To my experience, the majority of mathematicians are studying mathematics for its inherent beauty, not for its power in applications. For this reason, in general, it cannot be expected that a mathematician has any interest or, related, any capacity

of applying mathematics to problems from natural sciences, engineering. On the other hand, the majority of calculus students, at least in the US, majors in engineering, physics, business, biological science and social sciences, not in mathematics. Of couse, it is hard to produce good results, for such a mixture of teachers and students . The average calculus student is not yet capable of comprehending mathematical beauty, but would be more motivated learning mathematics after being exposed to its power in applications.

This problem has a long history. In case of your interest, I attach a recent talk in this matter. In short, the only reasonable way out of this problem I see is the adoption of a teaching method, similar to Otto Toeplitz's "genetic method," at least on the level of text books. The latter approach is also followed in my calculus book. Best regards, Horst Beyer

http://horstbeyer.eu.pn/Morelia2013.pdf

thank you both for your thoughts and ideas.Horst Beyer recognized the problem fully insaying that the majority of mathematicians are only studying it for its beauty not for its relation to applications in science and engineering.

I hope, present and future Maths teacher realize this point and concentrate on applying Maths to solve many of our technological problems and also explain fully to their students why particular Maths technique (i.e. differentiation, Integration, Matrices, Laplace transforms.....etc.) has to be used to solve certain applied science or engineering problem.

When we were students in secondary school or in a university, we were taught Maths and Calculus as an abstract subject, it has no life. That is to say there was little applications and problems solving using calculus. the teachers did not expose us to the mathematical approaches to solve physics and engineering problems as it should be. Actually, Math teachers should educate themselves about the engineering application by reading physics and engineering textbooks before embarking on the teaching profession.

To my opinion, Fakhri's remark that the average teaching of mathematics "has no life" is the key observation. Such teaching can only suceed if the students are already interested in mathematics. To my experience, the majority of students are not, but have to go through basic mathematical training for application in other mathematical sciences. A lack of interest results in a lack of energy for studying a subject and usually produces suboptimal results (if not hate for the subject).

Below, I add a citation from the introduction to Otto Toeplitz' "Calculus: A Genetic Approach," which indicates a way out of this problem. The way out of this problem is to "create interest," which, to me, appears to be the most challenging task in teaching.

"Regarding all these basic topics in infinitesimal calculus which we teach today as canonical requisites, e.g., mean-value theorem, Taylor series, the concept of convergence, the definite integral, and the differential quotient itself, the question is never raised 'Why so?' or 'How does one arrive at them?' Yet all these matters must at one time have been goals of an urgent quest, answers to burning questions, at the time, namely, when they were created. If we were to go back to the origins of these ideas, they would lose that dead appearance of cut-and-dried facts and instead take on fresh and vibrant life again."