I agree with Qefsere for the most part. I do think giving them a calculator on the other hand is a bad idea unless it is for verifying work. I find students get far too dependent on their smartphones and calculators rather than understanding the mathematics. When they get to the post-secondary level and that is taken away from their arsenal, they are stranded from the discussion.

Now for Calculus itself, as pointed out, the heart of the problem is what students come into a Calculus class with. I do not teach Calculus, but have tutored it, and have taught other theoretic courses that use a bit of it. The biggest problem is in the process of "simplifying" the Mathematics for younger students has made it more and more inaccurate as to what really is going on. Face it, as university educators, most of the time we have to reteach them everything. From functions, to what even is allowed in algebra comes up all the time. I know I have taught 2nd year courses where I would have to teach students what a logarithm or a summation is all over again so they can be discussed more mathematically. Secondary school rarely prepares a student for university level mathematical studies. Why? Some of the answers here on this question are great explanations of this. Ranging from how concepts are taught, to how to interpret some concepts. The biggest thing is a lack of foundation. You will likely find a high school student going out of high school not even knowing what a Mathematician even does, and what theorems are than think it is useful. There is a lot of work to be done on the secondary-school side to avoid inaccurately presenting Mathematics as a "chore". I find when I talk to high school students about Mathematics, they always say to me, "why don't they teach us this stuff?". Biggest things:

1) We need to enforce the ideas of theorems, and proofs. Why is something true!?

2) Why is it important? Why did the problem come up (historically)? How did they solve it?

3) What can they use it for? Most teachers don't know how to answer these questions because face it, most never have been a scientist or have investigated the literature. Though there is a minority that do, which is good.

Hope this helps!

I started to learn calculus when I was 18. The most important thing is, make the maths to be interesting.

The concept of derivative is fundamental and maybe not clearly taught by professors.

A central issue is how mathematics is taught prior to calculus. One issue is the conventional method used: rote procedures used to compute answers to problems posed in particular forms. Part of the reason I know this is having taught classes and tutored students to take pre-College standardized tests like the SATs & ACTs. The former includes a quantitative component which does not even include trig or pre-calculus. However, I've used questions from practice tests to tutor college students. What makes this possible is the ways in which "easy" math questions are asked: they test one's understanding of the underlying logic/concepts, not how one can apply "algorithms" they learned to calculate answers to standard formalizations of questions.

The reason this is effective is because students typically do not understand the underlying concepts/logic and are not often taught that such a thing exists. Nor are they often taught anything that appears to them to have any relevance. Many students I've tutored use pre-calculus textbooks that include a chapter or two on linear algebra. For anyone who has taken an linear algebra course in college, they'll know that this is a conceptually complex subject (the computations in the problems are usually simple arithmetic). It is probably more essential than two semesters of calculus (at least the way calculus is typically taught), but there is no possible way to teach anything useful to pre-college students who have yet to take any course (with the possible exception of proofs in geometry) that teaches mathematical concepts rather than question-specific procedures. Worse still, many such procedures are almost completely useless as they are designed so that IF the student takes calculus they can e.g., evaluate limits using algebraic manipulations if plugging in the limit would result in division by 0.

Then comes calculus itself. The intuitive notion of infinitesimals that motivated the better part of the development of the calculus was replaced with the epsilon-delta formulation of limits that Weierstrauss developed. This was essential at the time, because a sufficiently rigorous definition of limits was needed for analysis. However, for the past ~40 years Robinson's work on infinitesimals has provided a sufficiently rigorous and VASTLY more intuitive foundation for the calculus.

And, not only do we teach outdated integration methods, but we spend a great deal of time teaching how approximations of these methods (different geometrical shapes that can be used to "approximate the area under a curve"). Why?

"we teach the Riemann integral. Then, when the necessity of integrating unbounded functions arise, we teach the improper Riemann integral. When the student is more advanced we sheepishly let them know that the integration theory that they have learned is just a moldy 19th century concept that was replaced in all serious studies a full century ago.

We do not apologize for the fact that we have misled them; indeed we likely will not even mention the fact that the improper Riemann integral and the Lebesgue integral are quite distinct; most students accept the mantra that the Lebesgue integral is better and they take it for granted that it includes what they learned. We also do not point out just how awkward and misleading the Riemann theory is: we just drop the subject entirely."

(Thomson's The Calculus Integral; see classicalrealanalysis.com)

We teach a computationally complex method of integration that will mostly be replaced IF a student continues her mathematical studies, and a logically/conceptually difficult foundation for analysis, the e-d definition of limits, that was certainly justifiable years ago but has not been for several decades.

We teach procedures, not math, so that students can learn more math so that IF they go onto more advanced topics they will, after much re-learning of the same subject matter they were "taught", realize the fascinating, diverse, and innumerable mathematical applications and topics.

If we taught algebra through statistics, used logic and set theory which lend themselves to immediately relevant subject matters rather than logarithms (invented, after all, to be calculators before calculators existed) or matrices, then we would provide a firmer foundation for calculus as it SHOULD be taught and, at the same time, ensure students learned mathematics rather than be able to say that they took mathematics before college.

I think than could be important if the students known the history of the calculus, ¿Why is very important in the mathematics? ¿What is the objetive of the calculus?, calculus is not only integrate and derivate equations

I think there is a lack of geometrical intuition and insufficient practice with algebraic manipulations these days. Students have difficulty visualizing things in both 2 and 3 dimensions, there is a lack of knowledge of how to do "proofs" based on geometrical figures even in Euclidean geometry. In my view, it's not their fault. It's the system. Students were let down by a dissolution of basic fundamental ideas in geometry, the unsystematic study of Euclid's elements, the lack of proofs, and insufficient practice with basic algebraic operations. In my calculus classes I begin with a very quick review of plane trigonometry, then move on to limits (intuitively and sometimes rigorously) etc. All this can be gathered in my "ABC's of Calculus" book, pirated copies of which can be downloaded almost anywhere on the web. Students must learn or re-learn pretty much all of the appendices in my book in a "crash course" that can take them about a week. They do it, too. I have complete confidence that anyone can learn Calculus who wants to learn it, a lot of it is in the preparation, and a lot of it is in the motivation by the textbook author or lecturer. One can learn how to use the tools of calculus without really understanding it, much like one can drive a car without understanding how it really works. Anyhow, this is a complicated question, one that all teachers grapple with all the time but, somehow, we manage to get enough through to create future scientists and geniuses.

Many studies have demonstrated that students’ difficulty in understanding calculus are caused by their weak understanding of functions (Dubinsky et al., 1992; Tall & Vinner, 1981; Williams, 1991) and the inability to use functions to reason and represent relationships (Carlson et al., 2002; Monk & Nemirovksy, 1994; Thompson, 1994)

New learning technologies such as the graphing calculator have gained acceptance in the mathematics education field. Technology can help develop understanding of abstract mathematical concepts through visualization and graphic representation. This will increase students’ competence in obtaining sufficient knowledge of mathematics.

I agree with Qefsere for the most part. I do think giving them a calculator on the other hand is a bad idea unless it is for verifying work. I find students get far too dependent on their smartphones and calculators rather than understanding the mathematics. When they get to the post-secondary level and that is taken away from their arsenal, they are stranded from the discussion.

Now for Calculus itself, as pointed out, the heart of the problem is what students come into a Calculus class with. I do not teach Calculus, but have tutored it, and have taught other theoretic courses that use a bit of it. The biggest problem is in the process of "simplifying" the Mathematics for younger students has made it more and more inaccurate as to what really is going on. Face it, as university educators, most of the time we have to reteach them everything. From functions, to what even is allowed in algebra comes up all the time. I know I have taught 2nd year courses where I would have to teach students what a logarithm or a summation is all over again so they can be discussed more mathematically. Secondary school rarely prepares a student for university level mathematical studies. Why? Some of the answers here on this question are great explanations of this. Ranging from how concepts are taught, to how to interpret some concepts. The biggest thing is a lack of foundation. You will likely find a high school student going out of high school not even knowing what a Mathematician even does, and what theorems are than think it is useful. There is a lot of work to be done on the secondary-school side to avoid inaccurately presenting Mathematics as a "chore". I find when I talk to high school students about Mathematics, they always say to me, "why don't they teach us this stuff?". Biggest things:

1) We need to enforce the ideas of theorems, and proofs. Why is something true!?

2) Why is it important? Why did the problem come up (historically)? How did they solve it?

3) What can they use it for? Most teachers don't know how to answer these questions because face it, most never have been a scientist or have investigated the literature. Though there is a minority that do, which is good.

Hope this helps!

I believe the main issue really lies in the limit definition, most students find difficult to understand it, at least in our educational system.

On the other hand, the use of technology, even when can be helpful, privates the student of the neccesary habilities working with functions and algebraic expressions, and so the ideas developed in calculus lessons take longer to be accepted and incorporated to their arsenal of instruments for work.

As a student, having a positive attitude and interest in the subject helps a lot, i.e. "I can do it spirit". Of course, some of the concepts are hard to grasp especially to beginners, but once the teacher approaches the subject in a way that makes it interesting, fun and interactive to the students, then surely, understanding the subject content becomes easier. I agree that the limit of functions among other things is hard to comprehend, but once it's well introduced, then the subsequent material should be easier to grasp and a lot more fun. I hope this helps :).

In my experience teaching Calculus and other Mathematic tools at secondary and University levels, the main problems that many students have in understanding not only Calculus but other Mathematic tools are, among others, the following:

1- Bad preparation of the students in primary and secondary levels, particularly in Mathematics

2- Wrong teaching methods (Mechanical reasoning in the study of Mathematics) at these two levels

3- Prioritization of Mathematics study through the realization of multiple exercises instead of emphasizing the understanding of the main concepts, principles and properties of Mathematics by explaining why these concepts, principles and properties are important. The consolidation of the understanding of the main concepts, principles and properties of Mathematics should be done through the realization of a limited number of practical exercises that helps students in the fixation of these important elements.

5- Lack of qualified teachers at these two levels and/or the use of updated methods for the teaching of Modern Mathematics demotivating students in the study of Mathematics

6- Lack of interest in the study of Mathematics by most of the students at these two levels due to their misunderstanding of the main concepts, properties and principles of Mathematics.

7- Insufficient time dedicated by the students for the study of Mathematics at these two levels.

8- Use of non-appropriate books for teaching Mathematics

I think it is a matter of concentration; are they willing to learn it or not?

Some responses to others I hope will be understood as complementary, not contradictory:

First, with respect to the comment about understanding functions, this is absolutely true (I used to tell students that the practically the whole of mathematics can be thought of in terms of functions acting on sets). I'm not qualified (either with sufficient experience or familiarity with the literature) to comment on the efficacy of incorporating technologies into "traditional" curricula, but I have little doubt this can be effective. I think, though, that there are two problems that technology can't address adequately.

One problem is the way math is taught, including functions. Students are taught mechanically. Adding mechanical tools to mechanical instruction won't help until the pedagogical approach changes.

Currently, too much of pre-college mathematics is a mess of traditional approaches, maximizing exposure to mathematical subjects (rom permutations to matrix multiplication) that are utterly useless or nearly so, preparing students for later calculus courses they won't ever take, and accomplishing this by teaching how to solve problems in topics without teaching the topics. As most of this is forgotten by the time the students who do take college/university level mathematics do so, it's a waste.

That's why teaching formal logic, sets, and statistics is far, far more useful than teaching matrix theory that many college students find difficult in courses that dedicate a semester to what is covered in 2-3 chapters in some pre-calculus texts. Set theory leads immediately to probability and it nicely intersects (no pun intended) with statistics that enables covering more topics in algebra than most students will remember anyway. Add logic to the mix and you get a better foundation than pre-calculus can provide for understanding calculus (by understanding language like "....if, for every epsilon greater than 0, there exists a delta greater than 0 s.t., whenever...."

Second, it took some 200 years for the greatest mathematical minds from Newton & Leibniz to Weierstrauss to forumulate a sufficient definition of limits, and the cost was abandoning the intuitive sense of infinitesimals. That was necessary then. It hasn't been for ~40 years. Spending a long time to learn how to rationalize algebraic expressions simply so that if one takes calculus one will be able to determine the limits of functions one will never use would be ok were it not for the fact that the bulk of this process is a complete waste. Much of what students who take 2-3 semesters of calculus learn is at least outdated and even were it not, the amount of time spent on calculus is ridiculous. Students can still take several semesters and pass them with flying colors yet be unable to explain the underlying logic, principles or to extrapolate from what they've learned and applied it. Any calculus past single-variable that isn't also a course on linear algebra is a waste.

I can add a few comments from my experience of teaching mathematics in Turkey, for about ten years, but being formed as a mathematician in an East-European former communist country, Romania. In Turkey, there is a big (national) problem caused by the University Entrance Test which kills almost all skills and desire of students for understanding what they actually do. These tests contribute to the transformation of normal children into brain-washed robots that are supposed to guess answers instead of reasoning. Then, after having this really painful experience, they come to realize in their first year of university studies that mathematics, and calculus included, is something totally different than guessing the correct answer, they panic, their mind blocks, and feel lost. We have to repair their minds (and souls) but most of the times the damage is too big and hence we fail.

In found that the first day of class is usually wasted on administrative details: grading/attendance policy, syllabus, book to get, etc.

I used that rest of that class not to lecture on material they have yet to read, but

instead

1) to talk at a high level about the major topics that they would need to learn, and what they would be expected to know by the end of the semester, and

2) To cover some of the gaps in knowledge that students typically have that trip them up:

Functions, what are they, how do you tell something is a function, why is it important,

and very important that f(x+h) is not f(x) + f(h).

Some basic algebra mistakes, and faster ways of doing things

(Since first semester is single variable calculus, show them synthetic division, and the analogue for multiplication. which is just the usual tablue in order of powers

just with no carrying. (Multiplication in Base X, but you don't know

what X is so you can't carry.)

Later in the semester in the section on inverse functions:

The section on the inverse functions is always horrible. (Reflect about a 45 degrees, ridiculous! that is hardly intuitive.)

Here is how I present it:

Draw the graph of the function on a transparent sheet of paper. This graph represents a relationship between two variables x and y .

Draw a bold arrow showing the positive x and positive y direction.

Flip the paper over so the students can see the graph from the other side.

Check that this defines a function (vertical line test.)

You now have the graph of the inverse, although in a space where x is positive to the left, not the right.

This reinforces the idea that one just reverses the roles of the two variables,

and if that relationship is a function, then that is the inverse function.

This also take almost no time.

So you see, even though blame can be placed on poor preparation,

we also have the problem that much of calculus is being taught by

grad students, who do not have the breadth of experience with what students

have problems with to be able to guide them away from troublesome parts.

It is mainly due to the lack of skills in doing algebra. In my experience the students often make mistakes in doing simplification of large expression. Further they find it difficult to remember all the formula of differential and integral calculus. In India, especially in Tamilnadu, majority of the schools do not concentrate on teaching elaborately the calculus as without that the student can able to secure 100% mark. This is contributing a major setback when these students no escape situation in universities.

To understand calculus,it require some basic understandings of variation in y with respect to x and higher order derivatives.Some basic formula of derivatives of std functions and some rules of addition multyiplications and divisions to find derivatives.After some experience it become used to.All the best to learn it.

It my experience as a student and teacher, it was usually the teacher's inability to understand how many different ways there are for understanding calculus - if you can learn what style of learning the student uses, there is almost certain to be a corresponding style of explanation of calculus

The main difficult due to peer proving and few applied examples. the famous book of Thomas is better for education.

An attempt to analyze, and also answer the question can be found in arXiv:math/0609343. I would much appreciate well considered comments on that paper. One of the main related problems seems to be the following one : When in the late 1950s and early 1960s a sudden, unprecedented and massive expansion of science and technology students was implemented both in the Western and communist world, no one made absolutely any kind of thorough and widely enough ranging studies in order to establish what is the percentage in a normal population of those who in the first college year can, along other major subjects, learn Calculus satisfactorily. And this issue is not trivial, since Calculus - when compared with the mathematics learned in school - is indeed considerably more difficult, not to mention that it is extremely new, when seen in the context of the millennia long history of elementary mathematics.

In my opinion any new subject is difficult in the beginning and calculus is no exception. First year students find linear algebra, geometry and computer programming just as difficult.

So, unless question is specified as to in what way calculus is found more difficult than, say, algebra, it is not clearly stated.

Dear Dr. Ivanov:

I agree that any new subject presents difficulties Learning in general tends to, and if subjects taught both before and in college/university were so easily grasped and so intuitive then they would not require teaching anymore than learning to speak does. That said, even accounting for large variation, some subjects almost always present more difficulty than others. And when we look at subjects within some discipline (such as mathematics) general trends of relative difficulty among student populations are even more telling. The way linear algebra is taught, for example, is empirically simpler from a computational standpoint than pre-college algebra. Much of the computational component is simple arithmetic. Yet students who have excelled in algebra can find to their surprise they flounder here. This is because linear algebra presents conceptual challenges for students (who, after all, are for the most part taught rote procedures to solving as if they were calculators; hence the general hatred of "word problems" which require the student to first set-up the problem before applying the rote mechanisms to solve it).

The way calculus is typically taught is designed to trade the conceptual difficulties that courses in real and complex analysis present with "practice" and much more computationally-heavy courses. The idea, from what I can gather, is that rather than require students to attack limits, series, and other foundations of calculus and analysis in a rigorous and systematic fashion, we remove much of the conceptual component and hope that by throwing hundreds of problems which consist mainly of pre-calculus material, the students will acquire the necessary conceptual basis through practice. Also, many calculus textbooks are somewhat anachronistic. Few students who take calculus will ever go onto numerical analysis yet many courses require students to learn 3 or more ways of computing the "area under a curve" using rectangles (often both as an approximation and using Riemann integrals) and then have to do the same with trapezoids. Yet the days in which any serious application of integration required little or no help with calculators or computers are long past, and most students who take calculus (even more than one semester) will never use the trapezoid rule, Simpson's rule, or any other numerical integration method after college even if they go on to graduate school.

It is quite possible to take calculus, even several semesters of calculus, and fail to grasp much of what analysis consists of. In part this is because so much of what is taught is presented in textbooks as largely discrete topics with little integration rather than the continuity we find in e.g., Spivak's Calculus (yes, I used all those terms deliberately). The big name textbooks separate make a point of separating material as a marketing ploy. Not only can they point out in the "Notes for the Instructor" sections that duplicate sections and extra components exist so that instructors can choose one of two ways to approach them, but the relative independence of chapters is (so the authors and publishers claim) a benefit which allows the professor to present e.g., limits at infinity when covering limits in general or later in the context of integration or series or whatever. This is only part of the reason. The other reason for the disjoint presentation of material is so that one calculus textbook can be sold in as many as 6 different versions with barely any additional work by the author(s).

The first chapter of Spivak's Calculus is limited entirely to addition, subtraction, multiplication, and division, and limits are introduced in chapter 5. Yet student's often have trouble with that first chapter even thought the computations are very simple. This is because Spivak uses basic algebra to introduce a systematic treatment of proof and rigor. Most of the variations of Stewart's, Anton's, Larson & Edwards', etc., not to mention Thomas' Calculus, Tan's Early Transcendentals, etc., start limits in chapter 2. So why do we usually find that either the number of chapter 2's first page is either roughly equal to that of Spivak's chapter 5 or greater? Because rather than devote endless pages to graphics, various different restatements of the same thing in terms of various different geometric and analytic interpretations (not to mention bloating the textbook to allow instructors to cherry-pick material), Spivak builds the foundations from the start and progresses continuously. It is more akin to classics like Courant's or Serge Lang's. Most importantly (in terms of relevancy to the topic) it is difficult for student's in a way that is quite different from the popular university level calculus textbooks.

Calculus presents one challenge when it seeks to introduce the student to the subject with minimal use of analytic rigor and maximal use of practice consisting of mostly pre-calculus computations, and a very different kind of challenge when the emphasis is on an introduction to advanced mathematics. As result, the challenges for most students are largely needless: the level of conceptual understanding typically gained could be obtained through a teach-yourself book of some ~100-150 pages and the challenge lies in having students perform needlessly complex algebraic manipulations and the incredibly poor way in which integration is introduced:

“[T]he Newton integral is nothing more than the mean-value theorem of the calculus. Thus it is ideally suited for teaching integration theory to beginning students of the calculus. Indeed, it would be a reasonable bet that most students of the calculus drift eventually into a hazy world of little-remembered lectures and eventually think that this is exactly what an integral is anyway. Certainly it is the only method that they have used to compute integrals.

For these reasons we have called it the calculus integral. But none of us teach the calculus integral. Instead, we teach the Riemann integral. Then, when the necessity of integrating unbounded functions arise, we teach the improper Riemann integral. When the student is more advanced we sheepishly let them know that the integration theory that they have learned is just a moldy 19th century concept that was replaced in all serious studies a full century ago.

We do not apologize for the fact that we have misled them; indeed, we likely will not even mention the fact that the improper Riemann integral and the Lebesgue integral are quite distinct...We also do not point out just how awkward and misleading the Riemann theory is: we just drop the subject entirely"

(http://classicalrealanalysis.info/documents/T-CalculusIntegral-AllChapters-Portrait.pdf#page=8)

It is easy to bitch about calculus, but it is hard to do anything about it.

@Michael

What is so hard? We could, for example, stop teaching 19th century integration that is both unintuitive and provides quite little in the way of conceptual connections to any serious use of integral theory. We could recognize that despite the lack of rigor infinitesimals provided prior to the epsilon-delta limit formulation of Weierstraß "most of the results in Calculus were already discovered through infinitesimals" (http://arxiv.org/pdf/1108.4657.pdf), and therefore not require students to grasp that which they have neither the exposure to logic to understand (so long as we have already abandoned rigor anyway, at least until students are sufficiently advanced such that we can teach them what we did not before and taught instead methods and concepts both outdated and needlessly complicated). We could provide the necessary exposure to formal logic as several texts (such as Spivak's do) through formal languages students are already familiar with. We could trim the fat from bloated texts that contain much material useless to the vast majority of students taking calculus. We could stop relying on marketing ploys encouraged by publishing companies such that a "Dummies/Idiot's Guide to Calculus" ceases to be at least as useful as popular university calculus textbooks. We could stop teaching pre-college mathematics as a if math were a series of procedures so that students taking their first course in university calculus (whether they have already been exposed or not) understand something of actual mathematics rather than a series of seemingly pointless ways to implement rules that have no recognizable use and in many cases are indeed useless other than to teach an approximation of actual calculus concepts. And we could stop using excuses like "it's easy to bitch about calculus, but hard to do anything about it" while ignoring the fact that 10 minutes with google could reveal is wrong and possible solutions. There are in fact organizations of mathematicians who have not only stated that we should do something about it but proposed methods and shown solutions. But perhaps you are already familiar with the rigorous formulation of infinitesimals ~40 years ago, the "Dump the Riemann Integral Project", the research in everything from the cognitive sciences to mathematical education journals, and so on. In which case, it would enlighten me at least to have your rejoinder to the several thousand pages of criticisms and solutions to the current approach that warrants your evaluation. In particular, it would prove useful to future attempts of my own, as these attempts to teach the subject to students who lack the background they could have obtained to deal with the best textbooks (even if I could get the department to purchase these) and therefore require my own tests and trials to make out of these poor excuses for calculus textbooks something worthwhile using whatever supplementary material I can find.

@Andrew Messing, it looks easy until you try, and when you try you will understand how difficult it is, especially in our real world, not in the ideal world that you may be imagining while discussing this or that "improved" approach. And if you tell me how "dumping the Riemann integral" or "rigorous formulation of infinitesimals" would make introductory calculus more understandable, I will be all ears.