# Teaching calculus today in college

1. Sep 14, 2004

### 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?

Last edited: Sep 14, 2004
2. Sep 14, 2004

### Muzza

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

3. Sep 14, 2004

### 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

4. Sep 14, 2004

### 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.

Last edited: Sep 14, 2004
5. Sep 14, 2004

### 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.

6. Sep 14, 2004

### 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 in to 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.

7. Sep 14, 2004

### 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.

8. Sep 14, 2004

### 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!"

9. Sep 14, 2004

### 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.

10. Sep 14, 2004

### 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.

11. Sep 14, 2004

### 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.

12. Sep 15, 2004

### HallsofIvy

Staff Emeritus
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!

13. Sep 15, 2004

### 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.

14. Sep 15, 2004

### 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 learnt 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. Last edited: Sep 15, 2004 15. Sep 15, 2004 ### 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. Last edited: Sep 15, 2004 16. Sep 15, 2004 ### Hurkyl Staff Emeritus 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! 17. Sep 15, 2004 ### vladimir69 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

18. Sep 15, 2004

### 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.

19. Sep 15, 2004

### Tide

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.

20. Sep 15, 2004

### 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!