More Check Points

Some of my students does algebraic computation rapidly, but they are prone to make mistakes.
My only advice for them was to do it more carefully.
But I wonder what "carefully"  really means. What a clever advisor I am!

I can do computations more accurately than them. Of course! But what makes me so?
Here are the reason I come up:
1. I practiced much more.
2. I have more checkpoints for the validity of my answer.

"1" is the biggest reason and "2" is the consequence of it. I going to explain about "2".
The ability to compute more carefully depends on how many checkpoints ones have.
For example, when I solve some equation. I have the following ways to check.
A. Substitute my answer value into the original equation.
B. If it is a polynomial equation of degree N, I would check whether I already got exactly N solutions.
It's an advantage of knowing Fundamental Theorem of Algebra.
C. Check the answer graphically.
D. Make sure the sign of the answer makes sense.
E. Make sure the size of the answer makes sense. (size as a number of digit.)
F. Solve the same problem in different method or in different order.
G. Guess or Estimate the answer first, and determine whether my final answer is as expected or not.
    If not, I would think why not.
H. If it is an integer problem, I would consider parity or divisibility.
I. When recheck my calculation steps, I pay extra attention to where I know I often make mistakes.
    Experience is required to know such points.

Well, what I wanted to say is I have various techniques and point of view to test the validity of my answer.  Hence I can do "carefully". I think the students who caculates rapidly with poor accuracy don' t know how to do "carefully" because they don't know how to verify their answer.
Know I think that's the main problem. Now I have to think how to solve this.


PS. If you have checkpoint that I didn't list above, please let me know. Thanks.