> Doing the math out helps build an intuition for things like commutative properties and what not.

Well yes, but that's the things in boxes example. Doesn't matter if there's 3 boxes of 4 apples each or 4 boxes of 3 apples each. That takes like 1 lesson and most kids understand it intuitively anyway.

I remember we had 1 lesson of introduction and then a whole semester to learn multiplication tables and use them to solve simple word problems with multiplication. I'm not sure how that time could be better spend thinking about multiplication in abstract.

Of course you can, but it's not intuitive to most. Most notably it requires understanding that in 3+4 you start at 3 and move 4 spaces to the right. In 3x4 you start at 0 and move to the right 3 spaces 4 times.

Kids absolutely do not grasp commutativity immediately. That they can solve 3x4 is 12 and 4x3 is 12 is not the same thing as understanding with confidence that nothing changes between these swaps.

That's a weird way to describe it. Doesn't make it more understandeable and isn't useful for solving problems. So why bother? We were taught the definition of multiplication (it's just repeating addition). So if you have 3+4 it's starting at 0 then jumping by 3 and then jumping by 4. If you have 3*4 it's the same as 4+4+4 which is starting at 0 and jumping by 4 three times. Or vice versa.

I don't think we had that jumping on number line on that lesson, probably not since it's kinda obvious and provides little value when you know the definition. We had a lot of word problems and whoever was the fastest would explain how to solve it to others. So basically somebody was the first to realize you don't need to do 2+2+2+2+2+2 when you can calculate the solution as 6+6. Kid got reputation boost for being smart in front of others and others stole the technique to be the fastest the next time. This got me into math.

But mapping the relationship between multiplication and the area of rectangles is very visually intuitive. Geometric reasoning is powerful. Being able to use your understanding of multiplication rules to derive the formula for the volume of a cube, or a cylinder, or whatever is probably more insightful than realizing some arithmetic tricks.

Strong disagree. The number line is one of the best tools for understanding multiplication of numbers (not just whole numbers, but any decimal). The problem is that it is no longer taught. Example: American rulers have both metric (cm) and Imperial (inch) units. You can use it to visually multiply and divide by 2.54.

To generalize multiplying positive numbers x and y, (1) Mark x on a number line, (2) create another number line with the number 1 placed where x would be on the original line, (3) find y on the new number line, and (4) the corresponding point on the original line is the product x*y. The point: Multiplication is scaling (stretching, shrinking, whatever you want to call it). The Common Core tries to address this.

I always had better luck with just manipulating the numbers mentally.

But having that capability is pretty important for understanding other concepts. For example, many concepts in calculus.

But as a kid I never got the multiplication thing. Even now as an adult I understand what they are trying to do, but it isn't "obvious" to me.

Then again I am one of those people who doesn't read comic books because all the pictures get in the way of the words. Heck as a kid I read illustrated books and didn't realize until I had finished the book that it even had pictures.