Yesterday was the first day of class! I’m teaching a class entitled Foundations of Pure Mathematics for first year students. David was along for the first lecture, and will likely be along for more; we seem to be pretty good at improvising a good show.
For the first class, we wanted to hone in on the idea that good definitions are important, since definitions are usually where you start on any mathematical problem. Thus, we asked the students to define addition, and then worked with their definitions until we managed to get something really precise.
The initial definition we got was, “Addition is the process of adding quantities together to get a total.” We pointed out that using ‘adding’ in the definition of addition was circular. The class then suggested changing ‘adding’ for ‘combining.’ Counter example: four balls of clay in one hand, three in the other, and then I smoosh them together to get one big ball of clay! Thus, 4+3=1. This led the students to put the word ‘numerical’ into the definition, This brought us to “Addition is the process of combining numerical quantities together to get a total.”
We then took a vote in the class on whether 4+3=7. Amongst 18 respondents, we got 14 saying yes, zero saying no, and one note sure. From this we concluded that 14+0+1=18, there by combining numerical quantities to get a total. From this it was determined that we needed to write out the details of the ‘process’ in the definition. We did a bit of work on that, got it to something about adding place values (after an interlude in which we decided that we would need all of the fingers in China to perform certain sums), and then were pretty much there.
Then we did some addition of hours (Earth and Mars) and showed that something else would be needed (as 11+3=2 on the clock). The definition was then changed to specify that it was for positive integers, and that was a wrap.
Then we moved on to multiplication, asked the students for definitions, and got two kinds of answers. One was circular, like the original addition definition, while the other involved adding a number to itself some number of times. This one we slightly refined using suggestions from the class, and showed that it even worked fine for our clock-math case, with numbers modulo k.
Throughout, we also emphasized the need to speak up and ask questions when unsure of definitions or what’s going on.
In the evening, David and I played a whole bunch of Dominion, and did a little bit of work, too. I think we’ll need to invite people over for games to keep things interesting!
Inventing Situations. 