Pay 2 of my response to Prof Willingham...
Here is the link to his original post:
My reply (unfortunately my reply was duplicated several times. My bad. ..)
(a) 9+9+9+9+9+9+9+9+9+9 vs
Is their equality a coincidence?
On your grid paper, make 10 rows of
* * * * * * * * *
Which addition problem does this represent?
Should how you could represent the other addition problem without drawing any more stars. [Rotate paper 90°]
Now write both as multiplication sentences...
We can pontificate about all of this ad nauseam but in the end teachers have to be trained to provide an environment which BLENDS explicit and implicit instruction. I learned much about arithmetic and number sense from playing Monopoly but I didn't learn everything that way! Some concepts/skills/procedures had to be clearly demonstrated to me. I was observing my precocious 6-yr old grandson learning to play Monopoly. From playing a couple of times he decided to buy every property he landed on. When he ran out of money I told him he'd have to wait until her could collect $200. "No problem PopPop. Just let me be the banker!" Will he improve his understanding of the game without formal instruction? Of course. Will he also develop some misconceptions if not corrected and given a clear explicit explanation? Of course. INFORMAL LEARNING CAN GO ONLY SO FAR IN MATHEMATICS. This must be balanced with the child developing proficiency with skills/algorithms, attention to detail and recognizing the appropriateness of approximate vs exact results. I'm only scratching the surface here. But I do know that none of this happens by accident. CCSS are necessary to raise the bar but without the"heavy lifting" required to train/prepare teachers, it will be futile. But nothing substantive will occur until the education of our children is genuinely seen as an investment instead of an expense. When we truly put our money where our mouths are...