Heated, I both agree and disagree with a lot of what you are saying. I avoided this thread as it is so difficult to comment on another teacher's teaching without knowing the whole context.
I did just want to say though that while I can see you recognize the need for discovery learning and explicit instruction, it is highly possible that what you are saying you wish for-quickly explaining HOW to do the sum (which in my experience many tutoring places do due to lack of time for sequencing etc.) is what makes many children lack those vital early number foundations. I get this all the time in my class-parents teaching their children vertical addition etc. Before they have recognized the need to even do so. Sure they can now 'work out the sum' but have buckleys applying it in any real context and certainly cannot explain the inherent number connections within it. Another Maths classic for this are times tables and skip counting. The amount if children I have taught that can do both but don't get they how or why is so frustrating! What is the point in knowing this if you cannot apply it? See it as multiple addition first to build multiplicative understanding??? My point is, in MANY cases with Maths instruction, children can certainly be taught the how but without the why are likely to flounder again at a later stage.
Can I suggest you google the Maths misconceptions? These are a set of learning milestones for Mathematical understanding and tests at each level. They say a child cannot progress to the next level until the 1st is mastered. Level 1 is subitising, 2 is place value, 3 multiplicative thinking and 4 fractions. This should highlight to you where your children's 'gaps' are. They suggest activities to support learning at each of these levels. It's amazing how many grade 5 and 6 kids still cannot subitise given it should be mastered at Prep!
BTW I actually don't think this should be your responsibility at home but it should give you some solid evidence to discuss with the teachers precisely where your kids need support