So, we are almost done with our deep and quick tour of AB topics in my BC class. We use the Stewart text which has an interesting section at the end of each chapter. The section is called Problems Plus and I have been browsing through these sections for class examples. On Friday I picked a problem that looked pretty challenging. The set up is this – Imagine a square region with sides measuring two units. In the square a region is shaded. This region is the set of all points that are closer to the center of the square than to the nearest side. What is the area of this region? I did not try this problem first, I had confidence that we could work our way through it. In each of my two sections this was the second problem of the day. Each group dispensed with the first problem in about 5 minutes. Each class spent almost 40 minutes discussing/debating/arguing over this square problem. What thrilled me was that both classes (the small morning class of 8 and the large afternoon class of 18) stayed engaged offering ideas, questioning each other, thinking about circles, etc. We looked at GeoGebra to try and sketch some regions. We thought about the distance formula and circles since the kids were convinced that the region where the distance to the center and the side was equal would be somehow circular in nature. None of our ideas came to find a final solution. To me, this fact is SO tiny in comparison to the fact that they fought, they were engaged, and some of the afternoon kids stayed after to share new insights. I am so proud of this group for being willing to engage and not being at all angry or visibly annoyed when we did not come to a solution. I can’t wait until Monday to see what ideas they bring to the table.


So, I sort of pride myself on being the type of teacher who creates an environment in his classroom where conversations can flow. I have my kiddos at two large tables where they are elbow to elbow and talk regularly. Sometimes, of course, the conversation strays – bit there are often rich math conversations going on. I posted this quick story over at One Good Thing, but I want to share it here as well.

I presented my BC class (all in their second year of High School Calculus) with the equation of an ellipse centered at the origin and asked the following rather vague question – “Are there any two points on this curve where the lines tangent to the curve are perpendicular?” One girl, Chloe, immediately answered that the tangent ‘on top’ of the graph was horizontal and it would be perpendicular to the tangent on the ‘side of the graph’ which is vertical. I congratulated her and challenged the class to find some other more interesting points. A student asked what the slopes of these more interesting lines might be and then a boy, Sal, chimed in that any number you pick must be the slope of a line tangent to this ellipse. His argument was based on recognizing that between the two tangents that Chloe had mentioned the slopes range from 0 to positive infinity. In other quadrant the slope would range from 0 to negative infinity. If he had mentioned the intermediate value theorem I might have fainted on the spot from joy.

After I posted the story to One Good Thing I read Ben Blum-Smith’s most recent posting ( and I now realize what an opportunity I missed by simply congratulating Sal instead of getting others to join in and complete the thought process. Read Ben’s post. You’ll be glad you did. I intend to try and incorporate this strategy into my daily practice.


Victory is Ours!

We made the decision this year to start our precalculus classes (both honors and non-honors) in the study of trigonometry. This decision was made based on frustration with the traditional slow start of reviewing Algebra topics and based on the request of our physics teacher. So now kids can start off a little more productively if they are simultaneously enrolled in some level of physics and some level of precalculus. So, today I decided to try out an experiment with Desmos. I made a table of values of the average daily temperatures of my beloved former home (Gainesville, FL) and I both gave the students a physical table of values and displayed the plot of this table of values on on the board through Desmos.  Their job, in their pods of 2, 3, or 4 students at a time was to match a function to this data. I was SO happy with the work they did and with the conclusions that they arrived at. Here is a link with the data and the various equations

Note that above I said (separately) that I was satisfied with their work AND their conclusions. I am trying so hard to make that distinction for my students. To talk about the process, the thinking that goes on. One group pulled out an iPad and called up their own Desmos app and kept tweaking their work. We had great conversations about how to identify the amplitude, how to deal with phase shifts, what should the period be, etc. I’m not crazy enough to think every day will go so well, but I sure am happy and optimistic right now.

Calculus BC

We made the decision at our school to make AP Calculus BC a second-year course in high school Calculus so our kids in BC have all completed AP Calculus AB with some measure of success. I am pretty firm in believing that we are doing them a favor (and doing our teacher – right now it’s me) a favor as well. Our kids had been ‘succeeding’ in BC by the measure of the AP test, but they were exhausted and had no time for reflection. We probably have a bit TOO much time for reflection with it as a second-year course, but that’s the problem I’d rather have. So, we use Stewart’s Calculus text (I inherited it and I’m not in love with it but it’s more than acceptable) and he has these clever sections at the end of each chapter called Problems Plus. I spend the first four weeks or so reinforcing AB topics through the use of these problems. Our AB calendar doesn’t lend itself to too much of this so I feel like it is not simple repetition for the kids. We just finished chapter two on derivatives and I’ve been mixing in some AP FR problems for HW. I’m completely at a loss to explain the wide range of reactions from the kids and their range of performance on these problems. One of our more ambitious kids – a junior girl from China who earned a 5 and A’s all last year – was presenting an AP problem with a linear piecewise function. We were told that this was function f and a function g was defined to be its antiderivative. We were presented with a triangular area. She carefully explained how she found the line equations for each piece of the graph and then proceeded to anti differentiate them. I pointed out that her work was correct but perhaps it would be more efficient to simply calculate the area of the triangle bounded by the curve. She was not moved AT ALL by my argument. I’m torn between being happy that she knew how to convert into a function (I actually AM happy that she can do this) and troubled that she is so rigid in her approach to antidifferentiation. Rather than recognizing that it is also a geometric concept, she is locked into the function interpretation. I don’t want my students to spend so much extra time and energy on problems like this. I also don’t want them to think that I don’t respect their thinking. I don’t want them to think that this is all about doing it the way that Mr. Dardy wants them to do it. Figuring out how to respond in a genuinely positive way while also pointing out how much more efficient she can be was such a challenge. I need to work on this. So much work to do in this job, no matter how long I’ve been at it…

Nicholson Baker and the plight of math teachers

So, I have just started the Baker article in Harper’s and I’m already pretty annoyed. However, I’m more annoyed (I think) by the reality of the situation than I am by his argument about the situation. We are in day three of school here and one of my jobs as Dept Chair is to sort out math placements for new international and domestic students. I have had the following conversation (or some variant of it) at least three times in the past day and a half:

Student: Why am I in this class? I should be in a higher class, I’ve learned all of this already.

Mr Dardy: Well, your placement test indicated that this was the best combination of challenging you without putting you in a situation where you might fail.

Student: That test? that was unfair, I don’t remember all of that stuff.

So, my question (asked out of frustration) is this – Do other disciplines have the same plague of students not knowing things they claim to have learned already? More importantly, do they deal with some inherent assumption that this is okay to not know what you claim to know? I know I’m not asking anything novel here, but I needed a virtual place to vent a bit so I can continue to smile and deal with my young charges with good cheer.

More Handshakes

So – as I anticipated, this problem was more challenging than the more typical direction where the number of people in the room is the given information. What completely knocked me out was that I saw two different diligent solution paths. One student convinced his neighbors that the formula to work with was x(x-1) where x is the number of people in the room. Nice thinking, not exactly right but a good start. Another group was diligently building a table of values. We plotted those values and it was pretty clear that a parabola was emerging. This was nice since the other formula was quadratic. However, the table of values did not match the formula. This led to a quick adjustment and we decided to solve the equation (1/2)(x)(x-1) = 253

I was told to multiply the 2 over to create x(x-1)=  506 and this is where class got exciting. The students, predictably, wanted to solve the quadratic. We had already concluded that we wanted numbers somewhere in the twenties based on the product. One of my students pointed out that we were looking at consecutive numbers whose product ends in a 6. The student patiently explained that it had to be 22 and 23 or 27 and 28. I LOVE this. The type of number sense at play here was so refreshing. First day of school a winner based on this exchange.


So, one of my favorite opening day problems for Algebra II is the handshake problem. You know, there are 18 of us in this room and if we were total strangers and went around and introduced ourselves to everyone in the room, then how many handshakes would occur are we go around the room meeting each other? I know, I know, this is a sort of pseudo-contexty type of problem but it always leads to interesting conversations. So, this year my lowest class is a Precalculus Honors class and I think I’ll start off by saying “There is a group of strangers in a room and they all go around and introduce themselves to everyone in the room. At the end of this process 253 handshakes have occurred. How many people are in the room?” 

Is this any more interesting/challenging than the standard version? I’m not certain, but I think it is.

Questioning Questioning

At our school’s whole faculty meeting yesterday morning (we see our kids on Monday for the first time) we had an interesting PD session on listening/counseling/advising and I have some ideas rattling around the cave of my mind that I need to air out somehow.

We had an exercise where we were to pair up with someone that we do not know well. Our school has two campuses separated by 3 miles – so the pairing generally was one upper school member with a lower school counterpart. We were to then take turns talking about something important to us for 8 minutes and then switch speaker roles. When it was not our turn to speak we were to listen – not to ask questions, not to comment, just listen. After the activity the guest speaker – a therapist from the Stanley H King Counseling Institute – was gathering reflections and impressions from us about this exercise. One of my colleagues commented that he found it difficult to listen without asking questions and the therapist remarked that it is difficult to simply let another person’s thoughts go where they want to go rather than influence them to go where we want them to go.

This got me thinking about my questioning habits in the classroom. I have long thought that one of my strengths as a teacher is my willingness to trade telling for asking in many situations. After this conversation yesterday I am now wondering how much I need to try to trade asking for listening in classroom discussion situations. Am I just fooling myself into thinking that my asking is any less monopolizing an activity than telling is? I need to ponder this and I’d love to hear any reflections from the world outside my head.

Stanford’s How to Learn Math Course

I have enrolled in Jo Boaler’s online math course this summer as have about 20,000 of my closest friends. I was excited by the prospect of this class and I urged my middle school and upper school colleagues to enroll as well. I have one MS colleague and two US colleagues who took the bait and joined in. I have to say that at this point I am a bit disappointed by the course. Perhaps I was not really the intended audience since I have done a decent amount of Carol Dweck reading before the course. Much of what Boaler is presenting is through that lens. However, my main disappointments have to do with the problems created by the size of the audience. I submitted work over three weeks ago and I have not received any feedback yet. The assessment panel of the course indicates an estimated wait of three days for feedback and I am now well past that. I know that some work has been done to streamline the discussion forum but my forays into that area of the course have not been promising. Comments are not obviously tied to lessons and too often threads are one or two comments long with no sustained dialogue occurring at all. Perhaps I have just not been persistent enough (a little in joke for those who are engaged in this course!) to find the rich veins of conversation, but it’s been a disappointment to me overall. I will try not to bring that into my opening meetings this year as my colleagues may have a different take. I’m not sad I took the course, it has helped me get my thoughts focused as the new year comes zooming in, but I cannot say that I feel really enriched by the content so far. I also have to admit that this is my first foray into the MOOC world so perhaps the concerns I have about discussion forums and feedback cycles are simply a reality of this big, brave new world. I’d love to hear about the experience of others who may have chosen to join in as well.