Teaching calculus today in college

[mathwonk]

Nick, you are taking a quote that I applied to myself. I was not interested in getting A's as much as in learning as much as possible. You seem to think that failure is getting a low grade, whereas I thought of failure as not trying as hard as possible to learn at as high as level as one is capable of.

By definition an A+ means one has got all from that course that there was to get. Doesn't that make you want to see if there isn't a little more challenging course available somewhere?

I am not interested in fake awards that do not actually mean one is good. I think I told the story here once of wanting to learn to play snooker, and my method was to play against one of the best snooker players in my town every day for a year, losing every single game.

Finally I won one. After that I moved on to other even better opponents and found I had myself become one of the top players in town. Most people like "success" in the sense of winning every now and then. I didn't care about winning against patsies, to me that was not success, I wanted to beat the best, and I could stand the long apprenticeship that required.

In math getting an A+ in a non honors undergraduate class was fun for a day or two, but then I wanted to move up to the big time, and get an A in a graduate class. The truth was I didn't belong in that class I got the A+ in, except temporarily, until I got my feet under me again. In horse racing there is a concept called "dropping down in class". A horse that is used to racing in a different classification, can easily win in a lower one, even against horses with better records on paper. A professional athlete even one with no notable fame at all, will destroy amateurs at will. I wanted to elevate my classification by competing against better competition. If you go and listen to professional mathematicians talk about math, or go to lectures in a higher level course, but one in which you can understand something, you will soon be stronger than your peers who do not do this. If you read the books I recommend here, and challenge yourself as I suggest here, I believe you will soon be much stronger than you were before.

In yoga this is called the concept of fulfilling ones desires. One is motivated to go as far as his desires push him. Some people have few desires, some might say little ambition, others have much.

It seems to me you do have ambition to excel in math since you say you are taking classes now that are more challenging than the ones you took before. So I don't see you as disagreeing with me as much as you say.

 

[mathwonk]

wow moonbear, your dedication, insight, and positive attitude is an inspiration!

 

[buffordboy23]

I have noticed that a vast majority of my students still have atrocious study skills. I'm not talking about willingness to study, or laziness, or procrastination, but that when they finally sit down to study material, they really aren't studying or thinking, just reading it over and over and over hoping it will stick.

For students who try to do well, is this the only student behavior that you have observed? Or are there other common behaviors? I'm curious to know what specific study skills you will teach your students in the future, and why those methods are most important in regards to your subject. Since your an expert in your field, you have a unique perspective about how to approach your subject. I often obtained a lot of insight when the professor shared their thinking regarding their field, rather than just content knowledge.

Perhaps we could use your specific insights to give mathwonk more ideas to teach effective study skills to his students.

 

[Nick M]

I take failure as not understanding, which results in those poor grades. Not understanding can spawn from many things such as not having the proper prerequisite knowledge, poor teaching, or simply not making an effort. But only so much can be taught in any given semester. Just because a student succeeds in a course by getting a grade in the 95-100 range doesn't mean that they weren't challenged. I'm sure if you take this example to the extreme and say, taught the entire four semester calculus sequence that engineering/physics students take over four semesters (plus introductory linear algebra) and combined them into one semester you would wash out even the best students.

I'll be finishing up in the math department this spring with Diff-Eq (I'm majoring in EE). I've felt that we covered quite a bit in Calculus I-III and Linear Algebra. I've got all A's and A-'s, but never did I feel that I wasn't being challenged. Sure some things came easier than others, but you can only expect so much to be learned in a given time frame - especially when you have different rates of learning based on a given teaching style in a given classroom. Obviously a student shouldn't be in Calculus I/II if they don't understand how to model with transcendental functions - Nor should a student be in Multivariable Calculus if they can't comfortably work with single variables. But even with a class full of prepared and hard-working students with the same learning style and a teacher who is in perfect tune with them, only so much can be covered in a semester. What is the point of piling on more or increasing the difficulty and leaving students confused about things? I can see spending a few minutes to begin explaining something that peaks interest in material that goes beyond the scope of the course objectives, but why make the objective density so high that students have a weak understanding of 100 topics rather than a solid understanding of 50?

We used McCallum/Hughes-Hallett/Gleason for Calculus I-III and David Lay for Linear Algebra. We'll be using Blanchard/Devaney/Hall for Diff-Eq.

I like the preface from M/H-H/G regarding their vision...

"Our goal is to provide students with a clear understanding of the ideas of calculus as a solid foundation for subsequent courses in mathematics and other disciplines. When we designed this curriculum we started with a clean slate. We increased the emphasis on some topics and decreased the emphasis on others after discussions with mathematicians, engineers, physicists, chemists, biologists, and economists. We focused on key concepts, emphasizing depth of understanding rather than breadth of coverage"

The pre-calculus text from Wiley that follows the same precepts called "Functions Modeling Change" was also excellent.

Attacking subjects by developing an intuitive understanding of the underlying concepts really helps out. Then things are developed through language, and finally through exercises geared from theory and modeling. "The Rule of Four" - presenting problems verbally, numerically, graphically, and symbolically gave me a hold on things that I feel is rooted much deeper than the Calculus classes I took using another text that simply opened with proofs/theory and then hammered me with similar looking problems that just increased in mechanical complexity as the problem number increased.

I guess I have the opinion that quality is better than quantity.

 

[Moonbear]

For students who try to do well, is this the only student behavior that you have observed? Or are there other common behaviors? I'm curious to know what specific study skills you will teach your students in the future, and why those methods are most important in regards to your subject. Since your an expert in your field, you have a unique perspective about how to approach your subject. I often obtained a lot of insight when the professor shared their thinking regarding their field, rather than just content knowledge.

Perhaps we could use your specific insights to give mathwonk more ideas to teach effective study skills to his students.

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. That, and there are relationships among concepts that they are not yet making. And when the context changes...they need to use information in lecture one and relate it to lecture two in order to apply their knowledge for a clinical scenario...they can't make those connections. My clinical scenarios are probably your proofs...actually having to apply the fundamentals in a way that leads to a correct conclusion.

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.

I'm considering some things like teaching them to use concept maps, which make them identify the major concepts and show how they relate to one another. The other exercise I'm considering is to have them each write a set of multiple choice quiz questions based on a particular lecture or two (due to the nature of our material, exams are usually multiple choice or short answer, not problem solving). I'm hoping this might get them into the mindset of how an exam question is constructed and what they need to think about when writing one. Multiple choice questions are harder to write for a student...it's easy for them to write a fill-in-the-blank type question without integrating concepts, but if they have to think about all the possible wrong answers they could put as distractors, then maybe it'll get them to realize how much more they need to learn than just a list of possible right answers. That sort of exercise will also help me get into their heads earlier in the term to see what they think an exam would look like.

One limitation I have is that I team teach with another lecturer. Her lectures were in the beginning of the course and unfortunately set a bad tone for the students. Her lectures would be just fine for med students, which is what she has mostly taught before, because they are already more sophisticated learners who can extract the right concepts and information from a rather dryly given lecture. It doesn't work well for the undergraduates who are still not at that level of learning. They still need repetition, big bold text that tells them something is important, and questions that prompt them to think about the connections they need to make. I will have more leeway next year, since I will take over as course coordinator. She has been attending my lectures and seeing my style and how much more responsive the students are, so is going to try to adapt some of that into her lectures too, but part of it is that her personality isn't really as well suited to that style of lecture. So, she will give her lectures as usual (maybe with some improvement), but I will ask her to keep them 50 min long instead of a full hour, then I'll come in with their group exercises at the end, since I don't think she's comfortable facilitating those yet.

 

[SticksandStones]

What is going on in some high schools? teaching with pictograms?

Ridiculous, mind-numbingly simple regurgitation-styled questions that require no thought what-so-ever. Hour long tests that should take 10 minutes, tops (yet people still always needed extra time). Glossing over anything that might be the least bit interesting to make way for information that most of the class stands a chance at passing.

To be honest, I'm not finding college to be much different. Perhaps this is for the better though, as my study habits consist of studying maybe 5 hours a semester.

The way classes seem to go for me is this: I get into the class, get lost a bit. Study until something clicks. Class becomes easy. Rinse and repeat each semester.

EDIT: For the older people here, I have a simple question. How many hours per week during the semester did you work?

 

[buffordboy23]

To be honest, I'm not finding college to be much different. Perhaps this is for the better though, as my study habits consist of studying maybe 5 hours a semester.

The way classes seem to go for me is this: I get into the class, get lost a bit. Study until something clicks. Class becomes easy. Rinse and repeat each semester.

SticksandStones, I had the same experience my first time in college. The hardest part about succeeding was actually showing up to your classes and copying the professors notes. No studying was needed until the night before test day, and all you had to was memorize the notes to perform well on tests that were multiple-choice format for the most part.

My experience in returning to college (studying physics) has been very different. I have to work hard to perform well. Since I have passion for the subject, I often work harder to learn more than what is required because it will only benefit me later. A student's workload likely varies across colleges and majors.

I hear that with the number of college graduates nowadays, it is more competitive to obtain a job after college--I went on about 25 interviews and had about 5 offers for mathematics/science teaching positions. In contrast, I also hear that many employers state that newly graduated college students aren't prepared to enter the workforce. How do we rectify this trend? This underlies why students aren't prepared for Mathwonk's college calculus--there's not enough certified teachers, and of those, not enough that are effective, and academic objectives are watered-down so that intelligent students complacent with mediocrity can perform well with minimal effort.

EDIT: Consider this. During only one of my interviews for a math position, I was asked math content questions. One question that sticks out was, "What type of number is sqrt(2) and why?" Apparently, less than half of interviewees for math positions within this district were able to get the answer right.

One of the things I dislike about the structure of a typical college course is that grades are heavily weighted upon exams, and in a lot of courses, this is it. If these tests are multiple-choice, the student really learns little; he is just memorizing some facts for a test, and soon forgets them. Research projects tend to hold students to a higher level of accountability and requires them to synthesize conceptual knowledge in a unique way, which leads to a better overall understanding of the content.

 

[mathwonk]

heres another conundrum to muddy the water: suppose you find that your students never do anything you recommend unless you take it up and grade it, prefacing the assignment by saying "this is due thursday, and will count in your grade".

I.e. if on the other hand you say, "I will not take this up, but it is absolutely essential to understanding the topic for you to work through these problems on your own, and they will be tested on the exam", you find that most people do not do them.

What to do? You notice that those teachers who function essentially as personal trainers, making certain things "due", and worth "points", are more successful at getting their charges to hand in the work, than you are by simply telling them what is essential, and testing it on tests.Now here is the puzzle: it seems that those teachers who get performance by "requiring" it rather than recommending it, are treating the students like high schoolers and little children. As a result, although their students may score higher on short term tests in that course, but afterwards they cannot function on their own, after leaving the environment where useful work is enforced. Is that good survival behavior in life?

I.e. is acquiring maturity a useful outcome of a course in which one is treated as if one is a responsible adult?So what is better goal for a college course? to enforce participation in drills so as to get a higher score on a narrowly based test? or treat students as adults, to help them learn the consequences of self motivated and self disciplined learning?

If acquiring maturity is desirable, how does one encourage this, so as few as possible fail out before taking charge of their own learning? I am interested in how to produce more people with a desire and willingness to do what they know will help them, without it being enforced by artificial means like meaningless "points".This is partly because I do not believe it is possible to make anyone think deeply, or understand something they refuse to try to understand.

So how do we get our charges to begin asking themselves, not "have I got all the points available here?", but "have I really understood this? could I answer other different questions using this same idea?"

When I ask a student why he/she thinks a certain argument is a proof, I often get the answer "that's what you said" or "that's what it said in the book". I try to make the point that a valid proof is one that makes sense, not one that mimics almost faithfully (merely omitting the definitions, the logic, and other key ingredients), the words found in another source.

How do we teach that the validity of an argument is measured not by comparing the words with those in a book, but by considering the meanings of those words?the last paragraph of buffordboy23's previous post is key. Is it possible that for many of us this behavior is only acquired by leaving school? Thats what did it for me.

 

[mathwonk]

anyone curious as to the value of gre scores in deciding admissions will get some insight from the previous remarks. 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.

 

[Moonbear]

heres another conundrum to muddy the water: suppose you find that your students never do anything you recommend unless you take it up and grade it, prefacing the assignment by saying "this is due thursday, and will count in your grade".

I.e. if on the other hand you say, "I will not take this up, but it is absolutely essential to understanding the topic for you to work through these problems on your own, and they will be tested on the exam", you find that most people do not do them.

What to do? You notice that those teachers who function essentially as personal trainers, making certain things "due", and worth "points", are more successful at getting their charges to hand in the work, than you are by simply telling them what is essential, and testing it on tests.


Now here is the puzzle: it seems that those teachers who get performance by "requiring" it rather than recommending it, are treating the students like high schoolers and little children. As a result, although their students may score higher on short term tests in that course, but afterwards they cannot function on their own, after leaving the environment where useful work is enforced. Is that good survival behavior in life?

Yes, it feels like treating them like little children, but some really are still at that level of maturity. Some of it is also that they are trying to prioritize the workload of all their courses. Most of my students will eventually get to those exercises they are told will help but that are not collected, pretty much the day or two before the exam. Of course, if they worked on it when it was recommended, everything else afterward would have been easier for them to learn. This is all part of the problem of study skills, and perhaps time management...their inefficiency makes it more of a struggle than if they just studied a little bit at a time efficiently.

On the other hand, for the vast majority of jobs, this is perfectly fine survival behavior. If you're the boss or a project manager, yes, you need to have the work ethic to look into every aspect of a project if you're going to do well, but if you're just the employee, then all you need to be able to do is what your boss tells you to get done by the deadline they set. When you see a student going the extra mile on their own, they are the ones to encourage to consider grad school. For those heading out into industry, doing what needs to be done "to count" and by a deadline is really all they'll ever need to do.

So what is better goal for a college course? to enforce participation in drills so as to get a higher score on a narrowly based test? or treat students as adults, to help them learn the consequences of self motivated and self disciplined learning?

I prefer to enforce participation on things that will teach them the bigger picture. If they can grasp a few major concepts, even if it's through persuasion, trickery and bribery, then they will walk out of my classroom with the minimum skill set needed to go back and learn the details on their own later when they need them.

If acquiring maturity is desirable, how does one encourage this, so as few as possible fail out before taking charge of their own learning? I am interested in how to produce more people with a desire and willingness to do what they know will help them, without it being enforced by artificial means like meaningless "points".

If you can show your students how a particular topic is relevant to their goals beyond just passing your course, then it's easier to "hook" them into learning out of self-interest. This is why in a nursing course, I use clinical cases to reinforce major concepts. Whenever possible, I try to adapt real cases involving situations that they might encounter outside the narrow confines of whichever doctor's practice they end up working in. For example, I used one of someone stumbling and vomiting at a conference at a hotel to illustrate how their knowledge of nervous system anatomy would help them realize this person was NOT just drunk, but having a stroke, and that they could even figure out where that stroke was happening by his behaviors. So, even if they end up working in a podiatrist's office, they see the relevance of understanding what's happening in the brain to know better than the average person when to call an ambulance rather than telling someone to "sleep it off."

If you can relate the subject to their interests in life, you'll have them hooked. Heck, sometimes I just point out things like, "...and you should be sure to understand this well and remember it, because most of your physiology course next semester will be based on this part of the nervous system." Now they know they need to do more than remember it long enough to take my exam, but also need to remember it to do well in another entire course coming soon.

This is partly because I do not believe it is possible to make anyone think deeply, or understand something they refuse to try to understand.

You can make someone think deeply. You can't make someone understand what they refuse to understand. I don't worry a lot about those who make no effort whatsoever. If they have no interest at all in learning the subject, the best thing that could happen to them is they fail the course and wake up to the fact that they have chosen the wrong major. But, for those who are simply not yet sophisticated learners, we can challenge their thought process.

So how do we get our charges to begin asking themselves, not "have I got all the points available here?", but "have I really understood this? could I answer other different questions using this same idea?"

This is what I use the group exercises at the end of class to do. In fact, in my very last lecture, I gave them a case that had no answer (or at least not a complete answer until additional diagnostic tests are added that would go beyond my course content). This is much more the reality they will see in the working world. They may not have all the answers, but they need to know enough to recognize there's a problem and the general system being affected so they know when to call the doctor in, and what to do while waiting for the doctor to return their call, etc. I told them if they could answer all the questions I asked related to that case, they were ready for the final exam, because it made them use everything they had covered not only in the lectures included on this exam, but even topics that were covered in their very first lecture of the course. A few students grasped it well and showed they really understood what was going on. More of them realized how little they still understood about putting the concepts together, and that case prompted a lot of emails and office appointments. A few walked out with no interest in learning about the case...they are the ones who have been consistently doing poorly in the course and I'll probably see again next year.

Sometimes, I catch a few more students during exam review sessions. My format this time around was to first let them ask any questions they had. Then when they had no more to ask, I started asking them questions. Again, some students had grasped the concepts and were able to answer my questions readily, while others could do it with some thinking, and then there were some still sitting there just trying to write it all down because they were realizing they still had a lot to learn. I then let them ask questions again, in case my questioning prompted them to think of something else that had them confused. I've gotten yet another round of emails and requests for office appointments after the review session. It may have taken me until the very last day of class to finally get through to them what they need to learn and how they need to approach learning, but while it would have been nice for them to come in able to do this, I feel satisfied that I've done my job well if they at least walk out of the class with that knowledge in hand.

When I ask a student why he/she thinks a certain argument is a proof, I often get the answer "that's what you said" or "that's what it said in the book".

I get that too. In my review session, when I asked a student to give an answer and then asked them why they chose that answer, they said, "Because that's what's in your notes." So, I simply just don't let them off the hook at that point. I ask them why they should believe my notes, or why I would write that in my notes.

They've heard me repeat a phrase in many, many, many of my lectures. "You shouldn't believe me." This is the key to critical thinking. They should not take what I tell them on faith, they should prove it to themselves. If they can prove it to themselves, THEN they understand it. This is sophisticated learning, and fairly advanced expectations. I don't expect all of my undergraduate students will attain this level, but I want them to start having that seed planted so that by the time they're reaching their senior year, or considering graduate or professional schools, that seed has finally grown into understanding. Someone needs to plant the seed, but then I have to put faith in the other faculty they have after me that they will continue to water and nurture that seed or seedling until those students fully blossom into mature thinkers. This isn't a process that happens overnight.

 

[SticksandStones]

Now here is the puzzle: it seems that those teachers who get performance by "requiring" it rather than recommending it, are treating the students like high schoolers and little children. As a result, although their students may score higher on short term tests in that course, but afterwards they cannot function on their own, after leaving the environment where useful work is enforced. Is that good survival behavior in life?

Well, the way I see it from a student's perspective is this. High schools treat their students like elementary school children. They're more concerned with whether or not you're walking in the halls and using "Mister" and "Misses" when addressing a teacher than whether they are capable of finding the roots of a quadratic equation.

From what I've heard from people who attended high school in the 70's (and earlier), and from people who've attended high school in foreign countries (even recently), the high school education here is pathetic in comparison. If you want to know why you're students are not prepared for your class and why they have no work ethic it's because they've spent the past 13 years of their life being babied.

 

[Joskoplas]

If you want to know why you're students are not prepared for your class and why they have no work ethic it's because they've spent the past 13 years of their life being babied.

And the teachers chose to do it this way, eh? No pressure from anyone? No mandates to serve all and please everyone? No forced accommodation for everyone's long list of disabilities, long litany of excuses, or long line of helicopter parents?

It's easy to throw stones at the perpetual scapegoats, but I for one was inspired to a career in science by an excellent, no-compromises chemistry teacher.

 

[SticksandStones]

And the teachers chose to do it this way, eh? No pressure from anyone? No mandates to serve all and please everyone? No forced accommodation for everyone's long list of disabilities, long litany of excuses, or long line of helicopter parents?

It's easy to throw stones at the perpetual scapegoats, but I for one was inspired to a career in science by an excellent, no-compromises chemistry teacher.

Where did I say it was the teacher's exclusive fault?

I made a point, that you seem to agree with. That the High Schools are failing their students. Whether this is the fault of teachers, parents, administration, or all of the above is another matter.

 

[Joskoplas]

Don't read it as an accusation; I was inputting a second perspective. You may feel, as a student, that your world sucks. There's a big list of reasons why things came to be the way they are, and a Gordian knot to untie to undo the damage.

 

[SticksandStones]

Don't read it as an accusation; I was inputting a second perspective. You may feel, as a student, that your world sucks. There's a big list of reasons why things came to be the way they are, and a Gordian knot to untie to undo the damage.

What the hell are you going on about?

Nowhere did I say my world sucks. My world is great, thank you very much.

I offered one explanation for why mathwonk, et al, are finding their students being incapable or unwilling to do the work necessary to succeed and you come in with what appears to be an attempt to counter my explanation by countering points I didn't make with points that only go to further strengthen my explanation.

So, again, what are you talking about?

 

[Joskoplas]

Well, the way I see it from a student's perspective is this. High schools treat their students like elementary school children. They're more concerned with whether or not you're walking in the halls and using "Mister" and "Misses" when addressing a teacher than whether they are capable of finding the roots of a quadratic equation.

Like you, I read more into it than was necessary. I'm out now. So I guess I'm not 'the hell' going on about anything.

 

[CoCoA]

Well, the way I see it from a student's perspective is this. High schools treat their students like elementary school children. They're more concerned with whether or not you're walking in the halls and using "Mister" and "Misses" when addressing a teacher than whether they are capable of finding the roots of a quadratic equation.

I would never be a high school teacher, but if I were and a student called me by my first name instead of using Mr., I would probably throw the student out of class. (Not because he/she is a student, but because he/she is a non-adult addressing an adult - and an authority figure at that - in a disrespectful manner. I was taught better by my parents)

I have taught a lot of community college classes, and I never insist my students call me mr. - after all, they are adults by then - but invariably they do. I take classes at university, and I call my professors Dr., even though I am older than some of them.

From what I've heard from people who attended high school in the 70's (and earlier), and from people who've attended high school in foreign countries (even recently), the high school education here is pathetic in comparison. If you want to know why you're students are not prepared for your class and why they have no work ethic it's because they've spent the past 13 years of their life being babied.

When I went to school, I walked 15 miles through the snow, uphill both ways. Oops - got to go, I have to tell some kids to get off of my lawn.

 

[SticksandStones]

I would never be a high school teacher, but if I were and a student called me by my first name instead of using Mr., I would probably throw the student out of class. (Not because he/she is a student, but because he/she is a non-adult addressing an adult - and an authority figure at that - in a disrespectful manner. I was taught better by my parents)

I don't see how what year one was born in has any baring on what one calls someone, but that's just me and to each his own.

In my high school's case, given the number of students in my graduating class alone that couldn't read at a satisfactory level I think the disrespect towards the [horribly inept] teachers was warranted.

I mean, let's put it this way. They weren't doing their job. They were and are wasting tax payer money, wasting my time, and hurting a generation of students by being incompetent, lazy, and in some cases just down right stupid. Why on Earth would I call someone "Mrs. So-and-So" when their idea of teaching history is to hand out a list of words we need to memorize by the end of the month and then spend the 4 weeks sitting on a computer looking at clothes?

Now, let me clarify something. I don't think high school students should just off-the-bat ignore and disrespect their teachers. However, I think that it's the teacher's responsibility to do their job and show that they are deserving of respect.

Which, again, if you want to know why students in college are becoming more and more lazy the above is a reason.

I have taught a lot of community college classes, and I never insist my students call me mr. - after all, they are adults by then - but invariably they do. I take classes at university, and I call my professors Dr., even though I am older than some of them.

Most of my professors have insisted on us calling them by their first name, but on the first day I'll usually refer to them as "Professor" or "Doctor so and so".

When I went to school, I walked 15 miles through the snow, uphill both ways. Oops - got to go, I have to tell some kids to get off of my lawn.

Never did that in high school, but that's roughly my college experience. :)

 

[Tobias Funke]

I know I sort of complained about the students mostly in my last post, and I hope I don't come off as too negative (I'd offer helpful suggestions but I'm just a 2nd year teacher trying to learn from you guys), but I wasn't kidding when I said that most of the middle school teachers I worked with were embarrassing.

Most didn't even remember the triangle or unit circle definitions of the trig functions, many couldn't make the leap from 1+2=2(3)/2, 1+2+3=3(4)/2, 1+2+3+4=4(5)/2, and maybe even one more line, to guess 1+2+...+n , and some couldn't even understand the very concept of generalizing to arbitrary n. One (and likely many more) couldn't do basic exponent problems like (x^2z^(-1))/(2z^3) or solve a simple "real world" geometry problem, something about finding the cost of carpet to cover a room given the cost per square foot, from her students' book. Sure, teachers may be ridiculously underpaid and have to put up with a broken system, but that has nothing to do with the students.

So here we are, myself included, talking about our students' deficiencies, but are they really that surprising? I know exactly where they come from.

 

[Moonbear]

I know I sort of complained about the students mostly in my last post, and I hope I don't come off as too negative (I'd offer helpful suggestions but I'm just a 2nd year teacher trying to learn from you guys), but I wasn't kidding when I said that most of the middle school teachers I worked with were embarrassing.

Most didn't even remember the triangle or unit circle definitions of the trig functions, many couldn't make the leap from 1+2=2(3)/2, 1+2+3=3(4)/2, 1+2+3+4=4(5)/2, and maybe even one more line, to guess 1+2+...+n , and some couldn't even understand the very concept of generalizing to arbitrary n. One (and likely many more) couldn't do basic exponent problems like (x^2z^(-1))/(2z^3) or solve a simple "real world" geometry problem, something about finding the cost of carpet to cover a room given the cost per square foot, from her students' book. Sure, teachers may be ridiculously underpaid and have to put up with a broken system, but that has nothing to do with the students.

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! At the university level, we don't meet many middle and high school teachers, we just see the end product in our students. The few teachers we do meet are most certainly among the cream of the crop, because they're the ones arranging for their classes to visit us and to learn something above and beyond the usual curriculum. For example, we had a teacher from the local high school who teaches a first aid course bring her class into learn some anatomy from the anatomy faculty here. It was great fun for both us and the students, and that's certainly not a teacher slacking on her work.

So here we are, myself included, talking about our students' deficiencies, but are they really that surprising? I know exactly where they come from.

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?

And, more importantly, how do we fix it? In part, the things we're discussing in this thread are a start, not to solve the problem, but to address it once it falls into our laps at the university level. And, one of the visions we had for this particular subforum here was to also help with some of these issues, in terms of those of us with university teaching expertise to reach out and communicate with those with middle and high school teaching expertise to figure out how to bridge the widening gap students are needing to leap as they enter college.

Though, I wonder, what else can we do? Is there more of a need for outreach activities, of university faculty visiting high schools and giving a brief lecture about what we do and why it's worth learning?

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? When I was in grad school, one of the colleges at our university had an annual "Teacher's Day" in which teachers from around the state were invited to participate in seminars and workshops led by university faculty. Because I was doing some unique work with running study groups to teach students to use study groups, one of the faculty invited me to join him to give one of the presentations on using small group learning in science courses. He was taking a different approach to incorporating it into our lab courses. Afterall, our Freshmen weren't all that different from the high school seniors, so what we were finding worked with them was certainly applicable to at least a senior level high school class.

 

[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.