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The Most Common Errors in Undergraduate Mathematics (2009)

math.vanderbilt.edu

141–150 of 161 posts

Re: The Most Common Errors in Undergraduate Mathematics (2009)

#141
post #5

Bad handwriting. At age 40 I started taking a masters in stats and had situations where I had exponents on exponents. This lead me to buy a higher resolution tablet for reading pdf's with tiny math. I also bought a finer point pen and this helped me improve my handwriting a lot . Closing loops on "o's" or backtracing the upward line of a cursive "t" to not make a loop. With the finer point, I was able to see my impre…

In the Intro-to-C class I TA'd, we dedicated several minutes in lab to instructing students on how to properly draw an ampersand (&) character, and the other ways of drawing it so potentially confusing (especially since a + means something completely different). Personally, the +/t handwriting issue has never been an issue for me: make sure you get a good tail on the t, and it's pretty easy to distinguish from a +. I…

I blew several minds at my first job when I pointed out the ampersand is just a mirrored capital cursive S. Suddenly it wasn't such a mysterious character to any of them anymore.

Re: The Most Common Errors in Undergraduate Mathematics (2009)

#142
post #78

Earlier quoted context omitted.

Nobody’s saying it’s totally invalid. I think you’re reading that into what they’re saying and you’re talking past each other. The complaint is that students work backwards without understanding why working backwards is different than working forwards. So if they happen to do it correctly (e.g. by using reversible steps), then it’s only success via accident. Done correctly, and on purpose, with the different care tha…

When you are doing equality and such all steps are reversible. Working backwards is exactly the same as working forward. You develop reasoning where you see it easier until to figure out to connect sides.

Having equality signs at each step does not mean you can reverse the steps. You need bidirectional implication for that.

Re: The Most Common Errors in Undergraduate Mathematics (2009)

#143
post #28

Most of these appear to be errors caused by stress or tiredness.

That's what my (junior in high school) son keeps saying when he makes simple arithmetic errors that cost him points on tests. I disagree, actually - these mistakes are made by lack of practice. If you've practiced solving enough integrals, it doesn't really matter how tired you are, you're going to get the right answer just as you won't read words incorrectly if you're tired.

Sorry, but this is absolutely wrong and potentially harmful to your son if you eventually convince him. It is absolutely possible and not that uncommon for a neurotic person's mind to go blank when under considerable stress, at the point of forgetting otherwise completely trivial information.

I urge you to not dismiss your son's complaints but to instead investigate them further, preferably with the help of a professional therapist, to save him from the possibility of escalation of his problem.

Re: The Most Common Errors in Undergraduate Mathematics (2009)

#144

Earlier quoted context omitted.

Oh yeah, I'm with you there. Teaching quality at both my undergrad math program and my graduate compsci program was abysmal, with the exception of a handful of dedicated teachers. Both at well-ranked research institutions, which is the problem. The whole system is crazy. Why on earth do we allow tuition money to be spent on anything else but full-time teaching staff?!

> Why on earth do we allow tuition money to be spent on anything else but full-time teaching staff?! Because the culture of innovation that part-time researchers/part-time teacher bring is far more important to society in the long run than the short term gains gotten by full-time teaching staff. Most fields in the world are rapidly changing, and what's taught in them changes every decade or so. Most full time teacher…

Have you experienced student life at a large public research university recently? The well-documented trends of administrative bloat, publish-or-perish, exploitation of adjuncts and TAs for cheap labor, obsession with luxury campus amenities, etc etc. have all slowly degraded this "idealized" version of academia that many people have in mind (assuming such a thing ever existed in the first place).

I run into the same wall talking about this with my parents, who don't really seem to understand just what a racket the whole university system as become.

> Most full time teachers will atrophy because they don't have any real incentive to keep their knowledge and skills at the cutting edge.

Tenured professors also have no incentive to keep their knowledge and skills at the cutting edge. An oh boy do I have some stories about lazy professors, both as a student and TA.

> Because the culture of innovation that part-time researchers/part-time teacher bring is far more important to society in the long run than the short term gains gotten by full-time teaching staff.

Research and education are both important. Both have long-term gains. Research is hard, yes, but many people fail to recognize that teaching is also hard. We should fund both, and use a transparent pricing model.

If an 18-year-old takes on tens of thousands of dollars of personal debt, the benefit should be directly to them. The whole "coLLegE lOAnS aRe An InVEstMEnt in YoURselF!!" argument against free university education is naive from the start, but becomes completely absurd when that money isn't even used to directly fund education.

Yes, basic research is fundamental to the advancement of human society. So let's fund it publicly, through business or personal taxes on all members of society. Let's also recognize that higher education has massive long-term benefits to society as well, and make high-quality college education cheaper and more accessible by hiring dedicated teaching faculty who consult with research experts to design their curriculums. We can do both.

The current system is extremely inefficient at distributing the time and resources of research faculty. We don't need research faculty to regurgitate the same intro linear algebra lecture in-person over and over again every year.

The current system also forces research faculty (and graduate student TAs) to teach, who more often than not have no interest whatsoever in teaching. It is also extraordinarily difficult to get a well-paid college teaching job without a research PhD.

I would advocate for a system where research faculty are in charge of designing the curriculum and modernizing lecture materials as the times change. Rather than repeating the same lecture hundreds of times, have them make lecture recordings, which can be updated gradually over time. This frees them up to design better assignments and projects, rather than repeating the same lecture over and over.

It is well documented that students benefit from small classrooms. Hire qualified full-time teaching staff to lead discussions / labs / projects. Pay them enough that they don't need a second or third job, so they have the time to stay up to date. Personally, I would love love love to teach undergrad-level compsci or math if such an opportunity presented itself.

This is the role that TAs normally fill, but it is extraordinarily rare to find a graduate student who has passion and skill for teaching, as well as the luxury of enough time to do it well.

Re: The Most Common Errors in Undergraduate Mathematics (2009)

#145

> Lack of clarity often comes in the form of ambiguity This is sometimes useful. Mathematicians have precision ingrained but this often repeals pupils. When you start with 'f is a continuous function defined in R of x defined in R, and for each x ..." - well, I am lost. This was the introduction of differentials in my son's high school. I am a physicist, so I started the other way round, by talking about speed, how i…

Somewhat tangental but I am very jealous of a high school that introduces differentials. My high school didn't have calculus - lack of interest and no one qualified to teach it.

This is interesting - so what (roughly) did you have in high school?

In the few Europeans schools I know of, calculus is brought in though all the 3 or 4 years in high school because it allows, ultimately, to draw functions. (I am not a math teacher, just someone who went though French school and was very interested in the school system in some other European countries).

It comes in pieces: you get to learn differentials and their meaning, usually then used in physics curriculum to denote speed and force.

Then you get limits and how to calculate them for infinity or when approaching a non-continuous function from both sides. And l'Hôpital rule, etc.

Again, it has direct implications in physics, and sometimes in biology (I saw that only once, when the teacher introduced enzyme kinematics).

Then second differentials to analyze convexity.

All this takes a huge chunk of math in high school, and you cannot du much in functional analysis without using differentials - so I am genuinely interested what was the pressure put on in your school (would you mind naming the country?). Trigonometry for instance? Or probability? (these are the two others I can think of that are the other important pieces they learn).

Some of the schools ended by introducing integrals, usually though the concept of calculating surfaces. It was quite useful to have a rough idea of integrals before going to university (where they were introduced anyway)

Re: The Most Common Errors in Undergraduate Mathematics (2009)

#146
post #133

Earlier quoted context omitted.

I took Physics as an elective in University, but it really didn’t sharpen my Calculus skills. Which is sad, but probably because I did so bad in Calculus as an undergraduate. But, one thing I absolutely did take away from Physics was the idea of units and their conversions. It has come in handy so often though in and outside my degree of computer science. If I had to simplify it, it really helps you not to compare ap…

Absolutely! internalizing dimensional analysis has been one of the most useful things I learned in grad school -- even though converting units was something many of us learn, in one form or another, before college.

It is interesting how some of the things just feel like common knowledge when they are not.

I frequently see people comparing things that aren't really comparable and it always seems to me it should be obvious this can't be done.

Most common example is mixing watts and watt*hours. Which should be pretty obvious. Nobody mixes by accident the speed at which you travel with the distance you have to travel. Nobody says "My car can pull off 200kms" or "I have 120km/h to cover" yet the same happens for energy vs power on a daily basis, even in publications.

Re: The Most Common Errors in Undergraduate Mathematics (2009)

#147
post #132
post #131

Earlier quoted context omitted.

Getting students to seek consultation is a major problem. I advise them to come in small groups. This helps the shy ones. They don’t realize that it is very difficult to fail a student who asks for help. Any failure of such a student is implicitly also the teacher’s.

Only true for the teachers that care. A lot of my professors didn't give a damn about teaching and were annoyed that I was wasting their time showing up to these office hours.

University lecturers in particular are a deluded bunch. They mistake their role for one of power and forget that the students are their employers. Source: 100 years as a University lecturer.

Re: The Most Common Errors in Undergraduate Mathematics (2009)

#148
post #136

Earlier quoted context omitted.

I guess I just don't get your point. D 2 = 0 doesn't imply anything, it's just shorthand to make working on something less tedious for people who are already familiar with it. It seems perfectly expressive and sane: mathematicians have successfully studied differentiation using standard math notation. 2 and (λ (x) 2) aren't 'treated the same', that's just a strawman.

Funny how its also used in every textbook for people who aren't. This is getting to be apologia on the levels that I'd only seen in history books about why Roman numerals were superior.

[deleted]

Re: The Most Common Errors in Undergraduate Mathematics (2009)

#149
post #136

Earlier quoted context omitted.

I guess I just don't get your point. D 2 = 0 doesn't imply anything, it's just shorthand to make working on something less tedious for people who are already familiar with it. It seems perfectly expressive and sane: mathematicians have successfully studied differentiation using standard math notation. 2 and (λ (x) 2) aren't 'treated the same', that's just a strawman.

Funny how its also used in every textbook for people who aren't. This is getting to be apologia on the levels that I'd only seen in history books about why Roman numerals were superior.

The premise of your argument is that this notation is confusing people, or it's impossible to make explicit what it means. I don't buy it, you even explained the type elision using standard mathematical notaion in your original comment.

So if there is some superior notation that would benefit all of maths, that sounds very interesting but I don't think you've made that point very well. Maybe a better example is needed, because on its own, eliding repetitive information from notation does not seem like a problem.

Re: The Most Common Errors in Undergraduate Mathematics (2009)

#150

Earlier quoted context omitted.

> Why on earth do we allow tuition money to be spent on anything else but full-time teaching staff?! Because the culture of innovation that part-time researchers/part-time teacher bring is far more important to society in the long run than the short term gains gotten by full-time teaching staff. Most fields in the world are rapidly changing, and what's taught in them changes every decade or so. Most full time teacher…

Have you experienced student life at a large public research university recently? The well-documented trends of administrative bloat, publish-or-perish, exploitation of adjuncts and TAs for cheap labor, obsession with luxury campus amenities, etc etc. have all slowly degraded this "idealized" version of academia that many people have in mind (assuming such a thing ever existed in the first place). I run into the same…

I have not studied as an undergrad at a large institution, but I have taught at one, and I have taught as a prof at teaching-only institute. There is very little difference in the quality of teaching. Guess what, when teaching only staff is asked to teach 6 courses in an year, they teach the same as a research prof who spends half their time in research and half their time teaching 3 courses.

I think there are many things wrong with academia and with university structures, and teaching quality is one of them. But the problem is not what you think it is. The problem is capitalism, and money-optimizations being the final decision maker rather than quality of teaching or research (which is also much worse than 50 years ago).

This is the same reason school education is so mediocre. You overwork school teachers, and don't pay them enough. And so even the best can't do much good.

You need more money from the state to go into education. And yes, full time teaching staff [1] to conduct recitations and office hours would help a lot. Smaller classrooms would help as well.

I don't agree at all with lecture recordings. As much as you want people in society learning, you also want a lot of people in society learning to teach. The long term intellectual benefits to society where people learn to teach others are enormous. It creates a culture of intellectual inquiry, which is different from attending a pre-recorded lecture or engaging in research or all the other things students/faculty at universities do.

[1] I have been told in German unis these are people with Masters who are waiting to go for a PhD. But they are essentially kicked out after 2 years because the uni recognizes that long-term full-time teaching staff degrades in quality.

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