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.



This post is inspired by a twitter exchange with the awesome Joe Schwartz (@JSchwartz10a) and by a running exchange with one of my Geometry students. Joe tweeted out the following picture: 

The picture was accompanied by the question : “Do 3rd graders know the answer to this question? Truly curious.” It just so happens that my Lil Dardy is in 3rd grade. I showed her the question and (briefly) explained the equation written. I replied to Joe that she was not surprised to see it written that seven sixes is the same a five sixes plus two more. However, she did not know any vocabulary word to describe this. Joe replied, succinctly, “And she doesn’t need one…” It made me smile. It also made me think when I was reviewing for a test with one of my Geometry classes. We just finished a chapter on triangle bisectors and centers. Loads of vocab in this chapter. Very few new skills, just new words describing relationships. Thinking back to the exchange with Joe I found myself questioning my decisions in writing the book and in teaching this chapter. During the test review a student asked if there would be any vocabulary on the test. This particular student has asked this question before just about every test. I answered the way I do just about every time. I told him that he needed to know what these words mean to accurately interpret the questions at hand. For example, if I ask about altitudes to a triangle, he needs to know what that means. However, there would not be a question where I simply ask him to replicate the definition of an altitude. Thinking back on this exchange, and this way that I answer the question, I have a ton of questions that I need to ask myself and I will start by posing some of them  my readers out there.

  • My guess (an uncharitable one) is that the student asking about vocabulary is looking to avoid committing anything formal to his short term memory before a test. Admirable in a certain way, but what does this question say about what he thinks his job on a test is? Why would students who have been working with words day after day express any serious concern about being asked what those words mean?
  • Real people have real vocabulary that they use in their studies, in their work environment, etc. I recoil at the suggestion that I should do something objectionable now because someone will do it to my students later. But, I am beginning to wonder whether I am cheating my students a bit. Should I be more emphatic in urging them to be careful about vocabulary now so that they will better understand what they read or hear later? Am I being lazy when I let them casually refer to the longest side of any triangle as the hypotenuse? [Note: I have written about this before. I DO correct them, but in a pretty gentle, nudging way. I remind them every time that the hypotenuse is a specific name, but this habit has settled in with my students for a couple of years now.]
  • What are we communicating to our math students if we mark points off or hold them accountable in some ways to formal language if they can get their mathematical ideas across through their work? Are these skills dependent upon one another? Is it okay that my students can swing into action and write the equation of an altitude of a triangle but be uncomfortable and vague if asked to write a definition for what an altitude of a triangle is? As someone who is so comfortable with these words, I struggle to understand how someone can write that line without being comfortable that they can write a definition, but I’ve been teaching long enough to know that this is a real thing.
  • Is this another instance where students have been trained to think that there is one right way to answer a question and their job is to make sure that they simply regurgitate (if they can decode correctly) what that correct answer is. I, of course, hope that my grading policies and the way that I communicate in class convinces my students that this is not the way life is in my classroom. However, I know that I am battling impressions that have formed over years.
  • More importantly – Does it matter that my students know things like the altitudes of a triangle intersect at the orthocenter? Is there ANY chance that they will remember this in a few months? In the past few years I taught the course, I pretty much only mentioned the word centroid and avoided talking about incenters, circumcenters, and orthocenters. I am not at all sure that I made the right decision then or that I made the right decision this year in explicitly defining them. In my text the words centroid and incenter are explicitly defined. Circumcenter and orthocenter do not even appear in the text. A mistake then? A mistake now? I’d love to hear some advice/opinions.

Gotta get dressed for school now. More thoughts swirling and I hope I am disciplined enough to get them down soon.

Thanks to Joe for prompting this post!

As always, you can reach me here in the comments section or over on twitter where I am @mrdardy

Greetings, 2018

A non-mathy post for this morning. I feel like I need to clear my head out a bit here.


  • Thanks to Meg Craig and the #Fitbos gang for helping to keep me motivated this past year. I set two goals for myself with my trusty fitbit flex. I wanted to accumulate an average of 30 minutes per day at an ‘active’ level. I compiled a total of 207.45 active hours. Last time I checked my multiplication, this exceeds my goal! I also set a goal of walking 2017 miles in 2017. I ended up at 2050.16. I am pretty pleased, but time is still working against me, despite this level of activity I am more achy and a bit paunchier than I was this time last year. Have to ramp it up to fight against Father Time.
  • Thanks to connections that my wife has at her college I was able to score a gig as a DJ at the local college radio station. Almost every Thursday since June, I have had the great pleasure of spending two hours (from 4 – 6 PM ET on wrkc.kings.edu) playing pretty much whatever music amuses me on terrestrial radio. I have been compiling playlists over at Spotify where you can search me up as mrdardy. It has been one of the real joys of my life this past year.
  • Mostly a consequence of my DJ gig, I have listened to more new music released in 2017 than any year since the birth of my son in 2003. It feels great to be reminded of the pleasure of discovering new music again. I still feel a bit overwhelmed when I read Best of lists at the end of the year, but there is a better chance of me knowing a number of items on these lists than I have had in years.
  • At work we have had a couple of important changes. We moved to a new, rotating schedule. We have 7 periods, 5 of which meet each day. In a seven day cycle each class meets five times. Four of the meetings are 50 minute classes (every once in a while an assembly moves that back to 45 minutes) and meets once for a 90 minute block. This has been a great change in our daily lives.
  • In our department we adopted a test correction policy where all students are allowed to earn back points by reflecting on their work. We ask them to submit corrections in the form of pointing out where/what went wrong in the problem’s work and then correcting said problem. I am super excited about this project and I see students being really thoughtful and attentive in submitting these corrections.
  • My life at school has been a bit more hectic than I’d like, despite the change in schedule. I have five classes this year (more often than not, this has been my standard work load here) which is especially manageable in this new rotation. What has been tiring is that I have four different class preps. Keeping all these trains running in my mind, especially since my two Geometry classes are rarely ever aligned anymore, has been a tiring challenge. I think being 53 and having a 14 year old boy and an 8 year old girl in the house has an impact as well!
  • I was able to attend TwitterMathCamp for the fourth summer in a row. As an added bonus, this past year did not conflict with my daughter’s birthday. Another bonus was that Atlanta is the home of an old high school buddy who was also my first college roommate. I had not seen him in years and had a lovely night with him and his family on a warm southern night, hours spent on his porch catching up was a delight.
  • My time at TMC was followed by a trip to FLA that included a couple of nights catching up with friends in my old hometown of Gainesville. Had not been there in a few years.
  • My school is a PK – PG school on two campuses. We live on the upper school campus, the lower school is about three miles away. My son is in 8th grade and he and his pals will be in my hallways in 8 months. Exciting and scary at the same time!
  • My wife is nearing the end of her Master’s Degree program. It’s been fun listening to her talk about her school experiences. It has been ten years now since my course work last ended. I’m a bit jealous, I think.
  • Off to face the new day, the new year, I guess, now…


A Tale of Two Questions

This past week I had a quiz for each of my Geometry sections. The two sections are out of synch a bit due to our rotating schedule. They typically assess on different days with different versions of whatever quiz or test I recently wrote. This week’s quiz had two different forms of the final question. I present them below:

In the diagram below you see a triangle ABC and you see what are called the exterior angles of the triangle marked. What is the sum of the measures these exterior angles? Be careful to carefully show your reasoning. Mark any angles clearly that you want to refer to in your explanation.  

The problem above was presented to my class on Thursday.

In the diagram below you see a triangle ABC and you see what are called the exterior angles of the triangle marked. The sum of these exterior angles is 3600. Write a proof explaining to me why this is true. Mark any angles on the diagram that you refer to in your proof.

The problem above was presented to my class on Friday.

Both classes had the same first problem on their quiz. They were asked to prove that the interior angles of a triangle sum to 180 degrees. This proof was explicitly presented in class and in their text. My thought was that this challenging fifth problem should be a (somewhat) natural consequence of the first problem on the quiz.

The students who took the quiz on Thursday struggled on the first problem and it bled over to the last. They generally performed better on the last problem than on the first. In part, this is due to my decisions about partial credit. I was definitely more generous with partial credit on the problem at the end of the quiz since they had not seen any explicit proof of this fact. My colleague who also teaches Geometry felt that I might be reaching a bit with this last question. My Friday class performed better on the first proof than the Thursday crew and they did a MUCH better job on the last problem. I am trying to sort this out and there are too many variables at play. First, the class who took the quiz on Friday has performed at a slightly, but consistently, higher level overall during the first trimester of our year. Second,there is always the possibility that information about the quiz was discussed in a way that gave the Friday class some advantage. Finally, the problem presented to them gave an answer and asked for justification while the problem as presented to the Thursday class did not provide the conclusion. I was more strict with partial credit with the Friday quiz class since the conclusion was given to them and the whole burden of the problem was the explanation.

The main reason I am writing about this is that I am trying to make myself think clearly about what my goals are in a  problem like this one and to convince myself that I was trying to get at the same thing with both classes. Did I drastically change the nature of what was being assessed by presenting the conclusion already? I have thought out loud on this blogspace about a similar question here – https://mrdardy.mtbos.org/2017/09/22/a-quick-question-about-test-questions/

Did revealing the answer to the question fundamentally change the level of challenge inherent in the question? Is it THAT much easier to reason through the proof when you know what you are supposed to conclude?

Our Geometry course is the last course in our curriculum where there is no Honors option. Everyone who takes geometry takes the same course at our school. This means that there is a wider variety of interest and talent in this room than in my other classes. I think that there is a tendency in a non-honors math class to think that the students cannot tackle challenging or novel questions. I have heard several colleagues over the years say something along the lines of ‘I can’t ask that question if I haven’t shown them how to do it.’ These are terrific teachers saying this and they are coming from a good place, they want their students to succeed and they do not want them discouraged or dismayed by assessments. I think I am coming from a good place as well, it’s just a different place. I’d also say that in the case of the question above, especially in its first form, I do believe that I have shown my students how to tackle such a question. They know that the interior angles sum to 180 degrees. They see three supplementary pairs of angles so that sum is 540 degrees. The difference is the exterior angles. Half of the students in the Thursday group earned four or five points out of five on the problem. Those who earned four generally had sound logic with real flaws in the vocabulary explaining their answers. Maybe my docking them a point is an entirely different question about how I assess.

Another reason I am writing this is that I want to have a conversation with my department about questions like this one, questions that are not a simple transformation of what has already been practiced. I have students who imply that I am the first teacher they have who asks them questions that feel like they might be ‘from left field.’ I know that students (all people, really) will exaggerate their concerns in the face of feeling stressed. I think most of my students do a nice job of stepping up to challenges like this one, especially when points are riding on it on an assessment. But I also know that there is an instinct at times to simply dodge these situations. The same group of kids who took the quiz on Thursday were presented with a problem from Steve Wyborney’s website on Friday in class. I showed them the video of the duplicator lab problem.  When the video ended I asked them to begin talking about the problem with their neighbors – in this class everyone sits in groups of three that get randomly reassigned every fifth day. I was met with mostly silence. To be fair, this was about 8:10 in the morning. However, when I showed them the comments section with teachers talking about their fourth and fifth graders solving the problem, they suddenly started talking. So, I don’t know if they were shamed into action or they simply needed to suspect that they were more than capable of solving the problem before they moved. I have to feel that the struggle with the problem on Thursday and their reluctance to engage with a novel problem on Friday morning are related. I also fear that I have not done enough yet to create a culture where they jump into these problems. I am interested in how the conversation goes with my department on Wednesday morning and I would love to hear from any readers as well.


Persistence and Creativity

I spent six hours yesterday watching students take final exams. Three two-hour shifts. Sigh…

My last group taking an exam was my AP Calculus BC team. They had 24 multiple-choice questions and four free response questions, so they had to be efficient in their problem-solving. One of my students asked me about a multiple choice question that troubled him. It was an infinite geometric series question. This student joined our school, and our country, last year as a freshman. He place tested so highly that he started in AP Calculus AB. However, it has become apparent that there are a few facts/skills that he does not have at his command. This is rarely a problem since he is so creative. On this question, he did not have the formula in his brain for calculating the sum of an infinite geometric series. He could have listed out a handful of terms to look for convergence. He could have shrugged his shoulders and guessed since it was one of twenty-four questions. Instead, he wrote a short program on his TI-84 that gave him increasingly good approximations until he saw one of the answer choices emerge. He did this in the middle of a two-hour exam! He asked me afterwards and just laughed when I showed him how easy this problem could have been. I am convinced that he will remember this formula forever now, but I am also convinced that I will remember this story forever now. His ability to problem-solve in this situation is SO much more powerful than having a formula at his command. It is interesting that this happened on the same day that I sent our a question on twitter about the use of formula sheets on assessments. I am disinclined to use formula sheets, but I could be convinced otherwise. Here is a story that would have never happened if a formula sheet was present.

I am going to cross post this over at the One Good Thing blog space.



Another quick post. We are in exam week here at my school. I have ALL sorts of thoughts about term exams and why we do them, but those are for another place rather than this public forum. I have written about our department’s decision to move toward a test correction policy. I am so so so optimistic about our exams this week. I really believe that we will see largely improved results because the students have been more actively involved with examining their tests and reflecting on what went right and wrong on their tests. They have been talking to each other and comparing ideas. They have been talking to their teachers about how to fix their problems solving approaches. Our department exams are mostly on the next to last day of exam week. This will work against our students as energy levels start running low. Despite this, I am hopeful that we will see a different level of engagement on these cumulative exams. I will report back either way in a week or two, but I feel good based on what I have seen in the past week of reviews in class and from watching and listening to kids in the hallway as they work together.

Another Fun Problem Debate

A super quick post this morning. On my last problem set for AP Calculus BC I included the problem below:

A 10 m rope is fastened to one of the outside corners of a house, which has the form of a rectangle, 6 m long and 4 m wide. A dog is fastened to the rope. What is the perimeter of the region that the dog can access?

I have asked a form of this question a couple of times over the years. One year I did not mention that the rope was on the exterior of the house and I had a student assume that the dog was tied to a leash inside the house. I fixed that mistake.

A GeoGebra sketch below shows the image I have in my head for this problem.  

My answer to this question, and the answer that 11 of my 14 students had, was that the perimeter is 20π. Two students argued that the answer should be 20π + 20. Their argument is that the borders of the house, the sides of the rectangle in the drawing, are also part of the perimeter. I loved the debate that ensued and most students migrated to this point of view. I reflexively thought of perimeter as exterior, while these two students argued that perimeter is boundary. I think I agree with them and I LOVE the fact that they cared enough to debate this on a problem that counted for one point out of about 400 something for the term. I also LOVE that students who got their answer marked correct started arguing against the answer that they arrived at.