Category: Uncategorized

A Fantastic Day of Wrestling with Problems

imageA former colleague of mine, Lisa Winer (@Lisaqt314) tweeted a problem at me last night. Wednesday night I s my basketball night and then I curled up to watch some Netflix with my wife, so I did not see the problem until this morning. It has since been making the rounds a bit. It is called ‘The Hardest Easy Geometry Problem’ and you can find it at https://www.duckware.com/tech/worldshardesteasygeometryproblem.html

I started working on it on my side board today and it caught the attention of my BC students. One of them found a solution using trigonometry and I constructed the triangle in GeoGebra to confirm that he is correct. However, I still have not found a way to solve it geometrically. I reduced the problem to four equations with four unknowns but the matrix is singular and I could get no solution. I did, however, have fun playing with it and watching a number of my students dig in.

In Calculus BC today we talked ourselves into the area formula for regions bounded by polar curves and we had great conversations about it in both of my sections. In each class I had at least one student remember some area formulas for triangles that are rarely used and that help serve as the basis for the integral involved. I was pleased with each of those classes today.

In geometry we are working with quadrilaterals and a recent HW problem presented the students with a parallelogram and some algebraic expressions to deal with. Most of my students made an assumption regarding the intersection of diagonals for the parallelogram. They correctly assumed that they bisected each other. I was pleased that they made this assumption but I made sure that they felt comfortable with an argument supporting that fact and then a series of questions erupted that carried us through the end of the day. Do diagonals bisect each other for all quadrilaterals or just parallelograms. A couple of quick sketches at their desks implied that it was not always true. A quick visit to GeoGebra seemed to convince them. Then a student asked if a quadrilateral could have congruent diagonals if they do not bisect each other. A few more sketches and then the guesses started flying in. It did not take long to guess that an isosceles trapezoid would fit this bill. Again GeoGebra confirmed our guess. What next? How about the triangles for,Ed when the diagonals cross? Are they all congruent? Are they congruent pairs even? Quick feelings that the ‘side triangles’ are congruent but the top and bottom ones are not. Right again! But my favorite part came next. I did not plan on talking about area for a couple of days still, but the moment felt right. I asked if we could deduce an area formula for this trapezoid. Now, last year at this point I had a student suggest drawing one diagonal to find two triangles. Standard and clean. I also had a student suggest dropping two altitudes from the ends of the shorter base. Again, a nice standard solution. I had one student suggest rotating the trapezoid 180 degrees to create a parallelogram twice the size of the trapezoid. Not standard at all, but also kind of confusing for his classmates. This year, I had a student named James make a suggestion I had not head before. He asked me to draw segments from the end of the shorter (upper) base down to the midpoint of the lower base. This created three triangles all with the same height. I took a picture of the sketch we Made on the board. That is the photo on top of this post. I must say that I am completely delighted at this clean and clear way of looking at this area problem.

A pretty good day overall, I must say.

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.

A Day in the Life

Here we go -jumping on the MTBoS Blogging Initiative wave

 

5:30 AM Wake up to hear that Mrs. Dardy is already in the shower so I roll out of bed to grind and brew our morning coffee. I often make a point of going out for a walk or heading to the gym in the morning, but I am kind of dragging after having been out of town for a couple of days. Besides, it is awfully nice to have some peaceful time in the AM with the missus enjoying good coffee.

6:15 AM Start doing some last minute planning for the day. As I mentioned, I was away for a few days and as most of you know it is hard work to get ready to be out of school AND hard work to catch up when you return. Find a couple of good AP FR for Calc BC and I am ready to get going.

6:40 Wake up the kiddos. We usually head to our school’s dining hall a little after 7 for breakfast. On those days we wake the kids just a  few minutes earlier. Today we eat at home and give ourselves a little more time. The bus for my kids picks up in front of my classroom building. I live in a boys’ dorm at a boarding and day school. I am pretty spoiled by walking to work but I pay for it by being Mr. Dardy 24 hours a day on campus. There are some real perks like last week when we had a groups of students from China cooking in our kitchen all day preparing for our International Dinner or the way that I get to know about my students in a deeper way than I used to. I can talk to our boarders about much more than their HW (or lack thereof!) and test grades. So, anyways, we get the kids moving and get their breakfast planned while Mrs. Dardy makes us a lovely breakfast.

7:15 Start brushing Lil Dardy’s hair and getting everyone moving to get prepared for leaving the house on a chilly morning.

7:30 Walk over to the classroom building and stand in the cold air waiting for our crossing guard to call out the arrival of the bus. Some mornings it gets here as early as 7:40 and I get to my classroom with time to think before my 8 AM Calc BC class. Today, it is closer to 7:50 by the time it arrives. I head upstairs to start the day. I open up our computer lab, I fire up my old iMac with my music drive on it, turn on my AppleTV and connect my laptop to it to show off my AP FR questions and I turn on the heater.

Our school has a number of different schedules we run and today is what we call a T Day. Fifty minute class and the day ends at 2:55 PM. Let’s get started…

8:00 – 8:50 AM Bell One – AP Calculus BC

My morning class is a terrific group but they have been having trouble getting to school on time. It is a bit frustrating and today I start with three of my seven students there at 8 but we dive right in and by 8:10 they are all there. We have a great conversation looking together at a free response question on vector functions defined parametrically. We spend about 25 minutes untangling the problem together and conferring about technique. I then flip to another old FR and I sit quietly for about 10 – 12 minutes while they work. This group is really good at sharing ideas and picking each other’s brains. I like listening to them think.

8:55 – 9:45 AM Bell Two – FREE

I split my time between the faculty lounge chatting and running a couple of copies and sitting in my room reading and listening to music this morning. I catch up a bit on my email reading and I eat up some more articles online getting ready for my second prep of the day.

9:50 – 10:40 AM Bell Three – Discrete Math

Yesterday we spent most of our time talking about Powerball odds and financial strategies. Today I circle back to the group quiz they took on Tuesday during my absence. we compare notes and they feel pretty great about their decisions. I graded them later in the day and confirmed their good feelings. We spend some time talking about an unusual voting strategy called preference voting and we have a decent discussion about its benefits and pitfalls.

10:45 – 11:35 AM Bell Four – AP Calculus BC

During this second session of the class I always have to battle myself not to speed up too much. This group of nine students is pretty phenomenal and it is far too easy to just rush through the problems since I already talked about them and this group is quick on the pick up. I fight the urge pretty well this morning and we have a nice leisurely discussion. After working through one together I give them time to work on the second in their group and we get to look at the AP rubric together just in time before the bell rings.

11:40 AM – 12:10 PM Lunch Time

Normally I wait a few minutes, dash over to grab lunch and bring it back to the faculty lounge or to my room. Today a student who graduated two years ago came by to visit so I spent the first 15 minutes catching up with him. He took a summer Geometry course with me and year long courses in Precalculus Honors, AP Calculus BC, and AP Statistics s well. He is a terrific young man and he is engaged in an interesting project related to a Space X challenge. I excuse myself to grab a super quick lunch and another student stops me in the hall to ask for Calculus help. I run over and back in time to chat for about 5 minutes with the Calculus student who is trying to make up for lost time in school and lost time for HW since he was working on a more pressing assignment for another class. We do not accomplish too much in the brief amount of time we have to chat.

12:15 – 1:05 PM Bell 6 Discrete Math

A repeat of the earlier class and I let them out a few minutes early since I am subbing for a colleague next Bell and I need to get my room in some oder for my last class of the day.

1:10 – 2:00 PM Substitute for an English class

I have some time to finish up some Geometry grading while supervising an English class. They have a quiet reading assignment while I get some work done.

2:05 – 2:55 PM Bell 8 Geometry

I am going to write more about this class later tonight or tomorrow. We have been really struggling with basic Algebra in writing line equations. I am trying to keep some good humor about all of this but I am getting pretty frustrated. We start by projecting last night’s HW on the AppleTV but very few students have their work out while we are reviewing. I have to take that as a sign that many of them are not actually completing what I am asking them to complete. We do have a decent conversation about the problems which makes me think that they could do the work, many are just choosing not to. I need to figure out a meaningful way to motivate more HW completion without resorting to punitive measures. Need to think…

We then look at a problem set together that focuses on the midsgement theorem for triangles. I spend some time circulating and listening in while they do most of this work on their own. I return quizzes at the end of the class and another T day is now in the books. At least, the classroom portion of the day is…

3:00 – 3:30 PM Bell 9 Conference Time

Not much action in after school help today and my former student vista again. He poses a challenging problem for my consideration. One hundred passengers enter a plane and the first passenger sits in the wrong seat since s/he misplaced his boarding pass. I know this is not realistic in these days! Now, every other passenger sits in the right seat if it is open or random;y chooses a seat if someone is already sitting in his/her seat. What is the probability that the final passenger gets to sit in the proper assigned seat? I need to do some thinking on this one.

3:35 – 4:35 PM

Stand around outside chatting with passing students and waiting for my kiddos to come off the bus. We walk home and I make them nachos for an afternoon snack. Lil Dardy watches an episode of Lab Rats while her big brother plays his new PS4 game. I take a brief nap while Lil watches an episode of Chowder.

4:45 – 5:30 PM

Mrs. Dardy returns from work, my boy gets dressed for his basketball practice and I catch up with the missus for a few minutes before heading out to pick up another boy on the way to basketball.

5:30 – 6:00 PM

Jot down some HW notes for my Calc and Discrete kiddos. I post assignments to our LMS and send out group emails.

6:00 – 7:30 PM Family Style Dinner

Two nights each week we have a sit down dinner where boarding students sit at tables with faculty and their families. On Monday and Thursday we eat like this. Since my son has basketball, he is not with us tonight. Dinner is from 6:15 – 7 and my daughter plays with friends for awhile after dinner. Tonight she was excited to bring along her new doll. It is a bit of an America Girl knockoff. She was proud to show it to some of her friends and to some of the campus moms. She even let one of the students at our table hold her for awhile. While she was playing with friends after dinner I get to chat for awhile.

7:45 – 9:15 PM Grading

I FINALLY finish the grading that has been hanging around from my two days off and then I get started on my Day in the Life blog post.

Whew. Off to bed soon. Another blog post to come tomorrow…

 

 

 

 

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

Every year we all experience this, right? We have big goals for the new year, we make promises and many of them fall by the wayside. I am going to be modest about my new year’s aspirations (at least in public!) and I am making a vow to myself to try something new for my Geometry kiddos this new calendar year. Awhile ago I got into a terrific twitter exchange with Henri Picciotto (@hpicciotto) and some other folks ( I wish I could remember everyone involved, I am pretty sure that Julie Reulbach (@jreuhlbach) was part of this) about HW. I mentioned that I will sometimes include problems in HW that touch on ideas we have not talked about yet in class. This brought up a conversation about leading vs lagging homework ideas and Henri is particularly articulate about these ideas over at his blog space (http://www.mathedpage.org) We tweeted about the idea of letting ideas percolate either before instruction or after instruction. I spoke about my feeling that it is important to have students struggle with an idea a bit to (maybe) help them appreciate a new idea/definition/formula when it does arrive. This, I think, is sort of like Dan Meyer’s series of if ______ is the headache, then ______ is the aspirin essays. One of the results of this terrific, spontaneous twitter chat was that I walked away with a commitment to instituting lagging HW assignments in my class. I wrote out a careful pacing calendar with the optimistic idea that I have a solid sense of how long these conversations will take and a hope that mother nature will not interfere by dumping snow on us at some point. Obviously, this calendar is not set in stone. What IS set in stone is my commitment to the pacing of the HW assignments. What I did was write out a pacing calendar that delays by three or more days any HW that directly relies on reading/instruction of a new section of my Geometry text. I do still feel a bit of a commitment to daily practice out of class – I know, this is another conversation completely – so I wrote a series of review HW assignments that reach back to old ideas/skills but I tried to do so in a way that is thoughtful and leads to preparedness for the new ideas we are discussing. I also wrote a series of entrance slips that I will start class with the day after we first discuss a section together. These will be collected and marked, but not graded. I am hoping that these will provide both me and my students a way of monitoring their developing understanding of new ideas. After three or more days they will final have a HW focused on a certain section of our text. Meanwhile, in class we will still be following a pace similar to what we followed last year in our first run through my Geometry text so I have some sense that this is a reasonable pace for our students. However, instead of going home and immediately practicing some new set of skills, they will be looking back either weeks (at the beginning of the chapter) or days (after a few days in) to ideas that have, hopefully, been percolating a bit in their brains. They will have had time to think about these ideas, they will have had an entrance slip check in on their facility with the ideas at hand, and they will have had further class time combinations of lecture and discussion to play with these ideas and build them up together. A few days afterward they will go home with focused practice and their assessments will also be lagging a bit to line up with the HW practice. I anticipate that there will be some concern expressed by my Geometry team and by my students because, you know, change is a bit of a challenge. This is especially true once you have established a rhythm and pace that you are comfortable with together. One of my three colleagues has expressed a desire to try this as well but the other two have not commented yet and I have no expectations that anyone else needs to follow me on this path. As department chair I kind of want to test the waters on this so that I can report back on the inevitable speed bumps as well as the successes that we encounter. If this works the way I think that it will, it will radically alter how I think about pacing and how my students think about HW. My biggest wish for this endeavor is that this practice will enhance retention and help my students think more about connections in the ideas we work with in class. I feel more confident about taking this leap in our Geometry course first for a couple of reasons. I feel more intimately familiar with the contours of this course since we are using a text that I wrote a couple of summers ago. I also feel that the Geometry course lies a bit outside of the vertical tower that much of our math curriculum builds. If we slow down a bit and there are not topics near the end of the course that we reach, it feels that there are fewer consequences in terms of what future courses and teachers will expect of these students. Also, these students are younger than those in my other two courses (a Discrete Math elective and AP Calculus BC) and I am optimistic that these younger students might be more flexible with the idea of changing their habits.

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.

Talking Math Over Piggy Banks

Six year old Lil’ Dardy accompanied me on a bank errand and she asked the clerk for a new piggy bank – her third now. Two of them are actual pigs while one is a sheep. She decided to name the pigs Pig and Peg. Pig gets dollar bills and change while Peg and Sheepy both get only coins. We emptied all of her money out and sorted for awhile. She has accumulated tons of coins, she has a great eye for coins on the floor. After we filled Pig with all of the bills, we started filling all of them with coins. Because I have a touch of OCD about this sort of thing, I started by picking up three pennies at a time and giving each container a coin. Then I moved on to nickels, etc. Lil’ Dardy mentioned that she, too, was going around the circle giving each one a coin but I noticed that she was not giving each one the same coin. So, I thought this would be a good opportunity to have a lil’ math chat. I asked her if she thought that each container would get the same amount of money and she quickly said no. So far, so good. I next asked her if she thought that they would have the same number of coins. Here, I was already planning her discovered instead of letting her discovery happen by itself. I am a little annoyed with myself for this. But, when I asked if they would have the same number of coins she again said no. I repeated that we were each ‘going around the circle’ putting in coins, but she said that she had just had this thought- she did not start out this way. Have to say I was pretty proud that she point this out in the face of my question. Hooray for Lil’ Dardy on this counting task this morning.

Brief Notes on a Good First Week Back

As I wrote about the other day, I tried something brand new with my Geometry kiddos this week. I had found online somewhere recently a three day packet exploring reasoning and proof (you can find it here) and I had my students in small groups. I had three groups of three and one group of four (I know, I am very lucky (spoiled?) to have such small classes) and they all grappled with one of the open problems in the set and gave brief presentations on Tuesday. Yesterday we conducted a fishbowl discussion. I had never done this before as a teacher or as a student, so I felt a little anxious about it. Since I had not taken the time to ‘train’ my class, I left this as a pretty open exercise. There are two pages of definitions to grapple with in the handout linked above. I had seven students in the fishbowl for the first round and I joined the other six in the fishbowl in the second round. Every student drew a card at random as they came in to decide which group they were in. I instructed the outside group to just quietly observe rather than to take notes on the inside participants.  Both rounds went pretty well – in my opinion – but what was best about the day was the talk at the end looking back on the exercise. I have tried to make this first week back for Geometry rather open – ended. I wanted to try and make some important points about learning, and about classroom culture, about proof and logic.

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.

 

Getting Back to Business

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.

I am starting off my Geometry kiddos with a three day workshop on Reasoning and Proof. I found this somewhere on the inter webs recently but I cannot recall where. You can find the link here and if you recognize it, please let me know. I am pretty excited about this. I think that it will be a lively way to restart my classes and I am optimistic that the students may make some inroads into understanding the logical structure of proofs. We had a great activity with making peanut butter and jelly instructions for each other earlier in the year. I think that this serves as a nice follow up and I am happy that there is such time between them. My optimistic hope is that the students will make that connection on their own without me pointing it out. This unit has a similar idea with sentence strips outlining the process of making spaghetti. I do know that when I do the PBJ activity again in the future I will scaffold it a little more carefully in advance so that more of the students will have a solid idea how to approach that. If you want to read about our PBJ adventures you can look at this post or this one.

I am also already committed to a project for my winter break. Right before Thanksgiving I engaged in a lengthy and lively twitter discussion with Henry Picciotto (@hpicciotto), Elizabeth (@cheesemonkeysf), Peg Cagle (@pegcagle), Julie Reulbach (@jreulbach), Mattie B (@stoodle), and Chris Baldus (@Chrisbaldus04) We were discussing HW strategies. When to preview ideas, when to lag and let ideas catch up, how to possibly blend those strategies. It was an amazing conversation with people from all around and at least two of whom I am certain that I have never met. One of those great examples of why engaging with twitter has improved my practice. So, I am too weary to rewrite my HW sets that I wrote last year when we rolled out the Geometry text I wrote. But, I realize that the time before January will allow me to write a few more sets that I can use as buffers near the beginning of the year while I let ideas settle in and percolate for my students. The assignments that they would have been working on the night they were introduced to an idea will now come three or four days later. In the interim we will concentrate on in class discussion and practice and I will write some homework sets that concentrate more on helping to cement definitions and some new mechanical skills along the way – along with reminding them of highlights from 2015. I am excited to do this and I would not have had the motivation to do so without the urging of those virtual colleagues who took the time and care to share with me their ideas and experiences. I am a little anxious because change = bad for too many of my students, but I am convinced that the time off will allow me to think deeply about how to be as intentional and clear as possible with my students. The other fear I had and came to grips with is this – I am one of four Geometry teachers at our school. I am also the chair of the department and the author of the text. My ego keeps creeping in and wanting everyone to follow my lead because of both of my roles here. I came to peace (thanks Julie and Elizabeth!) with the idea that I do not have to have everyone on the same page AND with the idea that I can be a better leader in this process next year if I go through it myself this year. I will still share out my old (and new!) pacing guides and homework assignments. I will simply make it as clear as possible that not everyone needs to agree with this HW strategy and with the timing of assessments that this will entail. If the students are not doing homework concentrating on, say, section 6.4 until three days after we introduce that section in class, they cannot be held responsible for that material on an assessment until they have had time to practice. Consequently, assessments will lag behind where we are in class as well. I need to rethink my ideas about what review days mean and look like, but this kind of rethinking is one of the things that makes this job such a joy.