# Day 1 Debate – What’s the Best Number?

I know that by now, most schools have started. I think I’m one of the last that still starts after Labor Day – not that I’m complaining. A popular blog post out there is a first day activity, and I wanted to share the one that I’ve done the past few years. Yes, it’s a debate, but a totally informal debate, one that works at any level of mathematical background, students have a lot of fun with, that builds both competition and teamwork in a low stress way, and tells me a surprising amount about my students.

After normal introductions, I give students a simple task – to come up with a number. It could be a favorite number, or a really interesting number, or a number that has some personal meaning. I then ask them, once they’ve decided on a number, to come up with as many interesting things about that number as they can, and give them a couple of minutes. They can use calculators, they can use the Internet, they can draw, and if they get stuck they can ask me for help (though, to be honest, they rarely ask for help with this).

After a few minutes, when at least some of the students are feeling like they’re done, I have them get into pairs, and then in the pair they decide which number is better. They are given about one minute to make their cases and decide, and once each pair has decided, I have each pair find another pair, and decide which of the two numbers is better. It’s interesting to me that I never actually describe this to students as a debate, and in theory they are working together, but they do have something invested in the number that they came with. Inevitably, groups start to argue, but generally nicely, and the whole idea of comparing the best and worst qualities of numbers becomes a source of passion.

The process of finding another group and then discussing, then finding another group and discussing, continues until you have (hopefully) two halves of the class shouting at each other about whether 32 is a better number than 360 (because powers of 2 are more important than having lots of factors and describing the degrees of a circle), or whether 12 is better than 18 (but of course 12 is better).

In listening to conversations that happen, I can get to know an amazing amount about student interests, as well as which students feel very comfortable with what numbers mean and how they can be manipulated and described mathematically. On top of it all, having an entire class passionately engaged in a meaningless debate about which number is best, where you can catch every student having fun playing with math from day one, is a pretty great way to start the year in my opinion.

So, what’s your favorite number? Why?

# 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?

*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.