>I can walk into pretty much any first semester calculus class and ask students to write down an example of an equation with no solution. A large majority will fail to do so. It doesn’t occur to them that 0=1 is such an example.
It's interesting that you picked 0=1 as your example, because I'd argue it stretches the definitions of "equation" and "solution" into semantic triviality. It's more of a falsehood than an "equation", since the two sides are trivially defined as not equal, and there's no variables to "solve". Using that as example exists somewhere between sophistry and pointing out the absurdity that mathematical definitions for terms technically hold even in trivially untrue situations. That's not how normal human communication works, and not recognizing that divide probably goes a long way in explaining the "inability" you see in students.
In other words, maybe you should have just used "0x=1" as your example :P
It's interesting that you picked 0=1 as your example, because I'd argue it stretches the definitions of "equation" and "solution" into semantic triviality. It's more of a falsehood than an "equation", since the two sides are trivially defined as not equal, and there's no variables to "solve". Using that as example exists somewhere between sophistry and pointing out the absurdity that mathematical definitions for terms technically hold even in trivially untrue situations. That's not how normal human communication works, and not recognizing that divide probably goes a long way in explaining the "inability" you see in students.
In other words, maybe you should have just used "0x=1" as your example :P