We can teach math in different ways. At the moment we are testing something in our schools called common core math. I’m not an expert in common core math (not even close!), but I do a lot of math, so I feel justified in suggesting an article that is definitely worth reading. There is some guy, who is probably not worthy of all the press he is receiving, who wrote a check using common core math which is getting a lot of press right now.
And Hemant Mehta wrote a great article titled, “The Dad Who Wrote a Check Using “Common Core” Math Doesn’t Know What He’s Talking About”. The article relates how this guy is fundamentally dismissing something because he doesn’t understand it. And that is a bad tactic to take. Mehta also gives a introduction to common core and explains why it’s more useful than the memorization tactics taught in Elementary schools previous to this. Please take a look, it’s a well written article!
Social Mathematics 
I liked the idea that kids are being taught to search for patterns in numbers. This is often how I think about simple numbers and the only way I know of to make basic math like this interesting. I saw from trying to help my 7 year old nephew with his basic subtraction how tricky it can be to make this stuff intuitive and hopefully even interesting.
Yes! Math can be intuitive …or not depending on how you are taught. I remember spending a lot of time playing with the math blocks we had in my classroom. I just learned that they are called Cuisenaire Rods: https://en.wikipedia.org/wiki/Cuisenaire_rods. I think the blocks helped cement the ideas of subtraction and addition being related; a similar idea to the one discussed with the ten-frame in the article mentioned above. But building that intuition is hard.