Seeking My Role in Diversity in HS Math Classes

I want to be clear. I teach in a private high school in Silicon Valley. Many* of the students at my school are white and come from wealthy families. Most of them have gone to either very good public schools or very good private schools, where most students received a generally good education in mathematics. It may not have been perfect, and our students may have slipped through the cracks, or been told that they weren’t math people, or somehow may have received the message that higher math wasn’t going to be for them. Yet those students were still exposed to the important ideas that they were expected to see, from fractions through basic algebra, from area formulas through linear equations and graphs. When that group of students has “Algebra 1” on their transcript, and they received a B in the class, we have no reservations about putting them into a Geometry class.

Our school also has a large number of students of color, and many* of our black and brown students came from very different schools. While most of Silicon Valley is quite expensive, there are pockets and neighborhoods up and down the peninsula that are considered low income areas. In some of these neighborhoods’ schools, some of our students receive a very different math education. I have seen students who received an A in an Algebra 1 class who had never seen a parabola, who had never factored a trinomial, and who were not consistently able to solve a single variable linear equation. In most cases, this was no fault of their own, and it is not my place to fault their Algebra 1 teacher.

These two different experiences are not an accident. Make no mistake about it, this is systematic racism. As Morgan Fierst posted in a conversation on twitter:

She is absolutely right. But what to do about it? The obvious answer is to dismantle the system, but how does that happen? There is definite harm happening in some elementary and middle schools that serve primarily students of color, but one thing that has become clear to me is that, as a white male high school teacher, I have no right to go in and tell other teachers, especially K-8 teachers of color, how to do their job better. My role is to find the leaders among K-8 teachers and teacher leaders of color and support them, and back them up, and give them my power to dismantle the system.

And what about my school? One of the deciding factors in me taking a job at my school was the high retention and low turnover rate. In my four years, we have had a science teacher retire (and then pass away), an art teacher go to graduate school, and a sign language teacher decide to become a stay at home mom. We have hired three outstanding replacements for those teachers, but only one was a person of color. One third of new faculty hires being non-white is an impressive number if we were hiring 200 people or 40 people, but not when hiring only three. I am not in charge of hiring, and I don’t know how much of an emphasis was made on looking for non-white teachers to interview. We are a small school and don’t have a lot of resources for hiring, and we are not a target school for lots of graduates of teacher credential programs. Maybe we couldn’t have done any better.

Our Head of School retired this past year, and there was an exhaustive search for just the right candidate. Our search committee decided on three very competent finalists, and again, one out of those three was a person of color. My question to each of those candidates was the same: “Our school prides itself on the diversity of our student body, but our faculty doesn’t look the same. We have an amazing and talented group of teachers, but we are mostly very white. Without firing faculty members, how would you improve the diversity of our teachers and staff?” It was an unfair question, and one without an obvious answer, but it was also a question where it was clear which one of the members had given it a lot of thought long before I had asked about it. No surprise, it was Phil Gutierrez, the one candidate who hadn’t lived with white privilege, and I am very happy that he is now on board as our new Head of School. I don’t think he has the answer (because, really, does anyone have the answer yet?), but I do feel that he has the same goal in mind.

For me, in my closed world of math education, the goal is to make sure that the higher level math classes have the same diversity as our general student population, and that our students who choose STEM careers in more rigorous schools are a diverse group of students. However, the end result of those students who enter 9th grade not prepared for  success in Algebra 1 or Geometry (despite what their transcript may say) is that they don’t take higher math or attend rigorous schools or choose STEM careers at the same rate as their white peers. They end up either taking a Pre-Algebra class and end up “behind”, or they struggle to keep up in their Algebra 1 or Geometry class, doing lots of extra work and getting extra help to catch up to their peers on the fly. The extra work and extra help takes extra energy and time that they frankly shouldn’t have to put in. Yet, what options do they have? What options do I have? And what options does our school have?

Over the past few days, I have read and followed and participated in several Twitter threads about these questions of equity and diversity. One blog post by Matt Vaudrey could have been written by me (if I was a better writer, and maybe got a few more squares in privilege bingo). Two new people I found to follow on Twitter, Twila Dang and Morgan Fierst, pushed me hard to think more about the systemic racism that exists, and made me wonder where I and my school still have work to do. Because the fact is that the moment a student becomes my student, their background, their previous experiences, every math class that they’ve experienced in the past is a real part of them, but it cannot be an excuse for why I can’t help them to be the best mathematician that they can be during the time I get to spend with them. As I write this, I realize that I’ve taken great care to focus on students with disabilities and non-male students, to help these traditionally disenfranchised groups see their potential and embrace their abilities in mathematics. I am proud of the work I’ve done in this area, and receive a lot of positive feedback and accolades. I have to wonder, though, why I haven’t made the same concerted effort with students of color. After 16 years of teaching, it’s a difficult realization, but one that I’m glad I finally made. Maybe this is the catalyst for the next phase of my teaching career. I think I have more clarity on my goal for year 17 and beyond, but I welcome any suggestions and feedback.

*To be clear, there is a diversity of economic backgrounds within each ethnic group, and I don’t have the hard data. I believe it is sufficient to say that most of our students from wealthier families are white and many of our students of color come from families and neighborhoods that most would consider lower-income. There are always exceptions to these generalizations. I also acknowledge that I am only discussing white, black, and brown students, and leaving out other significant parts of our population. I also haven’t brought learning differences into this post, which would further complicate the discussion, but these should all be important parts of any discussion of equity in education. I guess that’s the difference between an informal blog post and the book I wish I had the time (and skill) to write.


Functions – Operations, Transformations, Compositions

Several years ago, I taught PreCalculus from the COMAP PreCalculus: Modeling Our World (1st edition), which was a textbook that I really appreciated. It was very focused on good applied problems, on building conceptual understanding, and on avoiding lots of drill and kill style problems so prevalent in so many textbooks. I still use some of its problems as sources in my classes, but I did find that its lack of clear structure to its units, as well as minimal specific “vocabulary/theorems/algorithms to learn/memorize” was quite unpopular with students.

One of my favorite parts of the text was that it developed the idea of functions as a set of tools for modeling data. Based on the data that you are given, you determine which tool may be your best option. This led to a natural desire to transform or combine functions to make more sophisticated models. Suddenly, we could look at a polynomial in two different ways – is it a product of linear equations, or is it a sum of power functions? Depending on the situation being modeled, maybe one approach makes more sense than the second. And what happens if we want to divide one function by another? Suddenly, we can end up with a rational function, which can drastically change our end behavior and get us talking about a limit. What if we want to sum up different sinusoidal functions to approximate graphs that we see on an oscilloscope? And voila, we are exploring Fourier series!

The great part of thinking about a toolkit of parent functions and the various compositions, operations, and transformations on those functions, is that it allows a student to generalize what happens for any function, be it a direct variation, a sine function, a log function, or other. Playing around with Desmos makes these connections so much easier to see!


Inclusiveness in Math Education (#TMC17 Theme?)

Twitter Math Camp (TMC17) is over for this year. It took two days to start this post, and over two weeks to finish it, and there is still so much to process. This was my first one, and I’m sure that some parts are always the same, but other parts are surely unique to this year. If I had to pinpoint one overarching theme for the last week, though, it wasn’t directly about math at all.

Make no mistake – I did a lot of math, and had a lot of fun learning about new problems, playing around with new ideas, and discovering new mathematicians who I hope will continue to teach me such interesting bits of mathematics. But that wasn’t the most important part of the experience. If I were to sum up the most important part of the week in one word, it would be inclusiveness.

When I first arrived in Atlanta a week ago, I got the opportunity to meet up with an old friend, Shebah. I taught with her in Oakland almost ten years ago, and she has long since moved across the country. She comes from a family of Mexican heritage, and is engaged to a man with a Puerto Rican background. As we talked about my career, and my colleagues in math education, and this whole community of Twitter teachers, she asked about the diversity of TMC. I only knew a lot of people from their profile pictures, and although I can point out some people of color, that just highlights the lack of racial or ethnic diversity. I did mention that there is a lot of gender diversity in terms of men and women, although I do not know (nor is it really my business unless a friend or colleague chooses to share with me) how many identify as trans/non-binary/genderqueer. Is there diversity of sexual orientation? My experience is that gaydar is not to be trusted, so except for people who mentioned the gender of their partner or spouse, peoples’ sexual orientation just didn’t come up. So how diverse is TMC? The only answer I can say with confidence is that it’s not diverse enough. And right away, I had the idea of inclusion in the back of my mind. Do people who are not white feel included in the Math Twitter Blog o Sphere?

Wednesday was the Desmos Pre-Conference day, a day in which I got to see some amazing new developments in store for Desmos users, including more control for scripting when writing activities, enhancements to the Desmos Geometry app, and some interesting transformations to play with.  There was a great evening activity put on by Desmos, and I went to sleep that night so excited to be in my little world of nerdy math teachers.

And then, on Thursday, July 27, Dan Meyer published a blog post, “Let’s Retire #MTBoS”. And as a result of that post, lots of people over the next several days became hurt, angry, and felt disrespected and dismissed. Again the theme of this whole episode boiled down to inclusion. Who feels included in the #MTBoS community? Who doesn’t feel included? What can be done to bridge those gaps, to make every math education professional, new or experienced, K-12 and beyond, coaches and administrators, all feel a part of this community?

Through the rest of the week, themes of inclusion and belong arose – from which teachers felt welcome in #MTBoS (whether due to its perception as cliquish, the relative youthfulness of the organization, or due to the smaller number of non cis-white members), why there was such a relatively small number of elementary school teachers or minority teachers at TMC, to how we can improve the status and inclusion of students of color and non-male/non-binary students. A whole Twitter thread (or variety of threads) on the topic of what can and should replace “boys and girls” or “ladies and gentlemen” made its way into a Storify Posting on #Equity. Some preferred the idea of using “y’all” or a variation, some liked “scholars”, “learners”, or “my little monsters”, and some defended the older teachers who used the traditional “boys and girls” because it’s hard to change. I don’t buy that argument at all – we don’t (or shouldn’t) accept when teachers from earlier generations maintain their stereotypes and outdated language.

I have memories of my great aunt, a schoolteacher, talking about some of her “colored students” with a surprised affection, like they were overachieving in her eyes because they would sometimes perform at the same level as her “regular” students. I didn’t stand up to her comments at the time, since I was probably about 15 years old, she had been retired for probably 20 years, and my mindset was that it probably didn’t matter too much what she said in the privacy of her own home. Still, I know now that it did matter. I may have silently disagreed with her, but other people who she talked to may have been swayed by her statements. I feel like I have come a long way, but sometimes wonder just how forcefully I would confront a teacher who, whether blatantly or subtly, whether intentionally or accidentally, spoke in a manner that was offensive towards a student or group of students. And then I realize that I have a very mixed record, and that I’ve let teachers slide, not saying something in the moment, because I don’t want to get into a confrontation that will take an important discussion on a tangent. I definitely swallowed my tongue on a few occasions with parents as well when they have said things that offended me greatly. My goal for the future – to take that stand, even when it may be uncomfortable, even when it may cause some unwanted ripples. To allow a message of exclusivity, whether it means excluding teachers from our professional community or excluding students from the class culture, to be voiced without objecting to it is tacitly endorsing something that can’t happen.

So, amazingly, despite all of the great mathematical discussions and ideas that came out of TMC17, which were definitely the most fun, the ideas of equity in the math education community are a far more important takeaway to me. In light of the events in Charlottesville this past weekend, this theme is more important than ever.



#MTBoS vs #iteachmath Debate

Wow – what an amazing experience #TMC17 was. Including the Desmos pre-conference, it was 4.5 days of cultivating relationships, strengthening friendships, learning from peers, sharing my own experiences, socializing during meals and games, and soaking in a positive experience. And yet…a debate broke out on twitter that was marked by very strong feelings from two different camps. The short story is that Dan Meyer suggested (through tweets and a blog post) that it is time to retire the #MTBoS name and start using the hashtag #iteachmath instead. The response from many in #MTBoS was swift and unrelenting. And some of us mostly stayed on the sidelines, trying to process just what #MTBoS means to us.

I don’t have any answers yet, but I do have some thoughts. First, one thing that I think may have gotten lost is that both sides are really coming at this from a positive place. Maybe it’s naïve of me to believe that, but it’s also important to me that I believe that. It’s not just that I hate conflict (though that’s true), but more that I need to believe that as educators, we are people who care about our kids and our craft.

The #MTBoS does not have any membership application or member fees or anything official. It is (as Peg Cagle said this morning at breakfast) an organization where one joins by participating. Participating could be through writing blogs, through tweeting, through reading blogs, or even by lurking on Twitter.  People who were otherwise introverted and reluctant to reach out in real life at conferences, or who were in small or isolated schools and school districts, found a home in this virtual community. It grew and became a family, a network, a web of relationships. And as it grew, it took on a life of its own. Its members, many of whom identify as introverts, found an avenue to become leaders in math education, and found a community that they could love and call home.

At the beginning of my teaching career, oh, how I wish I had found #MTBoS, but it was several years before Twitter came along, and the only blogging I knew about was on LiveJournal, mostly by people much younger than I was who used it as a personal (but public) diary.  I taught in a small school, was one of only two math teachers at most in the high school, and lapped up the opportunities I had to attend the CMC-North conferences in Asilomar every year, and the NCTM Annual conferences when I could get the funding. But I was mostly in a bubble.  When I heard a few people talk about Twitter and math education almost ten years ago, I just wasn’t sold. I mean – what can be said in 140 characters? How could that be at all meaningful? And who has the time to read and write blogs?

In April 2015, I somehow got around to joining Twitter for real. I knew about a few people, and knew a few others, and eventually found a small group that I felt comfortable following, reading tweets, and reading their blogs. By the time NCTM 2016 came around, I had an idea of what #MTBoS was, and sort of felt like maybe I was on the verge of belonging. To a degree, it seemed like an exclusive group, not intentionally perhaps, but a group that was close and had developed great bonds with each other, and I wondered if I would really fit in.

And so that brings me back to the two sides of the debate. On one hand, there are many, many math educators who either don’t know about #MTBoS, don’t see the value in #MTBoS, or don’t feel invited to participate in #MTBoS. I know that they would of course be encouraged to “join” and would be welcomed with open arms, but they don’t know that. It has been said that #MTBoS wants to increase its diversity, especially in terms of people of color and in terms of more elementary school level teachers. These are necessary goals, and worthwhile goals, and something that we really need to figure out as a community if we want #MTBoS to best serve all students. However, I don’t think that changing the name is the answer. Are there possible issues with the name? Sure – it’s yet another piece of jargon that can make the group seem exclusive, it isn’t intuitive what it means, it gives the impression that a teacher needs to be active in Twitter or blogging in order to join. That isn’t the biggest obstacle, though. Changing the name is less important than tweaking the culture. Mind you, I don’t have the answer to how to change the culture – I just think that that’s the question that we have to be asking right now.

If the organization was starting over, maybe it would have made sense to use the #iteachmath hashtag, but the #MTBoS is a part of the identity of this group now. A decision to change something so fundamental to the group’s identity can feel very much like a betrayal, and can seem divisive to those who have developed a tie to this hashtag that is completely tied to the blogs, the tweets, and the bloggers and tweeps.

I don’t have an answer to this, but I think I may have identified (largely by listening to very wise members of #MTBoS over the last few days, especially Anna Blinstein, Sam Shah and Peg Cagle this morning over breakfast.) the two main issues that arose in this debate.

  1. How can we create an atmosphere within our community where non-members, especially elementary school math teachers and teachers of color, feel welcomed and don’t feel like outsiders?
  2. How can we reach that goal while preserving the tightly knit community and the parts of its identity that are so important to members of #MTBOS?

I don’t pretend to have the answers by any means to either of these two questions. I just want to be sure that, like in math class, we are addressing the problem that is being asked, and not solving a problem that doesn’t exist.

I would ask your thoughts in the comments, but the conversation has largely been hashed out on twitter, and I suspect it will continue for a while.

My Favorite Year End Review Activity

It’s the middle of summer, and I’m so far behind on blog posts I intended to write. All that free time in the summer seems to evaporate so quickly! It’s 2/3 of the way through July, and my first moment when I don’t have a family vacation, a daddy-daughter day, doctor’s appointments, car maintenance, work around the house, or scheduled work-related or scheduled math activities to do, so I get to share my favorite review activity. A lot of students like this too! I call it Speed Dating, and it’s fairly simple to set up. Each student is required to prepare one problem in advance. I give them the answer, but they need to work on how to solve the problem, and should make sure to ask any questions about the solution in advance if they feel unsure. If your class is large, you can break them up into smaller groups, and give each group the same set of problems. That way, too, all students working on a particular problem can come together to discuss their solutions in advance.

On the review day, each group should be set up in two circles – an inside circle and an outside circle, where each inside student is paired with an outside student. Make one larger space between two sets of desks – large enough to be able to walk through. This space will serve as the pivot point (explained later), but also makes it easier for students (and you, as the teacher) to get inside the circle. They should bring their solution with them as reference, and a place to take notes on the other solutions that they will see. I also include a small whiteboard and two different colored markers at each desk. Then, the fun starts.


I set a timer for three minutes, and the person on the inside explains their problem and solution, using the whiteboard. The person on the outside can use their own whiteboard marker to make notes or diagrams on the whiteboard if they have questions, and they can take notes on their own paper/tablet. When three minutes are up, they reverse roles, and have another three minutes for the outside student to explain their solution. After six minutes, students rotate.

All students move to the left, except for students at the pivot point, where one student in each pair wraps around, so that their partner stays on their left. Basically, you end up with a closed loop, where, given enough time, each student gets paired with every other student. During this process, each student gets to hear the solution to a wide variety of problems. In addition, every student is able to work on their explanation for their own problem, and through the extra practice, becomes a true expert in their problem, understanding it on a deeper and deeper level. Through the comments and questions they hear from their peers, they are able to focus on the trickiest parts of a problem, and refine their own solution.

When students are finished, they get time to re-write their solution and their updated solutions can be uploaded to our Google Classroom page and shared with their peers. I especially like this approach when doing a cumulative review, such as at the end of a semester in preparation for a final exam.

Dandy Candies + a Spreadsheet!

I know it’s probably an unpopular opinion, but I really love spreadsheets. The way you can set them up to adjust one value and have everything change, or manipulate a formula to adjust a fairly large (but manageable) set of data, really makes the calculations much more interesting and less tedious, and lets one start to make sense of some nice patterns. Spreadsheets are also (at least currently) still a part of the professional world, and properly writing an expression in the formula bar is a good and basic introduction to the syntax of programming languages. I don’t do it as often as I would like, but when the opportunity to incorporate a spreadsheet into a lesson presents itself as a good tool to simplify the calculations and get at the interesting math and conclusions, I jump at the chance. Chocolate Candies

Dan Meyer’s 3 act lesson, Dandy Candies, is an excellent way to explore basic surface area and volume comparisons. DandyCandies 4 PackagesBut what if we take it a step further?

In case you aren’t familiar with the premise, the lesson starts with a short video of 24 candies being packaged into various boxes, all with integer dimensions and a volume of 24 cubic candy units (where 1 candy = 1 cubic candy unit). Students then need to calculate the surface areas of the boxes and the length of ribbon required. But that’s about where the original problem ends. And this is where mine begins.

Students develop formulas for both surface area and ribbon length, and then create a spreadsheet in which they enter various dimensions for length, width, and height. They can then play around with different combinations to try to find some patterns that minimize both surface area and ribbon length. The next step, of course, is to minimize the actual cost. Students then must research (or be given) costs for cardboard and ribbon. For simplicity sake, we assume that 1 cubic candy unit is 1 cubic inch, meaning that we are looking for square inches of cardboard and inches of ribbon.

DandyCandiesThis gives another opportunity for some more formulas to find cost of cardboard for each configuration, cost of ribbon for each configuration, and their sum, the total cost of packaging for each box. I have a screenshot of one part of a student spreadsheet, to give an idea of what this (basically) looks like. We used Google Sheets, but this can be done in any other spreadsheet.

As a result of the extra spreadsheet work, along with some additional Gene-Wilder-as-Willy-Wonka-in-Willy-Wonka-The-Chocolate-Factoryresearch time, we now take about two full class periods on this assignment, but the amount of practice that students get with spreadsheets, spatial thinking, and applying a variety of skills makes this extra time well spent. Plus, I show a few clips from Charlie and the Chocolate Factory just to remind my students of the genius of Gene Wilder.

Why Debate in Math Class?

I have found my niche for this part of my career – I’m a math teacher and I love to make my students debate. Make no mistake – it takes some time to do a debate in class, it takes preparation, and it takes up a class period that could be used for instruction or assessment or a project or an exploration or a computer lab, which are the things that take place most days in most math classes around the country.

In the (no longer very new) Common Core Standards, Math Practice 3 (MP3) is one of the 8 Standards for Mathematical Practice, which are standards that cut across grade/content levels. MP3 states that students should be able to “Construct viable arguments and critique the reasoning of others.” A debate is defined by Oxford Dictionaries as “A formal discussion on a particular matter in a public meeting or legislative assembly, in which opposing arguments are put forward and which usually ends with a vote.” It is easy to see how well these can fit together. A student debate about a mathematical proposition is one in which they put forward arguments in favor of or opposition to the proposition. Through this process, they are given opportunities to further develop and synthesize ideas and create examples to support their arguments, which requires them to apply ideas that they can understand (and remember). In sum, it covers all the bands of Bloom’s taxonomy (which has its own controversy, acknowledged, but that’s another conversation).


For these reasons, I have found a debate to be an excellent way to review a unit after it has finished. For example, after covering solutions of linear equations in my Algebra 1 classes, we had a debate in which students argued that either elimination or substitution was the best approach to take in order to solve a linear system. I did not create an experiment that is worthy of peer review, but as a case study, in my two classes, assessment scores increased on average from 2.4 on substitution and 2.1 on elimination to 3.6 on substitution and 3.4 on elimination. The pass rates for substitution increased from 67% to 95%, and on elimination increased from 38% to 90%. (My assessments are SBG assessments scored on a scale from 0 to 4. These results cannot only explained by the debate, as students also got the feedback from their first assessments, and not all students retook their assessments. But it is some nice anecdotal evidence.)

That should be enough of a reason to include debates in math classes, but I’ve skipped over the most important reason. It’s a lot of fun! Students get to collaborate, research, and then they are encouraged to argue, with some extra points (not extra credit of course) on the line. WODBCirclesBlankThere are a variety of formats that we can use for debates, and they can range from a short “Which One Doesn’t Belong” warm up activity to a formal debate, and everything in between. Students are able to understand the protocols fairly quickly, and get used to preparing for debates because they could come up at any time. In fact they will sometimes ask if we can debate a topic if it isn’t immediately settled in class. (Case in point- a student asked if parallelograms could be defined as a subset of trapezoids. In other words, are trapezoids defined as having at least one pair of parallel sides, or exactly one pair of parallel sides? I realized that I didn’t have a definite answer, and that it isn’t a settled definition, and lo and behold, students begged for a debate, which we did a week later.)

Why Debate NCTM2017

Are you sold on debates yet? For more information, you can take a look at my Google Slides document from my presentation at the NCTM 2017 Annual Conference in San Antonio.

Math Forum Debate at NCTM 2017

This year, I was fortunate to have my first proposal to speak at the NCTM Annual Conference accepted. Even more exciting, I was invited by the Math Forum to do a presentation on debates, which then turned into an actual debate. Two friends of mine from #MTBoS, Anna Blinstein (@borschtwithanna) and Mishaal Surti (@MrSurti), agreed to have a semi-formal debate, which went over very well (despite the small audience). It’s been a week now, so it seems like a good time to reflect on this experience.

For the Math Forum debate, I have a vision of it growing into an annual math-ed celebrity debate event. Where did I get that idea? It probably goes back several years, and originated with Professor Colin Adams (whose website was last updated in 2008, but looks the same as it did during my senior year in college in 1995). I saw him more than 20 years ago at an MAA conference, in a performance as his alter-ego  Mel Slugbate, in a talk about how to cheat your way to the knot merit badge. Always creative, he wrote an excellent book on knot theory, and later debated his colleague, Thomas Garrity, on the topics of “Pi vs. e” and “Integral vs. Derivative”. (On a side note, he received his Ph.D. in math from University of Wisconsin at Madison, where mathematician, author, and NCTM 2017 opening keynote speaker Jordan Ellenberg currently teaches.)  I’m sure that hearing about and then seeing videos of those debates first put the idea of debates in math class in my mind. It’s definitely what I’d love to see annually if I can find a way to do it. There are numerous topics that can be debated in the areas of math and math education. Once I put the word out and reached out to a few people, both Anna and Mishaal graciously accepted the challenge. And now we had to decide on a topic, as well as the format.

The format was easy – I suggested some guidelines for a semi-formal debate (opening statements, closing statements, and how questions would be answered). Both Anna and Mishaal were agreeable, and that was that. The topic of debate was not too difficult of a decision; I suggested a few topics in math, and a few topics in math education, and both Mishaal and Anna were pretty excited to debate the topic of traditional sequencing of secondary math courses (Algebra 1, Geometry, Algebra 2) vs. an integrated math curriculum (which is used, basically, everywhere else in the world). And here was the problem that ended up not being a problem: both Anna and Mishaal are fans of integrated math, and so one would have to debate a side that they opposed. In the world of debate, of course, this isn’t uncommon. (I often tell the story of how I was required to take the side of allowing teenage tobacco use for a high school debate. I won the debate, making the argument that teenage smoking should be required. Not my proudest moment…or was it?)


All three of us were quite busy with, you know, all the things that come up in the lives of teachers, so our plans for great preparation were not completely fulfilled. I did create a document of research sources, and a slide show (with very few graphics), and my own list of questions for each side. Anna and Mishaal downplayed how much they prepared, but their performances during the debate were outstanding – arguments and rebuttals were well thought out, they listened to each other and responded with eloquence and intention, and they both injected humor at appropriate times.

If you’d like to see the video, it’s on YouTube (complete with quickly added end credits, Creative Commons licensed music, and minimal editing).

One thing that I’ve noticed is that, as moderator and presentation operator, I personally find it difficult to pay as close attention and take notes during the debate as I would like. As a result, I general take video of all of the debates in my classes so I can watch them afterwards. It’s also really great to have some evidence of some of the very insightful and funny things that students say.

I’ll post again with a summary of my debate talk, but right now I want to put this dream out there. Would you like to see a debate at a future math conference? Maybe a celebrity that also happens to be a mathematician debate? (I was thinking maybe Danika McKellar vs. John Urschel would be a fun one to do). What about author mathematicians? (Simon Singh vs. Keith Devlin, perhaps?) Or maybe it would be more meaningful with celebrities of #MTBoS. Perhaps…even you? Tell Suzanne and Annie and Max at Math Forum that you know you missed a great show and would like to see a debate next year! Tell NCTM you want this to happen. And tell me if you want to take part. I could always use a debater, a researcher, another camera operator or two, a video editor, a stage manager, and someone to do something about my hair and wardrobe. Say…anyone want to write a grant to fund all this?

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.