One graph. Ten minutes. An important conversation.

At the beginning of class I showed this to my students:

Screen Shot 2018-11-14 at 10.09.40 AM

They came up with lots of interesting things.

  • There are three variables
  • They are functions
  • They are different colors
  • The units are millions and years
  • The scale for the millions is by 500,000’s and for years is decades
  • The domain of all 3 functions is 1920 to 2010
  • The range is 0 to 2.5 million
  • All of the functions are positive over their domain
  • The average rate of change for the red graph from 1920 to 2010 is positive
  • The average rate of change for the light purple graph from 1920 to 2010 is close to zero
  • The greatest average rate of change for all functions appears to occur from 1980 to 2000.

Then I asked them to predict what the graph was about. Most felt it detailed some sort of economic situation. Or population. Then came the reveal:

Screen Shot 2018-11-14 at 10.09.06 AM

They were shocked. We talked about possible causes for this situation, like the school-to-prison pipeline and the privatization of the prisons. More eyebrows raised. I brought up the question of what the racial breakdown of the prison system might look like. It was an important conversation.

And then we moved on to the regularly scheduled program: the lesson.

That was this week. While this class opener wasn’t directly tied to work that we’ve been doing in algebra 2 and was relatively brief, I felt compelled to have this conversation with my kids. This summer I began thinking about how to deepen the connections between social issues and math. Since I suck at projects, I thought about making these connections in smaller, bite-sized ways — comparable to problems found on a typical NYS Regents exam. In an ideal world, I would find (and write some) problems around social issues that are directly tied to the algebra 2 curriculum and discuss them with students. But this is really, really hard. Factoring by grouping doesn’t exactly lend itself to talking about racial inequities.

I was upfront with them. I said that its hard for me to relate some of the mathematics we learn to their daily lives, but we can do it in other ways. I told them that it was my responsibility to help you see how math can you uncover your world. Graphs are one way.

Through this graph of incarcerated Americans, I’ve myself learned that periodically presenting an interesting graph or data can be another way to build in time for important discussions around social justice and empowering students through math — even if the discussion isn’t wrapped up in a “problem” or directly tied to what we’re studying. This is not unlike What’s Going On in this Graph from the NY Times.

 

bp

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Posted in activity, mathematics, PBL | Tagged , , , | 1 Comment

I have trouble talking about my teaching

I have trouble talking about my teaching.

Part of the reason for this is that teaching is so damn complicated. This makes it hard for me to have conversations about how my students learn — especially to teachers that I don’t know. Sometimes I just avoid talking about myself because I fear the incoherent answers that I’ll provide to the questions about what I do every day with my students. This got much worse last year when I started teaching through problems.

To get better at talking about myself, my classroom, and my students, I’m going to simulate such a conversation. I’m speaking to a fellow algebra 2 teacher. The scene opens when I make a comment about how they’ve set up their units.



Me: I like how you’ve structured your units. After your review unit, you start the year with exponential functions? That’s interesting.

Teacher: Yeah, it’s only the third year we’ve taught the course with the new standards, so our math team likes starting with exponentials and then diving into logarithmic functions. We then go into polynomial and rational functions and end with stats, probability, and trig.

Me: I like that. The standards place a huge emphasis on exponential functions now…it’s good to get that off the ground early. This is also my third year teaching with the new standards. There’s so much content in algebra 2 that I’ve run out of time each of the last two years. I couldn’t teach it all.

Teacher: There’s so much! How do you do your units?

Me: [Feels uneasy] Umm…I don’t have units.

Teacher: [Confused look on face] What do you mean?

Me: I suck at doing it, but I teach through problems.

Teacher: 

Me: It’s confusing, even to me. All the topics that were in my traditional units are now all mixed up…but it a way that helps bring them closer together. Instead of having discrete units where topics are isolated from one another, the problems allow for the concepts to be easily interleaved, spiraled, and married in ways that I found hard to do when I had units. I’ve realized that a lot of what and how students learn in math class can be studied nonlinearly…and that’s what my classroom reflects.

Teacher: So, wait, are the kids just solving random problems? How do they learn?

Me: Sort of, but I think a lot about how I sequence the problems. I’m very intentional about which problems kids do and when they do them. So while on the surface the problems may look random, underlying themes and concepts from algebra 2 emerge for students through the problems over time.

Teacher: … [still confused]

Me: Here, let me show you what I mean. In a typical math class, the units are sequenced and taught linearly. [Gets paper and begins drawing] For example, take four units from the school year. Traditionally, when we finish with one unit, we move on the next. [shows drawing below]

Screen Shot 2018-11-04 at 6.12.03 AM

Instead of using that model, I interleave the topics, skills, and vocabulary from each unit to span the entire school year. My old units are now parsed. Think of the first unit in purple as broken up into smaller pieces and spread out over the course of the school year. [Shows drawing below]

Screen Shot 2018-11-04 at 6.20.49 AM

Then the same for the 2nd unit in gold. [Updates drawing]

Screen Shot 2018-11-04 at 6.22.44 AM

And so on with the remaining two units. Notice that some pieces are bigger than others. In the end, it might look something like this: [Updates drawing again]

Screen Shot 2018-11-04 at 6.24.29 AM

One of my goals in using this model is that, since students are frequently revisiting key ideas from the units, it helps with retention. With all the units mixed up, it makes it harder for the students to remember what they’ve learned…but that’s the point. It’s messy by design. That said, I build coherence by thoughtfully sequencing problems.

Teacher: Hmm…I’m wondering how your lessons look?

Me: Well, I typically assign them 3-5 problems for homework. The problems aren’t “practice,” as homework is traditionally viewed. They are more like puzzles or explorations that I ask the kids to do before class. It’s not expected that they understand and speak to all of the problems when they walk into class…I fully expect them to have questions. I also expect them to do individual research to help them figure out the problems. And because of how concepts are interleaved, the problems are usually all on different concepts — and have roots in different units. We don’t typically study one idea per day as is customary in math class. Instead, we study several ideas — and sometimes they are not directly related.

Anyway, we’ll spend the entire period discussing the homework problems in small groups and as a whole class. I have large whiteboards all over the walls that help with these discussions. Students are fully responsible for putting up problems and trying to gain a better understanding of them together. If they cannot (or do not) put up meaningful work to drive our thinking for the day, then they don’t learn. Also, I put few constraints on how the discussions look and feel. The kids typically move about the room freely.

Teacher: So where do you come in?

Me: Most days I help students make sense of the problems while in small groups. I also sequence student presentations of solutions for the whole class discussion. Equity of voice is important here — I keep track of who presents and how often. I also step in with direct instruction on the problems when it’s needed.

On other days, usually 1-2 times per week, things will look more like a traditional lesson where the problems focus only on one key concept. I consider these my anchor experiences that usually focus on high-leverage concepts (like sequence notation or logarithms). I also bring in Desmos Activities all the time.

Teacher: I wonder, where do you get the problems that you use?

Me: All over the place. I steal most of them from other teachers online, but I do write some myself. Those suck. I use Regents problems, too.

Teacher: How did you learn about all this?

Me: Two summers ago I attended the Exeter Math Institute. It blew my mind. As an immersive PD experience that pushed me beyond my comfort zone, it helped me completely reimagine what math teaching and learning can look like. It was different and challenging. It was led by a teacher from Phillips Exeter Academy who used one of their problem sets with us for a week. Exeter has pioneered the problem-based model that I’ve adopted…and they are well known for their problems — they’re tough, but they’re rich. I have included a couple of them in the problems that I give my students.

Teacher: This sounds interesting…I would love to see it in action.

Me: You are welcome any time. I must say, though, there are tradeoffs to using this model. Lots of them. First, students generally don’t like it…at least initially. Giving them so much control and disrupting what they know to be “math class” causes plenty of frustration and discomfort. And they are regularly confused and don’t always leave each day with a “clean” answer or understanding of a problem or concept. This can be hard for everyone — them, me, their parents. Last year, I wasn’t prepared for the amount of dislike and pushback I got. Second, since students learn content nonlinearly, it’s a mess for me to plan and sequence. Also, each day can be somewhat unpredictable because what we do each day is largely dependent on students’ independent work before class and the motivation to drive learning during class. Our discussions can suffer as a result of kids not doing their part…which happens A LOT. What makes this worse is the fact that I’ve never met another public school teacher using this approach…so I haven’t been able to critically bounce ideas off anyone. This makes it very hard to improve. I miss co-planning. There’s more, but, yeah…[awkwardly changes the subject]

 

bp

Posted in lesson, PBL, pedagogy, reflection | 4 Comments

Student as author and critic of mathematics

I’m hoping to improve my students’ journal writing experiences this year. After learning about problem-based journal writing from the work of Joseph Mellor and Carmel Schettino, last year I created/stole a fancy handoutrubric, and told the kids to go write.

I was hopeful for more, but the kids ended up only writing one journal entry. This is totally a result of me assigning in the late in the year, yeah, but mainly because I was too lazy to actually read through them all. I pitifully underestimated how long it would take to read what was essentially 120 essays. English and history teachers out there, I can now finally appreciate your workload. I feel for you.

Fast forward to this year. I’m ready to step my game up. I’m primed to better position my kiddos as authors of mathematics. I tweaked the handout, rubric, and my introductory talk with kids about writing and why it is important — even in math class. Through the journals, they will be formally reflecting and thinking about their own mathematical thinking in a deep-ish sort of way. Just like with the Mathography, I’m pretty sure they’ve never done this before.

One of the key differences this year is that instead of me being the authority figure on providing feedback and grades (and putting this onus on myself for reading ALL those journals), I am forming six editorial boards in each class. Each board will be a yearlong grouping of students who will peer-review the journals.

I got this idea after I read The Art of Problem Posing by Stephen I. Brown and Marion I. Walter this summer. After they’re turned in, I will distribute 4-6 journals to each editorial board, who will use the rubric to do a blind-review (I will remove all names of journals) to discuss, assess, critique, and give feedback on the mathematical writing of the authors. I will have final say on all marks, but I will fully expect integrity, honesty, and fairness from the boards. And by reading through and analyzing so many of their classmates journals, I hope that their own mathematical writing gets better over the course of the year.

I’m really hopeful that they’ll get to write four journals over the course of the year. What’s really cool is that after each round of submissions, each editorial board will select one journal that they read to be published at the end of the year. By “published,” I mean featured in a compilation that I will print out in a little booklet in the spring. It’ll look and feel professional…like this one that I came across at TMCNYC this past summer from Ramon Garcia who teaches at Borough of Manhattan Community College Adult Learning Center:

By the end of the year, I want every student to get at least one journal entry published.

I’m not 100% confident in any of this, but I am very excited. At a minimum, I know it can’t be any worse than last year!

 

bp

 

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The Mathography

During the first week of school this year I assigned a mathography. It’s essentially a math biography and details one’s relationship and life experiences with math. This was a new idea to me and my colleague Stephanie Murdock put me on to it in the spring. She learned about from Wendy Menard’s NCTM session in D.C. earlier this year. Here is Wendy’s handout.

After assigning it, I figured it would take me forever to read them all (~120). So instead of getting overwhelmed and trying to cram them all within a week and probably not remembering anything about my students, I promised myself to read them a little bit at a time in bite-sized chunks. I wanted to slowly digest them, to really savor them. Each day, I might read a couple in the morning when I got to school and also just before I leave for the day. Maybe I squeeze in another during lunch. And because I want the kids to know that I read them and that their story matters, I write a healthy, thoughtful comment on each one (thanks Google Classroom). My goal is to read and comment on each mathography by the close of the first marking period. I may not make that deadline, but I don’t really care now because they’ve been so interesting.

Now that I’ve read a good number of them, I failed to anticipate the closeness that I would feel with my students as a result of the assignment. I’m learning things about my students that I would have never found out before. When I look at my students (some of whom I even had last year), I actually see them through their relationship with math. I can welcome who they are in our classroom because I actually know who they are now. It’s wonderful. And not only is it what they write about that tells a lot about who they are, but it’s also how they choose to write. For example, a few students submitted poems and fictional stories as their mathography. They beamed with artistry, were pleasant surprises to the pool of personal narratives that I expected, and told me so much about those particular students in ways that beyond what their words did.

I’ve always tried to pride myself as someone who works hard to get to know my students. But I’ve never done it through formal writing, like this. What a huge difference! Come to think of it, because writing plays an important role in my personal life, I understand the power of reflection and written word…and it only seems natural that I experience it with my students. That said, I’m so disappointed that I didn’t assign this to students earlier in my career.

Also, it was clear from their writing and from their reactions to the assignment itself, that my students had never formally reflected or wrote about their relationship with mathematics. This has been refreshing for them and me. And exactly why it’s so important that I assign it again next year.

I’ll close by sharing excerpts from some of the mathographies that I’ve read so far.

  • Math is like an ocean. The deeper you get into it the more harder and challenging it gets. Although it has different layers just like math has different concepts, if you look at it in a big picture it is really just one concept all together as one.
  • I’ve always had a constant battle with math. Whether it was counting money, telling time, or measuring something, math never seemed to be on my side. Since I was a kid, I would classify myself as “not a math person”. I wasn’t terrible at math, I was actually quite good, yet I never enjoyed it. My teachers also tended to teach a certain way which didn’t allow me to find my own way to solve problems. Math only got worse from there.
  • In the 8th grade it was the best, my love for you could have burst through my chest. [line from a poem]
  • When first introduced with mathematics, I was not thrilled with the idea of learning through numbers. At the same time it was a new learning experience, so why not give math a try. I ABSOLUTELY HATED MATH. My brain exploded when face to face with math. There was simply too much combinations of numbers at once. I gave up on it and just turned my mind to Science and History during my elementary school days.
  • Being an Asian, we’re usually stereotyped with being good at math. Also known as a subject I can’t ever get a good grade in because exams stress me out to the point I fail or score really low on. I hope to understand all math concepts at one point in my life but right now it seems like a stretch for me.
  • In English, I can annotate and understand the central idea. In history, I can study the important dates and find out why they’re significant. When it comes to Math, you need to understand each concept thoroughly and if you miss a step it’s automatically wrong.
  • My earliest memory of math I would say would be in kindergarten. I attended school in Mexico. I lived with my grandma for 3 years. I was about 5 years old. I remember going to pick eggs every 2 days with my grandma and she would count with me every egg we picked in Spanish. Every chore I did with my grandma would require counting out loud. I have to thank my grandma because if her I leaned my numbers pretty quick.
  • Math isn’t just a subject, it’s an experience.
  • As time passed things just got harder. I got less and less star stickers on the board for correct answers as I watched people get every single one of them. I have always been jealous of those people that just understood math with no problem. How did they get it so fast? That’s the main question I always use to ask myself. There were times where I felt like there was something wrong with me or I felt like I was never going to understand. No matter how fast I ran or how much I tried to avoid math I couldn’t get rid of it.
  • The bane of every math teacher’s existence is when a student asks why. Why are we doing this? How does this relate to our life? How will it affect us? To this day I still haven’t gotten a clear answer and why is it that most teachers can’t tell me why. They all have the same answer “I don’t actually know. Search it up and tell me tomorrow.” It’s ludicrous to think that someone who has devoted their life to a job wouldn’t actually know why they’re teaching a subject. Then there are people who say “their job is just to help us pass the test or the regents.”
  • I don’t recall any specific positive memories with math from my early childhood. My classmates were angry at the attention I received, and some of the teachers assumed my family gave me the answers.
  • When I came to the United States at the age of 3, I only spoke, understood and wrote Spanish which is why ELA was difficult for me the first 5 school years. However, the numbers stayed the same, they didn’t change their meaning, one continued to be uno, two continued to be dos, three continued to be tres etc.
  • For most of my years, math has not been so much of a satisfying experience, it was thought of something that I just had to do. I can only hope that in the future, math continues to surprise me and that we can find peace with one another. Maybe one day, math will find its permanent and pleasant place in my life.  
  • To me the whole concept of math and what math is completely confusing. I understand that I’ll need math in my life to keep track of my money and all that good stuff but there’s some topics in math that I just don’t understand how I’ll ever apply what I learned in those classes in my life beyond school. Classes like geometry, trigonometry, and calculus make no sense to me to be completely honest. When will I ever need to find the circumference of a circle or the Cos off point A in a right triangle? You see where I’m coming from?

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Posted in reflection, Social Justice Math | Tagged | 1 Comment

I have fewer desks than students. On purpose.

There is a growing number of shared public spaces that are popping up all over the world.

 

Cars, bikes, motorcycles, and pedestrians are forced to govern their own behavior. They have to make eye contact and acknowledge each other’s presence. There’s an inherent faith that folks will slow down and pay attention to each other. I bet they even greet one another way more than they would otherwise. It all makes for a more trusting and human experience. And, from what I’ve read, the number of injuries in these spaces has even decreased significantly.

Why can’t we operate on the same principle in our classrooms? Can we somehow use shared space in our classrooms to create a more personal and humanistic learning environment? Call me idealistic, but I’d like to think so.

That’s why last week I asked the custodian at my school to remove several desks from my classroom. I wanted to ensure that I had fewer desks than students. I didn’t want each student to have their own desk. Instead, because I have the room set up in groups, I have intentionally removed 1-2 desks from each group — but left the seats. The result is 6 groups of 4 desks with 5-6 chairs each. Here’s an example of one group:

IMG_3291

It’s a small change (unlike removing all street signs from a busy intersection), but the idea is that in order to navigate their group’s space, students must purposely engage with one another on a regular basis. It creates a more communal learning environment and helps them take ownership of their workspace (and our classroom). It can get messy because they rub elbows, get in each other’s way, and have to constantly negotiate how they should use the space. But in the end, I 100% welcome these inconveniences. (Honestly, living with the ungodly congestion of NYC, my kids probably don’t even realize these things.) They create a greater degree of collective energy each day. Ultimately, my hope is that they will be more mindful of each other, to be more present.

A side note: This line of thinking is also reflected in the large whiteboards that I began using last year to de-front the classroom. These are communal spaces around the walls of the room that students used to publically display their thinking at any time — unlike having one greedy board at the front of the classroom that screams for attention (and a lack of optimal engagement).

In some ways, I see desks as imposing segregation on my students (and me). Despite being organized into groups, desks still create distinct social spaces for students to think individually. There’s a clear end to my space and a start to yours. Can this subconsciously establish a greater sense of independence from others in the classroom? Tables would probably be the best physical solution for helping create a more personal, humanistic classroom, but I doubt that I’ll ever convince my principal to get me those.

 

bp

 

 

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