Precalculus Projects

As the school year comes to a close, I’m finishing grading the third project of the year that my Honors Precalculus students completed. As always, I’m blown away by the reflections that the students write about how the project period went for them.  I also do need to put a grade on their work, so I’m looking closely at what they did, wrote and created.  I am so pleased to see how many kids did something really interesting in exactly the type of exploratory fashion that I hope to see.

As an example, two groups chose to study Markov Chains.  They had seen some probability and a little bit about matrices.  It was a hard topic to tackle and try to understand, let alone explore and push and do more with it.  One pair of kids started by looking at weather data and how likely we were to have a rainy day followed by another rainy day, versus sun or fog.  They pulled data, analyzed it, made predictions, asked more questions, and explored.  They asked hard questions and tried to answer them.  The other group studying Markov Chains started looking at games, as well as economic applications.  One student ended up using her programming knowledge to write a virtual Chutes and Ladders game that models your moves based on rolling a die and the probability of moving to a certain position on the board.  The game included the ability to be played in 100,000 batches, to collect experimental probabilities.  Another group was interested in probability so they created a video teaching about the basic probabilities in poker.  They did some additional exploration of the mathematical differences between five-card poker and seven-card poker.  The best part about that project was how much fun the kids had who did it; they spent more time doing this work just because it was what they wanted to be doing.

[As a total aside, this makes me think of a way to motivate homework.  If I can create learning experiences that make students want to do more exploration “after hours”, then I don’t actually need to ever assign homework.  The kids who want to do it, can.  I’m not sure where that nugget leads but it’s an intriguing thought.]

Last year, we did three projects as well and the timing felt fine.  This year, it was too many.  Due to schedule, we didn’t complete the second project until early March.  Then, we needed to start the final project in time to share out at our school’s Math Exposition Night.  Therefore, doing the projects one month apart (not counting vacation time) felt too close together.  I knew this was going to less than ideal and I even felt a bit apologetic as I announced we were starting yet another project.  It is a testament to the great group of kids I’m teaching this year that they were willing to dive in again just because I’d asked them to do it.

It is only now, as I read and have time to process some of what they students wrote, that even though I felt like this project was too much and wish I had made a different plan, my kids think differently.  One person wrote, in answer to how the structure of the project period work with your own strengths as a student:  

“The structure provided allowed us to go into depth with our topic because we had so much freedom. In previous projects I was wary of the freedom that we had because it was difficult to get started and go in to depth on the projects without instruction but this time around I knew what to do to be successful during the project period because of the past experience. The peer review day also helped cater to my strengths because I was able to take the constructive criticism and apply it to the project to help improve it.”

Another person wrote “I feel as though I didn’t take full advantage of the project structure until this project period. I always tried to make a project plan before starting to know what I need to explore but this time I didn’t make a plan and rather just decided what I wanted to explore as I did the project. I discovered and explored a much more diverse plethora of ideas through this structure and I ended up enjoying the project a lot more thoroughly because I was doing what I was passionate about learning more about rather than what I thought would be most useful in my project. This played to my strengths as a student because I wasn’t planning every step and I could explore what I wanted to, even if it was outside of my comfort zone, and this allowed for the product of the project (the artifact) to be much more detailed yet simultaneously broad with having covered so many aspects of [my topic].”

Someone else shared: “My first project dealt with fairly simple math and I was more reluctant to push myself beyond what I already know. I think that each project period I have been able to push myself further in thinking of new things to explore beyond what we have covered in class. I think this project period was the most successful for this because we chose a topic that I had never even heard of before.”

Lastly, a student wrote: “I feel as though I have gotten more comfortable going outside of my comfort zone and using the projects to enjoy maths explorative side. I was in a habit at the beginning of the year to plan everything out before I began, whether it was a project or approaching a problem. This project series has brought me to realize that when working a hard or long math problem it is not always the most effective strategy to plan your attack before beginning, often times success comes from exploring and trying many different ways to solve a problem before discovering the most effective way.”

In the end, it seems my “mistake” worked out fine.  I am finally seeing the growth and type of work that I am hoping to see by taking this time.  By dedicating such a significant amount of class time (in and out) to the projects themselves, I think my students can see the value I am placing on this type of exploration.  I am continuing to tweak how I do things but I feel like I must be heading in the right direction to see more groups whose work feels like a great future project exemplar.  Now on to writing down those ideas to save them for next year!



Although I use Google for many things in my life, I don’t use it regularly for the Groups feature.  However, some colleagues of mine began an area math department chair Google group, and there have been some great email exchanges.  I credit  @MrJoshSeamon for organizing the group meeting this year, and keeping the conversations going.  I must admit that as I woke up today, and contemplated adding in a 2.5 hour drive into my day for a meeting that I could attend for 2.5 hours, I was not feeling inspired. However, I’m so glad I did get in my car, and rearrange my schedule to make it work.

It is always refreshing to see others who do your job, and to hear from them about how things are going.  I liked starting the gathering by sharing things that had gone well this year for each of us.  That set a tone of positivity to the day, although there were certainly moments to commiserate as well. Fourteen of us were present, from area peer schools.  There were certainly some voices missing, but there was a solid representation, and enough of a critical mass to have more than enough conversation and sharing of ideas. One of my big takeaways from the day were some ideas from @MrAbend ‏about grading more holistically.  He was generous enough to share his rubrics and handouts via email, and I’m excited to think about ways to use this next year.  I also liked the idea that is a key part of the Andover structure around compartmentalizing trimesters of courses.  This model may work best in a trimester system, but I hadn’t thought of that until someone else brought it up again today.  There are some intriguing possibilities, if students are able to enter into a course at the spot where they need it most.  Maybe some kids only need part of a year of a particular topic?  Or maybe this is how choice can enter the curriculum more easily?

I found myself connecting with colleagues today in ways that made me want to talk more, keep meeting and listen a lot.  I also heard several comments over the course of the day that solidified my view that my department’s current focus on equity and access is more vital than ever.  I feel like my school could be a leader in this area if we make some deliberate, positive, informed changes.

Every time I leave campus to go to a conference or meeting with other educators, I am reminded at least once about how lucky I am, and how great my school is.  Even if I took nothing extra away from attending today’s gathering, the realization of the amazing caliber of students we have, and how lucky we are to have them, makes me excited to get to class tomorrow.  And to keep working for all of the kids in my school.

How did I get to teach the nicest kids this year?

I’m not in my first year of teaching, so one would assume that my rose-colored glasses have long since been removed.  However, in this year, my 19th year of teaching high school math, I feel like I am lucky to be teaching the 36 nicest kids at my school.  This would be impossible, right?  No teacher actually thinks that about every single student they are teaching.  Perhaps I have left out a few key words: I feel like I am lucky to have realized that I am teaching the 36 nicest kids at my school.  I’m just the one waking up to notice.

I do not mean to imply that all of my previous students have been less nice, or less enjoyable to teach.  Rather, I often would find that a few kids were harder to teach, or harder to get to know, and I would just let that be a thing.  I wasn’t going to be able to connect with that student, and I am realizing now that I gave up on my teacher-student relationship with that kid.  This is such a loss; I know it’s a loss for me, and I can only imagine that having another adult to talk to, who isn’t your parents, and who can find the good in you, is only a benefit when you are navigating life as a teenager.

Part of me feels sad that it has taken me so long to come to this epiphany.  Another part of me is ecstatic that I have reached this place.  Of course, it coincides with me handing two courses over to a colleague who is returning from sabbatical.  I have never felt so bereft as today when those classes met and I wasn’t there.  This is silly, right?  Yes and no.

I approached this year with the goal of helping my students know each other better, and for me to get to know them better.  I mixed up seating a lot and adjusted whether we were working in pairs, small groups or in a full class discussion.  We took time before going over homework to share our favorite (and least favorite) Halloween candy, so kids could have some low-key time together.  In soliciting feedback from the kids, instead of feeling like we were being silly, they were enjoying these activities.  I asked them directly if they knew everyone’s name in the room, and one person honestly shared that they weren’t sure if they did.  This part of building connections is my job; navigating who sits where is seamless when the teacher forms the groups and only fraught with social challenges (or just plain lethargy) when the teacher doesn’t get involved.

I asked students to journal to me in a shared Google doc (inside of their shared math Google folder) about themselves, about a math article we all read, and about how they prepared for a recent test.  They were open and honest in their writing, and I was honest as I wrote back to them.  One student needed to be reminded about my expectations for the length of their responses.  Another one wrote at length about their love of math and science.  As I wrote back to them (changing to another text color), I felt like I was starting in on a conversation with them.  I look forward to continuing this dialogue as our year continues.

So, in this 19th year, as I take the time to truly get to know each kid in the room as a person, I am so happy with what I am finding.  Each student has their own story, their own interests, and their own challenges.  In building this relationship much more deliberately than I have in the past, I find that I am treasuring each interaction, pondering it, and using it adjust my instruction almost on a minute-by-minute basis.  I plan for the unit, the week, the day.  In the moment as I circulate through the room to check in with students, it can become clear we need a bit more time to work on a particular problem, or more questions are raised about another idea.  My class plans guide us but we adapt and adjust as we go.  I have a vision of our class being on a sailboat together, and there are moments where we tip just a bit too far to one side as people are confused.  We have to pull together to right the boat, and be able to proceed safely.

I know this sounds corny, but this approach has made me a much happier person this year.  I was thrilled to have one student write to me recently, saying, “Thank you for being an amazing math teacher that always seems to be smiling.  I swear, sometimes I’m so upset with the day, and then I walk into your class, and I’m suddenly happy again.”  I think this is the same feeling I have as I go to class every single day, and it definitely has affected my outlook on life as a whole. I’m already looking forward to tomorrow!

Ungraded quizzes

A week ago, as I began thinking about what I would do this week in class, I thought it might make sense to have a short quiz on Friday.  I left that idea in the back of my mind for a bit, got to class on Tuesday, and realized a quiz did not make sense.  We had just returned from spring break, and giving a quiz would feel unkind and stressful.  In that moment, it occurred to me to give an ungraded quiz instead.

Yesterday, I wrote the quiz using the same mindset I would use if I had been giving a graded quiz.  As I gave this quiz to two of my classes today, the whole experience felt different.  A few kids needed to leave class a bit early; not a problem.  They could leave the work they had done with me, and take a blank copy of the quiz with them to finish the last two problems, to bring to class on Monday for me.  Kids who typically have extra time on tests could use that time, or not, as they chose.  I felt like I was helping them, and they seemed eager for the opportunity for true feedback.

Some students finished their quiz in 15 minutes, while others stayed for 45 minutes.  I checked in with each kid as they turned in their work.  Quite a few students said they felt good about the quiz.  At least six other students asked to schedule a time to meet with me, knowing we would be having a graded quiz in about a week.  Overall, I think I just liked the feel of the whole process.  Students didn’t get stressed out.  Instead, they had an opportunity to test themselves, see how it felt, and then gain feedback from me.  I suppose it would be even better if they got that feedback before they left the classroom but I don’t see how to do that without using significant technology.

I would like to keep thinking about ways that I can view my role as “serving” my students.  What else do they need that I’m not currently providing for them?  What do your students need?

Halfway points


It is part way through exam week as Semester One draws to a close, and I realize that many of my “New Year’s Resolutions” from this fall have faded from my mind.  I am about to pick up another class and want to spend the time to get to know them so we can work well together in the second half of the year.  As my mind goes there, I realize that I have not done some of the things I wanted to do this year.  In September (and even earlier), I was planning to find ways to make my classroom feel more like an elementary school classroom.  I tried a few days with “centers” in my Calculus class, but my activities were too long and made the rotation idea less realistic.  Although I was initially enthused, I didn’t work at it hard enough to make it viable.  Now, I’ve just stopped.

I have three classes with very different personalities.  One small group is extremely quiet.  What changes do I need to make so I hear their voices more often, and they get what they need out of class?  Another group is boisterous, eager to jump up and do problems at the board.  However, two or three students sometimes get left out of that vibe.  How can I reclaim some class time to make sure everyone has their “moment”?  My last section is my favorite.  I know you’re not supposed to have a favorite, but I do.  I’m crediting some of the time I put into the class at the start of the year to build the environment we want in the room.  However, a lot of it is pure luck of the draw with who is in the class and how they interact.  It’s a variety of freshmen, sophomores and juniors, all of whom are curious.  They have just submitted their second big project of the year, and have had two real opportunities to explore an idea of interest to them.  We have had multiple days where I have had what felt like the perfect class with people all over the place, all working on different things in slightly different ways, and me interacting with a few of them in significant mathematical conversations.  I want to make opportunities for more of that to happen.

I know I’m tired, and that it’s a cold, rainy day in January.  However, I want to find the energy to direct to each of my classes (including the new one) so that each student has the best experience in the second half of the year as possible.  But first, I need to grade some exam projects. 🙂

TABS and Time to Reflect

I just returned from a weekend away at The Association for Boarding Schools conference, presenting with two friends about how to handle having a challenging advisee/advisory group.  It was a great change of pace from the usual week, and I’m grateful to have had the chance to present.  Our session had been planned in advance, and I felt ready prior to departure.  Therefore, my time away did not require me working into the night, and my early prep work for the week meant that I knew what my classes were all doing while I was gone.  In a way, attending this conference was a bit of a vacation.

In this mindset, I decided to attend any session that interested me.  I was awe-inspired from the start, as I listened to Wanda Hill as she received the Ruzicka Compass Award for work as an educator. Hearing her story about helping minority students gain access to boarding schools in  New England, beginning in the early 1980’s.  She was incredibly brave in her work, and was vocal about what needed to change.  For example, she would not send any of “her” students to a school where students on financial aid were required to work on campus.  I recommend looking up her full story; it’s a fascinating one.

Charles Best was the official keynote speaker, the founder of  He was suitably stunned to have to follow the impressive Wanda Hill onto the stage.  Nonetheless, he told an incredibly compelling history of, how it works, and where he thinks it could lead.  It was a great start to the day.

As I moved into an initial session, I went to hear Andrew Watson (@Andrew WatsonTTB) speak about Long-Term Memory Creation.  Although nothing he said was new to me, the ideas and research have solidified in my mind in a way they clearly had not been before.  Andrew spoke about interleaving, the benefits of blank page review, and how to provide “desirable difficulties.”  He was incredibly concise as he spoke, explaining the extensive research in this area.  Perhaps because my mind wasn’t racing with what I needed to plan for class tomorrow, and what was in my bag waiting to be graded, I could finally process what he was saying.

I do “blank page” review often, but not often enough.  That’s an easy change to make.  However, I do not interleave on a daily basis.  I think about the concept regularly, and try to think about overlaps between units over the course of the year.  This is not enough.  The yearly approach isn’t a bad idea; it’s just nowhere near enough.  I need to mix up problems every time students are doing them, so they aren’t in Section 4.1, needing related rates, and therefore expecting to use that skill.  This is not a hard change to make because I don’t use textbooks in many of my classes.  However, I need to make a commitment to changing the type of homework I assign, in particular.  My students will learn more by doing 3 or 4 different types of problems than doing 4 identical ones.

I am eager to commit to making these changes!  Thanks, #TABS2016.


I have started doing some tutoring on the side for one of the college students who helps at  my children’s elementary school afterschool program.  The young woman [I will call her Joanne] who needed help has impressed me over the years with her dedication to her work, and to the children.  Upon seeing her talking to another parent about a hard math test coming up, I let her know I was a math teacher and I’d be happy to help her.  We didn’t get together until recently, but working with her for just an hour was fascinating.

Joanne is taking what is essentially an Algebra I class at her college.  She has attempted three other math classes and has either dropped them or failed them, and she needs to pass one math class to earn her degree.  Her advisor suggested she try a self-paced course, which happened to also be computer-based, using the ALEKS program.

As we began to work on expanding polynomials, I shared a few techniques that seemed to help, at least at that moment.  Based on some of the comments she made, I realized that she didn’t really understand what an expression like “2y^6″ meant.  If students don’t understand the building blocks, and why they are useful, how can we expect them to make sense of multiplying one polynomial by another?  Our assumptions can be risky, and mostly they are risky for the students in our classroom.

The other thing that has stuck with me that Joanne mentioned was about her former math teacher.  She sees her high school teacher at her job, and that teacher has chided her about her math struggles, saying” Joanne, you know this stuff!  We did all of this together in class.”  Again, this is an issue of assumptions (which include mine as well).  What did Joanne understand when she studied these ideas the first time?  If she did really comprehend all aspects of her work at the time, what makes it so hard the second time around?  If nothing else, it points out how rarely anyone is being asked to use some of the skills we emphasize in high school.  How can we make the work that we do more relevant?

All of these ideas coalesced in my mind this weekend.  I was out in the city, and overheard two people talking about the incredible sales going on.  One person was accurately estimating what the sale price of something would be if it were 40% off.  They talked through their technique, and reached a correct answer.  Then, the person turned to their friend, joking and saying, “Math!  It’s finally helping me!”  This makes me sad.  I’m not quite sure what to do with that feeling, beyond keeping it in my mind every day.


I’m in the midst of conducting a few teacher evaluations, so as the department chair, I’ve been sitting in on a bunch of math classes.  I find a week of time when I can visit one particular course, so I can get a sense of the flow of the class, how a given day feels, and look more longitudinally.  Last week, I also participated in Instructional Rounds, a new form of school-based professional development that we are trying out as an institution.  On that day, I visited a Religions class, an English class and an art class, for 20 minutes each.  On a somewhat related thread, I have been teaching a colleague’s class for about two weeks while they are out on short-term leave.  It’s sort of like observing a class because I’m trying to figure out who they are, and how I can help them.

Overall, these visits have been really enjoyable.  Visiting the art class was probably my favorite, but I did realize that even just spending 20 minutes in a class gives me plenty to observe and a lot to think about.  Here are a few of my takeaways after being in others’ classes:

  • It’s invigorating and refreshing to watch my colleagues teach, regardless of discipline.
  • I can easily empathize with students who might have an entire class day with teachers talking at them.  I also find myself thinking about the student who is less inclined to participate in class discussions.  That person could spend the whole day not speaking, and not engaging, depending on the activities planned in each class.  I need to make sure my class engages all of my students at least some of the time.
  • A one-on-one private conversation with a student is often the best first step to take to understand someone.  I found myself giving the advice on multiple occasions to talk directly with the student whose actions are frustrating to you; step away from the heat of the moment, and ask some questions about why they are acting the way that they do.  Then, be prepared to listen.  How does that kid who is dominating the class discussion want you to notify them that they need to take a break?  Empower the student to navigate whatever situation you are in, and work on a solution together.
  • Returning student work promptly is always a good idea, and will always be appreciated.
  • Teachers doing all or most of the talking does not result in a good class experience.
  • Teachers need to be paying attention all of the time as to what is going on in their classroom.  They won’t see everything but they will see a lot if they look.  This makes me of a think of recent post from @dcox21 on When the Activity Isn’t Enough.  He wrote about watching students actively doing a Marbleslides activity on Desmos, and observing what we can learn about how they approach the challenge of finding the correct equation.  We all need to keep watching, and keep learning.

I’m sure there are many more items to add to this list, but these are my highlights for now.  When can you find 20 minutes to visit someone else’s class?  If you can make this happen, post something you learned or were reminded about by conducting that visit.

Good days

I recently read Zach Cresswell’s post on an ideal classroom.  I feel lucky because I’m teaching in what we are calling  a prototype room, as we think about new spaces for the math department at my school.  We have begun to find furniture and lighting, but I realized after reading Zach’s post that my class was not yet feeling like the class he described, and I wished it did.

It was pure coincidence that we were about to begin an axiomatic trigonometry multi-day lesson, where students would be working together and I would not be a prominent voice in the room.  These three days (especially the last two) made me so happy!  On day one, the first few proofs to do were fairly straightforward, so the students were working quietly and diligently on their own.  I didn’t like how it felt because I was hoping they would be talking to each other, and they weren’t doing that.  However, I realized afterwards that they didn’t need that help yet; they were doing fine solo.  Once they reached a harder proof, they immediately started talking.  For most students, that began on Day 2 of our in-class work on these problems.  Then, I started looking around (but forgot to take a few pictures) and realized I was seeing much more of the classroom that I hoped to have.  Even then, I couldn’t observe it all because I was deep in conversation for about 10 minutes at a time with one student at a time.  I did see students up at the board (of their own volition), talking through how to solve one problem.  Others were huddled (standing) in a group of about 6 or 7 people, listening to one person explain how to think about another question.  No one was using our two beanbags that day, but three girls were at the stand-up desk, and others moved around the room when they wanted to do so.  I had music playing through the projector.

I was already excited before I walked in the door to Day 3 of this unit.  In many ways, this was a repeat of Day 2, with new problems to consider.  A few new things arose:

  1.  My two students who were in the play and in the midst of run week apologized for not making it farther through the packet.  As I had already said to them, this was an experience and not the type of assignment where every student needed to get to the last question.  Whatever they could do in class would be the right amount.  I have to believe that this shift of expectations helped to make their week more manageable.  It also made me wonder if I could make this same statement about other sets of problems I ask kids to do.  Is it ever about doing all of the problems?
  2. One of my freshman began to shine.  I had seen his energy before, but others in the class didn’t yet know him well enough to see his “light.” As I looked over his shoulder, he definitely did not need my help.  Instead, he wanted to show me the proof he wanted to type up for homework.  They were going home that weekend with the assignment below:

Using LaTeX, type up three of your Axiomatic Trig proofs.  Include:

(1) a proof where you felt great about your final product, but you really struggled figuring out the proof

(2) a proof where you knew what to do, did it, and wrote a great proof


(3) a proof that showcases something else about this week for you.  In your .tex file, be sure to explain why/how you chose the proofs that you did, and what your third proof represents for you, and why you selected it.  How did this week go for you?  As a student?  As a learner?  As a way to solidify prior trig knowledge?  I really want to know what you think!  This is a journal entry-type thing but your writing will be with your proofs a .tex’d document.  

This student was so proud of his work on a particular problem that he knew he wanted to use that proof as his “3” choice proof.  It was fun to hear his exuberance and to be able to share that with him!

3. This same student had a chance, at my suggestion, to help two others in the class who hadn’t made it as far, and were struggling with one particular proof.  I love seeing a freshmen teach a junior something, and have that be a real, fruitful conversation.


So – as I had these two great days, I want to try and help myself remember what led to this moment:

  • letting go of being at the front of the room for any significant portion of class time
  • encouraging students to help each other, particularly building connections across grade levels
  • having students progress through material at their own pace, and only set the amount of total time spent on the unit (here, three days)
  • recognizing that students working quietly and productively can also be useful

Mostly, how can I provide just enough of a structure that there is something to explore, and then get out of the way so the students can dive in?

Post trig test thoughts

I started the year in Honors Precalculus a bit differently, looking at vectors first.  With a colleague, we thought about how reflecting vectors could lead to a “bowtie” of sorts, and lead into circular functions in fruitful ways.  It is now the end of October and I’m wrapping up a big part of the trigonometry unit.  My students took a test which went well for many of them.  The average was a 92%.  The two students with the lowest scores earned 80%.

In some ways, those numbers represent what I want.  An A- average seems great.  Having two kids who still have some misconceptions about this topic still able to master 80% of the questions seems about right.

However, the numbers don’t tell the whole story.  In looking at some of the student work on the test, here are a few samples that caught my eye:


This student circled this problem, and put a question mark at the top of his work.  He included a clear calculation of the period of the function.  Then, there are calculations all over the page (the student did not have a calculator during this assessment).  What made this student doubt themselves on this problem?  What part of the process felt “off” to them?  I am eager to ask.

Another problem:


For the first time this year, I did not have my students complete a single-sided worksheet with about 40 problems like this on it.  Clearly, this student’s solution is not a particularly efficient one.  There are some lengthy calculations present because the student only wanted to think about the problem in degrees.  Also, it shows me someone who likely uses their calculator briefly in questions like this, not hesitating to multiply 23 by 180, and then divide by 4, and also by 360.  Because they haven’t needed to be efficient in this process, there is not an inclination to simplify the problem before multiplying and dividing.  How can I better build up my student’s number sense to avoid spending time on something like this on a future test?  Or should I instead provide a more basic calculator to each kid in case their working strategy involves computations like this?

Here’s one more solution that was interesting to me:


This student’s work is interesting because he had part of his answer incorrect.  However, I think it was only off because of a miscalculation of the period (or of the coefficient of x in the equation).

Mostly, I’m ending this unit with some questions for my students about their thought process. I’m eager to see what they wrote in their math journals this weekend about the experience, and to have them reflect on a problem (on the test or from another moment) that did not make sense to them right away, as part of their ongoing math portfolio.  I’m trying hard to see my students, and realize that as I look harder, I will only have more questions for them.