How Do We Help our Students Ask Questions?

I have told this story to a number of friends and colleagues over the years. One of my favorite former students – he graduated in 1994 – gave me what I value as the best compliment I think I ever received about my teaching. He was a brilliant kid, school seemed effortless to him. I taught him for four years in a row culminating in AP Calculus BC when he was a junior. He took a math class at UF his senior year. About ten years ago I was living in Jersey and he was living in NYC so I had the chance to see him a few times there. Once we were having lunch and he told me a story. He worked in a small business doing financial analysis and he was frustrated by a problem he had been working on. He told his boss that he was going to take a long lunch to clear his head. When he came back his boss had left some notes for him on the file he had been working on. He told me that his boss reminded him of me. “He asks questions that I would not think of asking myself.” I walked away so happy about this. He did not remember a trig graph or a derivative or the fundamental theorem of calculus (although he probably did remember these things!), instead he remembered that I asked him questions that he did not think of asking himself. I felt SO good after that conversation. I was thinking of it today after school during our afternoon conference time built in to our day. All of our teachers are expected to be in our rooms for about a half hour after the end of the school day and many students make a habit of dropping by to ask questions. I was talking to two of my  Calculus Honors kiddos. This is our non-AP class that takes a deep year-long dive into Differential Calculus. We were looking at some problems on one of the problem sets I wrote and these two young women were saying that they understood the problems when we talked in class but they did not know how to start them on their own. I pointed out that almost all of the ideas in class came from the students, I rarely flat out TELL them how to solve a problem, we work through the question together. What I work really hard on is to ask questions of the students that prompt them to see connections and realize what they know about the problem. I want them to feel the power of being the ones who generate the answer. One of the girls said that she does not know what questions to ask herself when she is home working on these problems. So, the challenge is to figure out how to help her, and others, across that bridge. Is it enough to simply model an inquisitive mindset? Is it enough to be a good role model in persistently asking questions? How can I explicitly help my students develop that instinct and ability to push themselves along a solution path by asking meaningful questions? I would love to hear any wisdom on this front. I am going to share a meaningful quote that I ran across in my days as a doctoral student studying curriculum and instruction:

Genuine enquiry is an important state for students to recognize and internalize as socially valid. Consequently it is an important state for teachers to enact. But it is difficult to enquire genuinely about the answer to problems or tasks which have well-known answers and have been used every year. However, it is possible to be genuinely interested in how students are thinking, in what they are attending to, in what they are stressing (and consequently ignoring). Thus it is almost always possible to ask genuine questions of students, to engage with them, and to display intelligent directed enquiry. For if students are never in the presence of genuine enquiry, but always in the presence of experts who know all the answers, then students are likely to form the impression that there is an enormous amount to know, and that experts already know it all, when what society wants (or claims to want) is that each individual learn to enquire, weigh up, to analyse, to conjecture, and to draw and justify conclusions.

 

John Mason

Why Do We Know What We Know?

In my AP Calculus BC class today I presented an activity from Christopher Danielson that I found through the MTBoS Search Engine

You should check out Christopher’s post. I asked the questions he proposed, but I had a new first one. I asked my students to discuss in their small groups what assumptions they were making about this function pictured here. I heard some good stuff. They talked about continuity, they talked about it being a polynomial function, hopefully one with an even highest power. They talked about the fact that circle C could not have a root since it is entirely above the x-axis (although one student raised the question of complex roots and this prompted a conversation about use of the word root versus calling them x-intercepts), they talked about the minimum number of critical values. In general, just some great recall. At our school, BC is a second year Calculus class so we were talking about ideas from last October/November. This led me to raise a question that I was a bit worried about. Earlier, we had made mention of the fact that polynomial functions with an odd highest power have all real numbers as their range. Sure, we know this. But why? Do we really have an idea why this is true? I was worried that this was too vague a question. I was worried that they would waive it away. We know this is true, Mr. Dardy. Why talk about this? Instead, we got some great GREAT conversations. I was told to think about limits, to think about derivatives. I jokingly asked if we should think about area between curves or optimization or some other time honored Calculus ideas. I was told to consider the limit as x grows without bound both for a positive leading coefficient and for a negative one. We discussed how all terms in the polynomial eventually become insignificant compared to the highest powered term. We talked about the derivative being even powered and what we know about those graphs. Man, I was just so pleased that they were willing to travel down this sort of hazy questioning path with me and reinforce what they know and WHY they know it. I say this every year, but this class absolutely spoils me.

 

Reflections – While thoughts are Fresh

In my two morning classes today we tackled the problem that I just blogged about (link here)

Some interesting observations first and then some questions that came from my learners.

When data is not presented as a table, there is a distinct extra layer of processing that has to happen. One student went straight to desmos to graph the points his group had. I liked that he wanted a visualization. A number of them, when I asked how to find the AROC between two pictures were flummoxed. Let me give you an example of what I saw from a number of students. Look at the picture below:

A number of students divided 1381.5 by 34.18 (more on that in a moment) and arrived at 40.418 as an answer. This matched the given information but they did not seem to notice that this did not answer any part of my question about AROC from point A to point B, from point B to point C, or from point A to point C. [This conversation makes sense (I hope!) if you have read the first blog post that is linked above]

I had to poke to get them to recognize that any time we talk of an average rate of change, we are talking about more than one data point to consider. A number of them were happy to enter a time like 30:56 as 30.56. This disappointed me a bit. For smaller minute data points I can see the mistake a little more clearly. For something as close to a full hour as 56 minutes, the willingness to enter .56 seemed more clearly wrong.

Once we ironed out the fact that we need to see time and distance as coordinates of a data point, then the slope idea for AROC fell into place more comfortably. Before talking about my challenging final question, I want to share a few questions and observations from the classes.

A student asked about the clock on the dashboard. Does it still count when the car is sitting? When I am sleeping? Great question, the answer was no. It only moved when the car was moving.

Conversations came up about the geography of Florida where I was driving. They guessed that some snapshots were taken in city traffic, others after highway travel.

Discussions came up about why the average on the dash did not change even when intervals had different AROCs. I relied on a baseball analogy. Late in the year a 0.250 hitter might have a great day and go 3 for 4 while not changing his overall average at all. That seemed to make some sense to them.

The final question I asked was more difficult for them than I had anticipated. I asked each group to consider the following situation. How long would I have to travel at 60 MPH to raise the trip meter average velocity to 42? In my mind I simply wanted to use the last data point as the jumping off point and add an unknown time ti to the x value (the time input) while adding a distance of 60t to the y variable (the distance output.) This idea did not organically appear in class and I pushed a bit more than I wish I had as I saw our 45 minute time together elapsing. I am often comfortable with questions being unresolved at the end of a class period. However, with this question at this time of the year (this was only our fifth class together) I really wanted a conclusion to the mystery. I will definitely revisit this experiment as we work more deeply on our ideas of rates of change and I will remind them of this conversation on a number of occasions. An unexpected bonus idea that came through loud and clear was the MVT even though we don’t have a name for it or a formula describing it yet.

Pretty pleased, I must say.

 

Exploring Rates of Change

One of the courses I teach this year is a course called Honors Calculus. It is a non-AP course and we made a decision about 6 years ago to make this a course in Differential Calculus. While the AP AB course completes a college semester in a high school year, this course completes an AP AB semester in a high school year. This allows us to remedy some bad habits, fill in some gaps in understanding or mechanics and, most importantly, really slow down and think about what we are exploring. I have an activity that I do on the first day of class where we explore motion. This year we went in the hallway and rolled a whiffle ball, a softball, and a lacrosse ball down the hall a fixed distance. We tried to roll each with approximately the same force and had a good conversation about what the data told us. We made some physics based observations that I did not plan for and we talked about what we knew and what we did not know about the rate at which the balls were rolling. My goal was to arrive at the conclusion that we could talk about average rates of change but not so much about the rate at a particular instant. The conversations went reasonably well, but we got distracted a bit by conversations about bounciness of balls and air resistance. In any event, I think I planted some decent ideas to consider as we embark on a conversation about  average rates of change of a function on an interval (our text calls this the AROC) and we are about to wrestle with the limitations of being able to know much about the instantaneous rate of change (the IROC)

This summer I rented a car whose dashboard gave information that I knew would work well for this class. A picture below will prompt your teacher brain as well I think.

I took six such photos during the course of my trip. Tomorrow, I intend to give each of my groups of students (I have them in groups of three) three of the pictures. I’ll scramble them up a bit so different groups should have different subsets of the data. I intend to ask them some pretty simple questions that should generate some good conversations. I want to ask them the following questions:

  1. What was my average speed between any of the two pictures? (So each group should have three answers for this)
  2. Can you determine my maximum velocity in that time interval?
  3. I want to raise the average to 42 MPH. How far would I have to travel at 60 MPH for this to happen?

 

These are not terribly deep questions, but they feel rooted in an example of real world data (I was inspired by Denis Sheeran’s wonderful book Instant Relevance for this data driven experiment) I also think that this will continue to scratch at the itch that will make the breakthrough of being able to find the IROC feel more meaningful.

I have all the photos together in a WORD doc on my dropbox. You can find that file here. I would love to hear any clever ideas about how to play with these images/this data.

 

Beginning of the Year Thoughts – Inspired by Christopher Robin

About three years ago I made a commitment to myself. I was still living with my family in one of the boys’ dorms on our campus and was living within a pretty strict clock regimen of dinners at the dining hall, study hall hours, lights out (ha!), and rotating dorm duties. I made a decision that I would close my computer when I left my classroom at 3:30 and not open it again until study hours started in the dorm at 8 PM. I managed to maintain that commitment (mostly) for the year. I was regularly up later than I really wanted to be due to dorm duties and noise. I was up early every morning catching up on work, but I had every afternoon free of staring at my computer. I hung out with my kiddos, we played on the dorm lawn, we ran around with friends on campus, I sat at the dining hall and caught up with folks. Part of what made it possible to carry out this goal was that I was teaching four classes that year, most years I have taught five.

This year I am teaching four classes again and, once again I made this commitment to myself. This past week was our last week before classes. I took the time to take each of my children to see movies. On Thursday I took my son to see the new Spike Lee film. It was terrific. On Sunday I took my daughter to see Christopher Robin. It was sweet and tender. A bit predictable, but filled with good nuggets. I got emotional on a number of occasions because I have become that dad. But I walked away renewed in my sense that I need to create space. I need to put that laptop away for awhile. Email will be answered eventually. My job as a teacher and a department chair is not THAT important that a few hours will get in the way of carrying out my responsibilities in a meaningful, humane, and productive way. I can just be dad and husband for a few hours. I can hang out on the hammock. I can sit and chat about music with my boy. I can watch my girl play with LoL dolls and her Baby Alive crew. I can sit on the porch and read. I can listen to some music and just be. If I do this the right way, then when I open my email and my files I will probably be a better colleague and a better teacher.

Sometimes ‘Doing nothing leads to the very best something’ as a silly old bear once said.

 

Some writing about teaching again soon. But it won’t be done between the hours of 3:30 and 7:30 PM!

Opening Day Activity – Evolving

I am not 100% sold on how I have divided the data, but I want to think out loud about how my opening day activity for Precalculus Honors is developing.

Last year, I took the advice of a number of online colleagues and divided my classroom into table sets of 3 for student groups. I anticipate that I will have five of these student groups in Precalculus Honors (PCH from now on in the post) so I set up five subdivisions of the data that I previously shared (in this post) and I will start off with the cleanest of the data sets. At the bottom of the post I will attach GeoGebra file links as well as the current status of my handout. I will present screen shots of the data subsets and discuss my hopes and dreams for how this activity will unfold.

Group #1 will have an image similar to this when they graph their data. I will share this exact one with the whole class. I set up five GeoGebra files with the same screen dimensions (at least I think the dimensions are exactly the same, hope so!) My hope is that this will feel largely quadratic to them. I do not think that they have previous experience with regression equations on their TI or with any online tool. I am suggesting that they use Desmos in class. I will present GeoGebra with the goals of exposing them right away to two of my favorite tools. We’ll discuss which one might feel better for different situations. In the past few years I have had a number of students adopt both programs as a natural part of their problem solving. On more than one occasion a student has written a note on a problem set along the lines of ‘Desmos agrees with my conclusion!’ A natural direction for them to go is to use point H as a vertex and develop a quadratic that fits this reasonably well. Hopefully, we can incorporate a discussion throughout the year of the messiness of real world data compared to mathematical models. I love quoting George Box here – ‘All models are wrong, but some are useful’

Group two will generate something similar to this image. Again, I am anticipating a quadratic guess. This time point E should be seen as the vertex. Playing a bit with my TI and anchoring my guess at point E yields the following promising picture. 

Now, we are getting somewhere!

Group three sees this data and has some decisions to make. This is where I am really questioning myself. I worry that there is too much here for a first day activity. I think I may tweak this data set so that it is also more deceptively quadratic in nature.

Group 4 picture above.

Group five data picture.

And, finally, the whole set together here –

As I write this and think about my goals for that first day, I am sure that I want to modify the data that group 3 gets. I don’t want this play to become frustrating on day one. Let’s save frustration for later on!

I am hoping to plant seeds for a number of interesting conversations to have over the course of the year. I want the students to think about decisions based on small sets of data versus larger sets. I want them to think about periodicity and where/when/why to expect that behavior. I want to present them with an intentionally open question on our first day together to set a tone for open questions together throughout the year. I want them to remember something fundamental about quadratics and I expect to present two forms (standard and vertex) on the board after a little nudging from the groups.

I worry, as I often do, that I am being too ambitious. We have about 30 minutes together on day one due to a whole school convocation that day. I am really debating whether to make this the activity and conversation for our first full class day together. In that case, I would pull out an old favorite problem to have as the conversation seed for day one and mix in a bit of the boring old syllabus, etc on day one. Any advice there?

Here is a link to a dropbox folder with the GeoGebra files as they currently are arranged as well as my Word handout.

As always, advice/comments/questions are welcome here or over on twitter where I am @mrdardy

Preparing for Precalculus

For the first time since the 2010 – 2011 school year, I will be teaching our Precalculus Honors course. Since I left the course we changed texts and five teachers in my department have arrived since then, including the colleague who I will be working with on this course. Now that it’s August, I sat down this morning with our text (we are using the Demana, Waits, Kennedy, etc) book and I have some thoughts/questions that I want to air out. I will understand my own thoughts better after writing and I anticipate some helpful wisdom coming my way through this site or over on twitter. We start off in Chapter Four of this text, doing right into trigonometry.

Things I know I don’t like

  • Any formula to convert angles to arc length. Just emphasize part/whole relationships!
  • Language of vertical or horizontal shrinks or stretches. I just want to talk about amplitude and period. It feels like this extra language just clutters things up.
  • Inverse trig function using the odd negative 1 power. I want to write arccos x or arcsin x. Pointing out where it is and what it looks like on their calculator is a necessary nuisance.

 

Things I think I don’t like

  • The text has an odd emphasis on the word sinusoid. I don’t know why I would want to use that word, not clear on any benefit.
  • DMS notation. Why? Not sure, other than in surveying, when they will encounter this.
  • Introduction of cosine as x / r and sine as y / r – I kind of want to talk about the fact that all triangles are similar and simply scale down to ‘unit right triangles’ with a hypotenuse of length 1. This feels like a natural lead into the unit circle.

 

Things I know I like

  • An activity my colleague shared with paper plates, strings, and discovery that an arc that is equal to the radius will be subtended by the same central angle no matter the size of the plate.
  • The opening day activity I am tweaking that involves the length of daylight hours as a function of days after January 1.
  • Conversations I am planning on having about why it is usually sufficient to solve a triangle by knowing three facts out of six (three side lengths and three angle measures) and when it would not be sufficient.

 

I have taught Precalc at each of the four schools where i have worked and I always enjoy the course despite its weird, buffet style curriculum. The kids are fun to work with, sophisticated enough to have serious math conversations. We do not have the AP calendar breathing down our necks and our new schedule that includes a 90 minute class once every seven school days really lends itself to some meaningful play time in this course. I’m excited.

As always, please share any opinions/advice/questions here in the comments or over on twitter where I remain @mrdardy

 

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.