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.

 

Exam Review Week

Our school operates on a trimester schedule. The fall term ends as the Thanksgiving vacation begins and the winter term ends when our spring break begins. Before our fall finals I have been taking some time to review and reorganize some ideas with my students. I have to say that the past two days with my Geometry students have been filled with a combination of two strong feelings.

The positive feeling is that I am really proud of how my students have been working together. My class is set up in three ‘pods’ of six desks each. I am spoiled and my largest class has 16 students. At the beginning of the week I uploaded five review problem sets for my Geometry students. My plan for this week is to have one problem set each day and to circulate through class during the week, putting some burden on my students to be actively reviewing ideas rather than sitting and listening to me. The students have really been going at it and I have enjoyed listening to them. A couple of good hearted heated debates have popped up and I get called in to moderate these debates. There was one particular problem that I asked that I am proud of.

The points M(3, -1), N (4,3), and P (0, 5) are the midpoints of the sides of a triangle. Find the coordinates of the vertices of this triangle.

I am not certain that I can remember the origins of this question. It is likely that I found this somewhere and if anyone recognizes where it came from, please remind me. What really pleased me about this problem was the response of one of my students. I overheard a number of students discussing this problem, so I took the opportunity to gather up the class and lead a conversation here. I drew a triangle on the board, not on a coordinate plane, and joined the midpoints of that triangle (rough sketches here) then I asked them to make some guesses about the relationships between this ‘midpoint triangle’ and the ‘parent triangle’. It was very quickly apparent that the students expect that the midpoint segments are parallel to sides of the original triangle and the guess that lengths are related came out as well. No questions were raised during this part of the conversation. This group chat convinced me that the question I asked is much more interesting than if I had asked the question in the opposite direction where I gave the parent triangle and asked for the midpoint one.

Even with an agreement in class about the relationships here, the work to translate this to the coordinate plane is a tough leap. Early in the year I gave a few questions where I would give a segments endpoint and a midpoint and ask for the other endpoint. Not a creative problem, but one that the students remember. Since I knew that I wanted to introduce vector notation early in the year, I would frame these midpoint questions by talking about this problem in terms of taking half of the journey here. So the movement from one endpoint to the midpoint is half of the journey and we want to repeat this path. This way, I try to get the students thinking in terms of component movement and not simply in terms of distance. So, I took this approach with the class on this midpoint problem. Time did not allow a conversation that was as deep as I wanted it to be and I plan on starting class today by revisiting this problem. I am unsure whether I will use the same coordinates or not, I am inclined to think I will not. I also know that I will start class today by simply trying to get a few students to explain the thought process we outlined, not to focus directly on the necessary calculations. I have been working hard at getting my Geometry students to frame a process of problem solving for complex problems. We recently worked on finding the distance between two parallel lines and I asked my students to outline their approach to the problem rather than calculating the distance, so framing a question like this in the terms of describing their problem-solving approach will make sense to many of my students. After we discussed this problem at the end of class yesterday, one of my students remarked out loud that this problem was really hard but really interesting. I was pretty pleased to hear her think out loud like that.

i started off by mentioning that I had two strong feelings so far this week and I started with the positive feeling. The not so positive feeling is a deep sense of frustration about the overwhelming lack of recall of some simple facts. At least five of my students were stymied by a simple question of finding the midpoint of a segment. They consistently wanted to apply a simple formula and they could not (would not?) stop and think about what kind of formula might even make sense. For any teacher who has taught this idea, it will be no surprise to you that the debate centered on whether to add endpoint coordinates or to subtract them. I reply by asking what midpoint means and every student quickly says middle. Then I ask what number is in the middle of 2 and 10. I try to convince them that they do not need a formula if they are willing to stop and think. I wish that I had gone to the coordinate plane quickly to try and tie together some physical sense of these points. Instead, I was visibly frustrated and asked a few students what their average would be if they earned a 90 on one test and a 100 on a second. Would their average be 5 or 95? However, thinking back on this exchange I realize that I was not effectively making a point here, I was simply showing my frustration. This phase of the year where students are trying to tie ideas together should be a time when I am happy to see the growth in my students understanding. More often, this ends up being a time of frustration with finding myself repeating ideas that I thought we had successfully conquered early in the year. I understand the pressure that students feel when they are faced with a week of long exams and stress related to trying to show mastery of twelve weeks of knowledge all at once. I have been at this for a long time now so I feel that I ought to have a better idea about how to navigate this challenging time of year. I have worked at four schools now and they all believe in final exams, so I have tried to work within that context. I also believe in the principal behind cumulative displays of knowledge. I just know that the way we do it creates such stress for most of our students that they do not feel like this is a show of knowledge, they feel it is simply more of a test of their endurance.

I would love to hear how any of you navigate this challenge.

Problem Day becomes Persistence Day

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

Find the point of intersection of the lines tangent to the curve  if those lines are tangent at x = 1and x = 5

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

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

Sometimes it is in the Sequencing

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

You should drop by there if you have not yet

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

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

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

Non-Mathematical Musings

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

Beginning to Grapple With Proof

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

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