Some MA students made an awesome documentary about teachers (and i’m in it). I should probably post this here!
Some more words from other people about social media, multitasking, slowing down;
“At saturation level, social media reduces the amount of time people spend in uninterrupted solitude, the time when people can excavate and process their internal states. It encourages social multitasking: You’re with the people you’re with, but you’re also monitoring the six billion other people who might be communicating something more interesting from far away. It flattens the range of emotional experiences.
When we’re addicted to online life, every moment is fun and diverting, but the whole thing is profoundly unsatisfying. I guess a modern version of heroism is regaining control of social impulses, saying no to a thousand shallow contacts for the sake of a few daring plunges.”
“Intimacy for the Avoidant“- David Brooks
“A restlessness has seized hold of many of us, a sense that we should be doing something else, no matter what we are doing, or doing at least two things at once, or going to check some other medium. It’s an anxiety about keeping up, about not being left out or getting behind.
I think it is for a quality of time we no longer have, and that is hard to name and harder to imagine reclaiming. My time does not come in large, focused blocks, but in fragments and shards. The fault is my own, arguably, but it’s yours too — it’s the fault of everyone I know who rarely finds herself or himself with uninterrupted hours. We’re shattered. We’re breaking up.
It’s hard, now, to be with someone else wholly, uninterruptedly, and it’s hard to be truly alone. The fine art of doing nothing in particular, also known as thinking, or musing, or introspection, or simply moments of being, was part of what happened when you walked from here to there, alone, or stared out the train window, or contemplated the road, but the new technologies have flooded those open spaces. Space for free thought is routinely regarded as a void and filled up with sounds and distractions.”
The Encyclopedia of Trouble and Spaciousness – Rebecca Solnit
I felt a bit of discomfort with the uninterrupted time afforded me over the recent break, never seeming to make use of it the way I’d like. On the other hand, when school is in session, I am constantly fighting the feeling of fragmented multitasking in response to a busy schedule. Do you feel this tension around being fully present in your work and relationships? Do your students? What seems to work for you to reclaim space for daring plunges?
Was going to make some lists of where I’ve been and what’s on deck. Typed and deleted a few things.
It’s hard to be fully present. I was going to add a lot of qualifiers to that sentence like “it’s hard to be fully present when an inbox full of emails demands attention” and “it’s hard to be fully present when your phone promises an easy distraction to even a moment of boredom” and “it’s hard to be fully present when your mental to-do lists are shuffling out your ears and littering the lawn” but I needn’t bother. It’s hard to be fully present. Period.
Last week at school a speaker came to talk to the students about smartphones and the effect they are having on teenagers’ development of social and emotional intelligence. Her talk really stuck with me, as I thought about all of the ways I distract myself from what’s around me, all of the ways I disengage with the challenge of being truly present, and all of the ways the things that I think help me stay connected (twitter, facebook, instagram, email, texts, etc.) — also harm me. To that end, I’ve tried to be more intentional about when and where I carry my cell phone with me, how many times in a work day I check my email (I never leave it “open.” I respond quickly once I’ve read a message, but I designate times to read and respond), and how much screen time I allow myself in the evening before bed.
I checked twitter two nights ago after a busy, packed, fantastic weekend with a dear friend, saw all of the good stuff you all have been up to, and immediately experienced so much guilt and self-doubt about how I choose to spend my time. It’s this strange, competitive (embarrassing) FOMO around public professional successes. “I’m not blogging enough!” “I’m not tweeting enough!” “I didn’t make the time to read that book/article/blog post yet.” “I’m not curating mathematical art shows.” “I didn’t apply to be a Desmos fellow.” “Dan Meyer didn’t give me a shoutout in his recent blog post.” Dumb dumb dumb dumb dumb. Dumb brain. Comparison is the thief of joy.
Here are recent successes that exist wholly outside the digital domain: I’ve slowed down. I’ve still had my share of disjointed, less-than-ideally paced classes this year, but on the whole the entire vibe in my classroom has been calm, focused, productive, positive. My interactions with students feel more deliberate, my conversations with colleagues less rushed. My classes are a story with a beginning, middle, and end (and – I am not the main character in this story.) In the lunch line, rather than checking my notifications on my smartphone, I try to find someone to talk to about how their day is going. As I walk around campus, I am comfortable. I check in with students I taught last year, and really sink into those conversations. I work hard as a team member on my shared courses, refining curriculum and executing lessons that allow students the chance to authentically engage in meaningful mathematics. I am busy, but I feel like I have a good sense of the work that is worth my time, and I put as much of my effort as I can into that work. Then, I put boundaries and parameters around the rest of it in order to preserve my mental and emotional health.
I know there will never be enough time. But the time that we have is a gift. Trying oh-so-hard to slow down, and enjoy it.
A quick post!
The school year is officially underway, with one week in the books. Because we are on a block schedule, this means I’ve seen each of my classes twice.
This year, I’m teaching 1 section of Geometry, 1 section of Precalculus, and picked up 2 sections of Algebra 2 Honors. I’m super excited about my schedule, the mix of old and new, and the opportunity to teach an honors class at MA.
I kicked off my Algebra 2 Honors class with an awesome “Think of 2 Numbers” puzzle that came from the teacher who previously taught the course. He generously shared his entire (amazing) curriculum with me, and I’m so looking forward to digging into the cool problems and explorations he has refined over his 10 years with the course. Here’s the puzzle:
After playing with this puzzle long enough for the class to make a conjecture about how to figure out the 2 numbers (and putting a student on the spot to figure out MY two numbers), we did brief introductions, then moved into our next activity.
I decided to use this NRICH “Algebra Match” activity that has students silently, in groups of four, trade away cards with mathematical expressions until each group member has a matching set of 4 cards. The rules?
Here is one of my classes busy at work on the activity.
After each group successfully matched the cards, I asked them to take a moment to reflect on the experience by responding to a few prompts. We discussed the first few together, and then I went home and read the answers to the last two (more personal) questions, collecting some of my favorite responses to share back to the class on the second day. Here are the questions I posed, and some of the responses I got:
What are some of the benefits of group work in math class? What are some of the challenges?
D Block responses
(p.s. not on the board, but on a student’s paper. “Group work is challenging when people are stubborn. It’s even more challenging when stubborn people are incorrect.” lololol)
E Block responses
(p.s. not on the board, but nearly half of the students in this class wrote that a challenge with group work is “being on the same page” or “getting everyone on the same page.”)
What do YOU need to be successful during group work? From your peers? From me?
D Block responses
E Block responses
What individual goals do you have for yourself as a mathematician during group work this year?
D Block responses
E Block responses
To be honest, I wasn’t sure how this activity or the debrief afterwards was going to go, so I was pleasantly surprised to see that the task was sufficiently challenging (mostly because of the “no talking” requirement, although it did give me a sense of which students might be a little rustier with their algebra or need to rely on scratch paper vs. working in their heads) AND that they really dug into the discussion afterwards in a way that I think helped establish some class norms for group work that will pay off throughout the year.
At the end of class, we returned to the “Think of 2 Numbers” trick, where (again, stealing Jamie’s idea) the students had to prove algebraically why the trick worked for homework.
That was all just Day 1! Day 2 was even cooler! I’ve already written quite a bit and want to get back to enjoying my Saturday (it’s a beautiful day in the Bay Area!), but here’s the problem we tackled to start class on Day 2 and OH BOY did the students take it even farther than I had expected.
This is going to a fantastic group, and a fantastic year. I’m in love.
With 2 trips immediately following both Descon and TMC, I know that if I don’t blog quickly with these few nuggets I’m in danger of never blogging at all. So, quickly:
Desmos has been working to make its graphing calculator fully accessible for visually impaired and blind students. As of now, if you enable voiceover on your device (command + F5 on mac), Desmos will read the expressions in a way that “sounds like math” (ex: reading sin as “sine” instead of s-i-n) and gives students clues as to where they might be in the expression (ex: subscript, superscript, reading the opening and closing of grouping symbols). It is also capable of “playing” a graph to give a pitched audio representation of whatever has been graphed.
[Predictably] I immediately set out to use the feature to write a song, ignoring all other directions about what we were “supposed to be doing.” James and I worked on it during our morning work time, with serious insight and help from Chris and Sam to get us over a few hurdles. I loved the experience of playing with something that I had no idea how it worked by experimenting (pitches compared to y-values, time compared to x-values, etc.), trying and tossing out ideas that ended up not working, and I ended up learning some new things about how to use Desmos (lists of functions, anyone?) along the way.
Here’s our final version of “Mary had a little lamb.”
Desmos also premiered a few new things which others have written about – a new layout on the teacher.desmos.com landing page, bundles of selected+sequenced activities with notes for teachers about how each lesson builds on the preceding ones, and the “labs” option which enables teachers to use the brand new marbleslides and card sort features for activity builder lessons. The MTBoS has already set to work creating quite a few activities to test out the new features, and I can’t wait to think about how I might incorporate these in my own lessons for the year ahead.
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.]
[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. 😦 )Given 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: 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)
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?
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!!
Rachel: WHAT THE HELL WHY ARE THEY PERF SQUARES
give me ANOTHER second
Rachel: holy cow what is going on??? so 4 triangles is 9/12+4/12+1/12???
WHERE ARE THESE NUMBERS COMING FROM
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. 🙂
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.“
“[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.
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.
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!”
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.
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.