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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)

#111

Post by konjin »

ubercow13 wrote: post #73
konjin wrote: post #10

Mathematical notation is terrible. I don't mean this in the usual "we need to invent new symbols to make it clearer" way. I mean it in the way that it has implicit typing that gets coerced constantly. Using Haskell types D :: (R -> R) -> (R -> R). Yet it gets used on things like D 2 = 0 which implies D :: (R -> R). What you've actually done is an implicit conversion of 2 :: R to 2(x) = 2 :: R -> R and 0 is not 0::R i…

I honestly never found this an issue. Shorthand and type coercion is used a lot but at least when taught, it's usually made very explicit what notation means and when things are being excluded. After that point, the notation is the least challenging part.

It is not an issue until you start trying to teach any sort of fix point theorem, at which point the wheels fall off because the notion can't deal with higher order functions at all.

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

#112

Post by BrandoElFollito »

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 agree with this as the father of two kids in high school.

I routinely force them to fully understand some things they learn, and I am more liberal with others.

Mathematics is one of these subjects where there is almost nothing you can afford not to understand. It will absolutely bite you someday.

When learning math there are things that are really new (such as differentials, or operations on fractions) that are conceptually different from other things you just learn (Pythagoras theorem for instance) and I spend a lot of time with them to help them understand the "why" of these operations. Otherwise i know they will have a very, very hard time to follow up.

Physics is lighter - you can have trouble understanding thermodynamics and it will not mean much when doing mechanics. There are parts of physics I never really understood despite having a PhD in physics.

This is in sharp contrast with, say, history where missing a bit does not impact the next things one learns (at least at high school level - and this is from someone who loves history).

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

#113

Post by konjin »

bunje wrote: post #20
konjin wrote: post #10

Mathematical notation is terrible. I don't mean this in the usual "we need to invent new symbols to make it clearer" way. I mean it in the way that it has implicit typing that gets coerced constantly. Using Haskell types D :: (R -> R) -> (R -> R). Yet it gets used on things like D 2 = 0 which implies D :: (R -> R). What you've actually done is an implicit conversion of 2 :: R to 2(x) = 2 :: R -> R and 0 is not 0::R i…

So you say we should specify when 2 means number two and when it means the constant function? How does it help?

Derivatives then are a higher order function that follow it's definition in all cases. You can't exactly apply it's definition as a limit in cases where it's applied to non-fictions.

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

#114

Post by BrandoElFollito »

> 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 it is calculated, how one can get more precise by shortening time and that, eventually, we get to the exact momentary speed.

My son started to ask all kind of question such as how to "get closer" on my wavely drawing, to which I told him "good question - this is possible only when we know the function d(t)", etc.

It is only when he understood the general reason for differentials to exist that we went back to the conditions (continuity, planes, etc.). He actually deduced the continuity constraint himself because he understood the "why".

My math told told us once "I will show you a neat trick that you will not understand this year, but when you understand it next year it will be way less useful to you. Just know that it works only when this and that".

So lack of perfection is sometimes useful for people to understand something at all and not wander off after the first two introductory sentences.

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

#115

Post by aspaceman »

BrandoElFollito wrote: post #114

> 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.

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

#116

Post by aspaceman »

dkarl wrote: post #101

But the part that I really want to discuss here is the other part -- i.e., the phrase "if k is any constant." To most teachers, that additional phrase doesn't seem important, because in the teacher's mind "x" usually means a variable and "k" usually means a constant. Most teachers get this right, using these conventions as a redundant booster to their verbal and written communication. Some are grumpy and lazy and wan…

I completely agree. A good mathematician says things like

"this variable x and this constant k interact with our Transfer function T" etc.

It's a lot of words but it turns into a sort of drill for the students about the notational norms.

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

#117

Post by hnracer »

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 did well in my college math exams (top 5 percent of class) by just memorizing proofs and memorizing the methods to solve common problems without understanding much of anything about what I was doing. The exams were very much hackable.

I wouldn't be able to pull this off in grad level maths but undergrad it's feasible.

It was a total waste of time though and I regret that approach, but I did what I was given an incentive to do (least effort for max reward)

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

#118

Post by carapace »

konjin wrote: post #10

Mathematical notation is terrible. I don't mean this in the usual "we need to invent new symbols to make it clearer" way. I mean it in the way that it has implicit typing that gets coerced constantly. Using Haskell types D :: (R -> R) -> (R -> R). Yet it gets used on things like D 2 = 0 which implies D :: (R -> R). What you've actually done is an implicit conversion of 2 :: R to 2(x) = 2 :: R -> R and 0 is not 0::R i…

Are you talking about this?

https://en.wikipedia.org/wiki/Q0_(mathematical_logic)

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

#120

Post by ubercow13 »

konjin wrote: post #111
ubercow13 wrote: post #73

Earlier quoted context omitted.

I honestly never found this an issue. Shorthand and type coercion is used a lot but at least when taught, it's usually made very explicit what notation means and when things are being excluded. After that point, the notation is the least challenging part.

It is not an issue until you start trying to teach any sort of fix point theorem, at which point the wheels fall off because the notion can't deal with higher order functions at all.

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.
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