Teaching calculus today in college

[CoCoA]

Why on Earth would I call someone "Mrs. So-and-So"

Don't ask me that one, ask your parents.

 

[tshafer]

How do you think it got so bad? Is it a lack of qualified teachers that students have given up on their teachers, or is it that the students had become so disrespectful that the teachers gave up on them first?

What's crazy is this isn't just freshmen. I've taken graduate mathematics classes as an undergraduate here, and you wouldn't believe how whiny some of these graduate students are. They have become accustomed to being handed some simple work which they can finish quickly and then get a decent grade with no extra effort. Attend a colloquium? You must be joking. No extra credit assignments? The horror. Read the book ahead of and after class to understand the lecture? Not happening. It's all really quite unbelievable.

Of course, I am not too dissimilar. Being a senior hoping to get into graduate school, it was only last year I would wager that I learned to buckle down and focus. I've gotten straight As all throughout my physics and maths courses with a minimum of effort -- most courses are rehashes of the texts -- and I assumed that I was learning everything I could. Up until last year, when I was finally convinced that going to graduate school was a viable option (frankly, it had never occurred to me!), it was easy to go through my E&M courses, Mechanics courses, etc., and do enough to get As without a deep knowledge of material. Now I am seeing the error of that way of operating and it has markedly improved my grasp of material this year. I just wish someone had smacked me upside the head as a freshman and said that grades weren't enough so that it didn't take until end of junior year to grasp it.

And then this year I had my first truly difficult class (graduate ODE with the one truly excellent professor I've taken) and I aced it -- but I had to earn it this time. Lots of work, lots of study time, but holy cow -- I know ODE better than I know most things I've been taught. One of the better parts of my undergraduate career.

 

[mathwonk]

i sent my kids to the best school quite possibly in the state. i borrowed money to pay tuition. the first day i was worried my son was not dressed up sufficiently. then i saw his teacher was wearing sweat pants. the kids all called he teachers by their first names. some years later this became the first school in georgia to win a national math contest.

being hung up on what name kids call you is a sure sign of not having even a clue about what matters in teaching, unless you are teaching military style, i.e. mindless, obedience instead of thinking. if you earn respect you will get it voluntarily. if you demand it, you will be made fun of behind your back. if your students love you they will want to work at what you recommend, if you enforce discipline, they will try to avoid it.

 

[buffordboy23]

Moonbear,

Thank you for the great feedback. Students should feel fortunate to have professors like Mathwonk and you. It's easy for anyone to be perceptive of the problems in education, but taking initiative in trying to find and apply solutions is another matter.

This is not the ONLY one, but it is a pretty big problem and seems to be fairly common. Since I think someone in this thread already mentioned Piaget's learning theories, these students are still often at the concrete operational stage, where they expect a list of facts that they will memorize as facts. Their study approach focuses on that, looking at a page of notes and trying to memorize what is written there, but without really understanding it.

Unfortunately, this basic level of understanding seems to be all that is required in a lot of educational settings. One of the previous posters talked about how they had to memorize vocabulary words for some class, which didn't seem to be on par with what typical learning objectives should be. This is the fault of the teachers no doubt. I often think that we (educators) focus more on obtaining the right answer rather the originality and creativity associated with an approach to solving a problem. The interesting problems are open-ended, but yet most of the focus is on the absolute.

The other issue is they are very much still passive learners, just sitting there listening to lecture without really thinking about what is being said. When I started lecturing in the course, I tried to remedy this by having a 10 min group exercise at the end of every lecture (so, in my hour of lecture, I'd give a 50 min lecture, and then a group exercise for 10 min). This group exercise forced them to immediately use the information that had just been presented...my reasoning is that if they have to discuss it with a group, they actually have to think over an answer enough to express it to the group, and can't just sit there not thinking and waiting for the few people who did to raise their hands and provide answers at the end.

Research shows that when students participate actively in a lesson, their overall retention of the concepts improves drastically. Closure activities, like what you incorporate, are extremely important as well and improve understanding. These results are in regards to K-12, but I am sure that they hold much weight as well in a college setting.

 

[buffordboy23]

That is really quite sad, what you've described, and what SticksandStones has described in his elaboration of his problems with his teachers.

Honestly, I had no idea it was so bad!

My teacher co-workers were very competent in regards to their content knowledge, so I don't know if Tobias Funke's comment is a local or global problem. His comment specifically targets the competency of math teachers. It is known that there is a shortage of certified math teachers. Certification does not necessarily mean competency to teach the subject matter either, so mathematics is a large problem area in our education system. The certification tests are produced by the same company that produces the GRE exams.

What happens in teacher's workshops and what sort of continuing education do they get? Is there a place for university faculty beyond the education departments to offer workshops to teachers to refresh and update their knowledge in our subjects?

I worked in the Pennsylvania education system. The last I knew, teachers are required to obtain 24 credit hours of continuing education within five years after graduation, or else their teaching certificate is removed until the requirements are met. Moreover, every five-years teachers have to complete 150 hours of continuing education, which can include workshops or college courses, or face the same penalty.

University/K-12 collaborations would probably be beneficial, and this happens in many locales. I have never participated in one so can't comment on specifics.

Another problem I see with the K-12 education with the mathematics/sciences is a lack of good resources. This is true particularly for textbooks. AAAS Project 2061 has given many of the middle school science textbooks in usage a failing grade; I think the same holds true for mathematics textbooks but can't say unequivocally. Unfortunately, this seems to be the backbone of a teacher's curriculum, since many schools (mine, for example) could not afford science kits backed by educational research. With all of the requirements placed on a teacher, especially new teachers, it can be difficult for them to find time to remove this crutch of relying on textbook.

EDIT: The workshops that I have participated in have gone both bad and good. A lot of it is dependent upon the presenter. I participated in numerous Learning-Focused Schools workshops:
http://www.learningfocused.com/
This education model has been successful in a number of schools. The metric for the research studies on the model has been the federally-mandated tests. The validity of these standardized tests seems to be questionable, so its hard to say how effective the model it really is. Moroever, the research studies were done by who else, but the organization who is selling the products. There used to be a link that showed the research results (it may still exist, but it's been awhile since I used their website and they updated their web-layout) and my initial thoughts were biased statistical reporting. Nevertheless, it the methodologies I acquired seemed to have an impact on my success in the classroom.

What did we do in the workshop? Well, first we had to take time off from our teaching duties, requiring a substitute for 3-4 days over the school year. We learned an overview of the educational model, with certain components delegated to certain days of the workshop. We would learn the methodologies from the presenter, who modeled with examples. This was followed by collaboration with fellow teachers and we practiced applying these ideas to our particular context in the classroom. It was recommended that the school administrators, some of who were present at the workshops, form committees with teachers and monitor progress with the methods learned until next workshop. Overall, it was the best workshop that I participated in. There was easy and direct application of these ideas into our lessons and accountability on the teacher. Such is not always the case with workshops.

 

[Moonbear]

I worked in the Pennsylvania education system. The last I knew, teachers are required to obtain 24 credit hours of continuing education within five years after graduation, or else their teaching certificate is removed until the requirements are met. Moreover, every five-years teachers have to complete 150 hours of continuing education, which can include workshops or college courses, or face the same penalty.

Is that universal, or does it vary state-by-state?

Another problem I see with the K-12 education with the mathematics/sciences is a lack of good resources. This is true particularly for textbooks. AAAS Project 2061 has given many of the middle school science textbooks in usage a failing grade; I think the same holds true for mathematics textbooks but can't say unequivocally. Unfortunately, this seems to be the backbone of a teacher's curriculum, since many schools (mine, for example) could not afford science kits backed by educational research. With all of the requirements placed on a teacher, especially new teachers, it can be difficult for them to find time to remove this crutch of relying on textbook.

That's consistent with what I've seen posted around here from parents concerned about the books assigned to their students, and looking for supplemental materials. The topics being covered in the books seemed incredibly inadequate.

What did we do in the workshop? Well, first we had to take time off from our teaching duties, requiring a substitute for 3-4 days over the school year.

Do school districts still typically schedule teacher's workshop days into the school calendar? I'm wondering if this is in addition to those days, or if schools are no longer supporting this requirement. What I mean is that when I was a kid, a couple times a year, we'd have a day or two off from school as a teacher's workshop day. That meant teachers could participate in these required workshops without getting substitutes. If schools are expecting all time for teacher's workshops be done during hours when school is in session, and that they need to take time away from their classes and find substitutes to complete those requirements, then this seems detrimental to the students and a poor message to send to the teachers about how much the district really supports their continuing education.

 

[mathwonk]

one of the best ways to improve knowledge of subject matter i know of is to attend the summer program at park city, the pcmi park city math institute run by the institute for advanced study. There, high school teachers, undegrads, grads, and prefessional mathematicians, i.e. college professors, all live and meet in the same locale, have lunch together, and attend workshops of their choice. people are encouraged to attend workshops oriented to others spoecialties and to converse about shared problems.

At these meetings I have sat in on sessions for teachers as well as undergrds and grad students, plus those for researchers in my area. There are also evening presentations intended to address a question of interest to everyone in a way all can enjoy. Then after the summer session, groups from the same geographic area go home to continue in some way the activity with followup during the winter.

I have also given teachers my notes on graduate algebra and galois theory, hoping some will find a way to work this stuff into honors level classes for bright students. I have also given materials to grad students studying for prelims at other schools and had some feedback that they were helpful. This is a great place to learn and to meet interested learners of math, probably the best i know of. the program this summer is on arithmetic of L functions, i.e. number theory at the highest level. the legendary john tate will be there, my calculus teacher from 1960, and still active in research.

The quality and level of the presentations is so high I often find the graduate classes are about right for me. They also issue books afterwards recording some of the presentations. Check out IAS - PCMI.

or this link:

http://pcmi.ias.edu/current/program.php

 

[mathwonk]

here is a blurb from the education section of the summer program for 2009:
The SSTP is structured around three goals:
All teachers should be involved in
• continuing to learn and do mathematics
• analyzing and refining classroom practice
• becoming resources to colleagues and the profession.

Each of these goals is reflected in the three strands that comprise the summer courses and activities.

Some Questions and Problems in Arithmetic (2 hours per day, 5 days per week)
This course will investigate questions like these:

In how many ways can an integer be written as the sum of two squares?
In how many ways can an integer be written as the sum of four squares?
What's the probability that an integer picked at random has no perfect square factor?
What's the probability that two integers picked at random have no common factor?
Which linear functions f(x) = ax+b (a and b integers) generate infinitely many prime numbers for integer values of x?
What is the probability that an integer picked at random between 1 and 1020 is a prime number?
The real goal of the course is to answer the following quesiton:

How are all of the above questions related?

 

[buffordboy23]

Is that universal, or does it vary state-by-state?

I don't recall. I'll have to research it. Tobias Funke, are your state standards similar to those in Pennsylvania?

Do school districts still typically schedule teacher's workshop days into the school calendar? I'm wondering if this is in addition to those days, or if schools are no longer supporting this requirement. What I mean is that when I was a kid, a couple times a year, we'd have a day or two off from school as a teacher's workshop day. That meant teachers could participate in these required workshops without getting substitutes. If schools are expecting all time for teacher's workshops be done during hours when school is in session, and that they need to take time away from their classes and find substitutes to complete those requirements, then this seems detrimental to the students and a poor message to send to the teachers about how much the district really supports their continuing education.

My district scheduled in-service days into the calender, which may not be synonymous with what you had in mind. At these events, we would learn about new district policies, other odds-and-ends, and occasionally, a presenter would lead a mini-workshop. I think it is common for teachers to get off from school to go to workshops, and I agree that it affects the students' education. However, the federally mandated tests (No Child Left Behind) likely affects students' education more--there is a two-week testing window (so throw all your lesson plans for those two weeks of testing out the window), plus the the objectives for a course often become objectives seen on the "big" test.

 

[yeongil]

I teach math and music (I know, interesting combination) at a all-girls catholic high school. I am certified in math only (not in music -- in fact, many of our teachers are not certified). I am having the same issues as Tobias Funke, in that our entering freshmen have problems with fractions and negative numbers. They are also way too dependent on their TI-83/84's for computation. I teach the honors section of Algebra I, and I actually forbade those students to use the calculator for tests and quizzes in most of the first quarter. In some cases the results were not pretty.

Our math dept. has been complaining for years that we're admitting too many students who are not ready for high-school level math, but obviously the reason is that the students who do well in math in middle school are going to public school or being taken by other higher-performing private schools. Furthermore, it looks like that the math skills possessed by the elementary/middle school teachers in our feeder schools are lacking.

I can't tell you the number of times students write the wrong answer because they inputed into the calculator wrong. Two of my favorites:
(1) students will say something like 80/40 = 1/2 because they divided 40/80 in their calculators (they switch the order of numbers). Too many think that a number divided by 0 is 0 because they'll type 0 divide by (number) in their calculators.
(2) students will say something like (4 + 8)/2 = 8 because they forget to type the parentheses -- 4 + 8/2 = 8. They're not mindful of the order of operations.

I wish that calculators were forbidden to high school students until maybe Algebra II and above, where the graphing features may be useful. But I think I'm in the minority amongst high-school math teachers.

I also teach honors Precalculus. Some more favorite errors I've seen:
(1) (x + 4)^2 = x^2 + 16. Students forget to foil.
(2) ln x - ln y = (ln x)/(ln y). Students forget to use only one ln for the division, as in ln(x/y).
(3) I still get students who think that (x^3)^3 is x^6 and not x^9!

I can go on and on, but I won't, because this post is already depressing.

Is that universal, or does it vary state-by-state?

I was certified 3 years ago for the state of Maryland. In Maryland I know that I have to take 6 credit hours within 5 years to maintain certification. I don't know if it's 6 credit hours every 5 years, of if I'll have to take more credits within the second 5-year period.


01

 

[mathwonk]

your stories do not either surprise or depress me. that is just a small verification of the 30+ years of examples i have under my teaching belt. i agree with everything you say by the way, calculators should be smashed and burned in a pyre, until the student learns how to use them.

 

[mathwonk]

I am up at just after 6 am to go in today and offer a free day (all day) of review to my classes. Seeing this title reminds me of the changes since the old days. When I was student there was in 4 years of college never a single hour of review offered by any of my professors. Moreover the last week of class this semester has been spent repeating things I have already taught 2 and 3 times before. Nothing in my classes as a student was ever repeated. Every lecture was new material, and that continued up through the last day. In the classes I teach many students persist in trying, and unfortunately now succeeding, to get me to do for them what they should do for themselves, i.e. look up and read basic material in the book, review class notes, make up practice exams. I try to teach my classes things no one ever taught me: how to study, how to review, how to make up a sample test and exam. I am not sure I am having any success. But we continue to try. It does work for a few, and maybe more slowly for some others. I recall I also was an unresponsive student for a long time, and now as an old man I appreciate things that teachers did, that they never realized I appreciated at the time. It just takes some of us longer to become responsible for our own learning. The key is not becoming discouraged when you see little progress from your efforts.

 

[Nick M]

My math teachers plan one snow day into our schedule. If at the end of the semester we have had no snow days, then we use the last day for review.

 

[Cincinnatus]

Originally Posted by buffordboy23 View Post

I worked in the Pennsylvania education system. The last I knew, teachers are required to obtain 24 credit hours of continuing education within five years after graduation, or else their teaching certificate is removed until the requirements are met. Moreover, every five-years teachers have to complete 150 hours of continuing education, which can include workshops or college courses, or face the same penalty.

Is that universal, or does it vary state-by-state?

Continuing education for K-12 teachers is in trouble in New York state. Due to the current financial crisis they are probably going to be making extreme cuts to these programs. In particular, the New York State "Teacher Centers" which are quasi-local state-funded organizations have been targeted to receive either major budget cuts or to be eliminated entirely. These Teacher Centers are responsible for coordinating the majority of professional development options for K-12 teachers in New York state.

If anyone reading this is from New York, you should write to your state senator or the governor's office on behalf of professional development and continuing education for teachers. They are supposedly working on the budget now and will announce it next week.

 

[Tobias Funke]

I don't recall. I'll have to research it. Tobias Funke, are your state standards similar to those in Pennsylvania?

I'm in MA, so probably. I don't know for sure because I'm in a private school. But anyway, "continuing education" isn't synonymous with learning actual mathematics. These teachers I spoke of were in classes like that, and they were basically guaranteed to pass. They learned a little, but it was material they should have known. But mathwonk's courses seem better than these, so I may be wrong.

 

[mathwonk]

University of Georgia is apparently very unusual. Our math ed department requires its students, in addition to their work in methods and theory of instruction, to take math classes in the math dept. In particular they take some of the same courses that are required for math majors, such as courses in proofs and abstract algebra.

This is part of the reason the UGA math ed. department stands out nationwide, according to a recent survey which found it the only "exemplary" math ed. dept out of 77 programs studied in the country.

"UGA stands out because the program has stringent requirements and stresses the importance of conceptual knowledge, according to Denise Mewborn, professor and chair of the College of Education’s department of mathematics and science education.

“The big emphasis for the past 10 to 15 years has been on developing conceptual understanding in children, not just teaching them procedures, rule without reason,” said Mewborn. “Getting them to understand why these things work so they’re not just playing Russian roulette.”

Prospective elementary education students at UGA are required to take three content and two methods courses, while many schools only require one method and two content courses.

Mewborn also pointed to the close relationship between the elementary math education program and the UGA mathematics department in the Franklin College of Arts and Sciences as another reason why the program excels in preparing teachers for the classroom."

http://www.uga.edu/columns/080902/news-math.html

 

[mathwonk]

Offering good continuing education is an ongoing struggle with many challenges, some financial some political. A number of years back I taught such a course for returning teachers in calculus. I believe I used Apostol, the famous book for honors calculus students at schools like MIT and Stanford. At the end of the course I gave copies of the book "How to solve it" by Polya, to the members of the class.

It was one of my best experiences all round in a classroom. One of the teacher/students gave me a poem about her view of teaching. This person later was named the top teacher in Georgia one year.

But the class was small, and a few students dropped out, some complaining bitterly. The enrollment was so small the tuition hardly covered my salary, so the class was not offered again.

To make quality programs work, someone has to require students to take them, so the numbers add up to those trying to make the budget balance.

That reminds me that in my youth I even taught some honors courses for free, i.e. on top of my usual load, just so we could say we offered them, but one cannot forever carry the program on one's back.

Come to think of it, I have been teaching free classes for years now, i.e. when I have free time I have often taught a course that was under-enrolled even though I did not get credit for teaching it. This is still going on by other faculty as well. But it costs a heavy price in time lost from research for the faculty member.

For the faculty member, doing and discussing our research is part of our own "continuing education".

 

[buffordboy23]

We want students who will try to get to the bottom of things, not merely ones who can compute accurately the area between two curves.

I agree with your view. I think the large problem lies in the fact that many students can do mathematics, but they can't think about mathematics.

Let me try to make this point clearer. From what I have seen with my own educational experience and in working with peers as a returning student, I see that students have greater success with their math/physics courses when it comes to computational problems. Using a previous example from an earlier post, there are only so many ways to compute a line integral, so for a typical problem a student simply needs to recognize the correct form of the given function (i.e. is the function given in cartesian coordinates? parameterized? etc.) and apply the appropriate method to compute the integral. Once one is familiar with the algorithm it is a trivial process to compute line integrals.

In contrast, how would these same students fair in proving the statement, "Show that the zero vector in a vector space is unique." While it may be obvious to the student from experience that the zero vector is unique, how does one show this in a logical proof? This requires the student to think about what the definition of a vector space is and apply it in a novel way. Now, there are common threads when it comes to constructing proofs, but I am not convinced that students are fully aware of such systematic methods.

Looking back on my experience in high school, I was only introduced to solving proofs by direct methods. That is, given the general statement, "If A, then B.," my proof would consist of showing that A implies A1, A1 implies A2,..., An implies B to complete the proof. Most of the proofs I ever did in high school occurred in geometry class using the rigid lock-step method. Looking at my college experience, I am now required to do proofs that required uniqueness, quantifiers, etc; ideas in which I never had formal instruction nor much experience.

While a student may be able to follow the condensed proof given in a textbook by filling in the missing details, many students probably do not think about what caused the author to think in this manner to construct this proof in this specific way. For example, in trying to prove the statement, "The square of an odd integer is an odd integer.", the student should first think, "What exactly is an odd integer?" Answer: 2k+1, where k is an integer. This is what I mean about thinking about mathematics. I own the books "How to Read and Do Proofs" by Solow and "How to Solve it" by Polya, and I must say that these books are indispensable to my current abilities to construct proofs and think mathematically. In general, I think there needs to be a greater emphasis in courses and lectures about how to think about the mathematics rather than doing the mathematics.

Since you have had experience working with public school teachers, have you ever had the opportunity to analyze their abilities to construct proofs? I would guess that many teachers have skills that are inadequate, so our students should have no greater expectations for their abilities. Perhaps, this is something to focus on in future workshop seminars for math teachers.

 

[harvellt]

Since you have had experience working with public school teachers, have you ever had the opportunity to analyze their abilities to construct proofs? I would guess that many teachers have skills that are inadequate, so our students should have no greater expectations for their abilities. Perhaps, this is something to focus on in future workshop seminars for math teachers.

I think the problem starts well before students get to high school. I read a short article in the New York times a few weeks ago about a state where elementary school teachers were not required to show they had even high school algebra skills. The articles main focus was on a school that decided to higher some math tutors for 5th and 6th graders, not to help struggling kids but to teach the entire math curriculum. The students started to do much better.

 

[mathwonk]

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?

 

[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.