Gratuitous and a distraction

Today made me happy. Today the Supreme Court said that part of the Civil Rights Act of 1964 that applies to sex discrimination encompasses discrimination based on sexual orientation and gender identity.

Today Brett Bigham shared this thread on twitter about his experience being a gay math teacher. He won Teacher of the Year in 2014 and that same year he experienced … a lot. I entreat you to read it all.

 

 

We are in the math education community, and we are not immune from homophobia. Even from our own kin.

For the past couple of years, I debated sharing a story of my own. A very small story, compared to Brett. And not a story I’ve ever shared in full to anyone but one or two people.

In 2018, I submitted a short article to a respected math teacher journal on a prompt asking why we teach, and why we continue to teach. For me, it was an article about identity, and how math teaching can be a bridge allowing us to form lasting and meaningful connections with each other, no matter how different we may be.

Here’s the opening paragraph. The opening sentence was meant to be jarring because of what came after. Obviously the sentences that follow, my life trajectory, the things about me that make me unique, different, contradict the opening sentence.

I’m not so different from you. First generation kid of immigrants from India. Had only a single friend for most of my childhood. Lived in an extended family household of nine for junior high and high school. Switched high schools after freshman year when my dad got transferred to a different state. Studied math in college. Came out as gay at age 19. Studied history of science in graduate school for four years before deciding academia wasn’t for me. I just turned 37 and am starting my twelfth year of teaching.

The article is framed around the idea of identity, and my search for belonging when for many years I never felt like I ever truly fit in. And for me, one place I found that sense of belonging was with math teachers who connected online. I wrote:

I started to belong. Even though I had very little experience, people showed me that my voice mattered and I actually did have something to contribute to the world of math education. I was accepted from the start – this awkward, neophyte, math-obsessed, showtunes-loving blogger. Over time, people in the community turned from icons and handles on the computer screen to friends, and now they are a chosen family. I’m not capable in words of expressing how I feel about these people in my heart. From times when I was the most down, when I felt worthless and wanted to leave the profession, they kept my spirits up. They saw me in a way I couldn’t see myself. They continue to help me be my best teacher self. When my mother was in the hospital with complications from chemo, they called and wrote and visited. They celebrated me when I shared a cool project one of my kids had done or I shared part of a nice card a student wrote me. And they come running up to me hugging me at conferences and make me feel loved. Yes, this online community of math educators changed my classroom, but they also gave me something more precious: acceptance and unconditional support and love. This community has become a place of support not only for the classroom but for the heart.

You see, I told you. I told you! I’m not so different than you! Because although our journeys are wildly incongruous, they each led us to a place where we can connect. You and me, we both relish our time working with budding mathematicians. Our hearts beat faster when we’re hearing their exhortations of delight when something clicks, or we see in slow motion the face of a kid morph into a toothy grin when we hand back particularly good test, or we see a spontaneous high five when a group conquers a seemingly-impossible problem. And no one else – not even someone freakishly just like me but who decided on a different profession – could ever get that part of me. They have to have experienced that racing heart. You have lived and breathed it. And starting from there, we can build a bridge between us.

I received news that the article was accepted provisionally. They wanted to publish it, but they only had room for a shorter version. And they mentioned that as they put together the journal, there was a chance the shorter version might not be published because of space.

Understood.

I cut it down. I sent it back. And I got this email in return:

Thanks, Sameer.  You neglected to add your signature.  We need name, email, school affiliation and location and date of the letter.

We will be taking out the sentence about your coming out; it is gratuitous and does not help the letter.  This decision was made above me, but I agree with it.  Such statements, while they are certainly important to you, don’t really belong in the journal.

If you still want us to publish the letter, please get me the signature info asap.  Thanks for turning this around so quickly.  For what it’s worth, it seems stronger to me in this form.

Let me remind you of the entire first paragraph, and highlight the one thing that was asked to be excised from it:

I’m not so different from you. First generation kid of immigrants from India. Had only a single friend for most of my childhood. Lived in an extended family household of nine for junior high and high school. Switched high schools after freshman year when my dad got transferred to a different state. Studied math in college. Came out as gay at age 19. Studied history of science in graduate school for four years before deciding academia wasn’t for me. I just turned 37 and am starting my twelfth year of teaching.

I received their email when I was at school. I felt so many things. Anger. Confusion. Defensiveness. Shame. My stomach was churning. I wrote this back before going to teach my next class:

I think coming out to be as important a part of my identity as my extended family and moving from one place to another — if not more. And the fact is this is a piece about identity and connection, and finding commonalities when none might seem apparent.  The fact that this is the one thing the higher up editor felt important to communicate about what needs to be cut is problematic for me. I am fine withdrawing this article from consideration for [Journal]

The reply:

I will try again but remember, this is an academic journal.  Your coming out may be important to you, but I see it as a distraction to our readers

I couldn’t focus on anything but the words gratuitous and distraction. I didn’t understand what being an academic journal had to do with this. The piece wasn’t about a project using algebra tiles, it was centered around the idea of identity. Soon after, I replied:

I am sorry, but I do not agree. I think asking for this one erasure — of this one thing that you want to take out but not mentioning any of the other things about me (moved, extended family, first generation) — is problematic. I don’t know why it feels more “gratuitous” than anything else in the first paragraph. And I certainly don’t see it as a distraction for readers. For this one piece to be singled out over all the others, and having it be called a “distraction” and “gratuitous,” doesn’t sit right with me. Especially for a piece that centers around the notions of identity and connection.

I know [organization] has been working hard on diversity and equity issues. In my eyes, the fact that this has even come up means that [organization] has a lot more work to do to live its mission.

I’m sorry I wasn’t clear. I am withdrawing this article from publication by [Journal].

After a few more back and forths with false pleasantries, the conversation ended.

My article was withdrawn.

I was still feeling all the things.

When the school day ended, I went to talk to a teacher friend who often gives me wise counsel. In my head, I could hear a voice saying “it’s just one sentence.” Deep down, I knew she wouldn’t ever say that but I was worried that she’d think that. I was feeling defensive, like I wasn’t certain that my feelings were valid and that what had just happened to me was wrong, full stop. I remember trying to explain to her the situation and giving her so much context. If I explain it just right, maybe she’d understand and agree? I made her read the article to understand. I gave her some of the background. I let her read the email exchange.

Credit to her, she got it right away and helped me see I wasn’t overreacting. She too, when she got to the word “gratuitous,” flipped out, and again at the word “distraction.” Her own righteous anger validated all my feelings.

My biggest fear when approaching her is her thinking “you’re making a mountain out of a molehill.” I actually fear that now. Like people won’t understand when reading this post how this was homophobia–period. Like people won’t get how being told that being gay is “gratuitous” and a “distraction” is like being told that part of you is invalid. Like people won’t understand why this is an episode in my life that I will get past but never be able to let go. That email exchange and that day — thinking of all the emails and all the days in my life that I forget — stick with me. My hands were actually shaking when I was opening the email thread to re-read it to write this post. It brings back a lot.

I think that’s why I wanted to share this. Partly to show people that homophobia still exists in the math education community. But I think it goes further. I wanted to recount this to remind myself and others that concepts of identity can seem academic, but when we talk about erasing and denigrating parts of people, it is anything but academic. It’s not just one sentence. It’s one part of me that you don’t like. It’s one part of me that you find uncomfortable. It’s one part of me that you wanted to excise. But it’s one part of me that makes me who I am, that I want to be proud of, that you are diminishing and making me feel ashamed about. 

Maybe that wasn’t the intent. I know I’ll never know the intent. But that was the impact.

 

PS. Because I did not want something like this to happen to anyone else, I did contact someone high up in the organization who was immersed in diversity and inclusion work, and huge props to them. They recognized how problematic this was and did not sweep it under the rug.

PPS. Even as I am about to publish this, I’m terrified about getting an angry email in my inbox from the editor who emailed me. If you are somehow reading this, please don’t email me. I don’t want an email from you. This post isn’t about you.

It’s Tuesday evening.

It’s Tuesday evening. A faculty meeting just ended. I am sitting here with lots of thoughts I need to process. For some reason, I feel less fearful about sharing my thoughts recently. I don’t know if anyone else is going through similar things, but I’m sharing this post to hold myself accountable to this work, but also in case others can see glimpses of themselves and their thinking and feelings in my experiences. Here it goes. Stream of consciousness. No editing. Just writing.

Thought 1: I ended my classes this week. Our last classes were held on Monday and Tuesday. I didn’t know how to end them. I wanted to close the year with some solemnity and definitely not frivolity. I know when I’m feeling the most dark and depressed about the state of the world, I hold two things close. First, the starfish story. Second, it’s showing gratitude to others. Sharing with someone how they’ve affected you, thanking someone for some specific action they took or some role they’ve been in your lives, for helping you grow, or showing you grace. So I talked about these things. And I had students read the starfish story to themselves, and then write an email to: (1) a teacher who you want to thank for their work in online teaching, because students often can miss the humanity of teachers (just as teachers can miss the humanity of students), (2) a person who they haven’t been in contact much during quarantine but they appreciate and want to reach out to and tell them so, (3) someone who called them out and pushed them but helped them grow, or (4) someone in math class who made them feel validated and seen. I wanted to end the year by thinking about community and gratitude. Now that it’s over, I wonder if I did the right thing. I debating having more conversations on recent events (in two of my classes on Friday, we talked… you can read about that here…), but it didn’t feel like a way to bring closure to the year. But after my last classes, it just didn’t feel like closure to me. It wasn’t what the kids needed. And I had the even more horrifying and insidious thought. Maybe kids left with a message I never intended to send but also never specifically disavowed: “when the world feels dark and hopeless, the way to fix things is just to simply be kind.” Since I’ve had this thought, it’s been like an anchor weighing me down. I spent a long time trying to figure out what to do for the last class — and I still don’t know what I should have done instead — but this maybe wasn’t it. Or maybe I’m overthinking things. It just felt off. But then again, everything about the last few months in school has felt off.

Thought 2: Last night I had a revelation/insight which I wrote down and shared with some friends: “The anxiety and inability to have heart stop beating so fast. The repeated pain that Twitter and the news brings me. The tears and exhaustion. The past week has done a number on me. I’m going to try to say something openly and imperfectly. I remember saying a few years ago that I could intellectually understand when I heard poc talk about ‘trauma’ but I knew it was only intellectual and I was never going to be a strong ally until I could understand the word more deeply–because it’s a strong word and it wouldn’t be used without a lot behind it. I think for the first time I’m finally getting there. The weight of oppression feels — not intellectually, academically, but viscerally — heavy. Terrifying. All encompassing. Repeated. Relentless. I’m not saying I’m feeling trauma or know what it’s like to be a black poc. A student today in a discussion used the phrase “radical empathy.” I hope this discomfort and general awfulness I feel stays with me — though I hate it — because it’s allowing me to finally experience radical empathy. <don’t mean to center me-just trying to archive this insight.>”

After typing that out, I remembered I had written this sentiment down a year ago as I was thinking about this work and where I was when trying to grow:

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This was in a much longer email to my friend Hema when we were planning last year’s Virtual Conference on Humanizing Mathematics. It took me over a year from recognizing this deficit of mine to actually being able to say I finally get it… I think. And that deficit was truly understanding one word.

Thought 3: I feel like I don’t neatly fit into a category when my school has conversations around race. I’m not white. I’m not black. I’m Indian, and first-generation American. My parents tried to Americanize my sister and me. I don’t find myself aware of daily slings and arrows, not a thousand cuts, not in overt ways, not often. I have a few examples that I can bring up, but not a whole bank of incidents. Because of the privilege I have had through most of my life, I often feel it’s easier to identify as white than a person of color.

I’m also not introspective — probably because it shines a light on things I don’t want to think about. I find it truly challenging to do identity work. I can think deeply about my teaching and my core values and my students, but somehow I’ve gotten away with not doing a lot of deep self-examination in my life. I know I’ve been deeply affected by the way I was viewed by society growing up, and I’m certain racism had a lot to do with it — but I’ve never done the work to think about myself through that lens. I literally only had one friend at school until late junior high. Surprise, surprise: he was also a student of color. I think maybe we were the only two students of color in our grade. I literally don’t think I ever put that together until now. That’s how little introspection I’ve done. I never got invited to birthday parties. But since that wasn’t a thing in my life, I never knew I should be missing them. I was bullied in junior high. I just didn’t know it was bullying because it was so normalized in my life by that time. I honestly wasn’t a sad kid. I just didn’t know any different, so instead of friends, I read books. I know I’m scared to do the identity work to see the ways that structural and insidious forms of racism and homophobia have played a role in my life. I’m a first-generation gay Indian living in America. How could they have not? I’ve clearly had a lot of privilege which has allowed me to get to this point without doing this identity work. But at the end of the day, doing this work, this introspection, forces me to contend with the inner core of who I am and how I see myself. To revise my own definition of myself. It’s easier to just… not.

Thought 4: I recently came off of a zoom meeting with all our school’s faculty (we’re a K-12 school). The anxiety I wrote about above hit me this afternoon. My heart was racing again and I had to lie down in my bed and take a nap. I woke up not feeling any calmer. For the first time, I turned my video off in a zoom meeting. At the start, I heard so many words. Kind words. Sympathetic words. And… I am so tired of words. And so as I heard everything all I could feel was cynical and angry. Give me time and space. Just cancel things. You want to help? Take writing narrative comments off my plate so I can work on healing. I don’t want to know that you know we’re grieving and how things aren’t normal. Just give me something concrete. Just give me something. And at that moment, I realized two things. I wanted them to fix me. I’m broken, and I just wanted someone to wave a magic wand, and I knew they don’t have the power to fix me. I also deeply understood, again at a visceral level, the repeated call that black people have had that sounds like: <<stop talking. stop talking. STOP TALKING. I’m so tired of all the nice sounding talking. So so so tired.>>

I also heard a lot of anger and white people calling for immediate change. Accountability. Demands that we all revise our curriculum this summer. I’m for that also. But I felt such cynicism. Because I just kept thinking “is this performative? is this virtue signaling?” You’re ready and now want everything to change immediately because you feel moved? What about all the people who were at the party demanding change before now? Their voices had been ignored and lost.

And the horrifying thing about my own cynicism is that it’s not like I’ve been one of the people demanding this kind of change for myself or for the school. I’ve been moving along in my personal diversity work, content with doing some work but never getting to the point where I was centering the work. I felt cynical and frustrated listening to people making these statements, even though I have no right to feel cynical and frustrated.

Although I mentioned I often feel white, it was at this moment that I realized all these things I was feeling were what those advocating, crying for anti-racist humanizing curricula, had been feeling for ages. Again, I don’t have any right to those feelings, but in the moment, I had those feelings of frustration and cynicism. I’m grateful — because even though I don’t deserve to have those feelings — I had access to something I hadn’t had before. I was hearing the well-intentioned words that people were saying but from a different perspective, through a different lens, a lens I didn’t have before.

I also was nervous that in the fervor, in the tidal wave of support and sometimes hope that we were going to turn our school into a shining-palace-of-anti-racist-work-so-show-up-teachers-or-get-out, that we were going to do harm. Yesterday I read a tweet that said something to the effect of “if you’ve never talked about black issues in your classroom before, just don’t just come into class tomorrow and bring up George Floyd. Just don’t.” The fear is that if we don’t do this work right, if we rush into things without being strategic and thoughtful, we can do way more harm to students. White saviorism wasn’t really mentioned, but I was thinking about it. Others were probably thinking it. A colleague brought this up, articulated it better than I could have, and I was grateful.

Thought 5: I wanted to reach out to this colleague who brought it up. I wavered and went back and forth on whether to send an email. I wanted to say thank you and I support what you had to say. (I did chime in, in the chat). I wavered because I’m hyperconscious of people who have been doing continual anti-racist work saying they’re tired of words. They’ve been given words over and over. Instead of signaling support, show your support by acting. I wavered but decided to send the email anyway. After writing it, I checked myself. I wanted to show solidarity and that my colleague was heard. But I realized what I wrote boiled down to “thank you” and “I heard you.” I wanted to do better. Without that, the email would have been more about me and virtue signaling. I wanted to demonstrate I really did listen and hear, and I am grateful. So I wrote down what I heard them say, specifically. I still don’t know if I did the right thing by sending the email. But I know I’m now asking the right question: how much of sending the email was about me, and how much about sending the email was about the person I was sending it to? In the weighing, I think I had my answer, and so I sent it.

Thought 6: I am broken. I am tired. I can’t stop thinking about what this all means for me this summer, when all I want to do is heal from this school year. We’re going to be required to do online professional development on remote/blended learning. I’m going to be writing 22 college letters of recommendation. And now I suspect there will be anti-racist readings and professional developments that we’re going to be asked to do, in addition to submitting ways we’re going to alter our curricula. I’m can’t tell you how intellectually and emotionally and physically tired I am. Writing narrative comments on all my students in the next week is a mountain I have to climb but carrying an anchor. But what comes after? That feels impossible.”

Thought 7: I think it’s time to end this post now. Believe it or not, I initially was intending to make this blogpost a list of actions I have started taking, plan on taking, or ideas I have and want to think through before I implement. In other words, concrete ways I’m going to do The Work. I fear that when all these immediate feelings fade away and I’m not being bombarded day and night on twitter and facebook and the news, I may just “opt out.” So I wanted to type some things down to be public in my work and accountable. Maybe I’ll do that post tomorrow. But for today, I’ll have to settle for writing about of my feelings and thoughts, so a month from now, I can try to come back here to a place where I feel destroyed… but ironically because of that, motivated.

At the end…

This is an archive of my day yesterday.

We’re nearing the close of classes with online schooling. It isn’t the end for me, with narrative comments and college letters of recommendations looming. But in terms of meeting with kids, there’s not much left, and I had my plans in order — at least well outlined in my head for some, ready to execute for others. But I felt anxious. On Thursday afternoon, I finished my last meeting, had finished my prep work for Friday, and was in a good place. But my heart was beating, racing, and I didn’t know what it was. That anxiety usually happens to me when I have something impending looming (like a stack of tests I need to grade, or a challenging email I need to write or conversation I need to have) and I am stressed about it. But I had none of that. But my heart was racing. I was a little short of breath.

I tried to relax, but I couldn’t focus on reading. My mind quickly wandered. TV worked.

A few different things had been thrown at me in the past few days. I wrote to teaching friends, as I processed and as a way to vent: “The problem about keeping boundaries in teaching is that boundaries I draw to protect myself often involves saying no to a kid who I can prop up by saying yes. I recently wanted to say no to four requests — some entitled, some respectful. I don’t have much left of me to give. I couldn’t. When it comes to me or them, I pick them. Again and again. As I slowly burn out. Teaching is hard. It will never not be hard. Because we give so much of our selves, emotionally, mentally. I don’t always want to do that. But I (honestly) don’t know how not to.” The same thing happened two weeks ago with college recommendations. I teach two junior classes. All but three juniors asked me for a recommendation. I felt utterly deflated. And honored. If someone could be in my head as I try to craft these, you’d understand the deflation. They not only take me forever, but they are emotionally draining because I want to capture my kids with integrity. I told my classes I had to limit them and I hoped they understood. I tried to be transparent and vulnerable with them about what doing this meant for me, and why I had to limit them, because they deserved that. But when I read their reflections, I couldn’t say no. I thought about the Giving Tree. I’m not that self-destructive. But for a moment I martyred myself.

The next day I woke up at 6:40. to an alarm. I quickly showered so I could check my kiddos nightly work. With online learning, I go through what my kids worked on each day before class to give individual feedback, but also know what we need to talk about together as a class. Although my heart wasn’t racing, I still felt… tight. Wound up. I was off. My kids were doing really strong thinking with hyperbolas and that made me happy. Some kids really struggled applying the quadratic formula for the first time and that made me wonder about my approach this year. I tried to address that by breaking it into two pieces and having them work on those separately — but maybe there was a better way? I woke up earlier than normal because I had an optional meeting I wanted to attend at 8:15am about the finances of our school. Our new head of school wanted to share where we were at — trying to be transparent about the decisions being made by the board and the school. I appreciated that, and her openness to talk about it.

Yesterday evening all advisors received an email saying that instead of meeting in our advisories, we would be meeting as an entire high school to give some time to discuss the recent tragedies involving Ahmaud Arbery  Christian Cooper  George Floyd  Nina Pop. I hadn’t heard of Nina Pop. Her name was misspelled in the email we received. Both of those facts together speaks to something about me, and something bigger than me. I entered the zoom meeting and scroll through 16 pages of faces. More, maybe. Tired, yawning. And then the meeting begins. It was planned to be a 20 minute meeting. It starts with Feel. Reflect. Act. Grow. Repeat. 

Two hours later, we’ve heard the voices of students and faculty. Anger and frustration about teachers not bringing up these current events, and in general black/POC issues, in classes. About the ability to check in and opt out depending on skin color, and whether things should be optional. Calls for people to go to march at Foley Square at 4pm. Kids requested breakout rooms early on, and I was assigned Breakout Room 22 with eight students, seven of whom I had never met, nervous about facilitating a conversation about race that I hadn’t prepared with strangers. My heart sang as my colleague and friend Monika was also randomly placed in the room. We launched the conversation with a focused question, but that didn’t work, so we went broader to “how are you feeling?” Kids started talking. At times, they were discussing with themselves. At times they were asking the opinions of the two adults in the room. One asked our thoughts on ACAB. I had never heard of that and with slight trepidation (should I know?) asked. “All cops are bad” two students responded. It’s a thing, they said. Reductionist. Monika was more eloquent and nuanced in her response. We talked about multiple narratives (it’s complex). The reasons for violence in riots/protests, and if it pushes things forwards or not (it depends). There are no pat answers and easy fixes, everything is layered, structural, cultural, historical. This is one slice, one slide, of a much longer movie. We end with each saying a word, phrase, or something we’re thinking about. I say “over-intellectualizing.” We move back to the main discussion room and I hear our high school principal talking. We’ve gone past first period. I’ve already emailed my kids in my second period class saying we’re not having class and to stay in the meeting. I look and see over 300 students and faculty are in the room. Faculty and students speak. I don’t know how they avoid talking over one another with 300 people, but somehow they do. Outrage. Questioning of the school curricula. Frustration with the statistics around black lives. Calling out hypocrisy. Asking questions about whose responsibility it is to educate kids on race — some say kids are responsible for themselves, some say its the school. Discussions about cultural appropriation. Sharing of feelings. The importance of being called out and of calling out. About the difference between being a liberal and an anti-racist, and being a liberal doesn’t make you an anti-racist. About the conflation between being liberal and being “good.” About being called out and accepting that as a gift. I think to myself that some of these kids must have read White Fragility. A concurrent conversation happens in the chat, twisting and turning with the vocal conversation. I keep silent. I like spaces for kids to talk. I don’t often know what they think about. This isn’t a “conversation” in the sense that we’re talking to find a goal. Right now, we’re talking because people need to talk because they have a lot bubbling up inside. We all have a lot of bubbling up inside. That… that was when I knew why my heart racing yesterday.

Our high school principal ended the meeting before our third period class. It was two hours from when we started. I open my zoom meeting for our Algebra 2 class. Obviously we aren’t doing math today. As the large meeting was winding down, I brainstormed what would make sense and gave my kids some options. They chose to continue the conversation. Unlike in the first breakout room I was in, this conversation was mainly had by kids, where I tried to give them space to speak and I listened. I took notes. They were in the same grade and class and knew each other. They didn’t turn to me for answers. I don’t have answers. I interjected sometimes, and then stepped away. When class was officially over, they were still talking, so I told them they could leave if they wanted or continue talking. Most wanted to stay and so we held court for 20 more minutes.

I was late to a book club I had scheduled during lunch. I emailed the student leader to start it up without me. We were reading Hannah Fry’s Hello World: Being Human in the Age of Algorithms. We were two adults and five kids. The conversation at the beginning stayed close to the book — which also had some themes on race — but very occasionally would go back to something from the discussions the school was having. It was fun and a lighthearted discussion with smiles. We had finished the book, and the scheduled time we had together was done, but we all continued chatting, and the conversation turned into questioning about what teachers think about lots of things. It morphed into an unguarded and vibrant conversation, one that I love having with my friends when they ask a lot of questions showing interest in what I do. I felt close to these kids, and appreciative for them. I don’t know much about what students think about, and looking back I wished I had asked students questions too. It was 12:50 and I had a class at 1:15 so we bid each other farewell. My eyes were literally vibrating at this point. I had been emotionally run raw and I had been staring at my computer screen since 8:15. All I had ingested was a cup of coffee. I heated up some frozen leftover macaroni and cheese and wolfed it down. I emailed a student I was supposed to meet at 2:20pm — an important meeting so I hated doing this — to reschedule.

I entered my last class. At this point I had very little more to give. I had very little left I could discuss. I told kids I had been on zoom all day, and I had been engaging all day, and I needed a 10 minute break to do math and recenter myself. So I did math with them for 10 minutes. And then instead of giving my kids a choice like I did for an earlier class, I made the decision that we’d read something together. I pulled up Francis Su’s keynote address to the Mathematical Association of America — his farewell address as president. A section of this address is on “Justice.” It is centered around this quotation by Simone Weil:

“Justice. To be ever ready to admit that another person is something quite different from what we read when he is there (or when we think about him). Or rather, to read in him that he is certainly something different, perhaps something completely different from what we read in him.  Every being cries out silently to be read differently.”

We read it popcorn style. Someone would read a paragraph or two and then stop. Someone else would jump in. I hadn’t read this in a while. I have the book based on this speech that Prof. Su wrote afterward, after his speech went viral in the math community, trapped at school, unread, sitting on my desk. As kids started reading, I remember hoping it would speak to the events of the day at school. I hadn’t time to review it to see if it did. Everything I was doing today was on the fly. The section did, at least to me. It was longer than I remembered. It took a while to get through, or maybe I just lost sense of time because I was exhausted. When I read, every so often, I forgot words or mixed them around. My eyes were jumping around the text, unable to focus. At the end, I told kids if they wanted space to talk about the events of the day or the piece we just read, we could have it. Or we could call it an end. I consciously tried not to make it sound, when giving the options, like I really wanted to end, so they knew they truly had a choice. We called it an end, and I was grateful. I was empty.

I posted the nightly work for my classes from today. I shared with them some articles, including  this piece on Edray Goins, which I remember having my classes read last year and gave us a lot to talk about. And this keynote speech by Marian Dingle about “centering.” I apologized in my google classroom post for not bringing in more conversations about race, gender, and other -isms into class, and said it wasn’t just the jobs of the history and English teachers (something kids had brought up in the conversations).

I had 45 minutes to lie on my couch and close my eyes. I thought about how much the world has changed since I was in high school. I thought about how much the kids we teach are aware and know and care… and I wondered if the non-POC kids and faculty really cared in their hearts or if they cared because they were supposed to care. I wondered if I cared in my heart or if I cared because I am supposed to care. I felt the weight of guilt — trying to parse what I’ve done to help my kids know they’re “being read” — especially the POC, especially the girls and non-binary kids. I found myself thinking of when I had kids read about a trans mathematician and a queer mathematician and we talked about that in class. Or when I had kids read about a black mathematician and we discussed. And then I thought: wait, maybe we did that last year and not this year? About the ways I tried to encourage the bright girls in my precalculus classes who were outnumbered by boys — to know they were valuable and their ideas were valuable. But maybe it was too subtle, or not enough, or nothing at all. About my intention to have kids listen to a podcast of latinx mathematicians and how I never made space and time for it. As I thought about these things to assuage and exacerbate my own guilt, I clung to certain things as I was hoping to justify I was “good.” Though I’m not white, this was definitely white fragility. And I thought about if and how today would change anything. For me. For our community. Because I’ve been around, and these sorts of days have happened before. And we as a school community get all riled up and passionate. And then a week later, we’re back to the status quo. Or have we? Have small changes been made that add up over time? I remember thinking, maybe one day the fire will catch and maybe we’ll have something sustainable. Days like these are the sparks. You can’t have fire without a spark.

At 3pm, we had a department meeting. It felt incongruous to the day because we had no mention of the happenings of the day. We toasted to colleagues (friends) who were moving on — one to another school, one to blissful retirement. And we did some work together. When the meeting ended at 3:45pm, I moved to my couch and laid down.

A little over an hour later, I hopped onto a zoom call with three colleagues and friends. We have this tradition every Friday. It’s nice to have this normalcy and camaraderie. These are people I need to vent to, but also laugh with, because otherwise I would feel very alone as a teacher in isolation. We talked about the day, processed, a few tears were shed early on, and laughed. We lasted two hours. And we imagined ourselves on a beach adventure together, in person, and committed to making that happen when we could. It dawns on me that this trip is literally the only actual thing I have to look forward to in my life right now. A hypothetical trip to the beach.

This is an archive of my day yesterday.

Advisory Games/Activities

A friend asked me what sorts of games might be playable over zoom.

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I was going to write an email with some games I’ve played with my advisory, but I figured I might as well just blog about it in case it’s useful to anyone else.

Some important context:  I have an awesome advisory group of 8 kids. They’re juniors now but I’ve been with them since they were ninth graders. So we’re comfortable with each other and they know a bit about each other. In quarantine, we always start our distance learning school day with a 15 minute zoom advisory. Honestly, it almost always is one of my favorite parts of the day because … well, if you met my kiddos you’d know why. Sometimes I lead advisory with an activity, sometimes I assign them to come to advisory with a plan. Here are some things we’ve done and things we may do in the future. Some of them are games. The ones with * are things they’ve suggested but we haven’t done or I’ve been thinking about doing it but we haven’t done.

  1. We’ve done a pet share. Kids brought their pets (or some sort of stuffed animal or other object) and introduced their pet to the advisory.
  2. We’ve shown our workspaces during digital learning to each other.
  3. We shared things that were difficult about online learning and problem solved together.
  4. I posted a list of 8 random items (a balloon, a jar of peanut butter, the playing card the 7 of hearts, etc.). I assigned point values to the items based on their obscurity.  Students took a photo of the list and had 10 minutes to gather everything and come back at a certain time. Then they tallied up their points, and we declared a winner. (I hope a student does this when leading their advisory… I wanted to play!)
  5. Played Kahoot (with pre-made Kahoots)
  6. Each of us bring baby pictures of ourselves and share them with each other
  7. We break up into two teams on zoom. We pick a word (e.g. “love” or “red”). Each team has 30 seconds to come up with a song lyric or title with that word in it. We go back and forth until a team can’t come up with another one.
  8. The same thing as the previous one, but we changed it from song lyrics to song titles, movie titles, book titles, magazine titles, etc. (media titles).
  9. Played Mad Gab. I projected a card and kids tried to figure out what the card was saying. We played this fine with kids just sounding things out on their own, but I was thinking they could mute themselves and then they could unmute themselves when they think they got it.
  10. Played drawphone. This is like telephone, but with words/phrases and drawings. Initially we started out with pre-written words from telestrations/pictionary, but recently we’ve done it where we ourselves write the initial words/phrases. This is one of my favorites! Caution: there is a cards against humanity version that you can play on this site — so you should always initiate the game, and ask kids the join (and not let kids create the game).
  11. Played pictionary/skribbl.io. Initially, we did this using the pictionary words built-in, but then one of the students (when they were leading the advisory) came with words associated with our school (people, places, terms, etc.). That was so sweet.
  12. Picked a Sporcle quiz, I projected it, and we collectively tried to see how good we could do working together. I typed what they told me to type. We did this today and we did a Pixar quiz (3 minutes) and a Harry Potter quiz (8 minutes). Also, one of my favorites!
  13. Projected some “would you rather” questions I found online and had kids discuss them.
  14. Since kids are doing a lot of cooking, I told them to take photos of their cooking process and their final foodstuffs… and then we shared those photos.
  15. *One person gets put in a breakout room alone. The rest of the people get assigned some weird feature (e.g. they can only speak in questions, all their sentences have to start with the letter T, they have to cough each time they speak, they have to mention a color each time they speak, they have to address each person they speak to by their name, they have to do a strange hand movement at the camera each time they speak, etc.). Then the person has to come out of the breakout room and talk with everyone and try to determine what the person’s quirk is.
  16. *Play Spyfall.
  17. *Play Evil Hangman.
  18. Update: We did this! (But I just found it online too!) We can do it with the annotation/whiteboard feature on zoom. Me (not playing) secretly emails everyone but one person the same object (e.g. A teddy bear) and one person gets an email saying they don’t get told what the object is. We go around and each person draws one line/shape on the board. A line is considered anything you draw without picking your pencil/pen up. The group has to decide — after two rounds of drawing — who wasn’t told the object being drawn. For example, let’s say everyone but one person was told the object was “teddy bear.” So the person who wasn’t told “teddy bear” has to be careful about what they draw… they’re always taking a risk when drawing… but the rest of the people can’t be too obvious either, because if it’s clear early on it’s a creature of some sort, the person who wasn’t told could draw an eyeball or something. If the group correctly guesses what the item is (watch the video to determine how the group guesses) then the fake person can still win if they can identify what the object was supposed to be.
    (This is like Spyfall, but with drawings!)
    Update: We did this today! It’s based on a game called A Fake Artist Goes to New York [video tutorial]
  19. Update: My advisory played scattegories online today! I admit to not really understanding how it worked at the beginning, but by the end I think I had the hang of it.
  20. Update: Today one of my advisees created a playlist on spotify. She shared her desktop with audio but had a random chrome page shared so we couldn’t see her spotify. She then controlled her spotify from her phone and played the music. The first person to identify the song won a point. I kinda loved this even though I definitely didn’t know many songs… and then someone else suggested next time we all rotate who shares a song so we can get lots of different genres and get to know other peoples’ musical tastes also. Without screensharing with audio, I tried just playing a song from my computer to see if kids could hear it, and they said they could hear it fine… so it seems like that would work pretty seamlessly.
  21. Update from an advisor colleague at school (Carla K): Go visit a museum tour virtually together!
  22. Update: We played “Charades” today! We started by each person just acting out whatever they wanted to and everyone else trying to guess. And then when a student had trouble coming up with an idea, I private messaged them on zoom to give them 3 choices to choose from. And then we played that way! No teams. Nothing serious. Super fun!

If anyone has anything they’ve done that’s fun and low-key like this with their kiddos in advisory or class, please throw down ideas in the comments! I’d love to add to our list!

Update: John Golden shared, in a recent culling of neat math/teaching things, this google doc which not only has games but also digital escape rooms!

Update: My friend Mattie Baker sent me this image which has lots of boardgames online! It includes some that I’ve listed above, and a lot more!
online board games

Distance Learning: Sorting Through It All

Technically I’m still on Spring Break, but this all ends next week, when we go back to school remotely. I’m one of the lucky few who didn’t have to get thrown into the fire immediately, so I’m using this blogpost as a way for me to sort through what I’ve done and what my take-aways are. I’ll be updating this as new things come my way, so I can keep track of everything useful in one spot.

My dear friend @rdkpickle

So my friend @rdkpickle had to start distance teaching already. She’s kinda amazing in all the ways, and so on twitter she shared out how she was doing her lessons — and noted that they’ve been going well. They are low-tech in that they use Zoom and Google Docs, and use a Google Doc as an anchor for the lesson. I love that the doc allows kids who have to miss the lesson for whatever reason (emotional/anxiety issues, having to take care of a sibling, etc.) have a way to keep up.

Before sharing it, I want to say: seeing what she did was the very first thing that I saw that made me feel like: “okay, I can do this. It’s doable.” BLESS. When talking briefly with her online, she was saying right now she couldn’t be all investigatory in the same way she was in class, almost like she was ashamed. BUT very little of we’re doing is going to be like what we do in class. The ballgame has changed (from basketball to some other sportsball!). Right now, for me, the question is can I give space and structure and community to kids where they feel they can learn a few things. And @rdkpickle’s low-tech approach allows for that!

Here’s a PDF her googledoc, which she said I could share. (And here’s the google doc.)

 

Zoom

Mike Flynn (helped by Sarah Bent) gave two wonderful webinars on distance learning that he has put online — March 11th and March 17th. (If you only have time for one, I’d watch the second one, but both are great.) They were some of the first things that made me realize distance learning was possible, by showing how to do it through his webinar. (Unlike, say, dry powerpoint lectures on teaching active learning strategies. Ahem. We’ve all been there. I just think over and over, “Physician! Heal thyself!”) My takeaways were both about distance learning and about zoom, so I’ll list them here. Fundamentally, though, the best way to learn zoom is to actually just get a few friends and all try it out together (each of y’all practicing being the leaders/hosts of the meeting).

  • If you can, start the zoom meeting 10-15 minutes early and let kids know you’ll be there. You can just have informal chats like you do before a normal class, and you can use that as an easy way to start building community.
  • You can record your sessions, but if you do that, don’t start recording during that informal chat time. (Right now, since Zoom is overloaded, it’s taking them a long time to get the recorded sessions on their website, FYI. But you can have zoom do a “local recording” on your laptop… so I was thinking if the file were small enough, I could just upload that to a google drive folder my kids could access.) Note that the chat box doesn’t show up on the recording.
  • Talk with kids explicitly about the weirdness of talking on Zoom. There are going to be awkward pauses because we can’t use facial cues and body movement to figure out if we’re going to talk or not (we’re all sort of trained to sort of check before we talk so we don’t start at the same time as someone else). So name that, and say that awkwardness is normal in zoom. You should also mention (and give) lots of wait time — just like we should be doing in our regular teaching.
  • It’s okay if you’re having kids use chat to stop every so often and take a few minutes (in silence) to read over the chat so you can respond to what you’re seeing.
  • The chat can be the “lightest lift” for interactivity, but it’s effective! One tip I got on twitter is that you can ask everyone to write a response to a question, but not press enter until you give the command. Then you’ll get a quick flood of responses that you can go through, and students can also read.
  • You can also set the zoom meeting to have the chat be private – so students are talking to you but not each other… then as you see the responses, you can say “Nice thinking, Jake!” or “If you’re thinking about a parabola instead of an exponential function, you’re going in the wrong direction!” This came directly from Michael Pershan’s experience teaching online this past week:
    ETfBLfsWkAM3His
  • If you have pre-determined questions you want to ask at a particular time during the lesson, have them written in a google doc/notepad, so when you want to ask it, you can just copy/paste them in.
  • Have everyone use their own regular names in zoom (and not emails or userhandles) to make life easier for you.
  • There is a way to include “polls” in your zoom meetings, but I couldn’t figure that feature out when trying it out!
  • You can divide your class into groups (either randomly or pre-determined) and send them to breakout rooms. You can visit any of those rooms and join in the conversations. Each breakout room is given a number when students join. You can have one person in each group (e.g. the person whose last name comes first alphabetically) to create a Google Doc in a Google Drive Folder you share with them in the chat window… And title it “Group 5, March 25, 2020.” Then all participants can write answers in their google doc and you have access to all of them in an organized way.
    • When students are first put into a breakout room, if they’re new to working with each other, start with a non-mathy but quick ice-breaker to get everyone talking (e.g. what’s your favorite pizza topping?) and build a tiiiiny bit of community before diving in.
    • SUPER COOL DISCOVERY: When I did this in Mike Flynn’s webinar, one person in my breakout room showed me a ridiculously cool feature. In any google doc, you can go to INSERT > IMAGE > CAMERA
      julie4
      And then you just take a picture of your work using the webcam, and it automatically inserts the picture in the google doc!
      julie3
      Bam!
  • Don’t go crazy with the new technology. There are so many apps and websites. Limit yourself to just a few, like two, for your own sanity and your students’ sanity. Keep it simple and easy — don’t go down the rabbit hole of looking for “the perfect way to do x, y, or z.” Be okay with the tradeoff of having “good enough.”
  • When designing online learning, start with the question “how do we want our students to learn?” Then choose your technology based on that.
  • Screensharing is awesome (so you can set up a google slideshow, and in zoom you can screenshare that slideshow to the kiddos… And you can show kids how to annotate so individuals or the whole class and write/type/draw on a screen you’re sharing (and you can save that).

 

Desmos Activity Builder 

Julie Reulbach led a webinar on using desmos for assessments, but basically she outlined all the ways we could create activity builders to actually teach content also, and bring students along with us as they navigate the pages, and we talk through what they’re doing. Her resource page is clipped below so you can see what’s there…

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But importantly, her page includes links to various activity builders where you can simply copy and paste! Here’s how you copy and paste screens from better activities that your own into your own! They can even have computational layer in them!

Some key tips for creating Activity Builders (but not necessarily for assessments in particular):

  • Steal steal steal screens from other activity builder assessments if you’re doing anything fancy (e.g. self checking, anything with computation layer), because there’s no need to reinvent the wheel right now. Julie has curated a whole list of activities that she takes screens from! And desmos has curated a bunch of starter screens(e.g. “graph how you’re feeling today?”) that you can take!
  • DESMOS NOW ALLOWS FEEDBACK – so you can write a note to individual students.
    https://www.youtube.com/watch?v=P-ebHOily6k&feature=emb_logo
  • Importantly, let’s say kids are doing a desmos activity or assessment, and you want them to show their work that they’ve done on paper. All you need to do is create a screen which has a blank graph, and instruct kids to insert an image (see sidebar instructions in the image below) and they can actually INSERT a picture of their work! The workflow is a little clunky because they have to take a picture on their phones and then email/airdrop it to their laptops, and then select that file. But it took me only like 20 seconds after I had done it once.
    julie2

Some key tips for using Desmos for assessments:

  • Have kids log in with their name, but “last, first.” That way when you grade their online assessment, you can sort the responses by their username, and that will match your gradebook.
  • Have a fun introductory “hi there!” screen
  • Have students fill out an honor code/statement screen first if doing a formative assessment
  • After you finish the activity, have two screens at the end. First, a feedback screen so you can find out how they felt it went. Second, a screen asking them if they have any questions or anything they want you to know.
  • If a formal assessment, you should PAUSE the activity at the end — so kids can’t go back and change their answers or share the class code with other kids

 

Michael Pershan’s blogpost

My friend Michael Pershan has been in the thick of online teaching. He wrote a detailed blogpost about what he’s discovered thus far. I highly recommend reading it! Big takeaways:

  • His school is using Google Classroom (like ours does), so he’s using that to create a system of organization for the kids, with instructions given day-by-day (within a unit):
    He noted: “The most important thing, though, is that each learning activity becomes its own “assignment.” During week 1 I was creating large documents that students were working on over multiple days. This was good in one sense, because I had to post only one thing. But it became very difficult to monitor the progress of kids through the assignment at all. And then it became tricky to modify the plan in the middle of the week by adding on other bits of classwork.”
  • He’s using google classroom to teach kids how to upload their written work. (Note: my kids always submit PDFs of their work on google classroom, so they’re very familiar with this!)
  • To give feedback on google classroom: “Google lets you comment on the work itself via highlighting and commenting, but I’ve found it more useful to give a quick written comment that appears under the assignment itself.”

 

Twitter

Lots of great things being shared on twitter. It was so overwhelming that I stopped looking at twitter for a while, but I did save a few things:

 

What Wasn’t That Useful For Me – But Here are the Nuggets I’ve Taken Away From These

What I have below doesn’t mean these aren’t good for others. It just means that for me, I like to jump in and these things didn’t quite pan out fully.

Alice Keeler had a webinar (“Oh Crap, I’m Teaching Math Online Now“) that wasn’t crazy useful for me because it was a brief overview of many things I already knew about. It was just super tech happy (look at Pear Deck! Look at Geogebra! Look at Desmos! Look at …) and didn’t give me the focus or vision I’m searching for.

Global Online Academy (GOA)’s 1 week course on Designing for Online Learning. Since this was designed to be “big picture” (so it can accommodate people from many schools and teachers of all stripes and many disciplines), I had trouble getting specifics that I wanted to latch onto. Here’s what I did get:

  • They recommended Loom for laptop screen recording, if you were going to be making videos from your laptop. It seemed pretty seamless and easy to use, based on this short video tutorial:
    https://www.youtube.com/watch?v=OvRVJ46ffoQ
  • The basics of good teaching are still important — clarity and being organized is paramount. Specifically for online learning, they highly recommend:
    • building routines early (just like with regular teaching!)
    • share the “learning goals” at the start of each lesson explicitly
    • don’t get over-excited and share too much… curate what you share and make it super easy to follow
      • using a lot of whitespace and images
      • don’t include anything that isn’t super important — focus on key ideas
      • not using too many fonts
      • everything you share with your students should be “crisp” and “clean” (not “busy”)
  • Be present for students. Create or adopt an online persona. Don’t leave them hanging, but show them continual engagement so they know you’re with them on this journey.

Part II of Machines: Helping Us Understand Inverses

Here is Part I. Read that first! Also, I’m trying to write and post this quickly, so sorry if it is incoherent, monotonous, etc.

Okay, so now kids understand machines have inputs and outputs, and they understand that the “rule” can take different forms: words, equations, tables, and graphs. Wonderful.

Machines to think about Functions vs. Non-Functions

So recall that our definition of machines was:

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So I had kids try to see what might go wrong with these machines…

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From our conversations over these problems, students were able to see which machines were “problematic.” At this point, I told them machines that worked were called “functions” and machines that didn’t work were called “non-functions.” Conversations we had:

  • We talked through what made something a function (every allowable input had a single output) and which made something a non-function (there was one allowable input that has multiple outputs).
  • I had kids look at the graphs and come up with a quick way to “see” if a graph was a function or not… so from this, they came up with the vertical line test on their own.

We did a lot of practice with this idea. Kids were asked to look at a bunch of representations and decide if they were functions or not. And if they weren’t functions, they had to provide a concrete example showing where they failed (e.g. for (g) below, an input of 2 gives two ouptut of 2 and 4.)

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By the end of this, I was very confident kids understood the idea of functions and non-functions.

Combining Basic Machines

Okay. Here’s where we start to get more abstract. I start telling kids that for now, we’re going to focus on four basic machines (machines with add, subtract, multiply, and divide)… but because I’m lazy and can’t make cartoon machines all the time, I’m going to come up with a simplified notation for them…image3.png

You, dear reader, might wonder why I’m using “blah” in these machines. That’s because it is helpful when we start combining machines:

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Yes! It’s like a conveyor belt. Each machine takes in an input and spits out an output… that then becomes the input of the new machine… So they could start figuring out questions like these, which made me happy!

image5.png

Now #17 was really tough for kids. But I let them struggle before guiding them. From this one problem, students could start seeing how equations and machines were related. By the end of our conversation, kids knew the line was y=2x+4. So if they have an input of x, they multiply that input by 2, and then they add four to the result. Which means the machine would be:

____[blah*2]____[blah+4]_____

And so students started plugging in various values for the initial input (x value) and saw they got the final output (y value). Then we substituted x into the machine… and got 2x for the middle blank, and then 2x+4 for the final blank! Seeing that really helped kids drive home the connection.

Some kids got the second way to write these machines by trial and error. But I was hoping they’d rewrite y=2(x+2). And then think if they have an input of x, they first add two to it, and then they multiply that result by 2. Which means the machine would be:

____[blah+2]____[blah*2]_____

Creating Machines from Equations and Vice Versa

We then became comfortable going from a machine representation to an equation representation, and vice versa.

If I gave students: y=2x^3-4, they would say: we cube the input, multiply it by two, subtract 4. So the machine would be _____[blah^3]____[blah*2]____[blah-4]_____.

Or if I gave them: y=-\sqrt{-x+3}, they would say: we take the input and multiply it by -1, we add three, we take the square root of that, and then we multiply by -1 again. So the machine would be _____[blah*-1]____[blah+3]____[sqrt(blah)]____[blah*-1]_____.

And also in reverse, students could be given a machine, and easily come up with the equation by substituting in x for the input, and come up with the output. So:

___[blah-3]___[blah^2]___[blah-4]___

would look like: x, x-3, (x-3)^2, (x-3)^2-4. So the equation represented by this machine is y=(x-3)^2-4.

Basically, students are seeing how the basic equations are built up and broken down. What’s nice about this is order of operations starts to really get emphasized and naturalized.

Creeping Up To Inverses

At this point, kids are comfortable with combined machines. And so I throw them a backwards question, something they’re used to (since they’re my favorite type of question to give kids). First I start off concrete…

image6.png

… where they were doing a lot of thinking about inverse operations. But then I had an activity where students were trying to create machines that would “undo” another machine. By the end of this activity, students were starting to create their own inverses. They could do problems like this:

I give you this machine which I will call machine M: ___[blah-5]___[blah*7]____,
You need to tell me what machines I could append to the end of machine M would make any input be the same as the final output.

So kids eventually saw the appended machine would be: ____[blah*1/7]___[blah+5]_____

So the big machine would be: ___[blah-5]___[blah*7]____[blah*1/7]___[blah+5]_____

And any input would also be the output (e.g. if we put in 1, we’d get 1 –> -4 –> -28 –> -4 –> 1.)

We called the machine created the inverse machine… and we named it machine M^{-1}.

So the inverse machine of ___[blah-5]___[blah*7]____ was ____[blah*1/7]___[blah+5]_____. Kids saw we “read” the original machine backwards, and did the inverse operations.

From this, we started going into inverses of lines:

image7.pngEventually, we got to the point where kids would be given y=x^3+4. To find the inverse, they would create the machine for this: ___[blah^3]___[blah+4]___. And then they’d create the inverse machine: ____[blah-4]____[cuberoot(blah)]___. And then they’d find the equation that this machine represents by substituting in x: y=\sqrt[3]{x-4}.

So questions like these didn’t really faze them:

image13.png

Inverses with more representations…

From here, I started having kids come up with inverses of tables. I reminded them that if we combine a machine with its inverse, whatever we input into the big machine should be the same output… So let’s see what happens…image8.png

And this was lovely… We’re combining two tables… and our goal is to create a large machine that when an input goes through both of machines, the output would be the same!

So look at the two tables above as rules. And we’re going to combine both tables to make a big machine. So kids saw from this that if we put an input of -3 into M, we’d get 5 as an output… and when we feed 5 into M^{-1}, and we must have -3 as an output (since the output after going through both machines have to be the same as the initial input). So the inverse table is going to look like the original table, but with inputs and outputs reversed.

Why is this so beautiful? Because from this, kids saw that for inverses, the domain and range swap. And they also saw that to create an inverse, you simply have to switch the x-values and the y-values with each other. You get all of this for free!

And then I gave them graphs, and told them (with no instructions) to come up with the inverses… But since they had done tables, and see how the tables just swapped the inputs and outputs to get the inverse, they had no trouble drawing the inverse graphs.

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And it’s lovely. Because they figure this out all naturally. I didn’t have to tell them anything but kids were accurately drawing inverse graphs. Putting the graphs right after them doing inverse tables was genius! And some kids came up with the fact that inverse graphs were reflections over the line y=x themselves!

Of course, sometimes inverses exist but aren’t functions… So I threw everyone some curveballs…

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And they saw how they could create the inverse… they could fill in the table or graph… But they saw why the inverse was “problematic” (a.k.a. not a function).

So now kids were thinking: okay, what’s the inverse? Is the inverse a function or not?

I drove this home with lots of questioning…  We had previously looked at these questions and decided if these each were functions or not. But now kids were able to decide if their inverses were functions or not.

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They immediately were looking to see if any outputs had multiple inputs associated with it. And they came up with the horizontal line test on their own. It was glorious.

Going The Very Last Step with Inverses

From all of this, kids learned so much. They saw how to graph inverses. They saw the inverse graph is a reflection over the line y=x. And then we drove home the idea that the inverse graph is the same as the original graph, but with every x-coordinate swapped with every y-coordinate. To polish everything off, we saw the equation for the inverse graph can easily be found by swapping the x variable with the y variable.

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So the inverse of y=x^2 was x=y^2.

So finally, my kids could answer questions like:

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Sorry this was so long and scattered. But stay tuned. My favorite thing is coming up… whenever I get a chance to write the next post!

Part I of Machines: A Useful Algebra 2 Representation

So last year, I started teaching Algebra 2 again after years of not teaching it. I worked with a colleague on the curriculum, and one thing we really wanted to make sure kids were continually exposed to were various representations of functions and relations. Of course this includes equations, graphs, tables, and words. 

But in addition to these representations, I was inspired to include a fifth representation. It has a few drawbacks, but I can’t even express to you how many positive aspects it has going for it. It is the “machine.” I remember seeing images of these machines in middle school textbooks, and they really emphasize the idea of an input, output, and rule. Here’s one I randomly found online:

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In this blogpost, I’m going to share how I introduced this representation, and how I subsumed the others in it. In future blogposts, I’ll share all the ways I’ve exploited this representation. It’s pretty magical, I have to say. So stay with me…

At the very start of the course, I introduced this machine representation also. Just not as fancy and cartooney.

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I thought a lot about whether I wanted the machine to allow multiple inputs or allow multiple outputs. In my first iteration of drafting these materials, I did that, but then I backtracked. Things started to get pretty complex with an expanded definition for a machine, and I wanted to start the course simply. And, of course, I really wanted to emphasize the idea of a function and a non-function. So I started with the definition above. And started with things like this…

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Notice these are “non-mathy” examples of machines. They eased kids into the idea, without throwing them into the deep end.

What was nice is that we got to understand and interrogate the idea of domain and range from this… where I described the domain as the “the bucket of all possible items that can be put into the machine and give you an output” and the range as “the bucket of all possible items that comes out of the machine.”

image4.pngSo for the sandwich one, we know the range is {yes, no}. And the domain might be {all foods} or {every physical thing in the universe}. We talked about the ambiguity and how for these non-math ones, there might be multiple sets of domains that make sense. But then for the math-y ones, we saw there was only one possible domain and range.

In fact, to really drive home the idea of inputs, outputs, domain, and range, I created an activity. I paired up the kids and one kid was the machine, and one kid was the guesser.

The machine got a card like this, with the rule:

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The guesser got a card with the domain and range:

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And the guesser would give words to the machine, and get a result. And their goal: figure out the rule. Then I would switch the machine and guesser, and give a new set of cards. It was crazy fun! I did it a long time ago, but I distinctly remember kids wanting to play longer than the time I had allotted. (If you want the cards I made, here’s a PDF I created Domain and Range Game.)

Next I showed how the “rule” in the machine could take a number of different forms — tables, words, equations, and graphs — and this is how I introduced the various representations. Kids were given these and asked to fill in the missing information…

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… and then they were asked to find the domain and range for these same rules…

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To drive home the various representations, I gave kids questions like these, where kids were given one representation and were asked to come up with the equivalent other representations.

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So this was the gentle introduction my kids had to machines. I’ll explain where we went from there in future posts… and I promise you it’s going to be good… I’m really proud of it!