Another quick post. We are in exam week here at my school. I have ALL sorts of thoughts about term exams and why we do them, but those are for another place rather than this public forum. I have written about our department’s decision to move toward a test correction policy. I am so so so optimistic about our exams this week. I really believe that we will see largely improved results because the students have been more actively involved with examining their tests and reflecting on what went right and wrong on their tests. They have been talking to each other and comparing ideas. They have been talking to their teachers about how to fix their problems solving approaches. Our department exams are mostly on the next to last day of exam week. This will work against our students as energy levels start running low. Despite this, I am hopeful that we will see a different level of engagement on these cumulative exams. I will report back either way in a week or two, but I feel good based on what I have seen in the past week of reviews in class and from watching and listening to kids in the hallway as they work together.

# Category: Uncategorized

## Improving a Lesson Plan

My Geometry classes have just finished a few days considering different translations on the Cartesian plane. We are working toward being comfortable with rotations (almost always around the origin), reflections (horizontal and vertical lines and the lines y = x and y = – x), and vector translations. Last week I had a particularly unsuccessful lesson where I tried to help my students discover a pattern for 90 degree rotations around the origin. I want to try and outline my thinking here and I would love love love any insight into why it did not go well and how I can improve this in the future, or even how to go back to cycle and revisit this with my current team of scholars.

So, my idea was this – try to pull together what we know about perpendicular slopes, our developing ideas of vectors as a physical object similar in nature to a line segment, and developing an intuition about the fact that a 90 degree rotation should result in a move of one quadrant in a certain direction. I asked for three coordinates from my students and drew a triangle. I asked them to predict where this triangle would end up after we moved it 90 degrees clockwise. Two of the coordinates given were in quadrant I and one was in quadrant IV. It seemed that my students were happy/comfortable with the idea that the two quadrant I points would live now in quadrant IV and that the quadrant IV point would move to quadrant III. This may have been a tepid agreement in retrospect. Next, I focused our attention on one of the quadrant I coordinates and I drew a segment from the origin to that point. We talked about the slope of the segment, we compared this segment to a vector, we talked about the length of the segment. I then asked the students to imagine a wheel and I told them that when I think about rotations I think about a bicycle wheel. In my mind I saw this segment as a spoke and I thought about distance from the point to the center of the wheel. Here is one place where I know I failed my students. I did not explicitly stop at this point and discuss distance the way I could have/should have. I also have a handful of students who are still struggling terribly with the idea of calculating distance. We have been talking about it since day two, I have coached them to think about Pythagoras, we have practiced it repeatedly. The combination of squaring, of square roots, of subtraction in one piece of ‘the distance formula’ and addition in another piece of it, comfort with mental arithmetic, all of these factors are working against my students being unanimously comfortable with calculating distances. So, the next step in my plan was to ask them what they recalled about perpendicular slopes. They all should know this and most recalled it pretty quickly. We had a segment in front of us with a slope of 4/3 and my students quickly agreed, maybe passively maybe enthusiastically, that a segment perpendicular to this would have a slope of -3/4. So, the question at hand was now whether the fraction was in the form or -3 over +4 or in the form of +3 over -4. I was convinced in advance of this lesson that this string of conversations would be a positive path to take. I felt that the combination of recalling past slope ideas, looking at the physical Cartesian plane, tying in ideas of line equations, etc. would gel together to make a lasting learning experience. I was wrong. When I prodded them toward the conclusion that -3 over +4 was the conclusion we wanted I saw some uncomfortable faces. When I mentioned the idea of a spoke as a visual to hold on to, I saw blank faces at this point. I got a bit frustrated and asked for my students to describe to me what they were thinking of when I mentioned the word spoke. Nothing. I pulled up a google image of a bicycle wheel and asked them to tell me what the spoke was in the image. By this point their reluctance to engage in this conversation was building, my frustration was increasing, and any positive momentum in building this process was falling apart. My fault for showing my frustration. My fault for stacking up too many ideas at once, I think. When I spoke to my Geometry colleague she felt that adding on the layer of talking about perpendicular slopes was the tipping point of discomfort for my students. I trust her instincts on a number of levels in part due to her experience in teaching our Algebra I course. She knows these Algebra kiddos and knows not only what they know but how comfortable they are knowing it. So, at this point it was clear to me that this was slipping away. We limped to the end of the conversation. Most students were willing to agree that the point (4,3) would end up in quadrant IV. They were split on whether it would land on (4,-3) or on (3,-4) and it honestly felt like many were mentally tossing coins to make this call. I showed them the conclusion on GeoGebra and we sort of ran out of time by this point. We have since gone back and tried to reinforce the conclusion we reached and I think that most of my students can reliably answer this question, but I am completely uncomfortable with how we got there. I would love any insight/advice about how to best structure this info. You can certainly drop a comment her or over on twitter where I am @mrdardy

## Test Corrections, Round One

I mentioned in an earlier post that our school’s math department has adopted a new policy this year. After our spring workshop with Henri Picciotto (@hpicciotto) one of the decisions we arrived at was to allow (almost require) test corrections from students. The goal was to encourage and reward reflection and communication on the part of our students. When I wrote about it originally Brian Miller (@The MillerMath) told me he would hold me to my vow to report back on this. Here is round one’s report.

First, I should remind everyone reading this what our new policy statement is. Here is what my students read on their syllabus this year.

Beginning in the 2017 – 2018 academic year, the math department is adopting a policy of expecting test corrections on all in-class tests. The policy is described below.

- When grading tests initially each question will get one of three point assignments

- Full credit for reasonable support work and correct answer.
- Half-credit for minor mistakes as long as some reasoning is shown.
- Zero credit (in very rare cases) when there is no reasonable support shown or if the question is simply left blank.
- When grading tests, I will not put comments, I will simply mark one of these three ways.
- You will be allowed to turn in corrections. Corrections will be on separate paper and will have written explanations of errors made in addition to the correct work and answer. This work is to be in the student’s words but can be the result of consultation/help. These corrections will always be due at the beginning of the second class meeting day after the assessment is returned. You will return your original test along with your correction notes. I will remind you of this every time I return a graded test to you.
- It is not required that you turn in test corrections.
- The student can earn up to half of the points they missed on each individual problem.
This policy does not apply to quizzes, only to in-class tests.

The first class to have a test this year was my AP Calculus BC class. This class has sixteen students of the highest math caliber at our school. They had a test in class on Tuesday and on Tuesday night I marked those tests. On a number of occasions I had to restrain myself from circling something or writing a note to a student. I went through and only marked each question as a 0, a 5 or a 10. There were six questions, so I graded 96 questions overall. Only one 0 out of all these 96 questions. Thirty four questions earned half credit and the other sixty one questions earned full credit. Of those thirty four, most mistakes were minor and in the past they would certainly not have suffered a five point penalty, but in the past they would not have had the ability to earn back points and they would not have had the motivation to think clearly about what happened. Due to quirks in our rotating schedule, the second class day after yesterday is not until Monday. However, I have six of my students in my room after school yesterday working on corrections. All of them spoke to me about problems and three of them were working with each other. Four of the six students there completed their corrections and turned them in already. This feels like success. I know that it is early in the year and students have a little more energy right now. I know that this is my most motivated (by knowledge, by interest level, and by grades) group of my four different class preps I have this year so I will not expect quite this level of engagement right away. Oh yeah, two of the students there yesterday only missed points on one of the six questions. The could have happily taken their 92% and gone home to worry about other work instead. I expect that I will see another one or two folks today and then get a slew of corrections in on Monday. The initial grading was a little bit faster and I could get them their tests back right away. Looking at the corrections will take a little time, but this is time I want to take and it is encouraging the kids to think about what mistakes they have made and (hopefully) not make those same mistakes again. I’ll keep updating on this experiment.

## A Fun Question Through Different Lenses

Today I had three of my five classes meet and they all met before lunch. So, now, I am writing a blog post after lunch!

Here is the problem –

Doris enters a 100-mile long bike race. The first 50 miles are along slow dirt roads, while the second half of the race is on smooth roads. Assume that Doris is able to travel at a constant rate of speed on each surface.

- If Doris’ speed on the first 50 miles of the race is 10 miles per hour, what must be her speed during the second half of the trip so that her average speed over the whole trip is 13 1/3 miles per hour?
- If Doris’ speed on the first 50 miles is12.5 miles per hour, what must be her speed on the second half of the trip so that her average speed over the whole 100 miles is 25 miles per hour?

A fairly standard question and I admit I stole it directly from a text I use ( a pretty cool Differential Calculus book for my non-AP class) and I originally intended to just use it with my Calculus Honors class this morning. After the discussion with them, I decided it was worthy of the attention of my Geometry section and my Discrete Math class as well. It is kind of fascinating to me to think about the different receptions that the problem had in each class. In all three classes I asked them to think about the first question and wait until we discussed it before moving on. I KNEW what mistake was going to be made. There was no doubt in my mind that the answer to question 1 would be 16 2/3 and the students did not disappoint me. They focused on the information given and processed it in a logical way given their use of the word average when considering two measurements. What they did not focus on was time and it is logical that they did not *because it is not explicitly mentioned in the problem at all*. My hope going in with my Calc Honors kiddos was that this would be an object lesson in weighted averages. What it turned into in my other two classes was a bit of a primer in how to think through a word problem instead of just automatically applying some mathematical operation on a set of numbers in front of you. In each class after the initial incorrect solution was offered I presented the following question. “I have a 90 on my first test and I score 100 on my second. What is my average? Now, I have a 90 average after four tests and I score 100 on my fifth. Is my average the same in each situation?” I had hoped, overoptimistically, that this would prompt thought about time but in all three classes I ended up explicitly urging them to think about time. In Calculus Honors and in Discrete this led most (maybe all) students to a correct conclusion. Not so with my younger Geometry students who were still a bit wary of the problem. In fact, one student asked me if all of my questions were going to be like this. I need to be a little more careful about these last minute decisions to follow my muse and approach a problem. I need to be more thoughtful about how appropriate the question is for the audience at hand. I still believe that this is a meaningful question for my Geometry kiddos to wrestle with, but perhaps I would have served them better by not making this the opening question on the second day of school…

Both groups of older kids arrived at the correct conclusion to the second question and they were amused by the result. Only two days in but of the eight classes I have had, I’d say 7 were successful with one uncomfortable miss.

## The Big Kids

This post will make more sense if you have already read Joe Schwartz’s (@JSchwarz10a) thoughtful blog post about two experiences he and I shared at TMC 17. I’ll wait here while you go read it.

Back? Good.

Alright, let me share a couple of reflections first. Joe is one of the many delightful folks whose acquaintance I would not have made if I had not taken the plunge into this online world of collaboration. He and I met in person in Minneapolis and had a really in depth conversation about parenting, especially with regard to tech use for children. His words have echoed in my ear this year and my wife and I took on the challenge of a smart phone for our 14 year old. My son does not know it but Joe is one of the reasons why I was able to sort out my protests and come to the decision to give him one. So, in addition to my life being improved by Joe’s friendship, my son’s life is improved by Joe’s wisdom. Anyways… This summer I got to spend time with Joe again at meals (especially a LOVELY dinner at the oddly named Cowfish) and at a session run by David Butler (@DavidKButlerUofA) called 100 Factorial. As Joe wrote, he and I were in a group of four with Jasmine Walker (@jaz_math) and Mauren (Mo) Ferger (@Ferger314) We worked on a problem called skyscrapers (you can find a cool online link here ) and we were all full engaged. Now, I knew jasmine and Joe already and knew Joe was a primary teacher. This fact did not cross my mind during the time we were working on the problem, but it sounds like maybe it did for Joe based on his blog post. That evening about a dozen folks all descended on Cowfish for dinner and I was sitting near Joe and Jasmine. I won’t repeat the story of our conversation, Joe covered it well. What I do want to do is think out loud about my perception of the conversation and try to get into Joe’s head a little bit as well as getting into my own head. Early in the conversation I mentioned to Jasmine that I had the impression that she might be ‘mathier’ than I am. I tend to be a little self deprecating in this area, I have three degrees and they are all from College of Education. I have no formal math degree but I took a load of math classes in college and have taught a load of them in my 30 years of teaching. I know a few things and I am pretty quick at making connections, if I do say so myself. However, I also know that I am TOO quick to make certain conclusions and this caused some trouble in the Skyscraper game and I am also a bit too quick to throw in the towel if I don’t see at least some sort of pathway pretty quickly. I don’t need to know an answer right away but I do need to have some sense of where to find the answer to help me be persistent. As Jasmine and I were trying to ‘un’ one-up each other (Edmund Harriss (@Gelada) was sitting next to me and he joked that this was the opposite of a pissing contest) I was also wrestling with the question Joe had out on the table comparing the Exeter problem sets with the puzzle we played with that afternoon. Looking back, I fear that the banter with Jasmine about who was less ‘mathy’ may have been somewhat hurtful now that I see the feelings Joe laid out in his blog. If that is true, I am deeply sorry. What I DO remember distinctly about the conversation was that I described different initial reactions to the lovely problem sets and the creative puzzles that Prof Butler laid out. In the problem sets there is a reassuring (or distressing, I guess) sense that these are MATH problems. That there is some MATH technique or formula that will be needed to nudge me down the road to success. With the Skyscraper problem, it was clear to me that this was an exercise in LOGIC. MATH thinking strategies certainly are handy and helpful, but this problem did not yield to an algorithm (or if it does, I am not nearly clever enough to know it) but it did yield to persistence and communication. Joe talks about wanting to overcome some old residual fear or discomfort to go ‘play with the big kids’ on the Exeter problem sets. What I hope he recognizes is that he WAS playing on that stage, it was just in the cafeteria with Skyscrapers instead. I have had conversations around Exeter problem sets with students and with other teachers. They have been great conversations but they were certainly not more memorable than the feeling of diving in and and conquering the Skyscraper problem. Joe was an integral part of that problem-solving team and he caught a couple of my mistakes when I jumped to quick conclusions. We are all on a continuum of comfort and confidence in different problem solving scenarios and Joe’s thoughtful and honest blog post serves as an important reminder to me to try and be more aware of these feelings in others as a new school year begins.

Joe told us this summer that he has retired from his daily gig and is now doing a variety of consulting jobs. He talked about how some folks collect baseball stadiums over the years, visiting ballparks around the country. He talked about the idea of doing that with classroom visits now that he has a more open calendar. I would LOVE it if he carries through with this plan, it would be great to hear his perspective. I would welcome him to my school with open arms but I would also be slightly anxious and a bit nervous about it. Would I still seem like ‘one of the big kids’ if he saw me in action? This kind of anxiety, I think, is probably a good thing for me. It keeps me on my toes. I want to make sure that my students have a meaningful experience in my classroom and one of the ways I can do that better is to imagine that I was also crafting an experience for someone like Joe.

## First Day Plans

This school year I will be teaching four different courses – Geometry (2 sections), Discrete Math, Calculus Honors, AP Calculus BC. My Twitter feed is being bombed with first day plan posts, so I will jump in here as well. Sitting by a pool, so this I’ll not be lengthy.

Note that our first day has 25 minute classes and a long community gathering.

In Geometry I have started the past three years with a dramatic introduction to the handshake problem. It generates some fun guessing and conversations right off the bat. We are also able to revisit this problem in various forms during the year. I think it is a winning first day activity.

In our Calc Honors class I will take students out in the hallway with some wheels chairs. I will have a segment of hallway measured for length and we will have some races pushing these chairs down the hall. This, I hope, will generate some conversations about average speed tat we CAN calculate and all sorts of instantaneous information that we cannot. This should be a basis for distinguishing between secant and tangents over the first days/weeks of the course. Plus, it is fun to run down the hall!

In Discrete I am going to use a fantastic quote that I read this summer (you can find it here ) I think that this might generate some fun research and some fun conversations about magnitude.

In Calculus BC I want to start with a deep dive into a conversation about linearization and approximations. I have gathered some fun ideas on twitter about how this conversation can unfold. I hope it leads to quite a bit of noticing and wondering about accuracy and when/why that accuracy falls apart.

## TMC17 Reflection Addendum

I am kind of embarrassed that I forgot one of the best highlights of the TMC17 conference. A while ago I received a tweet from John Golden (@mathhombre) asking if we could have a video chat about calculus. He was putting together an idea about a resource for his calculus students and wanted a variety of perspectives. Well, after a series of attempts we finally settled on a group chat on Saturday night. It was pretty loud everywhere on the lobby level so I offered my room as a quiet refuge. I had the joy of chatting about calculus with John, Jasmine Walker (@jaz_math), Edmund Harriss (@gelada), and David Butler (@DavidKButlerUofA) You can find our conversation here

I was SO flattered to be asked to do this and it was such a blast to chat with these four lovely and brilliant people. I told John on Sunday that I was jealous of his students. My apologies for having this wonderful experience slip my mind when I posted earlier today.

## How Important is the Silence?

At our church this past Sunday one of the members of the congregation gave a thoughtful sermon about what it means to keep the sabbath. At least that was the primary framework of her conversation. Much of the time was spent talking about learning to take care of herself and what that looks like. Is it listening to a sermon at church? Is it staying home to garden instead? Is it taking a long walk? Is it listening to an inspiring TED talk? Naturally, there is no universal answer for this, but it sure got my mind spinning thinking about what it means for me to take care of myself.

I have engaged in a number of twitter exchanges recently sharing podcast tips. I have become hooked on a number of them and I listen while walking/jogging outside. I listen while I am on my Airdyne in the basement or on the treadmill in the basement. I was listening to Marc Maron and Randy Newman talking to each other just half an hour ago while eating my breakfast here in the airport (I am heading to TMC17 today!) Last month I was visiting my last hometown in New jersey and I went for a long stroll in a park where my wife and I used to walk as a pit stop on the way to pick up our son from day care. At the urging of an old friend – again, through a twitter conversation – I unplugged for that walk. For about 30 minutes I was strolling through this park, remembering cool fall days walking with my wife, listening to the sounds of the park and the neighborhood. It felt energizing. However, I have to admit I have not unplugged like that for another walk or run since. As energizing as that silence felt, I also recognize that I draw a great deal of energy from taking in ideas/content/entertainment through my podcasts. I tend to have music on in my house most of the time I am there. My wife and kids bought me a hammock for Father’s Day. I always bring a book with me to the hammock. I wonder (worry?) if I am just hiding from silence and from being with *myself* this way. I justify it by recognizing how much I enjoy being tapped into a number of conversation. By recognizing the joy I find when something I hear about that seems brand new suddenly starts popping up all around me. I am excited that I have been spending more time and energy listening to new music again due to my summer DJ gig (which I hope will turn into a fall one as well!) However, I also worry that this is making my time and mind feel even more crowded. I worry that I should put that aside and be quiet. I worry that I should be goofing around with my daughter at home more often instead of curling up with a book while she plays in her room or watches a show on TV. I justify this by thinking that I am ‘taking care of me’ by indulging in books, music, podcasts, exercise so that I can be better at helping others – wife, kiddos, students in the fall, etc.

I am in the last few weeks of summer here and I have taken on a teaching overload for the upcoming year. I’ll be teaching five classes with four different preps. This is on top of being a department chair. I think that the looming concern about what this will feel like has also made this past Sunday’s sermon more meaningful. I mentioned earlier that I know that there are no universal answers to this question, what I am worried about is that I am not clear about what the answer is in my particular situation.

I work better when I set specific goals. Last fall I was waking up early three or four times a week and going for long walks before coming home to wake up my wife for coffee in the morning. I think that I want to commit this fall to picking one day each week, Monday feels like a good choice, to making sure that I go out on this walk with no earbuds. Take a long walk or jog with the silence of an early morning in my ear. Keep my mind clear thinking about what the upcoming week holds. By putting this in writing, I am convinced that I am more likely to carry through with this plan. At the very least, I will feel vaguely guilty or embarrassed if I cannot carry through on this commitment. Not a BIG game changer, but at least this feels like a start. I will check back in on this after the school year is in gear.

As always, feel free to join in the conversation through comments here or by poking at me over on twitter @mrdardy

## Brief thoughts on Graduation Weekend

Yesterday was graduation day here at my school. I am pretty sure that this was my 31st high school graduation ceremony – mine and 30 years as a teacher. I think that I did not attend my little brother’s graduation for some reason or other. At least, I do not remember it if I was there.

There are always waves of joy/sadness/pride/regret that run through me on graduation days. I saw some alums and had lovely conversations with them that made me happy. One joked that his Calc 3 class at McGill was easier than his BC Calc class with me. I think that this is probably a compliment in the end. Some students went out their way to find me to express gratitude while others certainly showed no inclination that I was on their list of people that they wanted to talk to on graduation day. Every year ends with the good feeling that there are students who I have made connections with in or out of the classroom. Young people who appreciate that I was part of their lives here. Every year also ends with the disappointment that there are some students I was not able to connect with. Students who were frustrated by my class, did not connect with my goals or my classroom strategies. Students who will not remember me fondly – if/when they remember me at all. This is both a cause for sadness/frustration and motivation to recharge soon when I think about next year and plan for how to reach a broader set of students where they are.

I had a conversation with a colleague recently that made me reflect on graduation feelings and helped me make sense of them. My family moved in the middle of August 2016 from the boys’ dorm where we lived for six years and into a house on campus that the school owns. Since school was already looming when we moved, we did very little in the way of yard work to make the place feel like our own. This past week, my wife and I were able to spend a notable amount of time working outside and trying to make the place feel like ours. I was talking to a colleague at brunch and mentioned that I felt satisfied about the work we had done that morning. While I do not find any zen-like sense of peace and serenity while doing yard work, I do find a sense of satisfaction in looking back after two hours of work and seeing a recognizable change in our flower bed. When talking at brunch about this I contrasted the work in our flower bed with the work we do int he classroom. It feels pretty rare that we see noticeable change in just an hour or two in the classroom. The sense of satisfaction and pride I felt on graduation day when reflecting on the successes I have had is certainly deeper than my satisfaction about the flower bed, but it takes a great deal more patience to get to that graduation day feeling.

## Fixed an Error

Apologies to anyone who tried to follow the links on my last post. I am still learning the ins and outs of google forms. I have gone back and corrected the problem, so you should be able to view the question form if you are interested in doing so.