Hijacking My Own Blog

I have used this space for years as a platform to talk about math teaching. Today I am going to use it for another passion of mine. Today I am thinking about music. Many of you (most? ALL?) follow me on twitter and you know that I have been fortunate enough in the past year to score a DJ gig at a local college. I have been posting all of my playlists on Spotify where I can be found (as on twitter) as mrdardy. I have been a bit obsessive about music since my middle school years. In college I worked in a record store. After college, in my early teaching years, I wrote for the local newspaper and then for a free news monthly. Later, when I moved to south Florida I wrote about music for another magazine there. I am pretty deeply obsessed.

As evidence – here are my CD and LP racks in my basement.


For reasons I do not understand, each image is upside down. Oh well, the point is made.

Part of what has come along with the music obsession is a good bit of snobbery. Long ago I stopped listening to radio to find music and dove deeply into music criticism and word of mouth. I can recognize labels and producers and often purchase something simply because of some arcane connection to something I already love. The I had a child. Then, six years later, another one. I stopped going to shows. I moved. I let myself wallow in my collection and got swept up a bit in the huge world of streaming music. First Pandora, then Spotify. I probably could have gone for the rest of my life without really digging in and learning new music again but the DJ gig saved me from that. Somewhat at the same time, my son (now 15) started really getting interested in music. In the past year I have taken him to see Gorillaz (I quite like them and was happy to go), Tyler the Creator (not quite as happy), and Kendrick Lamar (glad I went but much of the show was not my style). I have been really delighted to see him so animated by something important to me and it gives us a safe venue to talk which is not always easy with 15 year olds in the house. I don’t love the music he loves. I feel life an old fart being bothered by the language of much of the music he listens to, but I remind myself of generations of parents complaining about ‘that noise’ that delights their children.

All of this concert business with my son has made my soon to be 9 year old daughter (birthday Wednesday!) jealous. She has started listening to pop radio and has her own pretty extensive playlist on Spotify. Listening to this station of hers is trying for me. I am not a fan of the style and they seem to play only about 8 songs over and over again. On the other hand, it brings her joy and it is music. So, last Friday I took her to Philadelphia to see Charli XCX, Camila Cabello, and Taylor Swift. I think that all of you reading this know Taylor Swift even if you don’t know much of her music. The other artists, probably not so much. I had not knowingly ever heard Charli XCX but I discovered that I did, in fact, know a few of her songs. So, when she asked me to go to see her #2 favorite singer (Camilla) and her #3 favorite singer (Taylor) I kind of had to say yes.

The show was at Lincoln Financial Field – home of the super bowl champion Philadelphia Eagles. I am pretty sure that the last time I went to see a stadium show was when I saw The Who as a freshman in college (1982 or 1983) with Joan Jett and the B-52s as opening acts. I am used to seeing shows in clubs where I can be 50 feet from the performer or in nice old theaters where the acoustics are great and the seats are pretty comfy. We were FAR from 50 feet away from Taylor, Camilla, and Charli but that did not phase my lil one bit. From the time that Charli XCX came on at about 7:10 PM to a crown only about 1/3 of what it would soon be, my daughter was engaged. Charli XCX was energetic and passionate. She worked what crowd was there and she kind of won me over. Camilla Cabello had a tight, synchronized, and choreographed set. She played most of the CD my lil has and she mixed in bits of Frank Sinatra and Prince in the middle of songs. She is still pretty new to this business (she was previously in the girl group 5th Harmony) but she will likely have a long career ahead of her. Then Taylor Swift came on.

What a spectacle. There were probably 40 semi trailers in the parking lot that had been carrying the three stages for her performance in addition to the fire, the 40 to 50 foot video screen, the fireworks, the cables that transported her from one stage to another while she sang, the cameras, etc. etc. etc. I have to say I was there out of a sense of duty and a sense of wanting my daughter to have a meaningful memory of an adventure with dad. She knows I love music and I think that she wants to connect with me on this field (as does my son – I am flattered in both cases) but I was fighting my snobbery and my cynicism. Let me tell you, it melted away quickly. Her opening numbers were from newest album and I am not terribly familiar with them. I bought my daughter the 1989 album and know all those songs. I know some of the older country-ish songs and I know a couple of the newest ones. It didn’t matter. The show was so energetic and spectacular, the songs are so carefully crafted, and the JOY of over 50,000 people cheering and singing along is simply transformative. I found myself so swept up by the whole thing. My daughter was taken by it, the crowd near us, the whole damned stadium was in the palm of her hand.

Now, I do not imagine that I will be dialing up the music of any of these artists during my free listening time. I will hear them because I live in this world and because I have a young child in my home who is taken by this music. I will, however, hold on to the memory of this show. Not just because I hope my daughter will treasure this but because it cracked through my grumpy music snob exterior. It made me smile and sing along to songs that don’t particularly matter to me because I was in a crowd of about 50,000 people to whom these songs really mattered. Because this talented artist who was playing in her home town really cared about the fact that she was coming home.  Because she has crafted a show (hell, a career) designed to bring joy to large groups of people together. Because it was a summer night and a beautiful one at that. Because I was proud of myself for having made arrangements for this event. Because I just love music. I love being in its presence, even when it is music I do not inherently love.

I have seen hundreds of nights of live music in my life. I am totally lucky in that respect. I have distinct memories of dozens of those shows/nights. This show will definitely live in the small set of special nights for all sorts of reasons.

Thinking About Trig

In the fall I will be teaching a section of Precalculus Honors at my school. I have not taught that course in about six years so I have not spent a bunch of time thinking about teaching trig functions for awhile. Our Precalc Honors class starts off with a study of Trig and I am thinking about an opening day activity that might plant a number of seeds that we will need to germinate over time. I want to play with some data that is periodic in nature and try to generate some hopefully interesting questions. Thinking along the lines of Dan Meyer’s challenge of finding an aspirin. I tweeted a bit about this and have been engaged in a terrific conversation with Bonnie Basu (@GotMathHelp) about a similar activity that I used as a demo lesson for a job interview years ago. The context is different since those students had already been a bit immersed in their study of trig but they had not been graphing yet. My goals with that group were to uncover the periodic nature of the hours of daylight, to talk about why other functions that seemed appropriate to the picture (especially quadratics) were not appropriate, to build up some sense of important vocabulary for trig graph analysis, and to simply plant some important ideas that would be explored in greater depth after I was gone. Looking back on the activity I see all sorts of tweaks I want to make and Bonnie has been super helpful in asking questions/making suggestions. Anyone who wants to take the time to share ideas or questions would certainly help make the beginning of my precalculus class this fall more meaningful and successful. I thank you in advance for any tips.

After another round of conversation on twitter that included tips from a student who just graduated from our school, I have been playing with three different data sets. I want to play with average daily temperature, high tide level, and with daylight hours. The data for the first two came from wunderground and the third data set came from dateandtime. Below are pictures of the data and Desmos links to the tables.

This picture (above) shows the average daily temperature in my town during 2017 at 10 day increments. You can see the Desmos table here. There are some things I like about this picture. I like the fact that the general shape can be inferred. We can talk about why it fluctuates on a number of different levels. However, I don’t think that this is a great data set for beginning to develop an idea of periodic functions. It feels too noisy to me.

Here is the picture for high tides.

This table was built on data at 6 day intervals from the beginning of the year through June/July. You can see the Desmos link here. I would definitely ask the students to play with window sizing and I think that some powerful ideas about amplitude and vertical shift can quickly come out of such a conversation. I picked 6 day increments thinking that I would be slightly off phase with what I thought would be a period related to the full moon. A quick survey on google just talks about a 12 hour plus period so I may have been making this up in my mind. This picture feels more friendly with just a little noise involved. I might use this one early – maybe even on day one. The next one is much cleaner looking. Here is the data on length of daylight hours during 2017.

This is where my mind was when I created the demo lesson. However, this data is for our hometown here. You can see the Desmos link here. My thoughts about this data set go down two different paths. One thought is that this is clean and clear and easily explainable. One tweak I might make based on my conversation with Bonnie is that I might extend past one full year (say about 400 days or so) to make the periodicity visible not only intuitively meaningful. My second thought is that this might be too clean that it might lead my students into expecting such clean, clear periodicity in a messy world. I am probably overthinking this on the second train of thought.

I expect to have five table groups of students (groups of three in my classroom after long debating it, I accepted the wisdom shared by a number of MTBoS folks – especially Alex Overwijk (@AlexOverwijk) and my classes were better this year because of that change!) and I am thinking that each table group should have different data. I am playing with the idea of mixing up sunlight or tidal subsets of data versus simply subdividing one larger set. For example, if I go with the cleaner daylight data I can extend it to about 450 days or so and give different table groups subsets of about 35 data points each. I feel that they would benefit from seeing how the data ‘fits together’ and that individual table group decisions about amplitude, vertical shift, and period all match each other pretty well.


I would love some feedback/suggestions/questions and I thank Bonnie again for her valuable thoughts. You can drop comments here or over on twitter where I am @mrdardy

Some Fun Approaches

We have adopted a new schedule at our school and we are on a seven day rotation this year. At the beginning of each rotation, I give my AP Calculus BC students a problem set that is due at the beginning of the next rotation. These are just grab bags of problems that I find interesting. Some are calculus problems, but most are just fun stuff I have gathered over the years. On our most recent problem set (the last one of the year) I gave a problem that I think I found in an Exeter problem set. The heart of the problem was the image below. 

We are told that we are to start at hexagon #1. We are allowed to progress at each step to an adjacent hexagon as long as that hexagon has a number higher than the number we are currently on. So, for example, from 5 you can proceed to 6 or 7 but cannot go back to 3 or 4. The question is to determine how many pathways are possible from hexagon #1 to hexagon #13.

I did not know the answer to this question, but I was confident that I (and my AP Calculus BC students) could find the answer.  I approached this problem the way I do many problems, I wished it was smaller and I hoped to see a pattern emerge. I have advocated this problem solving strategy with my students but few pick up on this. I think that this has to do with their sense of freedom as mathematicians. I think that changing the problem feels like a privilege that they don’t think that they have. Need to work on this…

So, I built up a table and saw that if there was just one hexagon then there is just one path. A boring one of standing there. If there are two hexagons, there is also only one path. Hmmm, not promising yet. Three hexagons? Two paths, from 1 to 3 or from 1 to 2 to 3. Four hexagons? 1 to 2 to 3 to 4, 1 to 2 to 4, and 1 to 3 to 4. Now, I am confident that Fibonacci is hiding here. A quick check confirms this and I was pleased with myself for finding a fun problem that did not have an obvious solution.

I used the word obvious for an in-joke. One of my particularly clever AP Calc students will routinely refer to things being obvious in class discussions. His name is Owen and the way he marked his diagram was interesting to me on his problem set so I asked him to explain this in class. He started essentially the way I did but instead of a chart he simply wrote a 1 in the 1 box for # of paths and a 1 in the 2 box for the same reason. Now, his explanation gets interesting. Next, he mentions that it is obvious that if you get to hexagon 3 you have to have gone through either #1 or #2 so that the total number of ways to get to #3 is the sum of these two other numbers. Similarly, to get to #4 you have gone through #2 (one path) or #3 (two paths) and now Fibonacci is obvious. I was so delighted by his approach to this problem.

So I decided to present this problem to my other classes to see how they might approach it. In each class I explained my result after allowing them about 8 – 10 minutes to share thoughts about the problem with their small group partners. While none of my other students arrived at a conclusion in this relatively short amount of time, they did have some interesting approaches. One of my Discrete Math students tried to leverage what he’s learned about combinations by starting with the notion that a pathway along the odd numbers is six steps. Then he said that we could add one even number and this could be done one of six ways. We could add two even numbers to our path. This could be done in 15 ways (using combinations or Pascal’s triangle) and he wanted to simply add all of these up. A super cool idea but we started to see problems here. For example, if we add 6 and 8 as stops along the way in a row, then we have to skip hex #7 so we started trying to enumerate all of the path restrictions. Similarly, we realized that we’d need to individualize the number of odd hex visits in a similar way. Daunting, but a great example of trying to use knowledge he has gained this year. A group in Geometry recognized that the shortest path had six steps and the longest had twelve. They wanted to enumerate the number of pathways broken into these categories. A great idea and a way to get a handle on smaller cases to imagine. They quickly became frustrated by the daunting task of keeping track of these tracks, but I loved the idea.

It was a fun couple of days batting around these ideas. I have been really thinking about the distinction between ‘problems’ and ‘exercises’ and  problems like this one reinforce the ideas I am wrestling with. I am determined in each of my classes next year to have homework and classwork assignments labeled as ‘problem sets’ or as ‘exercise sets’ and I am hoping to help develop some clear strategies with my students to use when they encounter a genuine problem in math.

Persistence and Patience

Neither of the qualities in the title of this post are apparent in abundance at this time of the school year (exams here start on May 21!!!) so I am especially pleased to be able to write about today in my two Geometry classes. They each took a quiz with me in their last class and I asked them to read ahead to the next section (more on that later) before meeting again today. After a brief warm up we reviewed the quiz and I returned papers. Then I presented them with this image from our Geometry text


Now, at this point we have established that the measure of a central angle in degrees is equal to the measure of its intercepted arc in degrees. We have proven that the measure of an inscribed angle in degrees is half of the measure of its intercepted arc in degrees. I told them that my dream for today was to derive some formula relating the measure of angle EFC to some combination of the arcs BE, EC, CD, and DB. I reminded them of what we already know and suggested that using what we already know is often pretty helpful when trying to learn something new. I then stepped back and let kids toss out ideas. In one of the two classes I took a walk to the water fountain and popped in on a colleague for a two minute chat. I came back into a room with people debating and waving their hands in the air to show what segments they wanted to draw. I wish I saw more drawing at their desks, but it was good engagement. In both classes they wanted me to name the vertical angle pairs at the hinge point F. I was a bit surprised that neither group wanted to name arcs yet, but it worked out just fine. I dropped a hint that another segment drawn would help and this spurred some lively debate about what segments to draw. BE and DC were popular but I pointed out that the vertical angles were not included really in these triangles that were formed. Some folks wanted some radii drawn. One girl was sad when we named our vertical angle pairs. She saw that BFE and DFC were clearly equal but that they faced arcs that clearly weren’t. I was pleased by this observation. There was a real desire in each class to assume that BD and EC were equal and I wish my diagram was more clearly designed to discourage that. We finally settled on drawing the chord BD which created two inscribed angles that we called x and y creating arcs called 2x and 2y. The heavy lifting and guessing were done by that point. Arriving at the conclusion that we now need a relationship between one angle and two arcs came in a couple of steps. In both classes I made a clear statement about what we just discovered and they seemed pretty pleased with themselves, if a bit tired from the exertion. In my morning class we were on our 90 minute block and I gave them some practice exercises that we will revisit tomorrow. In my afternoon class that ended the day we simply left the discovery on the board. I start tomorrow morning with that crew and I am excited to pick this up again.

In each of my classes I have 14 students. One was absent today, so I had 27 overall joining in on this conversation. One of them, when I mentioned my dream for the day, said ‘In the reading last night it said that this angle is the average of two arcs.’ [Thank you Niko!]

One student. Now, it is entirely possible that other students did the reading I requested. It is possible that some read the section and were confused by it. It is possible that some read it and forgot the conclusion. It is possible that some who read it simply did not feel like saying anything out loud. It feels more likely that few (maybe only 1?!?) did the reading.

I could dwell on that, but I am dwelling on the discovery we made. I am dwelling on the persistence and patience of my students today. I am dwelling on what went right to end a day that started poorly for me.


Later, I’ll dwell on how I can help overcome the likely habits of not reading that I am faced with. I’ll save that for a less beautiful day than today.

Sad Little Girl

Two posts today, both kind of brief. They are about two aspects of my life. The first is my dad post, my Geometry teacher post is coming later today.


I have a 14 year old boy and an 8 year old girl. They are very different people. My son does not stress out about school in any visible way. While I hear parents in our community talking about their 8th graders spending hours on homework, I never see that. He is up an down in his school performance but he does not seem to judge himself by these exterior reports on his progress. Sometimes I wish he was just a little more concerned, but he’s doing just fine.

My girl cares deeply about these exterior reports on her performance. She wants her teacher to think highly of her, she wants to please us. I am charmed a bit by this but I also wish she was less stressed about these types of things. This morning she was unusually quiet and reserved before school. I thought it might be her allergies – she has pretty wicked seasonal allergies and spring has just exploded on us here – but that was only a small part of it. She was worried because in PE this morning she was due to take part of the annual fitness test. This made her super sad. She loves to run and play with her friends but she has already begun to identify some friends as ‘sporty’ friends and she does not see herself this way. She had tears in her eyes because of anxiety about a fitness test this morning. This brought tears to my eyes as I thought about other students crying in the morning because of an upcoming Geometry test (or an upcoming Biology test or whatever), I was shook up thinking about the impact on self-image, on feelings of self-worth, on just the general task of living through the day that I saw a glimpse of. I am so sad to think that I am seen as the cause of such stress in my students’ lives. I wish I had some insight into how to battle this for my daughter and for my students. I know with my daughter I can talk to her about how her time in running around a field has nothing to do with how much I love her or how I value her. I try, in an appropriate way, to let my students know that their grade on a paper is not an evaluation of them as people, just a snapshot of them as learners. I need to be more explicit with them more often as I was reminded this morning. I also need to be more explicit more often with my two lil Dardys at home.

Man, what a bummer of a way to start my day today. My next post will be about a much rosier ending to the day (at least the school portion of it.)


A brief post this morning. We are winding down in our AP Calculus BC class and the last topic of the year is a short unit on vectors and parametric equations. Many of my students buy a (slightly) different version of our text book so some do not have the vector chapter. I use a curriculum module from the AP site as the spine for our work through these ideas. I have to admit that I do not have a great amount of enthusiasm for this topic, at least at the level that we work with it. But on Wednesday we had a fun breakthrough in class. We were working on a fairly typical example of a parametrically defined function on the Cartesian plane and found its derivative. The kiddos asked for a picture so we graphed both the position and velocity vectors on Desmos. One of my students expressed disappointment that we did not see the order of the graphs so it was hard to move our eyes from one graph to the other to see how they related. I am more comfortable making GeoGebra jump through hoops so I moved on to GeoGebra and graphed both with sliders and leaving a trace on. The kids seemed to perk up a bit liking this visual better. Then one student asked me to change the velocity vector. Instead of having it rooted at the origin, he asked me to redefine it so that it was attached to the point, so that it would be a tangent vector. I made this adjustment (you can find my GeoGebra of it here) and the kids seemed so much more engaged immediately. The power of seeing the trace points move apart from each other combined with the direction and length of the velocity vector changing along really caught their attention. I want to tweak it a bit still, there was a request for adding the acceleration vector as well. At a time of year when energy is running low, it was a fun blast of energy and engagement here.


That’s all for now, just wanted to share something fun.

Looking for a New Teammate

My school is looking to hire a new upper school math teacher beginning in the 2018 – 2019 academic year. I am in my eighth year here as chair of the upper school math department at Wyoming Seminary. We are located in northeastern PA right across the river from Wilkes-Barre. We are a preK through post-grad school on two separate campuses. The young ones – including my two children – are on our lower school campus located about three miles away from our upper school. My son, by the way, will be at our upper school next year. I am equally anxious and excited about this development. Our upper school is a 9 – post grad school that is a mix of day and boarding students. We have three dorms on campus and a number of faculty also live in campus housing. I lived in a boys’ dorm with my family for six years before moving into a campus home. We have a wonderfully diverse student body both in terms of where they come from – we have over twenty countries represented in our high school student body – and what their interests/talents are. We have nationally recognized athletes, we have top notch artists, we have stunning scholars. We have kids who LOVE math and are in Calculus as freshmen. We have kids who dread it and are in Algebra II as seniors. I firmly believe that all of these kids have meaningfully experiences and they grow as students while they are here. We are losing one of our valued members of the upper school faculty and I am sad to see him go. However, this is the time for me to look ahead and dream about what a new colleague can bring to our school. If you are reading this and are contemplating a change of scenery for next year, please reach out to me in the comments here or through twitter where I am @mrdardy or simply write to my work email. If you are reading this and you know someone who might be a great fit, pass this info along.

Our school’s website is https://www.wyomingseminary.org/


What Math Teacher Do My Students Want?

At TMC16 one of the impressive experiences was listening to the keynote speech by Tracy Zager (you can find it here ) and I remember briefly mentioning it in a post soon after the conference. Sadly, I never went back and really dissected my positive feelings about it, but I do know that one of the habits I developed after her speech was to communicate pretty regularly with our elementary school teachers more regularly when I ran across interesting math tidbits. I also know that Tracy has written a well regarded book called Becoming the Math Teacher You Wish You’d Had. I have not read this book (yet!) but the title of the book has been haunting me. I am not speaking with any deep knowledge about the strategies that she is advocating and I may be doing her a disservice, for which I apologize in advance. But as I am sitting here at the beginning of my spring break, I am thinking about what has gone right and what has gone wrong so far, 2/3 of the way through my 2018 – 2019 academic year. I also think about the phrase from Dan Meyer’s TED talk, ‘Be Less Helpful’ and this morning I want to try and wrestle with each of these phrases that are resonating in my head.

I will admit up front that my interpretations of these two phrases are my interpretations. I also need to fill in a little background to explain how I understand Prof. Zager’s phrase. I had two high school teachers who were enormous influences on me. I had the same English teacher, Mrs. Myra Schwerdt, for my junior and senior year. The way her typical class went was this – we came into class, she tossed out a question prompted by our reading assignment from the night before and then she moderated the discussion. Tossing out another question if the conversation ran dry or probing someone on an opinion/interpretation that they made. She was sort of first among equals in these conversations. I never felt that she forced us to interpret anything in any way but I also felt that we better have some backing for what we had to say. In my senior year I took AP Calculus BC from Mr. Barry Felps. A typical day in that class started off with a quick look at a new idea with Mr. Felps offering an example or two of ‘how to’ after explaining some new idea. He would sometimes go over a HW question or two from the recent past (often after his go to joke, we’d ask ‘Mr. Felps, can you do problem #12?’ He’d look at the book and nod while saying ‘Yep’. Still cracks me up a bit thinking about this…) After this, typically 15 – 20 minutes total, he’d say ‘Okay, you have work to do and so do I’ He would sit and we would work with our neighbors and friends. If we got stuck, we’d go ask him but most of the time we felt that we wanted to figure it out ourselves. We also had a study group that would periodically meet on Sundays for a combination of football, pizza, calculus, and physics. So, here I have in my mind what ‘Be Less Helpful’ looks like (to my 17 year old self for sure) and an image of the math teacher I wish I’d had (which, I feel enormously fortunate to say is a math teacher I did have)

Fast forward from 1981 – 1982 academic year to the 2018 – 2019 academic year. This year I am teaching four different classes with pretty different groups of students. I teach Geometry with the text I wrote four years ago to mostly 9th and 10th graders, I teach a Discrete Math elective to mostly seniors, I teach a non-AP Calculus class to a mix of juniors and seniors, and I teach AP Calculus BC to a mix of juniors and seniors (with one brilliant sophomore in the mix) These classes all have different needs and different inherent investments from the students involved.

It is, of course, unfair to generalize too broadly, but I think it is fair to say that the general needs/wants of these classes differ. I try to account for that, but I know that there are some tendencies in my teaching that appear in all four subjects. I ask more than I tell. I redirect questions to other students to get their input. I have student in groups of three facing each other. I randomize these groups so that a group of three is together at most for five classes. I write problem sets that dip into past knowledge and ask some questions for which we have not been explicitly prepared. My quizzes are narrowly focused on recent information and they are weighted less than tests. My tests are all cumulative in nature but they are about 70% focused on what has happened since the last test. I want at least one test question to feel novel, to ask students to put together information in a way that feels new – a problem rather than an exercise, I guess I’d say. We have a test correction policy that we adopted this year (you can read about that here, here,  and here  ) and this policy seems to have helped reduce anxiety a good bit. All of this is to provide a little context into my classroom and my vision of what these phrases ‘Be Less Helpful’ and ‘Becoming the Math teacher You Wish You’d Had’ mean.

What I have struggled with, this year more than in the recent past, is the discrepancy between the math teacher I wish I’d had and the math teacher that my current students wish they had. A good number of my students seem to be happy wth how our classes run. I have posted a number of entries this year about student success. I have received some lovely emails from parents of students in Geometry, some nice remarks passed along from colleagues about students in Calculus Honors, some terrific conversations with students in Discrete Math, and an abundance of energy and creativity from my AP Calculus BC gang. In all four of my classes I feel I am reaching some students and making a positive impact. What concerns me, and what prompts this post, is the fact that a number of students are not buying what I am selling. They are frustrated when I respond to a question with a question. They think it is unfair to have a test question that does not look like something they have explicitly practiced. They feel that I am off loading my responsibility as the teacher when I ask them to work in groups to figure something out instead of lecturing and telling them how to figure it out. I have students across a wide range of abilities and a wide range of reactions to what I am trying to accomplish. It is important that I recognize this and do not simply bask in the glow of the students for whom this approach really clicks. Where I struggle, is trying to reconcile what I think I understand about teaching research, what I understand about NCTM’s recommendations, and what I valued as a student with the discomfort and unhappiness that I see in some of my students. I also struggle with balancing what is clearly working with some students in each subject. I don’t want to lose that energy and motivation that I see in students who value what feels like a different way to experience math. I know that this is not an either/or situation. I have two weeks of spring break to think and reflect. I know that there is a way to reach the frustrated kids without giving up the the facets of my class that are valuable to some of their colleagues (and to me!), I know that there is a balance to be struck between asking kids to step out and feel challenged and making sure that they still feel supported.

I think back to a comment from a student abut 6 years ago. About three weeks into the school year she asked for a personal conference to talk about her struggle in our Honors Calculus class. By the way, struggle for her meant that she had a B average instead of her typical A. She said to me ‘I thought that I needed to learn formulas and how to use them. That isn’t working here.’ She and I had a lovely conversation and in the years since I have run into her dad a number of times. He always tells me how important my class was for his daughter. She has sent me lovely thank you notes updating me on her progress. She is an example of a student I was able to help cross a certain threshold. I want that feeling for all of my students in some way or another and I have solid evidence that this is not happening often enough right now. I have some serious thinking to do.

A Guest Post

The last time I wrote, I was talking about how spoiled I am by my students. I have two students who did an enormous amount of thinking about a random walk problem I proposed. Just for your reference, the problem was

Starting at the origin, a bug jumps one unit either left, right, up, or down. He jumps once each second. List the possible locations (and probabilities associated with those locations) that the bug could be after 6 seconds.

After working on it for awhile with one of my students I waived this one off and told my students to feel free to ignore it. A number did not. Two in particular, Bobby and Matthew (both of whom seemed happy to have me name them!) collaborated over a chat line of some sort and delivered a lovely presentation on their work. I asked them to guest post for me and below is their post. I cannot emphasize enough how impressed I am by their ingenuity and determination. There are two other files they asked me to make available.  A 3d representation and a 4d representation generated by their code.


Bobby: Last week, a blog post referred to a random walk problem that our Calculus class worked on, and two students that took a coding approach. We are those two students, and we’re here to discuss how we solved the problem, and the far more interesting work that came after it. For those of you who don’t know, the problem was a 2-dimensional random walk of 6 steps.

Matthew: When I first looked at this problem in class, I thought it would give way far more easily than it actually did. My first attempt was to rewrite the problem as choosing from the four cardinal directions a set of six steps, e.g.  {↑ ,→, ↓, ,←,}. By doing so, my hope was to reduce every position on the grid into a set of the bare minimum moves needed to reach it, and pairs of blank spaces which I could fill with a pairs of steps which added to the zero vector. Since the grid had 8-fold symmetry, I was unafraid of solving each point individually, however when the time came to actually do the problem, I found a number of oversights in my initial work which resulted in some pesky double counting.

Bobby: For lazy Calculus students (read: myself), the obvious solution for any random walk problem is to make a computer do it, so i set out to approximate probabilities by running a billion trials of a random walk and counting up the results.

However, this brutish method is relatively unsatisfying, and by the time I finished, Matthew had abandoned his set permutation approach and decided to make a program that iterated through each possible set of 6 steps once and counted up the destinations. This clearly being the better approach, I did the same thing, and we exchanged our results (which matched) and exchanged our code to make improvements: I took some of Matthew’s ideas to make the program easier to scale up and down, and I’d like to think he took some of my ideas and stopped naming variables “DONK,” but I’m sure he didn’t.

This grid shows the number of ways to land at any given point. After a short time looking over our grid of results, we noticed that each edge of the square was the 6th row of pascal’s triangle. A little more poking around and Matthew noticed that the grid was in fact a multiplication table of that row of the triangle with itself, meaning every entry could be written in the form (nCa)*(nCb), with n being the number of steps and a and b being the relative coordinates of the entry in the table. Finally, we made the observation that this is the 6th layer of Pascal’s square pyramid, a 3d structure much like the triangle where each number is the sum of the four above it. Recalling that a 1d random walk is the nth row of Pascal’s triangle where n = # of steps, a pattern seemed to emerge that we hoped might scale up in dimensions.

Bobby:  We did a lot more work, particularly in 3 and 4 dimensions. Our program logic started by defining a variable to track which number walk we were on, and keyed in on the base-four version of that number to define a set of moves for our bug. For example, test 5 translates to 000011 in base 4, and the bug reads each digit individually, right to left, using it as an instruction for motion: 0 = right, 1 = up, 2 = left, 3 = down, translating 000011 to U, U, R, R, R, R.

This logic makes it easy to change the number of dimensions or steps, so we went to 3 dimensions, expecting to get 3-dimensional cross-sections of some 4-dimensional solid for each particular solution. The results did not disappoint, as we got the expected octahedron as a result. Obviously, this output was a little tough to format, but I think what we settled on is easy enough to read. Each grid separated by the others from a row of asterisks is a 2d cross-section of the solution octahedron. LINK DATA Stack those layers on top of each other and the visualization is pretty easy, I think.

At this point, we had to do four dimensions, and had a good idea of how it would look. The 2d solution is the shape of the 1d solution stacked on itself, the 3d solution is a stack of the shapes of the 2d solution, so we the 4d solution would obviously be a series of 3d octahedrons in the same x, y, z, but translated along our fourth spatial axis. It took me a moment to wrap my head around this data, but I’ve become very comfortable with it. LINK DATA Each section under the rows reading “NEW 3D SOLID…” is a series of grids that are stacked into an octahedron, and then the octahedrons are stacked on each other in the fourth spatial dimension. This, of course, is also a 4d cross-section of a 5d Pascalian solid, each nth 4d layer of which is a solution to the 4d random walk of n steps, but that’s about as far as I can go envisioning these shapes. Matthew tells me the 5d structure is “Pascal’s Orthoplexal Hyperpyramid” but I’m not sure I can trust that.

Matthew: Of course, the idea of layers of Pascal’s square pyramid as multiplication tables of the rows of Pascal’s triangle is a fascinating one, and one I sought to prove. Though my initial effort to prove it algebraically did meet with success, I was unsatisfied with my clunky proof and did some quick googling on the subject. My search turned up a blog with a simple proof by induction, far more elegant than anything I had done. Still, I find that proofs by induction are rarely enough to understand a result intuitively, and so I continued working with Bobby, looking for a better way to interpret the formula in terms of the problem, as well as potentially a way of generalizing the result to higher dimensional walks.

Eventually, I took a step back from the symbol soup and searched for a more intuitive interpretation of the formula we had discovered in terms of the original problem. After some while, I had an epiphany while placing the problem back into the language of sets. My initial idea of viewing the problem as selecting a set of 6 steps along cardinal directions could be reformulated in a way which made the formula obvious. Instead of considering steps along the cardinal directions like the original problem had stated, I realized that every unit vector along a cardinal direction could be rewritten uniquely as the sum of two diagonal vectors of magnitude √2/2, e.g. {} = {↖,↗}. With this realization in hand, the original problem can be rephrased as a random walk in six steps of magnitude √2/2 along the line y=x, followed by another random walk of six steps perpendicular to the first. Or, to give an example in my initial interpretation of the problem as sets of steps, the two dimensional walk {↑ ,→, ↓, ,←,} is the same walk as {{↖,↗},{↘,↗},{↘,↙},{↖,↙},{↖,↙},{↘,↙}}, which is the same as the union of the two one dimensional walks {↖,↘,↘,↖,↖,↘} and {↗,↗,↙,↙,↙,↙}. Once the problem is reformulated in this way, the reason why the table obeys such a simple multiplicative relationship in two dimensions becomes satisfyingly clear. Though I am pleased that we were able to transform such an originally dense and unapproachable problem into a setting where it is obvious, we have unfortunately as of yet been unable to find an application of this method to higher dimensions.

I’m Spoiled

This post is inspired by my AP Calculus BC class. At the beginning of each of our seven day rotations, I give them a problem set. These problem sets are pretty wide ranging, some Calculus questions, but mostly wide open fun problems to explore. Here is a problem from the set that was due today (actually due yesterday, but we were snowed out):

Starting at the origin, a bug jumps one unit either left, right, up, or down. He jumps once each second. List the possible locations (and probabilities associated with those locations) that the bug could be after 6 seconds.

I have given them variations of this where the intrepid bug could only move left or right. I thought that this would be an intriguing extension. During my library duty nine nights ago one of my students spent about 15 minutes with me bouncing ideas around. We came up with the GeoGebra sketch below:  

This seemed pretty daunting and I admitted to him that I may have overreached with this problem. We shared our conversation the next day with the whole class, including showing this intimidating graph. Trying to figure out how many of the 4096 pathways could lead to each point felt overwhelming. A couple of students started tossing out ideas but I kind of encouraged them to let this problem slide and blame me for poor planning.

When the problem sets came in today three of my students discussed some coding attacks that they took. Two of them were collaborating late into the night working on an approach to the problem and the output from their attack is below:

They took over the class conversation today pointing out that they recognized a pattern based on a multiplication table where the column header AND the row header were each the 6th row of Pascal’s Triangle. The numbers in their printout above are the products of this table. A long explanation invoking a three dimensional Pascal’s Triangle (I thought of layers like oranges in a conical pile) was presented with great enthusiasm. Some of the kids in class kind of glazed out, but a number were fully engaged. I was engaged, awed, and slightly confused. I have asked the two who collaborated on this if they would be willing to be guest bloggers here and they seem amenable to that idea. I hope to have a follow up, in their words, in the next day or two.

I am SO spoiled to be able to sit back and learn from students who could have easily left this problem alone, but were unsatisfied with that notion.