As a maths tutor, some students I have tutored don't just make the odd mistake in arithmetic (including fractions) and algebra: they make every possible mistake and regularly.
My go-to approach for helping these students is to show them how to correctly apply the rule they have just made a mistake on, as and when they make that mistake.
However, for these students, their mistakes are relentless, and whilst the % of errors may decrease over time, once I show them a correct way of doing it, this % may only decrease from 80% of the time to 20% of the time a few months later, which still isn't very good, as for arithmetic and algebra, you need to be striving for errors basically 0% of the time.
"But 80% to 20% is good improvement"
No. This is algebra/arithmetic. This is not proof writing, which is a skill that you improve with over time. If you stick to the rules of algebra/arithmetic, then you shouldn't really be making errors at all. And yes, I do go through various checking methods with my students.
Moreover, such students will make every possible error on most future topics. I believe this is due to their, "I don't care about maths: I just want to pass my exams" mindset, which is counter-productive, as the only way to improve on maths is to spend time doing maths (exercises). Anyway, the point is that I think these students need a radical change in mindset/approach in order to succeed in maths.
I wonder if, for such students, an entirely different overall approach is more productive; namely, the following "sceptical" approach, which is the one I have always gone by:
Don't use a rule until 100% convinced:
- The rule works
- You know when the rule works and when it does not work
and more importantly,
If you are not 100% convinced the rule works, do not use it (other than to test it), and keep questioning it's validity.
They can even write an "investigation into the validity of rule X", up until the point they are convinced the rule works.
Then, at some point, they will be naturally more and more convinced of it's correctness, as they will have run out of questions as to why the rule might not work.
There are many obvious benefits of this approach, including the fact that it forces the student to actively engage in the material.
However, I'm doubtful of this approach because I doubt these type of students' abilities to correctly apply scepticism and questioning of the method. Another criticism is that this approach takes more time to carry out at first; however I don't buy this criticism as the approach probably saves time in the long run.
I'm interested to hear what other maths educators think. Has anyone tried this approach with relentless error-ridden students, and did this approach work?