## Problem Day becomes Persistence Day

As I have mentioned before, our school has AP Calculus BC as a second year course in high school calculus. One of the main reasons we do this is that we feel it creates a space where we can explore, we can dig in to deep problems, and we can give our students the opportunity to really reflect in a way that the pace of a one-year (or 1.25 year) course does not allow. I have heard students say things to each other like ‘Last year, I knew how to solve those problems, but I really did not know why I was doing it that way.’ The pace of the AP curriculum creates such pressures that students too often can fall into that trap of just learning how to do something. One of the tangible joys of teaching the BC course under these conditions is that I can routinely take days like today where I just toss a problem set at my students and listen to them think. The problems bounce around – sometimes they are calculus based problems, but more often they are not. This morning our problem set started with this question:

Find the point of intersection of the lines tangent to the curve $y=ax^{2}+bx+c$ if those lines are tangent at x = 1and x = 5

This is not a challenging problem in theory for my students in this course. However, the algebra is a bit snarly and unpleasant. They debated amongst themselves and seemed happy at the end about their work. I stepped up and did the problem on the board, with their guidance, and we arrived at the conclusion that the x-coordinate of the point of intersection was x = 3. Hmmm, I wondered aloud whether there is something going on here that x = 1, x = 5, and x = 3 seemed to have a nice relationship. I quickly examined the boring old standard parabola at the origin facing up and we saw the same thing. One of my students commented that this was probably generally true but these examples certainly did not prove this in any way. I sat back down and got to my work and a few minutes later a student named Richard called me over to show me that he had confirmed that this would always be true by replacing the 1 and 5 from my problem with p and q. His answer, instead of 3, was the average of p and q.

I cannot accurately convey how pleased I am that he was persistent enough to do this. It would have been the easiest thing in the world to accept this conclusion and to move on to the next problem especially since his neighbors had moved on. So he removed himself from a stimulating conversation with his friends to satisfy his curiosity (and maybe to prove something to himself about his ability to push through) about a problem that likely will not have any future impact on his calculus grade. What will have an impact is his increased belief that he can solve a snarly problem. Pretty pleased, I must say that my Friday morning started off this way.

## Sometimes it is in the Sequencing

Note – This is being cross posted a bit belatedly from the lovely website https://betterqs.wordpress.com

You should drop by there if you have not yet

I continue to try and get a handle on the Discrete Math class I inherited mid August. One of the highlights has been the conversational nature of the class. Each section has 8 students and they all sit at one large conference style table. We have been wrestling with probability and there were three questions I asked on their latest HW assignment that I was happy with as I wrote. What I realized during class today was that I will be much happier next year when I rearrange them. The first question was in four parts. The context was that I had tossed a coin four times and I asked the following questions:

• What is the probability that all four are heads?
• What is the probability that all four are heads if I tell you that at least one is heads?
• What is the probability that all four are heads if I tell you that at least two of the tosses are heads?
• What is the probability that all four are heads if I tell you that at least three are heads?

A number of my students struggled to see why the changing information changed the probability. In my first class they dutifully followed my path of reasoning as I drew the outcomes and talked about the answer. It was not until we reached questions three and four that I saw light bulbs go off. In question three I told them that a friend of theirs reported having tossed a coin ten times and seeing a head each time. Their friend tells them that he KNOWS that the next toss will be tails. Their job was to convince their friend otherwise. In question four the conversation continues with their friend arguing that there is only a 1/2048 chance of eleven heads in a row. No way that will happen! Again, they were asked to address their friend’s misconception. While we discussed these two questions a number of students popped up and said that the first question now made sense. So in my afternoon class I swapped the order of the conversation and question one fell into place so much more easily. I am definitely changing this order for next year to help support my kids as they develop their understanding of probability.

## Non-Mathematical Musings

One of the great joys of my life these days have been long walks with my iPod Touch. Last winter I won a Fitbit Flex at a school raffle based on a wellness challenge. Since I got it I have been much more conscientious about being mobile and I am no fan of running so I take long walks. With my iPod earbuds in place I go off and I listen to podcasts. There is not enough time to listen to all that I want to. I subscribe to Radiolab, This American Life, Sound Opinions, The Memory Palace, 99% Invisible, The Moment, and Marc Maron’s WTF podcast. Early mornings, late nights, and long walks are filled with these voices and these ideas in my ear. The newest one that I added was The Moment and one episode in particular caught my attention. Brian Koppelman, the host, had Seth Godin as his guest. Now, I have to admit to being ignorant of Godin but I was intrigued enough that I dug through the podcast archives to find an earlier conversation with him. One thing in this world that makes me especially happy are when I find little synchronicities, little places where ideas converge or where I become aware of something then see it in other places. Well, one of those just popped up again. As I was in the midst of the podcast with Godin I got an email from one of my former colleagues, a woman named Gayle Allen. Gayle (you can find her here or on twitter @GAllenTC) has been a great influence on me as an educator since I first interviewed with her for the job that brought me north from Florida. Gayle is a dynamic thinker and was a fantastic resource when we worked together. She challenged me to grow, to be more reflective, and to expand my world. She is one of the reasons my blog exists. I feel fortunate to have her as a friend and a wise voice when I need advice. Well, Gayle wrote to tell me that she is launching her own podcast and her first guest is, you guessed, Seth Godin. I am so excited for this endeavor and added her feed to my iPod this morning. I cannot wait to hear her conversation with Godin and I am excited to have her voice back in my ear. You should check out her site and listen in to hear what she is up to on her podcast.

## Beginning to Grapple With Proof

Last year I opened the door to a conversation about proof by adapting an activity from Max Ray-Riek over at the Math Forum. I asked my Geometry kiddos to write out directions for how to make a Peanut butter and jelly sandwich. I swapped directions around somewhat randomly and asked the students to do their best to make a sandwich only following the directions given. It is a fun activity, the kids get a laugh out of it and we have some yummy afternoon snacks. Most importantly, I think that it makes a vivid point about how detailed you need to be at times when trying to tell someone how to do something. I will use this conversation and activity as a reference point over and over in the next few weeks as we begin to grapple with what it means to prove something and how you can explain to someone why you think that something is true. I took a few photos of the sandwich designs that I want to share here. In the first one, try to notice on the paper how scant the directions were. Last year I had such a thorough and detailed description written by one of my students that I posted her PDF document here on the blog. I have yet to dig through all of the directions that were turned in this year, so I do not know if there is a winner in that category again this year. Here are some of the fun photos from the day.

## 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…