## Beautiful Problem Solving and Odds and Ends

While most of my colleagues enjoyed a well-deserved day off in honor of Martin Luther King, Jr. we were at work here in our boarding school. We take advantage of these days as visitation days and we keep on counting the days of the year.

Last week I wrote about my frustrations with trying to find a way to help keep my students more aware of the benefits of daily practice in Geometry. This weekend I engaged in a lengthy and mind opening twitter conversation with Elizabeth (@cheesemonkeySF) and my mind is still buzzing with ideas. I noticed something today that I may be able to take advantage of. Tomorrow we have our next Geometry test. This is the second year that my school is using the Geometry text that I wrote. This means that we are still working our way through the strengths/weaknesses of the text and we have a storehouse of documents to draw upon. I decided earlier in the year that I would hand out last year’s tests as practice a few days before this year’s test over similar material. So, last Friday I gave a copy of the test from last year that covers through Chapter Six of our text. Today in class I saw more evidence than usual of HW completion. So, when the HW feels particularly helpful then my students are more likely to complete it. Pretty logical, right? What I need to do then is to make sure that I can get buy-in like this more frequently. I have a batch of quizzes from last year that I can easily give out mid chapter as weekend HW that both serves as a sneak preview of the kind of quiz questions I was interested in asking last year AND serves as good, focused practice that feels to my students as if it has more payoff. This will not solve all of the problems I have been wrestling with and I need to sort out Elizabeth’s sage advice and figure out how to incorporate it in a way that fits me, but this feels like progress. I am happier about Geometry than I was last week and I am optimistic about tomorrow’s test. I hope that I will be able to report on student success.

Last week I also wrote about a problem posed to me by an alum when he was visiting. I may not have reported the problem accurately, so here is a second take. One hundred people are lined up to board an airplane with 100 seats. Each person has one seat assigned. The first person boards the plane and randomly chooses a seat. After that, each person who boards will sit in his/her assigned seat if it is available. If the correct seat is not available then that person will randomly choose a seat. What is the probability that the 100th person will be able to sit in the correctly assigned seat? I broke this problem down after one of our boarding community dinners last Thursday and a colleague and I simplified it to two people (50% chance, no surprise!) and then three people. With three people – call them A, B, and C – the seating arrangements are ABC, ACB, BAC, BCA, CAB, CBA. Two of these arrangements have C sitting in the third seat and for the purposes of this permutation, I am treating that as the ‘correct’ seat. However, the arrangements ACB and BCA are not possible under these rules. If person A does not sit in seat B, then person B is obliged to sit in his correct seat. So we have two of four possibilities for a 50% success. This seems pretty suspicious and I try to sort out the arrangements with four people. I won’t bore you with the detail but this is also 50%. When I mentioned this problem to a number of colleagues one of them mentioned that her son had talked about this problem from a math competition. Her son is in my AP Calculus BC course and he is an extraordinarily talented mathematician. He explained the problem this way in class today and I probably will not be as elegant as he was. Here is his take:

By the time that person two sits down on the plane we know that his seat has a person in it. Either it is person one and then person two chooses another seat or his seat was available and he sat in it. Similarly, by the time person three sits down we know that someone is in person three’s seat. Either person one or person two is accidentally in that seat or person three sits in her proper seat according to the rules of this problem. We can extend this argument all the way to person ninety-nine. Now, we know for a fact that all seats from person two’s seat through person ninety-nine’s seat are all occupied. The only mystery is whether the other occupied seat is the first person’s seat or the hundredth person’s seat. It is not a stretch to see that these two possibilities should be equally likely.

What I LOVE about this explanation is that it does not rely on combinatoric wizardry or thorny algebra manipulations. It also make crystal clear sense once it has been explained but it did not make crystal clear sense before that. It seemed completely unreasonable to me that, with so many people involved, the answer would be so clean. In fact, my student’s explanation made it clear that the number of people on board is a complete red herring. It might as well be one thousand people instead.

While I might have enjoyed the day off, I also enjoyed the day on.

## Geometry Progress Report

I have a couple of posts that I want to make. It might be a busy weekend between writing midterm comments and airing my thoughts here. I promised to report back on my grand experiment with lagging HW. Now that we are three weeks into the term I think that I have some meaningful observations.

My first observation is that I need to find some meaningful way to regularly incorporate HW so that my students feel that it is a meaningful exercise. I think I am making strides by writing problem sets that reflect my book and our class conversations. I think that I have written problem sets that strike a decent balance between practice and challenge problems. I have been making class space for conversations about the current topics and trying to create some space for simple practice and check-in with some entrance slips. However, it is becoming pretty apparent to me that too many of my students are not in the habit of doing their HW on a daily basis. When we check in on HW at the beginning of class there are plenty of empty desktops and too much silence. It also seems clear to me that these are old habits and the reason I say that is that MANY of the problems they are struggling with now are related to writing line equations. Since we are juggling perpendicular bisectors of triangles, altitudes of triangles, medians of triangles, and angle bisectors it is kind of essential to be able to work with line equations. I know that these are skills that they have had and have displayed, but if the practice was not put in originally, those skills do not settle in and stick very well. I am reluctant to grade HW for a number of reasons. If all I am doing is checking for completion, then I feel I will be often rewarding sloppy and incorrect work and possibly helping some bad habits settle in. If I collect and grade it based on correctness I fear that I will be encouraging students to take some dishonest shortcuts. Instead, I am trying to use the entrance/exit slip idea to encourage attention during class with the hopes that that attention and the reminders of the skills necessary through the entrance/exit slips will (a) make the HW easier when it rolls around about three days after the class discussion and (b) allow me (and my students) to realize what they do or do not know.

My second observation is that this idea of HW lagging behind instruction will take some time for my students to get used to. They have been SO accustomed to trying their hand at something as soon as they begin to think about it and this new pace feels very different to them. I think that the old habits are working against them as they have expressed more confusion on some of the problem sets than I saw last year when I was using these HW assignments and assigning them the night that we introduced ideas in class. This, again. is something I need to address. I need to figure out how to help coach my kiddos to be able to deal with this process. I am too convinced that this is the right way to do this. Reading about it, thinking about it, I am sure that this is the right thing to do. My first time checking in on their progress right now (on this quiz on Tuesday) was a bit of a disaster. There were a number of scores hovering around 50% and for each of those students I returned the quiz with a practice assignment on writing line equations. I am trying to be positive and emphasizing that they know how to do this. I am convinced that this is true but I saw SO many mistakes on the quiz that it was a bit disheartening.

Conclusions? As I mentioned, I am convinced that this is a good way to weave in review, encourage reflection, and try to embed knowledge more deeply. I just need to figure out how to help coach my students so that they can realize the growth that I want to see for them.

## Progress?

This post is a few days late due to, well, you know, life getting in the way. When I last checked in with you here I was preparing to put into action a plan to lag my homework with Geometry and have a series of HW assignments incorporating more review of past skills. My kiddos took their first quiz of 2016 yesterday and I plan on grading them tonight so I will have some data to back up (or refute) my reflections at that point.

What I have noticed so far:

• Review assignments – at least the ones I have written – make my students pretty frustrated. I certainly do not want frustration to be the go to emotion for my students when I ask them to work, but I am willing to have that as a  stepping stone if I can help usher my students into a place where they are more comfortable with problem sets that do not depend on a small set of skills and ideas. I realize that I am combating years of habits and expectations.
• We have more time to practice some of the new(ish) skills that I am hoping that they develop. We spent days in a row visiting some linear equation writing skills and some  ideas about linear combinations.
• When we did  finally get to HW concentrating more on single sections of the text I was not receiving quite as many questions as I had been expecting. This I am taking as a positive sign. I will be more convinced that it is a positive sign if I see some stability on their quiz. We spoke briefly about the quiz today and I suspect that I will see some hesitant work. If the mistakes are more algebraic in nature I will feel better about the development of their Geometry skills and ideas.
• We took a day in our lab to play with GeoGebra and I realize that I have to do something more consistent next year to encourage/require my students to engage with GeoGebra more frequently. What should have been a productive activity drawing some connections about the ‘centers’ of a triangle that we are examining, too many of my students were either distracted playing with zooming in and out on various images or they were flummoxed by some of the commands that we ‘learned’ in the fall. One of my new colleagues and I are brainstorming ways to make check-ins with GeoGebra a regular and meaningful part of our life in Geometry.

Part of the way that I am organizing out of school work right now is by asking my students to read based on class discussions while they do not practice those specific skills for a few days. I am not at all convinced that they are reading as I request, but I also do not think that they were doing to reading under our old structure either, so that is a wash. I will write again tomorrow after I grade the quizzes and I will check in to see if the data backs up my observations in any meaningful way.

## MTBoS New Year’s Resolution

If you have not seen my Geometry book yet and want to take a look, you can download it from my Dropbox at this link. If you want to look at my Chapter Six pacing calendar, entrance slips, and HW assignments, you can find them all in this Dropbox folder

I am hoping that January will be a productive month for this blog space as I reflect and report on how this experiment unfolds.

## Brief Notes on a Good First Week Back

I wrote already about the frustration one girl expressed during the spaghetti exercise where she wanted A right answer to the exercise. I took that opportunity to talk about different approaches, to try and emphasize our desire for efficiency when we can find it, but, more importantly, my desire to hear their voice and thoughts not just an echo of my voice and thoughts. They get too much of that from adults already. Yesterday as we reflected on the exercise two girls shared really interesting observations. One said that when she was inside the fishbowl (I was outside at the time) she felt really anxious about saying something out loud that might be wrong. She said she was more relaxed when she was outside but she felt she understood definitions better when she was inside. This is HUGE. This kind of self-awareness is so important. I asked her to think about that and think how she can use that realization moving forward in our class. I hope that she decides that she understands better when she is more actively engaged in the conversation around her. The other girl remarked that she knew that she understood better when she talks and I seized on that and challenged her to make talking in class a real commitment.

It’s been fun to be back – our school’s last full day of classes before this week was November 12. I appreciated the rest (other than grading finals – a post for another time) and I am glad to finally be prepared in advance for all three of my courses, but I sure did miss the interaction of the classroom and I have been thrilled with how my Geometry students (my youngest class) have come back ready to go. I have asked them to deal with different situations than they normally do and they played along beautifully. I am so pleased and I hope that we have made some important points about our time together. I also hope that I can hold myself to the most important lesson I learned this week. Unfortunately, it is one I have ‘learned‘ numerous times – my students are better off when they speak more and I speak less. I need to make this my mantra – especially if I want to effectively integrate some other changes in my classroom in the upcoming new year. That’s right, I do not want to let myself wait until August, 2016. I want some serious changes as of January, 2016.

So, our school works on a trimester system with Thanksgiving Break (a full week) marking the end of the fall term. We also have fall term finals, so my last full day of classes was November 12. I set myself some lofty goals for the break and met about 80% of those goals. My number one goal, by far, was to do what I could to plan out our next fourteen days for all three of my preps. We have fourteen days of class until the long winter break begins.

I found out late in the summer that I was teaching a new course (around August 10) and I also have two brand new colleagues in my  department. I have not been able to spend as much time mentoring them as I had planned to. The combination of this disappointment, along with perpetually being only a few days ahead of my Discrete class made the fall term a pretty stressful one. I have three preps, five sections, and my chair responsibilities. Luckily, I have a pretty light student load this year.

So, I have my calendar mapped out for Geometry and AP Calculus BC and I have about ten of the fourteen days of Discrete taken care of. Overall I am pretty pleased. Add in the naps and the time with my wife and kiddos and it has been a good break with just enough productivity thrown in.

## Catching Up with the Past Week

So there are a couple of activities this past week that I want to write about. However, I have been swamped with meetings so I have fallen a bit behind.

In AP Stats we have finished our required curriculum as of 8 days ago. I am a big baseball fan and my favorite team is the New York Mets. They are having a pretty wonderful start to their year (or at least were until the last few days) so last Friday I posed the following question to my kiddos: Given that the most optimistic projection I saw for the Mets’ season had them pegged as an 87 win team, what is the likelihood of their current record (which, if I remember correctly) is 10 – 5? I liked this for a few reasons. First, it concerns baseball and likely would have a positive outlook for my Metropolitans. Second, it was not so focused on the most recent material at hand. My Stats students tend to know recent material well but struggle remembering other procedures that have not been practiced as recently. Third, it generated some nice thinking out loud about what approach to take. Being more of an algebra stream guy myself I immediately placed this in the context of a probability problem and was prepared to go down a Pascal’s triangle/binomial theorem path. Most of my Stats students don’t tend in this direction so their conversation focused instead on comparing proportions – the 87 – 75 projection with the 10 – 5 proportion. They suggested running a two proportion z test and looking at the corresponding p-value. This opened up the avenue for me to sneak in my approach and make a connection pretty visible to them. Turns out that we felt that we had enough evidence to reject the null hypothesis of the Mets being an 87 win team in favor of believing that they will exceed that win total. Their recent 5 – 5 run of games might adjust that but I do not want to know this – so I will not re-run the test right now! After we checked our trusty TI to find the p-value of this test I reminded them of the probability approach and we set up the appropriate term of the binomial expansion. Guess what happened? This calculation matched the p-value of the two proportion z test!!! This is one of those ideas that we discussed but somehow seeing these results side-by-side seemed eye opening to my kiddos. A triumph on a number of levels!

In my morning Geometry class we dipped our toes into an exploration of radians yesterday using the ProRadian Protractor designed by the fantastic Jennifer Silverman (@jensilvermath) and using an activity that she designed. I wrote a follow up HW assignment that my kiddos worked on last night. I also linked to a fabulous web site that allowed my students to explore radian measure and I shared these notes with my colleagues. There is also a lovely GeoGebra applet (also designed by jennifer Silverman) that is linked from the worksheet. I was totally excited to explore this activity with my students and I had a really nice chat with one of my teammates.

I handed out the radian protractors as well as our regular old angle protractors and we had a nice conversation about similarities between the two protractors. We had a lovely discussion about this but, looking back on yesterday , I think I allowed too many clues to seep into the conversation too quickly. Jennifer’s activity is a terrific one and I got in the way by loading too many conversations in at the beginning of the class. By having students come to my screen and try to identify where one radian measure would lie on the circle AND by having the protractors side-by-side I reduced the mystery element that I think should have been part of the classroom activity. I think I took away the opportunity to discover what was happening here. I did have one student give a GREAT explanation of why the quadrilateral radian measure was twice the triangles radian measure. She invoked a proportional idea and referenced our (n – 2)*180 formula. I had a number of students quickly see that the ratios we had been working with before ($frac{x}{360}=frac{arc}{2pi r}=frac{sector}{pi r^{2}}$) could be easily extended to add one more simple fraction of $frac{x}{360}=frac{theta }{2pi }$. That definitely felt like a triumph. So, the lesson I learned here – and I hope I remember it for next year – is to be a little more minimalist in front loading this conversation. I think that we can touch on all of these resources and really let the discovery sink in, but I feel I nudged them a little too much this time around. So the plan for next year is to hand out the radian protractor and work through the worksheet. Then hand out the angle protractor and talk about comparing them. Then, the next day after some time to think, show the web app and have them identify where one radian is. Let this unfold a little more slowly.

## A Quick Geometry Snippet

So, we were reviewing this morning in Geometry getting ready to finish up our circle unit. I was reminding the kiddos of the angle and arc relationships we have been discussing. I had been writing things like $x=frac{1}{2}left ( a+b right )$ where x is an angle formed by the intersection of two chords and a and b are the measures of the intercepted arcs. However, today as I drew my diagram for it one of my students suggested I mark is as seen below.

I think I am delighted by this and will write it this way from now on. It feels to me that it is a more natural way to think about this relationship instead of having a coefficient of 2 or a coefficient of 1/2 that might not seem at all intuitive. I then drew the following

I’d love to hear from some other teachers about whether this seems at all like an improvement over the more standard way of writing these equations.

That’s all for now, just needed to get that off of my chest!

## Highlights of a Stressful Week

So, there have been many scattered thoughts on my mind in the past week but there are also three things that happened that are just completely awesome.

1. My pal John Golden (@mathhombre on twitter) steered a number of his teacher training students over to a post on my blog and twelve of them chimed in with comments. Totally cool! One of them decided to follow my blog and I took the time to respond to each of them. LOVE the idea of new teachers in training dipping their toes into this rich world of teachers blogging and sharing. I am also flattered that John thought my virtual home here was worth a visit.
2. I woke up Wednesday morning with a message on twitter from a teacher in Louisiana who asked if he could use my Geometry book at his school next year. I am so excited by the idea that this work might be used at another school.
3. In my Geometry class this week we are talking about angle and arc relationships. One of my students stayed after class one day this week and she had this to say. “You know, I was thinking, when will this be important? I mean, when will I need to find an arc length like that? Then I realized that the work we are doing to find that length is what is important. Pretty cool.” Wow.

## Some Fun Geometry Action

On the heels of learning some right triangle trig I am really trying to develop more proportional logic with my students. Just this week we had a really productive conversation about the following problem.

Being a bit of a bull in a china shop sometimes, I proposed that we should find the height of each triangle, find each chord length and find the height of the trapezoid by finding the difference of the heights. Not elegant, I know. I was trying to make sure that we remembered some right triangle trig. that we remembered our area formula for a trapezoids, and that we try to develop some patience in solving multi-step problems. That was my plan, but as with many school plans, it did not quite unfold that way. One of my students who is a bright and quick problem-solver pointed out that simply finding each triangle area would be enough. I understand that his solution is pretty much the same as mine, but it certainly sounds more efficient. But as soon as I acknowledged that his idea was more efficient than mine another student trumped each of us. She pointed out that I had already asked them to consider the ratio of areas between the two triangles. So, if we know one area, we can automatically know the other area. If we know both areas, we find their difference as suggested by the first student who chimed in. I was so happy that she took my clue from within the problem and that she was clever enough to really save time and energy this way. I made sure to compliment her in class and I bragged about her work to two of my colleagues yesterday. Oddly, this morning when we were reviewing before today’s quiz I reminded her – and the rest of the class – of her clever idea. She had no memory of this conversation. Sigh…

I’m trying to process this and figure out what it might mean for my classroom practice. I understand that I should be more excited by my students’ ideas than they often are. I understand that I will remember context of conversations more easily than they will because I am not dealing with the cognitive load of trying to learn/understand the conversation. I am simply coming at it from such a different place. What I don’t understand is how a student can be so in command of an idea but then not remember the creative process that made her arrive at this clever conclusion. I discussed this in the faculty lounge right after Geometry today and one of her other teachers intimated that this might simply be modesty on her part. I am not sure how much faith I put in that reading.

So, while I am a bit frustrated and confused, I am choosing to focus instead on the positive energy of yesterday’s conversation, on the clever ideas that my students brought to the table, and on the fact that my students did a really nice job on their quiz today.