**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.

I was re-reading one of my favorite children's books, Karlson on the Roof (it seems to be virtually unknown in the US, but it's very popular in Russia) and came across something that reminded me about this post. There's a part where the little boy is about to celebrate his 8th birthday. His mischievous friend, Karlson, asks what kind of yummy refreshments to expect. The little boy says that there will be a cake with 8 candles. To which Karlson replies that it'd be much better if there were 8 cakes with 1 candle.

ReplyDeleteYes! This is my point exactly!!! :-)

ReplyDelete