Not only is it convenient to express the various ways of teaching calculus in an innovative way in which the student can learn and break the fear that mathematics inspires. It is also necessary to direct some recommendations to the different colleagues who teach this curriculum unit so that together we can take this knowledge to other forms of teaching and new forms of learning.
I just take a particular example to make you realize how important it is to keep the technicality and the terms in the right way, what you can vary are new ways to teach, other tools, but we must keep an eye out for the new forms of expression we want to adopt.
One of the ways we must acquire to teach infinitesimal calculus is to assume that the topics addressed by this branch of mathematics must be treated more analytically than systematically.
When I refer to teaching in an analytic way, it is that perhaps the traditional way of teaching calculus encompasses a direct way of addressing exercises to problems in the order of graph functions, limit and continuity of a real function, derived and integral, successions, differential equations, all these topics are addressed in many of our universities in a systematic and direct way, believing that with this form simplicity is achieved, And really what you get is repetitive and thoughtless thinking on the part of the student’s learning.
When Newton and Leibniz addressed the initial problems that led to infinitesimal calculus they did so in a thoughtful way, reason why they were alive and being the propellers of infinitesimal calculus they would like to see in us as teachers a way of teaching more analytic, without leaving aside the accuracy implicit in mathematics.
A clear example to develop a set of introductory ideas for when we as teachers want to start calculus classes, is to orient our introduction by telling our students that calculus is part of the mathematical and modern advancement with which many advances in today’s engineering and technology could be made.
When we say that calculation is lived in the most modern of our daily lives, we must not only limit ourselves to the advancement of engineering and technology, but also establish importance in the order of science, since for each investigative process there is a possibility to help us with the fundamental principles on which the infinitesimal calculation is based.
This analytic characteristic in which I want to enclose the teaching of calculus allows me to think that the cognitive characteristics that we manage to stimulate in the student is the one that in the end will give us the seeds to be able to break down the barriers of the traditional systematic that the different calculation topics are being addressed.
How can we observe that we are not really teaching correctly and that we are practically continuing a traditional and erroneous form of teaching?
I invite you to take this very simple analytic exercise: Suppose you explain the class of derivative, and you fellow teachers believe that your students have mastered all aspects because they have solved all the possible derivative exercises that exist in the exercises proposed in the final part of each chapter in the calculus books, or simply solve the exercises of guides proposed in the faculties of the universities where their classes are taught.
It is at this moment that a series of analytic questions must arise from you to know if you have really taught the subject correctly, these questions must be as simple as possible, for example:
 What is the derivative of a real function?
 It really is getting every time you derive a real variable function?
 What other possible applications can your students get to the derivative other than those in the books?
This practical exercise can be useful for any topic that is addressed, not necessarily because the student manages to solve a calculus exercise means that he/she is learning well, It is necessary to combine an exact learning of the calculation with analytic characteristics of that teaching process - learning
Finding other teaching alternatives can affect the student’s correct learning
Many of us make the mistake of wanting to please the student by trying to stick to a teaching that leaves aside the technicality, many times it is a matter of looking for a language more attached to the needs of the student, until it can fall to the point of an erroneous teaching. If the vocabulary of us as teachers is used to teach out of the laws of mathematics it is most likely that our students will learn badly.
It pains me to say it, but I accidentally walked through the halls of the university where I teach my calculus classes here in venezuela, and I saw how a fellow teacher for wanting to explain in a simple way in which students could understand him used a false terminology.
In that case he called the increase in the axis x (delta x) slope, and when he explained the derivative by definition he referred to slope rather than increase by x.
These are simply a series of considerations, recommendations and reflections on how we can break with the classical paradigms of calculus teaching, and how we can implement some strategies so that our students can approach learning from an analytical perspective.
I hope this content will be of the taste and benefit of all, hoping very soon to be able to continue sharing content of this type says goodbye to you your friend @carlos84