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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*Who needs a house out in Hackensack, anyway?