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The Most Common Errors in Undergraduate Mathematics (2009)

math.vanderbilt.edu

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

#91

Post by yonaguska »

Jtsummers wrote: post #32
ARandomerDude wrote: post #23

Earlier quoted context omitted.

They're great...just not for math. Use a decent pencil instead, like a Graphgear 1000 and some HB lead.

Pens are fine for math. I often used a fine point gel pen in college. However, you have to have a high degree of confidence in what you're writing. If you make a lot of mistakes (either actual mistakes or just wrote down something incorrectly or with a misspelling), then pencils are definitely better.

I got to a point where I would just write down all my solutions on a final paper to turn in for my math assignments.

I did everything in pen, bc of handwriting issues and pencils. I simply burned through sheets and sheets of paper with my preliminary work- then copy everything to something both legible and presentable.

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

#92

Post by vector_spaces »

v64 wrote: post #81
bpyne wrote: post #21

My biggest problems as a math major at the undergraduate level were proofs. I could muddle my way through abstract algebra proofs but real analysis just didn't click. The oddity is that I could read proofs for both subjects: the reasoning made sense. But I couldn't develop a proof.

I also strugged with analysis. I always felt like epsilon-delta proofs involved pulling some absolutely strange value for delta out of your ass that happens to work out in the end, and I never developed an intuition for that. Same with integrating by parts, oh it just works out so nicely if you rewrite u in this totally obtuse way.

The tricky thing with analysis that I don't think many professors are good at conveying is that the ordering of statements in the proof isn't the same as the ordering of steps the proof writer performs to come up with the proof.

Basically you sort of write the broad strokes of the proof up front, leaving the right hand sides of statements like "Choose epsilon such that epsilon = __" blank. Then you do a bunch of scratch work to figure out what epsilon needs to be so that your proof works out in the end.

Another challenge with analysis is that inequalities are central, so fluency in their manipulation is absolutely critical. And naturally most students aren't fluent with them by the time they take analysis, so they get bulldozed by Baby Rudin and learn to hate a pretty cool (and useful) branch of math

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

#93

Post by johnsonjo »

kkylin wrote: post #60
benrbray wrote: post #53

Earlier quoted context omitted.

I would agree, but qualify by saying that not all the blame lies with the student. Precisely because mathematics is so interconnected, beginning students really do require an experienced guide to navigate through the tangled web of concepts. A particularly bad or lazy professor can be worse than no professor at all.

Not arguing with your point, because it has merit, but do want to add that part of the problem with math education (at least here in the US) is structural: for most students, to start with what they know when they enter and to graduate in 4 years with a engineering / science / math / CS degree that means something, there is just too much to "cover." So we tend to cover "methods and tools," e.g., calculus first, with…

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 apples to oranges. At the beginning of Street Fighting Mathematics it brings up that point. For example it isn’t a good comparison to compare the wealth in assets of a given company to a countries GDP. Because, GDP is happening over every year in time whereas the wealth is merely effected by what point in time it’s observed. Units help one see that in order to be truly and more fairly comparable they have to be the same type of units.

I used an analogy of units in physics recently to help a teammate with types and generics, sadly I can’t remember exactly what I said haha other than don’t compare apples to oranges, but I think I had further insight than just that.

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

#94

Post by stephen_greet »

bitxbitxbitcoin wrote: post #30
bpyne wrote: post #21

My biggest problems as a math major at the undergraduate level were proofs. I could muddle my way through abstract algebra proofs but real analysis just didn't click. The oddity is that I could read proofs for both subjects: the reasoning made sense. But I couldn't develop a proof.

I had a similar experience doing my math minor. All the way through Caluclus classes and Diff EQ and up until the first half of Linear Algebra, everything is plug and chug. After the first half of Linear Algebra when proofs started making an appearance, I came to the realization that proofs are another type of mental activity entirely. Survey of Algebra, Basic Real Analysis, and even the dedicated proof writing cours…

I had a very similar experience during undergrad. I loved most of my classes through Linear Algebra but developing proofs felt like trying to learn a new language. Unfortunately, it never really clicked for me.

I'd love to know how to develop an intuition for writing proofs from scratch.

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

#95

Post by grayclhn »

milesvp wrote: post #89
lmilcin wrote: post #47

I studied theoretical mathematics. The absolutely biggest and most common error and really the only one worth to be on list is not internalizing and understanding entirety of material as the course progresses. Everything else should be basically to achieve goal of understanding the entire course material. It is ok to forget later, as long as you are sure you understood it at least when there was a lot of discussion a…

> the first couple semesters are foundational and if you don't understand something it is like a pyramid I was taking a math/cs hybrid major in college, and the upper level math courses just felt like they were getting harder and harder in ways that didn't make sense in my previous math experience. Tests more and more were weighted towards verbal proofs, and I hadn't yet developed the vocabulary to deal with the prec…

Those upper division math classes are pretty painful regardless. The material is just innately hard in a way that's dissimilar to most other hard things.

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

#96

Post by grayclhn »

lmilcin wrote: post #47

I studied theoretical mathematics. The absolutely biggest and most common error and really the only one worth to be on list is not internalizing and understanding entirety of material as the course progresses. Everything else should be basically to achieve goal of understanding the entire course material. It is ok to forget later, as long as you are sure you understood it at least when there was a lot of discussion a…

I sort of agree, but your advice is targeted at an entirely different group of students than the article's. If someone is confused about how to interpret sin^2(x) vs sin^{-1}(x), there's no amount of "understanding" that's going to help them (it really is just idiosyncratic, inconsistent notation), but a checklist like the article might clue them in.

I think students often underestimate what "understand" means in a math class. The progression you're describing ("the first couple of semesters are foundational...") is definitely not universal in US undergrad programs, though. There's just not that much coordination between professors.

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

#97

Post by [deleted] »

dr_dshiv wrote: post #41
nemo44x wrote: post #13

Earlier quoted context omitted.

> I would go to office hours and the teachers were seriously so helpful. This was the only way I survived. I learned that you don't have to be a 'numbers person' at all to be good at math. You just have to understand what's going on under the hood, and this may require lots of questions and outside help. But most importantly, you just have to practice I agree with all of this and it's how I got through extensive math…

> I made sure to get to the office hours and I was surprised that there was almost no one there most of the time. I figured the cost of tuition is best justified by using as many hours of one-on-one time with an expert in the subject to better understand it yourself Truth!

[deleted]

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

#98

Post by grayclhn »

voxl wrote: post #75
aesthesia wrote: post #54

Earlier quoted context omitted.

Well, it's a proof as long as you stipulate somewhere that all the logical connectives are "if and only if." This doesn't mean that a student who writes down a long series of equations beginning with some identity to be proven and ending with some known fact has proven the identity.

Sure, that's fine, but to say that it is an invalid method of proof is wrong. If I want to prove A, and I prove A B for some proven statement B, then I have proven A! There is no question! Feels like crazy pills to think otherwise. This is a kind of backwards reasoning!

It is a kind of backwards reasoning that students empirically screw up all the time.

Edit: so much so, that I would definitely recommend students rewrite these proofs on assignments as an exerecise to make sure it's correct. By the time they reach a math PhD they probably don't need to do that anyomre. :)

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

#99

Post by bmitc »

lmilcin wrote: post #47

I studied theoretical mathematics. The absolutely biggest and most common error and really the only one worth to be on list is not internalizing and understanding entirety of material as the course progresses. Everything else should be basically to achieve goal of understanding the entire course material. It is ok to forget later, as long as you are sure you understood it at least when there was a lot of discussion a…

The problem exists in the educational methods, in my opinion. It seems most, not just in mathematics, feel that learning is the same as if the student can simply be played a tape that cannot be paused or rewound but can sometimes be sped up. Nearly nothing works like this, and it’s just that the penalties in mathematics are greater.

I think of the ideal learning path as a sort of circular, recursive path that bends back and goes back over itself from time to time. Imagine a human painting a canvas. One does not simply do a raster scan, painting pixel by pixel in a linear fashion. By the time the painting is done, the piece has been gone over multiple times with multiple layers of finer and finer detail. A painting is a mishmash of broad and finer strokes, with some layers simply forming foundational layers for the later portions.

A pyramid is the wrong metaphor. In my opinion, learning mathematics should be like creating a painting.

One of the best examples of this I know of is the book Advanced Calculus: A Differential Forms Approach by Harold Edwards. The first three chapters introduce the material heuristically. The next three chapters circle back on the material, thoroughly proving everything introduced in the first three chapters. The remaining chapters greet the student with applications and extensions of the material. In the preface, Edwards states that he wanted the book to be able to be opened to any page and be read and make sense. It’s a wonderful goal, and he achieves it.

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

#100

Post by magicalhippo »

godelski wrote: post #45
sparklingmango wrote: post #3

When I was in undergrad, I declared a math minor quite late - just before my junior year. This led to me taking up to 9 hours of math credits in a semester. The reason I declared my math minor was because I was interested in going for a PhD in economics and my econ professors all recommended that I try to take as many math classes as possible to prepare. Reader, I was not a 'numbers person' at all. However, in all of…

I find this concept of being a numbers person weird to being good at math. Most math is about symbol manipulation and numbers are really only at the end.

> Most math is about symbol manipulation and numbers are really only at the end.

During the first semester at uni, the math prof was going through some non-trivial (at that stage) "real-world" example involving definite integrals.

After what felt like 10 minutes of manipulating integrals and such we got to the final "x = ..." equation, upon which he turned away from the blackboard and said "and then you can just plug in the numbers and calculate the result, but that's not really interesting". And so the lecture continued...

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