# Lovely Explanations

I have not been writing here as much as I want to because, for the second year in a row, I am creating daily HW problem sets on the fly for a new class. This summer, with about two weeks left before school began, I found out that a colleague had taken another job. She had a fantastic opportunity that she felt she could not pass up. One of the side effects of her decision was that I inherited an extra section each day and I inherited a new course – Discrete Math – meeting twice a day. I am still feeling my way through it but I have a wonderful group of students. One section has seven students and the other has eight. We have been sitting around a pod of desks together sharing ideas. I think that this has been a bit of a culture shock for some of the students but I am thrilled with their flexibility. This elective was introduced last year and we are feeling our way through it. The students in here are not necessarily great math fans and I wonder whether many of them are used to feeling that their mathematical opinions are particularly pertinent. I have been working hard to develop an atmosphere of trust where the students are willing to share their opinions, their answers to questions, and, most importantly, their questions for each other and for me. I have been thrilled by the level of engagement and today, in each class, students offered explanations to a probability idea that just knocked me out. We are just introducing the ideas of compound probabilities and talking about how to interpret AND versus OR situations. For example, I offered an example telling the students that approximately 10% of American adults are left-handed. I asked what was the likelihood of two randomly selected adults both being left-handed. Students quickly offered the idea that 10% of 10% was the way to go. A different problem outlined percentages of blood types and mentioned that type B people can accept donations from type O or from type B. Here, the students quickly offered addition as the way to work through the problem. I congratulated them for seeing this quickly, but challenged them as to why they had this gut feeling. One student in my early class said that he thought of the two left handed people as one event where one thing happened then another, while he thought of the blood type question as two ways an event could happen. I was so pleased to here this developing intuition. In my later class the same scene unfolded and here a student simply offered the idea that two blood types gave you more opportunities so we want to combine them in a way that increases the result while the left handed question seemed to necessarily narrow the likelihood of being satisfied with the result.

I find both of these explanations to be much more appealing than tree diagrams or a simple rule that AND = multiply while OR = add.

So pleased with how the class is unfolding. I just wish that the problem sets would write themselves…