Reflective –> Effective

Okay. It’s time I hopped back on the blogwagon. I have been in graduate school this past year earning a Master’s degree in Private School Leadership (yeah, that exists) but I haven’t felt like I had anything to contribute to the discussion around math TEACHING up in my ivory tower. I want to end the year by reflecting on some of the things that I have been thinking about, ranging from little random thoughts to unanswerable questions, all of which I am excited to test out next year in my triumphant return to a math classroom!

So that’s what the next series of posts will be. But first… I have wondered if it was a good/necessary move for me to take a full year away from the classroom to reflect. One the one hand, it’s great to have the space and time to really delve into issues from human cognition to the use of data to improve student learning. On the other hand, I haven’t been able to workshop any of the ideas running around in my brain. Regardless, it has driven home to me the importance of reflecting, so I thought I would share the small change last year I made in my lesson planning that helped me become a more reflective, and thus more effective teacher.

You don’t need to take a year off or find more hours in the day to journal. You just need to add a column to your lesson plans.

lesson plan2

This is an actual screenshot from my Evernote planning notebook.  I chunk my class into activities, so that’s what you see on the way left. The next column describes any other necessary details or files that the activity requires. The third column is initially blank, and this is where I reflected. Every day, before planning the next lesson, I would go back to the one before and jot down a bullet point or two about each activity. Sometimes I would have a lot to say and would write some notes for my future self, but I would, at the very least, note how much time the learning activity took. Before long, this became a habit of my lesson planning that took no more than a few minutes.

I initially conceptualized this as a way to keep notes for myself in the future, should I teach the same class again, but found that I reaped the benefits far more quickly. Just sitting down for even 5 minutes to think about what happened that day started a recursive process where my reflections allowed me to make decisions in a different way in the future.

So I hope that some of my reflections from grad school might be helpful (for both readers and myself) — but I am also looking forward to reflecting next year in the classroom in a way that has a more immediate effect on my teaching and student learning.

Observations: Socratic Dialogue or Socratic Monologue?

(This is from a short series of posts about some of the things I learned from a year of observing and being observed with a colleague)

(hey, I’m a math teacher, not an artist).

Has this situation ever happened to you? One thing that my colleague and I talked about over and over was how we engage in Socratic dialogue portion of our classes. What kind of questions are we asking? What kind of answers are we getting? What do these answers actually tell us? One thing that we both noticed was that occasionally you can get in a flow where you keep a learning conversation going by taking one word student answers and fleshing out their thoughts fully, or just finishing their sandwiches sentences for them. It feels so good because it feels like THE WHOLE CLASS is on this EFFICIENT AND WONDERFUL thought train going at just about the same pace as someone who is basically an expert in the subject. And as “duh” how counterproductive this is to learning, it’s something that I totally didn’t notice until I had someone in my classroom to point it out to me.

What’s the whole point of Socratic dialogue in the first place? Well, for you, it is a way for you to check for understanding. Are students listening and understanding what is going on? And for them, it is a way to get them to think. Are they just taking in what you are saying or are they turning it over in their heads and taking it to new places? If we finish their thoughts for them, we are not only robbing them of the opportunity to think and learn, but we are deceiving ourselves about what they actually know and understand.

Here are some strategies that we discussed to avoid this:

HOLD THEM ACCOUNTABLE: Turn that into a full sentence please. Try that again with better mathematical terminology.

FOLLOW UP QUESTIONS: How do you know that? What do you think of _____’s answer? (Crucial to do this for both incorrect AND correct answers).

ASK BETTER QUESTIONS: Explain…. Why… How… If the question can be answered in one word, it’s probably not a great question.

STRUCTURED RESPONSE: Think-Pair-Share. Quick written reflection with cold calling.

BE EXPLICIT: Be direct about what you value in student answers.

I grew more and more aware of this as the year went on and I think I got much better at not only asking better questions but eliciting better responses from my students. I think this heightened the level of mathematical discourse in my classroom, and also gave me a much better idea of where they actually were in the learning process. It didn’t mean that students wouldn’t try the age old SHOUT ANY MATH WORD YOU CAN THINK OF to answer a complicated math question, but at least they understood why I pushed them further.


Observations from Observing: Methods

This past year, I had the distinct pleasure of serving as a mentor to an excellent new math teacher who was an absolute pleasure to work with (and is definitely going to be a superstar). All of the teaching fellows at my school took a seminar with an administrator, and then each one had a mentor who observed them and generally gave them advice. Though we did some lesson planning together, and we worked a little more closely at the beginning of the year, our main form of interaction was weekly observations.

For the whole year, she came to my class once a week, I went to her class once a week and then we met once a week to debrief. Sure, she tells me she learned a lot, but this was also the best professional development possible for me too. I learned so much over the course of the year and engaged in so many excellent conversations about teaching – I grew so much from a commitment of a little over an hour a week.

It makes me realize that I should have been doing this all along with a colleague, apart from the whole school appraisal process. Though it seems so easy, and I of course exchanged the common “I’d love to come visit your class” with so many colleagues, it never happened before. I think the thing that made it work with us was a structured commitment, and the formation of a habit in our schedules (it didn’t feel like something ON TOP of everything else, it was part of what I did every week). Any observation program, even if an informal agreement between colleagues, needs to be structured and scheduled so that we don’t push it away for the million other things we can do with our time. It can’t be something like “Go and visit someone’s class in the department at least once this term.” In my experience, that just does not work.

As I pore over my notes from the past year, I am going to dedicate the next few blog posts to my major takeaways from a year of observations. But first…

How We Observed Each Other

There are a million different ways to do observations, but here is how we did it:

  • All visits were mutually scheduled beforehand, at the beginning of the week. We got so comfortable with each other that knowing someone was in the room was REALLY not a big deal, and wouldn’t change how we would plan our lesson.
  • We did this so we could touch base beforehand to see if there was anything that the observer should look for. This was helpful when we were workshopping a technique, trying out a new type of activity or just wondering about something that is happening in the classroom.
  • The observer would always take notes in the following format, which is something we developed based on other formats, and which we found simple and effective:
    blog-observation notes
    The time column really helped us pay attention to something that is often the last thing you pay attention to when directing a classroom. The middle column helped us talk about what happened and helped us remember everything that happened in a class. The column on the right was an acknowledgment that we aren’t just robots sitting there, but have helpful opinions and suggestions, but it helped us focus our subjective comments by basing them on what was actually happening in the class.
  • Then, we would find a time to exchange notes and just talk about everything that happened in a comfortable and frank manner. We would talk about the objective things that happened in the class and share ideas and observations based on those. It wasn’t hard to talk for 45 minutes about teaching, but we could share notes in 15-20 minutes if pressed for time.

What worked for you for observing? Any other way of taking notes or organizing a program?

The “One Cut” Problem

What can you do with this?

Today, we did what I thought was one of the coolest explorations I have seen in a while. It is called the “One Cut” or “Fold and Cut” problem, with inspiration from Patrick Honner (@MrHonner, this part of his website). The premise is simple: you have a shape drawn on a piece of paper. How can you fold the paper so that you can make one straight snip with the scissors and cut the shape out?


Weirdly enough, there is a theorem that says that no matter what shape, this is possible (concave, convex, numerous closed figures, as long as the sides are all straight) . Which totally blows my mind. Because it’s really friggin’ hard in practice (though, the star above is pretty easy). Check out some patterns for some other shapes here.

photo (4)With one group today, we started doing this problem as a way to see what vocabulary they knew with polygons and to talk a bit about symmetry. I printed out little shapes on 1/4 sheets of paper and the levels went as follows, getting a bit harder as the levels go up:

Equilateral Triangle –> Square –> Isosceles Triangle –> Rectangle –> Regular Pentagon –> Regular 5 pointed Star –> Scalene Triangle –> Arbitrary Quadrilateral

They were TOTALLY hooked. Every single kid was working on their own and kept either having the “YAYY, TEACHER LOOK!” reaction or laughing hilariously at the silly shapes that they made by accident. The kid in the picture with his hand over his face kept yelling out “AGHHH, TRICKY TRIANGLE, TRICKY TRIANGLE” because he couldn’t do the scalene triangle. But he wouldn’t accept a hint from me because he wanted to find out on his own!

45 minutes later, the classroom was a total mess, and I was wondering where all the time had gone…

Now, working with the regular polygons you might get duped into thinking it’s pretty easy. But try a scalene triangle. Or a non-special quadrilateral (i.e. most quadrilaterals). It’s actually VERY difficult. And I think the solution is pretty fascinating, because my solution to it (which I got after about 10 triangles and an hour) heavily involved the triangle’s incenter (the intersection of the three angle bisectors). Which made me think that this would be a super cool thing to do in a geometry class when talking about angle bisectors. Are there multiple “incenters” on shapes with more sides, even if you have to define the idea in a different way?

I won’t share the solution but just post a picture to show you that YEAHHH it’s possible.

picstitch (1)Geometry or other math teachers… where else could this go in the classroom?

(If you haven’t heard me bemoaning how much energy it takes to do math for 2.5 hours with 11 and 12 year olds at a summer camp, well then FYI: I am currently teaching at a summer camp for 6th and 7th graders from China who are here doing an academic program that is half ESL, half math. It’s getting more and more fun, but certainly has been an adjustment from teaching seniors in high school for only 45 minutes at a time).

Building a Better Review for the AP Calc Exam

Both years I have taught AP Calculus AB, I have kind of dreaded the couple weeks of review. They are hard to plan for and probably really boring for students. On top of that, last year, I felt like I squandered the review time. I mostly gave them free time in class to do whatever they needed to do, and I am not sure how effective this was. 45 minutes straight of studying really dragged and I felt like students didn’t really even know what they needed to work on. In addition, the lack of structure I think prompted some students just to look at answer keys instead of struggling through problems themselves.

This year, I was dreading review again, but it really went much better and I think was much more engaging and effective. A few things I changed this year:

1. We started reviewing in class earlier, even before we finished all the material.
2. Review was more structured by me at first and slowly led to more independence, with opportunities for students to see which topics they needed most work on.
3. I spent literally 1% of the time explaining at the front of the classroom and 99% of the time having them to do the work.

The learning structures I used for review:



For about 3 weeks before we started review (i.e. while we were still doing DiffEQs, volume etc), we started class with 5 to 6 quick skill based questions. I tried to put a range of topics, from evaluating limits to writing a tangent line to finding the average value of a function. Students pulled out their notebooks and worked on the questions silently (or as silently as I could get my 85%-chatty-bro class to work). They did the ones they could do and tried the rest. After the timer ran out, I would scroll down and show the answers and show what Standard that the question corresponded with. Then, after explaining anything that needed explaining, we would vote as a class on whether to retire a topic if they felt confident or keep it on for the next day. This took about 10 minutes at the beginning of class.
I loved
this because even though it ate up class time during the end of the year and forced the actual material to take longer, by the time we were ready to review, students had already brushed up on the skills and could focus on big ideas.
Next time, I will try to be more organized about it and have a booklet printed, or sheets for them to glue – I was improvising with this and I felt like it took students too much time to copy things down from the projector.

(this is sort of what it looked like below, but this is for integrals earlier in the year – I’m between computers right now and don’t have all my old files!)


At the end of  many units towards the end of the year, we would do a 12 minute timed Free Response Question, and this is something that we did almost daily during our review time. I would hand out a free response question on a little slip of paper, they would glue it into their notebooks and work on it for 12 minutes silently. If they didn’t know how to do it, they would just try as hard as they could, struggle through it and write down what they know. Then, after 12 minutes, I would hand out the answer key and they would grade themselves, AP style.
I loved that this forced them to struggle through a question and see what they actually know, and I loved that this got them used to AP grading (I had a much lower incidence of unit-forgetting and less-than-3-decimal precision). The trick for both of these benefits is in really holding out on the answer keys until the end of the time!
Next time, I will try to coordinate the 5 minute skill drill with this so that students can recall the topic before a tricky free response question, as I had some students who were so stuck that they didn’t really write anything down and got nothing out of the exercise.


I find multiple choice harder to integrate into class than free response, but one learning structure I liked for multiple choice was Jigsaw. For those that don’t know this (I assume it is fairly common), there would be a set at 12 questions and groups of 3-4 would all work on a third of the questions together (1-4, 5-8, 9-12). Once every group got through theirs, I would rearrange the classroom so that each new group had one person who had worked on each of the sections. Then, they would either work on the rest of the questions individually and then check with each other when they got stuck, or they would just take turns and teach the other members of the group their questions. Some students reported to me that the process of explaining a question out loud really helped them understand what was going on.
I loved the interactions that this activity prompted and I loved how efficient it was for getting through many multiple choice questions (students could do this much faster than working on them on their own).
Next time, I will try to deal with the awkwardness of groups finishing at separate times and weak students incapable of explaining questions to their classmates, though I am not sure how.


I didn’t trust my students to do free response questions at home. They would just look up the answers and get NOTHING out of the process! But we did do a lot of multiple choice questions at home, through Moodle. It is super easy to set up quizzes, so I would just upload images of the questions from a multiple choice collection I had and set the correct answer. I would do 15 questions in a quiz, and it would take my students about 40 minutes to do. We started this about a month and a half before the exam, and then all the homework during the review time become these online multiple choice questions. Before the test, every single student did about 130 multiple choice questions, which amounts to about 3 full tests, and then many did more questions on their own outside of that.
I loved that the work was immediately self checked and automatically graded, as I think this did a lot for their learning from these questions.
Next time, I don’t think I would do so many of these as I think they got a bored with them. Also, I felt like some students were just clicking through the questions, so I would try to think of ways to get them to take these learning opportunities a bit more seriously.



This is, of course, nothing original, but if you have the luxury of stealing a few hours from your students on a weekend for a Mock Exam, do it! Correct it for them, but don’t put a grade on it so that it can be a truly diagnostic tool. This was the most helpful thing for my students in prepping for the exam, because, on top of everything else, the Mock really helped them figure out their weaknesses so that they could really be productive when finally I gave them…


By the time I was giving them large chunks of time to work in class on their own, most students knew what their weaknesses were (from the Mock, timed Free Response, Moodle Multiple Choice etc). Whether they needed to improve their multiple choice or their free response, or they needed to work on specific topics (and could with a packet I gave them with AP Free Response questions split up by type), I felt like most students REALLY used the time well, to the point where a lot of students didn’t even bother studying the night before the exam. All the structure and diagnosing we did at the beginning, and all the work that THEY were doing instead of me talking helped them become far more independent and effective in the review process. I hope it worked – I will find out in a few weeks!

Any review structures you used that worked well?

Volume in Calculus: Conceptualizing before Formalizing

One of our PD sessions in the past was about how to support students with learning differences. One of the points that the presenter made was that most pedagogical tools that you would use the better serve these students are great tools to reach all learners. This struck me especially because I teach almost entirely students for whom English is their second language, and sometimes when I do something specifically to help students with the language of mathematics I come to larger and more general pedagogical understandings.

For example, this past week, I introduced solids of known cross-section in AP Calculus in a way that I thought would ease my students understanding of the tricky language involved in the problems, but what I ended up doing was really effectively let them develop their own conception of how these solids are formed and THEN interpret the AP problem language and integral notation in those terms. Conceptualize and then add mathematical formality to their own conceptual framework.

Here’s how it worked. I put 4 of these solids out around the room:

  1. First, I gave them 1-2 minutes to SILENTLY write down in bullet points how they would describe to someone else how the solid was formed.
  2. Then I gave them 2 minutes to share ideas in groups.
  3. Then I cold called on 7 or 8 students via a deck small cards with their names on them (which is by far my new favorite teaching tool). After I called on some students, I called for volunteers with any other ideas.
  4. LAST, I asked them to flip to the back of the paper and read the actual description.

During the “share” part, students said some of the craziest, random stuff, but most of the important parts of the description were said by various students. When it came time for them to read the description, at first they were like “whoa” because the language is still a bit daunting. But after a minute or so of close reading, they connected everything in that description with things that they themselves had said. So when it was time to do the actual integral, the intermediate notation I use made 100% sense:


So general pedagogical moral of the story? Letting students conceptualize something on their own before bringing in mathematical language and notation makes it more likely that the notation will aid in their understanding rather than provide another hurdle in learning. 

A teaching fellow (a first year teacher) was observing my class (and has been observing periodically throughout the year). Afterwards, she remarked that she felt this was one of the most effective 10 minutes of the year, and I agree! And I think 10 minutes on this (instead of just 1 minute reading the question) will save lots of time in the future. Next week, I hope to try the same strategy with solids of revolution!!


Whiteboard Experiments: Modified Mistake Game

I have used Mistake Game a lot in class. Students write up the solution to problems on whiteboards and purposely make a mistake in the solution. Then they present their solutions to each other, presenting their mistake like they meant to do it. Then, students ask thoughtful questions to try to find the mistake.

This works great with topics that are conceptually rich, but less so in topics that are more mechanical, where mistakes tend to be a bit harder to see and are less rich to talk about, like implicit differentiation for example. I did a modification of the Mistake Game that worked really well for this:

  1. In groups of 2-3, students write solutions to a problem on a large whiteboard. After checking their correct answer with me, they go back through and make a mistake in their solution.
  2. Students then flip over the sheet I gave them that had their answer and write what there mistake is, kind of like a mini answer key.
  3. Groups then rotate around the room and try to find the mistake in the solutions in front of them. Once they find the mistake and check their answer with what the group wrote, they move on to the next board.

I wanted to train them in the art of looking over a solution and checking its correctness, and I think that this did that well. Compared to the mistake game, I felt like more students were active at any moment, more students could carefully follow complicated work, and it took much less time (20 minutes as opposed to 40)… but we also didn’t have the great mathematical discussions that we normally have during mistake game. I guess it really depends on the topic at hand which version is more appropriate, so I’m definitely going to keep this one in my teaching toolbox.

Whiteboard Experiments: Practice & Reflect

I tried something new in class this week that I think solves a few random problems:

  1. Sometimes, when working on whiteboards, one student hogs the marker and does a lot of the work (and thus the learning)
  2. With whiteboard work, students don’t have anything in their notebooks to study later
  3. When we practice things like derivatives in our notebooks, I feel like their notes become almost useless because it is a mess of 15-20 examples.


Practice and Reflect

I put the whiteboards out on the desk and left them there the whole period. We were learning the derivatives of exponential and logarithmic functions of bases other than e so I wanted to go back and forth between the whiteboards and their notebooks a few times.

After teaching them a derivative rule as a whole class, I gave them a sheet of 15-20 problems that definitely got more difficult as they went along. I gave them 12 minutes to practice (I put a timer on), and they worked on the problems with partners on the whiteboards, which gave them a chance to discuss, erase mistakes and see problems in large format with different colors.

Then, I asked them to put the markers away and open up their notebooks and gave them 3 minutes to reflect. I told them they could do whatever they want with this – copy down a few problems that were tricky, write down some things that they want to remember, write down steps for the problem. With this, I feel like their notes were a bit more focused and useful. I also felt like the whole routine was efficient, in that it kept a vast, vast majority of the students moving and engaged. I’ll definitely try this again.

An Anchor Problem for Riemann Sums

I like to start most new units in Calculus with an “anchor problem,” a common sense, every day problem that motivates new techniques and serves as a base that you can constantly refer back to. Some that I have used in the past, to varying degrees of success, are Infection for Inflection, Your Speedometer and the Intermediate Value Theorem, and Predicting Stock Prices with Differentials.

For Riemann Sums, and integration in general, I use the question that really inspired integration in the first place: how do you find the area of an irregular shape? I tell my students:

You work at the glass company. You are given the task of replacing all the glass on the front of this beautiful building, the Duxford Aviation Museum. How much glass do you need? All we know is that the building is 90 m long and 18.5 m tall in the very center.

(This task was partly inspired by this post from Shawn at ThinkThankThunk).

duxford(isn’t this building beautiful??)

I have it printed on two sheets of printer paper for every group (so big enough to draw on and mark), and I give them 10 minutes to come up with an estimate. Every group writes it on a piece of paper, and then I put it in an envelope. About a week and a half later when we learn the definite integral, we calculate the actual area (using a parabola fitted to the top of the building) and the winner gets…. well nothing. But I announce it at least?

Most students struggle a bit at first and then eventually just start to try something. Some students try some sort of bizarre modified equation for the area of a circle (which I always find really interesting), some turn it into triangles, but most use the maybe-not-that-subtle hint that the window is broken up into square panes.

Right after they are finished making their predictions, we discuss. I ask them what their strategies were and how they could have made their predictions more accurate. I try to get them to come up a couple of points (that sounds manipulative):

  • We took an irregular shape that has no simple geometric area equation and turned it into a shape that does have a simple geometric equation.

  • We split up a larger shape into a bunch of smaller shapes to be able to do this.

  • The smaller our shapes are and the more of them there are, the more accurate our estimate would be. In fact, if we could use infinitesimally small shapes, we could be perfectly accurate.

I think that this activity really shows them how difficult the problem that we are trying to solve is, and primes them to know why we set up Riemann Sums the way that we do, but to be unsatisfied with this solution to the grand area problem. Prepped and primed for Riemann Sums, but with some foresight to know where we are going.

Whiteboarding Mode: Simultaneous Show and Tell

Side note: Simultaneous Show and Tell is a terrible name for this whiteboarding mode (because it kind of sounds like a lot of whiteboarding). Forgive me, I cannot think of anything better. So… propose a better name?

[update 11/25: Andrew in the comments suggested “Function Iron Chef” which is definitely the winner. That’s what this whiteboarding mode is called now]

Students are in groups of two at a whiteboard with a VERY LARGE set of 3 X 3 axes drawn up on the board. They are sitting in a U shape so that if everyone put up their boards, every student could theoretically see everyone else’s. I put up a prompt like this:

Draw a function such that…

  • \lim_{x \to -2}=3
  • f(-2) does not exist
  • \lim_{x \to 1} does not exist
  • and f(1)=-3.

I put the timer on. Students are given a few minutes to draw a function (any function, lots of correct answers!) that fit the prompts. Then, at the end of the time, everyone puts their markers down and puts their board up. We spend a minute silently looking around the boards to look at everyone else’s work. Then, after a minute is up I allow the students to ask questions of each other (i.e. not just say “THAT ONE IS WRONG”). If they don’t ask questions about some that are suspect (or some that are totally correct), I will ask questions at the end to talk about specific boards. We then do 5 or 6 other rounds like this.

POSITIVES: We have done this so far with limits, continuity vs. differentiability and will do it in a few weeks with graph sketching – I think that making them do things the other way around, making them create (instead of just identifying limits or whether a function is continuous) really forces them to think harder. I also like this because when students have to show their work to their classmates, they often put a little bit more focus into making sure they are proud of what they have (and just about every student is engaged in the process, especially if you make them switch markers). I also love times to showcase mistakes as part of the learning process – we try to be as open and supportive as possible in correcting the boards. Lastly, having a discussion in a math class is always a really nice change of pace.

ISSUES: Students can get a little crazy during the discussion process and some can phrase things negatively. Not all students are good at following along verbally when discussing, and will wait for others to point out mistakes in the board. A few times the whole thing has taken a long time with all the transitions, but it has gotten better every time. I’m not sure how the weak students feel about this activity (having their work showcased and critiqued). Also, I’m not sure that this type of activity would be great for anything but a topic where the students already have some fluency and mastery.