Letters of recommendation: the aftermath + excerpts

Two months ago, I wrote about letters of recommendation. The post meandered, but it was fun to think through. The seriousness with which I’ve begun thinking about my writing was at its core.

Since that post back in October, I’ve written 16 letters of recommendation. This is the first time in any year that I’ve counted, but that has to be a record for me. I had a goal to change how I think about writing these letters. Instead of producing them casually, I wanted them to be more personal and directed. I wanted to throw myself into them and elevate their status above all the other rudimentary writing I do for school purposes. I spent a few hours on each, some more. I stayed up late and woke early. I grew more patient with what and how I wrote. I steered myself away from the immediacy of copy and pasting — which can be tempting with this type of writing. I strived to really find each student and give voice to their journey. I also jostled with opening paragraphs a lot, trusting delayed ledes more and more as the weeks passed. I tinkered with analogies, searching for the perfect match. I followed up with students on the details that I knew would pack a punch.

While demanding, my change in mindset never felt burdensome. Mainly because the letters went from being a task on my To-Do List to being a unique writing opportunity and means of expression for me. I used the letters to offer my thoughtful perspective on students for the faceless admissions officers who might spend 90 seconds reviewing it, but I also wanted to be critical of my own writing. With each letter, I stretched who I was as a writer. I’m proud of that. Interestingly, in the midst of grinding out my little cornucopia of letters, I began reading Becoming by Michelle Obama. Because of its eloquence, I found myself flipping through its pages often, stealing words, adjectives, and phrases to use in the letters. To pick up such a well-written book, while at the same time thinking deeply about my own writing, was an unexpected blessing.

Despite my newfound allegiance to them, the letters still weren’t as sharp as I would’ve liked. I would have liked to nibble on many of them more, help make them better. Oh, well. These things called due dates run the show. Nonetheless, because writing them swallowed so much of my time over the last eight weeks of my life, I wanted to capture bits and pieces that stood out. I combed through the 16 that I’ve written so far and found an excerpt that I liked or found meaningful. I’ve included some of them here.

  • It is uncommon to find to young person so concerned with bettering the world through community service. ____ is academically driven and ready to dedicate her life to serving others, but she needs exposure to and experience in the workforce in order to fully blossom into the giving adult she wants to become. This is where the ____ comes in.
  • Yet as valuable as words are, few humans, let alone high school students, understand their worth. Most use words boastfully or neglectfully, never thinking twice before allowing whatever it is that is on their mind to spill out of their mouth. I say all this because I want to tell you about a student who challenges this norm.

  • Usually, it is the teacher’s job to inspire his students. But with ____, I think it is the other way around because she has motivated me in profound ways. Her grit, tenacity, and strong-willed nature has taught me a great deal about what it means to be a young person today, to search for oneself, and to do things that you never thought you could.

  • Doing good and helping others with intention, not for attention. That is what ____ is about.

  • The fact that she asked me — her math teacher — to write this recommendation letter is further evidence of her willing embrace of her troubled mathematical past. She is using her struggle as a slingshot to launch her into the next stage of her life.

  • You should ask him about Between the World and Me by Ta-Nehisi Coates. We bonded over the importance of that book after I loaned him a copy late in the year.
  • Sport plays a pivotal role in _____’s life. This is why she has worked tirelessly to become both the captain of her basketball and softball teams. She not only gets leadership, but views athletics as a great unifier of people.

  • If there were a such thing as a professional learner, she was in my class last year. Her name is _____.

  • Brazen in her ability to see things not the way they are, but how they should be, _____ continually embraced the discomfort that math caused since elementary school. Truth is, she holds nothing back when it comes to this, her greatest academic challenge ever. I wish more students had _____’s resolve.

  • She reconciled with vulnerability and failure, two parts of her life that, despite her beaming positivity, she largely ignored.

  • Sadly, there is no way that this abbreviated letter could ever come close to capturing ____’s qualifications and profound growth through the years. It fails miserably. But as she prepares for college and a career in photojournalism, I am left thinking about her transformation since walking into ninth period algebra. So much has happened. Her nervousness has evolved into readiness, her uncertain world now filled with direction.

  • This is supposed to be a recommendation letter, but instead, I want to tell you a story of courage.

  • Who am I? What do I stand for? Why am I here? Whether consciously or not, it is these questions that fill the minds of young people. While a lot of my students — mostly out of fear or neglect — ignore these questions, there are strikingly few who work relentlessly to answer them. These are students who are exceptionally self- aware and always seeking to reinvent who they are. One such student is ____.



Writing Better with Alex Martin and The Academy for Teachers

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Last week, I had the privilege of attending Write Better: Tools for Building Stronger Stories, a “Master Class” organized by The Academy for Teachers. The Academy of Teachers is a not-for-profit organization that partners with experts from around NYC to bring unique learning experiences to teachers. My class was taught by the chief writing editor of the Wall Street Journal, Alex Martin.

Several weeks back, John McCrann put me on to the class. He and I are co-facilitating some writing workshops at Math for America this semester and he thought I should apply. I never heard of The Academy before John, so I spent a Saturday evening poking around their website. On a limb, I applied.

The application took me a few hours. It was reflective. No lie, I was dismayed that it required a recommendation from a colleague. I get it, but I have been bothering my AP for these a lot lately. I felt bad asking her, especially since it was a last-minute request, but I did. Of course, she didn’t think twice and eagerly came through, as she always does. One day I’ll find a way to thank her. Words aren’t enough anymore. These last 4 years, her infallible support has helped elevate my thinking and advance my career more than I could have ever done on my own.

I heard that acceptance rates for Master Classes were low. (For mine, I later learned that one-third of those who applied were accepted.) I hoped that teaching math and being an active writer might stand out. I wanted to help be the voice of math in a room that was sure to be stuffed with humanities teachers. I’m not sure if any of that came through in my application, but they did invite me to attend. John got in, too. Of the 18 attendees, we were the only math teachers.

The venue for the class was the News Corp Building in midtown. The Wall Street Journal hosted us. It was swanky. Great good. But thanks to the 6 train, I was late. Running 7 blocks from Grand Central just to arrive ten minutes late, my sweaty behind sat down as breakfast wrapped up. I grabbed a coffee and some leftover fruit.

The morning consisted of a lecture from Alex followed by a discussion about some reading that we were asked to do in the weeks leading up to the class. As we all waited for him to begin, something unexpected happened. Instead of introducing Alex by listing out his achievements and awards, as is customary when you have such an accomplished guest, Sam Swope, the head of The Academy, introduced us to him. It was flipped. For a few minutes, Sam went on and on about who we were, our experience, accomplishments, and the adjectives that our recommenders used to describe us. He spoke in a way that put us in context for Alex, instead of the other way around. He gave us status. It was an uncommon start to a workshop and a really nice touch. I felt special.

Alex’s lecture was one that he gives to the writers and editors at The Journal. It was full of specifics on writing like being FACL (Fair, Accurate, Clear, Lively), maintaining a tight focus, reporting, structures for writing, and minutia involved with composing good sentences. For example, he told us to kill our adverbs. Choose muscular verbs. Instead of saying ran quickly, say dashed. That advice seems obvious after he said it, but I never thought about it before. Also, he talked about keeping subjects near verbs and using right-branching sentences. Ironically, he urged us to focus less on writing, and more on thinking. Good writing stems from good thinking. He gave us tips on what to do when we hit a writer’s block (lower our standards, skip the lede, write fast without your notes) and how to achieve brevity (an idea may be good in isolation, but it must do work in the story). Given his role at the WSJ, all of what he mentioned had a journalistic undertone, but his principles were universal. My notes sprawled. I gobbled up everything.

We then discussed the readings. It was a healthy list of 13 newspaper and magazine articles that Alex handpicked to highlight elements of his lecture. They dated as far back as 1966 and varied greatly in their subject matter. One was on Frank Sintra, another on the aftermath of the Exxon Valdez oil spill, and yet another was a feature on a small Long Island town during post 9/11. A more recent WSJ piece centered on the recent collapse of GE. They were full of emotion, detail, and craftsmanship and I gladly digested them all. Not stuff that I would read naturally, but I’m glad I did. They were great stories.

In the afternoon, we wrote. First, freewriting. We referenced freewriting in the morning as a means of hurdling writer’s block and, considering we all just inhaled a gourmet lunch, it made sense to start with it. As Chip Scanlan said, we tried to “sneak past that watcher of the gates before the watcher can say, Hey wait a minute. You suck. You can’t write.” Fittingly, we wrote about our favorite dessert. Next, we closed the day by rewriting some real-life ledes that were used in real-life articles. This was fun, but challenging. I think I set myself up for failure because, as I was trying my hand at the ledes, I couldn’t help but try to imitate Edna Buchanan. A Herald reporter who we read about before class, Edna is renown in journalism circles for her ledes. It’s needless to say, but I failed to live up her standard. I enjoyed it nonetheless.

As the class ended, I couldn’t help but be grateful for Alex. Throughout the day, I found him to be sincere, thoughtful, and amiable. His brilliance was matched only by his eagerness to improve our writing. He had a business-like aura about him, but he remained open to our questions and answered each of them in detail. We were sitting in a “U” for most of the time and, after someone asked their question, I noticed that he would always lock eyes and move slightly towards them as he answered. No matter the question (I asked a few dumb ones), he valued each one and took great care in responding. I appreciate the inherent respect that he showed us. When one of us shared our writing with the group, he had a keen way of analyzing it immediately after the last word slipped out of our mouth. His command of language was special — I found myself quoting him several times.

Aside from Alex’s expertise and willingness to help us grow as writers, the size of the group (18) really made it special. It was intimate. It felt private. It allowed for a real back-and-forth between us and him. There was space and time to go talk about details and share side stories — which Alex gladly did. This did a lot to make the class meaningful. It was also really cool to be in a room full of teachers from both private and public schools. We don’t rub shoulders enough.

I’m so looking forward to another one of these classes.



Algebra 2 curriculum dump #1

I’ve decided to reflect on the interleaved algebra 2 course that I’ve designed and taught over the last couple of years. My hope is that by taking a more outward-facing approach to the curriculum, it will help me make it better. This is the 1st post in the series.

In my last post, I overviewed the course. I’m going to dig into the units now. Yay! Here is the working problem set for the course.

Unit 1 (Problems 1-24)

  • Function notation & analysis
  • Domain & range graphically
  • Function arithmetic
  • Trigonometric ratios & circles
  • Factoring the greatest common factor

Ah, the opening unit. The point where summer fades away. Students often find graphs to be their mathematical friends, so I lean on them a lot, especially in the first few units. I pair that with some analytical stuff by getting into function notation, function arithmetic, and factoring with GCF. Over 90% of the kids come to me from geometry (the others are taking it concurrently), so I also like jumping right into trig. The ratios are fresh in their minds and they get excited when they see them in the first few problems (e.g. 2, 8, 18). The problems reintroduce sine, cosine, and tangent and we begin thinking about the ratios in the coordinate plane with circles. It’s not explicitly stated in the concepts, but I also go hard with using interval notation when representing domain and range; it sets up so much of our future work on intervals. It’s a small thing, but I like how I frame factoring as “rewriting as multiplication” (problem 7).

Two things that I’m left thinking about: I don’t focus on domain and range in any other representation other than graphically. For this unit, I think this is ok, but as of now, I don’t devote any problems to it later on, either. This is an issue. Also, sadly, the function arithmetic that we do sort of dies at the end of this unit, as I haven’t yet included problems that loop back to it in future units.

In the end, this unit is full of pretty straightforward concepts and the students perform well. This is by design. With such a big contrast in how they learn math as compared to prior years, I find it helpful to build success in for them at the start. It’s a needed confidence booster for the long year ahead.

Unit 2 (problems 25-50)

  • Intervals of increasing & decreasing
  • Extreme values
  • Visual sequences & sequence notation
  • Factoring the difference of perfect squares
  • Trigonometric ratios & circles
  • **Systems of linear equations (2)

By studying where a function is increasing and decreasing and its extreme values, we continue the work on graph analysis that we started in unit 1. Again, interval notation comes in handy here. This problem set also introduces sequences. The intricacies of sequence notation can be tricky for students, but the few problems that I begin with (37, 38) have combated that nicely through the years. We also continue our factoring journey as we study the difference of two perfect squares. Now that I think about it, I wish I had included some difference of squares expressions that had a GCF. And I still don’t know how I feel about obscure expressions like (2x-1)^2 + (3x+8)^2 and asking students to factor them, but hey, that’s why I march to the beat of the Regents exam.

This is an important unit in the development of trigonometry because the kids learn about a unit circle along with defining sine, cosine, and tangent in terms of x and y. We also explore the signs of the three ratios depending on where the terminal side of the angle lies when it’s drawn in standard position. (Annoying: so many kids at this point start using the mnemonic ASTC because of the DeltaMath problems I assign.) Speaking of the terminal side of an angle, lost in all this trig work is the explicit learning of how to sketch an angle in standard position. I assign them problems from DeltaMath, which are really nice, but there’s nothing in the problem set that targets this skill. This results in me creating an opening problem to discuss it with the students at some point during the unit. This is haphazard and sloppy.

**Throughout the course, we come across topics and skills that are included in the problem set, but are not tested on that unit’s exam. I’ve found that there’s no need to test on EVERYTHING that we study during a single unit. Certain concepts, and certain problems, are stepping stones to broader ideas. I simply use them to navigate us to much deeper, more important parts of the curriculum. In this case, wrestling with systems of two linear equations will, over time, eventually lead us to solve systems of three linear equations (along with non-linear systems) in later units.

Unit 3 (problems 50-77)

  • End behavior
  • Function translations
  • Recursive sequences
  • Difference & sum of cubes
  • **Trigonometric ratios & circles
  • **Experimental Design
  • **Systems of linear equations (2)

In this unit, we continue our work on graph analysis, sequences, and factoring. Naturally, I am able to spiral in our previous graphing work (domain/range, intervals of increasing/decreasing, extreme values) in almost all of the end behavior problems (e.g. 57, 65). Factoring the difference or sum of two cubes always seems so formulaic and boring to me, but I know of no better way of approaching it. That said, this year I changed problem #59 to include an example of using the formulas, and it really helped. The students also get their first taste of a reciprocal trig function when they learn about cosecant. I define it plainly, which I’m ok with for now. I fold the ratio into several problems in the set to help build familiarity with it, especially in relation to the other ratios. When doing this, we continue to discuss terminal sides of angles and sketching triangles in the coordinate plane. Submerged in this set is rationalization. I touch on it with one problem (61), follow it up with DeltaMath, and bring it in again with problem 74. This basic level stuff is good enough to be able to help us with future problems. We also mix in our first look at transformations of functions when we do problem 53. I use a Desmos activity for this that I like, but I know could be better — especially when it comes to horizontal translations.

The sequence notation that they learned during the last unit is put to good use when I contract a throat virus (which renders me unable to speak) on the day they must investigate recursive sequences in problems 67-69. It’s a fun, memorable lesson. I drop two more linear systems problems in the set for good measure (60, 73). On a related note, I really like problem 72 because it combines function transformations with systems of equations. The equation f(x)=g(x) gets some attention, which is a nice.

I’ve sprinkled in a couple more problems on bias during this unit, which seemed like second nature to the kids. I feel the problems are useful, but they seem to linger with no stronger connection anywhere in sight. This will soon change when we start getting deeper in statistics in upcoming units, but maybe I can adjust for this next year?



Algebra 2 curriculum dump #0

I’ve decided to reflect on the interleaved algebra 2 course that I’ve designed and taught over the last couple of years. My hope is that by taking a more outward-facing approach to the curriculum, it will help me make it better. This is the 0th post in the series.

Two years ago I decided to turn my algebra 2 curriculum upside down. I did away with the thematic units that I used for my entire career and instead structured the course around interleaving topics. This was quite a shift for me, and I’ve written several posts about it, including a recent one about the mess of it all and why I love it.

Up to this point, I’ve never felt the need to formalize the curriculum. In year one, I didn’t know what the hell I was doing and developed all of it on the fly. It was an experiment. Last year, I heavily adapted much of what I did during the first year and reached a place where I could actually talk about what was happening in my classroom. This year I’m refining it. I’m even trying to integrate a book study. Since my head is now above water, and things haven’t been a complete disaster thus far, I’m feeling the need to take a more critical and reflective look at the sequencing and structure that I’ve developed for the course. To thoroughly think through my clutter of concepts and numbers and spreadsheets, to piece together a narrative from this tangled mess of a curriculum, is what I need to do to help improve it. For me, this means that I have to write about it. (It also wouldn’t hurt to have a coherent representation of the curriculum that someone else could understand.)

For fear of going mad, I’m breaking up how I reflect on the course by marking period. At the end of each marking period, I’ll write a narrative detailing the units that we studied during that time, my thoughts about them, and how they fit together.

Since this is the initial post, I’ll attempt to write an overview that I know is rushed and incomplete.

Course Overview

The course is based on the algebra 2 Common Core standards and taught through many non-thematic units. This means that, instead of being topic-centered, with concepts being taught in blocks of related content, concepts are problem-centered. This places emphasis on problem-solving, yes, but it also means that concepts throughout the course are interleaved. Instead of being introduced consecutively, related concepts are spaced out and learned over long periods of time.

To accomplish this lofty goal, each day we explore problems in class. Some days, there’s a singular focus, where all of the problems are related to one another, but most days we discuss multiple concepts that may or may not be directly connected. Disclosure: This doesn’t mean the problems are especially good, interesting, or challenging. On the contrary, because I teach a Regents course, the problems are quite boring.

What the course does well, I think, is challenge the notion that math needs to be taught linearly (like it is with traditional, thematic units). The cumulative nature of a course’s sequencing is overrated and less important than I think most math teachers realize. For example, early in the year, students in this course learn about factoring, graph analysis, and trigonometry concurrently. While these concepts are not completely distinct from one another, they probably wouldn’t be taught at the same time in a standard algebra 2 class.

As the year goes on, topics are revisited many times. We take bite-size chunks of content, digest them, wait a little while, and then go for seconds, and thirds, and fourths as the year progresses. Trigonometry is the best example of this. As I mentioned, we kick off the first unit studying trigonometry and we’re still learning (not reviewing) new trigonometric concepts in April. By design, it takes a while, sometimes months, for big ideas to mature in the course. Naturally, this development occurs a lot in the spring as we approach the backstretch of the school year. We’ve been studying certain topics for a long time, and it slowly starts to come together through the dozens of problems we have solved. As topics are woven toegther and learned little by little, finding connections between problems a foundational aspect of the course.