The advantage of “a mistake is a mistake” approach is that it makes it much cheaper to evaluate. Imagine having to grade tests from 100 students—how much time would it take if you only check the results, and how much time if you also check the process.
It also depends on how much the teacher’s verdict is final, or how much the students are allowed to argue. Imagine that out of those 100 students, at least twenty want to debate you individually on whether their processes were only 60% correct or 80% correct (because a partial point here and a partial point there might together make a difference in their final grade). It would drive me crazy. Also, it would be unfair against the less “litigious” students.
A mistake of the same magnitude can have a different overall impact, depending on where you make it. If at the last step of long computation you accidentally do 20+50 instead of 20-50, but everything else was correct, it is obvious that you would have solved the problem correctly. But if you made exactly the same mistake at the beginning of the problem… it could have sent you on a completely different track. Like, maybe you needed to calculate a square root of that, and then you solve a task with real numbers, while all your classmates were solving a task with complex numbers. If the point of the test was to verify your knowledge of complex numbers, then this completely failed the purpose. And yet, it was the same mistake.
I think my preferred option (assuming “veil of ignorance” where I can either be the student or the teacher) would be: a mistake is a mistake, but you can take the test again (with slightly different questions).
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At math olympiad, both your process and your answer matter. The process matters in the sense that it has to be mathematically sound (but it can be completely different from what other people would use). If the answer is wrong because of some stupid mistake but the process is generally sound, you get partial points; so I think a typo in the last step would still get you 5 out of 6 points.
(You also submit your working notes, which is like potential extra evidence that can be used in your favor. This is optional—if the problem is so clear to you that you immediately start writing the official answer without any previous note, of course you will not be penalized for that. It’s more like… if the way you wrote something in the official answer is unclear, people will check your working notes for possible evidence how you actually meant it. You will not be penalized for also trying approaches that didn’t work in the working notes; that is a part of what they are for.)
However, if you are competitive, you aim for 6 out of 6 points, so we are kinda back to “everything must be perfect” again? I guess the difference is that if you get most of the problems 100% right and one or two of them are “generally okay, but made a stupid mistake that only had a local impact”, you might still win.
The other advantage of “a mistake is a mistake” is that it matches the real world. When solving problems because you care about the world, rather than because of a score, the process matters for repeatability, communication, and trust of others, but the answer is all that matters in terms of impact.
The advantage of “a mistake is a mistake” approach is that it makes it much cheaper to evaluate. Imagine having to grade tests from 100 students—how much time would it take if you only check the results, and how much time if you also check the process.
It also depends on how much the teacher’s verdict is final, or how much the students are allowed to argue. Imagine that out of those 100 students, at least twenty want to debate you individually on whether their processes were only 60% correct or 80% correct (because a partial point here and a partial point there might together make a difference in their final grade). It would drive me crazy. Also, it would be unfair against the less “litigious” students.
A mistake of the same magnitude can have a different overall impact, depending on where you make it. If at the last step of long computation you accidentally do 20+50 instead of 20-50, but everything else was correct, it is obvious that you would have solved the problem correctly. But if you made exactly the same mistake at the beginning of the problem… it could have sent you on a completely different track. Like, maybe you needed to calculate a square root of that, and then you solve a task with real numbers, while all your classmates were solving a task with complex numbers. If the point of the test was to verify your knowledge of complex numbers, then this completely failed the purpose. And yet, it was the same mistake.
I think my preferred option (assuming “veil of ignorance” where I can either be the student or the teacher) would be: a mistake is a mistake, but you can take the test again (with slightly different questions).
*
At math olympiad, both your process and your answer matter. The process matters in the sense that it has to be mathematically sound (but it can be completely different from what other people would use). If the answer is wrong because of some stupid mistake but the process is generally sound, you get partial points; so I think a typo in the last step would still get you 5 out of 6 points.
(You also submit your working notes, which is like potential extra evidence that can be used in your favor. This is optional—if the problem is so clear to you that you immediately start writing the official answer without any previous note, of course you will not be penalized for that. It’s more like… if the way you wrote something in the official answer is unclear, people will check your working notes for possible evidence how you actually meant it. You will not be penalized for also trying approaches that didn’t work in the working notes; that is a part of what they are for.)
However, if you are competitive, you aim for 6 out of 6 points, so we are kinda back to “everything must be perfect” again? I guess the difference is that if you get most of the problems 100% right and one or two of them are “generally okay, but made a stupid mistake that only had a local impact”, you might still win.
The other advantage of “a mistake is a mistake” is that it matches the real world. When solving problems because you care about the world, rather than because of a score, the process matters for repeatability, communication, and trust of others, but the answer is all that matters in terms of impact.