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Author Topic: Random Stuff  (Read 1162606 times)

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SirPeebles

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Re: Random Stuff
« Reply #3050 on: May 04, 2014, 06:59:06 pm »
0

PSA:  I bought some lupini beans on a whim at an imported foods shop.  I soaked them overnight and started cooking them.  After tasting one or two I decided to check online to see how long they are supposed to take.  Apparently there is an extensive two week soaking process that they must go through or else they are toxic!  I looked into things a bit more, and apparently I purchased "sweet" lupini beans, which have less of the toxin.  Hopefully I don't die.  I'm just going to toss these beans and go get a veggie burger at a pub.
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Re: Random Stuff
« Reply #3051 on: May 04, 2014, 07:14:07 pm »
+3

PSA:  I bought some lupini beans on a whim at an imported foods shop.  I soaked them overnight and started cooking them.  After tasting one or two I decided to check online to see how long they are supposed to take.  Apparently there is an extensive two week soaking process that they must go through or else they are toxic!  I looked into things a bit more, and apparently I purchased "sweet" lupini beans, which have less of the toxin.  Hopefully I don't die.  I'm just going to toss these beans and go get a veggie burger at a pub.

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SirPeebles

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Re: Random Stuff
« Reply #3052 on: May 04, 2014, 07:21:06 pm »
0

I'll be sure not to drink Coors Light then.
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Re: Random Stuff
« Reply #3053 on: May 05, 2014, 02:45:07 am »
0

May the 5th be with you all today.
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Re: Random Stuff
« Reply #3054 on: May 05, 2014, 03:08:47 am »
+3

Revenge of the fifth.
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SirPeebles

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Re: Random Stuff
« Reply #3055 on: May 05, 2014, 05:37:06 am »
0

Cinco de Veganaise.
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Re: Random Stuff
« Reply #3056 on: May 05, 2014, 07:24:04 am »
+2

So, I don't really understand your problem in the original example. Could you tried to explain it in a way that, say, a high school math student who was taught to do this would understand?

Quote
x^2 - x = 4/x
2^2 - 2 = 4/2
4 - 2 = 2
2 = 2

I would point to line 2 and asked them why that is true.  Probably they will say "I just plugged in x=2".  I would ask them if 3^2 - 3 = 4/3 is also true since it is just plugging in x=3.  Hopefully they would say that it isn't, so I would ask them again why it is true for x=2.  At this point they would probably either say that they don't know, or they would say it is because 4 -2 really does equal 4/2.  After they explain that, I would tell them that that it what I wanted them to say in the first place.

Now, it is true that one can treat a statement like 3^2 - 3 = 4/3 more abstractly as an object with a truth value.  Then you can string together a chain of logical equivalences, ultimately resulting in a true statement being equivalent to the statement "x=2 is a solution".  But that is quite abstract and subtle.  Better to just tell them not to include a statement in their argument if they have not yet demonstrated that it is true.

The students' method is perfectly valid.  Mentally, they're doing the exact same work that I do when I verify solutions.  Basically, treat every equality as a symbol which is "=?"  (meaning, "these possibly be equal"...at the last line, with their checkmark, they have verified that equality holds at all lines).

This is a perfectly valid method of verifying a solution, assuming each step is reversible.  I would use it in papers.  And take serious issue with any referee who objects.

The example I gave here was just for illustration.  The problem I gave them asked them to show that a certain function is a solution to a given differential equation.
A completely equivalent question is literally on the Calc II final that I'm giving my students tomorrow.  A solution similar to the one you wrote would get full points.  I beseech you to give your students full points.  There's a point in teaching how mathematical formality should work and I mention this in many of my classes.  But I wouldn't expect a student in a general class to be assessed on mathematical formality. If your class was titled "Methods of Proof", and the formality was the main issue, then sure, mark off.  But if your class hit a wide range of topics, then the main take-away is that they understand what it means to be a solution to a differential equation, and properly substitute the terms y' and y.
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DStu

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Re: Random Stuff
« Reply #3057 on: May 05, 2014, 07:31:16 am »
0

So, I don't really understand your problem in the original example. Could you tried to explain it in a way that, say, a high school math student who was taught to do this would understand?

Quote
x^2 - x = 4/x
2^2 - 2 = 4/2
4 - 2 = 2
2 = 2

I would point to line 2 and asked them why that is true.  Probably they will say "I just plugged in x=2".  I would ask them if 3^2 - 3 = 4/3 is also true since it is just plugging in x=3.  Hopefully they would say that it isn't, so I would ask them again why it is true for x=2.  At this point they would probably either say that they don't know, or they would say it is because 4 -2 really does equal 4/2.  After they explain that, I would tell them that that it what I wanted them to say in the first place.

Now, it is true that one can treat a statement like 3^2 - 3 = 4/3 more abstractly as an object with a truth value.  Then you can string together a chain of logical equivalences, ultimately resulting in a true statement being equivalent to the statement "x=2 is a solution".  But that is quite abstract and subtle.  Better to just tell them not to include a statement in their argument if they have not yet demonstrated that it is true.

The students' method is perfectly valid.  Mentally, they're doing the exact same work that I do when I verify solutions.  Basically, treat every equality as a symbol which is "=?"  (meaning, "these possibly be equal"...at the last line, with their checkmark, they have verified that equality holds at all lines).

This is a perfectly valid method of verifying a solution, assuming each step is reversible.  I would use it in papers.  And take serious issue with any referee who objects.

The example I gave here was just for illustration.  The problem I gave them asked them to show that a certain function is a solution to a given differential equation.
A completely equivalent question is literally on the Calc II final that I'm giving my students tomorrow.  A solution similar to the one you wrote would get full points.  I beseech you to give your students full points.  There's a point in teaching how mathematical formality should work and I mention this in many of my classes.  But I wouldn't expect a student in a general class to be assessed on mathematical formality. If your class was titled "Methods of Proof", and the formality was the main issue, then sure, mark off.  But if your class hit a wide range of topics, then the main take-away is that they understand what it means to be a solution to a differential equation, and properly substitute the terms y' and y.

I think there should be a bit more explanation during the proof. You can't just put down 4 equations without any hint on how they correspond to each other, or in which permutation they have to be read to give a valid proof.
If you perlude with: "We put it the assumed solution to both sides of they equation and verify that they equal", this is ok I think, but without any text I think it should made be clear to students that they should communicate their line of reasoning, and not just write down steps without showing how they have do be connected to get to the proof.

:e And here, you can't really claim that by default, each of the lines are connected by equivalences, as obviously the first two lines are not equivalent.
« Last Edit: May 05, 2014, 07:33:48 am by DStu »
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Kirian

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Re: Random Stuff
« Reply #3058 on: May 05, 2014, 09:52:34 am »
+1

It isn't a level of formality thing though.  I'm not looking for anything particularly formal or precisely worded.  But the underlying logical structure of what they've written down is

"I want to prove P.  If P is true, then 2=2.  2=2 is true, therefore P is true."

Maybe that's not what the reasoning in their head is saying, but I'm not sure.  Maybe it is clear to you, but many students approach these things quite mechanically and are unable to articulate the reasoning when pressed.  All I can go by is what is written down.  Again, I am not looking for any fancy notation or style.  But the underlying structure of the written argument needs to be valid.  It's not a trick question or anything. Nor is about level of formality.

Honestly I'm rather taken aback by the resistance I've found here.

Hmmm.  So that you know, I'm coming at this from several different angles.  Math was never my primary concentration, so I'm coming at the math itself from the perspective of a chemist, i.e., not that of a mathematician.  Meanwhile, I'm coming at the teaching aspect from the perspective of a licensed high school teacher, and therefore someone who unsurprisingly thinks college professors are given a shockingly low amount of training in actual pedagogy.  (Note that I've taught both HS and college courses.)  That's not intended as a slight against you, or college teachers in general, just a fact of life.  I'm betting you've had no formal training in educational techniques.

Also:  what I'm about to write probably applies more to middle and end of semester than to the start of the semester, but I assume you're near the end of the semester, right?

So, looking at the problem you've presented, the method of solution the students used, and the method of solution you want, I'm going to say that this most definitely is a level of formality thing.  The solution the students present follow a logical flow for a conditional proof:

1. Assume that x = 2 is a solution
2. If assumption 1 is true, both sides of the equation will evaluate to the same number when x = 2
3. Both sides of the equation evaluate to 2 when x = 2
4. Therefore assumption 1 is true.

Obviously what they've written is a shorthand notation for this proof.  But you're expecting (as far as I see it) an unconditional proof something like:

1. x = 2 --> x^2 - x = 2
2. x = 2 --> 4/x = 2
3. 2 = 2
4. Therefore, x = 2 --> x^2 - x = 4/x (HS)

I suspect that the trouble here is one of unstated expectations not being met.  Now, if these are students whose prerequisites ought to have included two previous classes where they should have shown the more formal form, then by all means dock them the points!  They ought to know by now*, much as I would expect a junior-year chem student to be able to multiply 3 x 10 without a calculator.  But if that's not the case, then you have to set your expectations for these sorts of problems (as suggested by Polk above, an example of "no credit" and an example of "this is fine" should be sufficient for a college student).

And by the way, I'm just marking them wrong on one problem.  It's not like I'm failing them and petitioning that they be expelled from the university or anything.  Sometimes a student needs the concreteness of their answer being marked wrong before they really start questioning what they think they already know.  I can go on in lecture as much as I want about the proper way of verifying solutions, but I find that students have this issue ingrained already and just zone out during such lectures, assuring themselves that they already know how to solve equations.

Exactly right.  If you've covered in lecture the correct way of doing these things, then there's no problem.  Dock the points, make sure it's explicit why.

I think the resistance you've gotten here is that all of us were taught to solve things the way you initially showed, and we were surprised by what you wanted instead.

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Kirian

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Re: Random Stuff
« Reply #3059 on: May 05, 2014, 09:54:08 am »
0

Also:  How many college instructors do we have in this thread, anyway?  I count at least three specifically declared (me, Peebles, shraeye).  Anyone else?
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Witherweaver

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Re: Random Stuff
« Reply #3060 on: May 05, 2014, 09:57:45 am »
0

Also:  How many college instructors do we have in this thread, anyway?  I count at least three specifically declared (me, Peebles, shraeye).  Anyone else?

I taught undergrad courses while I was doing my PhD, but I went into industry instead of academia. 
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Re: Random Stuff
« Reply #3061 on: May 05, 2014, 10:05:26 am »
+1

This is not a formality thing, it's a correctness thing.  The argument

A = B
...
therefore A=B

is a valid argument, but is not sound if A does not equal B.  This:

Quote
1. Assume that x = 2 is a solution
2. If assumption 1 is true, both sides of the equation will evaluate to the same number when x = 2
3. Both sides of the equation evaluate to 2 when x = 2
4. Therefore assumption 1 is true.

absolutely does not show that x=2 is a solution.  Polk already brought up the counterexample:

1) Assume -3=3.
2) Then (-3)^2 =  (3)^2
3) 9 = 9
4) Statement (3) is true, therefore the assumption (1) is true.

I mean, it's more obvious if you skip 2. and 3. in your proof:

1. Assume that x = 2 is a solution
4. Therefore assumption 1 is true.

It was already true!  That's what step (1) did.  The argument that the students are using, even when made formal, is not correct. 
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Awaclus

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Re: Random Stuff
« Reply #3062 on: May 05, 2014, 10:48:14 am »
0

1) Assume -3=3.
2) Then (-3)^2 =  (3)^2
3) 9 = 9
4) Statement (3) is true, therefore the assumption (1) is true.
That's why you check if both sides are non-negative before squaring them.
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Witherweaver

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Re: Random Stuff
« Reply #3063 on: May 05, 2014, 10:55:10 am »
0

1) Assume -3=3.
2) Then (-3)^2 =  (3)^2
3) 9 = 9
4) Statement (3) is true, therefore the assumption (1) is true.
That's why you check if both sides are non-negative before squaring them.

You certainly don't need to.  Step (2) is valid because of step (1).  If a=b, then a^2 = b^2.

Okay that was slightly flippant.  But to the practical end, what if the "flaw" is more subtle, so it's not obvious that you're doing something invalid?  I brought up the trig identity thing before, and I'll see if I can think of an example that actually does that.  I bet we could come up with one when verifying y=y(x) solves a particular ODE (or PDE) as well. 
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DStu

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Re: Random Stuff
« Reply #3064 on: May 05, 2014, 11:12:45 am »
0

1) Assume -3=3.
2) Then (-3)^2 =  (3)^2
3) 9 = 9
4) Statement (3) is true, therefore the assumption (1) is true.
That's why you check if both sides are non-negative before squaring them.

Yeah, and why do you check it's a non-negative? Because the resasoning presented does not prove what should be proven, but only a modification of it [namely, that you should read the whole thing backwards and/or that you disprove the negation]. That the modification is possible is implicitly checked by the non-negativity.
But this is of course only true for squaring, in other situations other conditions have to be obeyed, abstractly that the operation applied are one-to-one, but in reality this is real confusing.  One should just write the thing in such a way that it can be read forward, and/or state that you disprove the negation.

edit: ftfy
Quote
Quote
1. Assume that x = 2 is not a solution
2. If assumption 1 is true, both sides of the equation will evaluate to different numbers when x = 2
3. Both sides of the equation evaluate to 2 when x = 2
4. Therefore (not assumption 1) is true.
« Last Edit: May 05, 2014, 11:19:54 am by DStu »
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Tables

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Re: Random Stuff
« Reply #3065 on: May 05, 2014, 11:21:20 am »
0

I've mostly been avoiding getting involved in this argument up to now but I'll just weigh in:

I was always taught to do things how Sir Peebles described. Or at least, very similar to it but to never ever do it the way his students did, because that's incorrect. This is what I was taught during GCSEs (13-16 education) and what was also said during A levels (16-18 education), because the way his students did it doesn't prove anything. The method we were taught was that you simply evaluate each side. So I would have written something like:

When x=2: LHS = 2^2 - 2 = 2
When x=2: RHS = 4/2 = 2
As LHS = RHS when x=2, x is a solution to the equation.
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SirPeebles

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Re: Random Stuff
« Reply #3066 on: May 05, 2014, 11:24:33 am »
+2

So, looking at the problem you've presented, the method of solution the students used, and the method of solution you want, I'm going to say that this most definitely is a level of formality thing.  The solution the students present follow a logical flow for a conditional proof:

1. Assume that x = 2 is a solution
2. If assumption 1 is true, both sides of the equation will evaluate to the same number when x = 2
3. Both sides of the equation evaluate to 2 when x = 2
4. Therefore assumption 1 is true.

This is not valid reasoning, which is precisely my objection.  The fact that a true statement follows from my assumption does not imply that the statement is true.  For instance, I could propose that the sun is a brightly glowing chariot which a god rides around the Earth.  This assumption implies that the sun will rise again tomorrow and that it will be glowing bright again.  Sure enough that's what I find when I check tomorrow.  But I have not logically determined that the sun is a chariot.

In other words, it is not sufficient to deduce a true statement as a consequence of my hunch.  Rather, I need to deduce my hunch as a consequence of some true statement.  I don't expect my students to know the jargon I used in these past two sentences, but I expect them to grasp that the above reasoning did not prove that the sun is a chariot.  I do not believe that a student needs a course in formal logic to get this.

By the way, I am giving them partial credit and lots of it.  80% of the points if they wrote down what I've been complaining about.  They certainly have demonstrated an understanding of much of what I'm testing.  But even though they've hit most of the key ideas, they did not put these ideas together into a well reasoned argument.

And this is not just a math thing.  Surely the same shows up when a student does a lab report for a chem class.  They take the data properly and do the correct calculations for analysis, but when it comes to determining the conclusions their reasoning is shaky.  Hell, maybe they reached the correct conclusions, but you can see that their reasoning is incorrect.  You don't give them an F, but you don't give them an A+ either.

Nor is this just for science.  A student could be writing a feminist analysis of a novel for a literature class, or writing a paper about the economic causes of World War 2 for a history class.  They may hit upon lots of salient points, but if they flounder when collecting these points together into a well reasoned and articulated argument then they'll get what, a B?  Certainly not an A+.  And no one would dismiss this as "literary formality" or question whether history instructor specified that reasoning had to be valid.
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AndrewisFTTW

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Re: Random Stuff
« Reply #3067 on: May 05, 2014, 11:25:26 am »
+7

This stuff is totally not random.
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Re: Random Stuff
« Reply #3068 on: May 05, 2014, 11:32:40 am »
0

So, looking at the problem you've presented, the method of solution the students used, and the method of solution you want, I'm going to say that this most definitely is a level of formality thing.  The solution the students present follow a logical flow for a conditional proof:

1. Assume that x = 2 is a solution
2. If assumption 1 is true, both sides of the equation will evaluate to the same number when x = 2
3. Both sides of the equation evaluate to 2 when x = 2
4. Therefore assumption 1 is true.

This is not valid reasoning, which is precisely my objection. 

Technically, it is valid, no?  Statements (4) follows from statement (1).
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Re: Random Stuff
« Reply #3069 on: May 05, 2014, 11:33:47 am »
+1

This stuff is totally not random.

Let Z be apositive random variables on a probability space (\Omega, \P). Show that x=2 is a solution to
Code: [Select]
x^2 - Zx = 4/x - (Z-1)x
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Re: Random Stuff
« Reply #3070 on: May 05, 2014, 11:41:16 am »
0

In other words, it is not sufficient to deduce a true statement as a consequence of my hunch.  Rather, I need to deduce my hunch as a consequence of some true statement.
Wouldn't deducing your hunch as a consequence of a some true statement be the equivalent of ignoring the fact that we already know that x = 2 and just solving the equation normally?
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Re: Random Stuff
« Reply #3071 on: May 05, 2014, 11:42:56 am »
+1

By the way, I am giving them partial credit and lots of it.  80% of the points if they wrote down what I've been complaining about.  They certainly have demonstrated an understanding of much of what I'm testing.  But even though they've hit most of the key ideas, they did not put these ideas together into a well reasoned argument.

And this is not just a math thing.  Surely the same shows up when a student does a lab report for a chem class.  They take the data properly and do the correct calculations for analysis, but when it comes to determining the conclusions their reasoning is shaky.  Hell, maybe they reached the correct conclusions, but you can see that their reasoning is incorrect.  You don't give them an F, but you don't give them an A+ either.

Nor is this just for science.  A student could be writing a feminist analysis of a novel for a literature class, or writing a paper about the economic causes of World War 2 for a history class.  They may hit upon lots of salient points, but if they flounder when collecting these points together into a well reasoned and articulated argument then they'll get what, a B?  Certainly not an A+.  And no one would dismiss this as "literary formality" or question whether history instructor specified that reasoning had to be valid.

Fully agreed with all of this!  8/10 would be entirely appropriate given what you're describing.

I think maybe we're all talking around each other?
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Re: Random Stuff
« Reply #3072 on: May 05, 2014, 11:44:29 am »
0

In other words, it is not sufficient to deduce a true statement as a consequence of my hunch.  Rather, I need to deduce my hunch as a consequence of some true statement.
Wouldn't deducing your hunch as a consequence of a some true statement be the equivalent of ignoring the fact that we already know that x = 2 and just solving the equation normally?
No, solving a third order equation is not that much fun without looking up formulas.  On the other hand, the sequence:

Code: [Select]
- If both sides evaluate to the same number, the Assumption is true
- Both sides evaluate to 2
is checked quite easily.
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Re: Random Stuff
« Reply #3073 on: May 05, 2014, 11:44:48 am »
+4

I think maybe we're all talking around each other?
Never, this is the internet after all...
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Re: Random Stuff
« Reply #3074 on: May 05, 2014, 11:51:46 am »
+3

Okay here's another example.

Problem: Show that sin(x) - sin^3(x) = sin(x) cos^2(x) and is never zero for all real x.

Argument:

(1) Suppose sin(x)-sin^3(x) = sin(x)cos^2(x) and is always nonzero.
(2) Since the left-hand side is nonzero, so is the right-hand side.  Since a product of two numbers is nonzero, neither number can be zero.  Therefore, cos^2(x) is nonzero.  Thus, we can divide by cos^2(x).
(3) Then
[sin(x)-sin^3(x)] / cos^2(x) = sin(x)
(4) Then
sin(x) (1-sin^2(x))/cos^2(x) = sin(x)
(5) Then sin(x) [cos^2(x)/cos^2(x)] = sin(x)
(6) sin(x) =sin(x)
(7) Therefore, (1) is true.

Thus, sin(x)-sin^3(x) is never zero.

Statement (1) is false.  But assuming it's true let's me do everything else and end up with a true statement.
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