Modeling Good Behavior

I think about this all the time as a dad – my lil ones are a 12 year old boy and a 6 year old girl – and I often criticize myself for falling short. I think about this all the time as a teacher and as a colleague. Again, I often criticize myself for falling short. Don’t get me wrong, I think I am doing the right thing much of the time, I just wish it were easier – or more manageable – to do the right thing all the time. Our school Reverend delivered a chapel today that made me really dwell on this and I remember an important quote that I keep on my bulletin board. It is a quote that the wonderful Meg Craig (@mathymeg07) shortened for a poster in my room. I want to share the quote to help me stay focused and, hopefully, to help anyone else reading this stay focused as well.

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

Source on the web here

Eyes on the Prize

In Geometry today we were reviewing for tomorrow’s test. A student asked to go over a proof about kites that we did together last Friday. It got me thinking about which proofs are really essentially interesting in Geometry AND it got me thinking about some former colleagues in Florida. I worked with a history teacher who had an interesting habit. The day after any test he handed out the essay question that was going to be on the next test in a few weeks. He checked back in on the question during the unit and used this question as a guide to their discussion along the way. At the same school I worked with a photography teacher. At the beginning of any project assignment she would hang up the best photos from previous years and referred back to these as a guide for her students.

So, today it occurred to me that there might be ways for me to model for my students what the goal posts are as we move along through the course together. I think I wrote about this already, but one of the changes I have made this year is that I am handing out previous tests that I have written a few days before our test day this year. My students take this HW assignment more seriously than any other assignments. I wrote the Geometry book we use and I have written all of the problem sets we use. I hope that my students feel that this problem sets are meaningful and worth their time. However, I understand that there are calculations to be made about how to spend time and my students feel that available time is at a minimum. I think that I will not wait until the week of my next test, I think that tomorrow I am going to hand out last year’s test even though our next test does not occur for a few more weeks. I will check back in on this test every couple of days to give the students a sense that we are making progress. I also want to spend this summer compiling about twenty or so proofs that I think are particularly interesting. I will not include the proof itself, I will simply put together a set of diagrams and given information along with the conclusions to be drawn. I think that I want to hand this packet out at the beginning of the year next year and use this as a regular reference during the year.

I would love to hear any opinions about the benefits or drawbacks of these ideas.

Thinking out Loud…

A super brief post here – hoping to find some great advice from the world outside.

Two years ago our school launched a Discrete Math course. We realized that we had a number of students who were not being served well by our curriculum – a relatively standard one, really – that served to deliver most of our students to the doorstep of Calculus. I loved Calculus as a student and I have been happy to teach some form of it for most of my teaching career. However, we realize as a department that not all of our students need to see Calculus as the pinnacle of their high school math career. We also offer AP Statistics but we still saw a groups of students who were not being served properly. We have a vision of this Discrete Math elective being a lively, provocative course that exposes these students to more (and different) mathematical ways of thinking. We adopted the text For All Practical Purposes (9th Ed) and in the first year we had the course, the publisher released a new edition. I am teaching the course this year and I really like the students in the course and I feel that we are making some real progress in showing a different side of mathematical thinking, something other than algebraic reasoning and equation-based mathematics. However, I am not thrilled with the text. I am not sure that the level of the writing is suited well to my kiddos and I spent most of the fall term writing my own problem sets. Since the text is not available anymore I am faced with a choice of moving on the the 10th edition, finding a new text to serve as the center of the course, or going text-free and writing/borrowing unit notes and problem sets to support the students.

I know that there are people who visit my space here who have experience with math electives outside of the algebra to Calculus stream. I would love to hear some advice from them either in the comments here or through my twitter feed (I am @mrdardy over in the twitterverse) The ability / interest level of the students varies in this group. Some are taking the math course out of good conscience/concern about the college process. They know it is a ‘good idea’ to have a fourth math course. Some are taking it to fill out their schedule. Some end up in it after dropping back a bit from another math course that was more of a challenge than we/they expected it to be. This year we spend some time on elementary statistics/probability already. We spend a little time in the fall getting ready for a last swing at standardized tests. We are currently immersed in a unit on elections and voting strategies. We will visit some finance ideas and we will dip our toes into graph theory / network theory ideas. I am not married to any of these particular ideas, but many of them pop up in most discrete math text options out there. I kind of love Jacobs’ text Mathematics: A Human Endeavor but it seems not to be currently in print and I do not want to go down the path of a text I cannot reliably get my hands on.

So, dear readers, I would appreciate any wisdom you can share from experiences at your schools.

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.

A Fantastic Day of Wrestling with Problems

imageA former colleague of mine, Lisa Winer (@Lisaqt314) tweeted a problem at me last night. Wednesday night I s my basketball night and then I curled up to watch some Netflix with my wife, so I did not see the problem until this morning. It has since been making the rounds a bit. It is called ‘The Hardest Easy Geometry Problem’ and you can find it at

I started working on it on my side board today and it caught the attention of my BC students. One of them found a solution using trigonometry and I constructed the triangle in GeoGebra to confirm that he is correct. However, I still have not found a way to solve it geometrically. I reduced the problem to four equations with four unknowns but the matrix is singular and I could get no solution. I did, however, have fun playing with it and watching a number of my students dig in.

In Calculus BC today we talked ourselves into the area formula for regions bounded by polar curves and we had great conversations about it in both of my sections. In each class I had at least one student remember some area formulas for triangles that are rarely used and that help serve as the basis for the integral involved. I was pleased with each of those classes today.

In geometry we are working with quadrilaterals and a recent HW problem presented the students with a parallelogram and some algebraic expressions to deal with. Most of my students made an assumption regarding the intersection of diagonals for the parallelogram. They correctly assumed that they bisected each other. I was pleased that they made this assumption but I made sure that they felt comfortable with an argument supporting that fact and then a series of questions erupted that carried us through the end of the day. Do diagonals bisect each other for all quadrilaterals or just parallelograms. A couple of quick sketches at their desks implied that it was not always true. A quick visit to GeoGebra seemed to convince them. Then a student asked if a quadrilateral could have congruent diagonals if they do not bisect each other. A few more sketches and then the guesses started flying in. It did not take long to guess that an isosceles trapezoid would fit this bill. Again GeoGebra confirmed our guess. What next? How about the triangles for,Ed when the diagonals cross? Are they all congruent? Are they congruent pairs even? Quick feelings that the ‘side triangles’ are congruent but the top and bottom ones are not. Right again! But my favorite part came next. I did not plan on talking about area for a couple of days still, but the moment felt right. I asked if we could deduce an area formula for this trapezoid. Now, last year at this point I had a student suggest drawing one diagonal to find two triangles. Standard and clean. I also had a student suggest dropping two altitudes from the ends of the shorter base. Again, a nice standard solution. I had one student suggest rotating the trapezoid 180 degrees to create a parallelogram twice the size of the trapezoid. Not standard at all, but also kind of confusing for his classmates. This year, I had a student named James make a suggestion I had not head before. He asked me to draw segments from the end of the shorter (upper) base down to the midpoint of the lower base. This created three triangles all with the same height. I took a picture of the sketch we Made on the board. That is the photo on top of this post. I must say that I am completely delighted at this clean and clear way of looking at this area problem.

A pretty good day overall, I must say.

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.

Geometry Progress Report

I have a couple of posts that I want to make. It might be a busy weekend between writing midterm comments and airing my thoughts here. I promised to report back on my grand experiment with lagging HW. Now that we are three weeks into the term I think that I have some meaningful observations.

My first observation is that I need to find some meaningful way to regularly incorporate HW so that my students feel that it is a meaningful exercise. I think I am making strides by writing problem sets that reflect my book and our class conversations. I think that I have written problem sets that strike a decent balance between practice and challenge problems. I have been making class space for conversations about the current topics and trying to create some space for simple practice and check-in with some entrance slips. However, it is becoming pretty apparent to me that too many of my students are not in the habit of doing their HW on a daily basis. When we check in on HW at the beginning of class there are plenty of empty desktops and too much silence. It also seems clear to me that these are old habits and the reason I say that is that MANY of the problems they are struggling with now are related to writing line equations. Since we are juggling perpendicular bisectors of triangles, altitudes of triangles, medians of triangles, and angle bisectors it is kind of essential to be able to work with line equations. I know that these are skills that they have had and have displayed, but if the practice was not put in originally, those skills do not settle in and stick very well. I am reluctant to grade HW for a number of reasons. If all I am doing is checking for completion, then I feel I will be often rewarding sloppy and incorrect work and possibly helping some bad habits settle in. If I collect and grade it based on correctness I fear that I will be encouraging students to take some dishonest shortcuts. Instead, I am trying to use the entrance/exit slip idea to encourage attention during class with the hopes that that attention and the reminders of the skills necessary through the entrance/exit slips will (a) make the HW easier when it rolls around about three days after the class discussion and (b) allow me (and my students) to realize what they do or do not know.

My second observation is that this idea of HW lagging behind instruction will take some time for my students to get used to. They have been SO accustomed to trying their hand at something as soon as they begin to think about it and this new pace feels very different to them. I think that the old habits are working against them as they have expressed more confusion on some of the problem sets than I saw last year when I was using these HW assignments and assigning them the night that we introduced ideas in class. This, again. is something I need to address. I need to figure out how to help coach my kiddos to be able to deal with this process. I am too convinced that this is the right way to do this. Reading about it, thinking about it, I am sure that this is the right thing to do. My first time checking in on their progress right now (on this quiz on Tuesday) was a bit of a disaster. There were a number of scores hovering around 50% and for each of those students I returned the quiz with a practice assignment on writing line equations. I am trying to be positive and emphasizing that they know how to do this. I am convinced that this is true but I saw SO many mistakes on the quiz that it was a bit disheartening.


Conclusions? As I mentioned, I am convinced that this is a good way to weave in review, encourage reflection, and try to embed knowledge more deeply. I just need to figure out how to help coach my students so that they can realize the growth that I want to see for them.