Calculus Formulas

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  • #1
BasketDaN
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I despise taking notes in school, and never do, so I was wondering if anyone knew of an online list of all formulas relating to Calculus so I can just highlight the ones we're learning as we go along in school. (I'm taking AP Calculus BC)

Thanks
 

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  • #2
StephenPrivitera
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You should get used to taking notes. Math isn't about memorizing formulas. You can't get good at math by watching someone else do it. You have to do it yourself.
 
  • #3
chroot
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Since there are an infinite number of "formulas relating to calculus," I'd suspect you'd have a hard time finding a list of them on the internet. Do what Stephen tells you -- get used to taking notes.

If you just hate taking notes in class, you might be able to get away with just listening to the lectures, and then going home and taking notes while reading your book. Either way, Stephen is right: you can read a book or listen to a speaker and have no trouble follow along for the entire ride -- but you won't be able to remember a single thing unless you actually use it.

- Warren
 
  • #4
BasketDaN
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Taking notes isn't 'getting good at something by doing it yourself' though.. it's just copying down exactly what you see BEING done, and I prefer just listening, and listening that much harder b/c I dont have to focus on movements of my hand. Anyway it's all good 'cause I found a site with them
 
  • #5
Gale
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There's tons of sites, it just depends on your preference. Google it. Plus, your book should already have most of them, and if not, many teachers let you write them in on the covers, or in the pages near the back cover where nothing's written. Of course, writing them your self in your notes ensures that you'll understand them, and help you remember them as well.
 
  • #6
phoenixthoth
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i stopped taking notes and found it more helpful to try to prove the theorems on my own rather than listen to the trite old methods for proving them.

i believe dr. nash didn't even go to class at all, much less take notes.

it's kinda like becoming a mechanic. there are essentially two ways. one way is to go to school where you learn how to do everything except anything new. one way is to teach yourself; that way, your only experience is what is with what is new, new to you at least.

cheers,
phoenix
 
  • #7
mathwonk
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Here is a compromise I recommend: Go to every class and listen in class as closely as possible. Then immediately afterwards go to the library, take out some paper and write out everything you remember in as much detail as possible. This way you think through everything again, and also produce useful notes. You also remind yourself of what you forgot, and make a note to review it somewhere. If it seems you are not able to remember much, you might need to take notes in class, or listen more effectively.

Another tactic is to read the material the night before to make effective listening easier.

Anyone addicted to memorizing formulas may be interested in viewing the play "The Lesson" by Ionesco.

The main point people are making here though is the unavoidable truth that reading notes is not nearly as effective for learning as writing notes is. When I write out an outline for my class of the material we have covered, I learn it very well by doing so. They on the other hand learn much less by merely reading my outline, so I always advise them to write their own.

Anyone who seeks to learn math by imitating the behavior of Professor Nash by skipping class however, does so at his/her own risk. It may be that Nash was more talented, and more eccentric, than some of the rest of us. At least missing class seldom worked for me, especially since I usually spent the time saved by playing "hearts".
 
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  • #8
JasonRox
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BasketDaN said:
Taking notes isn't 'getting good at something by doing it yourself' though.. it's just copying down exactly what you see BEING done, and I prefer just listening, and listening that much harder b/c I dont have to focus on movements of my hand. Anyway it's all good 'cause I found a site with them

I find this to be true.

On the other hand you have to take notes of important things, like definitions and what not. Examples or metaphors to help you understand the math is not necessary to write down. You should sit back and try to understand, and not take most of your time writing notes. My notes are really messy, and are barely readable, but I re-write them neatly after class.

Note: I am fortunate enough (well not really) to have a hearing disability, which isn't really noticeable when I put my hearing aid. Because of this, I went for "Help for Disabilities" for the first time this year, and they hooked me up with a notetaker. Although I'm doing fine now because it isn't anything I haven't seen before, I am doing this because later on when I see new material I don't miss anything. Basically, I listen and write sloppy notes, re-write notes, read notetakers notes, read textbook, and try exercise questions.

Also, I find that understanding the material is the only way you can succeed in math. Memorizing will get you past the first term, and then you're toast. Obviously lots of people forget the "Quadratic Formula", but those who experienced on how to deduce it themselves will remember for the rest of their lives (until alzheimers kicks in). :biggrin:
 
  • #9
Chrono
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StephenPrivitera said:
You should get used to taking notes. Math isn't about memorizing formulas. You can't get good at math by watching someone else do it. You have to do it yourself.

Exactly. One of my teachers would always say that math is not a spectator sport.

Besides, I think it would take quite a bit of work to memorize the Taylor Series of Expansions.
 
  • #10
bross7
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Unless you are being taught very poorly the course shouldn't be entirely comprised of the notes anyway. You should probably be able to get away without the notes, but it is usually the concepts that are taught that are the important thing anyway. No copying down examples from class/lectures doesn't improve your skills, but it does give you an idea of how to solve certain types of problems so when you run into them, you may remember a certain mathematical trick or two.
If you are dead set against copying out notes then just go to class and listen, if there is something that you feel is worth remembering then copy it down, otherwise prepare to have the world's largest memory.

As for equation lists. Math, like every other subject, tends to like to make equations have insane amounts of subscripts and notations so I don't think you could pay somebody enough money to take the time to put them all in a nice neat organized order.
 
  • #11
robert Ihnot
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phoenixthoth said:
i believe dr. nash didn't even go to class at all, much less take notes.

Yes, but Princeton is its own kind of school, and the graduation rate of freshman is very high. (Ohio State was 50%.) Niederhoffer, writer of Education of a Speculator, said that he "Niederhoffered" Harvard, which was to say, when called upon to recite, he always said that he had not studied that yet because he was prepairing a special paper about a special project that he would present later to the class. The professors never objected to this.

But, at most schools there is a lot of testing, such as Ohio State U. Thus the person has to handle this note taking question seriously because something may come up almost immediately on a short quiz. I alway felt that if I understood the freshman and sophmore material that notes were not that necessary.

But, I know people who felt the very opposite, especially by the time they were a senor, and went to the trouble of actually memorizing notes and formulas so that when the test came up they did not have to try and recall anything. I guess that could work too.

But the real important thing is not the grade itself, but whether you manage to make it to the next level, and at that point, nobody worries about how well you did on last year's tests. Remember it has been said, and is at least a half truth, "Testing is for your own benefit," and if you pass that is mostly right.
 
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  • #12
mathwonk
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why am I not surprized this is the same guy who "forgot his book" the night before the test.
 
  • #13
JasonRox
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mathwonk, you burned this fellow bad. LOL.
 
  • #14
mathwonk
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well let me admit, i was much worse than he as a student. i once got a D- in calc 2 and when i reviewed my notes found that I had only been to class once per month that semester.

i am very tough on this behavior because it cost me 10 years of my professional life wasted, when finally i learned better. so i am like the "repentant" sinner who goes around blasting everyone else for sinning.

even now i have work i could be doing. so my apologies to basketdan. hang in there young man. please remember however that being smart is something of a curse.

i.e. you have succeeded so far without working because you must be very bright. but there will come a day of reckoning, when the "bottom end of the curve" has dropped out of the course, and all the competition is both smart and hard working.

as my basketball coach told one guy, "do your self a favor big fellow and play hard on both ends of the court."

good luck.
 
  • #15
mathwonk
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it is that time of year:

when I marvel that all my students know a vector basis is a set which is both "independent and spanning" but few know what either term means.

at least some know that a function is integrable if its "upper and lower sums" can be made arbitrarily near, but no one can define an upper sum correctly.

most know a riemann sum is a sum of products f(xi*)delta(xi) but few know what either (xi*) or delta(xi) mean.

most know the derivatives of x^2 and sin(x), but few know what a derivative is.

several can state the "limit test" for a global minimum, but none know what a limit is.

most can state the fundamental theorem of calculus (every continuous function is the derivative of its indefinite integral) but all claim that the continuous function e^(x^2) has no antiderivative, and most do not know how to define a continuous function.

some say they fully understand "the math", they just don't get the theorems, proofs, definitions, and corollaries.


it seems hard to change the things people care to learn, to include the ideas rather than just the numbers.

merry xmas!
 
  • #16
JasonRox
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mathwonk said:
it is that time of year:

when I marvel that all my students know a vector basis is a set which is both "independent and spanning" but few know what either term means.

at least some know that a function is integrable if its "upper and lower sums" can be made arbitrarily near, but no one can define an upper sum correctly.

most know a riemann sum is a sum of products f(xi*)delta(xi) but few know what either (xi*) or delta(xi) mean.

most know the derivatives of x^2 and sin(x), but few know what a derivative is.

several can state the "limit test" for a global minimum, but none know what a limit is.

most can state the fundamental theorem of calculus (every continuous function is the derivative of its indefinite integral) but all claim that the continuous function e^(x^2) has no antiderivative, and most do not know how to define a continuous function.

some say they fully understand "the math", they just don't get the theorems, proofs, definitions, and corollaries.


it seems hard to change the things people care to learn, to include the ideas rather than just the numbers.

merry xmas!

Thanks for the review! :biggrin:

Calculus I Exam next week and Linear Algebra Exam tomorrow.
 
  • #17
mathwonk
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you are a good man jason rox
 
  • #18
mathwonk
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here is another remark prompted by giving my final: on test 4 i announced in advance a certain proof would be asked. namely i would ask the class to prove the only function f with f'=rf and f(0) = c, is f(x) = ce^(rx). I gave the complete proof, emphasizing that the first step was to prove that the quotient f/e^(rx) is constant by taking the derivative and applying the mean value theorem, which implies that any function whose derivative is zero is constant. then evaluate the constant by setting x=0.

everyone did well. then one week later, unannounced, i gave the exact same problem on the final, with the hint: first prove the quotient f/e^(rx) is constant by the usual method. it seemed no one knew what that method was nor had any idea how to proceed, even after I said "remember this is a differential calculus course and we have a special way of recognizing constant functions".

I could not understand how anyone could have forgotten in only one week a simple idea like taking the derivative to show a function is constant, especially since we had emphasized it all semester. worse yet, to forget a proof which had been learned correctly one week earlier. i guess the only way is if someone just memorized the proof the first time without bothering to do any thinking at all about what the argument meant.


it is almost as if many students are dragging their feet so hard, trying not to do any thinking at all, that it is really challenging as to how to get them to benefit from what they do as exercises. so the main thing to remember is that as a student one must always work on oneself to try to understand the idea behind the discussion at hand. the whole point is to use a few sample problems that one solves in class as models for a wider class of problems that one will meet elsewhere. this cannot be useful unless one sees how to generalize the idea behind the given exercise to use it in a new situation. this is the goal of all exercises, not just to do as little as possible to slide by the course.
 
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  • #19
JasonRox
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mathwonk said:
here is another remark prompted by giving my final: on test 4 i announced in advance a certain proof would be asked. namely i would ask the class to prove the only function f with f'=rf and f(0) = c, is f(x) = ce^(rx). I gave the complete proof, emphasizing that the first step was to prove that the quotient f/e^(rx) is constant by taking the derivative and applying the mean value theorem, which implies that any function whose derivative is zero is constant. then evaluate the constant by setting x=0.

everyone did well. then one week later, unannounced, i gave the exact same problem on the final, with the hint: first prove the quotient f/e^(rx) is constant by the usual method. it seemed no one knew what that method was nor had any idea how to proceed, even after I said "remember this is a differential calculus course and we have a special way of recognizing constant functions".

I could not understand how anyone could have forgotten in only one week a simple idea like taking the derivative to show a function is constant, especially since we had emphasized it all semester. worse yet, to forget a proof which had been learned correctly one week earlier. i guess the only way is if someone just memorized the proof the first time without bothering to do any thinking at all about what the argument meant.


it is almost as if many students are dragging their feet so hard, trying not to do any thinking at all, that it is really challenging as to how to get them to benefit from what they do as exercises. so the main thing to remember is that as a student one must always work on oneself to try to understand the idea behind the discussion at hand. the whole point is to use a few sample problems that one solves in class as models for a wider class of problems that one will meet elsewhere. this cannot be useful unless one sees how to generalize the idea behind the given exercise to use it in a new situation. this is the goal of all exercises, not just to do as little as possible to slide by the course.

I barely study for exams at all. For now, everything is pretty easy. I'll go through the theorems and proofs again, but I understand everything.

One of the coolest things I think, for a Calculus I course, is how you can prove that if something is differentiable at point x, it is also continuous at point x. You can prove it directly through the definition of the derivative, which is kind of neat.

Yes, students don't think and don't like to think. That's not good for me because I like to just talk math/physics, but unfortunately no one else does.

I don't know if it was like this before, back in the day, but I do know that it really sucks today.

Math club... in your dreams.
 
  • #20
mathwonk
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deltay/deltax converges to some finite limit L if f is differentiable, as deltax converges to zero.

It follows that deltay converges to L times the limit of deltax, i.e. zero, hence f is continuous.

this is nice, but there is a lot more going on than this level of argument. if this is what impresses you now, you have not even scratched the surface. you could get a lot deeper with some effort.
 
  • #21
whozum
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mathwonk said:
here is another remark prompted by giving my final: on test 4 i announced in advance a certain proof would be asked. namely i would ask the class to prove the only function f with f'=rf and f(0) = c, is f(x) = ce^(rx). I gave the complete proof, emphasizing that the first step was to prove that the quotient f/e^(rx) is constant by taking the derivative and applying the mean value theorem, which implies that any function whose derivative is zero is constant. then evaluate the constant by setting x=0.

everyone did well. then one week later, unannounced, i gave the exact same problem on the final, with the hint: first prove the quotient f/e^(rx) is constant by the usual method. it seemed no one knew what that method was nor had any idea how to proceed, even after I said "remember this is a differential calculus course and we have a special way of recognizing constant functions".

I could not understand how anyone could have forgotten in only one week a simple idea like taking the derivative to show a function is constant, especially since we had emphasized it all semester. worse yet, to forget a proof which had been learned correctly one week earlier. i guess the only way is if someone just memorized the proof the first time without bothering to do any thinking at all about what the argument meant.


it is almost as if many students are dragging their feet so hard, trying not to do any thinking at all, that it is really challenging as to how to get them to benefit from what they do as exercises. so the main thing to remember is that as a student one must always work on oneself to try to understand the idea behind the discussion at hand. the whole point is to use a few sample problems that one solves in class as models for a wider class of problems that one will meet elsewhere. this cannot be useful unless one sees how to generalize the idea behind the given exercise to use it in a new situation. this is the goal of all exercises, not just to do as little as possible to slide by the course.


This is exactly why I dislike studying the way its done now. People study to pass an exam, not to know material, and I think the latter is so much more valuable.
Tests arent effective in guaging a students understanding for that reason.
 
  • #22
mathwonk
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well, if people were doing well on tests and then not really understanding later i would agree with you, but as it is, many students are not even doing well on exams when told in advance what to expect.

so i am saying the threshold is extremely low and many people are still stumbling over it. moreover my experience shows that studying to understand the material is the most effective way to prepare for an exam, but many people refuse to accept this.

For example, even without preparing, and over material I have not seen or taught for decades, and not in my area, I still outperform almost all PhD prelim takers every year on the PhD prelims written by others.

I.e. when I have to grade a prelim, I just sit down and take it myself first, even in real analysis (my worst subject) or complex analysis, or topology, or algebra, even if i have not studied it for years and years. I almost always significantly outperform all takers of the test, who have prepared for months and months on that specific topic. In particular I always pass, whereas most students fail. It is precisely because i understand the concepts and can apply them in various settings.
 
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  • #23
whozum
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The standard has been lowered to accomodate for a student's inability to learn material effectively, I think its just a downhill spiral from here.
For example, I got A's in all my E&M exams, and I cna honestly say I blow at E&M. The fact that I could get by without knowing this stuff is so wrong.
 
  • #24
mathwonk
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you are right. we are penalized for having too many students withdraw from our classes, so we make them easier and easier every year.

but the students can always drag their feet even more.

but to be fair some are woefully unprepared from high school, and have no clue what ti means to study or try hard at all.

most of my students wanbt to do well, and want to do what ia sk, they really have no clue how to proceed. the high schools are apparently teaching nothing at all in many cases.

so in college calculus i cannot assume algebraor trig, or geometry, or how to read a sentence, or how to reason, much less how to amkke a proof.

my favortite criticism of my teaching was "I can't understand him at all, he teaches with WORDS!":
 
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  • #25
whozum
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Yeah, and the same words apply not only to the college level but throughout. For example, even pre-high school, students who struggle are just shuffled along to the next grade in hopes they will cathc up, but in reality they wont. In high school, emphasis is placed on the wrong things. Students should learn that they are responsible for their own education, and it is only the teachers job to teach you, not to tell you what you need to know. A student should know all that his teacher prescribes, and should be prepared for anything on an exam.

If you were to make those assumptions on algebra and trig, and assigned an exam based on perfect understanding of these elements, its no doubt that many more students will fail the exam. For example, in Calc 2, partial fractions is taught as a method to evaluate certain integrals. Should they really have to teach this? It should follow immediately from their workings in algebra courses that the integrand is expressible in simpler terms and it should be intuitive for the student to make that judgement and solve the problem on their own.

STudents are instead dumbed down and are retaught things that they should already know, and the students see this as "If I need it, theyl'l show me it next time", and when that next time doesnt come, they get overwhelmed.
 
  • #26
mathwonk
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you may not believe this, but on day one, a few years back, in calc 1, i gave a pretest to see how much trig and algebra 1 people knew. the average grade out of 100 was 10.

so i gave it as a hw assignment to do as take home.

the next day several people did not even show up and the average grade on the take home was 15/100.

so i no longer assume anything at all in calculus 1 from precalculus.

many people think all real numbers are integers. e.g. they will say the domain of a function define on all positive reals is [1,infinity).

makes it hard to solve if the answer is 1/2, or 4/(2pi).

but i try not to get discouraged, thats what summer break is for. recharge batteries, get some vitamin d, go back and try again.

especially i try to learn something new myself, and maybe do some research. thus i am sharing the experience i expect from them, i.e. i am trying to learn something. I am not just going in and repeating the same stuff over that i learned 30 years ago.

This hearks back to a fine graduation speech i nread from a speaker at brandeis, years ago, where he said roughly: "at a research university your teachers are trying to do the same thing they expect from you, to learn something."

This is an advantage of going to a university where the faculty do research, which is almost all universities today.
 
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  • #27
robert Ihnot
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I want to say that maybe it is good idea not to overtest. When i was teaching or learning, when we had the Friday quiz thrown in, students are too rushed to learn the latest and do not get the material mentally assembled for the midterms. Too much testing interferes with the student organizing the material. Quizzes count for little on the final grade anyway.
 
  • #28
bomba923
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*Not until I took calculus did I realize that time spent on practice was inversely proportional to time spent finishing test problems.
I remember all my math classes before I took Calculus (I began w/second semester college calculus, not first--so I had to catch up during the year!), where homework and attendance were really quite pointless---seriously, it was so freakin' easy! (why do HS teachers grade on "work"--I don't understand--seriously, tests should=100% grade!). I'd just sit in class Adv.Alg./Trig class, solving easily whatever teachers put on the board (OK. I admit. I like math--always been interested in logic, reasoning, mathematics/physics). But anyway, I got graded down for not turning in work (which was stupid and tedious for any student who wanted to think and knew how to think--and well, like me, hated stupid "reviewing" with a zeal).
*When I accidentally skipped first-semester calculus (not even I know how that came to be!--but it's your job to decide how this impacts my story), I was overcome by the large quantity of knowledge needed to begin second-semester calculus, and got a low grade for the first two quarters. But I caught up with a zeal--really, a zeal, and got full scores on my AP BC test (yes--one test, "two" scores) ---though I agree, tests never gauge ability completely (but that does not mean giving 20-30% for homework/other work! Though neither "fully" gauges student ability--i believe tests at least measure student ability better than homework ever can, anyway).
**About those few first low grades--well, let's just put it this way:
"Without practice, a student might take five or more minutes solve a difficult (well, by second-semester calculus standards--i suppose!) integral on a test. Way too long (even with some conceptual-technical understanding). However, with practice, those integrals take less than a minute (or some amount of seconds)--and the student develops SkillS."----and that is when I learned the value of practice (no, not the kind where the teacher assigns various easy/tedious problems, but the kind that YOU decide where YOU need practice on, YOU need to improve---regardless of what the teacher says!). And well, um..that's my story i suppose.
(*rational individual responsibility--is...a responsibility! :biggrin: )
---------------------------------------------------------------
*When I think about it, BasketDan, do you dislike writing notes because you fear that by copying formulas/equations off of a board or book you will not learn ("internalize") the material--only memorize it (even if you wanted to learn in the first place)?
 
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  • #29
mathwonk
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peace on earth, good will to people.
 
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  • #30
bomba923
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Well, like I said,
bomba923 said:
(*rational individual responsibility--is...a responsibility! :biggrin:)
----------------------------------------------------------------
mathwonk said:
i did not mention that the one asian woman in my class scored 100 on the test the americans averaged 15 on. so i tend to think it has little to do with philosophy of teaching but lots with attitude of students.

Exactly how old are your students?

*From experience (I graduate HS in 2006), most HS students (not me!) will not really do the work or study hard (Not "just study"--but study hard). So, the philosophy of teaching plays a major role in HS learning, just as teachers motivate/inspire students to succeed, learn, and think.

*From taking classes at a local college, the teacher lectures a mature audience of adults that diligently study, work on their own (e.g., do homework even if it counts nothing for the overall grade--because they want to excel or because they know what they must practice and study), and academically assist one another. They are responsible for their own knowledge and capabilities----and here is where that
"attitude of students"
really comes in. In no way is the instructor obligated to provide a (for lack of a better work---"cushy"-->) "philosophy of teaching" to "motivate/inspire" students---like HS teachers often do (and often fail at it as well)

I may be slightly overstating the HS-adult contrast, but I believe you understand my point.
Anyway, just curious---about how old are your students? (or do the ages vary from minor<-->adult?)
 
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  • #31
mathwonk
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students are ages roughly 18-22 in the classes i was discussing.

older students are indeed much more responsible.

returned teachers have the best attitude. of course students vary, but when porerequisites are clearly stated, it is discouraging to have a large number who have no visible preparation at all.

Many people attempt calculus without knowing how to write an equation for a straight line, or what slope means, or what cos and sin are, or the area of a parallelogram. In fact this is typical.

but attitude is even more important. on the occasion when the average score was 15/100 on a prerequisite test, the test was given again as homework, overnight, with open book access, and the average only went up to 20/100.
 

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