In your typical undergraduate math textbook, it's fairly common to see phrases like:
Applying the division algorithm, we see that ...
What is almost never explained, though, is why the author felt compelled to use the division algorithm in the first place. The tendency for textbooks to omit the strategies involved in formulating a proof leads to students lacking a sense of direction when tackling proofs on their own.
Understanding the subject matter is not sufficient
Simply understanding the subject matter does not in any way guarantee success in completing proofs related to the subject. This is because the act of proving something in mathematics demands the execution of small arithmetic operations that aren't directly associated with what you're studying. For example, the division algorithm and its properties have very few direct associations with Group theory, but you use the division algorithm constantly in Abstract Algebra.
An example strategy: the division algorithm
Here is an example of an actual proof technique/strategy that students can apply to any proof when necessary. Consider the following problem, which comes up quite frequently in elementary proofs:
"I need to get some number in terms of this other number, but I can't find any obvious correlation between the two"
The division algorithm lets you relate one integer to another very easily.
Since for any integer
b there exists unique
r such that
a=bq+r, you can use the division algorithm to express a number in terms of
any other number.
This fact will probably have almost nothing to do with the subject area being explored by the student, but knowledge of this fact is necessary to complete the proof.
Teachers and textbooks need to make proofs less hand-wavy. There are specific reasons why authors call upon proof techniques such as the division algorithm - they should take the time to tell the students their reasoning.