### Mathematics only in a context

I tried a slight twist on the last laboratory of the summer session. I walked into the room empty handed and then began an abstract lecture on site swap mathematics.

I deliberately avoided placing the notation into a context, introducing the topic as an abstract system of symbolic notation and manipulation. Note that I labeled the sites as alpha and beta, referring only to objects by color names. "The purple will move three sites down from alpha to beta, then another three sites from beta back to alpha."

Perdania

I initially put up a 3 and then showed that one could swap two site to yield the 42 seen in the middle of the pattern above. I noted that sums and averages were invariant under a site swap. The students dutifully took notes and worked on drawing the diagram.

Jayann

With the above on the board, I went around the room to orally check for understanding. Seven of the fourteen students smiled and nodded yes when I asked if they felt they basically understood the site swap system. Three were non-committal, but a fourth student simply said, "That does not make any sense. Too confusing." A recheck found that the initial three non-commits were now willing to admit that they had no idea what the markings on the board meant.

Julie-Ann, one tennis ball out-of-shot high above

Part of the above effect is something I have come to see as part of the way of interacting out here - smile, nod, shake your head in agreement, even if you have no idea what the other person is really trying to say or talking about.

Cherish, three ball juggle under control

I then explained that mathematical models, mathematical systems, should never be introduced as abstract systems without a context. I noted my own position that mathematics as a subject should never be taught in isolation. Mathematics should be taught within the contexts within which mathematics arises. Such as physical science class engenders. Data that is related leads to graphs with linear regressions which are expressed as linear equations.

Barry masters three ball juggling, site swap pattern 3

I then revealed the context within which site swap mathematics arises, that the alphas and betas are right and left hands, landing sites for juggled balls, and then illustrated the meaning of the system. The notation now had a context, a meaning. While still confusing, the idea of a site swap could now be physically demonstrated.

Mathematics without context is meaningless. I realize that I am attacking the ivory tower of pure mathematics, a field that will surely point to meaningless abstractions that paved the way for scientific advances. Or argue that their abstractions are beauty in the purest form. There will always be a place for the 0.01432% who professionally engage in mathematics. For the other 99.98568% of the population, mathematics without context is meaningless holds.

The class wraps up with the students attempting to learn a 3, and those who master a 3 attempting a 342 site swap. Juggling is not useful per se from a survival perspective, I would not argue mathematics has to be useful, only that math should be taught from within the contexts in which mathematics is useful.