I can remember that one from high school.
The answer is quite simple which I will not reveal for now.

 

if a math teacher did not know this.... should not be a math teacher...

 

2 is = 1, I swear by all thats holy !

Is that why none of my circuits work ?

Regards, Dana.

 

if a math teacher did not know this.... should not be a math teacher...

Sadly, most of the teachers employed at my school should not have been teachers – but otherwise, they would have been unemployable.


Those that can, do

Those that can’t, teach

Those that can’t teach, teach teachers

I think those who could not teach teachers, were teaching at my school

 

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That is one of the classics paradoxical problems. There are a lot of them that all suffer from the same fallacy.

As others have said, any math teacher that cannot explain the problem with this proof has no business teaching math at any grade higher than about fourth or fifth.

Also as others have said, it is a sad reality that a large fraction of people that are teaching math in K-12 (particularly public schools, but the problem isn't constrained to them by any means) have no business teaching math. My office is mixed in with the Math Department and several of the math faculty have told me that it is all but given that the bottom students among the math majors are those that are in the K-12 Education program and plan to teach math in public schools. The real tragedy is that these same students usually have such poor math skills that they should not have been admitted into a math major, let alone be allowed to graduate. But, key, it's a public university (and, again, the bulk of private universities aren't that much better and a certain subset of them are actually far-far worse).

 

As others have said, any math teacher that cannot explain the problem with this proof has no business teaching math at any grade higher than about fourth or fifth.

I wouldn't be quite that harsh IF the teacher had trouble with the challenge at the chalkboard, not at his desk. Sitting in my armchair it doesn't take long to check all the steps and find... (I don't want to reveal any spoilers.) BUT, in the rushed environment of a classroom with dozens of eyeballs watching you, I can easily imagine being unable to spot it. A teacher should be comfortable enough in front of his class that the "public speaking" effect doesn't kick in, but it's a real thing. It's similar to what Jeopardy contestants suffer. Knowing the answer is one thing but recalling it on TV is a whole different thing.

 

Maybe. But any math teacher worth their salt would have seen this math puzzle before, or should have seen it before.
This is the most common math quiz I know of.

 

I got a divide by zero interrupt.

 

I wouldn't be quite that harsh IF the teacher had trouble with the challenge at the chalkboard, not at his desk. Sitting in my armchair it doesn't take long to check all the steps and find... (I don't want to reveal any spoilers.) BUT, in the rushed environment of a classroom with dozens of eyeballs watching you, I can easily imagine being unable to spot it. A teacher should be comfortable enough in front of his class that the "public speaking" effect doesn't kick in, but it's a real thing. It's similar to what Jeopardy contestants suffer. Knowing the answer is one thing but recalling it on TV is a whole different thing.

I grant you that, but I suspect that most math teachers would have seen enough of these proofs (long before they became math teachers) that they would know what the first thing to look for should be.

When I was in high school my best friend showed me a proof that 1 = 2 that did NOT rely on this flaw, but rather a flaw involving calculus. I didn't get spot the flaw in the one or two minutes that he gave me to look for it (and I doubt I would have spotted it even if I had taken all night). Naturally, he hadn't spotted it when it was shown to him the night before by someone, either. After he showed the flaw to me we presented to our calc teacher and he spent quite a bit of time looking for the flaw that the problem in the OP has and couldn't find it. He came in the next day and told us the problem and said that he was up most of the night being haunted by it and just could not get it out of his mind. Now THAT's someone I want teaching math -- he didn't just spend a minute at it, shrug his shoulders, and move on satisfied that it doesn't really matter so why waste time trying to figure it out.

A year later, my freshman year in college, I showed it to a friend of mine. He couldn't get it (in the minute or two I gave him). So we went up to the teacher and I said something very much like, "Hey, Dr. Kelly, I can prove that 1 = 2 using calculus." To which he got a reflective look on his face and said, "Using calculus... I don't think I've ever seen that. So let's see what you've got."

I wrote "1² = 1" on the board and asked him if he agreed, to which he said, "Sure."

I wrote "2² = 2 + 2 = 4" on the board and asked him if he agreed, to which he hesitated a moment and said, "Okay. Go on."

I wrote "3² = 3 + 3 + 3 = 9" on the board and he immediately said, "You might as well stop right there because I can tell you where you went wrong in the prior step if you plan to use calculus at some point."

I asked him where and he was exactly correct, but he said that he still had no idea how it would eventually lead to the claim that 1 = 2. After I showed him, he remarked that it was a clever little false proof and he would have to remember it to use in his calc courses in the future.

EDIT: Inserted missing "not" and fixed typos.

 

He came in the next day and told us the problem and said that he was up most of the night being haunted by it and just could get it out of his mind. Now THAT's someone I want teaching math -- he didn't just spend a minute at it, shrug his shoulders, and move on satisfied that it doesn't really matter so why waste time trying to figure it out.

exactly...

 

In defence of my former maths teacher, the problem highlights the potential issue of blindly following the rules of algebra – without adding that all important ingredient; common sense.

Once it is clear that the result is wrong (2 ≠ 1) close analysis of the steps to reach this should reveal the flaw.

 

In defence of my former maths teacher, the problem highlights the potential issue of blindly following the rules of algebra – without adding that all important ingredient; common sense.

Once it is clear that the result is wrong (2 ≠ 1) close analysis of the steps to reach this should reveal the flaw.

So..... how is that a defense of your former math teacher?

You seem to be saying, in essence, that it's okay that they weren't able to figure out the issue because they so lacked common sense that they were unable to comprehend that a result of 2 = 1 is clearly wrong.

I don't see that as doing anything but further strengthening the position of those that contend that this person has no business teaching math.

 

Most of my math teachers ... were not "math" teachers until I reached the final years of high school and then college. Before that they were babysitters and football coaches assigned to teach us math. I would not expect them to have seen the common math quiz type problems in their education curricula when they were in school. They probably went into teaching, in part, to avoid such things.

 

Most of my math teachers ... were not "math" teachers until I reached the final years of high school and then college. Before that they were babysitters and football coaches assigned to teach us math. I would not expect them to have seen the common math quiz type problems in their education curricula when they were in school. They probably went into teaching, in part, to avoid such things.

My first real math teacher was my Honors University Physics professor in my junior year of college.

He taught (hammered into!) me "first principles", "units matter", and "is it plausible".

All my other math teachers/profs pale in comparison.

 

Most of my math teachers ... were not "math" teachers until I reached the final years of high school and then college. Before that they were babysitters and football coaches assigned to teach us math. I would not expect them to have seen the common math quiz type problems in their education curricula when they were in school. They probably went into teaching, in part, to avoid such things.

I was very fortunate in that regard. I think every one of my math teachers, starting from sixth grade which was the first time we had different teachers for some of the subjects, was adequately qualified and prepared to teach the level of math that they were teaching. I would have to say that all of my high school math teachers were quite well qualified and the same was true for my science teachers (one of whom wrote the text).

But I also did become aware of the all-too-common phenomenon in which a teacher was hired based on their ability as a coach and then were assigned to teach a math or science class as a way of filling out their contract regardless of their background in that area. Usually they were assigned the lowest level classes, which was good for me personally, but a strong argument can be made that the best math teachers (and that means best teachers, not just best math person) should be teaching those courses.

 

So..... how is that a defense of your former math teacher?

You seem to be saying, in essence, that it's okay that they weren't able to figure out the issue because they so lacked common sense that they were unable to comprehend that a result of 2 = 1 is clearly wrong.

I don't see that as doing anything but further strengthening the position of those that contend that this person has no business teaching math.

Having said my former maths teacher could not spot the flaw in the equations; she actually dismissed the solution as showing that a = 0 (a = b – b) and analysed it no further.

But in the algebraic equation (a = b), ‘a’ can be any numeric value.

 

Having said my former maths teacher could not spot the flaw in the equations; she actually dismissed the solution as showing that a = 0 (a = b – b) and analysed it no further.

But in the algebraic equation (a = b), ‘a’ can be any numeric value.

While I am aware of the flaw in the math, the teacher does have a valid point.

If the premise is that a can have any value, the sequence of equations conclude that

a = b - b
hence a = 0
Therefore there is an inconsistency.

 

While I am aware of the flaw in the math, the teacher does have a valid point.

If the premise is that a can have any value, the sequence of equations conclude that

a = b - b
hence a = 0
Therefore there is an inconsistency.

0/0=1

 

Having said my former maths teacher could not spot the flaw in the equations; she actually dismissed the solution as showing that a = 0 (a = b – b) and analysed it no further.

But in the algebraic equation (a = b), ‘a’ can be any numeric value.

And therein lies a big difference between this math teacher and a "good" math teacher. Spotting that the "solution" implies a = 0 when a should be able to be anything is fine. But to then "analyze it no further" is to basically say that she doesn't care about the math enough to even attempt to understand why a seemingly valid sequence of steps yields something that she knows can't be. Would she like her doctor to be comparably good at medicine? Or the engineer that designed the building she works in, the car she's driving, or the bridge she's passing over or under?

There's absolutely no shame in being someone that has no business doing something -- there are LOTS of things that I have no business doing -- but it doesn't change the reality that that someone has no business doing that something.