one-on-one tutelage.  In so doing they established mathematical analysis and set it on a firmer foundation. 

Few students today will be lucky enough to find tutors like Johann Bernoulli, and we would like to boast that university education in mathematics has improved tremendously.  By examining not just the conclusions of the formative minds of mathematics, but how these were reached, might inspire students today not just to read mathematics but also to discuss it, argue about it, and be inspired by it.

Note 10. M.B.W. Tent imagined a discussion between a young Jakob Bernoulli and Hudde, in which Hudde used the tried and true example of credits and debts to show that negative quantities are just as ‘real’ in applications as positive ones.  [12, pp. 61–63]