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The most common errors in undergraduate mathematics

math.vanderbilt.edu

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Re: The most common errors in undergraduate mathematics

#61

Post by bo1024 »

krastanov wrote: post #54
silentvoice wrote: post #48

One of the most bizarre errors I've seen undergraduates make is in induction proofs. I saw it so many times that I decided a TA must have been telling them to do this. They would always show the base case correctly. Then they assume it's true for all positive integers up to a fixed positive integer "n." So far so good, now we need to prove it's true for "n+1." This is where weirdness happens, I have seen a hundred as…

Wait, I am missing something here. If the reduction to a true statement is done through operations that can be performed in both direction (which are most of the trivial operations undergrads use) then this is a perfectly fine approach, right?

Depends on what you mean by "perfectly fine" ... if the student understands that they deduced the desired conclusion from the true fact 0=0, by a series of steps of the form "our current statement is implied by this next statement, which is implied by the next, which is implied by ... 0=0, which is known to be true", then I guess it's ok other than being an obfuscated/roundabout proof technique.

Re: The most common errors in undergraduate mathematics

#62

Post by superuser2 »

scott_s wrote: post #52
lisper wrote: post #50

Earlier quoted context omitted.

> Have you ever attempted to write Lisp on a blackboard What's a blackboard? Seriously, blackboards are pretty obsolete technology. Designing a notation to optimize for blackboards in this day and age is kind of like designing roads to accommodate horses.

Lots of people, particularly people who do lots of math, still communicate with each other over a whiteboard.

Because their notation is optimal on whiteboards and onerous on computer screens.

Re: The most common errors in undergraduate mathematics

#63

Post by lisper »

DanBC wrote: post #51
lisper wrote: post #50

Earlier quoted context omitted.

> Have you ever attempted to write Lisp on a blackboard What's a blackboard? Seriously, blackboards are pretty obsolete technology. Designing a notation to optimize for blackboards in this day and age is kind of like designing roads to accommodate horses.

> Seriously, blackboards are pretty obsolete technology. What? Blackboards are excellent. You're just working in the wrong places.

I honestly can't tell if you're being ironic or not. But if you're not: what makes blackboards so excellent?

Re: The most common errors in undergraduate mathematics

#64

Post by duaneb »

lisper wrote: post #63
DanBC wrote: post #51

Earlier quoted context omitted.

> Seriously, blackboards are pretty obsolete technology. What? Blackboards are excellent. You're just working in the wrong places.

I honestly can't tell if you're being ironic or not. But if you're not: what makes blackboards so excellent?

Blackboards don't generally crash, or have parse errors, or have encoding errors, or have usability problems. If you have chalk and a blackboard and have some semblance of an ability to write, you can use it to its fullest extent.

Re: The most common errors in undergraduate mathematics

#65

Post by nmrm2 »

brational wrote: post #43
SIGALRM wrote: post #34

I'm surprised that the author made no mention about how big O,theta,and omega is commonly misunderstood.

Likely because the author is a math professor and those concepts are typically only covered in CS department courses.

This isn't true, bounding functions are really useful for many proofs. But see nilkn's response.

Re: The most common errors in undergraduate mathematics

#66

Post by LaikaF »

kaitai wrote: post #57
lisper wrote: post #50

Earlier quoted context omitted.

> Have you ever attempted to write Lisp on a blackboard What's a blackboard? Seriously, blackboards are pretty obsolete technology. Designing a notation to optimize for blackboards in this day and age is kind of like designing roads to accommodate horses.

Mathematicians still fight college building committees and IT people to get blackboards in newly-built classrooms. (Why IT people? Dust + classroom computer system/projector/etc.)

What about white boards?

Re: The most common errors in undergraduate mathematics

#67

Post by repsilat »

Grue3 wrote: post #58
silentvoice wrote: post #48

One of the most bizarre errors I've seen undergraduates make is in induction proofs. I saw it so many times that I decided a TA must have been telling them to do this. They would always show the base case correctly. Then they assume it's true for all positive integers up to a fixed positive integer "n." So far so good, now we need to prove it's true for "n+1." This is where weirdness happens, I have seen a hundred as…

All horses are of the same color. By induction, suppose there are only n=1 horses. Obviously they are all the same color. Suppose we had proven the statement up to n. For every n+1 horses, the first n horses must be the same color. The last n horses too, must have the same color. Clearly all of them have the same color. QED.

This is lovely -- very simple, but my error-finding intuition started me off entirely in the wrong direction.

Re: The most common errors in undergraduate mathematics

#68

Post by nmrm2 »

LaikaF wrote: post #66
kaitai wrote: post #57

Earlier quoted context omitted.

Mathematicians still fight college building committees and IT people to get blackboards in newly-built classrooms. (Why IT people? Dust + classroom computer system/projector/etc.)

What about white boards?

There are about a million exucses (er, reasons).

Some of them are legit: older professors often have better handwriting at blackboard; chalk is sometimes better for certain drawings; left-hand smear; etc.

I suspect the biggest reason is that, in math circles, blackboards have a much larger cool/nostagia factor.

Re: The most common errors in undergraduate mathematics

#69

Post by taeric »

Grue3 wrote: post #58
silentvoice wrote: post #48

One of the most bizarre errors I've seen undergraduates make is in induction proofs. I saw it so many times that I decided a TA must have been telling them to do this. They would always show the base case correctly. Then they assume it's true for all positive integers up to a fixed positive integer "n." So far so good, now we need to prove it's true for "n+1." This is where weirdness happens, I have seen a hundred as…

All horses are of the same color. By induction, suppose there are only n=1 horses. Obviously they are all the same color. Suppose we had proven the statement up to n. For every n+1 horses, the first n horses must be the same color. The last n horses too, must have the same color. Clearly all of them have the same color. QED.

I admit I laughed heavily when I saw that as the first warm up in Concrete Mathematics.

And then just breathed deeply with how quickly it got beyond me. :)

Re: The most common errors in undergraduate mathematics

#70

Post by lisper »

duaneb wrote: post #64
lisper wrote: post #63

Earlier quoted context omitted.

I honestly can't tell if you're being ironic or not. But if you're not: what makes blackboards so excellent?

Blackboards don't generally crash, or have parse errors, or have encoding errors, or have usability problems. If you have chalk and a blackboard and have some semblance of an ability to write, you can use it to its fullest extent.

Computers nowadays don't generally crash either. And when you say "Blackboards don't have parse errors" what you mean is that blackboard never tell you when you've made a mistake (because they can't). That's not a feature. It's easy to program a computer not to tell you when it detects an error. But there's a reason this is not often done: detecting errors automatically is tremendously useful, especially when you're doing math.
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