What are our Goals in Class?

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 now 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 Mya, 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. Mya 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.

Appreciating Appreciation

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.

Thinking About Friendship

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 three 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.