I agree that teaching mathematics too formally prevents students from making concrete connections. I like how the OP made the abstractions intuitive. And made t

I agree that teaching mathematics too formally prevents students from making concrete connections. I like how the OP made the abstractions intuitive. And made the jargon less intimidating. As far as the 'mathematician's lament' goes though, I think all that is needed is for our culture to foster greater respect (and potentially payment) for teachers. If someone can put passion into their work, these successes tend to arise naturally.