Teaching calculus today in college

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The discussion centers on the challenges educators face in teaching calculus to students who lack foundational math skills and study habits. Many students fail to engage with course materials, neglect office hours, and expect to pass without adequate preparation, leading to high failure rates in college calculus. There is a consensus that high school math education has shifted focus, often providing insufficient background in algebra and geometry before students tackle calculus. Educators emphasize the importance of active learning and problem-solving, encouraging students to attempt challenging problems rather than seeking direct answers. The conversation highlights a need for improved teaching methods and materials to better prepare students for advanced mathematics.
  • #91
buffordboy23 said:
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.
 
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  • #92
CoCoA said:
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.
 
  • #93
mathwonk said:
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).
 
  • #94
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.
 
  • #95
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?
 
  • #96
I can easily tell from your posts that you are an effective professor.

mathwonk said:
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.
 
  • #97
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.
 
  • #98
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.
 
  • #99
mathwonk said:
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.
 
  • #100
mathwonk said:
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.
 
  • #101
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.
 
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  • #102
wow moonbear, your dedication, insight, and positive attitude is an inspiration!
 
  • #103
Moonbear said:
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.
 
  • #104
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.
 
  • #105
buffordboy23 said:
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.
 
  • #106
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?
 
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  • #107
SticksandStones said:
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.
 
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  • #108
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.
 
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  • #109
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.
 
  • #110
mathwonk said:
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.
 
  • #111
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.
 
  • #112
SticksandStones said:
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.
 
  • #113
Joskoplas said:
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.
 
  • #114
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.
 
  • #115
Joskoplas said:
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?
 
  • #116
SticksandStones said:
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.
 
  • #117
SticksandStones said:
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.

SticksandStones said:
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.
 
  • #118
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. :)
 
  • #119
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.
 
  • #120
Tobias Funke said:
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.
 

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