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[Ozle]

Nothing that you couldn't buy cheaper off amazon for the shipping costs....

[Tables]

May the Third be with you all today.

...

Wait, am I doing it right?

[SirPeebles]

Ugh, it seems that every single semester students in my math class make this same mistake.  Let's say I ask them to verify that x=2 is a solution to the equation x^2 - x = 4/x.

They will write:

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

and then finish it off with a big, confident checkmark. 

Is this a consequence of how they were taught in high school?  Is it a bad habit from their engineering classes?  It isn't just the Americans; the students from China do it too.

[pacovf]

It's definitely ugly, but lines 2, 3 and 4 are equivalent to each other, so they are demonstrating what you are asking.

If I really wanted to do it this way for some reason, I'd say "let's assume that 2 is a solution (line 2). Equivalently, we have (line 3). Which is equivalent to 2=2, which is always right. Hence, our assumption was right."

I guess you would rather see "x^2 - x - 4/x = ... = 0" ?

[SirPeebles]

It's definitely ugly, but lines 2, 3 and 4 are equivalent to each other, so they are demonstrating what you are asking.

If I really wanted to do it this way for some reason, I'd say "let's assume that 2 is a solution (line 2). Equivalently, we have (line 3). Which is equivalent to 2=2, which is always right. Hence, our assumption was right."

I guess you would rather see "x^2 - x - 4/x = ... = 0" ?

It's not just ugly.  They are not demonstrating what is asked, unless perhaps they were to explicitly indicate lines 2, 3, and 4 are equivalent.  Unless the student indicates otherwise, the convention is that they are indicating that line 1 implies line 2, line 2 implies line 3, and so forth.  One student even put a "therefore" symbol in the last line, which is especially egregious.

What I would like to see is

x^2 - x = 2^2 - 2 = 4 - 2 = 2
4/x = 4/2 = 2
Therefore x^2 - x = 4/x when x=2.

But what they have written is most certainly not correct, and I don't know where this habit comes from.

[DStu]

It's definitely ugly, but lines 2, 3 and 4 are equivalent to each other, so they are demonstrating what you are asking.

If I really wanted to do it this way for some reason, I'd say "let's assume that 2 is a solution (line 2). Equivalently, we have (line 3). Which is equivalent to 2=2, which is always right. Hence, our assumption was right."

I guess you would rather see "x^2 - x - 4/x = ... = 0" ?

It's not just ugly.  They are not demonstrating what is asked, unless perhaps they were to explicitly indicate lines 2, 3, and 4 are equivalent.  Unless the student indicates otherwise, the convention is that they are indicating that line 1 implies line 2, line 2 implies line 3, and so forth.  One student even put a "therefore" symbol in the last line, which is especially egregious.

What I would like to see is

x^2 - x = 2^2 - 2 = 4 - 2 = 2
4/x = 4/2 = 2
Therefore x^2 - x = 4/x when x=2.

But what they have written is most certainly not correct, and I don't know where this habit comes from.

Most of what they have written is correct, eg 2 really equals 2...

[SirPeebles]

It's definitely ugly, but lines 2, 3 and 4 are equivalent to each other, so they are demonstrating what you are asking.

If I really wanted to do it this way for some reason, I'd say "let's assume that 2 is a solution (line 2). Equivalently, we have (line 3). Which is equivalent to 2=2, which is always right. Hence, our assumption was right."

I guess you would rather see "x^2 - x - 4/x = ... = 0" ?

It's not just ugly.  They are not demonstrating what is asked, unless perhaps they were to explicitly indicate lines 2, 3, and 4 are equivalent.  Unless the student indicates otherwise, the convention is that they are indicating that line 1 implies line 2, line 2 implies line 3, and so forth.  One student even put a "therefore" symbol in the last line, which is especially egregious.

What I would like to see is

x^2 - x = 2^2 - 2 = 4 - 2 = 2
4/x = 4/2 = 2
Therefore x^2 - x = 4/x when x=2.

But what they have written is most certainly not correct, and I don't know where this habit comes from.

Most of what they have written is correct, eg 2 really equals 2...

Ha, alright.  They have not successfully verifed what was requested.

[DStu]

It's definitely ugly, but lines 2, 3 and 4 are equivalent to each other, so they are demonstrating what you are asking.

If I really wanted to do it this way for some reason, I'd say "let's assume that 2 is a solution (line 2). Equivalently, we have (line 3). Which is equivalent to 2=2, which is always right. Hence, our assumption was right."

I guess you would rather see "x^2 - x - 4/x = ... = 0" ?

It's not just ugly.  They are not demonstrating what is asked, unless perhaps they were to explicitly indicate lines 2, 3, and 4 are equivalent.  Unless the student indicates otherwise, the convention is that they are indicating that line 1 implies line 2, line 2 implies line 3, and so forth.  One student even put a "therefore" symbol in the last line, which is especially egregious.

What I would like to see is

x^2 - x = 2^2 - 2 = 4 - 2 = 2
4/x = 4/2 = 2
Therefore x^2 - x = 4/x when x=2.

But what they have written is most certainly not correct, and I don't know where this habit comes from.

Seriously.  I think the habbit comes from, they do it like that in school (with equivalences), and then at university you tell them they should not use equivalences...

[Voltaire]

I work at a board game store now!

[Kirian]

It's definitely ugly, but lines 2, 3 and 4 are equivalent to each other, so they are demonstrating what you are asking.

If I really wanted to do it this way for some reason, I'd say "let's assume that 2 is a solution (line 2). Equivalently, we have (line 3). Which is equivalent to 2=2, which is always right. Hence, our assumption was right."

I guess you would rather see "x^2 - x - 4/x = ... = 0" ?

It's not just ugly.  They are not demonstrating what is asked, unless perhaps they were to explicitly indicate lines 2, 3, and 4 are equivalent.  Unless the student indicates otherwise, the convention is that they are indicating that line 1 implies line 2, line 2 implies line 3, and so forth.  One student even put a "therefore" symbol in the last line, which is especially egregious.

What I would like to see is

x^2 - x = 2^2 - 2 = 4 - 2 = 2
4/x = 4/2 = 2
Therefore x^2 - x = 4/x when x=2.

But what they have written is most certainly not correct, and I don't know where this habit comes from.

Algebra as taught to HS students (in the US) uses the format you presented first.  And all of calculus that I remember (a year of high school, plus a year of college including differential equations) also used that general format.  Similarly in all my engineering classes.

Is the problem that they are supposed to be showing a formal proof, and are instead working the algebra?  What you've given as preferred seems to be shorthand notation for a proof.

[Kirian]

Also!  Self:  When you're on tilt, stop playing Dominion.

[SirPeebles]

It's definitely ugly, but lines 2, 3 and 4 are equivalent to each other, so they are demonstrating what you are asking.

If I really wanted to do it this way for some reason, I'd say "let's assume that 2 is a solution (line 2). Equivalently, we have (line 3). Which is equivalent to 2=2, which is always right. Hence, our assumption was right."

I guess you would rather see "x^2 - x - 4/x = ... = 0" ?

It's not just ugly.  They are not demonstrating what is asked, unless perhaps they were to explicitly indicate lines 2, 3, and 4 are equivalent.  Unless the student indicates otherwise, the convention is that they are indicating that line 1 implies line 2, line 2 implies line 3, and so forth.  One student even put a "therefore" symbol in the last line, which is especially egregious.

What I would like to see is

x^2 - x = 2^2 - 2 = 4 - 2 = 2
4/x = 4/2 = 2
Therefore x^2 - x = 4/x when x=2.

But what they have written is most certainly not correct, and I don't know where this habit comes from.

Algebra as taught to HS students (in the US) uses the format you presented first.  And all of calculus that I remember (a year of high school, plus a year of college including differential equations) also used that general format.  Similarly in all my engineering classes.

Is the problem that they are supposed to be showing a formal proof, and are instead working the algebra?  What you've given as preferred seems to be shorthand notation for a proof.

What would verifying mean if not to prove it?  Those two terms are synonymous.  And it's not an issue of a preferred format.  What I wrote first simply does not verify the statement.  Here's another example. 

Let the equation be x - 1 = sqrt(x+5).  Show x=4 is a solution.  They would write

x - 1 = sqrt(x + 5)
4 - 1 = sqrt(4 + 5)
3 = sqrt(9)
9 = 9
Big checkmark

On the other hand, if they wanted to verify that x= -1 is a solution, they would write

x - 1 = sqrt(x + 5)
-1 - 1 = sqrt(-1 + 5)
-2 = sqrt(4)
4 = 4
Big checkmark

[Ozle]

I work at a board game store now!

Nice!
Please hide all of thier copies of Fluxx in a fire outside

[Awaclus]

Let the equation be x - 1 = sqrt(x+5).  Show x=4 is a solution.  They would write

x - 1 = sqrt(x + 5)
4 - 1 = sqrt(4 + 5)
3 = sqrt(9)
9 = 9
Big checkmark

On the other hand, if they wanted to verify that x= -1 is a solution, they would write

x - 1 = sqrt(x + 5)
-1 - 1 = sqrt(-1 + 5)
-2 = sqrt(4)
4 = 4
Big checkmark

Why would they write 9 = 9 and 4 = 4? In what you wrote first, they're inserting x=2 into the equation and then verifying that it's true, which does prove that x=2 is a solution to the original equation unless I'm mistaken. In these examples, they're doing the same thing until the last line where they just pull some random numbers out of their asses.

[heron]

[...]

What would verifying mean if not to prove it?  Those two terms are synonymous.  And it's not an issue of a preferred format.  What I wrote first simply does not verify the statement.  Here's another example. 

Let the equation be x - 1 = sqrt(x+5).  Show x=4 is a solution.  They would write

x - 1 = sqrt(x + 5)
4 - 1 = sqrt(4 + 5)
3 = sqrt(9)
9 = 9
Big checkmark

On the other hand, if they wanted to verify that x= -1 is a solution, they would write

x - 1 = sqrt(x + 5)
-1 - 1 = sqrt(-1 + 5)
-2 = sqrt(4)
4 = 4
Big checkmark

In this example, the problem is not the method used by the students; the problem is that squaring is not invertible so while -2 = sqrt(4) implies 4=4, 4=4 does not imply that -2 = sqrt(4).
However, in the original example, no transformations (invertible or otherwise) are applied to the equality, so the statements are all biconditional and it has been verified that 2 is a solution.

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?

[sudgy]

I've never had anybody tell me to write things like the way you are asking.  To know where I am in my math education, I'm taking precalculus, but have done a lot of calculus on my own time.

[Ozle]

Also, clearly wrong.

Everybody knows that X = 10
Duuuh

[SirPeebles]

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?

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.

[sudgy]

Actually, I've never had a problem where I had to prove some value of x is a root to an equation.  I'm always asked to just find the roots of an equation.

[Polk5440]

It's definitely ugly, but lines 2, 3 and 4 are equivalent to each other, so they are demonstrating what you are asking.

If I really wanted to do it this way for some reason, I'd say "let's assume that 2 is a solution (line 2). Equivalently, we have (line 3). Which is equivalent to 2=2, which is always right. Hence, our assumption was right."

I guess you would rather see "x^2 - x - 4/x = ... = 0" ?

It's not just ugly.  They are not demonstrating what is asked, unless perhaps they were to explicitly indicate lines 2, 3, and 4 are equivalent.  Unless the student indicates otherwise, the convention is that they are indicating that line 1 implies line 2, line 2 implies line 3, and so forth.  One student even put a "therefore" symbol in the last line, which is especially egregious.

What I would like to see is

x^2 - x = 2^2 - 2 = 4 - 2 = 2
4/x = 4/2 = 2
Therefore x^2 - x = 4/x when x=2.

But what they have written is most certainly not correct, and I don't know where this habit comes from.

You have not been clear that you are expecting a higher level of formality when presenting answers. And you should be clear that the point of these stupid questions is to practice writing in a higher level of formality, not to review basic high school algebra.

What the students write down makes clear that they plugged in numbers and did enough calculations that they convinced themselves that 2 is a solution. They have not formally written out the verification process, though. It's scratch work, not proof. The "bad habit" comes from not having to write out formal proofs for stupid things.

[Witherweaver]

Peebles is right, and it's a much bigger problem with trig identities, where you may be motivated to manipulate both sides.  So students are asked to show P=Q and they go

P=Q
[Do some manipulations]
...
P=Q

So P=Q because P=Q.   And it can actually be false that P=Q, but everything else was correct, so the argument is valid, given the (untrue) start.  So they "proved" something that is false to be true.

And yes, they get taught this in high school and possibly by informal college teachers.

[sudgy]

I think the problem is that they haven't done things like you think they have when they haven't.

[Witherweaver]

It's definitely ugly, but lines 2, 3 and 4 are equivalent to each other, so they are demonstrating what you are asking.

If I really wanted to do it this way for some reason, I'd say "let's assume that 2 is a solution (line 2). Equivalently, we have (line 3). Which is equivalent to 2=2, which is always right. Hence, our assumption was right."

I guess you would rather see "x^2 - x - 4/x = ... = 0" ?

It's not just ugly.  They are not demonstrating what is asked, unless perhaps they were to explicitly indicate lines 2, 3, and 4 are equivalent.  Unless the student indicates otherwise, the convention is that they are indicating that line 1 implies line 2, line 2 implies line 3, and so forth.  One student even put a "therefore" symbol in the last line, which is especially egregious.

What I would like to see is

x^2 - x = 2^2 - 2 = 4 - 2 = 2
4/x = 4/2 = 2
Therefore x^2 - x = 4/x when x=2.

But what they have written is most certainly not correct, and I don't know where this habit comes from.

You have not been clear that you are expecting a higher level of formality when presenting answers. And you should be clear that the point of these stupid questions is to practice writing in a higher level of formality, not to review basic high school algebra.

What the students write down makes clear that they plugged in numbers and did enough calculations that they convinced themselves that 2 is a solution. They have not formally written out the verification process, though. It's scratch work, not proof. The "bad habit" comes from not having to write out formal proofs for stupid things.

But it really does matter.  Like for verifying trig identities

[Witherweaver]

I think the problem is that they haven't done things like you think they have when they haven't.

The don't think it be like it is, but it do.

[liopoil]

Peebles is right, and it's a much bigger problem with trig identities, where you may be motivated to manipulate both sides.  So students are asked to show P=Q and they go

P=Q
[Do some manipulations]
...
P=Q

So P=Q because P=Q.   And it can actually be false that P=Q, but everything else was correct, so the argument is valid, given the (untrue) start.  So they "proved" something that is false to be true.

And yes, they get taught this in high school and possibly by informal college teachers.

I'd like to see an example of this, just to convince myself.