My puzzlement first began while I was building multiplication towers. What's the difference between two 3s and three 2s? The commutative property says, essentially, no difference. But I say, take another look:
3x2 | 2x3
What's the difference between...
#1: 3 green-3 red
#2: 2 green-2 red-2 blue?
Quantitatively, nothing. Qualitatively, everything. These are two of the multiplication towers I built. They both have exactly the same number of beads in exactly the same places, but they are qualitatively different from each other in some interesting ways. You can read the full post here.
1x2 | 2x1
Here's another example: We were out to breakfast one Saturday morning. The kid wanted more sausage and bacon. I told her she could have two more pieces of protein. Fair enough, right? But is it going too far to argue that 2 sausages do not equal 1 bacon + 1 sausage? If you're seven years old, it's a tricky choice. [Edit: Re-reading this, I realize this is not necessarily mathematically sound, but I think my defense is to say that if we focus on the value of '2', 2 means different things if you have 2 of 1 thing or two different things. Make sense?]
3x4 | 4x3
And, consider a job I have coming up. I have work in the city, which is a 3 hour drive round trip. I am working 4 days with 3 workshops each day for a total of 12 workshops. I inquired about doing 4 workshops a day for 3 days since that would mean I would have one less day of driving and incur less child care costs.
Just like in the sausage/bacon example it's important to compare units. Sure, 3x4=12 and 4x3=12, but when you factor in the driving time, gas consumption, and child care costs...
3 days of 4 workshops per day is qualitatively better than 4 days of 3 workshops per day, wouldn't you say?
Quality vs. Quantity: An Alternate Perspective
These ideas have been in the back of my mind for a month or more. As I was thinking about it again today the subject spontaneously came up as my daughter told me about her new invention. She had figured out a way to mechanize the production of what she calls "tape rope". A month or so ago she twisted it manually with her fingers for hours and got some big blisters. Today she harnessed the turning action of the electric pencil sharpener to help her twist.
Here's what she had to say about quality vs. quantity as she spun her tape rope. If you listen closely all the way to the end, I'm pretty sure her final conclusion does not support my argument here, but in the service of a balancing my own view point, here it is:
"The quality [tape] is tighter, the quantity [tape] is looser...but it still works as rope," she says.
I think she's on the side of the commutative property, but me? I'm still not sure what I think.