Errors & what to do about them
Errors are inevitable. You may think that you are so careful that you are immune from making errors. If so, you are wrong. You will make errors and the only unknowns are how many and how bad they are. You cannot avoid them and in this short piece, I shall try to explain why that is, and what you can do to mitigate the damage that errors can create. However, you should never think about your errors as a failure: it is an opportunity to learn something new. If you take that opportunity, and that is what you should do, then your learning will improve exponentially.
I am extraordinarily careless in all sorts of areas. I skim news stories and get the complete wrong end of the stick. My mathematics is filled with the damn things and most of them have to do with minus signs. When I write code, which I do a lot of, it is very rare for me to write a class or method which is 100% right first time I test the code.
When the great mathematician Andrew Wiles first published his proof of Fermat's Last Theorem, after those of us who were aware of the problem, started breathing again. (It was a truly breathtaking moment, if you were even remotely interested in number theory at the time.) I was teaching mathematics to secondary school students at the time, and my lessons on Pythagoras' Theorem (similar and related, but much easier to prove) were never going to be the same again. The euphoria lasted a couple of months, from June 1993 to August 1993, when an error was found in the proof which meant that Wiles had to go back to the drawing board and very nearly gave up in 1994.
In May 1995, Wiles published another paper with his corrected proof and no-one has managed to find fault with his new solution in over 25 years. Andrew Wiles is remembered for solving one of the most difficult and profound problems in mathematics, quite rightly, but behind the success, there is the failure. And the failure is an integral part of the success, because that is where it can lead to, if you are motivated to try.
Ever tried. Ever failed. No matter. Try again. Fail again. Fail better.- Samuel Becket
Everyone with an even vague interest in learning, should have this quote from the great Irish playwright, pinned up in their workspace, because it is the best advice there is.
Do not practise your mistakes!
This section is going to irritate some people. I don't really care, but it needs saying. Homework sucks! Homework might be the most counterproductive piece of stupidity ever devised by an unthinking Hobbesian mindset.
Train them to understand that life is hard. Work is hard! Get used to it and do your homework!. For my entire career, I have had to justify the setting of homework to all the students I taught, even while I knew the whole time just how useless it is to the child's education and to their development. And let me tell you, that is not a great way to live; doing something which you are fairly sure is counterproductive, takes its toll after a while.
If one wanted to crush and destroy a man entirely, to mete out to him the most terrible punishment ... all one would have to do would be to make him do work that was completely and utterly devoid of usefulness and meaning.- Fyodor Dostoyevsky
Let me set the scene for you. I have just set a class of year 7s (11 years old) a homework task to solve 20 simple linear equations (eg $2x-3=7$), following a couple of lessons in which I have introduced the topic and managed to do a small amount of practice in class. I have set the homework in order to allow the students to consolidate the work they have done in class. So 30 odd students go home and start doing their homework. They have mostly half remembered what was taught in the lesson and they want to get their homework done before 10pm, so they are not going to check their notes book because, if they do, they won't have the time to get into their online game of Fortnite with their chums before bed time. So they do their best and in every example they make the same error. Not only have they now got a poor mark on their homework, but they have also practiced their error, and they have become really good at making that particular error. They have internalised it and breaking that anti-skill is much more difficult than learning it right to begin with. Practising music is similar. If you play a bar wrong while practising, and it is not immediately corrected, you will become really really good at making that error and breaking the habit is hard.
Students need access to the answers, which they should check after every single attempt at practising a skill. But you can't do that if the student is to be assessed based on the homework. You can't do both. It makes no sense. There is a quote which comes to mind here about pig farming, though I do not know the source and this is a paraphrase.
If you want a fat healthy pig, weighing it every day will not help. Feeding it, on the other hand, just might.