Posted by: rdkpickle | 05.05.2016

“i did it a different way.”

Are there more exciting words?

I’m at the board after some time spent in groups working on the warmup, going over a few answers and trying to clarify big ideas. They’ve stopped their work to tune in – quiet moments with eyes on me are so much rarer these days. Some kind of magic has happened in this happy, focused A block Geometry class this year. They work hard. They persist. This has so little to do with me – no one has told them they can’t be good at math, no one has told them it’s boring. So they believe they are capable, they expect to enjoy math class, and they are and they do. (I hope!) I give them good problems and explorations designed by the Geometry team and they set to work. I am so secondary to the work they do in my room.

My inexperience teaching Geometry actually feels like a gift – this combined with 3 preps at a new school (which means I can’t obsessively overplan each lesson) – means that the questions they ask sometimes surprise me; the solutions they offer often delight me.

I’m at the board, I’ve solicited an approach to a problem and written it on the board; it’s what I predicted, I’m moving on. M stops me. I can’t even remember if she raised her hand but honestly it’s better if she didn’t, let’s all just imagine she shouted at me. “Rachel, my group did it a different way.”

[“Rachel – let me share this with the class. Rachel – I know we’re all smarter than you, you tell us that all the time, let me prove it for the 30th time this week.”]

I pause (we all pause.) It’s that moment in the music where the wind players breathe before the downbeat.

3 examples of “I did it a different way” from my students from this week:




[Note about that last one that could probably be its own post. In reflecting upon “their way” and why it felt so startling, I’ve come to the conclusion that I am deeply a product of 1) learning math through rote, procedural practice in my high school classes, and 2) of the technology that was available to me when I learned this particular concept. I had a TI-83 calculator in high school – both common and natural log were at my disposal, but I didn’t have the ability to evaluate logs of any base (without using change of base formula) – so I couldn’t type something like the example above in directly to get a decimal approximation. My students with TI-84s can. Of course the two answers are algebraically equivalent. I would never in a million years have used this approach.]


Posted by: rdkpickle | 05.01.2016

favorite problems 2 – triangles

[previously: favorite problems 1 – pirates]

Two summers ago at Twitter Math Camp 14, I saw Dylan Kane (@math8_teacher) present a “My Favorite” about a problem he had enjoyed pondering during his summer adventures. You can read his post about it on his blog here. I believe the original problem came from the Five Triangles blog (which is now password protected. 😦 )trianglesGiven three “lined up” congruent equilateral triangles, each with area 20 cm^2, what is the area of the shaded region?

Dylan was using the problem to tell a story, but I couldn’t help myself and immediately started sketching out ideas for solutions. If you’ve been to TMC, you know the sheer volume of good ideas, interactions with rockstar teachers, and mathy things to ponder is overwhelming, so after my initial attempt at a solution I got distracted and set the problem aside for the rest of the conference.

Once I finished driving home from Jenks, the first thing I did that night was sit down and start thinking about the problem again. I told my sister Lindsey about it and she got in on the fun too. Here we are, lounging in my room past midnight doodling triangles and getting stumped.


The week got busy as preparations for the 2014-2015 school year got underway, and I think I forgot about the problem for a bit. Lucky for me, I mentioned the problem to Dawson over brunch one Sunday, and by mid-afternoon he was texting me about his solution.

[Note: Don’t read ahead if you don’t want to get spoiled on the problem!]

Dawson: Solved the triangle problem (I think/hope)! It’s great!! Although I thought it was leading me toward a different answer than I ultimately got

Rachel: i think i did too
did you do it in terms of 60 units sq?
can we compare answers?
and i agree…. i was almost annoyed at what it turned out to be
(if i did it right)

Dawson: Heading out to run errands
Happy to discuss later but…

Rachel: okay no worries!

Dawson: With each eq triangle 20 (total of 60)
I got 50/3

Rachel: ME TOO!

Dawson: Thought I was headed to 20 and that would have been awesome

Rachel: haha, yes.
okay let’s chat later about methods. i’m relieved we got the same thing though!

Dawson: Me too! Talk soon
Extensions to the triangle problem are amazing
I knew part of what was happening after 4 triangles…but 5 triangles really clarified it for me

Rachel: oh goodness, dawson. i guess i have to do that now…

Dawson: No no would love to talk about whatever you have later on

[Note: Do you see what he did there?!?! Fast-forward to “later on.”]

Rachel: what did you get for 4, 5? did you do it in terms of 20 sq units per triangle? or just proportion of one whole triangle?
both are illuminating, but proportion of one whole triangle has a nicer pattern… i think… if i’m doing it right
(2n-1) / 6 * area of 1 triangle??

Dawson: Proportion of one triangle

Rachel: n = number of triangles

Dawson: Yes

Rachel: sweet

Dawson: Did you look at each individual small triangle?

Rachel: just one, then scaled down area
like my work for the first one was 2/3 + (1/2)^2 * (2/3)
and the second 3/4 + (2/3)^2 * (3/4) + (1/3)^2* (3/4)

Dawson: Ok…

Rachel: once i had thoroughly convinced myself the first shaded region was 2/3, i was good and set

Dawson: If you start with 5 eq triangles of area 20…
What’s the area of each individual part?
That’s what got me hooked

Rachel: when you look at the individual pieces is there something pretty?

Dawson: YES

Rachel: i just continued on… 4/5 + (3/4)^2 * (4/5) + (2/4)^2 * (4/5) + (1/4)^2 * (4/5)

Dawson: I think we did this WAY differently
At least in approach
Take your (3/4)^2 * 4/5 and simplify

Rachel: 9/20…. okay okay give me a sec

Dawson: You bet…I needed a long time!!

give me ANOTHER second

Dawson: Yes!!!!!!!
It’s awesome

Rachel: holy cow what is going on??? so 4 triangles is 9/12+4/12+1/12???

Dawson: Yes
It’s beautiful

Rachel: holy cow
and those are triangular numbers on the denom??
does that continue?

Our text conversation turned into an email thread that included Dylan, Justin Lanier (@j_lanier), Nathan Kraft (@nathankraft1), Glenn Waddell (@gwaddellnvhs), Mimi Yang, Mark Greenaway, Jed Butler (@MathButler), and Tina Cardone (@crstn85), all of whom sent out their own version of the solution. It’s completely fascinating to me how many different approaches people took. People employed so many different strategies from using similar triangles and scale factor of sides/area, using ratios of sides in special right triangles and the formula for area of a triangle, writing equations of lines and finding points of intersection, to using trig and law of sines.

Several folks tackled the extension Dawson prompted above, and some worked on generalizing the problem for congruent isosceles triangles, or even congruent scalene triangles.

I have a folder where I’ve saved all of the submitted solutions so far, and I’m still collecting more, if you want to share! Mine and Dawson’s are both posted below. Not quite as high-tech as Justin’s typed-up solution, or Jed’s impressive iMovie solution, but they get the job done. I especially enjoy Dawson’s for his use of colored pens, and the running self-deprecating commentary throughout. Enjoy. 🙂



triangle problem solution dawson 1

triangle problem solution dawson 2

One last thing: I used this problem as a “warm up” problem in a math department meeting a few months ago. One of my colleagues emailed me with his solution after the meeting, along with what might be my favorite quote in recent memory.

“It took me about 15 minutes beyond the mtg, and yes, after one sees it, one feels slightly ashamed, as with all good puzzles.

Indeed. 😀

Posted by: rdkpickle | 04.29.2016

i’m still unpacking from nctm

“[Math] operates with unearned privilege in society, in the same way that whiteness does.”

Rochelle Gutierrez’s #shadowcon16 talk is up – check out the video.


“We all contribute to the kinds of identities students develop, both in our classrooms, and long into life. So when you think about those reactions you get from people when you say you’re a math teacher, that’s been carried with them into their lives.”

“All mathematics teachers are identity workers.”

Blend that with Kaneka Turner’s talk, and ask yourself – who gets invited to the math party? Who gets the invitation to be “good at math?” How can I extend the invitation?

“Teaching mathematics is political, I say it’s because every day we make in the moment decisions that affect students not just in the classroom, but long into their lives. We hear others talking about our students, labeling them or creating policies that will affect them. And we ourselves, we carry out assessments that will tell them something about themselves. Not just something about what they’ve learned, but also something about their value or their worth in this society.”

I believe to my core that all students are capable of learning about and engaging in meaningful mathematics. It is imperative I examine the ways in which I inadvertently send the wrong message to students about their own ability or access to math, and the way that gets bound up with other cultural messages about identity and status.

Go listen and learn. Happy Thursday.

Posted by: rdkpickle | 04.22.2016

favorite problems 1 – pirates

Holt claims “vivid, vital, pleasurable experiences are the easiest to remember” – I’d add interpersonal to that list and be ready to go. All of my favorite math problems sit nestled inside the memory of the who-what-when-and-where they were solved.

I’ve had a post saved in drafts for over a year now, detailing favorite problems. Perhaps this should be a series. Let’s start at the beginning.

1) pirates

The first summer I helped Dawson teach at VSA [Vanderbilt Summer Academy], I walked in each morning fresh to the puzzlers he planned to pose. Realistically, I could write an entire post about Dawson’s puzzlers – he’s got a treasure-trove of good stuff for high schoolers. Instead, I’ll write about the first and most memorable time I felt the thrill of simultaneous discovery – solving a good problem, and instead of stopping, demanding more of the problem than it offered at first glance.

Here’s the pirate problem. If you’re Dawson, a meticulously arranged treasure box featuring exactly 100 “gold” chocolate bars sets the stage. ARRRRRRRRR!


There are five pirates who steal exactly 100 gold coins. They go to a safe haven to distribute the gold among themselves. Being democratic pirates, they decide to vote on how to do so. Each pirate in turn submits a proposal for parceling out the booty. Immediately after the first proposal, they vote. If the proposal wins a majority of votes (MORE THAN HALF), they distribute the gold according to the proposal. If the proposal does not get a majority, they kill the pirate for suggesting it and move on to the next pirate, who makes his proposal. The process continues until a proposal receives a majority of votes cast.

You are the FIRST pirate to make a proposal. What should you suggest to maximize your share and, of course, remain alive? Explain why this proposal would win more than half the votes. (Minimally, the explanation requires saying who would vote for the proposal and explaining why they would do so). You may only presume that a pirate will vote for a given proposal if it is definitively better than another proposal that he may receive.

Each pirate knows his own and everyone else’s position-first, second, third, etc.-in the order of giving proposals. The pirates are entirely “logical” and unemotional people. All they care about is maximizing their share while remaining alive.

Solving the problem isn’t so tough. I set to work. I’m a terrible collaborator during problem solving – I want to hide in the corner and try all of my own ideas without sharing, then return the group and get louder and louder as I share my insights in half-formed, broken sentences or SHOUT my questions at whoever is unfortunate enough to be working with me. This was nearly 10 years ago, and Dawson and I were coming off of a spring of student teaching together – 6 am carpooling commutes from campus to class forge a bond that sticks. Dawson had solved the problem before, so I did my thing quickly. I watched the students in the class navigate their way through the problem after I figured it out, and poked and prodded at those who needed a push over some of the sticky places.

The day ended; the students left. I was still puzzling over the problem. “What if…?”

What if… there were 6 pirates? 7? How long does that pattern continue? When does it break down? Does anything interesting happen after that?

This was the first time I felt like I had stumbled on something new and my own. I’m not really sure how to express the sense of ownership I feel over this particular problem, this particular piece of mathematics that we carved out together. We set to work, we filled up a whiteboard, we invented ridiculous notation for pirates who won’t propose. I shouted at Dawson, he shouted back. We followed the problem where it lead us, and stumbled upon a surprising and interesting pattern. Apparently, we made some stranger who happened to be wandering the halls of the Blair School of Music come in and take a picture of us in front of our work.

Afterwards, I did some poking around the internet and found that Ian Stewart had written an article about the patterns that emerge in the extension – OUR extension – for Scientific American in 1999.


Postscipt: When I think about problem solving now, ideas like “solve a simpler problem,” and “extend the problem” seem so obvious, but I think this was one of the first (or at least one of the best) examples of me doing just that, before I had the framework to understand what I was doing. It’s not uncommon for people to ask me if “I always knew” I wanted to be a math teacher, and the honest answer is yes. But the longer answer is that the story of me falling in love with math, of seeing the richness and vastness of this subject, of understanding the way it excites, ignites, and brings people together… that story is still being written. And the pirates problem is a major chapter in me seeing myself as a mathematician – someone with agency to tinker, play, invent, and discover.

Update 4/23/16: As usual, Dawson says it better than me:

“WHOA! I’ve actually been thinking about that very afternoon a lot in the last week […] this was also my first such day, and it changed my teaching career and what I value about the mathematical classroom experience more than anything else I can think of. THANK YOU for shouting at me!”


Posted by: rdkpickle | 02.21.2016

introverted teachers

After a week full of constant interaction – lunch meetings sandwiched between classes, emotional conversations followed by a room full of expectant teenagers – it feels appropriate to share this article I’ve had bookmarked in my browser for a couple of weeks.

Teacher Burnout is More Likely Among Introverts

At the very least, this short post serves as a lame apology to my new roommate for hiding in my bedroom all day today reading.

It used to drive me absolutely crazy. In moving to a brand new city and starting my career 7 years ago, I met a lot of strangers – introductions to friends of friends, crossing paths with other young people enjoying a night out in the city, striking up a conversation with the person sitting next to me on the Metro, etc. And inevitably when the conversation turned to “what do you do?” my response always seemed to elicit a similar series of reactions.

“I’m a teacher.” “Oh really? What grade?” (The assumption there being that I teach elementary school.) “High school.” “Oh! What subject?” “Math.”

Then it was one of a few things, most of which basically boiled down to this: “My math teachers didn’t look like you.”

I struggled with how quick I was to get offended by the immediate assumption that I taught younger grades – I can’t imagine many jobs more necessary, important, difficult, and physically/emotionally demanding than teaching elementary school (I certainly couldn’t do it), and yet that reaction always made me feel undervalued, somehow. In early years, it felt like a comment on my intelligence or capability – and I almost enjoyed the chance to “earn some credibility” by throwing my teaching of math in their face. But, inevitably, responding this way always made me feel like part of the problem.

I struggled with the way my woman-ness, and often my specific physical appearance, was commented upon – as though it is inherently strange or special for a young woman to have a passion for mathematics and still look like a woman. Men usually take it to a really uncomfortable place – “If I had a teacher like you, I wouldn’t have been able to focus in class.” “How do you handle the teenage boys in your class?” “I bet all of your male students have a crush on you.” I still have no scripted reaction to this category of remarks. It always leaves me feeling small, uncomfortable.

Often the person doing the reacting really meant whatever they were trying to say as a compliment of sorts. Other women, especially, are quick to move the conversation into a space where they can share their own discomfort or perceived limitations with mathematics, and express genuine regret that they didn’t have the opportunity to see it as an area of study that was open to them. I am developing a better understanding of how my presence in the classroom as a woman who teaches math has an impact on the way my students imagine both mathematics and mathematicians. But that’s not something I did, it’s just something I am. It’s something that (perhaps naively) only really became apparent to me as an important part of my work in the classroom once I started teaching. When someone thanks me for being a “role model,” I can speak with passion about working with young women as they develop the mathematical abilities within them, but I always feel a bit confused about what, exactly, I’m being thanked for.

When it comes to the “math” part of my math teacher identity – yes – it’s really important to make it possible for young women to see a future for themselves in these traditionally male-dominated STEM fields. (Not to mention – all of the desperately necessary work we have to do in allowing other marginalized groups access to these careers as well.) Representation is a big part of that. Being proud of my genuine, nerdy love of math and finding a comfortable home for that alongside my identity as a woman is significant. And yet, I worry that as we fight for greater access for women to male dominated spaces, we’re only fighting half the battle unless we simultaneously begin to value more greatly the work being done by those in traditionally female careers, like teaching. Why is it it that the praise for my work as a math teacher so often only focuses on the “math” piece?

I’m reminded of this really fantastic article from a year back, “Why I Am Not a Maker”

“I am not a maker. In a framing and value system that is about creating artifacts, specifically ones you can sell, I am a less valuable human. As an educator, the work I do is superficially the same, year on year. That’s because all of the actual change, the actual effects, are at the interface between me as an educator, my students, and the learning experiences I design for them. People have happily informed me that I am a maker because I use phrases like “design learning experiences,” which is mistaking what I do (teaching) for what I’m actually trying to help elicit (learning). To characterize what I do as “making” is to mistake the methods—courses, workshops, editorials—for the effects. Or, worse, if you say that I “make” other people, you are diminishing their agency and role in sense-making, as if their learning is something I do to them.


A quote often attributed to Gloria Steinem says: ‘We’ve begun to raise daughters more like sons… but few have the courage to raise our sons more like our daughters.’ Maker culture, with its goal to get everyone access to the traditionally male domain of making, has focused on the first. But its success means that it further devalues the traditionally female domain of caregiving, by continuing to enforce the idea that only making things is valuable. Rather, I want to see us recognize the work of the educators, those that analyze and characterize and critique, everyone who fixes things, all the other people who do valuable work with and for others—above all, the caregivers—whose work isn’t about something you can put in a box and sell.”

Last night was just a typical social Saturday night out, and over the course of the evening I had several chances to practice a range of reactions to others’ reactions as they found out I teach math. It’s still a work in progress.

Posted by: rdkpickle | 02.08.2016


Today I’m moving around the classroom during the warmup and as soon as I kneel down to listen to one group’s conversation, a student stops, points to her work on the first question and asks me “is this right?” All three group members pause and look up expectantly at me. I hedge. I usually hedge.

“Talk me through your process,” I say with that pleasant, unreadable grin.

“That means I’m wrong, doesn’t it?” She’s smiling but she’s creating an out for herself. “I probably messed it up, did I do the whole thing completely wrong?”

“I didn’t say that! You’re the first person I’ve seen to try it this way, speak a bit about choosing this approach.”

Another member of the group laughs. “Have you seen the TV show Parks and Recreation? You know Chris Traeger? He’s so positive and affirming that he broke up with Ann Perkins and she didn’t even know it. Rachel, that’s you, when we explain something.”

I break into genuine, surprised laughter, and the rest of the group follows suit. But it’s true. When students offer to publicly share their attempt at a problem, suggest an idea about how one topic relates to another, or ask a question that sheds new light on the math we’re investigating (for me directly, or in a way that more clearly helps me understand their thinking) – it’s almost always valuable. I’m not faking my reaction, even if their work is way off base: we won’t get nearly as far together unless we are willing to take that risk and forced to articulate and refine our thinking.

I hope my relentless Chris Traeger positivity makes it feel safer to persevere, collaborate, and listen in my classroom. I also hope my students see clearly modeled the habits of building on other people’s ideas and appreciating different ways of being smart.

She launched into explaining her process. The group refocused its attention on her, not me. I moved on.

Posted by: rdkpickle | 01.09.2016


Conversations that get me thinking very deeply about math = A++ way to return from a luxuriously long winter break (spent being sick. Boo sinus infections!)

So. My precalculus team was discussing trig identities today – pacing our unit and then just sharing bits about how we’ve experienced teaching this topic in the past. A colleague brings up the piece about working both sides independently, vs. manipulating the entire equation. He shares an idea I hadn’t considered before, neatly captured here as part of a larger discussion on the issue (click to read entire thread)

Screen Shot 2016-01-08 at 9.18.55 PM.png


I sat at lunch staring into space like a dummy, trying to more clearly parse the discomfort I’m still feeling with this approach. I think the beginnings of my difficulty lie here:

Screen Shot 2016-01-08 at 4.22.18 PM.png

So one place I’m feeling the discomfort is: students could start with a false statement (something that is not an identity) and operate on both sides to produce a true statement. Worrisome, but hopefully they could realize that they couldn’t work backwards, starting with true statement, to produce the false statement. Maybe I’m worried they’ll get so excited about true statement they’ll forget to really justify the proof they’ve GOT by showing it “backwards.” It just leaves me uneasy.

Second place I’m feeling the discomfort: are there true statements – identities – where it’s possible to operate on both sides to get two identical expressions… but then in trying to write the “work backwards” part, run into trouble? I don’t know how to more clearly ask this question. I should probably think about this more when I’ve had some sleep.

Really, you should all just read all of: this

Really, I should appreciate the return to school. These teenagers are endlessly energizing. Their insights surprise me. Their struggles challenge me to adjust my approach.

Really, I should appreciate that I have colleagues who want to dig in so deeply on these ideas. I crave more! More conversation! More chances for me to fail in articulating my questions! More chances for me to refine my thinking about how to engage with students who ask “why can’t I…?”!

Really, I should be sleeping. Happy Friday! Happy weekend! Happy start of 2016.

Posted by: rdkpickle | 12.13.2015

miles and months later

I talk less. I listen more. I laugh plenty. I collaborate with an incredible team of math teachers. I am mentored by colleagues who are generous with their knowledge and time. I have eased into relationships, into community, into my role here. I struggle with balance. I try to be a daughter, a sister, a friend as life changes come fast and furious. I say no to some opportunities, say yes to others, and remind myself that I’m committing myself to this work for years, not months – and it’s okay to have miles left to go.


Some moments:

  • Reconsidering and deepening my knowledge of trigonometry based on a sequencing that forced new approaches to concepts, and colleagues who thoughtfully construct student-centered explorations into big ideas
  • H’s question when we were studying Law of Sines about where, exactly, this sinA/a ratio showed up on the physical dimensions of the triangle itself, which led to this new discovery about the circumcircle and its diameter
  • Dance Dance Transversal. Will always remind me of something else, now, something about how the best way to honor someone else’s life is to keep living yours.
  • S’s observation about exterior angles summing to 360 – completely unprompted – in Geometry after discussing exterior angle theorem
  • My Algebra 2 crew. Alternating between amazing days and days that make me want to pull out my hair, this class is never boring. They’ve worked so hard all semester and many of my standout moments have been with them. From Bucky the Badger to sweet sweet Desmos activity builder love, they have come a long way since Day 1.
  • Interleaved homework in Algebra 2. This team is 4 strong, and I follow the lead of my colleagues for the most part on pacing, sequencing, and homework. Nice to see topics spiral back again.
  • Navigating 3 teams with 3 very different personalities and approaches to collaboration. Mostly this just leaves me appreciative of experience and a diversity of teaching styles. “There is no one way.”
  • Desmos activity builder! and Desmos activity builder! and more Desmos activity builder!


As the semester wraps, trying to honor and celebrate the good things that have happened. I’ll have more time to dive back into Global Math, attend conferences, and hang out with Bay Area tweeps at our meetups soon. These first few months were what they needed to be.

Screen Shot 2015-12-11 at 2.23.57 PM.png

Posted by: rdkpickle | 10.08.2015

looping loops

“The newspapers, even some in France, said it was the marvel of the age; better than the Eiffel tower – and it was a Ferris wheel.

That almost mundane sensation we have now of looking down from above and moving through space, up and out and down and around again, no one had ever felt those things before, and of course now we can’t really feel those things again. We’ve gone around too many times. We’ve looped too many loops.”

High Above Lake Michigan – The Memory Palace

There’s something profound here about rhythm and routine vs. the novel and once remarkable. There’s a transition from terrifying and thrilling to comfort and confidence. I have a few more loops to loops, but I’m getting closer.

(We listened to this episode in Precalculus this week: working on circular motion, angles, linear and angular velocity, etc.)

Turns out other folks have been thinking about loop-de-loops lately:

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