Teaching calculus today in college

[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.

 

[lurflurf]

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.

That is good advice. Although I might say pick the sin that makes the limit 1 because it is easier to remember.

 

[khemix]

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.

the problem is, with math in particular, that lectures are not very useful. its not the fault of the professor really, as its difficult to learn and understand math in the mere 2-3 hours alotted a week. math is simply too abstract to be digested by a mere lecture, and if you are lost in one step, the entire lecture becomes a textbook copying section. this contrasts say biology where the lecturer can show motion pictures and diagrams to illuminate the textbook readings. this is why math class attendence is low almost everywhere.

as for the attitude of doing only what is "due", this too is very logical. there is simply too much college homework for anyone to keep up with. if people were to do all the suggested questions in all their courses, they would be running on 5 hour sleep. because all that matters is grades, most people will only focus on getting assignments in so that they can succeed.

i personally think every math prof should write his own textbook for his class to use, and that this should be the students main source of information. additional examples could then be provided in lecture. i mean seriously, is there any reason to be copying lemmas and proofs word for word on the board?

 

[mathwonk]

one thing that can help is to actually listen and ask questions during the lecture. also if you pay attention and know what is coming up you can read up on it in advance, and then it is easier to follow the proof. it may seem there is no time to study this well, but in fact there is. and in fact it is the only way to learn the material well. i have often written a textbook for my classes but this does not prevent some students from refusing to read it. declining to ask questions, or come to office hours, or even to class. if time is important then never ever miss class. a professor can explain in class material that will take three or four times as long to learn independently, so every hour of class missed is 3 or 4 hours of time lost. and often it is simply impossible to recover what the professor gave.

 

[mathwonk]

let me give another example of how illogical it is to only do what is "due". on a test, if there are 4 problems required and one "extra" problem, some students will not even attempt the extra problem. but if there are 5 problems, and none is called extra, most will attempt them all. where is the logic in that? no matter what the problem is called it is still worth points.

but some students will simply not do anything they do not think they have to. so if i want to increase my students scores, instead of having 4 required and one extra problem, if i simply give 5 problems and score the test over 80 instead of 100, they will do better because they will try them all.

 

[khemix]

reading in advance defeats the purpose of going to lecture. lectures are supposed to give one a grounding in what will be read. this works in less abstract courses, but in math i personally come out with very little from a math class expect some historical facts.

i don't know if its a matter of intelligence, but very few people can keep up with a math lecture no matter how hard they listen. as you are reasoning one step, the prof is already on another, so either you be stubborn and try it out or not risk missing the new information. i sit there and day dream and take notes, because at this point the effort isn't worth it. i know ill get a real understanding when i read the book. because the book is patient. i can't stop and ask the prof hundreds of questions because its unfair to the class, and he has an agenda to go through. this is why even going to class is useless; the course is in the textbook.

you are correct in saying students will only do what is required. efficiency is key in undergrad, and you rarely have time to go beyond the requirements because there is likely another course that needs your time. also, any free time you can snatch isn't likely to be spent on more work - a break is very nice from time to time.

as for the extra problems on tests, i can only give my personal student prespective. if a problem is listed as extra, i immediately assume its super hard and is only attempted by the best students. this and extra problems tend to have stricer marking schemes.

 

[Moonbear]

the problem is, with math in particular, that lectures are not very useful. its not the fault of the professor really, as its difficult to learn and understand math in the mere 2-3 hours alotted a week. math is simply too abstract to be digested by a mere lecture, and if you are lost in one step, the entire lecture becomes a textbook copying section. this contrasts say biology where the lecturer can show motion pictures and diagrams to illuminate the textbook readings. this is why math class attendence is low almost everywhere.

Except students have the same attitude in biology classes. What the students don't seem to recognize is that if they get lost during lecture, they can make a note about where and now they can save time reading their textbook by going straight to that section rather than having to read the whole book end-to-end with no idea of what is important.

as for the attitude of doing only what is "due", this too is very logical. there is simply too much college homework for anyone to keep up with. if people were to do all the suggested questions in all their courses, they would be running on 5 hour sleep. because all that matters is grades, most people will only focus on getting assignments in so that they can succeed.

The workload in college today is lighter than when I was a student, and even lighter yet than when some of the older faculty were students. I have access to years of course materials and exams from the now deceased professor who used to run the course I'm taking over, and it's amazing how much has been cut out over the years. Yet, students used to be expected to learn it all...and did. College IS full time study. It doesn't help that students all seem to think the weekend starts on Thursday night now. Cutting out a day of studying certainly makes it harder to get everything done, but that is not the fault of those of us who are teaching them.

It's frustrating that students don't take any responsibility for their own learning. If I don't have mandatory attendance, or give quizzes in class, or require something be turned in every class period, they don't show up to lecture or try to cram the week before the exam rather than keeping up with the content all along.

i personally think every math prof should write his own textbook for his class to use, and that this should be the students main source of information. additional examples could then be provided in lecture. i mean seriously, is there any reason to be copying lemmas and proofs word for word on the board?

Do you have any idea how long it takes to write a textbook? Even more so, how long it takes to write a GOOD textbook? Why reinvent the wheel if there are good textbooks already available?

As for why to use examples directly from the book to write on the board, while this isn't my favorite practice, it sometimes makes sense. For example, to make sure all the logic of each step is explained clearly. And, when you complain about how much work there is, this is cutting down on the amount of reading you need to do outside class so you can spend the time working on practice problems. I had a student this week ask me what to read in their textbook (not math) because "there's a lot in the book" on the current topic. We spent 4 lectures on a single chapter. I pointed out that I covered more in my lectures than is in the book. If anything, the book I'm currently stuck with (will change next year...it was ordered this year before I was hired to teach the class) is very inadequate for the material covered and glosses over important topics, so I went into much more detail in my lectures.

Part of the problem I'm seeing, though, is that large numbers of students have already made it into their sophomore year with atrocious study skills. When I taught freshmen, I expected this...they all came from different backgrounds, and some could breeze through high school without studying. But, that should be caught in freshman year and corrected. Now that I'm teaching sophomores, I'm still seeing a lot of students who basically waste a ton of time studying because they aren't doing it at all effectively.

 

[mathwonk]

khemix, at least you are posting your views here, which helps the discussion, but some of your views are sadly incorrect, and they are holding you back. the idea that trying to learn enough by reading in advance to be able to actually follow a lecture you yourself think is almost impossible to follow cold, is ludicrously self contradictory. you are clearly intelligent, but opinions like that one, and your argument for not trying extra problems, sound as if you are trying to give yourself an excuse to be lazy.

 

[Moonbear]

reading in advance defeats the purpose of going to lecture. lectures are supposed to give one a grounding in what will be read. this works in less abstract courses, but in math i personally come out with very little from a math class expect some historical facts.

My advice to my students is to SKIM the textbook chapters ahead of the lecture. Know the overall content and direction of the assigned reading before walking into lecture, and maybe even have a few questions in mind of content that didn't make sense the first time through. That will help follow the lecture more easily, and ensure you know the right place to stop and ask questions if the areas you KNOW the book is not going to adequately cover for you are also not adequately covered in lecture. After the lecture, go back and reread the sections that were emphasized for detail and to ensure your lecture notes make sense and are complete.

i don't know if its a matter of intelligence, but very few people can keep up with a math lecture no matter how hard they listen. as you are reasoning one step, the prof is already on another, so either you be stubborn and try it out or not risk missing the new information.

If you spent some time looking over the book BEFORE lecture, you would have an easier time following the lecture. This is the problem of not even looking at the chapter before class. If you have already read the general content and know where the tricky parts are, you can switch your focus in class from just trying to keep up with taking notes to listening for tips and hints on how to get through the sticky parts. And, you know the right places to ask questions.

You seem to think professors got where they are without having been students too. We know what the undergraduate course load is like, and that it can be done.

i can't stop and ask the prof hundreds of questions because its unfair to the class, and he has an agenda to go through.

No, you can't ask hundreds of questions, but you can ask a few if he went over a point too quickly and everyone else is just as confused about it.

you are correct in saying students will only do what is required. efficiency is key in undergrad, and you rarely have time to go beyond the requirements because there is likely another course that needs your time. also, any free time you can snatch isn't likely to be spent on more work - a break is very nice from time to time.

As in my previous reply, the problem is that students aren't very efficient. They tend more toward hastiness than efficiency. If you are really efficient in your learning and studying process, it does not take as much time, and you would have time to do the extra work. In fact, that extra work IS necessary for learning and part of efficient studying. Just because you aren't told to hand in a problem set for a grade doesn't mean that you don't need to practice those problems to do well on an exam.

as for the extra problems on tests, i can only give my personal student prespective. if a problem is listed as extra, i immediately assume its super hard and is only attempted by the best students. this and extra problems tend to have stricer marking schemes.

Okay, so they're going to be hard and graded harder...why is that a reason to not even bother attempting it? If you've struggled with the entire rest of the exam and run out of time for that problem, then it makes sense to skip that problem, but just because it might be hard is no reason to skip it. That's just laziness, and is precisely the problem seen in students today...they aren't interested in challenging themselves to learn as much as possible, only as little as is required. If you have an hour for an exam, and there are still 10 min left at the end of finishing the required problems, why would someone choose to hand in the exam early rather than sit for the remaining 10 min and see if you can get a couple extra points on the challenge problem? When I was a student, we knew those problems were hard, so we didn't get frustrated if we couldn't get the solution, but there was no harm in giving it a try...in fact, we could only benefit from getting a bonus point or two even if we couldn't get the whole problem solved. It could make up for a silly mistake somewhere else in a required problem, or give an extra cushion if you get a low score on another harder exam, or a bad quiz grade on a day when you're not feeling well.

I think part of the problem is students take university for granted rather than recognizing it as opportunity. You have 4 years when your only requirement or responsibility is to learn as much as you possibly can to prepare you for any number of paths when you are done. Never again will life be so easy.

Then again, if a student really only cares to do the bare minimum to pass a class, it is their choice. They will receive the grade that reflects they did the bare minimum to pass, which in my courses is a C. If they want to be C students, I can't force them to do better. However, they also shouldn't be the ones showing up at the end of the year complaining that they didn't get a B.

 

[epenguin]

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 am losing it.
The patience I mean.
Mainly caused by the about half who post a question one responds to, and then never come back.
But I saw yesterday a post by someone who came back after a year and said he had solved it so maybe I am not patient enough!

 

[cdotter]

If you spent some time looking over the book BEFORE lecture, you would have an easier time following the lecture. This is the problem of not even looking at the chapter before class. If you have already read the general content and know where the tricky parts are, you can switch your focus in class from just trying to keep up with taking notes to listening for tips and hints on how to get through the sticky parts. And, you know the right places to ask questions.

This isn't directed towards you, but I really hate this kind of reasoning. I am paying tuition to be taught, and not have a professor regurgitate a textbook.