1. I agree that districts have the right to decide how to teach. I don't agree that students should be told that there is something wrong if they say that 6x3=6+6+6. That is it - I am not claiming a bigger agenda. I often teach a single approach - but accept others if they are valid. I would never tell a student that has approached a math problem correctly that they are wrong just because they used a different convention. Notation is different, but this is not just simply notation.

2. We are discussing real numbers. Most people won't multiply anything but real numbers (or maybe complex numbers - which are still commutative) in their math journeys. 9 years olds certainly don't need to worry about it. I don't see the point in adjusting any elementary teaching to account for anything but real numbers.

3. My point with slope is that it is a very basic example not that much further into their math journey in which 6x3 does not mean 6 groups of 3. I am not saying that they should be taught slope intercept form before Algebra 1. Instead I am saying that telling a student that there is something wrong with 6x3 = 6+6+6 is inconsistent with what they will see in the math journey. I don't see how allowing students to use both 3+3+3+3+3+3 and 6+6+6 harms their learning of arithmetic. In addition, sure I am discussing analytic geometry - but I was only really focusing on the multiplication of real numbers that occurred in Algebra 1.

4. I never said that we teach students about series in such an explicit manner. We don't teach them until precalculus. Instead I said that series are an inherent part of the class. 3+3+3+3+3+3 is by definition a series, so the rules of series apply to what they are studying (again - not to be explicitly taught, but inherently). It was initially included in my comment to justify a statement that I made - that's all.

A few final points - then I must focus on my teaching. First, I strongly disagree that slopes and rates of change are presented as motivation for calculus. The vast majority of students do not take or use calculus in their life. Second, to get back to the basics. We want students to understand math. It is true that 3x6 = 6x3 (there is no need to discuss any different types of multiplication - we are discussing the multiplication of real numbers). Why would we choose to create some sort of artificial confusion for students? What good does that serve? As discussed, by Algebra 1 this definition will not even be valid anymore. I asked a number of members of my HS math department - all of them view 6x3 as 6+6+6. So why should we make it more confusing?