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.

  1. 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?
  2. 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.

Another New Beginning

Tomorrow morning will be my 31st opening day of school as a math teacher. I am luckily pretty well past the days of nervous anxiety. Luckily I still have the experience of anxious excitement about the task ahead. As I begin a new year, there are three things in particular that I am really excited about. One has to do with school structure, one has to do with classroom policy, and one has to do with my life outside of school. I’ll highlight them in reverse order.

I mentioned over on twitter that I had the great pleasure of taking on a gig as a DJ at a local college radio station this summer. My wife works at a college across the river from where I work and she helped me land this fun gig. I just got the good news that I can keep my slot at least through the fall. I will be on (as DJ Calc – an old in-joke from my past) on Thursdays from 4 – 6 PM ET on wrkc.kings.edu where you can stream and listen online if you are so inclined.

One of the takeaways of the workshop my team did with Henri Picciotto (@hpicciotto) last spring was that we committed to a new policy of test corrections in our department. I teach four different courses this year, three of them are senior and junior heavy while one is freshman and sophomore heavy. They will all receive the statement below with only one tweak. My Geometry kiddos will turn in test corrections on the third class day after receiving their test while the others (Discrete Math, Honors Calculus and AP Calculus BC) will turn theirs in on the second class day. Here is the statement I crafted for a syllabus.

Test Corrections

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.

I will definitely be blogging throughout the year about this topic and I’ll be sharing my thoughts and experiences about this change in approach. The baseline message that I hope we will be sending is this : I want you to learn the material at hand and I want you to have an opportunity to show me (and yourself!) that you have learned this material.

The last thing that I am thrilled about is our new schedule. After being here seven years and meeting every class on every school day in the same order for the same amount of time we area adopting a very different new schedule. We are moving to a seven-day rotation schedule. We will meet five classes per day and each class meets five times during a seven day rotation. During that rotation each class meets in each of the time slots AND each class has one 90 minute block and four 50 minute class meetings. I am excited on a number of levels about this initiative.  I have taught at two other schools with rotating schedules and I noticed a couple of clear advantages. You know that sleepy kid in your 8 AM class? That kids is not usually so sleepy for an 11 AM or a 2 PM class. You know that athlete who keeps missing your 2 PM class because of travel obligations tot he team? That athlete is rarely traveling at 8 AM or 11 AM. You know that class that wanders in right after lunch with a mixture of twitchiness because they do not want to sit again or lethargy as they digest their lunch? You do not see them in the same state everyday. I have seen that different students emerge as classroom leaders at different times of day. Most importantly, I have noticed that students (many of them, at least) give a more honest commitment to effort on HW when they are getting ready for four academic classes in a day instead of six of them. This, too, will be a regular topic of conversation in my blog this year.


As always, drop me a line here or on twitter where I am @mrdardy

I’d love to hear from those who have experienced a major change in school schedules I want to have some idea of how to anticipate possible problems this year. I’d also appreciate any comments about our test correction policy. Anecdotes from experience will probably help me and my team as we make this transition.

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.

TMC17 Reflections

Beginning my reflections on the latest TMC experience (I am fortunate enough to have been for the past four years) I find myself focusing more on the personal experiences in ATL than the mathematical ones. That being said, I LOVED the presentation on base-8 math by Kent Haines (@kenthaines) and I am beginning to shift away from my strict aversion to multiple-choice questions based on Nik Doran’s (@nik_d_maths) advice in his morning session.


Last year in Minneapolis I allowed myself to dwell on the fact that there were social happenings that I was not part of. I KNOW that this is an inevitable fact when any large group of people gather together. It was especially true since we were housed in different places AND I was not equipped with technology that allowed me to tune in to everything going on around me. It was not until November of this past school year that I had a smart phone. Looking back, I KNOW how foolish this was. I had lovely dinners and chats with folks. I went out within hours of arrival to a lovely pub with Brian Miller, his school colleague Wilson, and Henri Picciotto. I had an amazing talk at dinner one night with Dave Sabol, who is kind enough (or crazy enough) to be one of the hosts for TMC18. I had fantastic math conversations and life conversations and came home a richer person than I arrived. However, I have allowed myself to dwell on what did not happen.


This year, armed with a smart phone (that I did not end up using much at all, really), going to a hotel where (almost) everyone was staying, and being in a city I knew, I went in with an agenda for myself. I knew I would be away on Friday night visiting an old high school buddy who was also my first college roommate. I made a commitment to myself. I was not going to hang around and see what happened about lunches or dinners. I sent out a tweet on Wednesday night inviting folks to join me at a restaurant I found called Smoke and Duck Sauce. Wednesday night ended up with a large gathering at Rose and Crown that was a great deal of fun. I sent out a call on the #tmcplans for Thursday night and had a great dinner with a fun group. On Friday night I had a lovely meal with my old friend and his family and returned to the hotel to stumble in on a deeply meaningful conversation with a fantastic group of friends. I was drawn over by seeing Brian Miller and Jasmine Walker (a couple of my favorite TMC pals) and ended up awake far later than I intended to be as a sprawling group of folks in a corner of the lobby bar really dug down deep on some personal and professional issues in a sensitive and vulnerable way. My had was spinning as I went to be. On Saturday night, I sent out another call on #tmcplans and ended up at Cowfish with a dozen folks. A LOVELY meal, great conversation, laughs as we celebrated a fake birthday, and a great sense of belonging and satisfaction as people piled into my rental car there and back on each evening. I went along to a breakfast at Waffle House based on an open invite. I had lunch with different folks every day at the campus of Holy Innocents. I had a quiet breakfast by myself the first morning of the conference enjoying southern grits and getting my head focused for the upcoming adventure.


I am not going to dip my feet into the mini controversies that came up during the week about hashtags and inclusion. I just want to say that I know that when I took it upon myself to be responsible and engaged in the community I enjoyed myself far more than when I was passive about it. Even though I also enjoyed myself then!


Of course, the social aspect and the connections are only part of the reason to come to TMC. There is also some sweet math to be experienced. My morning session with Nik thinking about hinge questions has me seriously re-thinking my bias against multiple-choice questions and recognizing their value if they are thoughtfully constructed and are treated as important data points in understanding what my students understand. His energy, intelligence, and good cheer made the morning sessions well worthwhile. I had two moments of mathematical epiphany during the week. On one of David Butler’s afternoon sessions he introduced us to some of his puzzles from 100 Factorial. I worked in a group with Jasmine, Joe Schwartz, and a new pal Mo Ferger on a fantastic problem called skyscrapers (you can find a link here!) We worked doggedly, and successfully, on this problem. On an afternoon session with Kent Haines I worked on some problems and pattern finding in base eight arithmetic. Again, working with some folks in the room (I wish I could remember who!) we poked around and noticed and wondered and fought the frustration that many of our students must routinely feel as we tried to find a comfort level in this realm of mathematics.


After a busy, happy, and rewarding three days with my #mtbos family in Atlanta, I am now relaxing with my (much smaller) family on vacation counting down the days to the new school year. I know I will still have some of this energy fresh in my mind in a few weeks. The challenge is to keep it fresh in my mind all year.