My First #ObserveMe Experience

First things first. It can be intimidating to have someone observe you. At least, when I first started teaching, I felt like any observation was a judgement (even when I was told it wasn’t). So for all you new teachers out there, if I just told you that it can be a rewarding and fruitful experience, would you believe me? No? Well, let me tell you what I got out of my first #ObserveMe experience. If you aren’t familiar with the #ObserveMe movement, it was popularized by Robert Kaplinsky as a way to help teachers take ownership to improve their practice. When he put out his call to action last August, I excitedly got on board, posted a sign outside of my classroom, announced that my door was open at several staff meetings, and then…nothing. To be fair, I’m in a tiny school, where there’s only one other math teacher, and non-math teachers probably felt that they wouldn’t get much out of observing someone outside their subject area. And this year, I couldn’t model observing (which had been one of my goals) because I have six classes this year and no prep periods. With that class load, I definitely did not want to write any more sub plans than necessary.

So summer turned to fall, which turned to winter, which turned to spring, and still no luck. And then I realized that I could go outside of my own school. I mentioned to a few people who I’d like to observe other teachers during my spring break (since I wasn’t going anywhere and was planning to work through it anyway). Then I mentioned it to Robert when I ran into him at the NCTM annual conference in San Antonio last week. And that did it – now I had to follow through! And that’s when I took to Twitter:

Soon, I got a response from Paul Jorgens, an 8th grade teacher in nearby Palo Alto. A few messages back and forth, and I was set to visit him to observe his Algebra 1 class. Paul got one of the Desmos fellowships that I’m so jealous of, and it sounds like it’s a pretty amazing experience.  Maybe I’ll be able to commit to it in 2018-2019 (assuming they have it again and I get accepted). He is also an experienced teacher who has been at the same middle school for longer than I’ve been teaching.

I arrived at the front office of his school, and then got a few minutes to chat with him in the staff lounge. When I asked him about what he wanted me to focus on during my observation, he mentioned the work of Schoenfeld, who researches out of UC Berkeley. He described the Teaching for Robust Understanding (TRU) Framework, something that wasn’t familiar to me, but which I’m eager to investigate. In his classroom, he handed me a paper with the table below, and suggested that I choose one area for my focus today. SchoenfeldObserveStudentEyes

I decided to focus on the “Equitable Access to Mathematics” row, which is similar to one of my own personal goals for this year. From a student’s perspective, “Do I get to participate in meaningful mathematical learning? Can I hide or be ignored?” Such important questions, and really, all of these questions are very student centered, and something I plan to bring back to my own practice, and to share with my school.

Then, we went to Paul’s classroom. Paul team-teaches this particular class with another teacher, Brian (whose last name I didn’t catch). Today’s lesson was a Desmos activity about exponential functions, and because Paul is a Desmos fellow, he had access to some interesting new features that I’m excited to try out when they get released to the masses.

Not only did he use Desmos, he started the Desmos activity with a “Which One Doesn’t Belong” activity. Already using two of my own favorite things to do in class. But of course, I did have a focus for my observation, and it’s important to remember that even with an amazing lesson plan, it’s really the implementation of the lesson that dictates its success. A great lesson in the hands of an unprepared teacher can go awry in the same way that a poorly designed lesson generally cannot be saved by even the greatest of teachers without abandoning the lesson plan.

In this class, all the students knew what they needed to be doing, and were well engaged. Each student had their own Chromebook, and technology problems didn’t deter anyone. A few Chromebooks had to be traded in for new ones, and there was some sharing, but there was no real disruption. There were a number of things that Paul said which made it clear that all students got to participate. I heard a lot of “Tell me more about what you said” or “Tell me more about what said.” Paul also gave students time to reflect quietly and then write down thoughts before he called on students. This meant that students could not hide, but they had the opportunity to prepare which makes participation less intimidating for those introverts out there. (Adding more time for students to write before answering is something I really need to work on in my class.)

I also really liked the technique of not asking “What do you think?”, but rather asking “Who heard something in their group that made sense? What did you hear?”. This is one of my favorite strategies in my classes to prevent the students who always know the answer from answering, and to give a chance for a student who maybe feels like they aren’t an expert the opportunity to evaluate what they heard from classmates and to share that information. It’s a great way to give everyone an opportunity to have a voice.

One other area that Paul mentioned he was working on was to give a stronger voice to the students from disenfranchised backgrounds, and so he called on those students a lot more often. I wondered if it was perhaps too often, at the expense of other students. There were definitely a few students that neither Paul nor Brian interacted with directly (that I saw), which is understandable in a class of 32. You probably can’t chat meaningfully with each of them in a 60 minute class period. However, how can you be sure that the same student(s) are not being forgotten or left out of the conversation?

Paul did say he’d rather err on the side of working with underrepresented students too often rather than not often enough, which I agree with, but I wonder if there’s a way to mix things up a little more. Perhaps with a random walk from table to table to check in with students? With the random variable groups that are created through cards at the beginning of class, students are never in the same groups two days in a row, which is great. I do wonder, though, if focusing on a group at a time, and within that group, a student from a disenfranchised background, may be a little more balanced, while still giving weight to those students who may be underrepresented in higher math classes later on.

One of my favorite moments came towards the end of the class, when Paul put up a long list of equations that students had written to try to model a particular exponential function to go through two points (I believe). He told students how much he was “excited to look at all of these equations <they> found to fit those points. Should there just be one equation? Could all of these equations match that graph?” What a great way to validate the variety of thought that students used to come up with their ideas, and to give them a sense to critique each other and think more deeply mathematically.

I was so thankful for this opportunity that Paul gave me to observe his class. I’m looking forward to setting up a time for him to #ObserveMe in the coming weeks, to give me some feedback on some of the areas that I’ve been working on in terms of student engagement and how they participate.

Again, I absolutely encourage you to get someone to observe you. Another math teacher at your school, another teacher of any discipline, a teacher outside of your school or district, and even a teacher outside of the grade levels that you teach. There are so many things we can learn from these observations, in a completely constructive and positive manner. My recommendation is to stick to one or two very specific and measureable goals, just like the kinds of goals we try to give our students. How do you want to improve upon or build up your teaching practice, and how can you know that it’s happening? Both getting feedback from another teacher and observing another teacher to give them feedback are meaningful and inspiring ways to continue your growth as a teacher.

Cited Work:

Schoenfeld, A. H., and the Teaching for Robust Understanding Project. (2016). The Teaching for Robust Understanding (TRU) observation guide for mathematics: A tool for teachers, coaches, administrators, and professional learning communities. Berkeley, CA: Graduate School of Education, University of California, Berkeley. Retrieved from:


How I Do Honors

In response to a tweet about differentiation, I connected it to my honors class. Benjamin Leis asked if I had any posts, and I realized I had been meaning to write one. So here it is.


I teach at a small independent school, and we don’t have tracks of classes. We don’t have A.P. classes. What we do have is in-class differentiation. We have the opportunity for students to take classes at what we call “Skills” level, for students who are taking the class purely for exposure to the material. This is often used for students who have some significant challenges with success in a class, due to learning disabilities or lack of exposure to prerequisite concepts and skills. The class does not count as a college prep class. Often, a student will take a class at skills level one year, then repeat it the next year as a regular college prep class, with greater success due to the extra exposure to the ideas.

Honors is different. Since we are in California, and many of our students are applying to UC schools, we make sure our courses meet UC a-g requirements. Image result for UC a-gThis also means that only certain courses can be classified as honors level. Last summer, I prepared the paperwork and got approval for an Honors Precalculus course. So what does that mean?

In my classroom, any student in the class can take the class as an honors class. There are two ways that the class is different for them, but as long as they take on the commitment, they get the honors distinction at the end of the semester. No student is turned away, and all students are encouraged to take on the challenge.

The first way that the class is different is in the way assessments are assessed. I have adopted a version of standards based grading (SBG) in my class. Assessments are scored on a scale from 1 to 4. A score of 4 means that a student can solve problems and is able to apply a concept (or the “how” of the math). Honors students are assessed on a scale from 1 to 6, where a 5 or 6 demonstrates a deeper and more thorough understanding of how that concept works and fits in with other ideas in math (or the “why” of the math). Make no mistake – a student who isn’t taking honors can still receive scores of 5 or 6 (which will improve their semester grade, as they’ve gone above and beyond), but to receive the honors distinction, they’d have to also meet the second requirement.

For the second requirement, each student must complete a research project for each unit and then share their findings with the class in a short summary presentation. For example, when we did our logarithms unit, the honors project was to build a slide rule, and record a short video in which they demonstrated how the slide rule uses logarithms to make arithmetic calculations. When we did our trigonometry unit, students had to develop and explain a trigonometric proof to the class. For matrices, students compared and contrasted using different methods (Cramer’s Rule, Gauss-Jordan elimination, standard algebraic solving linear systems by substitution or elimination) to determine which methods were better, and under what conditions, when solving by hand, with a calculator, or with a computer.

In both requirements, the honors students are expected to demonstrate a more sophisticated and thorough understanding of the math concepts, and to work on how to effectively communicate those ideas. That isn’t to say that the non-honors students aren’t required to understand concepts and communicate, but rather that they do what we have time for in class, whereas the honors students must take some time outside of class to deepen their understanding. For example, in class, we covered the trigonometric identities, and proved a few of them to demonstrate that it could be done. Honors students were required to prove all of the main ones, and demonstrate an understanding of how to prove a challenging one on their own.

Again, there is no restriction on who can elect to take the course with the honors distinction. If a student chooses to do the work, they will get an “H” on their transcript. I would like to see more students of color attempt to take on the challenge, but I would like to see more students of color in my PreCalculus class as well. This is a goal that we have to work on earlier, but that’s a separate blog post.

Three Days of Amazing PD, Bookended by Travel Problems (That Weren’t That Bad)

Staying up all night in an airport really isn’t my thing, but neither is trying to sleep on these uncomfortable chairs with the background hum of vacuum cleaners and occasional conversations. This is as good a time as any, I suppose, to reflect, at least a little bit, on the NCTM conference this week in San Antonio.

Let me get the travel part out of the way. I had set my alarm for 5am on Wednesday morning, but like a small child on Christmas Eve, I really couldn’t sleep. I woke up at 4:50, and was out the door by 6. My flight wasn’t until 8, and it’s really only a 15 minute drive, but I was so excited. Three days with math teachers, with my people, really means a lot to me, and since I found the #MTBoS community, I found a truly international staff lounge where I could share (and steal) great ideas. Seeing these fellow math teachers that I admire and respect in person is invaluable. But more about them later – back to the travel. I met up with Nicola (the other math teacher at my school) at the San Jose airport, and we had no problems getting on that first flight. For the first time, I had TSA Pre-Check, which was a nice novelty. Such a privilege to keep my shoes and belt on.

When we arrived in L.A. for our connecting flight, we found it was delayed. Then it got delayed some more, and some more. We were originally supposed to arrive at 3:30 for an early evening keynote by the author Jordan Ellenberg who wrote the book How Not to Be Wrong. I got to see him last fall at a Stanford Public Math Lecture, and was excited to see him again, and this time to get him to sign my book. Well, interestingly enough, I did read an interesting chapter.

Since we didn’t arrive until a few hours after his keynote ended, my book’s title page is still naked.

While waiting to eventually board our plane, I bumped into Jed, who I met last year at the #MTBoS booth at NCTM (I think?). I got to chat with him a bit about his talk on geometric transformations, which is his niche I think, the way that my niche is currently debates. Eventually, we did make it onto the plane, back off, and I took my first Uber ride with Nicola and Jed to attend the #MTBoS Game Night. Yes, it’s as geeky and as nerdy as it sounds – lots of math teachers, sitting around playing various games, including some fun Estimation 180 contests and a rock-paper-scissors tournament.

The following are the things that I attended. I’ll blog more about these after I get some sleep, which hopefully means later this week, but could mean the end of June. We’ll see.

  • Thursday morning: Math Debate with Mishal Surti and Anna Blinstein!
  • Thursday afternoon: Megan Schmidt – Statistics for Social Justice; John Urschel
  • Thursday night: ShadowCon; Mathalicious Party
  • Friday morning: Journal Writing; Josh Wilkerson – Service Learning; Mishal Surti – Depth of Knowledge; Jose Vilson – Math, Equity, What’s Not Adding Up; My Debates Talk
  • Friday night: Ignite; Desmos Trivia Night
  • Saturday morning: Chris Shore – Clothesline Math; Mark Couturier – Out of Classroom Experiences; Simon Singh – FLT to Simpson’s Math

Earlier today, I got a message that our flight to L.A. was going to be delayed, but no big deal – we’d still have an hour for our layover, and I’d still get home by 10 tonight. It also meant I got a chance to spend some more time with new and old friends from #MTBoS. Maybe we did an adult version of Which One Doesn’t Belong?

And then, another message, just as I was closing my tab. Our flight was delayed more. I met up with Nicola, and we got a cab to the airport to try to figure things out. However, our flight kept getting pushed back. We were told that we would receive a voucher in L.A., but we were considering just renting a car and driving back to the bay area. Sitting around, I found that some of the people waiting were also math teachers on their way back from NCTM…but not all. That got me thinking…how many of us on the plane would be math teachers? Seems like a good Estimation 180 task – how many passengers on the plane, and what proportion are math teachers? And who should show up, but Andrew Stadel, the developer of Estimation 180. We were chatting with a couple of other math teacher friends, waiting, and waiting, and waiting to board, and finally it happened!

On the plane we went! Seemed like a good time to pose that estimation question.

And so, after a fairly uneventful flight, we landed. Early! But that meant that our gate wasn’t ready. It took 40 minutes for us to arrive at our gate. Just before we pulled in, we did a big loop on the runway. I am pretty sure the pilot was just doing donuts.

So, we stumbled off of the plane, tired, and ready to grab our vouchers and go home. Except that the line for the Delta service desk was…about….40 people deep? That didn’t seem too bad, until we found that the people at the front of the line had been there for three hours with no resolution yet. We started exploring other options for hotels and car rentals and anything that would get us back earlier. Little by little, people in front of us gave up and got out of line, and soon we got to the front. (And by soon, I mean 2 hours). At this point, we’d tried calling hotels, but they were all booked. Also, I got a text that our flight tomorrow morning was delayed, from 8:10 to 11:30. Seemed strange, so I checked Google, and sure enough, there was a 3 hour and 20 minute delay. Then I checked the trusty Delta app, and it said the flight was on time. The gate agent also said the flight was on time. Not sure who to trust anymore, but since there was nothing more the gate agent could do except give us paper boarding passes and reassure us that we’d get compensated somehow eventually for our troubles, we went off to try to sleep or rest or something to pass the time.

And that’s where we are now…waiting for this last leg of the journey. I hope it’s swift, because I miss my wife and daughters and my bed and sleep. But really, I have to remind myself that as bad as it seems, it could be so much worse. There were people I met who had been stuck in the airport for three days, small children, persons with mobility based disabilities, and others who had every reason to be far more frustrated than I. Their difficulties don’t make mine any less, but they should change my perspective.

Teaching Stats in High School

An interesting thread came across my twitter feed, posted by the wonderful author Steven Strogatz. It was especially timely as I am hoping to find a way to add a Statistics class to our program here at Mid-Pen (while also hoping to reduce my classload from 6 to 5 classes – talk about needing a mathemagician! Anyone know a great math teacher who wants to teach 2 classes for free?) And it brought up an argument that I’ve thought a lot about over the last couple of decades.

I didn’t take statistics in high school. It wasn’t offered at my school at the time. When I entered Bates College, my first semester, one of the classes I enrolled in was BIO 244, an introductory class in Biostatistics. This was a long time ago, but I don’t remember being impressed by the class. I remember entering data into MATLAB, following recipes for statistical tests, and not feeling like it made a lot of sense. To be honest, I was also a very young first semester college student and probably didn’t take my academics as seriously as I should have. (Mind you, that had been true throughout middle and high school, and would continue through college. I finally learned how to be a student in grad school, but that’s another topic.)

MATLAB has come a long way since the 1990’s console window version!

My next exposure to statistics was MATH 215, a pure statistics class that had pre-requisites of MATH 206 (Multivariable Calculus) and MATH 214 (Probability). Here it was – the rich and theory based statistics that let you get your hands dirty with integrals and proofs and all the stuff that those non-math people couldn’t understand (or so I thought at the time, from my isolated and prestigious tower I imagined myself living in). Now this was the statistics class that I was looking for – nothing about data interpretation, except from a pure mathematical standpoint.

Finally, in my senior year in college, I decided to take a nice, easy class – and what do you know? There was PSYC 218, a statistics class in the Psychology Department. I figured it’d be an easy A – and if it was just about the math, I would have been right. In fact, I was that annoying student who had to show the professor that I knew the math better than he did, pointing out errors in the explanations of various tests and why they worked. I was also the student that lots of people came to in class when they didn’t get how to do the math. Unfortunately for me, there was a lot more in that class about how to design experiments, evaluate experiments, make conclusions based on the data, interpret results, and lots of other things that just didn’t seem to matter to me because they weren’t about the pure math. And (at the time) I was only about pure math. Applied math was a pointless waste for people who couldn’t live in my world of number theory and topology and elliptical curves.

Fast forward a few years, and I found myself teaching, which meant making the math relate to the real world for students, to make it meaningful and relevant. I was the only math teacher in a tiny high school program of 19 students, and was teaching Pre-Algebra, Algebra 1, Geometry, and Algebra 2…and U.S. History. (We had a math, science, and English teacher, and each of us had to take on a social studies class until the program grew.)

And grow it did…more than doubling in size over the next seven years that I was there. I started teaching calculus, and eventually was talked into teaching A.P. Statistics. And I was really wary of this – it didn’t require calculus, and how could you teach statistics without calculus? I had seen it go poorly1 for both biology and psychology classes at a very prestigious small college. But I decided to give it a try, and it worked well for students. A few years later, at a different school, I offered the same class, with some excellent results from students. And I realized that Statistics fits an important niche in the math education curriculum. It is a class that is taught exclusively as an applied course, which is different from how we teach Algebra and Geometry and Calculus. For those classes, even if we focus on how they can apply to the real world, that never seems to be our primary focus. And that’s okay I think. But statistics opens up the world of how to interpret data, whether in the sciences or social sciences. It’s power gives it validity and usefulness to many students I’ve had who, even when they come around to thinking that logarithms are cool and the Law of Cosines in vector form is amazing, have not experienced in other classes.

So, back to that thread in the beginning. Twenty-five years ago, I would have been firmly in the camp that calculus is required for statistics. And it’s true that knowing calculus may be required for understanding the theory of statistics. But a high school or introductory statistics class should be about how to use statistics, how to apply and interpret statistical tests, how to evaluate the use of statistics (especially in this era of fake news).

Florence Nightingale’s Coxcombs led to great advances in epidemiology.

It also offers some great historical stories about the development of statistics, and gives the opportunity to talk about Florence Nightingale’s contributions during the Crimean War.

Statistics can be a great class for the math student who has already taken calculus, and they can be encouraged and guided through some of the theory behind statistics. It can also be the perfect class for the student who is sure they are not going on in math or science but has finished Algebra 2 and wants a math class on their transcript. Make no mistake – I’m not calling this a math class for non-math people. What I am calling it is a chance for students to make a choice to take a math class that they see as relevant, that they will buy into, that they won’t question when they’ll use it because it is clear in every lesson just how it can be used. In the end, doesn’t this give them a greater appreciation for the richness of math?


1In retrospect, it is clear that my memories of those classes were colored by my biases about math as well as my poor student skills – they may have been outstanding classes, after all.

Working on Feminism in Math Class

My college adviser in math, Professor Bonnie Shulman, has been one of my biggest influences in my math education career. She inspired me as a teacher who could help s35dd7datudents find the excitement and joy that mathematics could bring them. I had entered Bates as a relatively young first year student, definitely over-confident, and with no real idea for what I would do with my life – probably become a doctor since I liked the idea of helping people, and it seemed like it would be a nice challenge. After my first semester, I realized that I really didn’t like chemistry and there would be a lot of chem required for pre-med, it would be hard for me to not major in music after having devoted so much of my life to writing and performing in various ensembles, and most significantly, that math wasn’t just easy for me – it was actually a lot more interesting and fun than I ever gave it credit.

Much of Bonnie’s focus was how to get more girls and women involved in math and science. As I started my teaching career, I made a conscious decision to give all of the girls in my math classes the most encouragement, the highest expectations, and the best future in STEM careers. And I failed miserably. Well, maybe not miserably, but I didn’t have them leaving my class and going straight into prestigious colleges and universities and graduating with math and science degrees. And I was sure the problem wasn’t me, because you know what I found? None of my female students had confidence. At least, that’s what I saw. When I asked a question in class, they didn’t raise their hand. If I called on them anyway, they would mumble, or say they didn’t know. If they did answer, they did so in a questioning voice. And what would I tell them? “Speak up! Speak confidently! Don’t answer with a tone that keeps going up!” And that fixed none of the problems. That is, maybe they would answer in that moment with a louder and less questioning tone, but their lack of confidence hadn’t really changed.

This isn’t to say that there weren’t issues of confidence that these girls had learned over the previous 8, 9, 10, or 11 years of classes before they got to me. And yet, despite my best intentions, all I was doing was pointing out their lack of confidence. Any progress that they were making in improved confidence was incremental, and probably being undone by my pointing out how they weren’t being enough like the male students.

I don’t pretend that I have all the answers now. I still have high expectations for my students, regardless of gender (male, female, trans, non-binary, fluid), and I make every effort to let each student know that I believe in them. But I don’t call on students when they aren’t volunteering. I don’t demand that they answer with confidence. In fact, I now relish the questioning, since doubt allows us to develop deeper and more interesting explorations into the mathematics. And with more time for reflection, I’m sure that I could figure out other things that I do that help.

One thing I do know is that I am surrounded by Dr. Johanna Nelson at SLAC's SSRL synchrotron facilityamazing girls and women in my life, many of whom absolutely love and appreciate math and science (including a large number of my previous students). I am also grateful for having my wife in my life, for many other reasons of course, but for the purpose of this post, because I find myself talking about a woman who is a successful physicist on probably a weekly basis in my classes.

Now, sixteen years after having started my teaching career, I have two (very different) young daughters, and a whole new lens to look through when I think about how I approach my teaching style for my non-male students, and also for my male students. I want to encourage all of the positive behaviors that students can have, regardless of their gender and regardless of gender stereotypes. I also want to remember to meet each student where they are, and to do everything I can to help them see the exciting things that math can tell us about the world and ourselves. Or sometimes, show them how we can just have fun with math.

Extra Credit? No More!

I used to give out extra credit. Not just at the beginning of my career, but right up through last year. I just thought it’s something that teachers do, even though I knew better, and even though it never really accomplished what I wanted it to accomplish. So finally, this year, I told my students that I was done. Those awesome challenge problems? No longer for extra credit. Attending a Stanford Public Math Lecture? No more extra credit for that. Predictably, students told me how unfair this was, how stupid this was, what could I be thinking?

Inspired by a tweet (shown above) by Brad Weinstein, I decided it was long past time to blog about this decision to tell others exactly what I told my students. So here it is…

What do we, as teachers, give extra credit for? To reward students for awesome and interesting things that they do that may not directly be a part of what we are normally grading them on. In other words, we are bringing up a student’s grade, not because they understand the thing that the class is all about, but because they did something outside of the class expectations. And if you have spent time in a classroom with extra credit, perhaps you’ve seen what I’ve seen. The students who do the extra credit tend to fall into one of two categories:

  • High achieving students who wants to bring their 99% up to 100% (because they are hypercompetitive or tie their own self worth to their grade).
  • Students who have not been doing classwork or homework and now realize that their D+ will get them in trouble with their parents and they want an easy way to at least get a C-.

So what is the result when a student does a ton of extra credit to bring their grade up by 3, 5, even 10 or 15 points? Their report card becomes a lie – or at least misrepresents their accomplishements. In theory, to me, a student who receives a C in, say, a Geometry class is a student who has demonstrated that they have a moderate understanding of the material presented, who may have never successfully and independently completed a proof but who gets the basic relationships between different shapes and figures, who has some grasp of parallel lines, congruent triangles, and how to find areas from formulas (even if they don’t memorize much and retain much in the way of details afterward). The main takeaway is that the student should have reached a certain conceptual understanding of geometry and spatial mathematics that will be useful in future math classes.

But what happens with a student who had a D, extra credit brings them up to a C, and certain assumptions are made about their conceptual understanding of geometry. This does them a disservice when they go to future classes, when their Algebra 2 or PreCalculus teacher get an invalid impression of their skillset in Geometry. This kind of grade inflation drives teachers crazy as well.

And what is it that we want students to get out of these extra credit assignments, anyway? The fascination and joy that can be found in mathematics, and the beauty, creativity, and surprises that arise in a really interesting problem. I want my students to get that, make no mistake, and I can incorporate a lot of that into my classes. When something comes up that doesn’t directly apply to a class, I can still bring it to my students because it is absolutely worthwhile for them to be exposed to all of this rich mathematics that doesn’t show up in a syllabus or in a typical high school textbook. Plus, when students see me get really excited about a math problem, it gets them excited, and that happens a whole lot more when investigating graph theory or topology than when solving linear systems by graphing or entering data into a table in my calculator to find a logistic regression function.

Plus, I want to reward students for their contributions to a positive math culture, especially by taking risks, by sharing their mistakes, by (respectfully and kindly) pointing out mistakes of others (including/especially me), by working well as a team, by helping their peers, and by bringing to class math that they find in the real world. So, to reward students for those contributions to our math world without artifically inflating their grades, I offer students points. To a degree they are subjective, which means that at times I will give a well meaning student the benefit of the doubt. It won’t affect their grade, just their current point total.

I announced at the beginning of the year that there will be individual prizes for the students with the most points at the end of the semester, and a class prize for the class whose students have as a whole earned the most points (representing a very positive math culture in a class). While there are some students who still grumble about not having extra credit anymore, many students are fairly invested in the idea of extra points. The semester ends this coming Friday, we have finals next week, and the following week, students will receive their prizes. The class prize will probably be a lunch – something simple, anyway, and food is always popular. The individual students will come with me to attend a showing of Hidden Figures.


One of the first big opportunities for students to receive extra points next month will be by attending the Stanford Public Math Lecture by Ingrid Daubechies on Mathematicians, Art Historians, and Conservators. These public lectures are a real treat, and often very accessible to high school students. The best part is, whether through a movie or a public lecture, students get a chance to learn more about, and be inspired by, real mathematicians. Isn’t that worth more than all the extra credit in the world?


Rethinking Vocabulary

In math, vocabulary is important. I mean, it’s really, really important. It’s important to those of us who can’t stand when someone points at an equilateral rectangle and a non-equilateral rectangle and insists the equilateral rectangle can ONLY be called a square, and it isn’t a rectangle. It’s important to those of us who know that the word unique doesn’t just mean unusual. Then we develop tons of notation and jargon, and we speak in our own funny language, and tell students that if they want to join the math party, they have to memorize all of these obscure words with different meanings than they’ve been taught before.

This has always been problematic to me, and I think I have a better idea of why now. As with so many revelations, this was largely born through thinking about Dan Meyer’s analogy of a problem being a headache, and math being the cure. But…these are always about the problem that a student is trying to solve without knowing the math required, and creating that need in the student to learn the math for a better reason than because we say so.

Well, what if we taught math vocabulary that way? That is, instead of telling students that the side opposite the right angle in a right triangle on a plane is called a hypotenuse, let them realize that it’s pretty tedious to always refer to the side opposite the right angle in a right triangle on a plane as the side opposite the right angle in a right triangle on a plane.

Polygraphs in Desmos I think are a great way to develop this need, where the vocabulary is the cure. I make a point in my class of how important it is to be precise in your mathematical communication, and I never test students on vocabulary. However, I make a point to use the relevant vocabulary regularly once we’ve defined it in the class. I also don’t take off credit if a student does not use the most efficient vocabulary. If she really wants to say the side opposite the right angle in a right triangle on a plane instead of hypotenuse, that’s up to her. As long as she is communicating her ideas effectively, eventually she will recognize the need to be more efficient. Generally, I don’t find that students in my class struggle with vocabulary even though I don’t test them. Both verbally and in written form, that just isn’t a common stumbling block.

My goal now is how to put more thought into making the extraneous words and extra writing and speaking enough of a headache that students are asking for vocabulary words to simplify our language. Does anyone out there do something like this – intentionally? Curious for those who do word walls – what’s your approach? Student created? Definitions/diagrams included? How much do you find it helps? Could the word walls become an aspirin for our challenging language?

Why Prove Things? A Geometry Debate

Earlier this week, my geometry students took part in a new debate format. As is often the case, the topic was student generated. Every year, students inevitably complain about proofs in geometry. It generally happens based on what they have heard from parents  or older friends who have horror stories about their experiences with impossible wastes if time, and it comes up before we do proofs. Once we start doing proofs, students question why we are proving things that were already proven. “Shouldn’t geometry be about measuring angles and lengths and areas and volumes?”, they ask. “Why can’t we just apply what others have proven before us? What’s the point?”

Of course, I have my reasons, but  

Pythagorean Triples For The Whole Class!

After introducing my geometry students to the Pythagorean Theorem, I introduced them to the idea of Pythagorean Triples. I showed them that the numbers 3, 4, and 5 will work in the equation: 32+42=52. Then I had students pick a couple of numbers, and we generated a brand new Pythagorean Triple with some fun algebra. At least, I thought it was fun, and students were curious where I was going with my algebraic manipulations. There are several methods, but the one I used went like this.

Pick two interesting numbers. Make them integers, not too big, but not too small. Did I hear 7 and 19? Perfect.

First, multiply those two numbers, and double your answer:
2 ∙ 7 ∙ 19 = 266

Next, find the (absolute value of the) difference of the squares of those two numbers:

192 – 72 = 312

Now, find the sum of the squares of those two numbers:

192 + 72 = 410

And just like that, we have a brand new Pythagorean triple:

2662 + 3122 = 4102

Go ahead – check it on your calculator if you’d like. Pretty nice, huh? I’ll leave it to you to do the algebra to see that it will always work. The most amazing thing about this little task is that students were immediately convinced (and rightly so) that even though it may be hard to find Pythagorean triples, there are an infinite number of them.

And then the game started. I gave each group of 3-4 students a stack of 24 numbers that can be turned into 8 sets of Pythagorean triples. (Of course, there’s only one way to get all 8 sets, but since there are some doubles, it’s possible to match up some that will throw you off).

For both of my geometry classes, all students (no exaggeration!) were engaged bell to bell, furiously testing and playing with the numbers, discussing strategies, and trying to find the eight correct sets. Not only that, although students were in separate groups, they soon realized that they’d be more successful working together. I heard so many great ideas about how to attack the problem that I wasn’t surprised when, even though it came down to the wire, one of the classes found all 8 sets!

This, to me, was a great low floor, high ceiling activity. Every student was able to work on calculations – calculating squares, calculating sums, etc. And some students really worked through some great strategies – what units digits can sum to the biggest number’s units digit? Can we work backwards from that? And of course, the little bit of algebra review in the beginning was helpful as I found later, when I had students repeat the process to develop their own new Pythagorean triples. It was easy for them to check their answers at the end and then look for any mistakes they may have made, and there’s something about big numbers that makes students truly proud (and rightfully so, even when they are using a calculator to do the calculating work).

Student Generated Problems Are Best!

Around this time of my geometry course each year, students have learned a whole lot of geometry – parallel lines, congruent triangles, basic proof techniques. It is the perfect time for them to come up with challenge problems for me. Believe it or not, classes never disappoint in this fashion. I’ve had some great questions in the past. (One of my favorites – can you fold an 8.5 by 11 piece of paper into an isosceles triangle where the entire triangle has the same number of layers of paper. I’m still not sure about that one, but I really enjoy watching students try every year.)

I tell my students that there are two things that make a challenge problem great:

  1. It is relatively easy to explain to anyone.
  2. It is challenging to find a solution (or even figure out if a solution is possible).

This year, a student came up with a problem that fit those requirements perfectly: Is it possible to take a square, and cut it into isosceles triangles that aren’t congruent? We then restated it to require a finite number of non-isosceles triangles (though the fractal version is a pretty cool visual and a nice approach).

We spent a few minutes on this at the end of class, but didn’t get anywhere right away. I sent this out via twitter, and almost immediately got a tweet back from Henri Picciotto:

What? That brought out some resolve in me to find a solution, and later that day I did. The next morning, before school, I had a student rush in to show me her solution, which was pretty similar to mine. I suggested to both my geometry classes that there may be more solutions – how many unique solutions or approaches may there be? Can we find an even number? Fewer than seven? I ended up waking up in the middle of the night thinking about this, and came up with even more solutions. So far, I’ve got solutions for 6, 7, 8, 11, 13, and 17 triangles. But the best part of this whole story is that a week later, I have students talking about this before and after school, at lunch, outside my classroom, totally invested in a problem that they made on their own.


When you are a teacher, these are the moments that you live for. Students of varying backgrounds and supposedly different ability levels collaborating and arguing with passion about squares and triangles. Sometimes, that question of “When am I going to use this?” or “How does this apply in the real world?” never come up because the math, by itself, is just fun.

So…can anyone find a solution with fewer than 6 triangles?