Fast forward forty years to tonight's graduate math ed class in which I have to teach probablity and proportional reasoning. Both are fairly abstract concepts althought they can be made concrete through the use of inquiry-based activities such as tossing coins and drawing. There are, however, some tricky abstract concepts one of which eluded my understanding for several hours today. For some reason I could just not understand, perhaps remember is more accurate, the difference between additive and multiplicative comparisons. I read extensively and figured some problem examples but it just was not working for me and I could feel the panic and fear rising. I even read about how some people cannot think proportionally but I knew this did not apply to me. I finally saw the light, so to speak, when I completed a problem in which a runner starts 6 laps before the second runner starts. If they are both running at the same speed the first runner will always be six laps ahead (Additive comparison) at all times. If the two runners started at the same time and one had completed 9 laps while the other had completed only 3 laps this would be multiplicative comparison. Ahaaaa, sigh of relief!
Why should lack of understanding invoke such panic when the inability to do something such as a back flip is taken for granted as quite normal? Therein lies one of the cornerstones of my beliefs about teaching.