Teaching calculus today in college

[symbolipoint]

i try to give my students practice in simple reasoning grounded in their everyday experience; for example:

1. dr. smith's students have placed their hopes in his teaching skills.
2. those who place their hope in weakness are in deep trouble.
3. dr. smith has some of the weakest teaching skills in existence.
conclusion?

Mathwonk,
You need to modify that presentation or many weaker or less motivated students will simply enroll in courses taught by other Mathematics professors and intentionally avoid taking further courses from you. You run the risk of students making informal counter-recommendations about you as teacher. This could result in less newer students enrolling in your classes. Students who are inadequately conditioned academically will not be able to properly appreciate the meaning of a professor expressing having weaknesses in teaching-skills.

 

[buffordboy23]

i try to give my students practice in simple reasoning grounded in their everyday experience; for example:

1. dr. smith's students have placed their hopes in his teaching skills.
2. those who place their hope in weakness are in deep trouble.
3. dr. smith has some of the weakest teaching skills in existence.
conclusion?

I understand your approach in trying to teach reasoning skills to students. You should be commended for your efforts, because I see that much of the focus is on teaching students to obtain content knowledge rather than skills to think about the content.

I think students need to learn such thinking skills. Moreover, I believe that there are common themes in thinking about mathematics, as shown through the texts written by Polya and Solow I mentioned in a previous post. These themes seem to be absent from a typical students' education experience, at least from what I have seen. Therefore, students do not have the skills to solve problems which rely upon these themes. Please correct me if I am wrong with this assumption.

 

[mathwonk]

hows this?

1. the best examples of teachers are those who are held up high by their own students.

2. the best examples of teachers are indeed paragons of greatness.

2. dr smith was carried out of town on a rail held high by his class members.

conclusion?

 

[mathwonk]

well indeed thinking skills, proof skills, and general argumentatiion are missing from most classes and hence from most students.

we have for some years now tried to remedy this by offering courses in proofs and logic in colleges at the junior/senior level, such as my current course.

this however is apparently too little too late. this need used to be approached in sophomore year of high school by teaching euclidean proof based geometry, but that has all but disappeared from many schools.

so today things are completely upside down. instead of geometry and algebra to 9th and 10th graders with real content, we teach watered down calculus to these same people who now do not even know what the letters QED stand for.

then after they do not learn that, we teach them calculus again in college which again fails because of a lack of algebra skills, and then we try to teach proofs and logic to juniors, and then finally euclidean geometry to seniors and graduate students.

'this hodge podge of remedial teaching is quite a failure all round. for the few lucky ones, we teach them calculus from a good book like spivak in which they are pre -taught algebra and logic and number theory, and calculus with proofs.

this is a brutally intense way to make up in one course for all that is missing beforehand from high school, but is much better than the non honors program. so the best students are very well taught today in some colleges, but the others are much less well taught.

Oh and spivak courses are no,longer offered at such "top" places as Harvard, since their students decline to take it, having had calculus in high school. their beginning course, and that at stanford, is a course from loomis and sternberg or apostol vol 2. this pace quite destroys even some of the most capable but presumably works again for a privileged few.

 

[mathwonk]

how does a average student cope with a situation like that at harvard today, where the description on their website of math 55 reveals that harvard itself does not offer a course that can prepare its own students to take this course?

i.e. this course can only be attempted by someone who has at the minimum mastered style course. it really is aimed at students who in high school have already taken numerous college courses before heading away to college. so harvard sees no need to try to help prepare the merely talented youngster for its own best courses

this is also true elsewhere. these are really elite schools, since without having prepared in very rarefied circumstances, they do not offer you access to their best courses.

 

[Count Iblis]

What are some ideas on how to improve this?

Just take a look at how we teach physics at university. I actually learned calculus myself at high school from engineering books. I mastered topics ranging from ordinary differential equations, partial differential equations, Fourier series, Laplace transforms, complex function theory (Cauchy's theorem, residue theorem). All that at age 16, all by myself and I enjoyed learning it.

Thing is that there is no way that I could have mastered these topics from pure math lecture notes that explain everything rigorously. Then the proof that any integral over a closed contour of an analytical function is zero would take ages.

This does not mean that you can leave out all proofs, but I think that by presenting things in a way that we in physics do, you can teach far more math in high school.

 

[mathwonk]

it really doesn't take long to understand the closed contour fact mathematically.

by the fundamental theorem of calculus, if you are integrating dF along a curve,

the answer is F(b) - F(a) where a,b are the endpoints of the curve.

All analytic differentials f(z)dz have the form dF. QED.

The physical interpretation is that of a force filed that has a potential, i.e. that is conservative.

So in mathematics we have a hypothesis that implies conservative. how does one in physics explain why some fields are conservative and others are not?

I.e. how does one use physics to explain why the integral of f(z)dz around a closed contour not containing any singularities of f, is zero, just because f is analytic?

by the way i agree with you that physics is a great vehicle for learning calculus. it so happens physics is not exactly the most popular subject with many students either. Indeed trying to use physics to illuminate calculus is usually quite challenging since most calculus students in college over the years have known no physics at all.

I used to read to them from galileo, show them the laborious geometric proofs he gave of his results on falling bodies, and try to make the point that calculus renders these easy enough for anyone to derive. sometimes the only thing that got their attention was me jumping up on the table and dropping something, or throwing the chalk in a parabolic arc neatly into the trash can. but we keep trying.

 

[EC21]

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

What are some ideas on how to improve this?

The problem of competitive (un)education

You have competition. It was there before your students entered your classroom and will probably be there after they’ve finished your class. My guess is that you are trying to teach your students good reasoning skills. For every hour of education in your classroom, however, they are likely receiving countless hours being uneducated.

For example: a student turns on the TV and sees Kobe Bryant drinking a bottle of Sprite. Perhaps he identifies the claim as Sprite being desirable. The proof? It’s because someone respectable vouched for it. Following this line of reasoning, a student might think a math proof is whatever you, an expert, said so.

Another example: a government official on TV is claiming that it’s time to go to war. The proof? It’s because now is the time for war. Following this line of reasoning, one might think it justified to prove 1+1=2 because 2=1+1.

Logical fallacies in popular culture or media may seem unrelated to math, but consider the effect that consistently reinforced poor reasoning has on the development of math skills.

Here are some ideas of approach:
1) Offer a token amount of extra credit for each student that prepares an abstract of their goals/interests/hobbies along with their picture and name. This survey let's you know what you’re up against.
2) Perhaps humorously, analyze some logical fallacies in popular media. The goal is to show the power that good reasoning skills have in discerning fallacies (wherever they appear). I think personal discovery is important. If a student has poor reasoning skills, a logical argument may be both ineffective and unpersuasive.


The problem of naïve student view on education

Learning begins with the acceptance of dogma. There is no inherent reason why counting should be 1-2-3 as opposed to 3-1-2. The role of an elementary school teacher is comparatively (and perhaps necessarily) authoritarian. Somewhere around high school and college, the roles change.

I would argue that in college, teachers and students are now engaged in partnerships; each party has their obligations. The problem is that there is no class in school or social cue that appears to facilitate this change.

A proposed solution is to explicitly declare the partnership relationship (which eventually will give rise to mentorship). The goal is to change the view of homework from “punishment” to “duty” and to make apparent the student’s own responsibility in their education. Other cues may be helpful, such as: “You decide your grade, not me.”


The problem of a lack of motivation

Here’s where the class survey would be helpful. Maybe your students are wondering such questions as: How will math make me a better scientist? What does math have to do with my career goals? If a computer can do all these computations, why do I even need to take this class?

Here are some ideas:
1) You could give a periodic digression on the uses of specific topics in industry, sciences, or their relation to higher math.

2) To demonstrate the point that math is useful in daily life, you could ask your students to describe their age without using any numbers. Their answer will probably be in the format a<x<b, where “a” and “b” are shorthand for landmark temporal boundaries. Derivatives are important in driving, knowing when to brake and how much gas to apply. By thinking about the math they already use in their lives, they will discover for themselves how important it is.


The problem of students only doing what is required

Besides developing motivation and a sense of obligation, here are some ideas:
1) Make homework collected but not graded for accuracy. Have it also be worth a negligible amount of points (e.g. 3% of total). The idea is for the experience to be a transition from the (possible ingrained) idea of homework as a necessary teacher appeasement to it being an aid to understanding the material.
2) Quizzes can be given which will use a homework problem, with the exception of slightly changed numbers. Calculator usage is permitted although no partial credit will be given for partial reasoning missing. If time/grading is an issue, you could give quizzes like drug tests, a guaranteed X per semester. I think the important point to be stressed is that quizzes are an important source of feedback, and not just another random hoop to jump through for points.

The problem of poor pre-requisite understanding

I don’t have any ideas to rectify this problem, though I’m interested in hearing them. Theoretically, the student should (and could) go back to learn such pre-requisites. Practically speaking, however, formal U.S. education seems to encourage only progression. Independent Study is always an option but the question is whether or not the student currently has the discipline necessary to set and reach their own goals.


By the way, these opinions are from a community college student who has just completed Multivariable Calculus. I’ve also attempted to pay attention to the various teaching techniques used by my past and present college professors.

-Eric

 

[Count Iblis]

Mathwonk, yes, I agree. What I meant was that complex differentiability implies Cauchy's theorem (and then you can show that the complex diferentiability implies that the function is analytic). That requires more work.

 

[symbolipoint]

EC21, from your message, post #148:

Most of what you say is very good. You also did not know what to do about the problem of poor pre-requisite understanding. This is what a school or a department can do about that one: Course registration could require proof of pre-requisite credit, as displayed on transcripts, or as demonstrated on some official/institution assessment test. Also, a DEPARTMENT can apply its own assessment for pre-requisite skills & knowledge regardless of any pre-requisite courses, in case students have not kept those skills & concepts and if those skills & concepts are needed in a course. If pre-requisite knowledge is not adequate, the department can require either student must drop if a full course or more is needed; or student enroll in a targeted remediatory course in order to remain enrolled in the current course - otherwise be administratively dropped from current course. Amazing that students who want to "get through" courses believe that pre-requisite course credit by itself qualifies them for another course, but several weeks after a prerequite course was taken and credit earned, the conditioning from that prerequisite can be lost.

 

[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.