Sunday, 26 August 2007

Science Sunday IV: Fun with Numbers!

When I first came to the UK, it seemed that the world worked differently. Ever so slightly, the laws of physics seemed altered. The weather seemed to be determined by different laws. Household appliances functioned through different properties. People's social understandings and interactions appeared out of phase. In ways that are hard to describe, I seemed to be in an ever-so-different parallel universe.

Of course, both the UK and US societies had developed different techniques and methods of doing things, ignoring many of the other potential options. We choose some paradigms and forget others. In some ways, this is so fundamentally obvious, it is banal to speak of it. But yet, when it is experienced first-hand, it still seems bizarre and surprising.

The same principle applies to our understanding of numbers and mathematics. Our schemas and paradigms of what numbers are, how they function, is very limited. I have already written about our common shortcoming in understanding repeating decimals, and this week, we have a video of how any multiplication problem can be solved with pencil and paper using only line drawings and no calculations:



3 comments:

Rudy said...

This is fun, but multiplying 999 times 999
looks like it would get pretty tiresome...
And isn't counting all the intersections "calculation"? Does only multiplication count as calculation?

Multiplication corresponds to "cartesian product of sets", a pretty easy operation. Adding sets is not nearly as intuitive; you have to provide labels to indicate which set the result came from ("disjoint union"; ordinary unions might smush together two copies of the same element).
So, multiplication is even easier than addition, at least for sets. Though not for collections of very small rocks; I guess that's why set theory came later...

Are you at U. of Exeter? My wife went to college there; we met at grad school here
in North Carolina.

Christopher Chandler said...

I find numbers to be fascinating stuff, but I must admit I'm no mathematician. I suppose counting does technically fall under "calculation", but most lay people would not consider counting to be a mental calculation. The cognitive processes are quite different.

And yep, I'm at the University of Exeter.

Question: Have you been a part of SAYMA? If so, we might know each other. When I lived in Georgia I was active with the Atlanta Monthly Meeting for a number of years.

Rudy said...

No, I live in Cary NC, near Raleigh. I once visited Atlanta meeting during a trip to Auburn for a conference, way back in the early 90's, I remember that the subway runs under the meetinghouse. But that's the only time.

Alexandre Borovnik, who has a fascinating online book, "Mathematics under the Microscope",

http://www.maths.manchester.ac.uk/
~avb/micromathematics/MM_0.918.pdf,

talks about the fact that we don't multiply time, but add time intervals or compare their lengths.

It also seems to me that multiplication is basically spatial, as in calculating areas.

So then it shouldn't be surprising that the two operations are cognitively so different.