## Working on the Holiday

So, one of the oddities about teaching at an independent school is that days off that are taken fro granted most places are seen as prime days for campus visits here. So, we were in session today. No need to feel sad about this though as we have plenty of vacation time as well. Just another Monday.

But…it wasn’t. It was a terrific teaching day and I want to make sure that I make note of it even if only for my own pleasure.

1st Bell – Geometry. I was looking forward to returning their excellent tests from last Friday but first I wanted to dip into our new unit. I passed out paper, rulers, and protractors and gave them a simple task. Draw some polygons. I insisted on not defining what a polygon was and I did not reveal why they had protractors. I tasked each of them with drawing six polygons and I saw some pretty great stuff. Complex, crisply drawn concave polygons. Some students stuck to the middle of the road and drew squares, triangles, etc. Then the fun began. I started asking for definitions of polygons and I framed the question this way: Explain to my 11 year old son what a polygon is. Brainstorming began. I heard about the need for line segments as sides, I heard about the limitation that there had to be at least three sides, I heard conversations about polygons that pointed inside versus those that did not. Someone offered up the word concave and I pressed for a definition. Everyone seemed happy about the inside-ness of some points on a concave polygon. I pressed them not to use the word point for where the line segments met and they offered up vertex as a better alternative. We decided that each vertex needed two (and only two) line segments and one boy suggested that polygons that were not concave (we agreed to convex quickly) should have angles larger than 90 degrees. He backed off of that but I will definitely revisit this idea soon. When he tossed out that idea he was greeted with references to equilateral triangles. Well played. Then the highlight of the morning came for me. I drew a figure on the board that was made of line segments, that had two line segments at each vertex but it was clearly not a polygon. The reason why is that my figure was not closed. One of my students used that exact language and I pressed, again with my 11 year old son in mind, what we meant by closed. Miranda said ‘Imagine it has water in the boundaries. If it’s closed, the water can’t get out.’ I thought that this was a lovely image. I then closed my crazy drawing but blocked off access to some regions while doing so. I was quickly told that the water needed to be able to get everywhere. I hope that this image stays with my students the way that it is staying with me. As we wrapped up class in a blur of vocabulary about quadrilaterals one of my students said to her neighbor, ‘What a great way to start the week. We got to sit and draw.’ I’ll count this one as a success.

Bells Three and Six – AP Statistics. My senior heavy Stats class did not come back from winter break with much of a sense of urgency. I did not want to just launch right into a new chapter on the heels of the disappointing chapter test we had on Friday. My Computer Science colleague had recently shared with me information about Sicherman Dice which are two six-sided dice that are not standard dice but their sum has the same probability distribution as the sum of two standard dice. I presented my students with a  challenge. Describe two six-sided dice that replicate the probability distribution of two standard die. No other directions really. I have a rudimentary handout I gave them and you can grab it here. I fielded questions as they chatted about this problem with their neighbors. Can the die have negative numbers? Can the die have fractions? Can the die have zero? I kept replying in the reluctantly affirmative and checked on their progress. Most of them had a pretty logical attack where they would transform one standard die in a certain direction, say subtract two from every face and then transform the other one in the opposite direction. Not very sophisticated, but it was nice algebraic logic. One student was working on fractions trying to balance combinations of 1/3 and 2/3 so that she would always get integer answers. Overall, it was the most focused energy from this group that I have seen in two weeks. I hope that this is an omen for our next unit. They were pretty surprised by the reveal and I am curious, in retrospect, that they never asked if any die could have repeated values as this is necessary for the Sicherman Dice to work their magic.

Bell Seven – AP Calculus BC

We’re just getting ready to start exploring the magic of Taylor Series. We took baby steps today reminding ourselves of the language of arithmetic and geometric sequences and series. I always think that this material is such fun to untangle. Tomorrow we’ll play with GeoGebra and I will try to tease out of them the key ideas about how to make a polygon behave like the sine function. Nothing much else to report here.

Tomorrow three of my four classes will be taking unit tests. I have always devoted the class day before a test to review. Over the past 5 – 7 years I have become more and more insistent that a review day should be a day where I am here to answer some questions that students come to class with and to help facilitate some meaningful conversation between my students. What many students seem to believe is that review day before a test is simply a time for me to tell them exactly what will be on the test. I always come to class on these days with some prepared questions in my back pocket and I always dream that those questions will stay there. That is not often the case, and it certainly was not the case today.

My Geometry class, the one I’ve been SO proud of recently, was in pretty good shape. We looked at our last HW together, they had some good questions about that but they could not really generate too many meaningful questions of their own. I displayed the review questions I had prepared and they perked up and were terrific in joining in the conversation. I just came away wishing that the class had been more about them and what was on their mind. In retrospect, perhaps it was exactly about what was on their mind. They are concerned about what I am interested in right now so that they can glean some important clues about preparing for tomorrow’s test. Sigh…

My two AP Statistics classes are in a different place emotionally than my Geometry class is. They are almost all seniors and the energy level that they brought back from winter break is distinctly different than the energy level I see in my Geometry students. I gave them class time yesterday to work on their own or with their neighbor on the review exercises at the end of their most recent chapter and my observation is that there were relatively small pockets of productive conversations. However, there were also quite a few incidents of aimless chatter, obsessive checking of their phones, silly debates, and general non-statistical conversations.

So, I feel that I am asking myself the same question I asked myself on these pages just a couple of days ago. How can I be less helpful in the standard sort of hand-holding way that my students want me to be while actually being helpful to them in modeling smart behavior about how to work, how to be metacognitive, how to be reflective, and how to be more self-aware. Trying to recall who I was when I was in high school is probably not the best exercise in answering these questions. I was a different person then than I am now. I am remembering through a distinctly tinted memory lens and I am not teaching four classes of teenage Mr. Dardys.

Gotta keep thinking and keep pushing.

## The Perils of Being ‘Less Helpful’

i am guessing that most people who will read this are familiar with Dan Meyer’s TED Talk. When I first saw this it crystallized some ideas that had been festering for awhile. It also articulated some thoughts about my evolving practice as a classroom teacher. I have shared this video with my department colleagues and have also shared it in class with students. As I have mentioned before, most of the teaching I have been doing at my current school (I arrived here in 2010) has been with AP Students. They are pretty well equipped to deal with the sense that I am being ‘less helpful’ with them. Of course, I want my students to feel supported, but I also want them to be pretty self-sufficient. We have a feature in our schedule where school ends after eight periods ( we call them Bells ) and every teacher is expected to be in their room for Bell 9 which is a time for students to come for extra help. We call it conferencing here. This background sets up my post for the night.

i blogged a number of times last week about my Geometry students and the success they had this week. I also shared the news with my Geometry colleagues. I teach this course with two terrific colleagues, but we do not have very similar principles about a number of items. When I popped in to visit one of these colleagues last Friday she said she was happy to hear that the quiz results were so good and she said that she had four of my students in her room Bell 9 on Thursday and she had to ‘talk them off the ledge.’ I’ve been sitting on this for a few days because I am not at all sure what to feel about this. There are, of course, no rules about who you can visit for conference help. I was working with someone Thursday afternoon and missed an brief, but important, meeting that afternoon, so I know that these students did not go to my colleague because I was not available. I have to think that they went there because they thought it would be more helpful to ask her questions. I honestly believe, I REALLY do, that it is vital to hear other voices explain ideas. This is why my room is set up in the form of two big conference tables. I also wrote last week about the great work they did in talking about HW with each other last week. I fear that my students did not feel that asking me for help would be very helpful. I am not at all sure of how to proceed here. It has been awhile since I taught a big group of students this young and I have spent so much time with AP kids that I fear I am developing a reputation as someone who is less than helpful. It’s kind of ironic that I tout the virtues of this video and share it with students and I am now bothered by the idea that this image might be following me a bit. I know that my Geometry colleagues have a different attitude about the type of help that students need/deserve than I do. I have to figure out how to reconcile my beliefs about how to teach and how to answer questions with my belief that it is okay – even necessary at times- to hear another voice explain things. I just wish my stupid ego did not get in the way.

Any wisdom out there for me?

## Success!

I have had a very active blogging week thinking about (and writing about) my Geometry class. I have three preparations this year, AP Statistics, AP Calculus BC, and Geometry. I’m not proud of it, but I know that my attention to each class varies at different times of the year. Iy’s not a simple matter of 33 1/3 % of my planning energy being spent on one class at any time. Do many of you go through this as well? By the way, how many preps do most folks have?

Anyways, I blogged in December about my discomfort with HW in Geometry and gathered some nice ideas. I blogged about my decisions about changing habits and it has felt like a raging success. Five to seven minutes at the beginning of class of students sharing their work with each other and correcting each other/reinforcing each other/ sharing their miseries, etc. It’s just been a really terrific week with them and I have let them know how much I appreciate their demeanor, their energy, their willingness to share with each other. Today we had a quiz (you can grab it from here) on Sections 6.1 – 6.3 of our text (you can grab that here) exploring centers of triangles. We’ve talked about perpendicular bisectors, altitudes, medians, and angle bisectors this week. We have played with GeoGebra and looked at how, in each case, all three segments have a common point where they coincide. We’ve talked about which ones could coincide outside the circle and those are not popular choices as the best center of the triangle. We had a great lab activity yesterday (you can grab that here) and it developed into an interesting debate where one group of students nominated the intersection of the angle bisectors as the best representation of the center of a triangle while the other three groups all felt that the intersection of the medians was best. As we had a healthy debate I found myself wishing that I had been clever enough to have physical triangles to manipulate. Next year, I want to be prepared with cardboard triangles of various types with these two candidates for center marked out. I dropped the ball on this one anticipating that everyone would feel best about the centroid. What really impressed me was that the group arguing for the angle bisectors had GeoGebra construct a circle that had this incenter as its center and showed that this circle touched all three sides. I was THRILLED that they thought of this argument.

So, this morning I felt confident as my cherubs asked their last few questions before the quiz and the results are in. I have 12 students in this class and 4 of them earned perfect scores with another 4 earning an A on the quiz. Their class average was 93%!!! I’m thrilled by this. I think that this is due to a number of factors.

• In general, my students have had more energy this week in January than they did in the few weeks leading up to our winter break.
• I believe that the HW strategy has made a positive difference.
• I believe that the extensive use of GeoGebra in class is finally spreading to the home. I have overheard a number of students this week make reference to looking at GeoGebra while doing their HW this week. I am a firm believer in the power of these graphing programs and, for my Geometry students at least, I think that this is the best of the bunch.
• I worked hard during break planning out this unit for me and for my Geometry team of two terrific colleagues. This thoughtfulness has paid off.

Oh yeah, one final thought. As a long-time Calculus teacher I have a strong preference for lines in the point-slope format. Every one of my students presented at least one of their line answers in this format.  Woo-hoo!!!

## On a Roll

Man, my Geometry students are on a roll right now. Today we went through our same new HW procedure again. I was quiet for the first 5 – 8 minutes of our 40 minute class while my students shared their HW with each other. They were asking each other good questions and catching each other’s mistakes. They are still a little shaky at times on their line equation writing skills and their line intersection skills, but the mistakes they are making are much more of the arithmetic and detail type rather than broad conceptual mistakes about what to do.

Today we were concentrating on medians and they seem convinced that the medians should always intersect inside the circle. Last night’s HW (which you can find here along with all our other HW assignments) was on Section 6.2 and they were finding medians and their point of intersection. I also asked them to find the perimeter of an original triangle and the triangle made by connecting the midpoints of the original. This allowed us to do a little noticing and wondering in class together. Everyone seemed pretty convinced that the perimeter of the interior triangle should always be half the perimeter of the ‘parent’ triangle. We displayed this on GeoGebra and I asked them to pick a certain type of triangle that they wanted to explore. One student suggested that a right triangle would be fun so we moved out vertices to the origin, the x-axis, and the y-axis. The interior triangle was still half the size and now the noticing began. They noticed that the smaller triangle not only was also a right triangle but that its acute angles seemed to be the same as the acute angles of the original. They noticed that the four right triangles formed inside were probably all congruent. They noticed that the centroid of the original triangle was also the centroid of the smaller triangle. Then Tara asked about yesterday’s peek at angle bisectors and whether they would meet where the medians met. I asked if there might be a special triangle they could think of where this is true and Miranda guessed that our favorite right triangle, the 5 – 12 – 13 triangle might be special enough. Sadly, it was not, but I was happy to hear a quick guess at this familiar old friend. Then Julia suggested that an equilateral triangle might fit the bill. I worried about how to manipulate our given GeoGebra sketch to match up and she cleverly told me to start a new screen with a regular polygon. Class concluded by seeing that GeoGebra was confirming that Julia’s guess was correct. What a great 40 minutes! I also made a point of telling them that they were on a roll and I hope it carries over to our GeoGebra lab day tomorrow. This is called Chapter Six GeoGebra activity in the dropbox file I had the link to above.

I should have dwelled a little more on my second stats class yesterday. I was really pleased with the three different formulas that those four groups generated. I was especially intrigued by the group that decided that the minimum number in their sample plus the maximum number in their sample should be a good estimator for the true max in the population. I discussed this idea with my morning stats class and we had a pretty vigorous debate over how appropriate this was. Playing with our TI and drawing random samples of 5 from a group of 342 (kind of like the German Tank problem!) convinced them that this technique actually turns out to be pretty accurate.

It’s easy to actively blog when it’s fun to relate what’s happening in class. I hope I can keep up a reasonable pace, if not daily, for the year.

## Beginning New Habits AND a Fun Activity

This morning in Geometry I started by not talking for the first five minutes while my students shared their HW with each other. They talked about their answers, they puzzled over why/where they differed and they talked about using GeoGebra on their own to explore the intersection of perpendicular bisectors of triangles. I was SO delighted I almost wanted to call the rest of the day off.  I did not, though and I’m glad I stuck it out.

We looked together at GeoGebra, reviewed (again) how to find the intersection of lines, we let GeoGebra confirm that we were right. We remembered from yesterday that these lines coincide on the hypotenuse for a right triangle, in the interior for an acute triangle, and outside of an obtuse triangle. After playing with another GeoGebra sketch we all agreed that this behavior made this point of coincidence a pretty poor candidate for the center of a triangle. I pointed out that one of our students had suggested – on his way out of class yesterday – that we should concentrate on vertices rather than midpoints of sides. Again, we let GeoGebra take over and looked at a compromise by constructing a line through a vertex AND through the midpoint of a side. I named this for them as the median. I also displayed that these medians seem to ALWAYS intersect in the interior of a triangle and I named this point for them as the centroid. We all agreed that this name was ‘center-y’ enough. As time ran out, at the suggestion of another student, we asked GeoGebra to construct angle bisectors. It does so, but draws an exterior line as well. They did not complain when I erased them, but I want to examine what is really happening there. It felt a little too much like I was waving away a distraction. We saw that these angle bisectors intersect in the interior as well – setting up a great debate for tomorrow about which center is the center-est. Just thrilled with how they hung together during the intro time and during the quick GeoGebra exploring. Need to commit to both HW review time tomorrow and to revisiting the blur of activity on GeoGebra. I am planning on a lab day for Thursday so that they can manipulate these ideas themselves.

In my AP Stats I tried out the German Tank problem using resources found here at the Stats Monkey site. My two classes dealt with this in pretty different ways. My smaller class (12 students today) worked in 3 groups of four. I made a mistake in responding to one of the first ideas I heard. One group decided to invoke the empirical rule and guessed that the # of tanks was their sample mean plus three standard deviations. I responded positively to them but this simply steered the other two groups into following this lead. In my other class I was smarter and quieter. Here I had four groups of four. One group invoked the empirical rule but they also pooled their three samples together. One group used the inverse normal function on their calculator seeking a point where the area was 0.999. One group added their sample minimum to their sample maximum guessing that they should be (roughly) equidistant from the extremes. The final group doubled their median guessing it should be halfway to the max. I was thrilled with the level of discussion and the variety of responses. A great step forward from yesterday’s disappointment where they largely ignored my Radiolab assignment.

I’ll count this day in the victory column for sure.

## First Day Back – A Tale of Triumph and Sadness

My first period class (we call them Bells here, rather than Periods) is my Geometry class. I started by sharing with the the NYTimes story about ‘The Interview’ and was pleased that they quickly attacked this as a system of equation. I had a secret plot for starting with this problem. We are getting ready to explore triangle bisectors of various sorts. I started out with this question for my students: ‘What does it mean to call a point a center for an object?’ Luckily, this prompted a quick recollection of centers of circles along with a nice attempt at remembering a sound definition for a circle. I then asked them to consider what would be the center of a square. One of my students, a freshman named Matthew, quickly proposed that the intersection of the diagonals would be his point of choice. I opened GeoGebra, drew a rather random square and tested Matthew’s idea. We saw that this was in fact equidistant from the vertices. I then asked about distance to the sides. This required a quick conversation to remind them of what we mean when we talk about distance from a point to a line or to a line segment. We quickly came to an agreement that the perpendicular distance was what we wanted. GeoGebra confirmed that this ‘center’ was equidistant from the sides of the square, but I pretended to be troubled that this second equal distance was not equal to the first equal distance. My students quickly overruled me and were comfy with this point as the center. Next, I asked what the center of a triangle might be. I had three students each volunteer and ordered pair as a vertex of the triangle. It turned out that they formed a right triangle. We agreed the idea of perpendicular bisectors (which we had JUST looked at for the square!) was the way to go. Some quick GeoGebra showed that these lines coincided at the midpoint of the hypotenuse. I was pleased that this raised questions. No one jumped to the conclusion that this would always happen and a student named Tara quickly guessed that this was happening due to the original triangle being a right one. I then moved one of the vertices so that the triangle was acute and, happily, we noticed two things together. First, the perpendicular bisectors still coincided as I moved a vertex. Second, they coincided inside the triangle. Matthew then asked to see what happened with an obtuse triangle and we saw the point of coincidence migrate outside the boundaries of the triangle. It was great to notice that they still met at a point, but the idea of a ‘center’ being outside the triangle did not make anyone happy. Matthew observed that this point did not feel very ‘centery’ to him. Awesome stuff. Finally, since we had GeoGebra to confirm our work, it did not seem that intimidating to go ahead and find the coordinates of the point of intersection for these lines. My secret plot of having them think about systems of equations at the beginning of class paid off. Overall, a wonderful way to start the new year. Tomorrow, I’ll try my idea of HW review at the beginning of class and see how that feels.

Unfortunately, the feeling of triumph dissipated quickly. I have two Bells of AP Stats this year and I had asked these students to listen to a Radiolab Shorts episode called Are We Coins? and I gave them a handful of question prompts. I asked everyone to jot down some reaction notes and to bring these notes to class today. In my first class of 11 students I had three who showed clearly that they had listened to the episode. I had zero students with notes. I asked everyone to take out their notes and a number tried to fool me by having a notebook in front of them, but none of these had anything to do with my questions. In my next class of 16 students, three of them had notes and one or two others showed clear evidence of having paid attention to my request. Sigh…

I’ll dwell tonight on the Geometry kids instead and get ready to really dig into this idea of ‘center-ness’ for a triangle tomorrow. A couple already asked, on their way out of class, about using the vertices as anchors instead of midpoints. Should be fun tomorrow. Lots of noticing and wondering and a concession on my part to their need for HW reinforcement. Hoping for another great start to a day.