Why high school teachers do not emphasize knowing the proofs of properties and theorems in math. In my 40 years of teaching prospective high school teachers, I rarely found students who can derive formulas they learned in school. For example properties of logarithms, changing from one base to another, Law of Sines and Law of Cosines proving major Geometry theorems. These "formulas" and theorems can be presented as problems to solve so with help, students can discover them on their own. Then they would "own" the theorems.
I'm going to propose two reasons: the problem of assessment, and the design of curricula.
The problem of assessment
One reason high school teachers do not emphasize proof and reasoning (outside of Geometry) is that it is difficult to assess. Take, for example, the proof that $\log(ab) = \log(a) + \log(b)$. A teacher can show the proof to students; a teacher might even be able to guide the students to discover the proof themselves. But how can the teacher assess whether the students understand the proof? The only real options are:
- Ask them to replicate the proof on a test -- but this may degenerate into a test of memorization, rather than comprehension; or
- Ask them to prove something different, but similar -- but the scope of available things to prove at the high school level is limited, and to a certain extent each one is sui generis.
Instruction tends to default to that which can be assessed. One of the reasons the Geometry course has developed the highly artificial, stylized kind of proof that is common in that context is precisely because it is easy to generate many, many problems of a similar level of difficulty. The two-column proof form in this sense "solves a problem of teaching", making it viable for teachers to hold students accountable for learning something. (See Herbst, 2002 for more on this.)
The design of curricula
A second reason teachers do not hold their students accountable for deriving, justifying, and proving formulas in high school is because for the most part the curriculum materials they follow provide little support for such instruction. Thompson, Senk & Johnson (2012) studied 20 contemporary high school mathematics textbooks and found that:
Overall, about 50% of the identified properties in the 3 topic areas were justified, with about 30% of the addressed properties justified with a general argument and about 20% justified with an argument about a specific case. However, less than 6% of the exercises in the homework sets involved proof-related reasoning, with developing an argument and investigating a conjecture as the most frequently occurring types of proof-related reasoning. (p. 253)
The last of the statistics quoted above -- that fewer than 6% of the exercises in the homework sets involved proof-related reasoning -- is obviously related to the problem of assessment I mentioned above.