TMC Reflections, Part Three

In this post I want to concentrate on a couple of the afternoon sessions I attended. The TMC program (you can find it here) was filled with so many interesting opportunities that I kind of agonized over some of the choices. One that I knew I would attend was the session run by Danielle Racer (@0mod3) discussing her experiences in implementing an Exeter-style problem based approach to Geometry this past year. Danielle and one of her colleagues (Miriam Singer who is @MSinger216) came back from the Exeter summer math program (it is called the Ajna Greer Conference and if you have never been, I suggest that you try to change that!) all fired up and ready to reinvent their Honors Geometry course. Danielle spoke eloquently about their experiences and shared out some important resources. We had a great conversation in the session about the benefits and struggles of problem based curriculum. This conversation tied in to another session I saw as well as some thoughts and conversations I have been having for years. First, the afternoon session that I think linked in here. Chris Robinson (@Isomorphic2CRob) and Jonathan Osters (@callmejosters) are colleagues from the Blake School in Minneapolis.  Chris and Jonathan spoke about a shift in their assessment policy that centered around skills based quizzes using and SBG model and tests that were more open to novel problem solving. I am simplifying a bit here for the sake of making sense of my own thoughts. I thought that their presentation was thoughtful and it generated great conversation in the room. Perhaps we (especially I) spoke out more than Chris and Jonathan anticipated and we ran out of time. Another sign of a good presentation, I would say. When there is more enthusiasm and participation than you thought you’d get, it probably means that you are tapping in to important conversations AND you have created a space that feels safe and open.

These two sessions had me thinking about some important conversations we have been having at our school and I am totally interested in hearing any feedback. The first conversation I remembered was with a student who had transferred to our school as a senior and was in my AP Calculus AB class. She was complaining about my homework assignments which were a mix of some text problems and some problem sets I wrote. She said in class, ‘You seem to think that AP means All Problems.’ A little probing revealed that she saw a difference between exercises and problems. A brief, but meaningful, description I remember reading is that when you know what to do when you read the assignment then it is an exercise. If you read it and you don’t know what to do, then it is a problem (in more meanings than one, I’d say). The next conversation I recalled was with a colleague who has now retired from math teaching. We were talking about homework and the struggles with having students persevere through challenging assignments. He also used this language making distinctions between exercises and problems and he suggested that HW assignments should have exercises and problems should be discussed in class when everyone was working together. He felt that the struggle and frustration of problems when you are on your own would be discouraging to too many students and would likely lead to less effort toward completion on HW. A similar conversation came up with another former colleague who was frustrated with some of the problem sets I had written for our Geometry course. She did not want to send her kids home with HW that they would not be able to complete successfully. I recognized that this was coming from a fundamentally good place. She did not want her students to feel frustrated and unsuccessful. However, I firmly believe that real growth, real learning, and real satisfaction are all related to overcoming obstacles. I have witnessed this recently with my Lil’ Dardy who just became a full fledged bike rider this summer. I heard it from my boy, my not so Lil’ Dardy, who made the following observation recently, ‘You know, I find that I like video games much better if they are hard at first. Why do you think that is, dad?’

I know that we can anecdote each other to death on these issues and I also know that there is not ONE RIGHT WAY to do this. But I am in the process of trying to make coherent sense out of my inherent biases toward problem based learning. I want to have deep and meaningful conversations with students, with their parents, with my colleagues, and with my administration about how to approach this balance and about what a math class should look like and feel like in our school. While I have been writing this I was also engaging in a meaningful twitter chat about some of this with the incomparable Lisa Henry (@lmhenry9) and with one of my new favorite people Joel Bezaire (@joelbezaire) so I know I am not the only one struggling with these questions. Please hit me up on twitter (@mrdardy) or start a raging conversation in my comments section sharing your successes/failures/theories about how to strike a balance between exercises and problems between challenging students while making them feel safe and successful and between running your own classroom with your own standard and fitting in with a team at your school. These are all big questions and I wrestle with them all the time. I want to thank Danielle, Chris, and Jonathan for sparking them up in my mind again and for creating lovely spaces for conversations in their afternoon sessions.


Coming soon will be my last entry in this series where I think out loud about the amazing keynote delivered by Tracy Zager (@TracyZager)

Problems / Exercises

I wrote about this earlier today and I want to spend a few minutes trying to organize my thoughts.

A conversation on twitter today with David Wees (@davidwees) reminded me of a conversation with a former colleague. It also reminded me of a class I took in my master’s program. The course had the vague title Problem Solving and my professor (who was my advisor) had a long background in studying problem-solving. I remember (not clearly enough) that we had a working definition for what qualified as a problem. The definition revolved around the idea that there are certain questions we encounter in math where we immediately know what we are supposed to do – what formula to use, what definition or theorem to call upon – while there are other questions where we do not immediately know what we need to do. The first group we classified as exercises while the second group were called problems. It is not necessarily that problems are harder. I have certainly dealt with many challenging math questions where I knew exactly what I needed to do, it was just really hard to do it. I have a real fondness for problems in mathematics and I have developed the habit of writing homework assignments for my classes that should probably be called problem sets. For years, I was writing these for Honors Calculus, AP Calculus AB, AP Calculus BC, Honors Precalculus, and AP Stats. A few summers ago I wrote a Geometry text for our school and I wrote all the HW assignments as well. These students are fundamentally different in many ways from the students I was working with in those other classes. I do not necessarily mean that they are inherently less talented or anything like that. What I do believe is that they are younger, less experienced, and less patient in their problem solving. So, they are more likely to simply shrug off a problem and figure that we’ll talk about it the next day. Over the course of the year most of them have become more patient and they are aware that we will discuss these questions and that they will not be graded on their HW. After my twitter exchange this afternoon, I am (once again) rethinking this strategy and I am nowhere near a conclusion. I did share the tweets with my class after we struggled through the question I wrote about earlier today. I asked them to honestly share their opinion about whether it is a valuable exercise to struggle with questions like this one. A few were upbeat and said that they liked thinking about these questions and that it is helpful to try challenging problems. One student said something really striking. She said that it is really frustrating to work through these problems alone and that she wishes she could get the opinions/insights of others when she is struggling with these questions. This is certainly in line with what David suggested in our exchange and with what my former colleague (who I wrote about earlier today) mentioned as well. I have some thinking to do here. I do believe that it is powerful for students to wrestle with challenging questions. I do believe that by not grading HW I am helping to create a safer environment to struggle. I know that a number of students work together on assignments either here in our dorms or libraries at night or during study halls during the day. I also believe that conversations in class are richer when they have some ideas already thought out to toss around. However, I also recognize that this is frustrating for some students and may simply push them further away. I recognize that if I am going to say that I value collaboration that I need to commit to making the time for that when we are together. I also recognize that what works to motivate seniors in college level math classes might not work as well with 9th and 10th grade students in a required math class.


Lots of thinking to do, luckily the summer will afford me some valuable time.

Thinking Out Loud

Super brief post here – class starts in 15 minutes. One of my Geometry colleagues asked me about a HW question I had written. I asked the students to find two cylinders that were not congruent but that had the same surface area AND the same volume. I thought it was a pretty interesting question, but I realized I did not have a coherent strategy for discussing it other than playing with numbers. I threw the question out to Twitter and engaged in a terrific conversation with Matt Enslow (@CmonMattTHINK), John Stevens (@Jstevens009), Dave Radcliffe (@daveinstpaul), and David Wees (@davidwees) Some good math was tossed around, but what really has my brain bubbling is an exchange with David Wees. He said he thought it was a great question but he would not have used it as a homework question. When I asked him why he said something that reminded me of a conversation with a former colleague. My former colleague once said that he sees a difference between exercises and problems and that he liked to keep problems for times together with the students where they could work together. I find that I feel (hope) that meaningful conversations can happen in class more readily about a problem like this one if the students have had time to think about it first. However, his words carry some weight with me as do David’s. I feel as if there is some conclusion I want to reach, but I also suspect that there is no right answer to this. I would love to hear some opinions about this in the comments or through Twitter where you find me at @mrdardy



A wonderful Problem

Today was our last day of school before a loooong spring break – we do not return until March 14. We were asked not to have any assessments today as some students have term finals tomorrow. So, I wanted to find a flexible problem that all 3 of my courses could wrestle with today. I teach AP Calculus BC, Discrete Math, and Geometry so this was a bit of a challenge. I found a lovely problem here : Screen Shot 2016-02-25 at 2.55.39 PM

I was so delighted by how my students engaged with this problem today. A little background first. My BC kids are on the verge of learning about power series so a series/sequence question is right up their alley right now. We have been talking convergence and divergence tests. I also had some competition problems in my back pocket because I knew this would not take them very long. My Geometry class just finished a chapter on similarity and we have spent the past two days playing with Taxicab Geometry. A blog post on that adventure is coming tonight or tomorrow night. My Discrete kids just finished their winter term where we studied patterns (numeric and visual) as well as some theory about voting and ballot strategies and they, too, have played with Taxicab Geometry this week.

I want to share a few of the insightful comments that some of the students made about this list of sequences. I prompted each class with one question first: Why does they say that these are related sequences?


In all of my classes students first focused on the rules for each arithmetic sequence and made observations about the pattern of differences moving from 2 to 4 to 8. In one of my BC classes a student instantly said ‘Each first term is 2^n and then you add 2^(n+1)’ Amazingly fast pattern recognition, but more than I hoped for right out of the gate. Most of his peers were taken aback and seemed happy to focus on smaller pieces. In each Discrete class and in my Geometry class I had students noticing the doubling pattern from one sequence to the next. Only my Calc students used recursive language technically, but all classes had students recognizing that pattern. It is interesting on reflection to see how formula driven (or is that formula comfortable) my Calculus students are compared to the other classes.

I tried to get a series of ‘what do you notice?’ comments going and the following popped up in every class;

  • The first sequence is the only one with odds
  • They are all arithmetic series (either by description or by use of the formal language)
  • The difference in each sequence is increasing by an twice as much as the difference from the previous sequence
  • The first term is a power of 2 (my Geometry kids needed prompting to remember about the 0 power)
  • All the sequences other than the first have only even numbers


After gathering a series of observations about the sequences, we directed our attention to the charge of finding where the number 1000 might be hiding. Luckily no one wanted to list all the terms of a sequence until 1000 arrived or was passed by. So the following suggestions came my way;

  • Subtract the first term from 1000 and divide by the common difference to see if 1000 is on the list
  • Divide 1000 by 2 repeatedly until we arrive at a term that is more manageable and more clearly on one of the lists
  • 1000 is 10^3 so we need to find 5^3 since 10 = 5 * 2 and we know that 2s are built up row by row


I was really pleased by the focus on 1000 being built up by factors of 5 and 2. One of the discrete classes built up to 1000 while the other kept dividing by 2 to bring it down to the 125 necessary. Once we were focused on 125 it was clear in all classes that the first sequence was the only place that 125 could live. My first class of the day is one of my AP Calculus BC classes and after realizing where the 1000 is there was no discussion of whether that 1000 could appear anywhere else. In my second class, one of my Discrete Math classes, they focused on the plural in the question and wondered whether there might be multiple landing spots for the 1000. We counted out 1 – 20 together on the lists and noticed that no number was repeated. We were pretty confident that this pattern would hold. In my second AP Calculus class – the one where a student generated a formula right away – he stepped up and showed a terrific proof that this had to be a unique solution. Writing each term as 2^n + (2^(n+1))*k where k represents some multiple of the number of differences in the sequence. By setting this equal to 1000 and factoring out a 2^n he made the argument that 1000 needed to be written as a product of a power of 2 and an odd number of the form 1 + 2k. Listing factors of 1000 it was pretty clear that only 8 * 125 satisfied the conditions of the problem.

Finally, my Geometry kiddos had the opportunity to dig into the problem and I was pretty darned pleased, I must say. It was the last period of the last day of school before a two week spring break. They are the youngest of all my students and they are the least experienced mathematically. What I saw today was real evidence that these students have been growing as problem solvers. They are more patient and persistent than they were in the fall and they are more willing to make guesses out loud than they were when we started the year together. I am so happy about the conversation we had. Other than the concern about whether 1000 exists in any of the other lists, they were able to nail all of the important pieces of this problem.


I discovered the problem at about 5:45 this morning and I could not be more pleased about the conversations I had with my students today.



An Old Favorite

Screen Shot 2016-01-26 at 8.54.16 PM

The image above is found on the Nrich math site at


I first encountered this problem in 2014 in Jenks at a TMC session run by Megan (@Veganmathbeagle)

In the past two days I presented this to three of my classes – my Geometry class and my two Discrete Math classes. Much to my delight the classes all solved the problem and they all solved it different ways. In one Discrete class the group locked in right away on the fact that squares are worth two more than triangles. One of my students made a quick decision to attack this by a guess and check method and he, luckily, guessed correctly on the first try. We had a pretty good conversation about the strengths and weaknesses of relying on lucky guesses. In my second Discrete class there was a bit of debating about what clues to focus on. While they were tossing some good ideas around one student told us that none of our ideas mattered. Well, he was nicer than that but he did manage to circumvent all of our clever ideas by simply asking if he could add all the sums indicated in the right column and compare that to the sums indicated in the bottom row. It too a little convincing for his classmates to believe him, but they came around to his way of thinking. Interestingly (at least to me) some of the students still wanted to know the individual values of the shapes. In my Geometry class the students also focused on the difference between a square and a triangle. Before we went much further in that conversation, a student pointed out that the first and third columns only differed by a square turning into a triangle. Since we knew that squares were worth two more than triangles (again, they found this using the third and fourth rows) we can know that the question mark should be replaced by ______ (no spoilers here!)


I loved listening to the ideas bubbling out and I especially liked that they moved forward quickly in all three classes with nothing more than the visual prompt above. It’s great to hear the interactions and it is instructive to hear what they are focusing on when engaging with a problem like this. Fun problem solving in these classes. Later this week I intend to write about our Calculus exploits and revisit my ideas / frustrations with homework in my Geometry class.

Beautiful Problem Solving and Odds and Ends

While most of my colleagues enjoyed a well-deserved day off in honor of Martin Luther King, Jr. we were at work here in our boarding school. We take advantage of these days as visitation days and we keep on counting the days of the year.

Last week I wrote about my frustrations with trying to find a way to help keep my students more aware of the benefits of daily practice in Geometry. This weekend I engaged in a lengthy and mind opening twitter conversation with Elizabeth (@cheesemonkeySF) and my mind is still buzzing with ideas. I noticed something today that I may be able to take advantage of. Tomorrow we have our next Geometry test. This is the second year that my school is using the Geometry text that I wrote. This means that we are still working our way through the strengths/weaknesses of the text and we have a storehouse of documents to draw upon. I decided earlier in the year that I would hand out last year’s tests as practice a few days before this year’s test over similar material. So, last Friday I gave a copy of the test from last year that covers through Chapter Six of our text. Today in class I saw more evidence than usual of HW completion. So, when the HW feels particularly helpful then my students are more likely to complete it. Pretty logical, right? What I need to do then is to make sure that I can get buy-in like this more frequently. I have a batch of quizzes from last year that I can easily give out mid chapter as weekend HW that both serves as a sneak preview of the kind of quiz questions I was interested in asking last year AND serves as good, focused practice that feels to my students as if it has more payoff. This will not solve all of the problems I have been wrestling with and I need to sort out Elizabeth’s sage advice and figure out how to incorporate it in a way that fits me, but this feels like progress. I am happier about Geometry than I was last week and I am optimistic about tomorrow’s test. I hope that I will be able to report on student success.

Last week I also wrote about a problem posed to me by an alum when he was visiting. I may not have reported the problem accurately, so here is a second take. One hundred people are lined up to board an airplane with 100 seats. Each person has one seat assigned. The first person boards the plane and randomly chooses a seat. After that, each person who boards will sit in his/her assigned seat if it is available. If the correct seat is not available then that person will randomly choose a seat. What is the probability that the 100th person will be able to sit in the correctly assigned seat? I broke this problem down after one of our boarding community dinners last Thursday and a colleague and I simplified it to two people (50% chance, no surprise!) and then three people. With three people – call them A, B, and C – the seating arrangements are ABC, ACB, BAC, BCA, CAB, CBA. Two of these arrangements have C sitting in the third seat and for the purposes of this permutation, I am treating that as the ‘correct’ seat. However, the arrangements ACB and BCA are not possible under these rules. If person A does not sit in seat B, then person B is obliged to sit in his correct seat. So we have two of four possibilities for a 50% success. This seems pretty suspicious and I try to sort out the arrangements with four people. I won’t bore you with the detail but this is also 50%. When I mentioned this problem to a number of colleagues one of them mentioned that her son had talked about this problem from a math competition. Her son is in my AP Calculus BC course and he is an extraordinarily talented mathematician. He explained the problem this way in class today and I probably will not be as elegant as he was. Here is his take:

By the time that person two sits down on the plane we know that his seat has a person in it. Either it is person one and then person two chooses another seat or his seat was available and he sat in it. Similarly, by the time person three sits down we know that someone is in person three’s seat. Either person one or person two is accidentally in that seat or person three sits in her proper seat according to the rules of this problem. We can extend this argument all the way to person ninety-nine. Now, we know for a fact that all seats from person two’s seat through person ninety-nine’s seat are all occupied. The only mystery is whether the other occupied seat is the first person’s seat or the hundredth person’s seat. It is not a stretch to see that these two possibilities should be equally likely.

What I LOVE about this explanation is that it does not rely on combinatoric wizardry or thorny algebra manipulations. It also make crystal clear sense once it has been explained but it did not make crystal clear sense before that. It seemed completely unreasonable to me that, with so many people involved, the answer would be so clean. In fact, my student’s explanation made it clear that the number of people on board is a complete red herring. It might as well be one thousand people instead.

While I might have enjoyed the day off, I also enjoyed the day on.

Bragging About My Students

Two things I want to share tonight. One of them has multiple parts.


One of my international students shared a lovely gift with me yesterday. It’s a food treat that her mom sent here for her to share. I have a few food allergies so I was concerned but did not want to tell her because it felt rude. Luckily, there are a number of boys in my from who can translate the ingredient list for me. Pretty cool. Oh yeah, I’m allowed to eat it – no nuts.


I blogged a couple of days ago about a problem on my calc BC final. Here is the problem

For your final problem on your final Calculus test, we will play with number bases. Consider the following passage from Lewis Carroll’s Alice in Wonderland:


“Let me see: four times five is twelve, and four times six is

thirteen, and four times seven is — oh dear! I shall never get

to twenty at that rate!”


 Explain, in terms of your knowledge of number bases what is happening in this pattern. Explain how four times five is twelve, and how four times six is thirteen.    Guess what she will say four times seven is and make it clear to me why she won’t be able to get to twenty.

Twenty-four students took this final and a number of them really did a wonderful job in explaining their reasoning. I’m going to present a handful of the best responses here.

  1.  4 x 5 = 12  is number base 18. 4 x 6 = 13 is number base 21. 4 x 7 =   she will say 14 in number base 24.   4 x 12 = 19 number base 39, 4 x 13 = 20? number base 42. If following the pattern, as one of the numbers remains constant 4, another increasing by 1 each time, we get the product increasing by 1 each also. This is possible for number base 18 , increasing 3 each time. This 4 x 13 = 20 in number base 42 accordingly. However, 4 x 13 in number base 42 is one full round with another 10, which in base 42 as there exists another symbol for that suppose A, thus 4 x 13 = 1A and will never equal to 20 this way.
  2. She is using number base to calculate it. Every time 4 times initial number plus 1 and number base that is increased by 3. It will never get to twenty because the number base is always growing as number grows. Number base is growing faster than the number we multiply by. (Note – this one was accompanied by meaningful, but scribbly, calculations)
  3. (This answer starts with all the calculations hinted at in the first answer i presented)  Because the base in continually increasing by 3 and the answer is only increasing by one, the answer will never be able to get out of the ones digit.
  4. The base is increasing and in order to get to 20 the result of the calculation must be EXACTLY twice as big as the base., which is not possible.


All of these were accompanied by calculations on the side. We spent about two and a half days talking about number bases and I must admit I was really impressed by the patience that my students had with this problem. Nice way to end the year!


Thinking About How to End a Year

So, this morning both of my AP classes took their final exams. I have some questions for the world about this process, but first I want to share something fun from the Calculus BC final. You should know that my students understand that the word fun, when I use it in class, means that a problem is challenging, thought-provoking, unusual, or some other words that they might use but I won’t type. I believe that I mentioned that we ended the year after the AP test with a quick tour of some interesting topics that many high school students don’t get to see. I included a small unit on different number bases. I always start this by writing a series of addition and multiplication facts on the board. However, they don’t know that these facts are base 8 number facts. I usually reveal the secret by showing a picture of Lisa Simpson and tell them that these facts are how Lisa would compute. It’s a fun conversation to have. So, for our final exam I told my students that there would be ten problems. Nine of them would be Calculus problems taken from old tests. they have all of their old tests (all ten of them) so they could be very well prepared for that. I also told them that one problem would come from our last two weeks. After discussing this with my friend Richard – a former math teacher – he sent me the following passage from Alice in Wonderland

Let me see: four times five is twelve, and four times six is

thirteen, and four times seven is — oh dear! I shall never get

to twenty at that rate!    

I played around with this for a while and fell in love with this as a final problem for the year. This is how I presented it to my students:

For your final problem on your final Calculus test, we will play with number bases. Consider the following passage from Lewis Carroll’s Alice in Wonderland:

“Let me see: four times five is twelve, and four times six is

thirteen, and four times seven is — oh dear! I shall never get

to twenty at that rate!”

            Explain, in terms of your knowledge of number bases what is happening in this pattern. Explain how four times five is twelve, and how four times six is thirteen.    Guess what she will say four times seven is and make it clear to me why she won’t be able to get to twenty. 


I will sink my teeth into grading finals tomorrow since I am on dorm duty tonight. I did browse through four or five of the test as they were turned in and two students really nailed the problem and provided beautiful, detailed answers explaining the pattern. I won’t spoil it here, I’ll let you work through it if you wish to do so.


I don’t know how your school works, I do know a bit about the four schools where I have worked. All of them have been independent schools that emphasize the idea that we are college preparatory schools. Each school I have worked at has had a statement in their handbook about the importance and significance of final exams as a college preparatory experience. However, I know that tonight many of the seniors in the dorm will tell me that they do not have any finals left. Today was the first of three and a half days of final exams and many seniors won’t have any more after today. There is a pretty common feeling that final exams will not be pretty and are of questionable usefulness with our seniors who are days away from graduation. This is not just a feeling at my current school. But I really wrestle with this. We say we believe that taking a final exam, preparing and organizing a large body of information for a one-day thorough examination, is a useful skill AND one that is important for college. However, it is those students who are closest to college who are the most likely to have been excused from a final exam. In some classes the final experience is a paper or a presentation that happened last week. But our school, and others where I have worked, carve out quite a bit of time for final exam administration. I wonder whether we could use our time in a more meaningful way. I wonder whether the idea of a final exam makes sense only in certain disciplines or for certain age levels. Is it reasonable for us to ask our freshmen to take exams under the same circumstances that we ask of our juniors and (sometimes) seniors? I don’t see AP scores for my students until July. I like the idea of some capstone where I check in with them in one last, broad examination of ideas. I feel pretty old-school in that regard.

I want to make my assessments meaningful for me and for my students. I am really beginning to doubt whether exam week is such a positive way to do this. I would love to hear from others about how they deal with this question. Are many of you bound to a policy that your school or your department has mandated? I want to be smarter about this and I’d love your help.


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.