
Common Errors in Undergraduate Mathematics (2009) - vinchuco
https://math.vanderbilt.edu/schectex/commerrs/
======
tgb
These are primarily for algebra mistakes. Teaching a proof based course, the
most frustrating error I saw was people doing a fine proof but in reverse
(this is related to the article's point about irreversible steps). If I wanted
them to prove that the square of any number was nonnegative, they'd start by
assuming that x^2 >= 0 and then driving equations from that until they got
something obviously true like 0>=0. This was the proof written "upside down"
if every step were reversible. They should have just taken the last step and
started from there until they derived their first step.

So how was I supposed to grade that? It showed a significant misunderstanding
but also often literally had every step necessary for a correct proof. Usually
I just wrote overly long notes about how they should write proofs in the
future that I'm sure the students never read.

The other fun one was unnecessary proof by contradiction. They'd start by
assuming not P. Then they'd prove P directly, not using that assumption. But
they, that contradicts the assumption of not P. Hence not P is false, so then
they conclude P. Nothing was wrong there, they just had a couple extraneous
sentences that overcomplicated things.

~~~
jancsika
> So how was I supposed to grade that?

Short digression: It's been interesting to read the criticisms of Uber on HN.
Every single one has been accompanied by harsh criticisms of the taxi
medallion system. Now I'm quite certain that a tiny proportion of HN have any
working knowledge of that system (where working knowledge would be something
like your reading, teaching, and genre literacy in proofs), but the general
uninformed opinion seems to be that Uber's disruption of that industry is
welcome and has no significant drawbacks to speak of.

With that in mind, what do you think is going to happen when someone in
Silicon Valley disrupts higher education accreditation and/or secondary
degrees? Don't get me wrong-- I think there is a lot of value in university
education and university life. But if people who have never even heard of taxi
medallions are quick to write off the costs of it being replaced by an
unregulated contractor, they are going to be even more willing to say goodbye
to a system whose instructors (who themselves are mostly independent
contractors) cannot or will not accurately assess the correctness of a
student's work?

I don't mean to single you out here-- I've certainly done the similar. The
point is _everyone_ on HN has their own stories of experiencing this problem--
where the intellectual environment advertised by the university was violated.
Coupled with the absurd debt students today carry (at least in the U.S.), that
is more than enough ammunition for a disruptive company to put an enormous
dent in the undergraduate degree. My fear is that will strip the money needed
for grad programs-- where the _real_ learning and research happens-- and the
modern uni will fall like a house of cards.

~~~
throwawayjava
_> ...whose instructors... cannot or will not accurately assess the
correctness of a student's work?_

Have you graded proofs? GP's approach is entirely sensible.

Proofs are written for humans. Writing a proof upside down is _wrong_ , even
if the derivation is correct.

The underlying cause of that incorrectness could be a simple matter of poor
communication, or the student might have a more fundamental misconception
about how implication works (and/or quantifiers). Sometimes the instructor
knows enough about the student to determine which is the case. Sometimes it's
impossible without talking to the student. Hence the long note and (perhaps
implicit) invitation to office hours.

I can't find anything in GP's post that indicates he "cannot or will not
accurately assess the correctness of a student's work", aside from the obvious
impossibility of projecting a multi-objective assessment of a fundamentally
human task onto an integer in [0,100].

If anything, GP provides an excellent example of why auto-grading for upper
division courses -- especially courses that are trying to teach students how
to write for a human audience -- will probably require fundamental scientific
advances in AI in addition to innovation on top of existing tech stacks. My
money's on self-driving cars happening first.

 _> With that in mind, what do you think is going to happen when someone in
Silicon Valley disrupts higher education accreditation and/or secondary
degrees?_

 _This already happened!_ University of Phoenix has been around for a while.

 _> The point is everyone on HN has their own stories of experiencing this
problem-- where the intellectual environment advertised by the university was
violated_

If you want to know how people really feel, look at where they send their kids
rather than they say/say they will do in online debates. Again, University of
Phoenix et al have been around for a while.

 _> My fear is that will strip the money needed for grad programs-- where the
real learning and research happens-- and the modern uni will fall like a house
of cards._

Masters programs are cash cows.

Top Ph.D. programs usually aren't appreciably funded through undergraduate
programs. And when they are, it's almost always backed by Ph.D. student labor.
Which is crazy cheap, and which startups are going to have a _very_ hard time
competing with.

 _> Short digression: It's been interesting to read the criticisms of Uber on
HN._

The same is true of higher education :-)

~~~
jancsika
> Have you graded proofs? GP's approach is entirely sensible.

GP wrote characterized the mistake as a "significant misunderstanding" of the
material. GP use of the word "just" strongly implied that the assessment
inside the long, probably unread note was not reflect in the grade. If that is
indeed the case then the student's "significant misunderstanding" of this
material wasn't reflected in the calculation of their final grade, which is
the textbook definition of grade inflation.

Again-- this kind of grade inflation happens often. (If the class size is
greater than 75 and fulfills one of the uni's writing requirements, I'd argue
it's almost a requirement that it happens.) It's a truth about an impossible
situation with only bad choices, not an ad hominem.

> This already happened! University of Phoenix has been around for a while.

By disruption, I mean Uber/AirBnB style disruption. Disruption resulting in
customers who leverage the technology saying things like, "Dude, it's changed
the way I go out in Atlanta." I seriously doubt the existence of University of
Phoenix has ever made an employer say anything close to that. (E.g., "it's
changed the way we hire employees.")

I mean an accreditation service that basically tells an employer: look, do you
_really_ want to know if that employee is worth hiring?

I doubt this even requires any testing whatsoever-- just require incoming
Freshman to get a Chromebook. But the end of their studies there could be a
new Google "Fasttrack" app. You could make it almost like a digital scratch
ticket-- giving Google the relevant permissions and you can see whether you
_really_ graduated with honors.

Edit: added example

~~~
throwawayjava
_> If that is indeed the case then the student's "significant
misunderstanding" of this material wasn't reflected in the calculation of
their final grade, which is the textbook definition of grade inflation._

I don't know; it depends. I typically write my rubrics so that there are
"style" points. In a really easy proof, style might be up to half the
available points. In a very difficult proof perhaps only 1/10 or less.

But even that doesn't totally solve the problem. Some mistakes are on the
border between style and substance. Grading and assessment aren't a perfect
science.

 _> I seriously doubt the existence of University of Phoenix has ever made an
employer say anything close to that._

Call me old-fashioned, but IMO the actual education is still the hard part of
the Education Industry.

My comments have all assumed the goal is student understanding. IMO the
obsession over grades (by students and by external observers), testing, and
accreditation is completely mis-guided. Focus on exceptional student outcomes
and the rest is easy.

------
ezequiel-garzon
I think it ultimately comes down to the level of primary school teachers, and
good luck reversing that. I taught at a community college some classes for
future elementary school teachers. When I found things like 1/2+1/3=2/5 I
tried my best to get everybody's attention as to how important fixing this
misconception was, especially given they'd be teaching children in a few
years... Unfortunately, I mainly received questions about how much of an
impact an error like that would have on their grades. Major sigh...

~~~
throwawayjava
_> Unfortunately, I mainly received questions about how much of an impact an
error like that would have on their grades. Major sigh..._

"You will fail the entire assignment on which the error was made."

If only... :)

------
okket
Previous discussion:
[https://news.ycombinator.com/item?id=8181101](https://news.ycombinator.com/item?id=8181101)
(3 years ago, 110 comments)

------
Kenji
_For instance, contrary to the belief of many students

sin(x + y) is NOT sin(x) + sin(y)

(x+y)^2 is NOT (x^2) + (y^2)

sqrt(x+y) is NOT sqrt(x) + sqrt(y)

1/(x+y) is NOT (1/x) + (1/y)_

Wow, seriously? Especially the one with the fraction is painful. I'm not sure
if university is the right place for people who still make mistakes like that
at that age (if they had more than a decade of public school behind them).
Certainly not STEM fields.

~~~
cmrx64
I have a story that is too long here about my struggles with math in high
school, and while I'm maybe not the model CS student for a variety of related
reasons, I think that I turned out mostly fine...

None of the symbolic manipulation really made sense to me until I started
learning programming (and, much later, mathematical foundations) and how to
_define_ these operations. The answer to "Why isn't sqrt(x+y) = sqrt(x) +
sqrt(y)" shouldn't be "It's not in the table of 'laws' we're allowed to use".
Sure, you can (easily) provide examples where that purported identity is
false, but trying to understand why was really hard for me, and I wouldn't
accept it until I understood why.

/ramble

~~~
dmurray
> The answer to "Why isn't sqrt(x+y) = sqrt(x) + sqrt(y)" shouldn't be "It's
> not in the table of 'laws' we're allowed to use".

This is the most rigorous and correct answer I can imagine! It's not true
because we have not proven it to be true or learnt it from someone who has.

~~~
posterboy
By the same line of reasoning it would be not false until we have proven it
false.

~~~
dmurray
Yes, we should consider it neither true nor false, and not use the rule (or
its negation) to prove whatever it is we actually want to prove.

------
madez
The author claims

> And for some purposes, an ellipsis is not just a convenience, it's a
> necessity. For instance, "1, 2, 3, ..., n" represents all the integers from
> 1 to n, where n is some unspecified positive integer; there's no way to
> write that without an ellipsis.

This claim is wrong. Let n be a positive integer. If one wants to denote the
set of integers between 1 and n, then one can do so by just formally writing
what one wants: {k in Integers | 1 <= k <= n}. If one wants the (ordered) list
of integers between 1 and n, then one can write (k)_{k=1}^n.

~~~
posterboy
> For some purpose

e.g. without going into set theoretic fundamentals to build up ordered sets.

~~~
madez
That is not necessary, at all. One can always just literally write "the list
of ascendingly ordered integers that are greater or equal to 1 and smaller or
equal to n". The introduction of formal notation is just a shortcut.

------
tankenmate
One point that wasn't noted is that division is anticommutative of the
multiplicative variety. It always intrigued me as to why commutativity is seen
as being applicable to all operators and functions that provide the necessary
property, but seemingly only applies to the prefix arithmetic inverse for the
anticommutative variety.

------
bluenose69
It seems unlikely that students who need this advice will be willing to read
all the words in this essay. But, so long as they continue to pay tuition...

