Can you explain? My understanding is we're all working on the same hard math problem, once a certain number of us have verified the answer (ie distributed consensus) one of us is given a reward to encourage further problem solving.
Nope, any bitcoin client can verify any transaction very easily. (Bootstrapping by verifying every transaction in the blockchain does take a while though.) Proof of work is only to prevent a 51% attack where someone spins up enough bots to hijack "consensus" and rewrite transaction history. That's why the difficulty can be dialed up or down - most of the work in mining is not related to verifying the transactions.
Yes. People often misunderstand that verifying transactions with blockchain is intrinsically difficult which is not the case. The hard part is making a ledger that can't be rewritten.
And mining is actually less akin to solving math problems and more akin to, well, actually mining. That is, digging around enough until you find some gold.
This is very much not what distributed consensus is. You don't "work on the
same hard math problem" and you don't get "a reward to encourage further
problem solving". Consensus is agreeing on a value among (almost) all protocol
participants, and "distributed consensus" is proper technical term, not
a vague definition by hand waving like "object oriented programming". Read
about Byzantine Generals problem first.
