Teaching calculus today in college

[mbisCool]

I agree with this, at least to some extent. I think referring to the text instead of writing everything out implies that the lecturer has referred to the text also, and supplies an example for the student to do so.

On the other hand, not writing things on the board can easily be taken too far. Thinking about mathematical ideas takes time. If a professor talks too much about several things in a row from the book without writing down enough stuff on the board, the professor will just talk faster than students can absorb. In a sense, the most important part of writing is to control the pace of the lecture.

Ideally the lecture would compliment the text and the text compliment the lectures. I personally feel the act of writing my notes helps me remember them very well. I assume this and the reference they provide are why lectures are presented on the board...

 

[mathwonk]

khemix you inspired some heat because you complained that your professors were being paid to teach you, but when we also tried to teach you something you apparently rejected it, revealing that you are not so easy to teach. you are at least partly your own problem and you seem to be blaming someone else. you remind us of why it is so hard to help some people. but we piled on a bit. my apologies.

and if you look at my notes for math 8000, abelian groups are defined on page 1.

even in the math 4050 notes the qords are used but no knowledge of them is required. don't be afraid of words which are explained in context in the sentences where they appear.

those notes state explicitly that the abelian groups being discussed are products of Z/n 's, which presumably all elementary algebra students have seen.

 

[Moonbear]

Some students also just learn better by listening than through reading, so lecture serves them far better than a textbook ever will. And, as we all know from the various misunderstandings that occur on forums such as this, speaking to someone carries a lot of non-verbal information that helps convey context, importance, relevance, and understanding in a two-way direction between both lecturers and students that cannot be accomplished in any written text.

I give my students my notes...I say a lot more than is written. I had one comment this week that there is a LOT in the textbook, and what should she focus on while studying (this is a student who actually attends lecture). I was a bit taken aback, because my impression is that the book is incredibly skimpy on the subjects we were covering in this unit. I suggested focusing on my lecture content, because there was a lot more covered in lecture than in the book. They don't realize that my hour of talking covered more material than 3 hours of them reading.

Of course, reconsidering what khemix has been saying, perhaps he just has bad experiences with bad lecturers. We do know they exist, the lecturers who just practically read the book to the students and provide no additional information or insight, and don't take time to help them connect concepts or make associations between related materials. Perhaps that is khemix's only experiences. Of course, that is not the intention of this thread to complain about the bad lecturers, but to focus those interested in improving their teaching on the GOOD lecture techniques.

So, I'm going to ask that we get this thread back on topic. I really don't want to lock it when there is possibly still valuable information to discuss, but non-constructive complaints are not going to keep this thread going.

 

[buffordboy23]

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.

Many never ask questions, and those who do, often ask things that could be found immediately by looking them up in the index of the book. People who ignore office hours for weeks expect me to schedule extra help sessions the day before the test. Questions more often focus on "what will be tested?" instead of how to understand what has been taught.

Everyone seems to have taken calculus in high school, but most also seem not to know anything about algebra or geometry or trigonometry. With the advent of calculators some also do not know simple arithmetic, like how to multiply two digit numerals. (I have had students who had to add up a column of thirteen 65's on a test, apparently not knowing how to multiply 13 by 65.)

Many think that having taken a subject "2 years ago" is a valid excuse to have forgotten the material, and to expect the teacher to reteach the prerequisites. Appparently no one ever dreams of reviewing the prerequisites before the course starts. Books like "Calculus for cretins" are apparently more popular than books like "Calculus for science majors".

When I was in college students like this were just ignored, or expected to flunk out, but in today's setting there are so many like this that they form the primary market. With all good faith to teach these stduents, the failure rate is still about 50% in college calculus across the nation, in my opinion. What are some ideas on how to improve this?

I realize that this is an old post, but I saw that it has been resurrected. I am an ex-teacher who returned to college to study physics, so I have recent experience regarding both educational windows, as a student and as a teacher.

I see what you are saying in my upper level physics courses. Basically, many students don't perform well due to many reasons, so grades are curved and a large majority of students pass the course although maybe a significant number of them shouldn't. What is the overall benefit to the mathematics or physics student for such behavior from an academic institution? Because they get away it, these students do enough to get by, but not enough to gather a deep understanding of the material and to develop a serious discipline in regards to their learning. So, the student never matures. Eventually, they will graduate with a degree, but they are not adequately prepared for graduate school or to enter into their job field.

If I were a professor, nothing would be more easy than to take the hard line with students; if you don't understand the material, you fail. However, like you said above, the common college student entering university will not likely meet these expectations, so we must look at alternative strategies.

Here is my basic plan:

Don't get rid of the hard line approach completely. Enforce it with strict discipline with junior and senior level mathematics courses. Let this knowledge be transparent to freshman and sophomore majors, so that they are fully aware of the expectations ahead, while at the same time supplementing their education with objective tasks to transition them towards these higher expectations. This will give the student a few years to develop an academic level of maturity to be successful in the higher-level courses, or at least enough of warning to switch to a new major, like physical education, without having to stick around at college for another 2 years.

The objective tasks for freshman and sophomore students could include "gateway" examinations. For example, you said students don't have basic trigonometry, algebra, and geometry skills. Well, have your department make them learn these skills and prove it by taking a rigorous test at some point, maybe in their first semester. If they don't pass, they don't take higher level courses until they do. Of course, many students will need to review these concepts, so professors teaching intro courses should develop tools to help students accomplish this task. As another example, after the third calculus course, students are required to take a comprehensive calculus test, which will force them to go back and learn concepts they forgot.

When I returned to college, I had to take a calculus III course. One of the components of the course was a weekly lab using Maple software. Often, I found myself confused with the code as did many other students, so assignments were difficult to complete although the problems were basic in design. Graduate students ran the course and there were about 40-50 students in the lab so it was difficult to get assistance. Outside of lab, it was even more difficult, since the graduate students are extremely busy themselves. Overall, I thought the labs were a waste of time, although they had potential to be great learning tools if designed effectively. The code should already be developed so that it is easy to manipulate, which allows the student to focus more so on conceptual understanding than programming the correct code. It would also be of benefit if the labs were developed around interesting phenomena associated with the students current background of study (physics, engineering, etc.).

Hopefully, there will be more to follow when more comes to my mind and I have the time.

 

[buffordboy23]

Here is another thought that comes to mind. In your intro courses, assign a lot more problems as homework. Practice makes perfect, right? In a typical section of calculus books, there might be numerous ways of applying the concept. For example, there are numerous ways to compute a line integral. I see that professors assign about 10 problems per week in a math course as homework that span a few sections of the textbook, which is only a couple problems per section. So, the student doesn't get much practice solving line integrals unless they take they initiative to do it themselves. Therefore, many students probably forget how to solve these problems a short while later because they have so little experience except for the one or two examples, and likely won't even care to go back to relearn it.

So, as a professor I would assign more problems. Many of these problems would be easy munching to the student and take little time, yet they would reinforce the concepts and hopefully enter into the student's long-term memory storage. Solutions should be available to a decent number of these problems pertaining to each concept, so that students have a target to work towards and don't get easily discouraged. Hints should be given from the outright to teach supplementary ideas not necessarily discussed in the book that will appear in a certain problem and offer simplification. In this case, it may be necessary to design your own problem sets since the problems in the book may be sparse or their integrals time-consuming to calculate.

 

[buffordboy23]

Another idea for intro courses. We know many students don't read the textbook, so hold the students accountable. Assign sections to be read for each class. Every so often, give students a short 2-3 minute quiz about very basic ideas in the reading at the beginning of some classes. This would likely increase the number of students who are prepared for lectures and promote discipline with their studying.

 

[mathwonk]

those seem like good ideas. the reality we face is very serious however.

e.g. what would you suggest when faced with the following data that actually occurred to me?
My department does not give placement tests to my knowledge, so at the beginning of a calculus class, in an attempt to measure readiness, for my sake and the students', i gave a precalculus quiz, mostly covering facts on equations of lines and circles and the definition of sine and cosine.

the average score was 10%, although one Chinese transfer student got 100%. So I gave it back and made it a take home test. The next day, several students did not come back at all, and of those who did and turned in the take home test, the average score this time was 15%.

I was so discouraged I never did this again.

The only thing I know of that seems to work, is to send students to the board for presentations and detailed discussion and critique of their work, but this takes about 4-5 times as much time as is allowed in a semester.

my calculus classes think i am teaching a very theoretical class if i ask them to learn the statement of a theorem, including hypotheses, or to state a definition, although i essentially never ask them to prove anything, or even to learn a proof.

A typical incoming student has never apparently seen a definition. Even if I write
DEFINITION:...

in front of every one, almost no students, even the brilliant ones, can answer a question asking for a definition to be given. only a few think hypotheses are part of a theorem.

I am going to try to make this point this week by making up a list of series that do NOT converge even though they may appear to if you fail to check the hypotheses of the tests we have learned.

But I am worried that my students will simply give up if they are confronted with the fact that theorems are not true unless the hypotheses are true, and the need to verify this. Math has been dumbed down so much for them, some of them seem saddened by the thought that this level of precision is part of the subject.how is it possible there is such a disconnect between what an incoming student thinks of as theory and what i think? i have even been criticized for teaching "with words".what else should i have used to teach with?

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

 

[mathwonk]

as to gateway exams, we have turned to them in an unlikely setting: for graduate students in our phd program. we have admitted so many grad students who do not have even basic college math skills that we now have a remedial program for graduate students. and presumably it is working.

in america apparently failing students is never an option, so our educational system is built around remedial programs at every level, since students arrive in every grade without any reason to be there, except time spent in previous grades.

in your posts b23 the main thing that strikes me is that you are a returning teacher. that is a universal qualifier in my experience. i have never had a returning teacher who did not have the dedication to succeed that i look for. so maybe we could kick students out sooner, and put them in practical settings where they will learn quickly that they do need to actually know something to perform in their jobs. i.e. maybe work - study is the answer.

 

[mathwonk]

we tried labs for years, and they were universally considered a waste of time (as you found them) except by the people who designed them. students simply refused to put in the time needed to benefit from them. the time needed to deal with the software and hardware was overwhelming to them, and the program was a huge time sink for the few faculty who lavished enormous effort trying to make them relevant, interesting and potentially beneficial.

my feeling is that there are many things that would work on a more motivated population. so motivation is the key.

how do we motivate students to work hard, harder than they have had to in high school? to ask themselves to understand what they are doing rather than just memorize procedures?

 

[Tobias Funke]

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

I'm a high school math teacher and I ask the same thing about middle school. Like you said, these students are never held back when they should be and thus you have the problem all the way up to grad school. The problem doesn't start in high school, it starts with fractions and maybe even earlier.

I bet you won't be surprised since I've read a lot of your horror stories involving unprepared students, but about 90% of my freshmen algebra 1 class can't add fractions unless you stand over them and tell them every step. I'd say about 20% can't consistently do computations like -3-10. They simply can't remember which of 1/0 or 0/1 is undefined (and I try to explain to them why, but you must know more than anybody that they don't want explanations, just facts to remember). How can I possibly bring them from that level to the algebra they need to know in a year?

I've worked with middle school math teachers too. I was told we were covering prob and stats and I was a bit worried because stats is not my strongest area. I soon realized I had nothing to worry about because I was absolutely shocked when I saw the level these people were at. Only a few of them actually wanted to learn and the rest just suffered through the course to get more credit for a masters (in education. Many math teachers don't even seem to have a degree in math). It was embarrassing.

Of course there are the good students who make it all worth it though. A few of my calc students were genuinely excited when we took a day off of related rates to prove the formulas for the sum of the first n squares/cubes in several different ways (no boring induction), something of no relevance to the class. They actually enjoyed going through the proofs, and hopefully it will inspire them to try their own proofs.

 

[buffordboy23]

e.g. what would you suggest when faced with the following data that actually occurred to me?
My department does not give placement tests to my knowledge, so at the beginning of a calculus class, in an attempt to measure readiness, for my sake and the students', i gave a precalculus quiz, mostly covering facts on equations of lines and circles and the definition of sine and cosine.

the average score was 10%, although one Chinese transfer student got 100%. So I gave it back and made it a take home test. The next day, several students did not come back at all, and of those who did and turned in the take home test, the average score this time was 15%.

Wow. These scores really are dismal, even when given a second opportunity. Are these students mathematics/science majors?

EDIT: I think this would be reason enough to require placement tests. Your students are in a hole before the course even begins. In a sense, your quiz is kinda similar to a placement test. You have determined that the majority of the class is not adequately prepared for calculus, yet you are stuck with them the whole semester.

The only thing I know of that seems to work, is to send students to the board for presentations and detailed discussion and critique of their work, but this takes about 4-5 times as much time as is allowed in a semester.

This is good because it holds students accountable, but like you said it consumes a lot of time. I currently have a physics class that does something similar. Two times during the semester, each student has to give a brief 3-5 minute review of the previous lecture. The students seem to handle it well and understand the content, but, overall this consumes a total of about 2-3 whole classes throughout the semester. This further supports the idea that students are capable when they choose to be.

my calculus classes think i am teaching a very theoretical class if i ask them to learn the statement of a theorem, including hypotheses, or to state a definition, although i essentially never ask them to prove anything, or even to learn a proof.

A typical incoming student has never apparently seen a definition. Even if I write
DEFINITION:...

The thing I like about mathematics is that it is a subject that is in black and white. The basic rules are given, from which other rules follow. You always win when you play by the rules. However, I think that, in general, students don't understand the rules and why they exist. I would argue that many students find proofs very difficult, and I would wager that this is due to their high school education because they probably aren't exposed to the rigorous aspects of the mathematics.

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

Probably. =)

 

[buffordboy23]

Since my return to college, I have been very critical of the manner in which professors run their courses. I find that many of them do the basics and cover what is exactly in the book--write the theorems/definitions on the board, talk about the gist of the concept, do some examples. For a student like me, I often won't go to class like this because I read the book and learn nothing new from their lecture. I can't blame them though, since they are catering to the average student. Plus, it takes a lot of time on the part of the professor to be original with his presentation, since professors often exist in a publish-or-perish environment and have exhaustive workloads themselves.

In light of this, I thought of an alternative approach in offering a course to students. I am sure that you probably have quite a few sections of calculus each semester, with multiple professors carrying this workload. I wonder if it would be beneficial for these professors to record video-taped lectures and develop notes for each concept, so that their students can view them online. As a supplement, students can post urgent questions on a forum, where another student or another professor can respond. Essentially, it would be just like going to class, but instead, students can watch the lecture videos from their dorm and at their convenience. Of course, there needs to be student accountability somewhere in the mix for this to potentially work. There would still be regularly scheduled class to attend.

These are the immediate and foreseeable benefits:
1. Students often have trouble following the professor during a lecture, and many are scared to ask questions, so they can review the lecture over and over again as need be. This also gives more time for reflection, and the student can come to class with good and quality questions about the content.
2. Many students choose not to go to class for one reason or another, so they miss out. With this idea, students have no reason for missing a class because they can go anytime they want.
3. In the long run, this will likely reduce a professor's workload. If the video-lectures are designed effectively, they can be used semester after semester, freeing the professor from putting basic content together.
4. During the regularly scheduled classes the professor has a lot more freedom, since they don't have to cover basic content. They could discuss ways of how to think mathematically, elaborate on key ideas/theorems, introduce interesting applications, solve some examples, dedicate more class time to holding students accountable, etc.

I don't know of any college that has tried such an idea. The closest thing that comes to mind is the online courseware offered by MIT. Perhaps, my thinking is too idealistic, but it would be interesting to see how such an approach works out.

 

[buffordboy23]

how do we motivate students to work hard, harder than they have had to in high school? to ask themselves to understand what they are doing rather than just memorize procedures?

I think this is the fundamental problem. I think many students get discouraged easily because the work is difficult and requires time. But if they get passed through the course just by going through the motions, is intrinsic motivation necessarily an issue for them? Personally, I think many college students still need scaffolding to be successful. Should it be this way? Probably not, but this is the current state of our education system.

 

[mathwonk]

the video model is hard for me to imagine. i try to learn my students names and lecture with them in mind. i call upon them i class, and if i see a puzzled face i stop and ask what is bothering them. if students seem receptive i am encouraged to go beyond usual bounds and present things that are more advanced. i model the lecture on their response, and plan the next one based on questions from this class. without an audience i would be very bored and have trouble knowing what to say without the stimulation of the faces.

 

[will.c]

If you can expect the students to watch your lectures on their own free time, why can't you expect them to crack the damn book?

expecting students to prepare outside of class in any way is, unfortunately, not tenable - not the way basic math and science classes are expected to be taught anyway. More unfortunate yet is that these will always remain to be service classes, and so deviating from student expectations is dangerous.

 

[buffordboy23]

If you can expect the students to watch your lectures on their own free time, why can't you expect them to crack the damn book?

You raise a good point. I am throwing out ideas, even radical ones, to spur discussion. If one of my ideas is of any use, then I consider my time well spent.

To answer your question. Students are not held accountable to read their book. If they come to class without having read their book, then the only immediate consequence they suffer is not understanding the lecture. I believe accountability is a necessary prerequisite to improving student performance. Having the lectures already recorded will free up more time to focus on accountability. You can still find numerous ways to hold students accountable with a traditional course (e.g. presentations), but do you think more things would need to be sacrificed to do so in comparison with the video-lecture idea?

 

[buffordboy23]

expecting students to prepare outside of class in any way is, unfortunately, not tenable - not the way basic math and science classes are expected to be taught anyway. More unfortunate yet is that these will always remain to be service classes, and so deviating from student expectations is dangerous.

Yes, people fear change. When you take them out of their comfort level they have no choice but to adapt to a new situation. Unfortunately, to obtain some goals does require change. The widespread availability of technology in our society today has changed our society in ways unimagined in only a relatively short period of time, so why shouldn't we incorporate this in our education system? Some public schools have chosen to give their students a personal labtop computer; I don't yet know if any schools have replaced traditional textbooks with digitized copies.

EDIT: Some teachers are now required by their districts to run online chat/forum sessions with students outside normal school hours.

EDIT2: I agree that deviating from expectations can be dangerous, but only if such expectations are not conveyed clearly and maintained with consistency. This was a large reason why I saw students having difficulty in some of their classes when I was a teacher. The students did not understand their teacher's expectations on an assignment. They sometimes would ask me for assistance, and so I would look at the assignment given to them and I did not understand the expectations either.

 

[buffordboy23]

the video model is hard for me to imagine. i try to learn my students names and lecture with them in mind. i call upon them i class, and if i see a puzzled face i stop and ask what is bothering them. if students seem receptive i am encouraged to go beyond usual bounds and present things that are more advanced. i model the lecture on their response, and plan the next one based on questions from this class. without an audience i would be very bored and have trouble knowing what to say without the stimulation of the faces.

My original intent with the video-lecture idea was to still hold classes. If students are held accountable by completing short questions at the end of a video-lecture and submitting them, you now have foresight about misconceptions and can structure the next class meeting to rectify their misunderstandings. By holding students accountable before they arrive to class, more of them should be prepared to engage in meaningful discussion and permit you to have greater probability of success in introducing any interactive class activities. Students will still need some form of accountability sometimes during class meetings or there would be no reason to go to class. So you would still have your audience.

 

[CoCoA]

i gave a precalculus quiz, mostly covering facts on equations of lines and circles and the definition of sine and cosine.

the average score was 10%

This is bad enough to wonder about a systemic problem - then I saw a web log (can't find it now) from a prof in state of Wash with a child in high school algebra using a book with essentially NO algebra in it! Students "solve" problems by looking up values in tables and creating graphs.

California has a "algebra for every 8th grader" initiative (perhaps failing to legel challenges, perhpas not) that just has to water down the curriculum severely to accomplish this. Every level of our educational system just seems to be underminig the teaching of math, no wonder colleges can't get good math students.

 

[CoCoA]

I wonder if it would be beneficial for these professors to record video-taped lectures and develop notes for each concept, so that their students can view them online.

I took an line course a few years ago with recorded video lectures online. It was actually pretty good.

On the other hand, about a year ago I was "talked into" (forced into?) teaching a course onlilne that I thought wasn't that amenable to online delivery. It turned out to be pretty bad.

 

[CoCoA]

YThe widespread availability of technology in our society today has changed our society in ways unimagined in only a relatively short period of time, so why shouldn't we incorporate this in our education system?

I think schools have a wide range of effectiveness in their technology initiatives. For example, some schools have implemented well-developed online courses, others (such as one I work for, that I will not name) has several online classes that I candidly believe are nothing more than glorified independent study courses - students read on their own and simply turn in assignments to an online system. If anything, it is just another way to excuse students from attending a class. Sometimes it's justified, when some students can know the material in their sleep and just need to complete the course for the credit (it does happen in computer courses), while others need every possible interaction with the instructor to learn the material.

 

[buffordboy23]

California has a "algebra for every 8th grader" initiative (perhaps failing to legel challenges, perhpas not) that just has to water down the curriculum severely to accomplish this. Every level of our educational system just seems to be underminig the teaching of math, no wonder colleges can't get good math students.

I agree. One of the things I saw in public schools was the use of gimmicks to accomplish learning goals in mathematics.

Here's some examples:
1. When assisting a student with homework that required them to multiply positive single-digit integers, the student would use her index finger to tap 5 imaginary dots while performing some algorithm, called "touch-math" or something like that. I thought how strange. Why not memorize your times tables rather than some lengthy algorithm?
2. To isolate an algebraic variable, students--some of these were honor students too--would say some mnemonic catchphrase aloud or inside their head and perform some weird maneuver. I briefly tried to show them how I accomplish the same task. They were very confused and thought I was strange.
3. One of my relatives needed help with an arithmetic assignment. They had to do the problems with some gimmick method not even discussed in their textbook.

I guess the thinking that comes with these gimmick methods is that being able to compute an answer is equivalent to understanding how we could and did arrive at such answer.

 

[Moonbear]

The only thing I know of that seems to work, is to send students to the board for presentations and detailed discussion and critique of their work, but this takes about 4-5 times as much time as is allowed in a semester.

This is really a great teaching strategy. Getting students to DO something in a class really helps to increase learning and retention. Sitting and listening to lectures is entirely passive learning, and not very effective at all. Of course, they do need the lecture to have the material presented to them, but then taking it a step further and making the use what they have just learned is what solidifies that knowledge.

Does your course have a recitation section? If so, that is a great use for it. Instead of having a TA solving problems for students, which is really just continuing more lecture and puts them all to sleep, have the TA facilitate while the students go to the board and work the problems for one another. Then the TA will just need to watch for errors and prompt with questions when a student gets stuck (much like we do here in the HW forums here).

 

[mathwonk]

As an example of how much time it takes to teach, using student participation, it can take a whole period to guide a good student through a proof that the composition of two injective functions is injective.

the hard part is to get the students to connect up mentally the definitions they memorize with the steps in the proof. many students regurgitate a definition for something like injectivity, and then two seconds later have no idea how to begin a proof that some particular function is injective. (answer: begin by assuming the "given" part of the definition, i.e. take two arbitrary points in the domain, then assume either that they are different, or that their values are equal, then ...).

it is very challenging to teach learners to use quantifiers or to use them properly. letters are simply written down without defining what they mean. this is a very basic problem: the same letter is thought to mean the same thing, whereas this holds only within the same quantifier. the idea of a "variable" i.e. a letter that can have several interpretations is quite foreign.

if an injective function is defined as one such that for all x,y in dom(f), assuming x different from y, implies f(x) different from f(y), then for many students this definition cannot be readily applied in a case where the arguments have names other than x and y.

e.g. in the case of the theorem above, if f(g(a)) = f(g(b)), this does not trigger any response from the definition of injectivity of f, to conclude that g(a) = g(b). for other students this makes sense only if they rename g(a) = x, and g(b) = y, thus recovering the same names they have memorized.

somehow the teaching of algebra, i.e. the use of variables with multiple interpretations, and the corresponding understanding of quantifiers to keep book on what those variables mean, has apparently disappeared in a "saxon" high school curriculum where "algebra" means multiplying x^2 times x^3 and getting x^5.

the excellent algebra book by harold jacobs i believe, or maybe some older 60's books, treat this problem by substituting place holders like [ ], or ( ), for a variable. then the student simply fills in the box with the relevant value. this seems to help teach that anything can go in there, but two boxes of the same shape must be filled in by the same value in any given setting.

 

[mathwonk]

we must try to somehow maintain focus on what seems over the centuries to matter, the ability to analyze problems, to store and use prior knowledge, check ones hypotheses, and employ useful analogies.

these abstract skills seem to me what is missing, not just the rules of exponents, or the many other topics on a subject list for a specific math course.

what are some ideas for inculcating the ability to understand and use language in analyzing problems, including precise mathematical language such as variables and quantifiers?

sometimes I discuss variables as pronouns, which require antecedents just as variables require quantifiers, i.e. x is like "he", but who is he? must be specified. I take my cue here from some great old 1960's algebra books from the university of illinois i think, some of the excellent products from the 60's math revolution, like smsg books.

i still recall coming home from college as a freshman and reading these junior high books and learning the distinction between a number (abstract idea) and a numeral (concrete symbol for a number. the illustration was to imagine writing the word "milk" on the board and asking whether or not there is milk on the board. answer no, not milk but "milk" is on the board. i thought: my word - where were these books when i was in high school!?

are such books actually in use anywhere? it seems to me we do have the beginning of a solution to our problems in the existence of these wonderful materials from the 60's.

what we need next is a commitment from someone to use such materials in the schools. the political problem is how to be allowed to set a standard that not everyone may meet, or at least not without rising to it.without the ability to require this standard, the key is motivation, how to get the child to want to learn what is actually beneficial.

my first chairman had several suggestions for motivation, something like: appeal to the beauty, or the applicability, or the historical significance, or the reliability, or the power, of mathematical results.

how much time do we spend convincing the students they will benefit from learning our subject? what are other approaches?

 

[buffordboy23]

I can easily tell from your posts that you are an effective professor.

what we need next is a commitment from someone to use such materials in the schools. the political problem is how to be allowed to set a standard that not everyone may meet, or at least not without rising to it.

I think there is an issue of practicality here in terms of designing a mathematics curriculum. The average student who graduates from high school will not need a rigorous background in mathematical formalism and its abstractness to be successful in the real world. However, any exposure is likely to improve their skills of logic and reasoning.

I think the larger focus of this effort should be on more academically inclined students, especially those planning to pursue the mathematics/sciences in college. But then we need effective mathematics teachers and educational settings that offer the required environment. (Before I resigned as a teacher, the administration decided to do away with honors level classes in the middle school, with one of the reasons for doing so because it distinguishes some students, the "smart ones", from others, the "dumb ones".)

It is the responsibility of the university to develop these effective teachers, and with the current level of the mathematical background of the average incoming college student, your task is a very difficult one. It may be necessary to design and add new courses to the college curriculum requirements, which focus entirely on the formalism and abstractness, since it is missing from the student's background. In conjunction, large efforts to reform our public education system are necessary to create the environment to work towards the goal of preparing these students for college. It's a cyclical process that has decayed to the current state over time.

 

[mathwonk]

maybe you are right that what is needed for average students is something like an old (really old) fashioned logic and rhetoric course, in which students learn and practice reasoning and argument without the obstacle of formal mathematical symbols and language.

I try to convince my students that proofs are useful in real life, e.g. in arguing for a raise with your boss, or convincing the iRS that you deserve a tax break.

e.g. the description of the conditions under which the break is given is the definition. then the theorem you set out to prove is that you qualify for it. to prove this you must address every requirement in the definition, which as you may know from experience involves lots of logical connectives like "all of the following must be satisfied, and one of the following as well..."

i may have made an error in assuming that learning to prove via rolle's theorem that a function with never zero derivative is injective, will lead to the ability to obtain a tax break or a raise.

such real life applications could help motivation. there is a psychological difficulty with offering such courses in university which i think were traditionally high school courses even in nineteenth century america (i still have my grandfather's books from the 1880's), but maybe they could be reintroduced in high schools.

 

[mathwonk]

im not sure how effective i am. the attention span of an average calc student is pretty short, and making calculus entertaining is tough day after day. after giving the series expression for arctan(1) = pi/4, i actually calculated it for a couple decimal places, showing how many zillion terms were needed to get good accuracy as an attempt to teach use of the error term in "taylor" series.

then i got euler's works from the library and showed how he had adapted this series using addtion formulas for tan to get over 120 places of accuracy. interestingly i also noted he made a mistake in the 112th? digit. (I actually checked them all.) that was fun for one day, but then it was what next?

and after all was said and done maybe one student could use the error term in taylor's series to show the series for e^1 converges to e, on a test.

 

[Nick M]

I felt that those courses in which one earns an A+ are not sufficiently challenging.

I couldn't disagree more. Most courses have learning objectives that students are expected to reach. I get straight A's because I don't miss a single lecture/lab, I complete all the readings, I do all the homework, and I take the time to think. I bust my @$$ to achieve a level of understanding that allows me to complete the work with those grades.

Why would you continuously set the bar higher and higher to the point where good students who apply themselves can't cope and begin to fail at learning? A student needs to get a B or C and leave the class confused on some subjects in order to be sufficiently challenged?

There is no excuse for students who don't show up or study, but there is also no excuse for setting someone up for failure when you are supposed to be a guiding force.

One thing I learned from attending both CC's and University is that a Ph.D. isn't synonymous for teacher. While students hold a great share of the responsibility for their own success, a "teacher" that can't connect, can't excite/inspire, and can't develop a sensitivity for the state of mind of their pupil's is really just a fogged in island of knowledge at the board. When this is layered with low expectations and tertiary factors such as language barriers, it's a wonder so many students do actually manage to progress.

And of course it starts when the person is young. I was taught statistics in high school as "Pre-Calculus". My first attempt at engineering placed me straight into Calculus I where I got a B (without any knowledge of transcendental functions), and then washed me out in Calculus II. I went to office hours, sought help, and dedicated a significant amount of time, but I had a shaky foundation that was checked off as satisfactory by those who were supposed to be my trusted advisors.

I'm now in college experience 2.0 after starting fresh in Pre-Calculus taking mathematics courses at a local CC that are at least 50% more rigorous than the ones at UMass, but are taught by people who actually know how to teach. One thing for sure is that I no longer trust any of my professors or those in leadership positions - I verify everything myself and essentially act as my own advisor and coordinator.

 

[Moonbear]

maybe you are right that what is needed for average students is something like an old (really old) fashioned logic and rhetoric course, in which students learn and practice reasoning and argument without the obstacle of formal mathematical symbols and language.

When I was in college, I took a course on symbolic logic, offered through the philosophy department. I had no idea what I was getting into when I registered for it, it was just one of those options to fulfill core requirements in philosophy and the only one offered that fit my schedule that term, which was usually good enough for me when choosing electives.

While I will never in my life need to remember any of the symbols used...who cares what the backwards C or upside-down U meant (if those were even the symbols used), the formal learning of how to structure an argument and to find the logical flaws, missed steps, and false conclusions was an exercise that benefits me every day. From careful designing and interpretation of experiments, to teaching material to students in a careful, logical, step-wise fashion, to arguing for or against various things people are proposing to change, etc. I think every student could use a course such as that, and perhaps something like that would be especially good preparation before taking math courses that require a lot of proofs. Basically, while they're still using words and before getting bogged down in mathematical terms, get the concept across of how, in general, a proof is supposed to function to logically demonstrate that one concept derives from another.

I know I keep injecting discussion here that's not specifically related to calculus classes, but I think it can be helpful to recognize common themes present across curricula that pertain to the modern-day students, since it will affect them in any class they take. But, it took me too long to recognize this in my own students this year, so I'm going to share it here for other's benefit, and I am going to actually take some time out next year to address it. 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. These are sophomore students too. I had assumed that since I was teaching a sophomore level class and the average GPA of students admitted to this program was a 3.6, that these would be students who figured out how to study in their freshman year if they hadn't come into college already knowing that. But, somehow they skated through their first year still without acquiring those skills.

So, I'm going to take a little time out of my first few classes next year, and teach them how to study for my subject (and of course study approaches do vary a bit from one subject to the next, so even those with good study skills may need to hone them for my course). I'm already going to incorporate a team-based-learning module into the lecture, so will use those teams to first teach them the value of a study group, when a study group is done right. It takes time away from the content I can deliver, but I'd rather cover a little less content but have them learn all of it well than to cover a lot of content and have most of them only grasp a small portion of it. I'm already sitting down and outlining the learning objectives for the course for next year (nobody has done that yet for this course) and will make sure that content delivered focuses on those objectives. I'm also changing textbooks, so more of what I don't have time to deliver in lecture will be available in their textbook (it's really rough when you're stuck teaching from a book that is entirely inadequate for the course, especially when I have to point out sections that are completely irrelevant to the subject or that flat-out state things incorrectly. I also have problems that the textbook and lab guides are using different terminology for the same things...the med students can handle that there are more terms than structures because names have changed over time, but the undergrad nursing students cannot...it just confuses them at this stage of their learning). All of these are things any course can consider.

 

[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.