- the rear break can be useful, but only when it's very slippery, eg. in snow and ice,
- the rear brake contributes very little stopping power even at low speed,
- the convention of having the left hand control the front brake lever is important because it's consistent - it avoids nasty surprises, and
- it's very important to brace yourself against the handlebars when braking hard.
I'm pretty sure the last point was not addressed in my CanBike courses - it'll go into my own notes for sure, if I ever teach a course. The winter 2008 issue of Bicycle Quarterly addressed brakes in detail, and is described on Jan's blog. His complete response to me is well worth reading. Here it is:
The reason the rear brake does not contribute even at low speeds is that almost all the weight is transferred to the front wheel. Furthermore, focusing on one brake only is easier. When you brake truly hard, the rear wheel will lock up even under very little braking, and the rear wheel will slide, which makes controlling the bike difficult. It takes a lot of slipperiness to make the rear brake useful - snow and ice, or rain after a long dry spell are common scenarios. There, you cannot brake hard anyhow, and the weight transfer is limited.
We discussed the issue of which hand should control which brake in our "Brake Special" Vol. 7, No. 2. I don't think a general recommendation is needed, but consistency is useful, so riders don't accidentally actuate the wrong brake.
The most important result of the test for us was how hard one can brake with correct technique. We commonly got the smell of burnt brake pads rising from the front brake after we came to a stop. The risk of going over the handlebars only occurs at very low speeds, because at higher speeds, wind resistance pushes the rider backward. The rear wheel rises slowly, and riders easily can open the brake to counter this. Most "over-the-bars" incidents appear from the rider flying forward as the bike slows down, similar to car occupants flying forward into the windshield if they don't wear seatbelts. Bracing yourself against the handlebars is crucial during hard braking. That is perhaps the most important part of the technique to be taught.