Teaching calculus today in college

[mathwonk]

The biggest task I have seems to be helping students learn how to learn. Some fail to come to class, others never look at the notes they take, and many seem not to even open the book.

Many never ask questions, and those who do, often ask things that could be found immediately by looking them up in the index of the book. People who ignore office hours for weeks expect me to schedule extra help sessions the day before the test. Questions more often focus on "what will be tested?" instead of how to understand what has been taught.

Everyone seems to have taken calculus in high school, but most also seem not to know anything about algebra or geometry or trigonometry. With the advent of calculators some also do not know simple arithmetic, like how to multiply two digit numerals. (I have had students who had to add up a column of thirteen 65's on a test, apparently not knowing how to multiply 13 by 65.)

Many think that having taken a subject "2 years ago" is a valid excuse to have forgotten the material, and to expect the teacher to reteach the prerequisites. Appparently no one ever dreams of reviewing the prerequisites before the course starts. Books like "Calculus for cretins" are apparently more popular than books like "Calculus for science majors".

When I was in college students like this were just ignored, or expected to flunk out, but in today's setting there are so many like this that they form the primary market. With all good faith to teach these stduents, the failure rate is still about 50% in college calculus across the nation, in my opinion. What are some ideas on how to improve this?

 

[Muzza]

When I was in college students like this were just ignored, or expected to flunk out

How many people took calculus way back when you were in college (compared to today)?

 

[shmoe]

I hear your pain. My favorite are students who fail the course, then beg to be passed when they've managed to miss half to term work and I have absolutely no idea who they are apart from a name and number on the class list. Where were they all year? Somewhere along the line students developed a sense of entitlement, they (or mommy and daddy) are paying big bucks for tuition so they somehow deserve good grades no matter what they do.

The most important thing I try to drill into students heads is the only way to learn mathematics is to do mathematics. The first step to this is giving problems an honest attempt before giving up. This means trees will die. I'd rather see a student come in with a page of nonsense that failed to work than a blank page and expression. They're often afraid to make mistakes so they don't even try. This is rubbish, the number of mathematics mistakes I can make in a day is limited only by the number of hours I'm awake. I try to lead by example here and show them two of my favorite textbooks, which are just problem books (one algebraic number theory, the other analytic) and explain the mounds of paper I've burned through over the years.

It takes work, but if they are willing to put in the time to understand the course, I'm willing to put in the time to assist. I have nearly infinite patience for students who are obviously doing the work. Others, not so much

 

[mathwonk]

Muzza, I do not have figures, but it seems many fewer took calculus then, and many more took algebra, etc, in high school. When I started college I only knew algebra and geometry, not even trig. I believe the change in high schools from teaching precalculus subjects thoroughly, to offering too many people watered down calculus today without adequate background, is a big part of the problem, but I am not trying to place blame, just think of solutions.

In fact I was one of the students in college who flunked out from poor study habits myself. (I was in a lecture class of 135 students that met Tues, Thur, and Sat, at 9am, not always including me.) In my case working in a factory helped give me an attitude adjustment. When I got back in college, I was not allowed to repeat anything, but had to pick up where I had left off. (We were admitted for 8 semesters, no more. The philosophy was: either graduate, or leave without a diploma so someone else could have the spot you are wasting.)

I got a D on my first test back, in diff eq, after not learning calculus. When I complained to the teacher I was being penalized for stuff from the previous course he just said "well, mathematics is cumulative". So I got a Schaum's Outline and began burning up the scratch paper as recommended by Shmoe. I ended with an A+.

The next year I asked the professor teaching honors advanced calc what I needed to get into his course. I listened, got a copy of Widders Advanced calc and read what he recommended over the summer. I managed a B+ and an A- in a course that covered Banach space calculus, infinite dimensional spectral theory of compact self adjoint operators, and differentiable manifolds.

By senior year I was in graduate real analysis and holding my own.

 

[mathwonk]

One thing I think does work, is the patience shown by people on this forum, at helping people without doing their work for them.

 

[Tide]

Mathwonk,

The part that gets me is how many students will flat out refuse to read the textbook. Some who claim to have read it will simply declare it makes no sense as if it's YOUR fault and then expect you to magically impart the knowledge into their brains.

Along the same lines I can't count the number of times when posing a problem some student will simply demand that you tell them the answer or show them how to do it without even the slightest effort on their part. It's as if when you finally get around to testing them they expect the same question to be asked and all they need to do is provide the answer they've already seen. It's a shock to them when they encounter new problems on a test and then they complain bitterly that you never showed them how to do that!

Of course, none of that changes their mindset as the course progresses.

 

[mathwonk]

Tide, you remind me of the student who complained that I asked him to maximize the volume of a closed top box, when in class I had only shown how to do it for an open topped box.

This may suggest again that I need to be understanding how to help my students broaden the scope of what they are "learning". Simply handing out a syllabus that says "you will be tested on your ability to use what you have learned in new situations" is not sufficient.

Perhaps we should accept that the frustrating experience of hearing the complaints about our tests is actually a learning experience for the student, as painful as it is for both of us, and just stick to our guns.

 

[modmans2ndcoming]

you know why no one knows their algebra? because Math is not taken seriously enough and the methods used do not teach vocabulary so it is like trying to use a hammer and nail to put two boards together but no knowing what the name of either are. heck, even if you said a name of an algebraic tool to me today, there is a good chance I will not know it from that descriptor, but I do algebra like it was second nature.

usually after calc 1 and 2 the students who are asking a lot of algebra questions have dropped and the ones that are left either know what they are doing or have low confidence so they ask. I found it helpful when my prof said "it is just paper, if it does not work out...ERASE IT!"

 

[modmans2ndcoming]

if you ask me, all students entering college need to be required to take Trig and pre calc there even if they test into calc. then the math department can know what to expect from the students in calc.

 

[modmans2ndcoming]

umm, isn't the volume of an open top box going to be the same as a closed top box if all parameters for the rest of the box the same?

seems high school needs to teach common sense as well.

 

[modmans2ndcoming]

Tide,

in the defense of many students, some of the calc books are just plain badly written. my Real Number Analysis book was more interesting than my calc book.

I think that calc book writers need to use less brevity in examples because I know that a lot of students tend to get lost in the details because they cannot figure out how the writer went from step one to step two. they could at least have an appendix with a full description of the example, step by step. that way the bigger picture will not be lost.

 

[HallsofIvy]

Tide, you remind me of the student who complained that I asked him to maximize the volume of a closed top box, when in class I had only shown how to do it for an open topped box.

I can top that! On a Calculus III test, I gave a question right out of the book (but not one that had been assigned) on finding the maximum temperature on a plate given the temperature as a function of x, y. When a student complained that we hadn't covered "that kind of problem", I pointed out that we had done a number of problems on finding the maximum of a function of two variables. He protested "they didn't have anything to do with temperature!"

On the other hand, once, when a student protested after a test that I hadn't taught them how to do "that kind of problem", I was able to point out that, not only was that specific problem one of the assigned homework problems (I do that with 1 or 2 problems on each test), but that a studeng had asked about it in class, we had gone over it in class, and I showed him where he had the complete solution in his notes!

Give us strength!

 

[Dburghoff]

As a freshman college student (in Calculus III), I'm kind of offended by the blanket statements flying around here. I spend at least a half hour every day studying from my notes and reading the next section (so that I may be able to participate intelligently in the next lecture), and that's in addition to my homework. I've never missed a single class, and I'm always prepared, as are most of the people in my class (as far as I can tell). I just think that many of you are displaying classic "kids these days..." syndrome. It's always easier to judge your juniors more harshly than you judge your peers.

On an unrelated note, the fact that many of you are teachers comforts me greatly. I often just read topics in various forums, and I'm always disturbed when you talk about things which I haven't yet covered, whether it be in physics or math. It makes me feel like I am behind, even though I know that I'm not.

 

[mathwonk]

The quality of calc books is another important point raised by modmans. Actually many calc books are excellently written when they first come out, but publishers push for dumbing them down, to raise sales, and they tend to decrease in quality as the later editions come out.

Everyone knows what the good calc books are; well written, and authoritative: Courant, Courant and John, Apostol, Spivak. These are the time - tested, great contemporary calc books, and they have held this position for many years.

Engineering problem solving is probably still best learned from the original book of George B. Thomas, now sold as the "aternate edition".

By the way Modmans, here is the closed / open top box problem: given say 6 square feet of sheet metal with which to build a rectangular box with a square bottom, find the dimensions which maximize the volume if the box is to have a top, and also with no top.

Thus clearly you should be able to make a larger box with no top, than with a top, but also the dimensions are different, interestingly. See if you can imagine why. If you think about it and have some intuition, this does not even require calculus, but calculus does work on it.

Here is a recommendation of a good cheap, short, paperback calc book, the one by Elliot Gootman, selling new for about $15. Of course it does not contain all important topics, but it is well written with excellent clear explanations froma master teacher. And it is better to actually learn a few key topics, than walk around with a thick book one does not or cannot read.

If you really want to learn the subject thoroughly, then get one of the classics recommended above, and spend time with it.

 

[mathwonk]

I was afraid of what DBurghoof has said. Even though the statements here have clearly referred to "many" or "most" or even only one example, he takes them as directed at him. This is of course not the case, but it is unavoidable. I think if you will reflect on it Mr Burghoff, you will find that either you are at a very elite school, or you are a very unusual person at your school, and that indeed many students are not doing what you are doing. But nonetheless I apologize if you are offended. We are not worried about the future of students such as yourself. Those of you who actually go to class and prepare the lesson are the ones that make our job worthwhile.

Notice for example that all the professors on this site are donating their time, with nothing whatsoever to gain, largely because it is so rewarding to tutor interested students like yourself.

I might add however that 1/2 hour a day is not much study time for a genuinely challenging course. Most people agree that 2 hours per class hour is minimal. Perhaps you are one of the fortunate few who learn quickly and easily. It is also possible your class is too easy for your abilities. In the example I gave above of a class in which I went from a D to an A+ in one semester, notice I left that program afterwards and entered one in which I could not so easily earn such a high grade. I felt that those courses in which one earns an A+ are not sufficiently challenging.

 

[Hurkyl]

On an unrelated note, the fact that many of you are teachers comforts me greatly. I often just read topics in various forums, and I'm always disturbed when you talk about things which I haven't yet covered, whether it be in physics or math. It makes me feel like I am behind, even though I know that I'm not.

That feeling never goes away. No matter how much math I learn, I always encounter some new topic about which I feel my education cannot be complete without learning it!

 

[vladimir69]

One thing I think does work, is the patience shown by people on this forum, at helping people without doing their work for them.

i second that. the people here have always kindly replied to my questions in a non smart a$$ fashion and i greatly appreciate that - it gives me a warm fuzzy feeling all over

 

[mathwonk]

Well, perhaps this shuold have been obvious, but the key suggestions emerging seem to be:

1. Be patient.

2. Be clear.

3. minimize criticism.

 

[Tide]

As a freshman college student (in Calculus III), I'm kind of offended by the blanket statements flying around here. I spend at least a half hour every day studying from my notes and reading the next section (so that I may be able to participate intelligently in the next lecture), and that's in addition to my homework.

I'm certain no offense was intended by anyone here. Classes and students vary. I'm sure you've seen enough postings here to recognize that student capabilities and commitment vary immensely. You must recognize from what is written here that many participating students demonstrate extraordinary abilities but just as many are clearly over their heads in the courses they take. Both categories comprise your typical classmates but it's the latter group that has caught the attention of this thread.

 

[mathwonk]

Here is a very positive experience that happened recently. I gave a test and as a extra question asked students to pose and answer the most interesting question about the course material they could. This was a beginnning calculus course. Normally students either omit the question entirely or ask something trivial.

This time one student tried to solve the "ham sandwich" problem, that given any triangle and any line in the same plane, some translate of the line bisects the triangle by area! It blew me away. The student's understanding was very partial but did contain the essential idea of using the intermediate value theorem applied to the comtinuity of the areas with respect to the position of the line.

Afterward we chatted enjoyably about it and created another more elementary solution allowing one to actually solve for the position of the bisecting line, in principle.

This is a delightful experience which has happened only a few times in several decades, but is still wonderful. The moral dilemma now is whether to kick such a student out into a more advanced class or simply continue to enjoy their presence.

Of course sometimes students appreciate our dismissing them. I remember one of my students writing back after a several decades and thanking me for my classroom guidance, as apparently I inspired him to drop out of school and become a cartoonist!

 

[modmans2ndcoming]

With a top, the dimensions will be a cube 1ft x 1ft x 1ft

with out a top, each of the 5 sides should be 6/5ft x 6/5ft x 6/5ft.

this is just of the top of my head, no calc, just quick thinking. so if it is wrong, it is because I did not put to much thought into it.

 

[HallsofIvy]

As a freshman college student (in Calculus III), I'm kind of offended by the blanket statements flying around here. I spend at least a half hour every day studying from my notes and reading the next section (so that I may be able to participate intelligently in the next lecture), and that's in addition to my homework. I've never missed a single class, and I'm always prepared, as are most of the people in my class (as far as I can tell). I just think that many of you are displaying classic "kids these days..." syndrome. It's always easier to judge your juniors more harshly than you judge your peers.

Yes, I can see how you would be offended. We have many good student, even more mediocre students and a few that just drive us crazy! Sometimes you just have to laugh about them to blow off a little steam!

On an unrelated note, the fact that many of you are teachers comforts me greatly. I often just read topics in various forums, and I'm always disturbed when you talk about things which I haven't yet covered, whether it be in physics or math. It makes me feel like I am behind, even though I know that I'm not.

I've been teaching math for more years than I care to count but I see topics here that I am not at all sure about! Mathematics (not to mention physics) is a broad area and there is always more to learn.

 

[matt grime]

Such is the nature of teaching that you often tend not notice the good students until too late, so for the freshman calc major, sorry, but those who shout the loudest get the most attention. And they usually piss off the lecturer too.

Seeing as we're into our anecdotage:

A colleague from a previous workplace of a colleague at a place I no longer teach at decided to experiment with his classes. In one he taught as normal, in the other he assigned all the questions as homework that would eventually appear on the final, in the third he gave out the answers to the final before hand. The marks in each class were almost identical.

I had one student threaten to sue me because when I said:
"Yes, you failed the midterm by 2%, I don't see why you can't pull that round and pass the course though" took that to mean he could fail the final by 2% and still get a C. Apparently I was jeapordizing his future earnings.

Anyone see the old Onion article about 'new, principled teacher offended by older teachers ridiculing <some student's> Hamlet essay in staff room'?

Some times we do need to blow off steam. My bete noir is when students make the exact mistake on homework that I told them was the common error and if it seems strange that's where you've gone wrong. (Commonest one, working out the length of i-j, say, to be zero).

 

[mathwonk]

Modmans, your thinking is right on for a cube as the mbox with top, but for the topless box, it is not a cube. A cube does have somehting to do with it though! Think about this: if you make the biggest possible topless box from 6 sq ft of material, then two of them put together at their open ends, should give the biggest possible box WITH top, made from 12 sq ft. If you believe that, what does it imply?

 

[modmans2ndcoming]

Modmans, your thinking is right on for a cube as the mbox with top, but for the topless box, it is not a cube. A cube does have somehting to do with it though! Think about this: if you make the biggest possible topless box from 6 sq ft of material, then two of them put together at their open ends, should give the biggest possible box WITH top, made from 12 sq ft. If you believe that, what does it imply?

the only thing coming to mind is that it should scale down so you get a similar effect with 2 boxes made from 3 sq. ft.

 

[mathwonk]

well the thing it implies to me is that putting the two together should make a cube of area 123 sq ft, i.e. it should give the answer to the closed top problem. So the answer to the open top problem should be the result of cutting a cube of total area 12 sq ft in half, so the answer will have base sides twice its height. The base will then have side of length sqrt(2), and height half that. so the base will have area 2 and the 4 sides will each be rectangles of area 1.

what do you think? does that make sense? try it using calculus and see what happens.

 

[mathwonk]

By the way, in the spirit of helping people learn calculus, we have often heard an appeal for good books. Hence also in the spirit of providing a list of answers to frequently asked questions, undertaken by Matt Grime and others elsewhere, we might compile a list of highly recommended calculus books.

These should probably be sorted and commented on by a moderator here, or other teacher, to make clear which ones are user friendly but low level books preferred by non math types, and which ones are genuinely deep treatments, for those desiring to get to the heart of the subject.

It might be useful to replace the word "good" by more descriptive ones, such as: "rigourous", "example oriented", "brief", "guided problem solving", or other terms.

Under "rigorous, authoritative and masterful", I will repeat the names of the calculus authors Apostol, Courant, Courant and John, and Spivak. There was also a terrific book by Joseph Kitchen, long out of print. All but the last of these authors have written books covering both one variable and several variable calculus. For a quick, rigorous, modern, introduction to the essentials of several variable calculus, Spivak's Calculus on Manifolds, at 140 pages, is probably unmatched. For a longer treatment, with Lebesgue integration, Wendell Fleming's Functions of Several Variables is excellent. A nice feature of that book was a 20 or 30 page summary in the appendix of an honors level introduction to one variable calculus, for those lacking the appropriate first course. Used copies for as little as $10 are available on the well known site abebooks.com

Books like those of Sylvanus P. Thompson, and Gootman, probably belong under the category "limited in scope, but highly user - friendly".

Stewart 2nd edition, and Thomas and Finney 9th edition, are thorough, standard, clear, well laid out, not overly theoretical.

The Schaum's Outline series books have been thought useful for years, but I suggest getting as old an edition as possible.

Books with titles like calculus for nitwits, should perhaps be taken at their word and avoided.

For todays web priented stduents, there are also excellent free books and tutorial sites online, like http://www.karlscalculus.org/calculus.html#toc .

 

[Gokul43201]

I did high school and college in India. Just to provide a frame of reference, here's a couple of randomly selected problem from the math book I used during my last 2 years of high school :
(1) (VECTORS)Prove using vector methods, that cos(A+B) = cosAcosB - sinAsinB
(2) (COMPLEX NUMBERS) Find the other vertices of a regular polygon of n sides, whose center is at z0 and one vertex at z1.

As a Physics Grad student in the US, I was TA for an introductory physics course. That was my first, and only, culture shock. There were college students that were adding fractions by adding numerators as well as the denominators !

Later I found that the high school syllabus for Physics entirely avoids using the word (and the concept of) "voltage", because it is too abstract for someone in high school. I'm not sure if this is only for this state (Ohio), but I was shocked (no pun intended) !

I strongly believe that there's not enough math and science being taught in schools in the US.

 

[modmans2ndcoming]

in Michigan, at least when I was in high school taking physics (1997-98) (yes I am still in college, but that has more to do with my kids and my wife's work schedule than being lazy) we certainly talked about voltage and amps. I do not see how they are to abstract for students in high school... I mean, just tell the students that voltage is to electricity as hight is to, well, a brick. they tend to get it.

I mean, voltage is no more abstract than electrons or protons, or forces.

 

[modmans2ndcoming]

well the thing it implies to me is that putting the two together should make a cube of area 123 sq ft, i.e. it should give the answer to the closed top problem. So the answer to the open top problem should be the result of cutting a cube of total area 12 sq ft in half, so the answer will have base sides twice its height. The base will then have side of length sqrt(2), and height half that. so the base will have area 2 and the 4 sides will each be rectangles of area 1.

what do you think? does that make sense? try it using calculus and see what happens.

yes that does make sense. using thought experiments makes it a very out of the box (pun intended) solution. but analytically, it is very apparent.

 

[Pyrrhus]

I am a student [a sophomore] at a University, and I've seen many of those students you all teachers has described. I've always been considered by many teachers ones of the few that actually care about learning, I've always studied the topics before they were taught, so i make sure i understand them well.

In my opinion the best way to learn is by teaching yourself, Teachers are just merely guides that can help you in case you didn't understand properly an idea.

 

[mathwonk]

I may be wrong, but I pride myself when I teach a course on trying to present more logical, or more insightful, or deeper versions of the material than are found in most books. I.e. I try to actually provide or suggest ideas that are not in the text.

Of course this is largely possible because the books we use are not the best available, hence there is plenty of room for improvement. But it is also true that in college, teachers usually know a bit more about the topic than is in most current books.

There are rare exceptions, but if your teachers cannot offer you anything beyond what is in your texts you might consider seeking better informed teachers, or taking more advanced courses. For example, if you are a calculus student, do you know that all monotone functions are Riemann integrable? Can you prove it? This simple fact was known to Newton, and is far easier to prove than the integrability of continuous functions, yet is omitted from most beginning calculus texts. On the other hand it is found in good ones like Apostol.

Did you know that if a function f is Riemann integrable on [a,b], even if not continuous everywhere, then it must be continuous "almost everywhere", and moreover that its "indefinite integral": F = integral of f from a to x, has the property that F is continuous everywhere on [a,b] (even if f is not), and F is differentiable everywhere the original function f is continuous, and that at such points F'(x) = f(x)?

This is a version of the fundamental theorem of calculus which is more precise than that in the most commonly used books. If you have not seen it you might enjoy proving it for yourself.

Did you know further that this is not sufficient information to recover the indefinite integral F from f? I.e. given an integrable function f, and a continuous function G which is differentiable wherever f is continuous, and with G'(x) = f(x) at such points, it need not be true that G(b) - G(a) equals the integral of f from a to b?

Can you think of a function F which is continuous everywhere on [0,1], with derivative zero almost everywhere (i.e. on a collection of disjoint sub intervals of [0,1] with total length 1), and yet with F(1) - F(0) = 1? Such a function does not obey the mean value theorem (F(1)-F(0) does not equal the value of the derivative F'(x) anywhere in [a,b]), and F cannot be an indefinite integral.

There is however a stronger version of continuity satisfied by indefinite integrals, stronger even than uniform continuity, which does suffice for this purpose. Can you discover it? If you can do any of these things without having seen them in books or courses, you are well on your way to bering a mathematician.

If you are more advanced than this already, I apologize for these elementary challenges. I could not resist trying to provide soemthing that may not have been contained in your calculus course. Even if you are already at the graduate level in mathematics, as some sophomores are, there are people here who can suggest topics of interest to you.

 

[modmans2ndcoming]

Mathwonk,

how about the lack of any formal Proof at all in many calc curriculums.

 

[mathwonk]

that is a big mistake in my opinion. formal and informal proof are the strongest features of mathematical science. everyone benefits from learning this, and so I try to include it in all courses i teach.

here is my blurb for my students:

One of the main benefits of a mathematics course is in learning to make logical arguments. (This can actually help you in arguing with a judge, or the IRS, or your boss, for example.) This means knowing why the procedures you have memorized actually work, and it means understanding the ideas of the course well enough to be able to adapt them to solve problems which we may not have explicitly treated in the lectures.

 

[matt grime]

If only all calc teachers took that attitude. I know one who refused to allow a question to appear on the final (multiple teachers for the section) because they hadn't taught one exactly like it. Some one asked if they'd taught how to do the preceding two questions, since the third was just doing those two questions sequentially. they had, but still refused to allow it on the final. that person won lots of teaching awards (based upon student evaluations).

 

[HallsofIvy]

I am a student [a sophomore] at a University, and I've seen many of those students you all teachers has described. I've always been considered by many teachers ones of the few that actually care about learning, I've always studied the topics before they were taught, so i make sure i understand them well.

In my opinion the best way to learn is by teaching yourself, Teachers are just merely guides that can help you in case you didn't understand properly an idea.

Exactly. The most important thing you can learn in school is how to learn by yourself. You won't always be in school, but, hopefully, you will always be learning.

 

[mbisCool]

Im currently in calculus and the entire course emphasizes computation. From what i have read this is quite common. Students are not developing thorough understandings of calculus concepts. I notice people do things such as plug and chug whatever rule they just learned when if they just stopped and looked for a second they could easily simplify the expression and the answer becomes trivial. In my class personally, the entire capter on epsilon delta proofs was skipped over with mention. I recently got a copy of spivak and immediately began working the problems. They are so much better at testing your understanding of the concepts than simple computation problems over and over. Even if the answer is derived from simple computation the exercise will be how to approach the problem not the answer itself.

Because of this i think students should work out problems that are not merely computation. Who cares if the students can calculate problems 3-32 if they don't understand what's going on or why they are doing what they are. Problems that emphasize understanding and not computation skills. Although from my experience most students at least in highschoo/cc think of math as computation and nothing else.

 

[mathwonk]

have you studied taylor series? have you ever noticed that most books omit much mention or detailed discussion of the series for tan(x)? It turns out that knowing the coefficients of this series is equivalent to knowing the sequence of Bernoulli numbers.

these numbers are extremely interesting, as they determine the values of the riemann zeta function at the even integers (in a formula due to euler), the order of the image of the "J homomorphism" in the theory of homotopy groups of spheres, the number of diffeomorphism classes of exotic spheres of dimensions 4k-1 which bound parallelizable manifolds, the criterion for a prime number to be "regular" in the sense of kummer, who proved fermat's last theorem for those primes, and last but not least, they determine the todd polynomials which arise in the statement of hirzebruch's general riemann roch theorem!

now why would this taylor series normally be passed over in silence, since it seems to be by far the most interesting one?

 

[montoyas7940]

Here is a recommendation of a good cheap, short, paperback calc book, the one by Elliot Gootman, selling new for about $15.

Thank you, I just ordered a copy. It was $4.00 used on Amazon.

 

[ice109]

you're fighting a losing battle. these kids come in from high school with no experience thinking about mathematics, only memorizing. you're only going to frustrate people by forcing them to do otherwise. you want to improve the state of affairs? disallow students from testing into classes above algebra and then teach those classes in your university in a challenging fashion. they will have enough facility with the formal procedures in those classes to not be frustrated by creative problems. by the time they're done with the algebra/trig sequence taught in this way their memorization habbits will be broken and they'll be comfortable with a challenging calc course.

 

[john16O]

To the OP, I finished calculus in the fall of my freshman year(2007). I noticed one thing, the people who wanted to do well in the class did well and the others just did what they had to do to pass, and some just failed. I understand what you are trying to do, sometimes you just have to let the people who fail, fail. I mean they are in college now. If they still have yet to take education seriously then they are in serious trouble and there is probably nothing you can do. All i can say is help those who WANT the help and let the other fall where they may...You are a very nice individual though. I would have loved to have you as my Calc teacher!

 

[john16O]

you're fighting a losing battle. these kids come in from high school with no experience thinking about mathematics, only memorizing. you're only going to frustrate people by forcing them to do otherwise. you want to improve the state of affairs? disallow students from testing into classes above algebra and then teach those classes in your university in a challenging fashion. they will have enough facility with the formal procedures in those classes to not be frustrated by creative problems. by the time they're done with the algebra/trig sequence taught in this way their memorization habbits will be broken and they'll be comfortable with a challenging calc course.

good point, IMO that is the problem with the schools in america. They do not teach the students how to become critical thinker/analytical thinkers. They only teach and test them on how well you memorize the material. Luckily I had a very strick father who would sit down every night and go over countless problems with me...I hated him back then but now I realized why he was doing it..

 

[phyzmatix]

I've scanned through most of the posts here and it seems that the general idea is that the teachers/lecturers should do something different or that crap students might only be crap because of something their teachers are doing wrong and not because they might just not have what it takes. Where I agree that a teacher can obviously make a difference, I personally believe that the problem lies with students mostly and their general lack of maturity when it comes to their approach to life (and studying).

Honestly, with the exception of a few, how many 19 year olds (well, I don't know how old they are when they hit university in the states, but over here that is the average age) know what they want from life and/or have learned the value of an education where your ultimate goal is the accumulation of knowledge and not just the receipt of a piece of paper?

Mathwonk mentioned something about working in a factory and I had a similar experience. Totally screwed up my first attempt at Uni and after seven years of menial jobs and mind-numbing employment as an unskilled worker I just had enough...

This time around, a lot of conscious thought went into my choice of degree as opposed to the first time when it was pretty much a coin toss and "seemed like a good idea at the time". Some things are more difficult being a student now than it was when I was younger, but some are easier, e.g. the motivation, determination and the understanding of what and why I'm doing what I'm doing is on a different level than before.

Oh yes, and of course the predominant idea of what it means to be a student no longer involves alcohol, exotic substances and finding someone to play with mini-me...

 

[mbisCool]

I remember when my class was first learning limits in calc. I wasnt sure how to calculate the limit of sin(x) / x, as x goes to 0. When i asked my teacher how to approach this problem he told me "the limit is 1, its just something you memorize."

I then began to question his expectations of the class.

 

[Moonbear]

Where I agree that a teacher can obviously make a difference, I personally believe that the problem lies with students mostly and their general lack of maturity when it comes to their approach to life (and studying).

Honestly, with the exception of a few, how many 19 year olds (well, I don't know how old they are when they hit university in the states, but over here that is the average age) know what they want from life and/or have learned the value of an education where your ultimate goal is the accumulation of knowledge and not just the receipt of a piece of paper?

That is part of the overall picture of things educators need to consider. The maturity of students hasn't drastically changed, at least not in the time I've been teaching. Yes, I see slight variations from class to class in how seriously they take their studying, but overall, this doesn't change much. So, if someone teaching an undergraduate course expects their students to have the same level of sophistication and ability to work independently as students in a graduate level or professional program, they're not doing the best they can to teach those students.

Likewise, not everyone has the same learning style. I think because different teaching approaches in some fields have self-selected those learning styles among the educators (the ones who learned well with the old teaching methods will be the ones who move on and eventually teach themselves), it takes some real mental stretching on the part of the educators to address different learning styles and ensure part of the class isn't left out simply based on their predominant learning style.

 

[physics girl phd]

The maturity of students hasn't drastically changed, at least not in the time I've been teaching.

My questions: Is there a change in the preparation/preexisting skills of students?
I have no data/insight here. I see prepared students and unprepared students.

Is there a change in the way they approach classes?
I again have no data here... but via a conversation with my husband yesterday: when I was a student, I considered attending classes and doing all the assigned problems "my job", and I also thought it impolite to skip classes, regardless of the skills of the instructor. I also did all problems the professors recommended, checked my results with the professors hand-writtten taped-on-the-wall solutions and went to office hours if I didn't understand them, even though I wasn't given credit for them towards my grade (a few professors would collect problem notebooks and give a bit of extra-credit at the end of the term if you needed a slight curve to get to the next grade).

I went to a private undergraduate institution, while I now teach physics at a public university; but my husband, who went to a large land-grant university for his undergraduate degree also said he would never have considered skipping classes as an undergraduate. I also note that students don't do the recommended problems and often don't even come to class unless homework and attendance are built into the grade. An undergraduate student working for me this term noted that she skips ~ 1 class per week "without even blinking" at doing so.

While I need to do a literature search, I think part of the "dirty-little secret" to clickers and some techniques that I use (like team learning) are that these techniques are ways of building attendance into the grade. Any peer instruction may be just a secondary bonus.

Maybe I'm just down-and-out today. Even though I announced my lecture was covering a chapter that was removed from the present edition of the text, I had 10 students (of a class of about 110) decide to leave my class yesterday since I was giving a lecture and not having a graded team-learning "in-class work" (it's difficult to do "activities" covering some thermodynamics topics). While I confess to not enjoying the process of lecturing, it was still a good lecture.. . I had tons of demos (including a big flame when I discussed and performed the process of lighting a Bunsen burner... with a large propane tank, since there is no gas hook-up in our demo hall and our "campstove" burner is broken and I need a high-heat source for my linear expansion demos). I had the students laughing when I talked about buoyancy and overfeeding my fish when I was little... increasing their mass so that they couldn't fill their air-bladders up enough to be buoyant). I could tell by looking around the room that I had the students engaged. Some of my students have complimented me on my lectures... saying they are still interesting. I know they are better than many of the lectures I attended as a student... I can only recall ~5 demos ever being used in a lecture when I was a student (including both my physics and chemistry classes!)!

Is this a problem with internet tools?
Do these students who leave think all the information they need for a test is online on the posted slides and pre-class quizzes? In my case, it really isn't... I think I'm using the tools well. My demos and extra information (in both lectures, feedback to the quizzes, and in-class works (where I make sure to visit all groups & ask extra questions, etc.) are often a starting point for my test questions... which are VERY applied. My husband's view is that these are students that probably really shouldn't be in college... this is the "death of the university."... and even the death of our society... and we need to provide other sources of training for jobs that should be done in the US (note: we are still disturbed that the machines that mint our coins are made in Germany). Also: for math based classes... is this a problem with online homework systems? I'm still trying them out, and haven't really been pleased by what I've used yet... my students had better results when I had them turn in problems on paper and I manually graded them.

Well, this turned into a minor rant despite the fact I felt I had a few important questions to get some other viewpoints on! ... sorry...

 

[Moonbear]

I really don't think students are that different. When I was in college, on the first day of class, the lecture hall was packed to capacity. After about the third week of class, about half the seats were empty. There were always students whining for extensions on assignments (I HATED that, because if they were granted, I thought it was unfair to those of us who had gotten our work done and turned in on time). Though, one difference is that the lecture halls used to be designed in a way that the lecturer could hear the students talking in the back, so could ask them to leave. I've realized that I cannot hear the students talking in the back of the classroom in the modern classroom, which means I cannot stop them when I think it is getting to a level that is disrupting other students.

Of course, when I was a student, even though I attended lectures, if a lecturer was boring, I'd often be sitting in the back working on a crossword puzzle rather than listening. I'd look up every so often to copy the next board-full of notes.

Though, do students NEED to attend lecture as often today? If we record our lectures and they are available for them to listen online, does it matter if they are sitting in the classroom if they can get the same information online? For some students, yes, seeing the non-verbal cues, facial expressions, gestures, etc., helps them to focus on what is important. But for many, they might do better just listening online. I wish I had that available when I was a student. No matter how hard I tried, about halfway through any lecture, no matter how engaging, I'd start losing focus (our lectures were 120 min long, and a double lecture was a full 3 hours...I couldn't even get through that without needing a restroom break somewhere along the way). It would have been nice to be able to listen at my own pace to the lecture online, complete with all the figures, rather than trying to decipher my notes as they drifted off to a squiggly line while I nodded off. If I missed something, there was no going back, and those old tape recorders didn't work very well.

Though, to me, 100 students in a class is a SMALL class. I was usually one of 300-500 in lecture courses.

But, on the other hand, when they show up at my office all teary-eyed because they are struggling to pass the class, and I've never seen them in lecture, I don't feel the least bit troubled by telling them there's nothing I can do for them other than to work with them on improving their study skills.

As for demos, when I was a student, the chemistry and physics courses were FILLED with demos. Well, actually, only one of the two lecturers for the chemistry course gave demos. I used to go to the lecture for the one who did demos, because it was just so much easier to learn when I was staying awake to watch demos; or, if I thought my own lecturer would notice my absence, I actually attended BOTH sections with both lecturers...that was probably the best thing I ever did, even though it doubled my time in lecture, because I got to hear the same material presented two different ways. I'm not sure I got a lot out of the demos themselves so much as it was a break from the monotony of lecture that woke me back up and regained my focus on the remainder of lecture. The biology courses didn't have much in the way of demos, but that's because they were all combined with lab courses, so we got plenty of hands-on experience without demos.

 

[cdotter]

When I took calculus in high school, my textbook used a lot of examples. We covered some proofs...while I don't remember any of them, some were interesting but most weren't (to me). I'd much rather do practice problems using real life examples and not only get the right answer, but understand WHY I am doing what I am doing. But as for the underlying principles of mathematics, I don't care for them.

I guess you could think that for some, calculus is just a tool. A carpenter uses a hammer, but he does not care about the underlying principles of how and why the hammer works.

 

[mathwonk]

i am inclined to agree with comments putting the explanation for poor performance on how young college students approach their courses. I have recently had some of the smartest, least successful classes I have ever had. Some of the students do not attend regularly nor do any of the extra work recommended to do well.

The attitude of doing only what is required or "due", seems to explain the poor performance of these very talented underachievers. So it is not only that understanding has not been expected, but that independent work has also not been expected. This may not differ from past years, but there is more pressure today to excuse it, rather than letting people fail, because there are so many who would fail.

 

[fluidistic]

Hi,
I'm a freshman and passed calculus I and II. Here in Argentina and more precisely in my University I think I was taught calculus from a different manner than it is usually taught in the US I think.
Theoretical part (a big classroom with about 100 people): The professor writes on the blackboard definitions, then lemmas (with their detailed proof), then theorems, etc. Everything that can be proved is proved on the blackboard so that we can take notes of the proofs of the theorems and lemmas. The lecture lasts 2 hours. After this we enter into another classroom. It's 2 hours of practice (there are about 4 helpers for 25 students). Here we buy the sheets of problems or they give it to you. Basically there are many different kind of problems and some ask you to prove relations. As there is no "Introduction to demonstrations" course, many students find this part the most difficult. But it is a very good training.
Tests are made of a theoretical part (about 30% of the test) and a practical one. To pass the test you have to pass BOTH the theoretical part and the practical one. (you have to score more than 40% on both parts in order to success the test.)
The theoretical part consists to demonstrate 2 or 3 theorems/lemmas or relations you never dealt with before. So this is clear : you cannot success it by memorization. It's impossible.
Furthermore the final exam which is the only exam that count for your grades and the only exam that can make you pass the whole course is much harder than any test. The professor gave us a list of 45 demonstrations (He gave only the name of the theorems/lemmas and not the demonstrations. We have to find them in our notes, books, etc.) that could get into the final exam. What to say about the practical part of the exam? Well it can be any kind of exercise... so you have to have dealt with all kind of problems.
And one more thing I can say : we don't use any calculator in any math course. So integrals are calculated handily and so are series and whatever you can imagine. This means you cannot check out the result you got for an integral with a calculator.

I want to add that the professor said as advice to use the book from Spivak even if not alone. He clearly told us to study hours at home everyday not to get lost with all the new stuff coming fast. (3 months to cover calculus I and 3 months to cover calculus II).
All professors including the Physics' ones too told us that University is VERY hard home working... you have to study at home by your own, checking out books and doing a lot of exercises. I think it's clear that University is hard for almost everyone and it should be told to freshmen. They can't ask "what will be tested?", they should understand the chapters covered and have done many different exercises. In one word they must be prepared and it's not the job to the professor but their job. The professor and helpers are there to help the student to success, not to make him success.

Having said that I liked calculus and the way it is taught here. You cannot pass if you don't understand the matter. That's why more than a half of the students give up during the first year.