A Portrait of a Math Teacher as an Aging Man

This past Monday was my 37th opening day as a classroom teacher and I forget every year how tiring the first week can be. I suppose that it is also possible that it is more tiring as I get a bit older!

For the first time in over a decade I am teaching an Algebra I class. This year on opening day I presented a handout on the handshake problem. You can find it here

We had a pretty terrific conversation about this problem. I have eight students in this class and I made sure that they understood that I count as a person in figuring out this problem. The question at hand, at least the initial question, is how many handshakes will happen if each person introduces themself to everybody else with a handshake. I asked one of my students to tell me how many handshakes he would need to make and he easily answered that he would have eight handshakes to make. Immediately, a student to my right then guessed that there would be a total of 64 handshakes. Her classmates did not see right away where her guess came from, but I could tell that they sort of stopped thinking much about their own guess since one was on the table. I asked her to explain her guess and she was a bit reluctant. Totally natural, it’s day one and she is a new student in our school. I asked if it was okay for me to try to read her mind. She granted me this right. I guessed that she thought if there were eight handshakes for one person, then eight squared felt like a natural conclusion. She agreed that this sounded like her thinking. Another student jumped right in then and said 72 would be a better guess. Since there are nine people total, if each shook eight hands, then there would be 72 handshakes. Everybody seemed pretty happy with this guess and I have to admit that I, too, was pretty happy with this even though I knew it was not right. I already had two students think out loud and share their guess. A good start to the year.

I pivoted and asked if it was okay to try and check this guess. I asked how many handshakes would happen with only one person in the room. Confused faces tried to process this odd question and we all agreed that no handshakes happen here. How about two people? One handshake. We were confident about this. Three people? I turned to the person on my right to imitate a handshake, then the person to my left. We all agreed on three handshakes for three people. Four people? Someone immediately said six handshakes. Much of the class quietly took in this guess without saying much. I was happy that a different student asked why that would be true. I backed up and asked if we all agreed that three people have three handshakes. We all did agree. Then, I asked if a new person enters the three person room, how many new handshakes would be needed now? A – ha, three new handshakes! Then, the student who guessed six immediately says ten for five people. Now, we are picking up steam. We have gone from 0 to 0+1, to 0+1+2, to 0+1+2+3, to 0+1+2+3+4. This piece, by the way, was my addition to the conversation. Looking back, I wish I had helpd back because I think it would not have taken much to get a student to say this. Sigh…

After this, I took over a bit and led them down a more algebraic thinking path to arrive at a more general formula. We have thirty minute classes on the first day and I feel like we packed in some pretty great math talk on this first day. Is this a problem that you’ve used in your classes? If so, I’d love to hear a little bit about what class and how the conversation unfolded. You can share here or reach out to me over on twitter @mrdardy

Happy 2024 – 2025 academic year!

This year I had the great pleasure of taking part in the grading of the new Advanced Placement Precalculus course. I have been teaching AP for over thirty years now and I was on a waiting list for AP Calculus grading for about seven or eight years. When I had the opportunity to throw my hat in the ring for this course I jumped at it and I’m pleased that I was able to be part of the inaugural grading for this new course. I’ve been back a couple of weeks now and I should have written earlier. I want to thank Dan Meyer (@ddmeyer) for a tweet this morning that prompted me to make myself write again.

I want to think out loud here about my experience and what impact I hope it will have on my students’ lives moving forward.

I graded question 2 on the Free Response exam.

The first thought to share is that this is hard work. Over the course of my time there and with a little bit of home grading thrown in after I got home I graded more than 2000 student responses to this question. Concentrating carefully to help ensure consistency in the process requires a great amount of attention and focus.

The question was worth six points with each sub response being worth either one point or zero points. I don’t really want to go into any detail about the student work I saw, it’s not really my place to do so. What I do want to go into are my thoughts about how this experience will impact my teaching moving forward. This year I will be teaching Algebra I (in addition to AP Precalculus and AP Calculus BC). This is the first time in over a decade that I will be teaching this foundational course. I am really excited to take this course on for a number of reasons. The last few years I have almost exclusively been teaching higher level math classes which means I often do not meet students until their junior or senior year and they are enrolled in a class with high curriculum pressure at a time when they are focused on college acceptance and a load of other heavy duty courses. These older students also have a pretty deeply developed sense of who they are as math students and what works for them. I am looking forward to meeting some students at an earlier point in their trajectory, at a time when the curricular pressures are not as intimidating, and at a place in their learning arc where maybe they are not quite so set in who they are as math students.

It may not make sense to others, but the AP FRQ above had me thinking about my prospective Algebra I students more than about my prospective AP Precalculus students for next year. It had me thinking, especially the last two answers required in that AP question, about how students express what they know. I hope that my experiences in grading those last two answers will guide me this year in how I talk to my Algebra I students and what work I ask of them. It occurs to me that this might be an important opportunity to help students earlier in their math arc to learn how to articulate what they know. How can I write in a way that (a) helps me sort out my thoughts and (b) explains to my teacher what it is that I know about the math I am engaging with? I know that there are some smart resources out there on the web that teachers have shared that will help me here. One format that I like is to present an incorrect solution and to ask my students to identify where the mistake is and what the correct step is instead. I think that the hard sell will be to get my students to write in words, not just in mathematical equations.

I know that I have written before about how I want to write more. I hope that this time I not only mean it, but will follow through with it. I think that having this new course added to my plate will help to be a motivator for that. If you have any ideas to share with me you can find reply here or find me over in the land formerly known as twitter where I am @mrdardy

A post by Michael Pershan (you can find it here – https://pershmail.substack.com/p/quizzes-are-better-than-tests?utm_source=post-email-title&publication_id=695673&post_id=138715944&utm_campaign=email-post-title&isFreemail=true&r=2ligi&utm_medium=email ) got under my skin (in a good way!) and I have been stewing about this for a while now. I finally have some time to breathe and to get my ideas down about this.

Michael refers to an idea by Henri Picciotto about lagging assessment. This is an idea that I like and one that I have out into practice (with my own tweaks) in my Honors Calculus class. We have regular time built in our school schedule for student conference time where students can come in for extra help. This is in our schedule Mondays through Thursdays. What I do in our Honors Calculus class is schedule a quiz for three to four class days after we wrap up a section. The idea being that this allows students to make sure they are caught up with their daily HW practice (not collected, so it is not graded) and make sure they have time to conference before a quiz. Sadly, fewer students are taking advantage of this conference time the last few years. COVID has had all sorts of effects on our habits and conference time used to be regularly pretty well attended for our department. Not nearly as much anymore. So I have implemented that idea in this course. Early in the year it is confusing to students and they initially are frustrated that we are learning new material before a quiz but they seem to figure it out. I like this practice for a number of reasons, the primary one being that I think it sort of discourages the idea of short term cramming of information in a way. I also have a standing policy in all of my classes that tests are announced as something like a test through Chapter Two not a test on Chapter Two. Quizzes are short, generally 20 minutes and just three or four quick questions, and they are targeted to a certain section (or a few sections in the case of my two AP classes) while tests are always cumulative. Tests take a whole period, they are worth more points, and they cover more material. This is where Michael’s post really got me thinking and I want to use this space to think out loud about this.

I certainly do not hope to contribute to stress and tension in my courses. I want students to think hard and to think deeply, but I also want them to build connections. By asking them to continue to think about ideas and skills from the past I hope to show them that what we are learning now is dependent upon ideas from their past. That past might be the last chapter or it might be their last course (or two) I prime them for this with in class conversations reminding them of previous ideas, old skills that are needed on daily HW practice, and old skills that are needed on Problem Set HW assignments that are collected and graded. Do I stay with this quiz AND test model simply because that is how I have always done it as Michael seems to imply? Do I stay with this quiz AND test model because I want students to have practice thinking and concentrating for longer stretches of time than they do on quizzes? Do I stay with this quiz AND test model because that is what they will likely be seeing in future classes and I want them to be in that rhythm? Do I stay with the quiz AND test model because that is what they are used to and what they expect? Should quizzes be so focused on recent material, while tests that count more invoke a wider set of skills? Should I flip this so that the less stressful (meaning, mostly, less impact on their grade) events have the wider set of skills? Would there be less stress if the weightier tests had the more tight focus on skills to prepare for? Would students place less emphasis on brushing up on earlier skills if I changed this focus? Lots of questions and I am unsure of the answers. I would love to hear from some folks on their practice. I am going to do more thinking and talking about this and I will write again as I come to some focus on all of this. As always, I can be found over on the land that used to be known as twitter. I am @mrdardy over there.

I am flattered and excited to be part of a pretty cool event happening next month in California. A little history here to explain this. In 1995, and again in 2008, I was fortunate enough to attend the Ajna S. Greer Annual Conference on Secondary School Mathematics, Science and Technology at Philips Exeter Academy. Looking at those dates, I think I need to make my way back again before too long!

Each of my summers there were fantastic in a number of ways, but in terms of todays post the particular fantastic thing to point out is that I met Carmel Schettino there. My memory is that we met the first time I was there, she may remember otherwise, but the important thing is that I met her. Carmel is a thoughtful educator who is committed to Problem Based Learning in the math classroom. You can read up on some of her work here ( https://www.carmelschettino.org/ ) She currently works at a pretty amazing school called The Avenues School (you can see what they are about here – https://www.avenues.org/ ) and I have had the pleasure of visiting their New York City campus and taking part in a summer workshop there a few years ago (pre-COVID) and Carmel and I have been in touch off and on for quite some time now. I was VERY flattered when Carmel reached out to me in January to ask if I would be a presenter for what she calls a Jam Session at an upcoming conference she was organizing. I jumped at the opportunity and was able to work it out with my administration and I am totally happy to share that I will be in San Jose, CA at the newest campus of The Avenues School for what is being called a PBL Math Teaching Summit at The Avenues Silicon Valley campus from April 27 through April 29.

While Carmel has worked at schools where there is a more intensive focus on problem based learning, she knows that I have been working at schools that have a more traditional view of their curriculum. However, even at the three schools I have worked at in the time I have known Carmel, I have been able to regularly carve out some space in our curriculum for problem based learning. I have stolen ideas (and some problems!) from a number of sources over the years. I imagine that anyone reading this is familiar with the Exeter Math Curriculum and their public problem sets (you can find them here – https://www.exeter.edu/mathproblems ), I have also used math competitions that I have found on the internet, I borrow liberally from the bookshelf of texts I have in my room, I have borrowed from The Park School curriculum, and a number of other schools that have been generous in sharing their work out on the web. The proposal I put together for Carmel for this event is that I am going to speak out the social learning aspect that I see my problem sets creating in my school. I am going to concentrate on the school where I currently work. I have been here since the fall of 2010 and in four of my courses here I have made take home problem sets a part of the curriculum and a part of how I assess my students’ progress. I live and work at my school. We are a PK – PG day and boarding school and I lived for six years in one of our student dorms. In my time here I have regularly seen groups of students gathered working on my problem sets. In the dorms, in the cafeteria, in the library, after school in my classroom during our extra help conference time. I routinely see students gathered and working bouncing ideas off of each other. The real joy is that they often ignore me, there are times when even though I am in the room with them, they simply pick each others’ brains. Sometimes students who aren’t in my class come by because they hear an interesting conversation. Sometimes students who had been in my class the year before drop in to share their memories or some tips to help move the conversation forward. I have been reaching out to a number of our alum to ask them to reflect on their experiences and I have been receiving some terrific replies from folks and I am excited to share in depth when I am in CA in April. I will be setting up a shared drive with resources and I will be sharing some of the particularly valuable reflections that my students’ have shared. Mostly though I am looking forward to spending some time in the company of other math teachers who are able to share their experiences in this approach to teaching and meeting a number of math teachers who are looking to make the leap and expand their toolbox.

I will be sharing my experiences after the conference in early May when I get back and get settled in. If you are interested in more information about this conference, there is contact information on Carmel’s website and you can reach out to her that way.

As always, if you want to reach out you can comment here or find me over on twitter where I am @mrdardy

Our school has a two week spring break that happens at a time in the year that does not feel much like spring in our part of the world. My Precalculus Honors class is due to start a chapter on systems and we will learn to work with matrices. However, for the intro to the chapter I knew I wanted to revisit old skills in a new, hopefully more interesting, way. I had the germ of an idea last Thursday (IIRC) and I sent out a tweet to my math team.

I thought that this would be a fun sort of open way to get my students thinking again about systems. As the #MTBOS is inclined to do, I immediately got helpful feedback. Among the suggestions were to make sure that the line was not horizontal or vertical as this would be WAY too easy. Noted! Nice suggestion.

A favorite early suggestion came from a teacher I had not interacted with before on twitter (at least I do not remember any extended interaction)

This got my mind working at a much deeper level. I had kind of thought of just throwing out a quick challenge question. This suggestion, and the resulting conversation, got me thinking along the lines of a more organized activity leading to this challenge question. A few more exchanges with Pete and with a couple of other twitter pals and I landed here with a great suggestion from Karen.

After a few more tweets back and forth, and with school looming finally, I crafted an activity (you can find the finalized product here )and shared this link out to three twitter pals who were pretty actively engaged in this conversation. I was so touched by the fact that all three of the folks I shared the doc with took the time to look it over and share some ideas/questions/insights into how I constructed this. Years ago when I first started blogging then took the leap into twitter this kind of collaboration and sharing of ideas was exactly why I took that leap. I feel remarkably fortunate to have found this community of open, curious, and generous minds.

I decided to jump right in on day one back and present this activity to my small Precalculus Honors class. I have seven students in this class and they normally work in two small groups. One group of three boys usually work together and another group of four students (two boys and two girls) usually share their ideas together. I specifically asked them to start this work on their own and I spent the first ten minutes or so of class just sitting quietly and trying to spy on the work that they were doing. I have small whiteboards at our big Harkness-style table and most students do their thinking on those whiteboards rather than in a notebook. After a little while I began taking notes on our GoodNotes app and asked the students question by question to share their ideas. You can find our class notes on the hyperlink in the previous sentence.

Next, I asked them to sketch their graphs and identify how many intersections there were. They were asked to find the coordinates of the solutions to their system. I suspect that few of them recognized right away that the point (3, 4) had to be one of their solutions, but that is fine. Not too much time or energy was wasted here. Now comes the payoff for the activity.

As soon as Ava shared that she remembered that a circle and a line that intersect once has this special name from Geometry (with a little prompting from me) her neighbor Abby immediately remembered the perpendicular slope notion. Once Abby said this out loud a couple of her classmates nodded in recognition that they, too, knew this at some point. Now we were off to the races. The link I shared above to the GoodNotes file has a number fo Geogebra sketches for the circles and lines that my students developed. We had a great conversation about whether it was easier to think first of the circle or the line first (we seemed to agree that the line as a starting point was more efficient) and everyone was pretty engaged in the conversation. I am pretty thrilled about how well this went and how my students were able to dissect this in our 45 minute class today. I want to thank Pete K (@PAKalenik9), Karen Campe (@KArenCampe), and Benjamin Leis (@benjamin_leis) for their thoughts and this activity was so much richer because of their contributions.

Yesterday was the last day of our winter trimester and we are now on a two week spring break. My students this year have been asking me on a number of occasions about having a group quiz. They know that I have done this in the past, it’s a small school and word spreads, and they wanted a piece of that action. While I emphasize collaboration and I want to let students know that I value what I say I value, I have struggled a bit with group quizzes. Too often, not always of course, I witness one or two students simply sitting back and letting their peers do the heavy lifting in these settings. With the end of the term looming I wanted to try and satisfy this demand from my students while also re-thinking it in some way. I tried something yesterday and I think I am pretty pleased with how it played out.

I wrote five different versions of a quiz for my Calculus Honors class. We are working on the chain rule and there is an ocean of variants for any of these problems where it feels easy to ask for the same skill under a slightly different disguise. So I decided that I would have a group quiz but the members of the group did not have the same quiz. I told them that they should spend the first minute or two looking over the quiz and sizing up what they needed to know and that they should lean on their teammates for insights, hints, or reminders as necessary. So each person had their own four problems to do BUT each of their problems was a direct analogue of the ones their teammates had. I want to report out some of what I saw and heard during these three classes.

The first thing I want to note is that a number of groups, not all of them but enough of them that I noticed it, spent the last five minutes or so swapping their quizzes and checking on each others’ work. I LOVED this. We often talk about students checking their work but it is way too easy to look at a mistake of your own and not notice it at all. If you wrote it in the first place it is because you believe that is what you should do. Seeing a mistake from another person is much easier. The other thing I heard was students reminding each other of problems we had done in class. Thinking out loud, saying something that you remember, is much easier than conducting an internal monologue to try and refresh your memory.

As they were finishing their work and turning in their papers, I asked a number of students whether this felt like a meaningful exercise. They were pretty positive about it and a couple of kids made an almost identical remark that really pleased me. They noted that they felt like they got a lot more practice at the skill of analyzing chain rule derivatives. Someone said it was almost like a 16 question quiz that they had to think about even though their own quiz only had four questions. I think I am pretty pleased with this tweak on the group quiz idea and I will probably pull this idea out again soon.

I have spent the past 25 years teaching at schools that go all the way from kindergarten through high school. I often tell my students that I have a crazy dream that it would be great experience to work with a group of them starting in kindergarten and move up together all the way to high school graduation. We’d REALLY get to know each other and be able to reference ideas/habits/experiences together. I am in no way equipped to teach younger children and I know this is a crazy mental exercise anyway but I do think often about habits and language that my students arrive at my classroom with that I wish they did not have. For example, when faced with the distributive property I always hear what I refer to as ‘that four letter F word’, when faced with a fraction on both sides of the equation I always reflexively hear ‘cross multiply’ [even when there are other factors in a problem] These are just two silly examples of fights that I am too often willing to wage in the classroom. A more interesting one, and one that has deeper implications, popped up in my class last week.

I often have a problem projected on the board when class starts. It is rarely a problem directly linked to the day’s tasks, I just want to get the brains moving and, hopefully, get some conversations started. This year in my Calculus Honors class I have noticed a discomfort with some of the classic word problems of the sort that pop up in Algebra II regularly. This is not a big surprise, these kind of word problems don’t get practiced regularly and there are all sorts of gut feelings that don’t always yield correct solutions. When I throw up a class opener problem I usually make a quick reminder of ‘Hey, look, there is an interesting question on the board!’ On the day I am thinking of, the question I posed was this one:

The first minute or two of each class involves a couple of gentle reminders to attend to the question that is posted and this day was no different. Another minute or so later I heard two remarks almost simultaneously that caught my attention. One student, a boy named Parker, said ‘I know the answer’. He did, by the way, know the answer correctly. Another student, a girl named Maya, said ‘I know how to do this.’ She did, in fact, know how to do the problem but had not arrived at the solution as quickly as Parker did. These remarks instantly caught my attention and I said to the class ‘Did you hear the difference between those two remarks?’ I repeated them both but did not follow down a conversation because I was not sure yet what I really wanted to say. This blog post is an attempt to unpack what I want to say.

I remember back in 8th grade Algebra I having a debate with my teacher Mrs. Hart. I got back a quiz where I lost some credit on a problem despite the fact that I had a correct answer. I asked her about this and she said I did not show her how I got the answer. I offered to show her right then but she told me she wanted to see it when I took the quiz, not at that time. I was pretty unhappy about that, I guess I kind of still might be, but I understand her point. I try to emphasize to my students that I am interested in their thinking in addition to their conclusions. I want to understand how they arrive at certain conclusions. Parker got the right answer to the questions pretty quickly and he briefly explained his logic. I think I understand how he approached it, but I wonder how generalizable his approach was. He is a student with good number sense and quick reactions. He does not always back up his thinking as algebraically as I wish he would. Maya is a more step by step student and is getting noticeably better at explaining her thinking out loud and on paper.

This particular problem is not a deep one but it does remind my students of some system of equations strategies that I know are valuable in their study of Calculus. I also think it is just a good idea to brush off some of those skills every once in a while. Mya had equations written down for this, as did a few other students. Some used three variables. Some expressed the mother’s weight AND the nurse’s weight in terms of the baby therefore only having one variable. I reminded them that matrix operations sometimes help with three by three systems but they (wisely) rejected that memory here. I posted this question on my twitter account (@mrdardy) and had a nice conversations over there with some folks. My favorite conversation was this one:

I LOVE the point Susan made here. She is 100% right that the question asked for an answer, not a process. In my defense, and I suspect Susan knows this, by this time of the year we have established pretty well that process and explaining/defending your answer is an important part of what is going on. The question I have, and what I am trying to unpack in my mind, is this one: To what degree is it fair to ask my students to accept the idea that process is at least as important as conclusions? How often in their life is this message actually meaningfully communicated to them? I am sure that they have had teachers, and other adults, routinely tell them about the importance of process but I suspect that they are also routinely awarded for answers. What is the message here? I think that students equate grades with what we value and I understand that instinct. So, if I routinely award full credit for a correct answer without support then I am sending the message that it really is all about the answer. I try to send the message with small point deductions that process matters. What I probably need to do is make more of a commitment to something like a rubric where the actual answer to the question is a small percentage of the question’s point value. This brings other problems with it. I think back to a student in the second school where I taught. She joined our school about a month into the year and she was in my AP Calculus AB class. About two weeks into her tenure at our school she announced ‘You seem to think that AP means All Problems’ She was making a distinction between problems and exercises and she was unhappy about the prevalence of problems on my assessments. I have a working definition, from my Master’s advisor Prof Mary Grace Kantowski. She said when facing an exercise we know what to do right away. We might make a mistake in process but we know what to do. When facing a problem we are not sure right away of the process called for. If I present my students with mostly exercises, then there is not a ton of work to be shown much of the time and I do not learn much about them as learners. If I present my students with mostly problems, then there is process to be presented but some students might not have the time and energy to show me enough of what they know in the time allowed. None of these are new problems for teachers, but they all came flooding into my brain when I heard these two remarks the other day in my class.

I think I want to follow up with that class later this week to try and unpack the conversation and try to probe a little on where they land, and where they want to land, when faced with problems in this class. While the goal, understandably, is often ‘I know the answer’, I hope that there is at least some recognition that ‘I know how to do this’ is a pretty good place to be.

One of the nice traditions we have at our school is that we host alum at our school early in January and they work together to run what we call College Panel. Our College Guidance Office toss some question their way and then we open up the floor for students to ask questions. These alum discuss the whole college application process and how to adjust to life at college. This is a nice tradition, but the part I appreciate best is getting to talk to some graduates after the assembly.

This year we had two different assemblies, one for our current juniors and seniors and the other for our freshmen and sophomores. In the second assembly, one where I was not present, my name came up in response to a question about which high school class was most challenging. I heard a couple of different versions, but basically a couple of kids said my class (both referring to AP Calculus BC) was one of their most challenging classes in high school but that it was also a great preparation for college coursework. It is, of course, gratifying to hear that but what I want to write about here is the follow up. I saw each of these two students during lunch and had lovely conversations and they each sent me an email today. I am going to pat myself on the back here in public on a Friday afternoon. I hope you don’t mind.

The first student I am thinking of majors in English and Philosophy at an Ivy. Why did an English/Philosophy major choose AP Calculus BC? This is a natural question and, in fact, this student was challenged with this question by a classmate last year. This girl was also taking AP Physics and when she mentioned that she intended to major in Philosophy she was challenged. Why take these classes? She calmly replied that she had had high aspirations for her college admission and felt that she needed to present herself as a student capable of excelling in a number of arenas. This makes all the sense in the world and the fact that I know that she has no deep love for math makes the following remark from her email even more meaningful to me.

While you know that I am not a math person, your class really was the academic highlight of my senior year, and the way you taught us to think creatively about problems has helped me tremendously in college. Two teachers here have fundamentally changed the way that I think about the world — you and Ms. _____ ( I don’t know that it is smart to name the other teacher, but I am sharing this fact with that teacher) and I’m so grateful that I was fortunate enough to be taught by you both. I never thought that I would be excited to go to a math class each day, but your passion for teaching and your encouragement for us to think for ourselves made the Calc BC an incredible experience that I cannot even express how grateful I am to have had.

I have a story about a student from my first school (I left there in 2001) who I am still in touch with off and on. He paid me what I think is the greatest compliment I received in my career. He was talking about a boss of his and he said that this boss reminded him of me. He said his boss would ask him questions that he had not thought of on his own when faced with a problem to untangle. He said that reminded him of me. The compliment above and one I am going to mention next rank pretty close in my heart.

The second student I am thinking of wrote to me this morning. She references another one of our great traditions at our school. Every fall we host alums to come and talk about their experiences in STEM related fields.

I think quite often 0f a moment during my sophomore year where we had a conversation at my locker in your hallway. The STEM alumni panel had been the night before, and after our discussion on the content, you commented “how does it feel to know you will be back here doing that in a few years?” This unwavering belief and confidence in me before you even had me as a student helped me beyond words, so once again, just thank you. 

What is striking to me about this email is that I do not really remember this conversation but she certainly did. Those of us who have been in the business for a while know how important these kind of interactions are and it is an important reminder to be present and to be attentive to our students. I need to remind myself of this more regularly and this is one way to do so.

This has been a tough couple of years teaching high school. Virtual teaching in the spring of 2020. Hybrid teaching for 2020 – 2021 (the year that the second student was in my BC class) and we are still feeling the sting of all those experiences. Hearing such gratitude from students who went through these years is especially gratifying.

This post is going on both of my websites, my math teacher site and my music fan site. Although it is not a piece of writing about math teaching or about being a music fan, my thoughts on what friendship means/looks like have been deeply influenced by my community of math teachers and of music fans/musicians. I am hoping that 2023 will see me writing more and this might be the trigger that starts this up.

I’ve been thinking about friendship for some time. Although it is a thread throughout my life, my current thinking kind of started this summer. This summer my family spent time in Florida where my in-laws host family at a gulf shore condo every July. Two of the four friends I still have from my high school days both live in Florida and they came to spend the day with us at the coast. While we were debating music and movies and sharing anecdotes about the current state of our lives we got into a conversation about the origin story of our friendship one of my friends thought out loud about his wife and how/when they met. He had graduated from UF where we all went to school and we were roommates at the time. He was already committed to head to Orlando to work for Disney, amazingly he STILL works for them, and he mentioned what a small window of time there was where he and his wife could have even met each other much less had time to fall for each other and make the move to Orlando together. That sent me down this long tangled rabbit hole of thoughts and reflections for the past few months as I have tried to sort out what I think/how I feel/how fortunate I have been with regards to friends.

This will be a bit rambly but I need to just get these thoughts out of my head. I have a number of thoughts/experiences/stories I want to share here. I’m not sure how meaningful it will be to anyone else, but this feels like an important exercise for me. [Author’s Note – I have been writing this over a couple of weeks when I have the time and energy to devote to it so some of the time references (like the beginning of the next paragraph) are not accurate anymore]

This past weekend I had an experience that sort of pushed me to the edge of needing to get these thoughts out. A few months ago a college friend reached out to me. He and his wife were hosting some old friends at their place for a weekend. All friends (and their spouses) with roots from University of Florida. My friend Van who reached out to me is the first person I met in my dorm my freshman year. A high school friend of mine was my roommate freshman year and he had moved to UF a couple of days ahead of me. I was lingering at home intimidated to leave and grabbing a few extra days to be around my girlfriend. When I arrived at UF and starting moving in to my room Van stepped out of his room to introduce himself. He started talking to me and noticed the ocean of music I was moving with me. He remembers dropping a cassette case – or maybe it was a box of albums – but in any case he and I hit it off immediately. This weekend talking to him and his wife I remarked how fortunate it was that he was right across the hall. If he lived a few doors down it is likely we would have been friendly dorm hall mates but much less likely that we’d be friends 40 years now after we met. Total chance at play, right? However, I think there is more than that here. I was open to meeting people as I moved in, he was open to reaching out and sharing observations immediately about the music I had. I think about this these days as my son struggles to feel connected to his university. He has not made the kind of friendships there that I did in my dorm. It’s a different time, the dorm physical structures are different (his dorm room door has a spring that automatically closes it while my dorm room door was pretty much open if I was in the building other than when I slept), he is still in touch with high school friends and friends from back home in ways that I was not able to be, and he has a world of entertainment options in his pocket that I could not have dreamed of. I have nothing new or wise to add to the conversation about connectedness among the younger generation but I do know for my son that actually being physically present with people matters. He loves his PS5 and he spends time gaming with folks but he really misses actually being with people and doing things with them. Speaking of video games, he recently asked me if I knew the abbreviation NPC meant. I kind of did after he explained it and I made a comparison to the recent movie Free Guy. He casually remarked that many people at his university are NPCs. This is a funny, but mean, observation and I cannot help but think that if he was a bit more open to his experiences there, if he had met his version of Van on moving day that he would have a different opinion and a totally different feeling about his time there. 

I also keep thinking about a conversation I had with my son when he was a freshman in high school. He had some good friends at a local Catholic school and he said that he thought it would be fun to go visit that school on one of their visitation days if it aligned with a day off from our school. I suggested that this would not be such a great plan since visitation days usually imply that you are thinking of attending that school. He said he thought it would be fun because he could finally meet some friends he hadn’t met yet. This phrase really struck me. My first reaction was that it made little to no sense. I later started realizing just how many friends I have that I haven’t met yet. I have two twitter accounts, one for my life as a math teacher (@mrdardy) and the other for my life as a volunteer DJ (@DJCalc) at the college where my wide is employed. In both of those worlds I have a wide network of people who I would likely refer to as friends even though I have never been in the same place as them. Friends who listen to my radio show from all around the country, friends who have shared lesson plan ideas or given me inspiration for classroom practices, friends who have shared their music with me, the list goes on. Heck, I even co-host a podcast ( https://www.buzzsprout.com/1460920 ) with three friends none of whom have ever been in the same place as one another. We have been doing this for over two years now and I no doubt consider them friends. We have never met in person. 

In 2007 I moved from Florida to New Jersey. Shortly after arriving in New Jersey I joined an education message board and dove into some conversations over there. I received a direct message soon after that from a friend named Steve who had started teaching at the same time at the same school as I had in Gainesville, FL. We worked together for a number of years there before he moved on and I had totally lost track of him. This was in the mid 1990s when it wasn’t quite as easy as it is now to track folks. Anyways, he recognized my name on the message board and seemed sure it was me but he reached out to confirm. That was 15 years ago and we still keep in touch regularly. He now lives near Boston and I am in NE PA but we see each other a couple of times a year and speak on the phone a few times a year as well. Again, like my experience with Van AND like my high school buddy and his wife, a remarkable coincidence. He happened to be frequenting the same message board and happened to notice my name. If he had not visited that board that week my message would have been buried and we never would have reconnected. 

When I started teaching way back in 1987 I was told that I needed to meet a young history teacher who had just started at that school a year or two before I did, his name is Chris. A friend who hung out at the record shop where I was working knew him. During the first week of school he was not there, he had injured his knee recently, but I made a point of getting to know him as soon as I could. He invited me to a party at his house, this was mid-September and the baseball season was winding down. I had made some plans to watch my beloved Mets play that night at a local dive bar but I felt it was important to make this connection. I went to the party wearing a Mets cap and someone named Doug eventually came over to introduce himself. We started chatting and he invited me over to his house, he lived right next door, to watch the game rather than stay in the backyard mingling. He is still one of my dearest friends 35 years later. If I had not chosen to wear my Mets cap that night there is a very good chance we never would have spoken that night.

Another story about Steve. At one point he was working on the west coast of Florida. I received a phone call from him one Friday night. A hurricane was coming through the Gulf of Mexico and he was fleeing his home on the coast and heading back into the state toward Gainesville. He called ahead to a number of hotels and found that there were no rooms available that he could find. He knew I still kept in touch with a number of friends from my days there, I lived in Gainesville from 1977 – 1979 and from 1982 – 2001, so he called me seeking help. Steve asked if I knew anyone that might be able to put him up. I called my friend Bill who immediately offered up a bed for however long Steve might need it. Bill had never met Steve, but he is such a good friend, and a good person in general, that he immediately treated my request for helping Steve as if it were a request for help for myself. I feel SO fortunate to have friends like this and to have been able to help out Steve in such a time of need.

Today I am at a math competition with some of my students. In the faculty room are people who I only see two or three times a year and only at math competitions. We have some natural common interest in kids, in math, in teaching but we are not a part of each other’s lives other than on these days. Some of them just go their own way or grade papers today but there are a few of them who are happy to see me and I them. We talk math, we try to solve some of the problems together, we share stories of how school life is going and it all flows very naturally as if it were just last week that we saw each other. 

All of these little vignettes are simply a way for me to remind myself how incredibly fortunate I feel to have so many relationships in my life that matter. Relationships that genuinely improve my quality of life. I have friends who have been important to me for 40 something years and I have friends who I dearly love who have only been in my life for three or four years. I have lived in four cities in my adult life and I have valuable friends from each of those places. I have taught in four schools and still have contact with friends who were colleagues from each of those schools. I guess all of this is just to say that I feel kind of blessed and with Thanksgiving still visible in the rear view mirror, I want to take this space to say thank you to all of these friends in all of these settings that have made my life as rich as it is.