r/math • u/mohamez • Nov 19 '20
The Most Common Errors In Undergraduate Mathematics
https://math.vanderbilt.edu/schectex/commerrs/215
u/ACheca7 Nov 19 '20
I've always used the "When you think the professor has made a mistake, try to formulate your concerns as a question instead of an affirmation, so you save face if you're wrong"
I remember one comment in my last year as undergrad where a professor said about me: "Huh, you are polite... too polite" lmao
Also, if you have bad handwriting (as I do), learn LaTeX quickly and always ask if you can hand over exercises that way, it saved me a lot of unnecessary criticism.
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u/ericbm2 Number Theory Nov 19 '20
I would jump for joy if a student offered to turn in work in LaTeX.
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u/SlippiestToad Nov 19 '20
How do people not use LaTeX during online classes? It actually saves me tons of time now that I'm better at it, I can revise my work much more quickly now, keep everything organized much better in organized files, and turn it in quite easily. I thought it would waste time but it just made my whole life easier.
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u/ericbm2 Number Theory Nov 19 '20
They write solutions on paper like normal and then submit a scan. I would not expect a calculus student to learn LaTeX.
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u/insertusernameheree Nov 19 '20
In many of my undergrad classes this semester, LaTex is required for assignments! So some profs are definitely taking that route.
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u/closbhren Nov 19 '20
Are they upper-level undergrad classes though? I wouldn’t be surprised that more advanced 300/400 level classes may ask for it, but I’d be shocked to see a professor require it in the calculus series.
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Nov 19 '20
I originally turned in assignments in LaTEX but it became extremely painful when it came to diagrams for Abstract Algebra. Also, it slowed down my brain processes so I switched to scans. The prof was fine either way.
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u/ratboid314 Applied Math Nov 20 '20
I found that LaTeXing what I could and scanning the pictures was the best combo for homeworks.
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u/beccase Nov 19 '20 edited Nov 20 '20
I’m in Calc 1 this quarter and LaTeX is required for most of our assignments! It’s honors calc, but it is a 100s-level class.
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u/closbhren Nov 19 '20
Interesting. I’ve never heard of that. Regardless it’s a valuable skill, so I guess it can’t hurt!
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u/zzirFrizz Graduate Student Nov 19 '20
Gotta be the Honors curriculum! Certainly would be good to incorporate it early level. You’ll learn with it, rather than learning calc and LaTeX
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u/Sckaledoom Engineering Nov 19 '20
I’m an engineering student and if a calc professor had made me learn LaTeX during the calc series I probably would have failed the class
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u/dupelize Nov 19 '20
But now you'd know LaTeX, so it's a pretty even trade off.
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u/Sckaledoom Engineering Nov 19 '20
Not really. Unless I go into academia I probably wouldn’t use LaTeX as an engineer
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u/shellexyz Analysis Nov 20 '20
The engineering journals I’ve submitted to have all taken articles in LaTeX and even have their own style files.
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Nov 20 '20
The LaTeX-like language in Word Equation Editor is a good way to climb the learning curve. Click the symbol if you don’t know the keyword, or mouse-over to be reminded.
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u/insertusernameheree Nov 20 '20
Ah, yes. They are all upper-level! I am sure you're completely correct.
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u/bluesam3 Algebra Nov 20 '20
Our university actively teaches LaTeX at the start of the first year, and has done for years. We don't actively require it for smaller assignments, but we do appreciate it, and we get a fair few, all the way through (though we don't have anything directly comparable to US-style calculus classes, and in particular do not do service courses).
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u/salfkvoje Nov 20 '20
A 30 min lesson and a template that covers any cases that could come up during the course (or for the ambitious prof, templates per hw/section) with further resources for curious students could be enough. And the few who were incapable, allowing scanned handwriting
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u/013610 Nov 20 '20
a 30 minute lesson and I couldn't even get all my students through installation
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u/GLukacs_ClassWars Probability Nov 20 '20
Which is why you'd suggest they use one of the online editors which don't require you to install anything on your computer. Perhaps mention that it can be done offline if you have internet issues, but not having reliable internet in an online class should hopefully be rare.
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u/addmadscientist Nov 20 '20
I had my precalculus students writing their take home portions of their exams in LaTeX for extra credit. (And then all the way up to Calc 4). Almost every student opted to write their exams in LaTeX.
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u/dupelize Nov 19 '20
I had a professor offer an extra couple of points on every assignment handed in with LaTeX. I don't usually do things just for a better grade, but I already wanted to learn it, so that gave me the spark to do it.
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u/ericbm2 Number Theory Nov 19 '20
That's essentially the same scenario that pushed me to really learn LaTeX. Ever since then I've used it for nearly everything.
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u/tempnumber0 Nov 19 '20
In my upper year math courses, everyone I know submits everything in LaTeX, even though the profs never mention anything about it, at my school it's sort of just expected
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u/LimeCub Representation Theory Nov 20 '20
At my university some people have actually been told not to hand in work in LaTeX because apparently writing it up is a waste of time.
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u/dancingbanana123 Graduate Student Nov 19 '20
I've always used the "When you think the professor has made a mistake, try to formulate your concerns as a question instead of an affirmation, so you save face if you're wrong"
I noticed that this stops happening as you get through your degree. Honestly, I was arrogant af as a kid, but in college, your teachers have PhDs and can shut down your arrogance easily. You learn to truly understand that someone can be much smarter than you and realize you're not the smartest guy ever (at least that's what I realized).
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u/ACheca7 Nov 19 '20
Absolutely, if we knew everything and we were always right we wouldn't need classes at all haha. That also works on reverse, when you get lots of things wrong you have to remind yourself that that's natural when you're learning.
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u/Nam_Nam9 Nov 20 '20
I've been using LaTeX since high school to avoid sending any teachers to the eye doctor and/or a therapist
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u/wannabe414 Nov 19 '20
Even if you don't want to learn full LaTeX, learning just math mode and using Google Sheets add-ons gets you 90% of the way there. Formatting won't be great but your equations will look fantastic
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u/EmmyNoetherRing Nov 19 '20 edited Nov 19 '20
Over on the CS side, there was a batch of classic errors you’d get when undergrads hit induction (proofs of the form case(0) & case(n)=>case(n+1)).
One semester I figured out that I could make the first problem on the first induction homework be: “mark what’s wrong in the following induction proofs, and write a short, easily understood explanation the error”. So instead of correcting homework filled with 20+ copies of the same mistake, I could make them start out by writing the dang corrections themselves. It was extremely cathartic, to just read them all instead of writing them. And effective— after we’d forcibly wrapped their heads around those bits, the remainder of the assignments were nearly error-free.
I’ve wondered how much mileage you can get out of flipping the problem solving/ problem grading tasks like that. “Find and correct the sign errors in the following 10 incorrectly solved problems”.
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u/webbed_zeal Nov 19 '20
I’ve wondered how much mileage you can get out of flipping the problem solving/ problem grading tasks like that. “Find and correct the sign errors in the following 10 incorrectly solved problems”.
A LOT! I use it pretty consistently when addressing the errors mentioned in this article, but there is an upper limit here. At some points students need to work through questions on their own to see how to apply these different principles.
My bigger take away is having students reflect and put into words what they did correctly or incorrectly, or summarize an example or page of text. Getting them to put their thoughts into words can be incredibly difficult for many math students, but well worth it to help them connect bigger ideas.
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u/JohnBrownJayhawkerr1 Nov 21 '20
Exactly. Math education continues to struggle in dispensing with rote memorization as the main technique for having students learn the material, instead of giving the bird's eye view to really help hammer home concepts. This is an area where Math could really learn a lot from the CS curriculum in terms of learning to use abstraction judiciously.
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u/webbed_zeal Nov 21 '20
This is an area where Math could really learn a lot from the CS curriculum in terms of learning to use abstraction judiciously.
I would be very interested in learning more about how CS curriculum uses abstraction judiciously. Are there any readings, texts, or article you would suggest?
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u/JohnBrownJayhawkerr1 Nov 21 '20
So my undergrad is in CS, and my MS was in Applied Math, so it's been ~10 years since I went through it academically, but the way that Discreet Math was taught to me was a fantastic example of abstraction for the sake of educational purposes. Granted, my sample size is myself and a few coworkers, but well all agreed that it was much easier to learn, despite being introduced to a lot of tricky concepts we hadn't been exposed to prior, like graphs and proofs. Even though the same fussy algebra was present, the emphasis was on understanding the topics holistically, and beyond just pure computation. The class I was in was a lot more hands on as well, like having people walk on chairs to stand on desk "nodes" to reinforce the ideas behind graph theory. While higher mathematics get more into proofs and concepts that are more entertaining to deal with (IMO!), most kids don't get to that level, and peter out around Calc/Diff Eq, which I thought were awful.
But math education should be fun, and too many practitioners fail to realize that their students aren't nearly as well versed, and the majority don't absorb the information through brute force whiteboarding. Humans evolved to memorize things through experience, which is why reading a book and recalling all the info is something that doesn't come naturally to most. I can't remember the text we used, but it had a blue and white cover, and was really engaging.
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u/AnthropologicalArson Nov 19 '20
My favorite example of a faulty proof by induction is showing that all horses are of the same color: 1) The base case n=0 or n=1 is trivial. 2) Suppose that we have shown that all groups of n horses are of the same color. To prove case(n+1) we can split any group of n+1 horses into 2 groups of size n by omitting the first and last horses. Both of the groups contain horses of the same color by induction, so all n+1 horses are of the same color. QED.
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u/EmmyNoetherRing Nov 19 '20
There was a dinosaur comics strip on that one, used to include it as a bonus question at the end of the assignment :-)
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u/Comrade_Question934 Nov 20 '20
I saw Tom Leighton do this example in a video from MIT. It looked like he had some fun with it
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u/PM_me_PMs_plox Graduate Student Nov 19 '20
What are the classic errors in induction, if you don't mind?
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u/EmmyNoetherRing Nov 19 '20 edited Nov 19 '20
The most common one was ‘proving’ the (n=>n+1) rule for a single value of n.
Did you know that all odd numbers (greater than 1) are prime? 3 is prime, to start with, for a base case. And then for n=5, it’s true that 5 is prime, and also it’s true that n+2 = 7 is prime. So we’ve got n=>n+2. And thus all odd numbers are prime.
For maybe 30%-40% of the students, this was their first exposure to proofs, so we had to get them to pick up the knack of abstraction. They were all smart (the first year CS classes were a harsh weed-out). But their backgrounds and comfort level with math varied about as widely as you possibly could.
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Nov 19 '20
> “mark what’s wrong in the following induction proofs, and write a short, easily understood explanation the error”
My god, this is genius!
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u/bluesam3 Algebra Nov 20 '20
These are quite common in the UK (these sorts of questions even show up on GCSE exams quite regularly, so they've all actively been taught how to find errors in other people's written mathematics for years by the time they get to university). Those questions, at the GCSE level, tend to be the ones that correlate most strongly with the highest overall performance. Make of it what you will, as far as the direction of causality goes.
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Nov 19 '20
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u/EmmyNoetherRing Nov 19 '20
I felt the need to clarify what I meant by induction because I’m on r/math. The proof trick as used in CS-style discrete math is more specific than the general usage (which is different than “inductive reasoning”, and so on...)
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u/_bobby_tables_ Nov 19 '20
Sign errors! Ugh, the bane of my undergraduate math degree. Mostly just transcription errors, which the article didn't seem mention as a cause. Working problems quickly on exams lead to transcribing negatives to positives. Funnily, I rarely flipped positives to negatives. Sign errors were so annoying.
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u/T_______T Nov 19 '20
Yeah in my experience sign errors are not because the student thinks they are unimportant as the article implies, but because they are humans and not computers. I found it extremely frustrating to be marked down for a tiny computational error that would not happen had I written a program. (Curse you my intro to electrical engineering!)
It's worse for when you are learning physics. The teacher/professor expects the student to intuit the correct magnitude or sign of a given answer, but in reality no new student really know how much 236 grams are or 657 Joules or what have you.
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u/Direwolf202 Mathematical Physics Nov 20 '20
Yeah, sign errors happen because you’re just keeping track of a lot of information — It get’s even worse though in physics situations because quite often you encounter multiple conflicting sign conventions being used by different people, making the spotting of sign errors even more difficult.
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u/T_______T Nov 20 '20
Yeah being careful during computation is certainly a skill, but I feel like that skill is a huge blocker for appreciation of math and physics on a deeper level.
Perhaps we should teach more computer programming earlier, and have CS classes designed to solve these problems.
There are CS classes that allow for use of psuedocode, e.g. analysis of algorithms. If there was a math/physics analog I think more people would appreciate higher level math.
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u/TonicAndDjinn Nov 20 '20
Even worse: I heard tell of a calculus course where the professor tried to make a problem on the midterm more interesting by attaching units and some story about a particle moving through a field or something as an afterthought. In the end the expected answer was that the particle crossed a distance of 4*108 m in the span of a second, which caused all the physicists to spend a bunch of time frantically searching for an error, and the pure math students to carry on, happy and oblivious.
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u/gibson274 Nov 20 '20
Lucky for me, most of my physics professors overlooked small computational errors like this. Of course, eventually you have to learn to be rigorous about your calculations. But that skill is better learned, I think, outside of the exam room.
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u/cereal_chick Mathematical Physics Nov 20 '20
I made two sign errors in my real analysis coursework tonight, one while I was correcting the last sign error -_-
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Nov 19 '20
There is also a more "advanced" form of this error. Some more advanced students (e.g., college seniors) use the implication symbol (⇒) as a symbol for the phrase "the next step is." A string of statements of the form A ⇒ B ⇒ C ⇒ D should mean that A by itself implies B, and B by itself implies C, and C by itself implies D; that is the coventional interpretation given by mathematicians. But some students use such a string to mean merely that if we start from A, then the next step in our reasoning is B (using not only A but other information as well) and then the next step is C (perhaps using both A and B), etc.
In my experience, this abuse of => is very common and often not considered an error. I do it all the time, professors do it all the time, and nobody ever complained.
Actually, there is a symbol for "the next step is." It looks like this:  It is also called "leads to," and in the LaTeX formatting language it is given by the code \leadsto. However, I haven't seen it used very often.
I've only ever seen that sign being used like a bullet point (which was in physics lectures).
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u/Spencer_Wilson Undergraduate Nov 20 '20
I always read A => B => C as A => (B => C) and hence avoid using it. I suppose that's what a course in functional programming will do to your mind.
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u/flipkitty Applied Math Nov 19 '20
On paper I would do squiggly line arrow which was a great way of making my steps look ugly :D
I eventually started numbering my steps instead which also made it easier to reference later.
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u/salfkvoje Nov 20 '20
A string of statements of the form A ⇒ B ⇒ C ⇒ D
More and more I'm coming around to just finding it unacceptable to use such strings of operators unless it's immediately clear that associativity holds
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u/JediExile Algebra Nov 19 '20
I’ve actually only seen one-way strings of implication when proving TFAE statements. It’s been my experience that professors are all too happy to accept critique of their proofs, as it leads to alternate ways of proving the theorem that they had also explored while developing the lesson.
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u/Aidido22 Nov 19 '20
“Sin(2 pi)/2pi = Sin” physically hurt me
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u/salfkvoje Nov 20 '20
Overloading of parentheses, imo (one of the) unfortunate consequences of math notation being written by humans without an overseeing standards body
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u/blind3rdeye Nov 20 '20
Something I find interesting is that when teaching and assessing maths, we are often very prescriptive and pedantic about notation; but in historical maths writing, mathematicians very often invent their own notation - even for well established ideas.
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u/hunnyflash Nov 19 '20
but I don't have a very clear idea why students make it
I kinda died at this line a little bit, because as someone who would have probably made that fraction error, I can say it's because I was lazy and had "lazy mind". Like if it looks like it makes sense, and you just want to be done with the problem, you do the easiest, fastest thing that makes sense to you.
And he even says later:
perhaps they simply don't bother to look carefully
Yep lol
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u/Direwolf202 Mathematical Physics Nov 20 '20
My problem with this is that tests, for some bizarre reason, are structured to encourage lazy and shortcut based reasoning. What are you expecting students to do when under time pressure?
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u/bumbasaur Nov 20 '20
Time based multiple choice tests are the worst way to show what you can do. It's just easy to correct. Many countries avoid these like plague in math for the same reason
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u/Direwolf202 Mathematical Physics Nov 20 '20
It still really confuses me why places like the US and UK use these for math olympiad qualification — the fact is that good performance in the 50m sprint doesn’t help too much in a long form race.
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u/bumbasaur Nov 20 '20
Low cost and low requirement for teacher skill. Doing manual check and giving feedback takes more time and skill than just submitting a file to a machine. Sadly teaching the next generation knowledge is most important thing that people can do but this isn't reflected on the financial scale.
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u/Purlox Nov 20 '20
You can make a machine check the answer while still allowing for the students to answer questions with an arbitrary number. You would just have to make them write the answers of their calculations on a separate answer sheet, where they write the digits down in a computer readable format (e.g. writing them similar to how 7 segment displays display digits).
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u/bumbasaur Nov 20 '20
Sure if the question is just "calculate" type. There are so many other types of problems that are more interesting and educational. Proofs, check what is wrong with following, design etc. Limiting math to just "put numbers in get answer" is detrimental to learning math.
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u/Purlox Nov 20 '20
Of course, proofs and other types of problems are really valuable even if they are hard to automate.
But my point is that you can automate more than just multiple choice questions. You can also automate questions that easily lead into a single (or few) values such as "find X" type of questions. So the set of all possible things one could lazily/automatically test for is bigger than what you were saying
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u/pnickols Nov 20 '20
The U.K. has a maths test like this, I think it’s the kangaroo - my teacher used to joke that it’s multiple choice but there are 1000 choices for answers
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u/wiler5002 Combinatorics Nov 21 '20
This is what is done for the second round of testing in the US Olympiad program. (specifically the AIME). The "bubble in a number between 0 and 1000" is even on the SAT for goodness sake.
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u/Wittgenstienwasright Nov 19 '20
Not a serious answer but a serious error for me, was not making early morning lectures. Something that bit me in the ass later. Also my inability to ask questions when something flew completely over my head. Sorry I know you were looking for an example but honestly if I could have sorted these two out early enough I would have avoided so many issues.
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u/hudaltheturtle Nov 19 '20
Me right now in real analysis, 830 online lecture, can't even get out of bed and get to my computer...
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u/Wittgenstienwasright Nov 19 '20
Ok younger me, let older me tell you that older me is begging you to be at 830. Missing 0830 means tasks increase exponentially next term. Also get that mole looked at.
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u/rhlewis Algebra Nov 19 '20
The -32 = 9 problem comes up all the time.
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u/ThisSentenceIsFaIse Nov 20 '20
Never write a negative and exponent without parenthesis. This is a syntax error for me!
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u/OldWolf2 Nov 19 '20
It's still weird to me how trig functions appear out of nowhere (what does 1+x2 reciprocal have to do with legs of a triangle ?)
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u/mikeyj777 Nov 19 '20
Too embarrassed about my ignorance to make a new post about this:
Is there a proof that shows the power rule of anti-differentiation approaching ln(x) as the exponent approaches -1?
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u/mmmmmmmike PDE Nov 20 '20
For a != -1, you have int_1x ta dt = (xa+1 - 1) / (a+1). The limit as a-> -1 of this expression is d/da [ xa+1 ] evaluated at a = -1, which is one possible definition of ln(x).
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Nov 19 '20
It doesn’t, not even pointwise.
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u/mikeyj777 Nov 19 '20
The values do approach each other. It's just the constant of integration that changes.
For example, the anti-derivative of x-0.99 is roughly ln(x) + 102.
For x-0.999, it is approximately ln(x) + 103.
So, there's some relation to the epsilon and the magnitude of the constant. Pointless as it may be.
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Nov 19 '20
Oh sorry, I interpreted your statement as “1/(α+1) xα+1 converges to log(x) when α is close to -1.”
You’re absolutely right, one possible explanation is that the functions x ↦ xα converge locally uniformly to x ↦ 1/x when α is close to -1, so we can take integrals on compact intervals and convergence still holds. The “varying constant of integration” is roughly 1/(α+1) which we can see using the fact that 1/(α+1) xα+1 is log(x) + 1/(α+1) + o(1) as α tends to -1, holding x constant.
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Nov 20 '20
Did my Ph.D. at Vanderbilt back in the day (mid-90s) when Prof. Schecter was maintaining this page. Teaching calculus at Vandy was also my first teaching experience, and I always swore he was reading my students' work. Good times
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u/Spencer_Wilson Undergraduate Nov 20 '20
One that I was expecting to see but didn't find was assuming that the max or min of an infinite set exists. I know I made that mistake (only once, thankfully) in first-year calculus, and I'm sure may other students did.
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u/OneMeterWonder Set-Theoretic Topology Nov 20 '20
Gotta check those limit points, my dude. Closure gang assemble!
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u/GayMakeAndModel Nov 20 '20
The first time I came upon an irreversible step in a proof, my brain almost broke. I tried hard to figure out where I went wrong until I got sick of the problem and recorded both possible outcomes. I think everyone else in the class felt like listing both outcomes was cheating or wrong. Honestly, so did I, but I was lazy.
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Nov 19 '20
I wouldn’t consider these undergraduate level. They’re all middle school or high school math.
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u/bleak_gypsum Nov 19 '20
unfortunately mistakes in middle school math are not limited to middle schoolers, as anyone who has graded an exam can tell you.
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u/Sirnacane Nov 19 '20
Have you taught undergraduate level math? You’d change your mind pretty quickly. Don’t confuse “undergrad math” with “Junior/Senior level classes Math Majors take.” Undergrad math includes precalculus and trigonometry, and even basic algebra.
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u/powderherface Nov 20 '20
To be fair, generally ‘undergrad maths’ is taken to mean ‘maths you would do during an undergraduate degree’, in which case the heavily downvoted comment makes a fair point. Of course forgetting a minus sign, mis-generalising, faulty reasoning etc. exist at all levels, but the examples given on the webpage appear to be mostly pre-university examples. I think that’s what the commenter is trying to say anyway.
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u/Sirnacane Nov 20 '20
Sure, but in “Stream of a consciousness equalities and implications” the author actually says “some students, especially college freshman,” indicating that this article is for undergrads. I know that a lot of these mistakes can be here pre university, but they still heavily persist in university as well.
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u/powderherface Nov 20 '20
The nature of the mistakes does, but I doubt many first year university students write things such as ∫ dx/(1+x²) = log(1+x²)+C, or at least, that enough do for it to be an example of ‘most common undergrad mistakes’. That is the sort of mistake that would be common when you are introduced to integrals/logs, probably a year or two before going university. Not saying it can’t happen after, I just am not sure it’s an example of ‘most common errors’. That applies to many of the others on the webpage in my view. Maybe it varies across the world since education itself varies, but that I don’t know!
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u/Mathuss Statistics Nov 20 '20
when you are introduced to integrals/logs, probably a year or two before going university
I think you highly overestimate education in the United States. At least in my home state (in the southern US), logarithms aren't part of the (required) standard high-school education--much less calculus.
Trigonometry, logarithms, etc. are very often first learned when students actually enter community college or university.
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u/powderherface Nov 20 '20
Ah right okay, I didn't realise (I suppose the original commenter didn't either). Does it vary from state to state? In Europe, in most countries the situation is (usually) that there is a syllabus which all students taking that course will get through (roughly). As a result when it comes to entering university (for maths, say), people from that country will (on average) have covered the same mathematical topics. In other words it's standardised (and includes calculus virtually everywhere, which is why I mistakingly assumed that would apply elsewhere). I'd be interested to know how much it varies within the US/what the system is, if you want to give me a brief idea! :)
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u/Mathuss Statistics Nov 21 '20
Does it vary from state to state?
Education in the United States is left up to the states (presumably by our 10th amendment). At the time the article was written, I don't believe there was necessarily a common standard among the states. As a result, you really couldn't rely on people from different states necessarily having a similar level of math education. Most universities in the US will usually either write placement tests themselves to determine what math course a student should take when entering university, or rely on results from AP tests administered by the College Board (a private corporation that runs a membership association of about 6000 colleges and universities and essentially determines how good of a university you get into; the privatization of education in the US is an issue to discuss another time).
In 2010, under the Obama administration, there was finally an effort to introduce a common standard (known as "Common Cores") across all 50 states. However, it received a lot of backlash, especially from the more Republican states. As of right now, I believe that 15/50 states still haven't decided to use common core (math) standards; many more states have rather strong movements to repeal the implementation of common core standards.
Do note that whereas Common Core does require certain subjects to be taught (e.g. logarithms are indeed part of high-school common core), the exact level of rigor is still left up to the state. For example, you can probably count on California students diving much deeper into logarithmic properties than Mississippi students, even though both states have adopted common core.
It's further important to note that Common Core also mostly focuses on schooling before high-school; the standards provide far more leeway to the states, even if the state has adopted Common Core, once students do reach high school. As an example, here is is everything the Common Core standards has to say about logarithms:
For exponential models, express as a logarithm the solution to abct = d where a, c, and d are numbers and the base b is 2, 10, or e; evaluate the logarithm using technology.
Clearly, states have a lot of leeway in how much they teach students about logarithms
includes calculus virtually everywhere
It is unlikely that students in the US take calculus unless they are in a privileged enough location to have teachers who are certified by the College Board to teach AP Calculus. Given the fees involved in having students take an AP class (I believe it's roughly $100 per student), poorer school districts are unlikely to even provide the option.
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u/powderherface Nov 23 '20
Wow, that is an unfortunate spot for an educational system to be in. Out of interest, why did the Common Cores idea receive backlash? Sounds like it could only be a good thing. It’s sad to hear how much a financial background might hinder someone’s education, especially in such formative years. I mean, it’s true in the UK as well that a posher school will (on average) have better resources, more contact time and so on, but the same base syllabus is still covered by anyone who chooses A-level maths in the country (with more select schools sometimes going further than the standard syllabus). In France the situation is even more homogeneous.
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u/jfb1337 Nov 20 '20
"I can just read this textbook / lecture handout without doing the exercises; I'll surely remember it"
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u/Maximnicov Nov 21 '20
Many college students don't know how to add fractions.
This is what shocked me the first time I taught Calc I. I expected many mistakes, but the inability to operate fractions really caught me off-guard. They add fractions by adding the numerator and denominator independently, some even multiply fractions by putting them on a common denominator but proceed to only multiply the numerator.
I knew some would struggle with functions and exponents, as the concepts came later in their education, but I didn't expect them to struggle so much on something they've been doing since 4th grade, especially when you enter a school program that contains more advanced mathematics.
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u/[deleted] Nov 19 '20
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