A Teacher’s Imposter Syndrome

I’ve recently begun to realize that I suffer from a bad case of imposter syndrome. I don’t know if it’s because my first year teaching was such a struggle, or because my first principal was so critical of me, or if it was because my last supervisor kept telling me they thought I needed to grow in the same area year after year, but I find it hard to believe in my skills as an educator sometimes.

When I’m talking with new or inexperienced teachers, or especially with pre-service teachers (i.e. my student teachers), I do feel a sense of expertise. In those situations, I know I have something to share – even if it’s just a message of hope “it gets better.” I can legitimately tell those folks with less experience that I remember being in their shoes, and I remember struggling so hard that I was in tears, and literally, if not for Math for America, I might not have even made it through year 1 of my teaching career, let alone be in the midst of year 8 now. When I’m surrounded by newbies, I am willing to acknowledge my experience as something that makes me somewhat of an expert – by comparison. (notice how I had to qualify that statement!)

But when I’m surrounded by excellent teachers who’ve been teaching as long, if not longer, than I have been a teacher, I begin to question and doubt my ability to contribute to them. It’s ironic, actually, because as my partner pointed out to me last night: I truly believe all students, of all ability levels, have something to contribute to classroom conversations and communities. And I acknowledge that all of the students in my classroom are different individuals with unique styles, personalities, and strengths and areas for growth, with various potential ways to contribute to the classroom. I even believe that students who have skills in math have something to learn from students who struggle with math.

However, somehow, while I can and do believe that of young people/ my students, I don’t seem to hold the same beliefs about myself and other adult learners. I question sometimes whether other teachers will value my ideas or my contributions. Whether my ideas are anything new or if I’m just sharing “knock off” models of other people’s ideas. For example, one of my favorite instructional routines is one I gleaned from Annie Fetter at the Math Forum: Notice/Wonder. I not only use that on an almost daily basis, I’ve shown the video to everyone: My colleagues at my school, student teachers, teachers attending PDs that I’ve facilitated, etc! But I wasn’t the first one to discover it, nor was it a “Kit original,” so therefore, despite the fact that I’m sharing this awesome technique that transformed my practice, I rarely give myself credit for actually pulling this strategy off on a regular basis.

In fact, most of the pedagogical techniques I use in my class are ones that I’ve learned either by observing people or by reading about them or occasionally even watching videos. It is very rare that I’ve created anything truly unique, and it’s mostly been built up on the ideas of people who came before me.

Because of my perceived lack of originality perhaps, I don’t consider myself as worthy as other teachers of recognition. I look at nominations for awards pages and I think to myself “How am I different than all of the other teachers out there who have access to the same resources I can find on the internet?” or “Why should I deserve this award? I’m not doing anything particularly special.” And I’ve never nominated myself for a single award – which by the way, seems like an awkward thing to do!

This imposter syndrome is so pervasive for me that it wasn’t until last year that I finally proposed facilitating a PLT at MfA. Before that, I was always content to attend other people’s workshops because I didn’t think I had anything worth sharing with other teachers. And then, suddenly, I ran this PLT with a co-facilitator, and we had such positive feedback from our participants. And in fact, techniques we had taken for granted as “101,” we discovered were actually new to some of our colleagues. Now, again, I didn’t write anything new or discover anything brand new, but we were putting together a bunch of related ideas and linking them concretely in a way for these teachers that they hadn’t seen before. Some of those teachers had been teaching for longer than I had been. Now, I was mostly sharing other people’s techniques with them, so again, I didn’t feel too original or unique in my sharing. The fact that our session filled up both last Spring and this fall was due entirely to the fact that our group was so small, right? Never mind the fact that it filled up in less than an hour after registration opened.

That co-facilitator connected me with a summer PD that I helped facilitate, and I began to really stretch my facilitator muscles even more, including outside the MfA community. I constantly question whether I belonged on the facilitator side of the room or the participant side – because I haven’t taught these particular lessons before. And it was at this PD that someone shared a particularly powerful quote with me. They said “never compare your beginning to someone else’s middle” and then, because this was at the facilitator meeting, they shared the reverse for us, “don’t compare our middles to someone else’s beginnings.” And I began to recognize that I might actually be a “middle teacher” and not a beginning teacher anymore (in my 8th year of teaching now!). I began to recognize that while I might not have taught that specific lesson from 7th grade, I’d taught most of the 8th grade books, and I knew the lesson structure and I knew the routines, and I know the teacher moves. So it became clear that I had skills to share with these other teachers.

I think because I made the transition right from MfA’s Fellowship into the Master Teacher program, I didn’t quite notice when I’d crossed the line between being a beginning teacher and being an experienced teacher. But then, last year, I changed schools. And while there’s been a heck of a learning curve, because I’m learning two new curricula to teach, I’ve felt confident in my ability to teach because I know the basic pedagogical moves I plan to use, and I know the classroom management techniques, and i know how I plan to build relationships with my students or handle difficult behaviors, or what I value from my classes. I’ve implemented certain systems that have helped me streamline my day in some ways (I’m still searching for better systems for certain tasks!), and I’ve got my stock set of routines, procedures, policies, etc. So while the adjustment was challenging, it wasn’t as overwhelming as being a first year teacher all over again.

Now, it took me until my 7th year teaching to propose a PLT to MfA that I felt confident facilitating. So there was no way I thought I felt ready to give a talk at MfA’s annual “Master Teachers on Teaching” (M-T-squared for short) – an event that essentially has teachers giving TED-style talks for 10 minutes to an audience of up to 200 ish teachers. I’ve never even bothered proposing a talk before, because really, what did I have to say about teaching that was so important that other MfA teachers would be interested in hearing me say it?

And then I got an email direct to me, asking me to submit a proposal. Now, it seemed like a form letter, honestly, probably sent out to lots of master teachers to encourage us to apply, but the letter combined with this year’s theme actually inspired me quite a bit. The theme this year is “Truth Matters: Lies, Trust and Logic in the STEM classroom.” I shared my ideas with my trusted friends and asked them if they thought I had the kernel of a talk in there. They both agreed I did, and encouraged me to apply. I decided to go for it, and spent the next week writing and revising my summary of the speech I hoped to give. I was at MfA to facilitate one of my two PLTs this fall and I pitched the idea to one of the program officers, still not entirely convinced my idea was what MfA was looking for – and I got quite the opposite feedback! She loved how I’d tied my idea to the theme, and she thought it was particularly important in fact! She encouraged me to submit my application that weekend.

After writing, revising, and getting my editor-mom to read over my application, I submitted my idea. A long week later (and a postponed notification deadline!), I finally heard: they chose my speech as one of 8 teachers to speak. WOAH. I’m not sure how many of the 1000+ MfA teachers actually applied to speak, but I’m sure it was competitive (that was the reason for the delay in selection, anyway). I began to wonder – maybe I do have something worth sharing?

In between submitting my proposal and hearing confirmation, I went out for coffee with a friend and mentor who encouraged me to potentially publish my idea, whether or not I gave a speech on it. And she helped me coin a new term for the skill I plan to talk about. Adapted from an idea in therapy called “Therapeutic Use of Self,” she coined “Pedagogical Use of Self.” To learn about what I mean, you’ll have to wait until I finish writing my speech (don’t worry – I’ll post the final version here after December 14th when I present!). She even expressed surprise to me at the fact that I considered myself a “middle” teacher – don’t I think I’m past middle? Into the arena of expert? After all, I am a Master Teacher…

And I wasn’t quite buying it, even still. That pesky imposter syndrome keeps telling me that if people came into my classroom, they would discover that I’m actually a fraud and my skills in the classroom with real students don’t measure up; I fear that I talk a good talk, but I don’t walk as good of a walk (which, intellectually, I don’t think is entirely true – I do think I’m a good teacher – what I think is that I’m not as good as I strive to be – but I might be striving to be perfect?).

Then last night happened, and I began to doubt my imposter syndrome. See, last night was MfA’s annual Fall Function (my 9th!). I saw lots of friends there – people I know from when we were fellows together, people I went to Pride with in June, people I went to grad school with, people I played softball or poker with or board games with, people who’ve been in PDs with me. But, most significant towards attacking my imposter syndrome were a few people in particular who expressed excitement about me talking at the MT^2 event (because my name and the other 7 people chosen to talk were announced on November 1st, in the monthly exponent – MfA’s email newsletter). The fact that they were excited to hear me talk made me nervous but also excited to share with them. The first person I’m thinking of is someone who’s been a Master Teacher for as long as I’ve been in MfA, perhaps even a year or two longer. He ran a workshop for us right before school started and shared his welcome letter and first day activity with me back then; a few years ago, I attended a workshop with him, and he shared other resources with me again. And now, here he was, telling me how excited he was to hear ME speak! Woah.

The second thing that happened was when we were in the banquet hall itself. Besides the fact that I had the best seats I’ve ever had, I noticed a table with a bunch of my fellow former 2009 Fellows (now 2014 Master Teachers!), and I went over to say hello. As I was chatting with them, someone who I admire and respect came over. I thought I was getting out of his way so he could go see someone up front, and he stepped aside, commenting to my partner that he was getting on the Kit line. The what line?? Wait, he was coming to see me specifically? I didn’t even know if he knew me well enough to match my name to my face, but apparently he very much did!

I guess this whole post is me, trying to wrap my mind around the fact that I’m someone whom my colleagues respect and value to such a degree that they’re seeking me out at Math Prom and they’re joining my PLTs and they’re coming to my MT^2 speech. People have said to me that they enjoy being in PDs with me because they know I’m going to enhance the discussions. I think there’s a part of my brain that’s still convinced I’m the teacher whose first principal told him it was “doubtful” that I would be a good teacher, and that perhaps I should look for a different career, because grad school was supposed to give me all of the skills I needed to be successful, and since I was clearly struggling with that, maybe I just wasn’t cut out for it.

I was watching a TED Talk tonight by Brene Brown about Shame and Vulnerability (research for my MT^2 Talk!), and in it she talks about how at a TED convention, everyone who gets up on stage has failed multiple times – and that’s part of what made them great. And it made me recognize that my desire to overcome the failure of my first year is part of what’s driven me towards excellence. It’s shaped everything I’ve done since then, though it’s also probably the root cause of the imposter syndrome feelings I struggle with. Luckily, with my Sarah Lawrence College background, I was skilled in the “define the problem, do some research, create a project to solve the problem” method of studying, so I was able to overcome this initial failure. Now I’m in a place where 120 MfA teachers are going to come and hear me talk about teaching. I’m still in shock and disbelief.

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#DayofSilence Breaking my Silence

Friday, April 21st was GLSEN’s annual/national Day of Silence, intended to highlight and bring attention to the bullying (and suicide) of LGBT teens. It started in 1996, and I remember being in middle school and later high school, and my friend from space camp, Heather telling me all about it. I can’t remember now if I participated or not: it’s very possible that I did. But as an adult, working with my colleague in supporting the GSA at my new middle school, it became really important for me to plan a special lesson for Friday, to support the kids in the GSA and the LGBT students at my school.

We started with a writing prompt: four questions about silence. When does it feel good to be silent? When does it feel bad? When might you choose to be silent? When might you be forced? and why for all of them. I gave the students about 10 minutes to write, and then they passed their papers to the right in their groups of 3 or 4. The next student had to read an annotate: either checking off what they agreed with, putting a # next to anything that angered them, an ! next to anything that surprised them and a ? next to anything that confused them.

Then I posed the question, “Why are we silent today?” on the board & revealed the Day of Silence palm card with their explanation of WHY we are silent that day. At the bottom, I posted a green sentence that said “please give me a thumbs up when you’ve read the whole slide” and I waited until everyone was giving me a thumbs up. It was fascinating to see different students’ reading speeds and I used my laser pointer to focus kids who weren’t reading on what they should be doing silently.

The next slide asked “What are the statistics?” Since I teach math to my middle schoolers (and I know they’ve ALL done percents at this point!), I wanted to have them do some percentages and some calculations. I know that just reading statistics like “65% of LGBT students heard homophobic remarks like “fag” or “dyke” frequently or often” doesn’t mean very much to kids – “Ok, so like more than half of them… but what does that mean in terms of actual numbers here, in our community?” So our calculations included predicting about how many students in our middle school probably identified as LGBT (or would someday) – which is about 57 out of 541 students total and comparing it to the 100,000 out of 1.1 million NYC school students. Then we used those two amounts as the basis for how many of the LGBT populations in NYC and our school were bullied or rejected or feared going to school. It obviously wasn’t based on SURVEYED information of OUR specific kids – but it was extrapolated from the GLSEN national school climate survey to apply to our communities. I think it was a great way to raise my students’ awareness, because many of them felt like “Oh, I didn’t know anyone at our school was LGBT” or that there was so much harassment. Now, it’s entirely likely that at our particular school (located in Chelsea, NYC), most of the LGBT students are NOT bullied (since even my kids who wear hats saying “Make America Great Again” wrote that they would stand up for LGBT kids being bullied), but I felt like this was the most concrete way to connect the idea to the students.

After they had about 5 – 8 minutes to work on the calculations, I revealed the amounts so they could see the facts (even though they hadn’t had a chance to finish calculating – but that was ok). I then played for them the Todrick Hall “It gets better” video (which is a music video). It was a great choice, and I’m thankful I chose it instead of some of the “talking heads” It Gets Better videos.

Then I had the students write a reflection: “How did today’s lesson make you feel?” and “How can you help end the silence of LGBT students due to bullying and harassment?” and gave them a few minutes to write. I advanced to the next slide, and I broke my silence (because at this point, I hadn’t spoken AT ALL during the first 30 minutes of class, communicating completely non-verbally with my students – using hand gestures, facial expressions, the projector, and clapping to gain attention and give directions). I said “Now we’re going to break out silence by discussing what we can do to help end the silence of Lesbian, gay, bisexual and transgender students due to bullying and harassment. Talk to your partner,” and I gave them a few minutes to share.

Then I brought the whole class back together with my clap and asked if anything had come up during the class or their conversation that they wanted to share with everyone. In all of my classes, it took a few moments of patience and wait time for someone to feel confident in sharing. I called on about 3 – 5 students to share their ideas (only choosing on volunteers).

And then I ended with my own reasons for why the day of silence was so important to me. I wrote my “script” in advance, but I didn’t want to read it aloud and sound stilted, so I “ad-libbed” it a bit each time. This is what I shared with my students:

“I wanted to share with you why the Day of Silence is so important to me. Silence has played a big role in my life. As a kid, I was silenced by the bullies who harassed me in school. But I was also silenced by a lack of vocabulary to describe how I felt inside. I think at age 11 and 12, you know words that I didn’t learn until I got to college.

“As an adult, I’ve felt silenced in a different way. I’ve felt silenced by my fear of being mis-pronouned, or not being seen as a man. I’ve felt silenced by my fear that a parent might complain to my principal that they didn’t want their child in my class, not because they didn’t like me as a teacher, but because they didn’t want their kid to have a transgender teacher.

“But I realized that by being silent, I have made myself invisible to you. So today, I want to break my silence and share with you that I am a transgender man. What that means is that when I was born, the doctor said “It’s a girl!” – but they were kind of wrong. I was raised by my parents and went to school as a girl, but when I got to college, I met other transgender men and I recognized my own experiences in their stories. I realized that the only way to be truly happy and feel comfortable was to medically and legally transition to living my life as a man.

“I also identify as queer. Now, some of you may have heard that term in a derogatory manner before, and it can be an insult. However, some members of the LGBT community have reclaimed the term to mean that they are attracted to people of more than one gender. Unlike the word bisexual, which implies two genders, queer acknowledges that gender is on a spectrum with many options.

“Today, I’ve broken my silence by sharing this with you so that you are aware that you know someone who is transgender. And I can tell you: it does get better. If you have questions or things you want to talk about, I’m happy to answer them or speak with you.”

And then I opened it up to questions in two of my classes (where we still had class time – the other two classes, I had to dismiss because we were out of time!). A couple of kids asked questions about what it meant or how it felt/how I knew, and I tried to explain as best I could in age-appropriate language. I had one student ask me my birth name, and I told him that was private, and not something I would share with them – but I explained why it could be an insulting question and why I wouldn’t answer (because I don’t want you to call me that name and I’m worried if I told it to you, you might accidentally).

In my two seventh grade classes, some of the students already suspected – which is part of why I decided to come out. I didn’t want my “status” as a transgender person to be something that needed to be discussed behind my back in rumors. I wanted the students to know that I’m not ashamed of being trans, but am in fact, proud of it. I wanted students to know that it’s okay to be LGBT and I wanted to be a role model for them. I wonder how differently my life might’ve been if I’d met a trans man when I was in middle school, and felt comfortable coming out as trans and transitioning as a teenager instead of waiting until adult-hood. I envy the teens who are able to  do just that, but I know that I can play an important role in their lives (especially since I feel like even in the public discourse, there are many fewer out trans men than there are trans women – and it wasn’t until I was an adult that I even knew trans men were a “thing” – even though I knew about trans women!).

After class, in both of my seventh grade classes, I had a few of the boys come up to me and shake my hand. A few students thanked me personally for coming out and being brave. One of my students said, “Mr. G, thank you for sharing this with us. I want you to know that this doesn’t mean me or any of the other students think of you as less masculine.” which I thought was really sweet.

It felt so powerful to come out to them, and I’m so glad that I did it. Of course – let’s see what (if any) are the repercussions next week and in the following months. I don’t think there will be too many negative ones – after all, I do live in NYC and the school is in Chelsea (also known as the gayborhood!). But, one never knows. (However, if I helped even one of my students feel more comfortable at exploring their own gender identity or sexuality, then I will know that it was even more worth it!).

Instructional routines as a theme

So I’m doing a lot of thinking about (instructional) routines this year: what they look like, what they mean, how to use them, etc. I feel like I’ve begun to develop a much deeper understanding of all the ways I can use routines in my classroom. I’m actually doing 2 PDs right now at MfA, both about using routines. One is with David Wees, from New Visions and one is by two of my long-time favorite facilitators, Kara Imm & Rhonda Bondie. They compliment each other, I think, as we’re doing different things with routines. Plus, this year, I just discovered Amy and Grace’s book, Routines for Reasoning (and first learned their other routine, Contemplate, then Calculate, from Jasper & Constance at MfA last semester). Last summer, I read Pamela Weber Harris’s Developing Numeracy books and the Making Number Talks Matter book, and this year, I’ve finally started integrating number talks (which I launched with quick images) & problem strings into my practice. 

I’m still marinating on all of these ideas about instructional routines, but these were a few quick ideas I wanted to document and share. 

Number talks vs Problem strings 

People sometimes talk about these interchangeablely, but it’s more useful to distinguish between the two. David shared a great descriptor tonight, which matches how I have been thinking about it. Number talks focus on ONE problem and MANY strategies, whereas a problem string uses a carefully sequenced SERIES of problems that focus or highlight ONE strategy. This made me think about when you’d choose to use each one and I had a realization of how I want to use them/how I’ve already started using them. 

At the start of a topic where I plan to use NT/PS, do a number talk to uncover student strategies, reveal misconceptions, and determine how many strategies the students already know. Think of it as a pre-assessment. Then do a series of problem strings designed to highlight/reinforce specific strategies: bonus points if you refer back to which students shared those ideas in the initial number talk. Hopefully, you also manage to incorporate some strategies that ate new to everyone here. Finally, conclude the unit with a number talk to see which strategies the students have integrated into their toolboxes and perhaps even evaluate which methods are “best” for given problems and why. 

I just realized I call BOTH of these “number talks” with my students, and i’m now wondering if it would help guide our discussion of they knew we had slightly different focuses in advance, before we started: many strategies vs target strategies. 

I’ve done several number talks and problem strings this year; some I’ve liked better than others. I’ve done it in both 6th and 7th grade with all of my classes. Sometimes, I’ve had a context and sometimes I haven’t (though, to be honest, I think the context has been extremely important in supporting student success). However, one thing I’ve noticed is that my 6th graders (who value listening to each other more) do better at it than my 7th graders and they all seem to struggle when the number talks/problem strings go on “too long” (what qualifies as too long can vary, depending on day, time, class, or particular content/context/problem). I realized that one thing I find lacking in the routine is that it seems very whole class focused, and while there’s individual think time (I use thumbs to show when students have an answer), there’s no partner talk, so it can be a long time of sitting and listening. 

Which brings me to the instructional routine I’ve been learning about/experiencing this year: contemplate then Calculate. It is VERY structured (which I like & appreciate!), and includes several specific times for students to talk/share with partners. I think this phase is incredibly valuable and I want to figure out how to integrate the partner component in to my number talks/problem strings. 
 

#WhyIStay #MfAProud

So this afternoon, I had an experience that reminded me why I love to teach math to middle school students. I’m a pretty patient guy, and I am a big believer in Jo Boaler’s ideas about depth and not speed and growth mindset. I have no issues with students who need more time to master ideas and concepts, and I’m often at school late and available to help.

In my 7th grade classes, today we took a quiz on operations with rational numbers: fractions, decimals, and mixed numbers were being added, subtracted, multiplied, and divided (including in yucky complicated fractions with expressions in the numerators and denominators). One of my students is a little bit anxious about math – his mom confessed to me at parent/teacher conferences that he often can’t sleep the night before a test or a quiz. Today, when the quiz ended, he seemed upset and asked, “Can I see you after school today?” and I clarified, “Do you need more time on the quiz?” and he said no, I just want to talk to you about it because I don’t think I did very well. I said okay, and he went off to 8th period.

Meanwhile, I had a prep 8th period, so I decided to start grading the quizzes from his class, and I was in the middle of grading his quiz when he returned after dismissal. I finished grading it, and we reviewed the mistakes. Some were silly mistakes – like 21 – 13 = 7 (which he admitted rushing through) or losing a negative in a few places. We checked out a minor calculation error in one of the problems, and then we addressed a major conceptual misunderstanding of dealing with subtracting involving going from positive to negative when there were fractions involved. For example, if we do 5 3/4 – 9 1/2, he was getting – 4 1/4, which is not correct. It should be -3 3/4. I drew a number line and we did it in stages:

start at 5 3/4, go back 5 to 3/4. Then go back 1 to -1/4. Then go back another 3 to – 3 1/4. That’s -9, and now go back 1/2 to -3 3/4.

Alternatively, we talked about how you could subtract the whole numbers and the fractions separately, but keeping their signs:

(5 – 9) + (3/4 – 1/2)

-4 + 1/4 = -3 3/4

From this, I even shared with him that I did a problem like this on the board in the other 7th grade class and I actually messed it up because I always get a little turned around right by the 0 (when we switch from having + 3/4 to having – 1/4), so that’s why on my number line, I jump down by the whole number part to +3/4, then jump by 1 to cross the 0, and then jump the rest of the whole number part.

He seemed thankful that I had explained things clearly to him – and he was clearly understanding the math better – but he was still nervous/upset about the grade on the quiz (which by the way, was an 18/25, a 72%). He said “What can I do about the quiz grade?” and seemed near tears. I decided I would let him redo the quiz right then and there. So I handed him the other version of the quiz (I always do two versions to avoid students copying) and he confidently redid it. His new score is a 23/25 because he only had one minor error – everything else was completely correct (He left without seeing his new score, but just knowing that he felt more confident about it).

I came home to find this email in my inbox:

Dear Mr. G,

I would just like you to know how much it meant to me today when you let me retake our rational number quiz after school. This year has been really different from last year because of the many times you have sat down with me after school, and helped me with what I have been having trouble with. Last year I had a math teacher who wasn’t the nicest person. So with that, and math not being my strongest subject, math was kind of a struggle. I would just like you to know how much I appreciate all of this, and how much this is affecting me in a positive way.

Thanks, <student name>

The positive impact I had on him from something so simple as spending 15 minutes explaining a concept to him one-on-one, the fact that I’m sending him the message that we’re not done learning math when the first quiz happens, the fact that I’m starting to change his mindset from fixed to growth – all of these are why I stay. I feel appreciated by my students this year, and I love them all so much.

Depth of Understanding vs Coverage of Content #MTBOS

I mentioned in my twitter that I’ve got about 5 different blog posts in my head that I haven’t managed to put to “paper” yet – and the one I’m taking the time to jot down tonight isn’t even one of them! (I’ve also got like two or three different drafts saved on here of ones I began but haven’t finished!).

Anyway, the one that I’m thinking about right now is about the constant battle that I find myself in: do I ensure that my students have a deep rich understanding of the topics at hand or do I make sure I cover every topic before the state exam? Without even the slightest hesitation, I tell you that I always opt for depth of understanding and mastery of the skills over coverage of content.

One thing I’m enjoying about my new school is that the other 6th grade teacher (there are only two of us) has been teaching the same content for many years (I actually observed her teaching in the same classroom during my graduate school placement back in 2009!), and she’s admitted she feels somewhat burned out and is excited to be working with me for all of the new ideas she gets from me. I receive some of her resources (and her advice/notes about the topics/order of the units/skills), and in turn, share with her my ideas for revisions and for next year. Some things she’s taken to more quickly than others, but I’m enjoying collaborating with her a lot, especially because she’s a mostly enthusiastic recipient of my ideas.

Our current topic is order of operations. She shared with me a bunch of her resources of things she’s done – some I really like/already knew (like Four 4s and Bowling), some I think are useful so I don’t have to make ten thousand expressions of my own (i.e. practice handouts), but some I think are repetitive and dull and not really helping students to grow their brains. I’ve been researching the problem with PEMDAS. I told my students it was banned, and if they HAD to use something, they should use GERMDAS, which I wrote as follows:

  1. Grouping Symbols (and gave examples, like the parentheses, brackets, braces, and absolute value bars, as well as the vinculum or the fraction bar)
  2. Exponents and Radicals
  3. Multiplication and Division
  4. Addition and Subtraction

>>>>>>>>>>>>>>>>>>>>>>>>>>> (from left to right)

We discussed that the paired operations were inverse relationships, and should be done from left to right. I thought I had done a pretty decent job of training my students to go beyond the basics that trip up high school and college students (stuff like 10 – 2 + 5, where they do 10 – 7 = 3 instead of 8 + 5 = 13). And then I kept reading and digging, and realized there are a couple more things I want to get my students to understand.

I want them to understand that the order of operations isn’t truly arbitrary! And that we can rewrite expressions to be more clear and less ambiguous by eliminating subtraction and division and rewriting them as addition of the additive inverse and multiplication by the reciprocal (also called the multiplicative inverse – in fact, I’m not honestly certain what the difference is between a reciprocal and the MI). So I decided tomorrow’s lesson is going to give the students some expressions that are intentionally ambiguous to uncover some of the potential problems they might still face AND also some problems where we can discover/use the “Boss Triangle” to talk about the hierarchy of operations and why it goes in that order.

I’m taking ideas from a bunch of different places:

http://mathforgrownups.com/the-problems-with-pemdas-and-a-solution/

https://www.google.com/url?sa=t&rct=j&q=&esrc=s&source=web&cd=16&cad=rja&uact=8&ved=0ahUKEwjBo5jolfDQAhWqBcAKHaT6CxoQFghrMA8&url=https%3A%2F%2Fwww.montana.edu%2Fem%2F2016%2F20160624PEMDAS.pdf&usg=AFQjCNHbNMIwtEuLjoc8bixwHapldUxHjQ&sig2=lzu3DKaIMf3V8pOPgSwTPg&bvm=bv.141320020,d.ZGg

http://www.mathematicalgemstones.com/gemstones/amber/pemdas-is-broken-and-how-we-can-fix-it/

 

I would way rather my students understand when and how we can ignore the “left to right” rules and the arbitrary order (and in fact that we can ignore the order of operations in certain cases and get the same answer because different parts of the expression don’t affect each other!), than for them to robotically repeatedly simplify expression after expression after expression…

Priorities for the Year

The first two months of the school year are almost over. I’m at a new school (new for me; it’s been around for many years), and I’m very happy with my move, in general. I’m finding teaching two totally new curricula to be a bit of a challenge, but I’m confident that I’ll do an even better job next year than I’m doing this year.

In talking with some of my math colleagues, I’ve shared different things I’m trying out this year, and some of them have been impressed/shocked at “how many” things I’m doing with my students. So I wanted to articulate for myself some of the ones which I want to prioritize and ensure I don’t lose/forget about as I go from unit to unit. I also realized there are some things I want to emphasize more that I haven’t focused enough on yet, and I need to do something to change that soon.

Things I’m already doing that I want to keep doing:

Visibly Random Groupings – mostly daily. Sometimes, we need to finish a project in groups, so we stay in the same groups as yesterday, but for the most part, students enter each day with new seats, and new partners. They’re mostly enjoying it so far, I think, but I did give assigned seats the other day (for one grade, based on an exit slip for a leveled activity, and in the other grade, for a partner project that will be graded).

Notice/Wonder – I’ve done these regularly, but I worry that my students aren’t “wondering” enough – and I’m not sure yet how to model for them what they should/could be wondering. I’m wondering if anyone has any good problems to use this routine with. I worry that I’m overusing it on things that aren’t worthy of it; in one class, I had a student express that he didn’t feel like the task was worthwhile (while in another class, a student expressed that she liked sharing her answers for notice/wonder, where there wasn’t “one right answer”). I’m not getting rid of this, but I definitely want to step it up a notch.

Partner talk daily – students talk with their mirror partners (the kids they sit across from) and their elbow partners (the students they sit next to), as well as in whole groups. Students have become proficient at talking, but I worry that the kids still don’t know each other well (especially in the 7th grade classes – or even the 6th grade class that’s not my homeroom). I want to do some middle-of-the-year ice-breakers. I need to decide which ones would be good!

Share/Check, Discuss – this is a small group routine I created to share answers to a handout and discuss misconceptions/mistakes in small groups (rather than as a whole class). I find it particularly useful with handouts that have a lot of small exercises/problems for students to work on that would be tedious to check as a whole group, but where misconceptions might cause disagreement about certain ones. I have students go around and share their answers for each problem in rounds, while the listeners circle ones they disagree about and check ones they agree on. After reading all of the answers, they go back to the ones they circled and discuss those.

Things I’ve started doing, but I am not doing consistently enough yet:

Naming routines & their structures  – and then using them repeatedly so students can get used to doing “a share/check, discuss” and not need the directions renamed every time. I learned this idea from Rhonda Bondie at MfA, and it’s reinforced in the Instruction Routines book I’ve been reading, but I want to be more and more thoughtful about the routines that I name for the kids and doing them repeatedly.

Number Talks – every class has done several dot talks. The two sixth grade classes have done several subtraction number talks, including a few with integers. The seventh grade classes haven’t really done any more number talks – one did a percent talk, but I didn’t get far with it. I want to incorporate more of these, but I’m struggling with choosing which number talk to do when – and with ensuring that there’s ACTUALLY enough time to do it!

Which One Doesn’t Belong – I almost want to include this in the category of “want to being but haven’t yet” because I’ve really only done it once or twice in each of my classes. I think this is a valuable task, but I haven’t been able to work out when to do it – perhaps as a starter instead of a warm-up once per week? I wanted to find WODB’s that matched our content (like ones with numbers or pictures, etc.), but I haven’t been working on creating those yet. Maybe I can make incorporating this my November goal.

Convince yourself, convince a mathematical friend, convince a skeptic – I need to create a new poster, so I can remember to reference it in class. I’ve definitely said “who can convince our skeptics?” a bunch, but I want this idea to be internalized by the students, and I’m not referring to it often enough for that to happen yet.

Things I want to be doing, but I haven’t yet:

Would you rather? – I want to incorporate these, but I haven’t had an opportunity to do so yet. I think it will work out really well in my ratios/proportions unit, but I might even be able to incorporate one or two of the probability ones in my probability unit (especially as we turn towards review for the end-of-unit).

Fraction Talks – I think my next unit in each grade involves fractions, so I’ll make sure to incorporate some of this in there!

Problem solving – we haven’t done nearly as much problem solving as I would like to!

Visual Patterns – I did ONE in the 6th grade (the one from Jo Boaler), and I haven’t had any opportunity to do one with the 7th graders yet! I know we’ve got a unit on expressions and equations – I plan to use these heavily in there and in the linear unit!

Vertical, Non-Permanent Surfaces – I heard about this at the same time as the visibly random groupings, but I need the right kinds of problems to have students working at these surfaces AND I need to ensure that I actually have the right kind of space for this! My white boards are way to small for this idea, and I haven’t figured out the logistics of enacting it yet.

Naming the classroom mode we’re in – for example, “whole class instruction/discussion” and what the expectations for them are right now vs “table talk” and how those expectations are different. I want to create posters that explain each of these in detail, but I need to finish formalizing it in my mind first. I got the idea from Rhonda Bondie again, and I mentioned it in more detail in a much earlier post, but I haven’t gotten around to this yet (though perhaps this is something I should begin sooner rather than later).

Open Middle problems – I’ve used these as an extension, but I haven’t had a whole class discussion about more than one or two of these yet, and I really want to encourage my students to do this kind of thinking.

*Deep Breath* I know there’s a lot here, and I have to keep reminding myself not to try to do too many things all at once. I would rather do only three of these things, but do them REALLY WELL than to attempt all of these things, and do none of them well. So I need to keep reminding myself of that as I move forward with my classes – it’s okay if I don’t do all of them this year; there will be future years.

 

 

 

Probability Practice @alittlestats

On Friday, I gave my 7th graders a quiz on two topics related to probability. In the first set of problems, I gave them a table with the unequal probabilities for some outcomes and students had to determine the probability of certain events: one which was just one of the outcomes and one which was two or more of the outcomes (requiring the students to add the probabilities). Most students got this section correct, but a few thought they were supposed to multiply the probabilities instead of add.

The place that concerned me more, however, was in the second section of the quiz. I gave a scenario that was a two-step chance experiment: drawing two marbles from a bag with replacement. In the bag, there are two, four and six of each color marble (red, yellow, and blue). Students had to draw a tree diagram and label it with the probability of each outcome, and then list the 9 possible outcomes (preferably with the 9 different probabilities, which they could find by multiplying along each branch of the tree diagram). They had to find some probabilities based on the tree diagram which required them to add some of those probabilities.

Many of the students got confused in this section and used the probability of 1/9 for each of the possible outcomes instead of the actual ones, like 1/36 or 1/12, based on the probability of the two marbles individually. A few kids tried to add the probabilities instead of multiplying.

I have to confess: it’s been years since I took a statistics and probability course, and honestly, other than a HORRIBLE one in grad school, I don’t think I took any courses past HS that included probability. I’ve been keeping up on the material by teaching myself a few days in advance, and then implementing the Engage NY lessons modified by my department head (and tweaking them for my own classes).

In the past, I might’ve interpreted the results of my exam as meaning that “I did a good job with some of my students and that some of these kids just didn’t want to study or put in the effort” – but now I recognize that the kids who did well on this probably already were familiar with this to some extent, and that many of the kids who did poorly had questions they weren’t asking in class – which concerns me. I also recognize that in retrospect, I probably should’ve given an exit ticket that I collected before giving a graded quiz.

Now I need to give back the quiz to the students and figure out a way to reteach the material to the students who didn’t get it the first time, but do an extension for the kids who did get it – OR do an activity where I have the students who got 100% help teach the kids who were struggling? I’m not sure exactly how I want to handle it. Nor do I know what a good activity would be! Guess I’ve gotta do some research for this week. Any advice?