Teaching calculus today in college

[mathwonk]

I also liked the suggestions from EC21. Time to implement them is of course the next challenge. I am grading exams now and have more questions.

My students have learned from frequent repetition that there are at least two simple properties, each of which imply that a function has an integral, namely it is sufficient for the function to be either continuous, or monotone.

The problem is this fact is useless to many of them, because some students do not seem to know how to recognize monotonicity, nor to understand the difference between "and" and "or", nor between "necessary" and "sufficient".

I.e. immediately after stating that each of the properties above imply the integral exists, some students claim that a function which equals 1 for x between 0 and 1, and equals 2 between 1 and 2, is NOT integrable, "because not continuous", or even "because not monotone".When asked to state a theorem "with hypotheses" about half seem not to grasp that this means to include the "if" part, the part that tells you when you can use the conclusion.

The "solution" adopted by some is to essentially avoid the use of words, statements of theorems, or arguments of justification for claims. just present computations, and even allow calculators for those, so that none of the rules of computation are even internalized, nor any computational power developed.

To me this is adding to the problem, i.e. that is why many high schools have stopped doing the job of teaching these things- because it is hard to accomplish. But if everyone cops out of trying to teach the use of language and reasoning in discussing concepts, it just gets pushed further and further down the line.

As suggested above, this results in inverted teaching in college. I.e. we continue to teach calculous first, as we did when entering students already knew algebra, geometry and reasoning, but now we teach those prerequisites afterwards.

I.e. calculus is now a 2000 level course, but reasoning and proof is a 3000 level course, and algebra a 4000 level course (this is where students now learn about polynomials and rational numbers), and euclidean geometry is a 5000 level course!

To teach calculus this way, one apparently assumes that students will ignore all parts of the book except the (easier) exercises, never read the explanations, nor even the worked examples, much less the theorems and proofs, and one then spends the class time merely working example problems instead of explaining phenomena and concepts.

But if this model is accepted, wouldn't make more sense to teach from a book like "calculus made easy" or schaum's outline series? instead of stewart or thomas or even better books?

Do you think it could work to re order the courses in college to reflect this change, teaching reasoning, geometry, and algebra first, and calculus later? This would perhaps be resisted by the students who want calculus for other majors, but don't even applied students need to understand how to apply the math correctly?

 

[mathwonk]

here is a simple conundrum. frequently on a test students have correctly stated in some form, the FTC "part one", i.e. that a continuous function always has an antiderivative and that antiderivative is given by the indefinite integral.

then on the next question, most or all of them have stated that some specific continuous function, like the absolute value function,or e^(x^2), does NOT have an antiderivative, even though all apparently knew these to be continuous.

it is hard for me to communicate with student who sees nothing wrong in making contradictory statements. i can only assume such a student does not know what his statements mean.

It seems many students simply do not realize that the indefinite integral i.e. the integral of f from a to x, is a function of x. Without this they cannot understand the FTC, which says this function is an antiderivative of f, when f is continuous.

Without this understanding, of making a function from a definite integral by letting the endpoint move, calculus is just a process of memorizing rules for areas and volumes without knowing why they work or when they work.

Has anyone succeeded in teaching what the FTC says, and why? does it help to give it a name, like the "moving area function"?

well three students got it right yesterday, so i guess that is progress!

 

[mathwonk]

the prerequisite problem is raising its thorny head again. in integration, there are 2 methods, substitution and integration by parts. substitution was supposed to be covered in the previous course so we only reviewed it, then taught parts in detail and repeatedly.

on the test most people are getting tedious parts questions correct and many are missing easy substitution questions. these are entering students who took the course previously in high school. maybe we could do some kind of intervention for these students, i.e. stronger than a placement test.

maybe we could have a summer session to help prepare incoming students for college level expectations. but how could we get high school students who think they are above average take a summer remedial course before college?

maybe we should just bite the bullet and admit that essentially all entering high school students are remedial in some way, and simply start out all freshman courses in a remedial way. it seems tricky.

but it is almost impossible now to cover traditional syllabi, when nothing can be assumed as understood from before. one big adjustment seems to be from a learning style where all a student has to do is the required work, to one where the student has to take responsibility for learning the material, doing whatever is personally needed by that student.

i like some form of moonbear's ideas on students presenting or at least talking in small groups. we all know we learned our stuff best when we prepared it to present to a class.

the trick is how to allow students to practice presentations without inflicting a lousy student presentation on the whole class.

i have tried letting students practice the presentation on me in advance but many still did poorly, for one reason or another, usually refusal to practice.

maybe talking in small groups avoids this problem. somehow i feel this uses time in a way that college students should be more grownup than to need, but I am also seeing the need. But ideally it seems these groups should be outside class and supplemental to it

The best model I know of is still Uri Treisman's where there were guided outside problem solving sessions, but we have not been able to provide the support and get the participation for these in the past. maybe we could get a grant for them. this is essentially the vigre model, which is working for grad students, but still in formative stage for undergrads. also it only supports research oriented behavior not basic learning.

 

[Count Iblis]

but how could we get high school students who think they are above average take a summer remedial course before college?

Let students take voluntary test exams after such courses. If they do well, they pay less tuition fees. They can participate in these exams without following the remedial course, if they think if they don't need it.

 

[mathwonk]

here's another problem that surprised me: several students think the inverse tangent function is 1/tangent, i.e. cotangent.

the book does write it as tan^(-1), but i explicitly pointed out the possible confusion, and always wrote it as arctan for that reason.

these are again students who have had calculus in high school and passed AP tests high enough to exempt first semester college calc, and do not know what an inverse trig function is. we did not treat them from scratch but have repeatedly used and calculated with them, using the defining property that tan(arctan(x)) = x.

maybe these students are among those who confuse function composition with multiplication. it is hard for me to mentally orient my expectations for students who are supposedly strong and advanced calculus students, but whose gaps in knowledge are those of very weak or beginning precalculus students. i just don't know where to meet them. if a course has certain prerequisites and the students have been placed there it seems natural to assume some of those things.

again i think outside discussion sessions might give more opportunity for random ignorance to surface and be corrected. indeed when i was in college we had problem sections with TA's leading them, and i felt more at ease there, but there is no money now for those. maybe we should go to larger lectures, but with problem sections as well. maybe a lecture will not be much worse if larger. Again, qualified TA's and funds must be found for them.

 

[mathwonk]

when i look back on my own classroom career as undergrad student, i did not master anything there, but i was fired with enthusiasm and excitement for some of the topics, which i then mastered later.

in this vein i try to show my students some of the connections that i have noticed while teaching the material in their course, such as the link between work and volume.

I.e. if you look at a uniformly massive plane region below the x-axis and compute the work done to raise it to the level of the x axis, assuming that is ground level, you get an integrand like yL, where L is the length of the horizontal slice of your region at depth y.

If you think about it, this is the same as the integrand used to compute the volume generated by revolving this region around the x axis, by cylindrical shells, except for a factor of 2pi.

thus if we are doing a work problem with a solid region, such as pumping the water from a swimming pool up to the surface, it follows that this is the same except for a factor of 2pi, as the 4 dimensional volume generated by revolving this solid around the x,y plane, in 4 space!

consequently, if the swimming pool is a hemisphere, and one computes this work, one can obtain the volume of the 4 sphere easily by multiplying by 2pi, which turns out to be pi^2/2 R^4, where R is the radius.

I.e. just as a 3 dimensional ball is generated by revolving a half disc around a line, so a 4 ball is generated by revolving half a 3 diml ball around a plane.

I hope this sort of thing will magnetize some of them to want to understand more mathematics, more effectively than learning to integrate tan^3.

it turns out that the volume of an n diml ball of radius R, equals (2/n)pi R^2, times the volume of an n-2 diml ball of radius R. Maybe some will have fun trying to puzzle out why?

this is essentially archimedes' discovery for the three ball and the one - ball (line segment), since he knew the ball occupied 2/3 the volume of the smallest cylinder containing it. i.e. (2/3)(piR^2)(2R) = (2pi/3)R^2 times (2R) =(4/3)pi R^3.

I guess this sort of thing is more my forte than mindless drill.

 

[Tac-Tics]

maybe these students are among those who confuse function composition with multiplication.

Function composition is hard because it doesn't come up very much explicitly. The notation for functions also causes problems. I don't think any freshman in college (nor many that graduate) are able to distinguish between the function "f" and the evaluation of f at a point x, "f(x)". In algebra, students learn "y = f(x)", but the it leaves it unclear notationally whether y is a real number or a function. Most students can get the correct answer without being able to make this distinction, but it doesn't lead to a solid understanding.

here's another problem that surprised me: several students think the inverse tangent function is 1/tangent, i.e. cotangent.

Why do they even teach those weird trig functions? I don't even know a geometric situation in where any of the reciprocal trig functions come up. I always found them to be particularly useless, and in high school, I refused to memorize them. I'd rather work out the derivatives with the chain rule at the start of each test.

I do agree that the notation is bad. Personally, I always write out arcblah.


It seems like you're really frustrated with some of your students and their level of preparedness (or competence in some cases). Does what you've written here reflect the majority of students, though? What are the best students' abilities like? What kinds of misunderstandings or gaps in knowledge come up with them?

 

[buffordboy23]

Mathwonk,

Have you ever consulted the educational literature on teaching mathematics and improving student achievement?

As a former teacher, I had the opportunity to read some articles. Some were beneficial to my effectiveness as a teacher.

 

[mathwonk]

I have extensively read articles on teaching methods for decades, and tried many different tactics. the definition of "improving achievement" however is not even clear agreed upon. E.g. raising scores on standardized tests is a common goal. the booiks by john saxon were written with this in mind,a nd for a while studies showed they succeeded.

they succeeded however at the price of de emphasizing thinking skills and totally focusing on rote drill. thus student achievement was raised in a narrow even harmful sense, while understanding was completely undermined and sacrificed.

these books were used for years at my son's school until at last they concluded "after using saxon the students didn't understand anything".

my situation is less of someone who is not familiar with the suggestions out there, but of one who has spent some 50 years trying them. some one has said lack of success is about the teacher, but this is not the experience i find. indeed it is almost all about the students in many settings. if we have a teacher we want to promote, it is easy, we just assign that teacher to honors classes with the best students.

these students actually show up, do the work, and appreciate the teachers efforts, and they say so on evaluations. bingo the teacher looks good.

 

[mathwonk]

here is a specific flaw in algebra the "saxon way", or the mechanical way.

letters in algebra are used properly to represent a range of values. they must be manipulated in a way that would be valid for any of the range of values they may assume.'

hence they are just placeholders for any of a set of values. this is amde clear in jacobs's books, where blank boxes are sued in place of letters at first.a student who assigns meaning to the letter it does not deserve thinks the same letter must always mean the same thing. e.g. such a student can solve a separable ode

of form dy/dx = 1 + y^2, by rearranging it as dy/(1+y^2) = dx, and integrating both sides.

but if you ask that same student to solve the ode f' = 1 + f^2, he cannot do it. to him the letter f cannot be treated like the letter y.

there is no understanding that a variable is a variable is a variable, no matter what it i called. this loss of the grasp of how to properly use variables creates another gap in understanding for many college students today, who may even have used calculators to "solve algebraic equations" the meaning of the statement represented by the equations is totally lost upon them.

 

[buffordboy23]

I see. The cloudiness that exists for the definition of what constitutes "improvement" is definitely a problem.

How does your definition of improvement and teacher expectations compare with other colleagues in the university setting?

How about compared to those of organizations like the National Council of Teachers of Mathematics, who define standards for K-12 mathematics education? What about the state of Georgia's standards for math education? Do their standards seem appropriate for college expectations?

 

[Tac-Tics]

some one has said lack of success is about the teacher, but this is not the experience i find. indeed it is almost all about the students in many settings.

It's a societal thing. Unless they have a personal interest, there is no motivation at all to do it. American society teaches us that math is nerdy, and if you like math, you're socially inept. You can't blame the students. You can't blame your teaching methods either. It doesn't matter if your class is the sturdiest rung in the ladder if the rest of the rungs are damaged.

e.g. such a student can solve a separable ode

of form dy/dx = 1 + y^2, by rearranging it as dy/(1+y^2) = dx, and integrating both sides.

but if you ask that same student to solve the ode f' = 1 + f^2, he cannot do it. to him the letter f cannot be treated like the letter y.

This seems like an excellent technique for testing understanding as opposed to memorization. If the student isn't able to recognize the underlying concept, he or she is in trouble, because you can't apply any technique to solve the problem until you know what kind of problem you're dealing with.

there is no understanding that a variable is a variable is a variable, no matter what it i called.

The idea of a variable is not a simple one. Learning about functional programming languages and formal logic helped out tremendously.

 

[mathwonk]

I am a bit out of touch with the official definitions of performance and expectations. As I posted earlier, here at Georgia, we have one of the best math and science education departments in the US, indeed it was the ONLY department considered exemplary in a survey of 77 departments nationwide.

Nonetheless the performance of Georgia students on nationwide standardized tests is almost at the bottom of the nation, e.g. in SAT scores. What to make of this?

A long time ago, a friend who was involved in developing materials, asked me to evaluate some of the material then in use for measuring high school teacher qualifications in the state. There was an official list of topics to be covered in schools at various levels, and there was a test to gauge teachers mastery of these.

My role was to evaluate the practice and review materials offered to the teachers who were preparing to take the test. It was simply abysmal.

The syllabus was far too optimistic for one thing. There was every topic in the world on there, and it is completely hopeless to expect any high school teacher to know all that. i still don't know all those topics after 35 years in the field as a professional.

When you see something like that you know it was made up by someone who knows even less than you do about the topics, someone who just took a list and decided that our teachers should know everything anyone might ever want them to know, or we should pretend they do.

then the review materials of course covered only a tiny fraction of those topics since someone had to actually know something abut the topic they were pretending to offer a review of. so the syllabus was a fake.

But the review questions were also sadly inadequate in most cases since they were clearly prepared by people who did not understand in the least the topics they were trying to test. Most of the questions either were unrelated to what they were supposed to test, or were trivial, or were actually wrong. Many of the multiple choice questions contained no correct answers at all, even though the review book said one of them was indeed correct. The reasons offered for the correctness or incorrectness were also wrong.

in reviewing the calculus materials i actually solidified greatly my own understanding of the process of finding volumes, since i needed to really understand it well to recognize wrong questions quickly and to be confident i was in fact right, when the answer book said otherwise.

When you see something like that, you immediately hypothesize that it was prepared by someone who had some kind of tenure in lieu of qualifications. My friend confirmed that there was indeed a grandfathering mechanism in place whereby the teachers already in place were deemed qualified by that fact, and were then asked to test the candidates.Things have presumably changed enormously since those bad old days.

Nonetheless, there is a lot of politics involved in education, and no matter what we teach our teacher candidates, they still have to go out and please some local school board that likely as not is focused on standardized test scores.

After all that's all we have to go by in many cases. that's all that is telling us our students here perform badly. I had a blessed opportunity once to teach in a local private school to a highly selected class of their best high school students and teach them whatever I wanted out of a book i chose. i taught linear algebra and vector calculus out of marsden and tromba, a book once used at berkeley. and i did it for free.

still the results were mixed. several of those students who took my class because they wanted to, did excellently, went to ivy league schools and obtained phd's in math and physics. others who were there to please their parents hated the class and felt justified afterwards when their college courses at state schools were in fact easier than mine. they thought that proved i was teaching a bad class, because they didn't need such a hard class to prepare them to pass in a mediocre school.

If I hadn't been required to give grades it would have been fine, since the misfits would not have been threatened and the best of the good students would still have worked hard.

l believe that in the introduction to one of the most famous physics texts in existence, by one of the most famous and celebrated professors in history, the feynman lectures, he admits that the actual course he taught was only successful for a few happy natures among his students. teaching is a collaboration between student and teacher, and there has to be an appropriate match for it to work.

 

[mathwonk]

One of the reasons the UGA math ed dept is so well regarded is they focus on student understanding,a s opposed to rote learning.

in part, they use materials developed by dr. sybilla beckmann that are really quite well done. It is no trivial matter to successfully teach these materials however, but they are widely praised for their potential value in improving learning.

We are interested in identifying people who are dedicated to helping teach these courses and wish to make a profession of developing outstanding teachers. This of course involves taking students who have been taught up to now in the traditional ways, and trying to produce graduates who value understanding, and are committed to teaching for understanding.

If someone wants a graduate math, or math ed degree, focusing on that aspect of the profession, they are invited to apply. Contacting Dr. Beckmann is a good way to begin.

 

[mathwonk]

here is another specific challenge to teaching basic calculus today, arclength.

the formula is simple enough: integrate sqrt(1 + (y')^2).

the problem is that it is hard to integrate a square root.

there is a trick that is usually used to make doable examples. namely

the simple fact that (a-b)^2 + 4ab = (a+b)^2, can be used if we rig

our arclength example so that (y')^2 has form (a-b)^2 where also ab = 1/4.

I.e. then we get (1 + (y')^2) is a perfect square. the problem is however that it is

almost impossible to give an example that today's students can simplify correctly.I.e. after carefully explaining this algebraic trick, most in

my classes can still simply not simplify 1 + (y')^2, when y = say x^2/4 - ln(x)/2.

here we get (y')^2 = (x^4 -2x^2 +1)/4x^2, so that adding 1,

changes it to (x^4 +2x^2 +1)/4x^2, a perfect square. this is just too hard.sometimes i try an easier version, like y = (2/3)(x^2 + 1)^(3/2),

where we get (y')^2 = 4x^4 + 4x^2, but still less than half of a typical class can

see that then 1 + (y')^2 is a perfect square.I suspect this problem is part of the reason that in the recent book by Rogawski,

arclength is set apart from volume and work,

in a chapter called "further applications of the integral"

as if it is somehow more advanced, and may be skipped.When a typical calculus student cannot recognize, even with specific instruction,

that (a-b)^2+ 4ab = (a+b)^2, something seems amiss.

 

[mathwonk]

the persuasive discussion above almost convinces me at last, that it is better to eschew such artificial arclength examples, and focus instead on natural ones, like finding the arclength of a parabola like y = x^2/2, where the difficuloty instead is to integrate

the integrand sqrt(1+x^2), which yields to a trig substitution,plus some tedious integrations by parts.

maybe todays classes will actually find those difficulties more palatable than the algebra of (a±b)^2.

 

[maze]

I think the parabola example will be much better received, and here's why:

If you use contrived examples based around a trick like using perfect squares, your students will get the wrong impression that finding arc-length is all about perfect squares, when really the two topics are quite unrelated except for in this relatively unimportant class of examples. The example will also seem terribly unmotivated - there are an infinity of different possible curves, and you are choosing a random one for your own seemingly nefarious purposes. It reinforces the negative view that math is all about applying a sequence of memorized manipulations to transform symbols from one form to another.

Better to provide a difficult but well-motivated example. "what is the arc-length of a parabloa"? That's a simple example that anyone would want to know. Some students in the class probably already asked themselves this question. People will naturally work harder if they genuinely want to know the answer.

 

[Count Iblis]

When a typical calculus student cannot recognize, even with specific instruction,

that (a-b)^2+ 4ab = (a+b)^2, something seems amiss.

Then your students haven't been exposed to math at school enough. I don't think it matters much exactly what they learn, as long as they are working with math formulas and doing non-trivial manipulations they shouldn't have such problems. But math education in school is limited to learning a few specialized techniques and practicing them over and over again, making math in school very boring.

It's a bit like trying to teach language to children by first letting them learn the alphabet then words and then grammar, sentence construction etc. etc.. Only if they have mastered all that perfectly will you think of letting them read books. I think that by that time they would have dropped out and those who haven't would have great difficulties learning to read and write.


In case of math eduction, we need to start changing things in primary school. We need to spend far less time on teaching children arithmetic. Because today we have calculators and teaching things like long division is a complete waste of time. It is useful later, if you learn algebra and want to divide polynomials. Then you also understand why the long division algorithm works at a deep level. Most children in primary school do not really undertand why the algorithm produces the correct answer.

We have to focus more on teaching things that are relevant. Today we work a lot with computers, so it would be a good thing to get rid of a lot of arithmetic in primary school and instead teach logic, computer programming etc. Children then get used to working with undetermined variables. They get to see what they are doing is working when they compile the code they have written.

Then learning algebra will be much easier for them. One can always return to letting them learn more arithmetic later. Then it won't be a trick they have to use, but they can understand much better why it works. Topics like modular arithmetic, Euclid's algorithm, Chinese Remainder theorem etc. should also be taught.

 

[Count Iblis]

Example of arithmetic using algebra:

If you understand that (N-X)(N-Y) = N(N-X-Y) + XY

You can simplify multiplications in your head. It does not offer any speedup from an algothmic point of view, but you can transform numbers to simpler numbers that your brain can handle more easy.

So, if we have to multiply X, and Y and one of these numbers is close to some round number N, then putting X' = N - X and Y' = N - Y, we have X*Y = N(X - Y') + X' Y'

Suppose then that you want to multiply 89 with 92. Then you choose N = 100, and we have 89*92 = 100*81 + 8*11 = 8188.

You can also use this iteratively. The great advantage of this is then that you can choose your N so that you get numbers that are easy to work with.

Multiplication in the tradional way is useless in practice if the computations you have to do are not simple. You either have a calculator or you are in a place where you quickly need to compute something in your head. Rarely will you be somewhere where you have a desk with paper and pencil and no calculator or computer.

Another example.

87*73

If we take N = 100, then we have:

87*73 = 100*60 + 13*27

If we take N = 30 to compute 13*27 we get:

13*27 = 30*10 + 17*3 = 351

So, 87*73 = 6351

 

[EC21]

Without this understanding, of making a function from a definite integral by letting the endpoint move, calculus is just a process of memorizing rules for areas and volumes without knowing why they work or when they work.

Has anyone succeeded in teaching what the FTC says, and why?

I am not a teacher and can’t share anything that worked. I can, however, tell you personal and observed difficulties in trying to understand the FTC. You may already be aware of these roadblocks, but if not then perhaps you can come up with some remedies.

The FTC was taught early into my first calculus course. I perceived an integral to simply be the “area under the curve” and the FTC was a coincidentally cute trick to finding it. Why? One reason is because other students told me so. Another was because the book’s statements used a level of abstraction beyond my current level of comprehension and consequently I did not understand them. Additionally, colloquial usage of the word “theory” implies uncertainty (e.g. I have a theory about wheat bread, theory of evolution, etc.) and so I didn’t take the FTC seriously. Finally, I asked around for the applications of calculus and was generally told that it found the area/volume of a variety of “nice” regions.

I also didn’t know what a function was. Some unlearning was involved as I had incorrectly thought f(x) to be an exclusive synonym for y, the dependent variable. Elementary notation increased my confusion. If a function is to be thought of as a sort of taxi between sets, (f o g)(x) seems much less intuitive than ((x)g)f.

 

[mathwonk]

with regards to jeff foxworthy, you may be an algebraist if ((x)g)f seems natural to you.

this is indeed the algebraists' rule. but they are the apparently only people who refuse to bow to tradition in this matter.

 

[mathwonk]

some people question whether any theory should be included in a practical calculus course. I still feel some grasp of theoretical aspects is needed to use the material correctly.

here is an example. the integral test for convergence of a series does not work unless the terms a(n) of the series can be extended to functional values a(x), such that not only is the improper integral of a(x) finite (from x=1 to x=infinity), but also the function a(x) has to be decreasing.

yet none of my students bothered to verify the decreasing part when using it, although we gave examples in class of series which diverge although the function has finite integral, when this property does not hold.

one didactic technique for this type of thing was introduced in the book in another test, the alternating test, which the book called the "3 condition test", to remind the student how many hypotheses there are.

this trick has limited use however, since it is hard to call every theorem the "n hypotheses theorem".

 

[mathwonk]

if you have a sudden loss of hearing or ability to concentrate, or your wife tells you you are becoming really boring, stop taking calculus immediately (due credit to cialis ads).

 

[mathwonk]

after all is said and done this semester, one overall problem I still see is the tension between a professor as teacher (or one who gives insight), and as "personal trainer".

For the student, it is crucial to attend class and do the recommended work regularly, to read independently, formulate and ask questions, and visit office hours for help with difficulties. It is a truism in the profession that the most successful student is the one who displays these behaviors, not the one with the quickest mind.

But what to do when faced with the realization that many students do not do all or sometimes any of these things? Then the professor feels pressure to "force" compliance with these good habits, i.e. to become a personal trainer rather than a teacher. This occurs especially in high school when teaching is measured by student performance on standardized tests, and it is hard for many students to begin to behave differently when entering college.

In the short run, indeed the techniques of personal trainers get better results on tests.

In the long run, however, do these techniques allow students to delay taking responsibility for their own learning? How can one enable sincere but naive students to acquire basic information, in spite of their poor study habits, and yet also encourage students to begin to assume responsibility for learning?

The professor is essential for sharing insight acquired over decades into a difficult topic, but for acquiring skill at a technique that has already been explained, only self discipline is needed. How much practice is it essential for the professor to enforce, and how much should be expected from the learner?

In a nutshell, if it takes three repetitions for a skill to become habit, some students will expect it to be repeated three times in class. But in college it is more usual to present it once, and then expect the student to do the other two reps at home. How to get this lesson across, so the professor's time can be spent more profitably explaining new topics, and deeper aspects of the subject?

One approach is to do the repetitions in class, making time for them by omitting the deeper aspects of the subject. This is called dumbing down the course. Creative ways to avoid this approach are needed.

Perhaps best is the time honored one of making advance assignments, then sending random students to the board to display what has been done. This is very time consuming, but experience shows it is helpful for many students at all levels, even into graduate school. Indeed since even professors use this method in learning seminars, perhaps the realistic thing is to reduce the syllabus of undergrad courses to allow more time for this activity.

 

[Troponin]

To put my very, very late for the seminar point of view here...

I am returning to school at the age of 29, 7 years after completing a degree in kinesiology, to study math and physics.

I spent my entire life believing I hated math and was horrible with numbers. I was told I wouldn't be good at math because I don't have the focus to follow the structure and order that maths require.

I was a good student throughout my life, and even did well in math up through geometry. The following school year in pre-algebra (this was early high school), I grew to dislike mathematics more than any other subject.

It seemed that every single thing we did was an arbitrary rule to follow for no other reason than...that is what you did.
Any question of "why" a rule was as it is was quickly hushed to make more time for practicing arbitrary rules.

Any question that didn't follow the order and rule set for that type of problem was marked wrong...regardless of the correctness of the answer.

I could continue with examples and frustrations, but I'm sure it will be nothing that hasn't already been covered here.

As a child, I loved science and drawing. I loved reading about the stars, looking through a microscope...even loved doing those math books you get at the drug store. I had always wanted to be an engineer...for no other reason than I was once told they were scientists that "drew things." lol

When ever I would mention that career goal to an adult, they would undoubtably give me the same response, "you know..that's a LOT of math..."

After pre-algebra, I avoided math classes for the rest of my studies. I took no math electives that weren't absolutely required in college and accepted that I hated math. Years later, I got the urge to read up on cosmology and astronomy again...like I used to love when I was a kid. After about a dozen "made for the public" physics books, I decided I should learn some math so I could read something with a bit more detail.

I purchased a "teach yourself calculus" book with the belief that it would be a torturous process I had to slave through for the better good of my reading.

Turns out I LOVED it. When left to my own, I could spend the time learning the theory of each concept and seeing what it could do from top to bottom. I poured through the differential portion of that book in about a week.

I'm now back in school and should have a second degree with Applied Mathematics major and Physics minor completed next winter. I hope to move on to a PhD program in Physics (hopefully theory...) following that.

I don't have any real advice to offer the situation, but it frustrates me deeply to look back at my grammar school years and remember how math was presented to me.

Math was for the socially inept nerds...organized accountants that love nothing more than spending their day doing long division. I'm happy to know that I'll be able to present mathematics to my children in a manner that allows them to see what it really is (or what I think it is). But, it frustrates me to know there are probably many more science nerds like me that were taught to hate mathematics.

 

[Tobias Funke]

Troponin, it's nice to read stories like yours and I only wish they were more common.

I don't have any real advice to offer the situation, but it frustrates me deeply to look back at my grammar school years and remember how math was presented to me.

Yeah, it's sad when my 9th graders tell me that their teachers marked them wrong if they left their answer as 3/2 instead of 1 1/2, or how they were forced to write 2+(-3) instead of 2-3, for example. Stuff like this is very common in elementary and middle schools and I'm glad you recovered from it. Good luck!

 

[Troponin]

Troponin, it's nice to read stories like yours and I only wish they were more common.



Yeah, it's sad when my 9th graders tell me that their teachers marked them wrong if they left their answer as 3/2 instead of 1 1/2, or how they were forced to write 2+(-3) instead of 2-3, for example. Stuff like this is very common in elementary and middle schools and I'm glad you recovered from it. Good luck!

Oh yes...my first semester back in school involves a story about that. I went up to my professor and asked a situation similar to that...he looked at me like I was insane. To my relief, he said "it doesn't matter...it's the same number...I don't know what you're asking?!?"

I'm sure he was even more confused when I seemed extremely happy with his non-answer...lol

 

[mbisCool]

When i was younger my teacher marked all of my answers wrong because instead of putting a semi colon between my answer and restriction for what x couldn't equal, i simply put a space...

She also marked me wrong when the end of my square root symbol didnt fully extend over the last number. :/

 

[Troponin]

When i was younger my teacher marked all of my answers wrong because instead of putting a semi colon between my answer and restriction for what x couldn't equal, i simply put a space...

She also marked me wrong when the end of my square root symbol didnt fully extend over the last number. :/

I have a differential equations professor that is FANTASTIC. He doesn't care about ANY of that. He wants you to "understand" what is going on...and almost prefers you to not follow the "plug and chug" list of operations for each type of solution.
He's a very difficult professor so a lot of students seem to really dislike him, but I absolutely love him.

All the arbitrary accounting style order and rules that I've always felt had more to do with memorization than understanding the materials are considered just that...arbitrary memorization that doesn't prove if you understand the material.

 

[descendency]

What are some ideas on how to improve this?

More abstinence from people who shouldn't be parents!

By the time you get these students in Calculus I, you have gotten a student who has developed years of **** poor mathematics. I can't say I was a great mathematician taking calculus I, but I can say I knew what the basic terminology was.

Knowing what a plane, a denominator, and other very simple concepts are is fundamental to have any hope of passing calculus.

Unfortunately, teaching fundamentals and refusing to work with students who won't meet you on a fundamentals level is the only hope you have.

If they don't understand what a number line means how will they understand a delta-epsilon proof? or a limit? or handedness? or derivatives? or integrals? or techniques of integration? or optimization? or applications of integration? or anything else?

edit: As for marking off points for sloppy work, you deserve them. When an instructor sets out a standard, you deserve points off for not being able to follow a standard. Half-***ing it is just not acceptable nor should it be. Not drawing a division bar long enough shows how little you care for the work you are doing. You feel your time is being wasted. If that's the case, don't waste your instructors time with your test/homework.

If the instructor themself can not follow a standard though, then you have a gripe. However, drawing arrowheads on your graph to indicate direction of increasing is not too much to ask. Or labeling the axes.

 

[uman]

I would consider it unreasonable that a teacher would mark off for using a semicolon instead of a space.

 

[Tobias Funke]

Teachers should only set reasonable standards. What if a teacher wanted you to write a...@b instead of a+b for the semester? Sure it's an extreme example, but it's no more silly than requiring 1.5 instead of 3/2 (unless the point of the exercise is to get practice with decimals). Descendency, it seems like you're thinking of something like the difference between \sqrt{a}b and \sqrt{ab}, which definitely deserves some correcting. I'm talking about much more trivial things.

Also note that these experiences all seem to stop in college. Professors aren't going to care if you write A-B or A\setminus B, but too many 1-12 teachers would and it's just plain stupid sometimes.

Actually I guess the equivalent in college would be those annoying computer homework sites that only accept the answer in a certain form that doesn't seem to be any better or neater than your own. Luckily I've never had to use them, but I tried helping a friend of mine and was very frustrated.

I feel like ending this on a positive note though. Today one of my students got a 100 on a math test for the first time in a year, and another student said "this is actually kinda fun once you get it" about solving systems of equations! That just makes the day so much better

 

[qwerty2x]

We had our year 12 end of year exam for specialist math(for the whole state) and we were allowed to bring in a bound reference. Everyone came in holding a large book with LOTS of papers glued into it that had problems and questions with the working out, while i came in with a lousy one-paged A4 sheet of paper with almost nothing on it as i didnt know what to write on it(its a math exam what am i suppose to put on it??).
Anyway the exam started and I am doing the problems, i look up and i see people flicking through their papers "trying to find a similar question" to what was being asked.

I ended up getting the highest in the class and in the top 7% of the state, i guess the way the other students learn the material is really the problem these days(mind you their parents are payed **** loads for private tutiton).

would you say that the rate of drop-outs for engineering students in higher than that of most other disiplines?

 

[symbolipoint]

qwerty2x, you made a nice observation about test-taking and possibly study methods, and then after that, you made a statement which seems unrelated. Readers may have clearer responses or comments if you would show a transition to that last statement.

One reason ignoring quantity for why any university or college student would drop from their formal education is to work, do an employed job, or start a business. You are probably trying to address something opposite to that. Engineering may be difficult to study well. Students might drop because they need to work to earn money/more money, or change major field due to difficulty of coursework, or drop intending to return but never do (but possibly work for many years after dropping from Engineering study).

YOU might be either more talented than the other test takers, or you have better study methods than they, or you may have studied longer and harder than they. Good Work!

EDIT: How much did this test involve Calculus? Could you tell us which topics from Calculus were in the test's content? Did the test rely on multiple choice answers, were showing written steps a main feature of your responses to the questions? Were proofs involved? Were essays involved? These are useful questions in this thread-topic since this could relate to the title, "teaching calculus today in college".

 

[qwerty2x]

the exam was made up of 22 multiple choice questions and 5 extended answer question.
Here is a link to part 2 of the exam(cant find part 1)(not uploaded by me).

http://trinon.info/exams/VCAA_2008_Specialist_Mathematics_Exam_2.pdf

EDIT:yeh sorry that statement does sound unrelated, just wanted to get a feel of the difficulty of engineering at uni.

 

[buffordboy23]

the exam was made up of 22 multiple choice questions and 5 extended answer question.

There are many questions on that test (particularly complex variables and differential equations) for which my high school education would not have prepared me. And, I was an accelerated mathematics student in my school, which was a very small school in a somewhat rural setting.

Who takes this test? Where is it administered? What is its purpose?

 

[qwerty2x]

There are many questions on that test (particularly complex variables and differential equations) for which my high school education would not have prepared me. And, I was an accelerated mathematics student in my school, which was a very small school in a somewhat rural setting.

Who takes this test? Where is it administered? What is its purpose?

Every year 12 student that takes the subject in victoria(approx 6000 people i think, small compared to other subjects) take this subject, it is like advanced math and i think its the second hardest math that a year 12 can take in victoria. administrators are VCAA.

Part 1 of the exam was much harder even tho it was only 10 questions.

 

[kbaumen]

I feel like ending this on a positive note though. Today one of my students got a 100 on a math test for the first time in a year, and another student said "this is actually kinda fun once you get it" about solving systems of equations! That just makes the day so much better

I definitely agree to the last sentence. Of course, I'm not a teacher, just a 12th grader but sometimes (quite often actually) I help other students from my class and it really feels nice to hear them say "wow, now that you explain, it seems quite interesting, hey if I substitute this with that, you get this and that's how you prove that, so this is another application of that, wow..." I can imagine, that being a teacher even if 99 of your students think that you suck as much as the subject does, it's still worth sucking it up if later you get a "thanks a lot, this is very interesting" from one student.

 

[Moonbear]

Teachers should only set reasonable standards. What if a teacher wanted you to write a...@b instead of a+b for the semester? Sure it's an extreme example, but it's no more silly than requiring 1.5 instead of 3/2 (unless the point of the exercise is to get practice with decimals).

In elementary school, when they make you do exercises like that, it is indeed the objective of the exercise to practice manipulating fractions or converting fractions to decimal points. It's not just a teacher being nitpicky, it's a teacher trying to ensure the students have the fundamentals correct. Part of that is making sure a student understands that 3/2 is the same as 1 1/2 and isn't confusing it with 2/3. When learning fractions, confusing the numerator and denominator is commonplace, so forcing students to do added steps that demonstrate they comprehend the distinction is necessary. The point of homework and exams is for the student to demonstrate they have mastered whatever topic they are learning, not that they have provided some reasonable semblance of an effort enough to get away with others guessing what they are thinking. And, when so many mistakes in math happen because of carelessness and sloppy handwriting, it's good for teachers to enforce those rules early so students develop careful habits for when the math gets harder. It's the same reason teachers grade notebooks; it makes students develop good habits.

 

[maze]

But it also causes kids to feel like math is nothing more than pedantic manipulation of symbols according to arbitrary rules. This couldn't be further from the truth, and it directly contributes to the dislike of math by the general population.

 

[buffordboy23]

But it also causes kids to feel like math is nothing more than pedantic manipulation of symbols according to arbitrary rules. This couldn't be further from the truth, and it directly contributes to the dislike of math by the general population.

Are you suggesting that making students accountable for the quality of their work and meeting teacher expectations is the reason for the dislike of math?

In adding to what Moonbear and Tobias Funke stated previously, I think the most important question to ask regarding this matter is, "Does such and such requirement align well with the lesson or overall course objectives?" For third or fourth graders, requiring them to carry out each step when it comes to fractions is acceptable in my opinion for the reasons Moonbear mentioned. But for a typical eighth grader, forget about it since this skill should have already been obtained.

I agree with what you said on why math is disliked by the general population: it's nothing more than pedantic manipulation of symbols according to arbitrary rules (and a nice catchy mnemonic tune every now and then). I think the fundamental and underlying reason for this is the teacher and the methods of instruction employed. Usually, these instructional methods are simply the pedantic manipulation of symbols.

 

[vsheng]

I am a student in university, recently i am facing problem to deal with my calculus lecturer,
well, actually they are certain similar point with your problem with the problem between me and my lecturer...

"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. "

Most of my friend do not ask question because they can't even understand what the lecturer talking about, for those understand who are able to ask question, they are often in a ;hald understand' condition, when they are in that condition, don't expect them to think normally as a normal person.


"Questions more often focus on "what will be tested?" instead of how to understand what has been taught."

Every student are worry and emphasing the result of their academic, if not, they are not actually concern about their future..

"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"."

Student have to take a lot of subject beside calculus, and they have to join activities too,
further more, as a student, when they are in holiday, many of them will just let their mind relax in a this short period,and they of sourse, they do not study the previous subjest that they have taken...so, sometimes if they forgot some important steps, it does not show that they are not a good student...

 

[Tobias Funke]

snip

I agree with you since you were specifically talking about elementary school. But what I'm seeing with my high school freshmen isn't comprehension and easy conversion between number representations. They're just confused. They didn't seem to understand that 22/5 is a number and they insisted on changing it to 4 2/5 or even wasting time doing the long division. Their previous teachers have forced them to write fractions in a certain way and it was hard for them to change. Like maze said, I'm telling them one thing in opposition to their middle school teachers, probably leaving them to stop thinking about the subject and just following my orders.

They're improving their abilities to memorize facts which to them are meaningless and regurgitating on their quizzes. That's not the kind of teaching I had in mind and I'm not sure I'm qualified to teach the low level math that they need. I chose high school for a reason, after all. Next year I'm going to ask to separate the students in the first week or so into algebra 1 and pre-algebra. Whether they officially do so or not, if I teach the same class next year it's going to be a reintroduction to fractions, half eaten pies and all, for about 3 months.