The single most common mistake I have made, and that I have seen when tutoring others, is sloppiness. Especially losing track of a negative sign, either not noticing one in the original problem or forgetting it when writing out intermediate steps. That is why I have started a few years ago trying to be very careful when working things out. As I wrote in a comment a few years ago (I don’t remember where, here or OB or HN, it was about math for engineering), when studying math or anything else that may be safety critical, don’t settle for an A, do your best to get a perfect score, make it a regular habit. Even if someone else is checking your work when it is critical, don’t trust them, they can make mistakes too.
One mistake I noticed when tutoring a high school student was what I might call “failure to take seriously the rules.”
We were studying Geometry, and many times the student would make a big assumption (e.g. the angle is 90 degrees) without noting or questioning whether it was true.
When I’d ask him about it, he would say “Look at it, it must be 90 degrees!” or “If it’s 90 degrees, then I can solve this other part over here and be finished.” When I’d explain “You can’t assume it’s 90 degrees,” or “You’re assuming what you’re trying to prove,” he would grudgingly go along.
So, I think there is a class of math mistakes that come from “a failure to realize that rules in math are not like ‘rules’ in your everyday life—they are ironclad and irrevocable.”
The single most common mistake I have made, and that I have seen when tutoring others, is sloppiness. Especially losing track of a negative sign, either not noticing one in the original problem or forgetting it when writing out intermediate steps.
I find that most of the work where this is a problem is work that should be done with a computer algebra system. Those do produce a pretty dramatic reduction in error rate.
As someone who’s often struggled with this, I’d disagree. “I listened to the lecture but I just can’t understand how this works” is a different category of math mistake than “I added 36 and 9 and got 43.” (I made this mistake on a test recently).
I think math mistakes in schools break down to two categories:
Not understanding the concepts (or understanding them as magic, and blindly applying rules even where they don’t fit).
Making stupid arithmetic mistakes (which seems to come from going too quickly or being tired or distracted).
Making stupid arithmetic mistakes (which seems to come from going too quickly or being tired or distracted).
More like not bothering to actively think about how to optimize reliability of problem-solving, as opposed to thinking about how to solve the problem. “Try harder” or “be more careful” is advice of very limited power, while there are many creative ways of ensuring reliability of results (for any given problem) that are much more powerful.
“Be more careful” is meta-advice, most people don’t actually start trying to be careful until they recognize the need. Worse, and I don’t understand why, but they often need to be reminded of it again in different situations, that is the need to be careful or to pay close attention seems to be context dependent.
No, sloppiness (or carelessness, if you prefer) is a particular category of mistakes resulting in not paying enough attention to what you are doing while you are doing it. Either because you are distracted or are rushing to get done.
The latter is particularly common with homework that you are not interested in doing in the first place, the worst thing about it is that like many behaviors it can become habitual, and you start doing it even when the result is important to you.
In early education this was by far my most common source of error, so much that my parents rewarded me (with books :)) based on the number of math papers I turned in without any “careless errors” rather than based on anything involving good grades, absolute scores on math papers, effort, etc. Incorrect reasoning was fine—I was already plenty motivated to fix that. But dropping a minus sign? What did I care, I got the underlying reasoning right! ;)
I don’t have any explicit techniques now. Apparently then, “most of your careless errors were in math, so I had you solve each problem and then re-work each backward.” was the only technique my mom remembers. That clearly doesn’t scale to timed situations.
The single most common mistake I have made, and that I have seen when tutoring others, is sloppiness. Especially losing track of a negative sign, either not noticing one in the original problem or forgetting it when writing out intermediate steps. That is why I have started a few years ago trying to be very careful when working things out. As I wrote in a comment a few years ago (I don’t remember where, here or OB or HN, it was about math for engineering), when studying math or anything else that may be safety critical, don’t settle for an A, do your best to get a perfect score, make it a regular habit. Even if someone else is checking your work when it is critical, don’t trust them, they can make mistakes too.
One mistake I noticed when tutoring a high school student was what I might call “failure to take seriously the rules.”
We were studying Geometry, and many times the student would make a big assumption (e.g. the angle is 90 degrees) without noting or questioning whether it was true.
When I’d ask him about it, he would say “Look at it, it must be 90 degrees!” or “If it’s 90 degrees, then I can solve this other part over here and be finished.” When I’d explain “You can’t assume it’s 90 degrees,” or “You’re assuming what you’re trying to prove,” he would grudgingly go along.
So, I think there is a class of math mistakes that come from “a failure to realize that rules in math are not like ‘rules’ in your everyday life—they are ironclad and irrevocable.”
I find that most of the work where this is a problem is work that should be done with a computer algebra system. Those do produce a pretty dramatic reduction in error rate.
“Sloppiness” just means “tendency to make errors that you later notice”, doesn’t it?
This doesn’t seem to actually be an explanation, just a relabeling.
As someone who’s often struggled with this, I’d disagree. “I listened to the lecture but I just can’t understand how this works” is a different category of math mistake than “I added 36 and 9 and got 43.” (I made this mistake on a test recently). I think math mistakes in schools break down to two categories:
Not understanding the concepts (or understanding them as magic, and blindly applying rules even where they don’t fit).
Making stupid arithmetic mistakes (which seems to come from going too quickly or being tired or distracted).
More like not bothering to actively think about how to optimize reliability of problem-solving, as opposed to thinking about how to solve the problem. “Try harder” or “be more careful” is advice of very limited power, while there are many creative ways of ensuring reliability of results (for any given problem) that are much more powerful.
“Be more careful” is meta-advice, most people don’t actually start trying to be careful until they recognize the need. Worse, and I don’t understand why, but they often need to be reminded of it again in different situations, that is the need to be careful or to pay close attention seems to be context dependent.
+1 for admitting a mistake.
No, sloppiness (or carelessness, if you prefer) is a particular category of mistakes resulting in not paying enough attention to what you are doing while you are doing it. Either because you are distracted or are rushing to get done.
The latter is particularly common with homework that you are not interested in doing in the first place, the worst thing about it is that like many behaviors it can become habitual, and you start doing it even when the result is important to you.
In early education this was by far my most common source of error, so much that my parents rewarded me (with books :)) based on the number of math papers I turned in without any “careless errors” rather than based on anything involving good grades, absolute scores on math papers, effort, etc. Incorrect reasoning was fine—I was already plenty motivated to fix that. But dropping a minus sign? What did I care, I got the underlying reasoning right! ;)
What techniques do you have for reducing careless errors? How do these scale in stressful/timed situations?
I don’t have any explicit techniques now. Apparently then, “most of your careless errors were in math, so I had you solve each problem and then re-work each backward.” was the only technique my mom remembers. That clearly doesn’t scale to timed situations.