Breaking it down: How do we address student attrition? By analyzing systems that disenfranchise students experiencing trauma #TMCNYC18

I should preface this post by saying that I’ve been thinking a lot about what Sara Van Der Werf calls the Subaru Effect: “One day you go and buy a new car.  You decide to buy a Subaru for the first time.  The next week as you are driving your new Subaru around town you see Subaru’s everywhere you go.  You wonder why you are seeing so many Subaru’s when you would have sworn last week there were very few Subaru’s on the road.  The ‘Subaru Effect’ is your brain being awoken to seeing something around you that was always there, but you just had never paid attention to it before.” In her post, she mentions how once white privilege was given a name, she started noticing what had previously been invisible to her. I have felt similarly, especially since I expanded my twitter PLN over the last year to include far more teachers of color and writers of color who shape my perspective and help me to notice the inequities and underlying structures and systems that reinforce and codify those inequities. So this post below about my experience today at TMC NYC was my most recent “subaru effect” observation, but one that I think is vital we teachers begin to address better.

The first session I attended at TMCNYC on day 2 really made me think about the ways in which we, well-meaning teachers (often white, but not always) refuse to examine the structural causes of disenfranchisement and instead, perform “tokenized” “culturally responsive teaching.” The name of the class was “ Cultural Responsiveness in Math: A Student Research Projectand I attended with the hopes of learning some ideas about how to use a project to make my class more responsive to my students. (The link above goes to a blog post about essentially the same session, but at a different presentation; that author has given a good synopsis of the class, but is not the focus of my blog post on the topic).

To start, Ramon put up the following “problem” on the board to launch his presentation:

“Teaching Problem: For three consecutive semesters, an adult education teacher began classes with roughly 36 students and ended with roughly 12 students. What can the teacher try that will help to reduce attrition?”

Now, he works with adults in a CUNY program that sounds like it tries to help people who’ve been out of school and are now adults get their GEDs/high school equivalencies, and perhaps even prepares them for community college classes.

On the other board, he shared out some of the things he learned from his students about what’s going on in their lives. I’ll summarize, since I didn’t take a photo:

  • one student’s twin died, and now this person had to take care of their sister’s three children
  • Another person was suicidal
  • Another person was living in a shelter
  • Another person was a single parent of 6 children and struggling to get by
  • one person had lost something like 6 people in their life in the last year

I don’t recall the exact nature of all the statements on the board, but I think you get the gist – these are all real life challenges that adult students in his program were struggling with in terms of coming to class, in terms of being prepared, in terms of passing his classes. And our first discussion was to think about ways we could design a class to reduce the attrition rate, given these challenges.

And at my table, Nancy, Kristen and I (and someone else, I think?) discussed some of the ways to keep students who were having trouble attending engaged in a class. We discussed how if (in a semester of “remedial math”) each class is essential, then if a student misses one or two classes, it may feel hopeless to catch up, whereas if the topics cycle throughout the semester, and students know them in advance, then they can figure out how to make up material they’ve missed. We discussed how that involves pre-planning in advance, but didn’t get into specifics of how to do it in a way that would still make ALL classes relevant to people who were able to attend every class (which, I fully believe is possible, but is NOT the focus of this post, so I’ll save that idea for a future discussion). We also came up with some other ideas that I recorded in my google doc:

  • Figure out why students are absent and ask them what you can do to bring them back, perhaps connecting them with college equivalents of guidance counselors or other mental health professionals.
  • Normalizing problems – people have a lives outside of class, things come up, life happens, emergencies/urgencies happen, and you need to set your goals and think about how to access the materials and problems when you’re unable to make class so that you can keep up with the instruction. (thinking about using google classroom or moodle or whatever online college system they have as one way to share the problem sets, the lectures/notes, the slides, etc.).
  • Potentially starting classes with the same reflection each week, especially focused on re-engaging students who missed the previous class or two, asking “What was the last class you attended? What’s been going on? Why did you come back?”
  • We had a question that went unanswered, but I think would also have helped inform our discussion: “How many of the students who drop out in a given semester from this class ultimately drop out from the college as a whole vs. return in a future semester?” Understanding whether the “drop out” is temporary or permanent might help figure out the obstacles that are keeping these students from engaging in the class.

(I didn’t even get a chance to talk about my bigger idea that if

Now, when Ramon called us to share out, Nancy shared the idea about meticulously planning the calendar to support students who’d missed a class in being able to cycle through the topics and revisit the ideas so that it didn’t feel like an impossible task to catch up.

Ramon’s response was to shut down this avenue of thinking. First, he said that there was a pretty strict policy about three absences = an F in the class (though he later recanted and said that he knew that policy wasn’t always strictly enforced in the college in all of the classes, just maybe in this class? it was unclear to me). And he said that one of his goals was to teach his students to BE students – part of that meaning showing up to class, being prepared, bringing their binders (even though they might not want to appear like students). At this point, he said that these skills were important for both college and jobs – not acknowledging at this point that it sounds like some of his adult students might actually BE employed and have jobs – and that might be part of the conflict for some of them!

I jumped in at this point to try to highlight the fact that coping with real life is sometimes quite challenging, and that he wasn’t acknowledging the role that was playing in the student’s “choice” to not attend class (as he framed it). I pointed out that when my uncle died, I was able to take a day off from work to attend his funeral – why wouldn’t it be reasonable to assume a student might need to miss a class or two to deal with taking care of real life issues.

He acknowledged at this point that some part of the absences were due to legitimate issues that are “out of his control” – but then he said “these issues come up more frequently for this population of students” [than the people in this room] (who were mostly white educators vs his students, who he described as primarily Latinx).

And this brings me to my biggest issue and why I wasn’t really engaged with the rest of the presentation:

It felt like he was refusing to acknowledge that this system is set up to disenfranchise students who are experiencing or who have experienced trauma and was refusing to interrogate the structural policies that might be keeping students from passing his class (and thus explain the attrition rate – because if you know you’re going to fail anyway, why would you keep attending?). I don’t know how many times his class meets in a semester (I will start with that acknowledgement), but I personally think any policy that is so numerical and removed from the PEOPLE (i.e. “three absences = F”) is one that is going to reinforce the status quo of inequity because it doesn’t take people’s real lives into consideration. I would much rather have a policy that says something like, “I understand you all have lives outside of school, and I encourage you to contact me as soon as you know of an issue that may conflict with your ability to be here in person so we can work out the details of keeping you up to date and in good status.” and then personalizing your response to each of the students’ individual situations (36 students is NOT that big a number). You can make the judgement call then with the students whether the fact that this person missed two major classes because they were at their child’s parent teacher conference, discussing their child’s class struggles should be enough to fail them, or if that’s a reasonable conflict and you can work with them to create a plan to get caught up.

I would also argue that there was some racism at play here, with the implication that “these issues come up more frequently for this population,” but not acknowledging WHY his student population might be MORE LIKELY to experience trauma in comparison with white educators (I would estimate everyone in that room today had at least a bachelors and probably one masters degree if not more higher education). I’m going to compare some of the situations mentioned on the board to my own life, acknowledging that I’m a relatively privileged white trans man.

First, after two years of college, I decided to take a year off to figure out where I was going to transfer because I didn’t want to be an engineer anymore. During that year, I worked at Space Camp in Huntsville, AL, for both summers. I also went down there to work in January of 2006. However, the Sunday after I graduated from training, I got a phone call from my mom that my dad had died from a heart attack. This news was unexpected and devastating. I immediately spoke to my boss and told her I had to fly home to New York to attend the funeral. Despite the fact that I was originally scheduled to work that week, my boss told me to go and take as long as I needed and to not worry about it. When I got home, my mom asked me to move home for a bit and help her after a hip replacement surgery scheduled for a few months later. So I emailed my boss and explained the situation, and asked if I could quit now and return in late May/early June for the summer session. My boss was flexible and agreed. Now, keep in mind – when I worked in Huntsville, I had to move down there. I was able to live on the campus of Space Camp, but now I was occupying a room there that I wasn’t planning to keep. My bosses let me keep my room until the following month when I was able to fly down, pack everything up, and have a friend with a pick-up truck drive me and my stuff back to New York!

Why do I bring up this story? Because I was coping with the loss of a parent, it impacted my job, and yet – I was able to return to working at Space Camp that summer – my bosses held a space for me. If I had needed to keep working, they would have allowed me to return in one, or two, or three weeks – or whenever I was able to return. Yes, the structure of the camp allowed them the flexibility to have space for me to return, but many jobs (especially at larger companies) do have this flexibility. So merely being absent “three days” was not cause for dismissal from my job!

Second story from my life: In grad school at Sarah Lawrence College, while getting my Master’s in Child Development, I got the flu the week before my thesis was due. While much of my thesis was written, I wound up flat on my back for a whole week and a half. My mom even paid for a cab to take me home to Queens so she could take care of me and I wouldn’t be alone in my apartment! This wound up meaning I missed the deadline to graduate officially in May because I hadn’t submitted my thesis in time. However, my professor gave me an extension until the deadline for the August graduation, and I was still allowed to walk at graduation and participate in the graduation ceremony – my diploma just didn’t get mailed to me until August when I “officially” graduated. I obviously missed my classes that week, including my field work at the Early Childhood Center (who in fact said “don’t bring the flu to our young children!), but again, was not penalized for missing class – just had the expectation that I would catch up on the material I missed (and probably was discussed in my conferences with each professor).

Yet again, I experienced a (more minor) trauma – a serious illness that kicked my butt for almost two weeks at a critical moment in the semester. However, I didn’t fail any of my classes, and despite missing my deadline to submit my masters thesis, I still got my degree! And now? Ten years later? No one cares whether my diploma was awarded in May or in August or the reason behind it!

I could go on, naming multiple other experiences where I’ve experienced a problem in my life (family member’s death, a hospitalization, an illness, etc.) where the boss or professor to whom I was accountable was understanding of my situation and able to work with me to accommodate my needs, support me while I was recovering/grieving/etc., and then to help me catch up once I was better. Now, I need to acknowledge that being raised by my mom, I was taught to advocate for myself, to reach out to professors in advance whenever possible, etc. Those are LIFE SKILLS and STUDENT SKILLS that are necessary for people to know how to “work the system” and get the support they needed – and yet, I would bet that these are skills that Ramon’s students might not all have mastered. Why isn’t this self-advocacy skill one that he’s willing or interesting to teach them, rather than the static “they have to be in class with a minimum of absences”?

Plus, Nancy pointed out that the skill to manage time is another executive functioning skill that people need in the “real world” of college and work. Thinking about how to schedule your time so that you have enough time to do your HW or accomplish projects for work, even while making your “social life” plans or balancing your children/family’s needs with your school needs are an important work/life balance skill that many of us struggle with – me included. And here again, there’s an opportunity to talk about how to set up routine in one’s schedule to try to have the time to accomplish the HW, and estimating how much time it will take outside of class, and figuring out a back-up time of how to get caught up if something comes up with the first time. But again, this was a skill that he wasn’t interested in talking about.

Instead, Ramon then shifted into his “main point” about cultural responsiveness, where he laid out the situation as follows:

“problem: severe attrition, potentially due to student cultures not being acknowledged, recognized, or incorporated enough,”

So his possible solution was to recognize student’s cultural values, as this might motivate students who were “teetering” on the decision of whether to attend to choose to attend his class, despite the issues coming up in their lives.

Thus, he planned and executed a project with the students, where they chose a “cultural artifact” and researched the math behind it, then writing a poem, or writing a math problem, or producing some other kind of written product about it. In flipping through the book he’d made of his student work, some of them wrote poems, some of them wrote problems, and one of them analyzed a game called Skellzies which we had an opportunity to play.

However, I felt like this was a superficial bridge between people’s selves and their schooling, hence my use of the word “tokenization” in my opening. Student cultures were “incorporated” in a research project that wasn’t directly tied to the math they had to learn for the class, and it isn’t addressing the underlying structural policies that are perpetuating the attrition issue in the first place. It seems disingenuous to try to “incorporate student cultures” to make the class more “responsive to them” when it’s not also supporting them in accessing the class itself.

Now, I should note here: I don’t have an issue with the IDEA of incorporating research projects that encourage students to connect math to their identities so they can see themselves as mathematicians – in fact, I think we need to do much more in our classrooms, in general, to posit students, especially oppressed groups, to see themselves as mathematicians, but I think BOTH avenues are necessary. I confess I haven’t done enough research into trauma-informed educational practices, but it seems to me that the attrition rate might be further reduced by addressing the trauma students are experiencing differently. A quick google search leads to this link about how to maintain school engagement in students experiencing trauma. I don’t think it’s enough to merely state that incorporating student cultures into math class will keep them from dropping out. Rather, I think that’s one of two (or more) major components – the other is an analysis of the systems, policies, and procedures in place that act as gatekeepers, potentially preventing students from succeeding.


NYC’s mini Twitter Math Camp #TMCNYC18

Have you seen the schedule for TMC NYC yet?

I’m facilitating twice about two different topics. Each day has a theme, so on the first day, I’m going to be leading a session about the distributive property through the grades. I had a brainstorm about it because of my work with the Illustrative Mathematics curriculum this year, but I’m kind of excited to explore this topic with other math teachers.

On the second day, I’ll be evangelizing about one of my favorite teaching techniques: an instructional routine called Connecting Representations, created by the fabulous Grace Kelemanik and Amy Lucenta. You can read more about it in their book, Routines for Reasoning or check out their website,, if you can’t attend my session.

I was probably one of the hardest converts initially, the most resistant to actually trying out the routines in my room, for fear that I would get it wrong – and now I’m a true believer and planning to launch my year with three CthenCs (a different routine for reasoning).

Anyway, there’s tons of other awesome sessions to check out, and the best part? The conference is free! (And local, so if you’re in NYC anyway, there’s no need to pay for lodging or more transportation than your typical metrocard!). Registration closes on August 20th, so get in now, before it’s too late!

#ClearTheAir Troublemakers Week #1-2

I couldn’t join the live chats for the #CleartheAir Troublemakers weeks 1 or 2. It’s probably for the best, as I don’t do great with twitter chats (I’m still figuring out how to follow them on tweetdeck), but also because I haven’t had a chance to read the book yet. However, the organizer, Val Brown, posted the questions in a twitter moment, so I decided to copy them here, and respond to them as a blog post. Which, for my loquacious self, is probably for the best, given how many words I’ll need (main reason I was resistant to twitter for so long – what can you even say in 140 characters??)

Indifference is a lack of interest, concern, or sympathy, according to the dictionary. What it may look like in my classroom is not asking students about themselves enough – do I have an interest in who they are as a person? Or do I only have interest and assign value to them as a student (in particular, as a math student)?

I think I’ve been doing a better job over my 8 years, of shifting from my view that the content was the key to figuring out how to balance interest in my students and their lives (all of them!) by taking a moment to talk to them and ask them about themselves or chat with them. I know the last two years, using the name tents, has helped me to get to know my students better and learn about their lives – but I still could be doing better.

I think one goal I’m going to try for this year is tracking which students I’ve spoken to “socially” (either in the hallway, or after school, or before/after class, or even during a break), and try to ensure that I’m equitable about my attention, ensuring that I check in with everyone at least once a month or so. Maybe I can use my attendance/seating chart to check off when I’ve spoken with a kid in a given week/month, and if I notice there are no checks next to a name, I can be sure to talk to them next.

I also know that as someone with ADHD, I often struggle to focus or listen to other people. I’m a big interrupter and I often center myself in conversations. Decentering myself and asking about the other person (and truly listening to understand, not just waiting my turn) is a skill that I’m still working on developing. And this question makes me think that cultivating that skill in my classroom is important (as well as in my personal life), so I’m going to try to be more intentional about listening to understand my students. (And knowing my strength as a writer/reader, I wonder if I can somehow re-use the idea of the name tents later in the year, to continue a conversation with my students – maybe we can do it on index cards or something – have a “silent” conversation).

Well, just thinking about who’s actually IN my classroom at the various schools I’ve been at shows the most direct connection. For example, my first school in the Bronx had NO white students, few Asian students, and mostly Latinx and black students, so that was who I taught. My second school on the Upper East Side had a big mixture – except that in the honors classes (not mine) were the mostly white and Asian students, so my classes had a wider variety. However, the academic (or “low level”) class had more black and Latinx students than my other classes, solidifying the tracking by race (even if not consciously doing so).

Envisioning what my classroom would look like if it were free from that is hard because every time I think I understand the racism in this country, I uncover yet another aspect that I was previously blind to. I think ultimately, if it weren’t implicated, every school’s demographic make-up would match the city’s demographic make-up. Every class would be of equal quality for ALL of the students in it. All students would be empowered to advocate for themselves and each other. Beyond that, I’m not sure yet. Clearly, this is something I’ve got more reflection to do (maybe I’ll hunt down the twitter chat and read other people’s responses to this question – I’m not sure how to do that yet).

White values are assumed by many white people to be universal, probably in part because so much of “white culture” is reflected in American Television as “universal.” We see the same expectations reflected in so many places, we begin to think they really are ubiquitous without recognizing that it’s a cultural value, and that other cultures may have other values, but are not deficient merely because they don’t value the exact same qualities white culture does.

I keep thinking about the difference between white church and black church and the analogy someone made between white and black student participation in classes (was that on twitter? In a blog? A conversation in person? I can’t remember now). My first year teaching, I responded so horribly every time a kid called out an answer. I would shut them down, and tell them that’s not how to participate in class. I often would have one-way conversations with the class, reminding them that to participate, we needed to raise our hands – because that’s what I’d been taught at my predominantly white teachers college.  Whereas this most recent year, when one of my enthusiastic black girls would call out, I would often cheer her on and direct attention to her contribution. Sometimes, I had private conversations with her to remind her that I wanted to give other people a chance to participate also, but I was excited by her enthusiasm. I think in that response, I was better able to convey to her WHY it was important some times not to call out (because I value think time and hearing a variety of voices), rather than just silencing her. I think the white value of “docile and quiet” held me hostage my first year teaching, because I didn’t recognize the ways students were acting was based on a different cultural value – just one I wasn’t tuned in to see.

The last school that I worked at was predominantly white and Asian, and many of the teachers would complain about a lack of participation by the “quiet Asian students.” I’m now re-viewing those moments through this lens of white cultural expectations, and I wonder if the white value that everyone is extroverted and verbal ignores different cultural values from my Chinese students’ backgrounds. This seed was planted when I read the book, Quiet, but I still haven’t quite figured out how to integrate this new recognition with my actual teaching practice. I’m thinking about how to ensure all student voices are heard even from students who are quiet, and I’m still wondering about ways to get their voices “heard” through written responses that get shared (anonymously?).

How do we stop perpetuating the status quo? That’s a tough one; I feel like I’m still figuring out what the status quo is because so much of it is invisible “universal expectations” that I’m finally becoming more and more aware of.

I have a confession to make: sometimes, when I think about how to practice academic freedom, I don’t know how to do it as a MATH teacher. I can envision it as a humanities teacher (man, the things my students and I would read and write! I’m so inspired by all of the humanities teachers I see doing wonderful and empowering things!).

I’m still reflecting on how I would need to change my class to imagine mathematical freedom, to practice mathematical freedom, and prepare for mathematical freedom. Some of it, I think, is in empowering my students to think for themselves – which I think I’ve done a pretty good job at over the last few years. I think some of it is about empowering my students to see math as a tool for making sense of their own lives. I also wonder if some of it is about empowering students to ask the questions that we investigate and figuring out how to modify the curriculum so we can explore their ideas more. I’m not sure.

I think one of the biggest things that I need to change immediately is the way I deal with student behavior. I’ve already grown so much from the first year teacher I once was who thought every student’s behavior needed to be corrected with detention by recognizing that most challenging student behavior is communicating unmet needs or lagging skills and it’s my job to help ensure those needs are met and those skills are taught (by me or elsewhere) (this is the model from Ramapo for Children & Dr. Ross Green). So I’m definitely much further along this growth path than I was, but I think I still need to be more intentional about it. The last two years, I’ve been at a school that had minimal behavior problems because it screened students heavily (based on grades and performance on a three-part entrance exam), but at my previous school, I relied on suspensions for especially challenging behaviors (like pushing over desks or “insubordination” along with cursing at me). I’m hoping that my new school’s focus on mindful breathing and circles during advisory will allow me opportunities to learn more about restorative justice practices, and figuring out how to focus more on building the relationships with my students that are vital. I checked out that website (I forget the link) that tracks data on suspensions by race, and my new school as well as the previous two, disproportionately suspend black and latinx students. I want to be more cognizant of that, and interrogate my practices – if I am thinking about how to discipline a child of color, would I interpret their same behaviors differently if a white student were doing it?

I’m also thinking about this post by Jess. In particular, I’m struck by this quote, “We are raising our students to put compliance above justice, to follow blindly without asking if what they are following is fair or just. We are raising the humans that perpetuate these conditions. But we do not have to. We can change that narrative. We can start to use our students’ noncompliance, not as a reason to punish, but as an opportunity to learn. To learn what we need to fix.” I’m wondering how I can have a similar conversation with my students about the rules. (As a MS math teacher, with three classes, and a more limited amount of time for each class, I always question how I can devote time to conversations like these that seem “out of place” – but then, as a disruptor, I question how I can avoid having these important conversations with them!)

In most classrooms, the rules, procedures, etc. are formed by the teacher… often before the students even arrive! So in many ways, I think ageism impacts class rules, procedures, etc. because we assume the adults can (and SHOULD) make all of those with minimal input from the students – and even when we solicit student input, they usually come up with the same general ideas – often regurgitated from last year’s teacher’s rules!

I often feel stuck about this topic of including students in doing more/creating more responsibility for them (probably in part because I’m nervous about giving up control to my students, and partially because I’m bad at improv!). I think I’m of two minds when it comes to student agency: I want them to be empowered to be agents of their own learning, but when we’re in our classroom community, I also want them to be a member of our community… But I know that some of the expectations I have for math class (as a constructivist teacher) are different than what they’re “used to” – so I don’t know how to balance my experience and knowledge about how to best make sense of math vs some of their desire to just “be told what to do.” Maybe some of the answer lies in having a conversation with students about how learning works and getting them to buy into constructivism in order to turn over more agency to them?

Otherwise, in terms of rules and procedures, I feel like I basically have one big rule at this point: our goal is to do math and think like mathematicians for the entire time we’re together, and anything that distracts from that needs to be set aside and saved for another time. I do use hand signals for the bathroom in part because I want to make sure that students are not going when I’m about to say something super important (unless it’s an emergency), and also because most of the schools I’ve worked at have required teacher signatures on bathroom passes of some kind. I try to take myself out of the process as much as possible, though, teaching kids the non-verbal ways I’ll respond to them so they can go. As for getting pencils or other supplies, I have cards on the desks, and I’ll usually call out a suit to go do something, “All of the hearts, go get rulers for your group” or whatever they may need.

I definitely need to reflect more about this one, though. I don’t think I’m doing a good enough job at it yet.

I’m definitely seeing a connection between the snippet of Sean I’m getting here and the civil disobedience movements – people who are not a part of them often react negatively to them and want to squelch them (such as the NFL’s new rules about standing or the all/blue lives matter response). I see that as the teacher who wants a student like Sean to become compliant and to “fall in line.”

However, now I’m thinking about those movements as drawing attention to something that has been ignored or gone unheard by the general populace – such as police brutality and the connection to student disruption. I wonder if we can expand viewing the challenging behavior as communicating unmet needs and lagging skills to viewing riots as the ultimate communicate of unmet needs and lagging skills.

I first learned the practice of exclusion as a student. I was excluded by my teacher in first grade, and later by my peers in middle school and high school. The exclusion taught me that I was unwelcome, that there was something wrong with me, that I didn’t belong. It didn’t meet any of my needs, and it honestly didn’t change any of the things about me that made those people exclude me. It wasn’t until I was an adult, and in a community where I felt accepted for me, that I was able to share my stories of exclusion with people who helped me to make sense of the behaviors that I might have been doing that caused others to exclude me. So, ironically, it was through inclusion in another community, that I learned how to mature and evolve, not through the exclusion from my initial community. I now recognize that some of my exclusion was due to my social awkwardness (a side effect of my ADHD and the way I used verbal communication – talking fast and about me/my ideas, interrupting others, not listening to their ideas) and some of it was due to being queer and trans at a time when my peers and parents didn’t have that language and I felt misunderstood.

As a teacher, I think about exclusions as class removals/suspensions (where we’re excluding students from the learning experience in our classroom communities) and being prohibited from going on a field trip. The longer I’ve been a teacher and the more experienced I’ve become, the more I try to limit my use of exclusion, as I’m recognizing that it merely removes a student from the situation, but it doesn’t actually make anything better in and of itself. I’ve already thought about restorative justice as the next thing I need to learn about and become proficient in, but now, making the connection between my own feelings of exclusion as a kid, and knowing what little I do know about RJ, I think it’s the way to combat the isolation that is felt by someone who is being excluded, and instead, to rebuild the connection to the community, so that more positive behavior will happen.

I’m not even 100% sure what being “fully human” is because I feel like it probably happens so rarely. As with any human, I am multi-faceted, but some of the aspects of myself have communities that seem to be incongruous or incompatible with each other. For example, I am ever cognizant of how being an openly sexual human makes some people (especially religious people) uncomfortable. Plus, several branches of Christianity think because I’m queer (and have sex outside of marriage) and/or because I’m trans (and I used medical science to change my body to alleviate my dysphoria), I’m a horrible sinner and I’m going to hell or what have you. Because I’m aware of the overlap between my teacher circle (where I can be fully a teacher and talk about edu) and religious folks (I notice Christ or Jesus in the profiles of MANY of the people I follow on twitter or my followers), I’m often uncomfortable and hold back. I don’t know how those people feel about me – if they’re going to judge me for my trans-ness or my queer-ness, or the fact that I’m anti-assimilationist (i.e. I’m not exactly like you cisgender, heteronormative, monogamous, religious folks and I don’t want to live my life according to YOUR values), it makes me uncomfortable to be myself around them (and even here, typing these words into my blog, I’m carefully considering how to phrase everything just right so that I don’t offend or alienate someone unintentionally).

On the other hand, I think about my tiny queer/trans circle of fellow educators – both the NYC community at large and the MfA community that I’ve been cultivating over the last year, and I think about how much easier it is to be authentic with those folks because I know they’re less likely to judge me for my transness or my sexuality.  But simultaneously, I also know that I can be fully my educator self, because they understand those aspects of me too.

I also think about my tight circle of friends (I have two romantic partners and a few really close friends), where I know I can be myself and I will be accepted. I won’t be judged, though I might be challenged or corrected. It will be done from a place of care and loving, with the goal/intention of helping me improve myself – not with the intention of “putting me in my place.” Some of it, honestly, is because I know that most of these people either have these “traits” in common (i.e. they’re also queer and/or trans) or I know their views about them to be positive and affirming.

I think the key to being fully human is probably knowledge that there’s unconditional acceptance on the other side of the relationship. That no matter if you come out as trans, or queer, or poly, or anything else, that person is going to respect you, love you, care about you in exactly the same way as before (or maybe more).

I’m honestly not sure yet how to replicate that in my classroom. I struggle in general with being authentic and not robotic (and not just with my students) whenever I’m nervous (and I always get stage fright at the start of the school year!), so I’m still figuring out how to convey my acceptance for my students. And I know that it’s not just enough to be accepting of students who are like me (i.e. queer or trans), but it’s also about making it safe for all students to be vulnerable, even ones who are different from me. So while it’s easy to become the GSA advisor and help create the safe space for the queer/trans students, it’s less straightforward to me about how to do that for ALL students in my MATH class – I think part of it is about being comfortable being vulnerable with the students (modeling how to be vulnerable and how to react to discomfort from being vulnerable) as well as how to value their vulnerableness when they’re willing to share that with me/us.


Just to #cleartheair, typing this up and making it public is an act of vulnerability too. But I think part of the purpose of these chats is to be open and reflective so we can improve ourselves. And it can be nerve-wracking to acknowledge that we’re not perfect, that we still do things that are perpetuating the status quo that we so vehemently disagree with, but hopefully, by hitting publish, I’m taking another step in my evolution as a better educator.


Thank you to Val Brown for organizing the clear the air chats. I appreciate the space to reflect on myself as an educator and a human.


My Feelings about the SHSAT & Specialized Schools #iteachmath #MTBoS

I have a lot of feelings about the SHSAT, the exam in NYC used to screen students for the specialized high schools. I’ve taught 6th, 7th and 8th grade at three different middle schools so far (in my 8 years of teaching) – a new and small school in the Bronx, a large comprehensive middle school in District 2, and a selective, screened middle school in District 2. I’ve taught a variety of students in my classes, from a wide variety of backgrounds and prior math experience. I noticed several things that I don’t think gets talked about enough in this on-going debate about the Specialized High Schools, so I’m going to talk about some of those issues here.

The first thing I want to mention is that I’ve noticed is Saturday School. Many of my Chinese students told me they went to Saturday School since elementary school where they often learned how to speak Chinese and sometimes where they received additional math instruction. Depending on the teacher and the specific program, some of those students received very procedural-based instruction, but some of them did gain a better depth of understanding or mastery of the content. Sometimes, they were very familiar with “famous problems’ sophisticated solutions” like the formula for Gaussian addition or the locker problem, which meant they had a mental framework for recognizing when new problems were similar to something they had seen before. Assuming a 4 hour class every Saturday, for about 30 weeks per year, for the 8 years of K-7, that’s an additional 960 hours of math experience. Never mind the fact that some of these schools ALSO gave the students HW to complete and many students may ALSO go to class after school, during the week as well. So some of their improved performance in math class may be  due to gaining more experience with the math they had to do in school. (I have a whole post brewing where I’m convinced that much of what we test for with “giftedness” screening is ACTUALLY just “experience” screening).

The second thing I wanted to mention about the SHSAT is the content that’s covered on the exam actually necessitates test prep. Supposedly, the version given to 8th graders only includes math covered by 7th grade standards or “below,” but based on the types of things I’ve seen my students learn in SHSAT prep, I’m doubtful. Let’s take a look at some of the questions that might necessitate more study than just your “standard” 7th grade math curriculum. These questions below are all from the 2019 SHSAT handbook’s sample exam:

“62. The sum of two consecutive integers is -15. If 1 is added to the smaller integer and 2 is subtracted from the larger integer, what is the product of the two resulting integers?”

Theoretically, operations with integers is covered by 7th grade, however the word “consecutive” is not NECESSARILY covered (though it could be). So one part of this is about the vocabulary. If I’m a teacher who’s familiar with the kind of vocab on the SHSAT, I can be sure that I’m giving students access and exposure to hearing, seeing, and using that word in class repeatedly. If I’m a less experienced educator, then I might not know to give those kinds of problems to my students. Furthermore, if I am just problem solving this, without experience with other consecutive number problems, I’m more likely to take longer to figure it out, whereas if I’ve gone to test prep, I probably have a formula memorized for dealing with consecutive numbers or I have had enough experience that I know I can say the consecutive numbers are x and x + 1 and I can write an equation to solve. x + x + 1 = -15 so 2x + 1 = -15 so 2x = -16 so x = -8, and therefore x + 1 = -7. Then, I can change them so that 1 is added to -8 becoming -7 and 2 is subtracted from -7 so it becomes -9. The product therefore is +72, which you would then have to grid in.

There’s a problem requiring students to have memorized geometric formulas (including for surface area of 3D shapes like prisms and pyramids), another one requiring students to understand how to graph a compound inequality (which is NOT technically included in 6th or 7th grade, though some teachers might teach it).

“97. In the set of consecutive integers from 12 to 30, inclusive, there are four integers that are multiples of both 2 and 3. How many integers in this set are multiples of neither 2 nor 3?”

Once again the phrase consecutive integers is used, but now an additional vocabulary word, “Inclusive” is used as well! Neither are necessarily part of the MS curriculum by default.

“91. There are 6 different cookies on a plate. Aiden will choose 2 of these cookies to pack in his lunch. How many different pairs of 2 cookies can he choose from the 6?”

This problem is technically a combinations problem, and technically, formal combinations aren’t taught until high school. An MS student who’s never seen a problem like this can probably work it out (especially with numbers this small), but a student who has memorized a formula can work it out more quickly. Factorials aren’t even technically part of the math curriculum up until this point, so while students who’ve done the Four 4’s activity might be familiar with them, it’s not something that’s “safe” to assume. By the combinations formula, 6 C 2, you would do 6! / 2(6-2)! which is (6 * 5 * 4!) / 2(4!) OR (6*5)/2 or 30/2 which is 15. With a number such as 15, you could also make an ordered list, if you called the cookies A, B, C, D, E and F, and then showed AB, AC, AD, AE, AF, BC, BD, BE, BF, CD, CE, CF, DE, DF, EF, also getting an answer of 15. But if you weren’t familiar with this type of problem (which is not precisely required by the math curriculum up until 7th grade), you might not know about it. Plus, the counting principal (which helps you “discover” factorials) is also not a necessity. Again, a student who’s been exposed to problems like this one can easily solve it, whereas a student who has only ever had “school math” (especially if taught by a teacher who DOESN’T know what’s on the SHSAT), might need to spend more time “problem solving” what appears to be a novel problem instead of applying a familiar strategy to an exercise they recognize.

“92. For a presentation, Deion can create 5 slides in 20 minutes, working at a constant rate. Kyra can create 3 slides in 10 minutes, working at her own constant rate. What is the total number of slides the two of them can create in one hour?”

Now, again, problems like this technically are within the scope of the 7th grade curriculum, but once again, this is a “type of problem” that is super familiar if you know this structure, but very complicated and prone to misconceptions if it is novel.

“99. A box contains 5 strawberry candies, 3 banana candies, and 2 orange candies. If Braden selects 2 candies at random from this box, without replacement, what is the probability that both candies are not banana?”

When I taught 7th graders probability, we only covered “with replacement.” WHOOPS! I’m not 100% clear that without replacement (conditional probability) IS in fact a 7th grade standard.

Screenshot 2018-08-11 01.04.57.png

“105. In the infinitely repeating decimal above, 7 is the first digit in the repeating pattern. What is the 391st digit?”

Although this problem would APPEAR to be covered by this standard: “CCSS.MATH.CONTENT.7.NS.A.2.D Convert a rational number to a decimal using long division; know that the decimal form of a rational number terminates in 0s or eventually repeats,” solving this problem efficiently actually requires using modular arithmetic (or at least thinking cyclically). For example, if I recognize that there are 6 digits in the repetend of the 13ths, then I know that to get out to the 391st digit, I need to repeatedly write those same 6 digits. Therefore, I can do 391 / 6 to get 65 and 1/6 or 65 remainder 1. That tells you that the 391st digit is the same as the 1st digit, which is 7. If I’m familiar with this style of problem, then it’s a relatively straight-forward division and remainder problem. If I’m only familiar with the actual standard, then this problem might induce me to tears, thinking about having to write 391 digits to find out what number is in that place!

“86. If w – 1 is an odd integer, which one of the following must be an even integer?”

A) w + 1, B) 2w + 1,  C) 2w – 1, D) 2w – 2

While writing algebraic expressions is technically part of the 7th grade curriculum, if you have increased familiarity with odd and even numbers being represented algebraically, then you can recognize that if w – 1 is odd, the next odd integer will be w + 1, whereas w itself must be even (due to the odd/even alternation). Thus, A can be eliminated. Looking at the remaining options, if w is even, then 2w must ALSO be even, so D must be correct, as that takes an even number and subtracts 2 getting to a smaller even number (whereas subtracting or adding one from an even number will give you an odd number). I don’t “mind” this question as much as many of the others, but I do think it requires a level of sophistication with algebraic notation that is NOT normally expected from the 7th grade math content standards and may or may not be gained from your typical math classroom instruction, as dealing with algebraic notation often doesn’t begin until middle school for most students (SOME get it in 5th grade, but not before), whereas dealing with even and odd numbers often happens in elementary school – and rarely to the degree of generalization required to solve this problem.

There’s also some geometry vocabulary that’s not normally used in MS:

“92. R, S, and T are midpoints of the sides of square MNPQ, as shown above. What is the sum of the areas of the shaded triangles?”

While midpoints aren’t unreasonable to infer the meaning of, they’re not typically talked about much in middle school – unless of course you have a teacher who knows to cover them! Midpoints aren’t officially covered until the high school geometry standards!

From 2017-2018:

“74. A large circular dinner plate has a radius of 20 centimeters. A smaller circular plate with a circumference of  centimeters is placed in the center of the larger dinner plate. What is the area of the part of the larger dinner plate that is not covered by the smaller plate?”

To solve this problem, you must know the relationships between circumference and area of circles (and their diameters/radii) without having those given to you. Dealing with pi also often adds an extra challenge for students, though that doesn’t necessarily rule this question out as doable.

In 2016-2017 or 2017-2018, the SHSAT got updated to better reflect the 7th grade standards (though I confess I haven’t actually seen a REAL SHSAT, because those are sealed and not released by the city/state). Prior to then, there were more questions on the test that required additional instruction (also, it included more 8th grade topics, but since it’s given in October of 8th grade, depending on your school’s sequence, you might not have covered those topics by then!). There were a bunch of questions about prime factorization (not officially a standard, though it often gets covered in 6th grade or before) and several involving scientific notation (which was often covered in later 8th grade).

Actually, now that I’ve looked over the SHSAT materials in more detail, I’m actually wondering why so many of my students were taught the Pythagorean Theorem by the test prep classes, as well as operations with radicals – those DON’T seem to be covered on the exams (anymore – I only went back to 2015-2016 for this, so maybe they used to be?).

Also, as an aside, I wonder what the questions themselves are actually screening for. Here’s one that I’m not convinced will tell you who’s going to excel at the specialized high schools:

What is the greatest common factor of 2,205 and 3,675?

A. 147 B. 245 C. 441 D. 735 E. 1,225

To be calculated without a calculator! Now, thinking about the steps you would need to take for this one, you have a few options: you can just try dividing the two numbers by each of the 5 options… But the long division by hand is bound to be a pain in the ass. Alternatively, you could try using the “ladder method” or prime factorization to find the GCF of 2,205 and 3,675… but starting with 3 and 5, you might not easily get to 735 – certainly not quickly! So there’s a potential waste of time.

What is the greatest prime factor of 5,355? F. 17 G. 51 H. 119 J. 131 K. 153

Another one where I’m not convinced it’s a good use of time. 51 is not a prime, neither is 119, nor 153. For 51 and 153, they can “easily” be identified as divisible by 3 (IF you know the divisibility rules – which I don’t think are required by the standards) by adding up the digits to get 6 and 9, both of which are divisible by 3. 119 is not divisible by 3, so it might take a little bit more work to verify that it is not prime: If you happen to know the rule of divisibility by 7, you might recognize it, or you could just try dividing it by 7 (personally, that’s my go-to, as I can’t remember the divisibility rule for 7s). Still, you’ve wasted a bit of time ruling out the non-primes, and you still have 17 and 131 to check. I suppose you can start with 131, as that is the bigger of the two numbers. Dividing 5355 by 131 will take some time, but you can see that it will give you 40 with a remainder of 115. So by process of elimination, it must be 17, and you can skip the extra division to confirm that.

Now, I’m going to go back to my original claim: I don’t think the standard mathematics curriculum IN MIDDLE SCHOOL CLASSES will prepare a student to successfully get into one of the specialized schools without going to extra math classes (and ELA, probably, but I’m focusing more specifically on the math, because that’s my area of expertise). Now, when I consider the ways in which teachers are prepared to teach and focused on their curriculum development, I would say that most middle school teachers (unless they’re teaching an SHSAT prep course) have never done an in-depth analysis of the kinds of questions on there, and are not looking to embed those questions into their curriculum or deploy them as extension/challenge questions for students who finish early (even though there are actually some questions that could be tackled by kids as young as 6th grade! And I may steal some this year!). So there’s an opportunity gap here: knowing what the exam is and recognizing that you do actually need to prepare for it. The DOE’s website is misleading about it (bordering on lying, in my opinion): “The test measures knowledge and skills students have gained over the course of their education. Keeping up with schoolwork throughout the year is the best possible preparation.” I don’t disagree that you can’t do well on the SHSAT if you haven’t done these things, but I honestly don’t think that’s sufficient. And to tell the students that it is sufficient is to rob them of the preparation time.

Now, some of the test-prep classes are very procedural and focus on memorization of formulas and “drill and kill” on a variety of the “most common” types of problems, showering kids with tons of worksheets and packets to practice skills they are shown in classes on the weekend or after school. Sometimes, these courses include content beyond grade level and may “ruin” discovery-based lessons (as was the case at my last middle school, where most of my 7th graders ALREADY KNEW the Pythagorean Theorem – which is technically an 8th grade standard!!).

The second big issue that I have with the SHSAT (and testing in general, to be honest), is the amount of money we pour into it that COULD be going to bettering school programs. Consider for a moment the fact that the testing industry is a $2.5 billion industry in the US (in 2015), and yet we can’t find the funds to fix the heating system in Baltimore schools, the water system in Flint, or any of the host of other problems that affect student learning(often in poverty)? New York City is currently paying Questar $44 million for 3-8 ELA and Math state exams (when they won out the current contract over Pearson), and according to the NYC Data website, our current budget spent $458,923 in FY 2015 and $542,515 in FY 2016 on the SHSAT alone (and another $100,000+ on the gifted and talented screening). Consider what we could do for our students with an extra half million dollars! And that doesn’t even include the money being spent on offering “free” tutoring – it’s free to the students, obviously, but not free for the city to provide – that’s gotta be budgeted for somewhere (though I couldn’t find the details on that cost, as it’s broken up into too many different programs that I don’t know all of the names for – there’s the DREAM program, among others). Plus, there are private companies that do test prep as well – some free or application-based while others for profit. All of the money that’s flowing in those pockets could be going towards improving ALL of our schools!

There are a variety of articles about specialized HS admission in general, the $13.4 million that the city pays Pearson over 6 years for administering/creating (scoring?) a test that no one knows the validity of, the current debate over eliminating the SHSAT, the reaction of Stuyvesant Students to the proposed changes, the culture of test prep, the monetary bonus that the specialized high schools get (which is part of why they can offer so many extras – that and their size in general!), the problems with (un)fair funding of schools in general, the new Chancellor’s reaction to screening/admissions policies, but I’m going to focus on some of the issues I think are missing from these debates. Or at least, the ideas that have occurred to me.

Now, let’s put the SHSAT aside for a moment, and handle my other concerns about the specialized high schools in general. Something that I don’t think is talked about enough is that the culture of those schools is very competitive, and not every student thrives in that kind of environment. I know for myself, when my parents asked my 8th grade self if I wanted to apply to the specialized high schools (and leave Catholic school), I declined. At the time, I said that I would rather be a big fish in a little pond than a little fish in a big pond (but not every 13-year old knows that about themselves!). I recognized even then that a school like Stuyvesant takes the students who were at the top of their class in middle school, and then immediately, half of them make up the bottom 50% of the class! I knew that level of competitiveness would be demoralizing for me, and I chose a more supportive school environment – and I’m glad that I did.

So this leads me to one of my big questions: what is it about these schools that makes them “specialized”? Supposedly, some of it is the culture of the students who “worked so hard to get in” that they continue to push each other forward – but I’ve never actually attended their classes, so I can’t really comment on it (though I did observe at Stuy when I was in grad school, I remember being bored by the lectures and wondering if the school catered to kids who thrived in that particular kind of classroom set-up). I know they’re able to offer more higher level math classes – but to some extent, that’s because they expect all 9th graders to come in having completed AT LEAST algebra 1 (which is typically 9th grade math in NYS, but many 8th grades offer regents courses to “accelerate” students). To some degree, because of the size of the schools (and having enough money to pay veteran teachers), they’re able to offer a variety of courses to students – things that are more specialized or specific than the general offerings of most NYC high schools – but that’s about access to resources. I know some of the educators who work at the various specialized high schools (mostly through MfA), and I don’t know that there’s anything “magical” about their curriculum choices or their pedagogy that other good teachers can’t also implement at their own schools. But, I do know that many of the teachers at the specialized high schools have MANY YEARS of teaching experience under their belt. And while it’s true that years alone is insufficient to guarantee a good teacher, the fact that so many of them are MfA Master Teachers speaks highly of their pedagogy. And this brings me to an article about segregation of the NYC schools in general that I was reading recently. When it comes to school segregation, “New York City is among the worst offenders. Among the city’s 1.1 million public school students — the largest school system in the nation — children of color have an 80 percent chance of attending a school where the student body consists of fewer than 10 percent white children. Fifty percent of white students attending New York City public schools are concentrated in 7 percent of the schools” (from the article I just linked to).

Continuing to look at that article sheds some light on the funding inadequacies that set some schools up to have skilled practitioners in front of every student with money spent on professional development to enhance their teaching quality, whereas schools in neighborhoods with children of color and people of lower economic backgrounds struggle to recruit and retain quality teachers. Many teachers at those struggling schools are within their first five years of teaching – I taught at a school that was struggling in the South Bronx my first year teaching, and among our teaching staff, there was an incredibly high number of first year teachers (something like 13!?). A student could go through 12 years of education and have a first-year teacher EVERY YEAR. While it’s true that some first year teachers manage to overcome the obstacles to do good work, I think even exceptional first year teachers become better with experience, and their later students benefit even more from their skills. Additionally, PA (parents’ associations) at schools raise very different amounts of money to support the school, depending on the financial capacity of the parents – at both of my last two schools, the PA’s managed to raise thousands of dollars to pay for arts instruction and partnerships beyond what the parents at my first school could ever have afforded.

With all of our “school choice” initiatives in NYC, we have created schools that are “dumping grounds” for the students that “no one else wants” – for example, in District 13 (Unison’s School’s district), 25 of the 27 publicly funded schools (including charter schools) require participation in an application process. “But as I see it, what the city described as competition turned into a segregation filter: Choice was only an option for those with the time, literacy, and determination to navigate a complex and nonstandardized admissions system. These are major hurdles for the most vulnerable families. The burden of their failure to navigate such a system hurt no one more than their children, who no rational person would argue should be able to manipulate this complexity at the age of 11.” So in effect, school choice has reinstated and reinforced school segregation in NYC. With charter schools expelling (sorry, counseling out) students twice a year (in October, after they get the money for kids with disabilities and in April, right before the state exams), there’s an influx to the schools with open seats – which are often the schools few people who understand how to “work the system” would ever choose.

Now, you might have noticed that I’ve zoomed out from just focusing in on high schools to looking at the quality of schools across the grades, from PK (and 3K!) to 12th. A big part of that is because you can’t just start trying to “equalize” the playing field in high school – it needs to start in pre-school. Otherwise, you might be setting up a student for failure. Going back to my own experiences, when I got to engineering college, I struggled in my math classes because I’d never developed the types of math study skills I needed to succeed, as I’d always been a “good student.” At the time, I had a fixed mindset, and I thought my struggles meant that I had reached the limits of my capabilities and that I didn’t belong there (especially when compared with my peers who seemed to get it easily). I now recognize that if I had taken the time to form study groups, go to tutoring (not just office hours), and to try to make sense of the material conceptually (even though it was more often taught procedurally to me), I might have done better (though no guarantee that I would have enjoyed it any more!). I also couldn’t easily develop these skills in college, having never needed them before.

So this brings me to the next part of this debate. This interesting article talks about expanding access to the gifted and talent program in elementary schools (I would bet some districts, more heavily populated by Black and Latinx people, have fewer GT programs in general) and maybe even creating a GT program for middle schools (since right now, it ends in 5th grade, but then the specialized HSs are essentially thought of as the GT of HS, so there’s a weird 3-year gap where some middle schools have either tracks or just general acceleration that take the place of GT unofficially). Now, this article also raises some points about why this increase in access to GT programs wouldn’t actually address the underlying segregation issues, two of which I’ve copied here:

  • “Gifted programs are already deeply segregated, so an expansion isn’t likely to spur more diversity absent other measures to increase integration.
  • Gifted programs would still rely on an entrance exam to determine admissions — and create a new test for middle school students. Integration advocates trace the diversity problems in gifted — and specialized high schools — back to admissions tests. Critics say that the tests advantage students with the time and resources to prepare.”

My last school was one of those middle schools that unofficially thought of itself as a GT school – it was certainly a feeder school for the specialized high schools, as many, many, many of our students went to them. And this leads me to the other reason you can’t start trying to equalize the playing field in HS – if the ability to solve math problems well is partially related to experience solving math problems (i.e. when you recognize that a new problem looks familiar, you’re more easily able to find a strategy to solve it), then more experience solving more complex math problems will result in a student who is more prepared to solve other more complex math problems. What one 8th grader knows at one school in NYC is NOT uniform across the board. My last two schools showed me that. I was teaching my 7th graders at the screened school things I didn’t even teach my 8th graders in the non-honors program at my prior school! We taught them about the Pythagorean Theorem in 7th grade (instead of 8th) and we included instruction about the “special” right triangles (30-60-90 and 45-45-90), and solving problems involving composite figures, inscribed figures, and circumscribed figures. And if they were in the accelerated class at my last school? Well, they even learned about trig ratios! Now, while none of that is officially on the SHSAT, if you think about the skills needed to solve those problems (comfort drawing in lines that aren’t there, comfort reasoning about diagrams’ angles and side lengths, etc.), they’re certainly honing that skill, and more comfortable deploying it on a problem on the SHSAT.

Now, this brings me to a small aside that I need to talk about. And I say this as someone who was in a GT program in elementary school (both for the 1st and 2nd grade that I was in public school, and the enrichment program in 3 – 8th grade at Catholic school). I’m not 100% convinced that “giftedness” exists or can be tested for or that it MATTERS. I have begun to think that much of what we think of as “giftedness” is actually related to a combination of experience and deliberate practice (though the 10,000 hour rule popularized by Malcolm Gladwell has been “debunked,” I still think there’s something to it). I’m really not convinced that there IS such a thing as “giftedness” that requires a DIFFERENT school nor that “giftedness” is actually a meaningful predictor of… anything. And I’m not alone! There’s research and articles debating whether or not giftedness exists and whether it matters. And our GT programs screening tests are norm-referenced, which means the students get scored against each other and then are placed based on being in the top ten percent… Or top three percent, depending on the space in the program! There is much controversy surrounding the history of IQ testing and race and eugenics, which I don’t think is talked about often enough by the people who tout IQ tests and giftedness tests as somehow being able to see the potential inside someone and give them more resources so they can reach their full potential (rather than recognizing that IQ testing began as a way for us to judge some humans as “more worthy” than others). I think there’s a fundamental, unanswered question here, which is “Is every human child in our school system equally worthy of our time, money, and resources?” And, sadly, the answer is often “No, some kids deserve more,” though the actual measure we use to determine WHICH kids deserve more varies person to person (and I should say, as a person committed to equity and justice for all, I acknowledge that some kids need MORE of the system’s money to support them in overcoming obstacles that other people don’t have to face, so please don’t misinterpret my statement to mean that I think we should spend the same amount of money on each kid – in fact, I think the kids who are struggling the most probably need more of the right resources dedicated to supporting them in achieving their dreams and being successful).

Now, I read articles like this, that insist on giftedness, and I think it’s important to specify here that I DO believe in neural diversity. I have recently discovered that I have ADHD, and my stepdaughter is on the Autism spectrum. Both of these things mean that our brains work differently than neuro-typical brains. This means we might react differently to stimulus than other people. This means that we might benefit from different types of activities or instruction than others – but it doesn’t mean we should be isolated in a room with only people who have the “same kind” of  brain we do (I put same kind in quotes because I don’t even think that another person with ADHD is necessarily “the same kind” of person I am – there are too many factors and variables for that one feature to necessarily link us).

I’m going to share an anecdote with you from my own personal life experience. My mom is a professional reader – no literally! Before retiring, she was a book editor. She has her PhD in literature from Harvard. So she has a lot of experience reading and a LOVE of reading. When I was a baby, she read TONS of books to me. I started pre-literacy stages as young as 2 (I was turning pages and reciting the words to my favorite book, “Chicken Soup with Rice by 2), and by kindergarten, I read over 200 books as part of a school “read-a-thon.” In first grade, I read the Lion, the Witch, and the Wardrobe, independently and had no idea it was “technically” a fifth grade reading level. I continued to read well above grade level throughout my life (though I often chose/choose to read what my middle school ELA teacher, Mrs. Smith, called Brain Candy, such as romance novels from my mom’s publishing company or genre fiction, which I still think is highly intellectual, but often scoffed at by literary folks). So let’s examine this situation: what’s the cause of my above-grade level reading? Is it the habit of going to the library every week as a child that was instilled by my mom? Is it her genetics of “good reader” that got passed on to me? Is it the sheer quantity of words and books I had read by the time I was 2? I’m not sure – but I’m also not sure that it matters.

I think that what’s MORE important is the way in which we (teachers) react to students who are presenting like I did. When I was in first grade, I went to my local public elementary school. I remember we used to use these readers where we had to read a short passage and then answer some multiple choice questions about it (presumably to assess our comprehension). I don’t recall anything specific about them beyond that, though I do think they seemed ancient and old fashioned to me at the time. I remember getting bored because I would devour those readers, and be done with five of them in the time that many of my classmates needed to read one or two of them. So I would often start talking, to engage with other students, to be stimulated again, and to enjoy myself. My mom tells me that my first grade teacher (a second year teacher) moved my desk into a corner by myself because I was too chatty (until my mom came to the school and protested, using her white, middle class privilege to demand that I not be in trouble because of my speed at reading). Now that teacher was feeling overwhelmed because she didn’t know how to handle me (and the multitude of other challenging behaviors the 30 first graders were giving her, especially when the previous year, she’d only had 25 kids). I sympathize much more with her now, having been in her shoes as a novice teacher. (I should also point out that I was in my school’s GT program at the time, but it was a small pull-out session, once a day? a few times a week? I don’t recall, so I was mostly in the gen ed room for most of the day). I only lasted in that public elementary school for two years before my mom decided to move me to Catholic school (where I flourished – but that’s a post for a different time).

However, what I think was lacking there, and often lacking from the discussion of “gifted education” in general is differentiation. I honestly think that if my teacher had been skilled in differentiated instruction (and probably had a smaller classroom size and more adults in the room!), the situation might have been very different. I think differentiation done right is one of the TRICKIEST things to do and yet, also, one of the most important things. I think there is intense value in NOT segregating kids by “ability” – I think we need to value each other, and the community that we can build when there is true diversity in the room. I won’t lie and pretend like I’ve got it all figured out: I’ve been teaching kids 8 years now, and I still struggle with differentiation at times – how to plan it, how to enact it, how to keep it manageable. I’ve done work around improving my differentiation with Rhonda Bondie, and I definitely think she’s on to something there. I also think that rich problems provide ample opportunity for self-differentiation (my preference, as a teacher!), where students can choose what avenue to explore, can deepen their understanding at its particular edges, etc. I think being given the right environment, where you can explore and ask your own questions – and try to answer them (even in math class!) is far more valuable than being mindlessly zoomed ahead through a “standardized curriculum,”

The reason I bring up all of these issues in a discussion about the specialized high schools is because I want to get to the heart of the matter. This question about how to ensure black and latinx students are equitably represented at the specialized high schools is really a different question: how do we ensure that ALL students in the ENTIRE city, no matter their zip code, etc., have a high-quality education that prepares them to be a successful adult? And that’s a harder question with no easy answers.

ALL students (not just the 18,000 served by the specialized HSs) deserve a quality education that prepares them for adulthood. ALL schools should be able to provide a quality education for ALL students so that school choice becomes an irrelevant detail. Instead of focusing on the “quality” of the schools, school choice (if we continue that system) could allow students and families to focus on other aspects of a school’s culture – perhaps the special focus (i.e. choosing to go to the academy of film and TV because of an interest in becoming a camera operator) or choosing a school that is known for trips to museums or whatever other features of a school make it unique – but ALL students are assured that the quality of their education would be equitable, no matter the school they select. To do that, I fully believe we need to tackle integration (which we’ve been discussing for YEARS, probably even decades, as this article about how to tackle segregation is from 2015), equitable school funding, teacher recruitment, retention, and training, especially focused on practical differentiation, and incorporating WAY more culturally responsive/trauma responsive teacher training (and expanding restorative justice and the focus on community building within the schools). Honestly, if students from EVERY NYC HS felt confident that the quality of their education was as good as if they had gone to the specialized high schools, then there wouldn’t necessarily be as much demand over the 18,000 spots (in a school system that educates 1.1 million, that’s a tiny sliver).

Day 1 and Week 1

So I’ve been doing a lot of thinking and reflecting about the first two weeks of school for this upcoming year. I’m not sure yet exactly what I’m going to do now that I’m changing schools. I originally had my plan for the first two weeks (since there’s only 6 days of instruction) when I knew my school, but now that I’m going to be at a different school, and I’m looking at the prior year’s opening weeks, I’m not sure what to do.

Let me lay out what my original plan was and then I’ll think about which parts to keep and which parts to change.

I used to have my students fill out a questionnaire on day 1, but after a few years of never doing anything with that information, I realized my strength was NOT in reading those papers (and making meaning or being able to USE that information in anyway), so two years ago, I did away with the questionnaire. Instead, I replaced it with the name tents from Sara Van Der Werf as both the opener and closer on day 1 (and the whole first week of school we use it to communicate). I learned WAY more from that than from the long questionnaire, and because it was so short, I was able to respond to each and every student I taught, and it built a much more meaningful first day conversation. So I definitely plan to continue to use that in my new school.

The main activity that I’ve used on the very first day of school has been basically the same exact activity since 2012, my third year teaching. I did some googling and reading on the internet and I found two websites that talked about an activity “Numbers about Me” that the two teachers used almost as a “quiz” to get to know them. They also talked about the quality of responses the students told them about important information about them. And I will confess – some years, I’ve learned more important stuff than others (like “7, the grade I was when I stopped cutting myself” – though that student later cut again in 8th grade). I like it in many ways because it gives me an excuse/fun way to introduce myself to the students using numbers and I like it because it’s original – none of my students have done a similar activity in their other classes, and activities where they write the rules all wind up feeling the same after a while (especially when you ultimately have the same rules every year!). I have shared with them the following facts every year (in multiple choice quiz format and then there’s a “reveal”). I ask a question along the lines of “The number 1 is important to Mr. Golan because…” and then three reasons, two of which are usually humorous and one of which is true.

    1. Number of children Mr. Golan has.
    2. Number of dogs Mr. Golan has.
    3. Number of states that Mr. Golan has lived in.

I then share my answer with the kids (in this case, it’s the number of dogs I have).

Screenshot 2018-07-23 13.28.38.png

I go through that process of asking a multiple choice question and then revealing information about myself about 4 more times. Last year, I shared information such as the following with my students:

  • 36: Number of weeks Mr. Golan attended Space Camp.
  • 2: Number of siblings Mr. Golan has.
  • 109: Number of board games Mr. Golan owns.
  • 5: Number of dollars Mr. Golan spent to buy his bike.

I then ask the students to write about 3 numbers that are important to them and why. Most of my sixth graders wind up only getting through one or two numbers if I make them do it in class, in about 5 – 7 minutes of writing time. And then I always inevitably have the problem where some kids write faster than others, so one kid is done WAY early and some kids still haven’t even finished writing about their first idea. Some kids choose to make it multiple choice and others choose to tell me the narrative about why that number is important. Both ways tell me a lot about the kid in some ways, but I also don’t have a great way of tracking this information, so with my memory, I wind up forgetting it as I get to know the kids. I have the kids use this activity to do their first turn-and-talk, where they partner up and share what they’ve written with their elbow partner. It gives me an opportunity to introduce some norms for the first day of school about how we talk in partnerships.

I feel like there are many pros and cons to this first day activity, and I’m on the fence about whether or not to continue using it as DAY ONE. I enjoy being able to share about myself with the kids, and I think it builds nicely into the name tents activity at the end where I invite students to share something with me. I’ve debated whether to have students finish it for HW or not to add more to it. I’ve even had students choose one to make a “poster” of in the past, but other than putting it up on a bulletin board, I rarely wind up using that. I feel like it’s not something I get super engaged with learning about the students from, even though that’s my ultimate goal. I think it’s like that classic fire hydrant in the face – I am getting too much information to take in. The last two years, I’ve had students complete the “Who am I” handout from Dan Meyer, and I’ve retained that handout much into the school year. I feel like that information I’m much more likely to go back to, though there are a few that I want to change. Graduating class always confuses my sixth graders (and graduating from which grade? 8th? 12th?). I also feel like there were things I wanted to know about students that weren’t included in there, but I can’t recall them now, off the top of my head.

Recently, I was reading twitter and I came across a new blog from Jess. I really like what she described as her ideas for the first day of her classroom in her blog, but I’m not sure whether it makes as much sense coming from their math teacher in middle school… So I might also see what the other teachers are planning for their first day activities.

The rest of my day 1 tends to be collecting summer assignments and giving out the HW and supplies list/welcome letter and then giving students enough time to respond to the name tent activity. Although, that’s been in a 44 minute period in the past, and this year, the new school I teach at has hour long periods, so I might be able to do a little bit more on day 1… The biggest downside to day 1 as I see is it that it’s very me-focused in terms of teacher talk (the kids do a turn and talk about their important numbers, and I’ve typically asked a few kids to share out at the end – often about something they’ve learned about their PARTNER, as opposed to sharing their OWN ideas, but I haven’t done that in a few years). Otherwise, they really don’t have an opportunity to talk; they just do a LOT of listening! And, there’s also no math on day 1! So these are the two reasons I’m questioning my choices – while it IS a fun activity and helps build relationships (the number one priority in all of September), I worry that it sends the wrong message for the first day of class.

Anyway, we’ll revisit day 1 ideas after I discuss the rest of the first two weeks.

On Day 2, I’ve typically done a Growth Mindset introduction the last few years. The kids read over their name tent feedback and set them up, they watch some videos from Jo Boaler, and I have them do an exit ticket about “I used to think… but now I know…” regarding ideas around intelligence. This, combined with the posters that say “change your words, change your mindset” make for a great beginning of school year bulletin board. With this activity, I do a lot of turn and talks after each video we watch, and I have whole class share outs to discuss some of the ideas we’re hearing that may feel new or different. I really find this to be a valuable activity for the students, though I’m also wondering about combining it with the Talking Points activity that James Cleaveland created and shared here. I think considering my new school uses mastery-based grading (and Jump Rope) and my 6th graders may never have been assessed like that (and I’ve never done assessing like that), and it fits in line with some of the ideas about the way our intelligence can be grown (and not comparing ourselves with each other, but only with our past selves, etc), I think it might make a nice conclusion to day 2.

However, once again, day 2 concludes with no REAL MATH. I’m on the fence about this delay. On the one hand, I value the importance of community building and norm building and relationship building. I think that we can’t do any real math, I can’t ask my students to be vulnerable and trust in each other and me if we haven’t done the pre-work to set up that type of community… At the same time, how many days without math do we need for that? Is there a way to accomplish some of those same ideas WITH math embedded? I don’t have an answer for that.

Day 3 is another homage to Sara Van Der Werf, as I use her 100 numbers activity.  I found that it’s a great way to take photos of my students and discuss what good group work should look like and sound like. I think it’s especially important because one of my personal focuses this year is on the tension and balance between independence and interdependence (see my previous post), and I realized that sixth graders don’t always know how to listen to each other and do successful turn-and-talks! They don’t know how to share the air (or that it’s important that we SHOULD!), so I think it’s important to spend some time discussing these norms. I just noticed that Sara also includes print outs of the photos she takes, and I LOVE that idea – and I think I will try to use it this year! Once again, though, we have day 3 with minimal math. At least this day, there’s MUCH LESS of me talking, and way more of them talking. I also use this activity to discuss what mathematics IS (I think I had my students do the tweet “#mathis” activity last year  that I got from Sarah Carter’s blog, Math = Love).

Over the course of these three days, I elicit from my that math is the study of patterns and that mathematicians make sense of math by convincing themselves (through independent think time), convince mathematical friends, and convince skeptics. We use that word skeptics A LOT over the course of the year, and I have two posters to match those two ideas. In the past, I’ve used the next week of classes to further set up our problem solving and listening community by using math tasks from Jo Boaler’s Week of Inspirational Math. This year, my original plan was to use three contemplate-then-calculate tasks around area because my previous school’s first unit was going to be a modified Illustrative Mathematics first unit, which made area its first unit. My original plan was to do three days of CthenC, starting with the same pattern David Wees and I created to launch it this last year (basically the number of squares arranged in a rectangle, where the rectangle’s area is 2n + 2), then using the circles set up in an array with a “hole” in them from Illustrative Mathematics’ grade 6, unit 1.6.1, and then using the visual patterns #43. In thinking about each of those three tasks, I felt like they each leaned a bit more towards one of the three different types of “structural” thinking – the first one feels CONNECTED to area, the second one feels like they’re most likely to SUBTRACT the missing CHUNK (or possibly to rearrange), and the final one seems most likely to be CHANGED. That would give us some structural language to use from the get-go.

The BIG difference is that now, instead of my first unit being about area (and thus directly using those skills I was going to introduce from CthenC), my new school is using the CMP3 curriculum still. Our first unit is Prime Time, which deals with factors, multiples, LCM/GCF, prime factorization, the locker problem, even and odd numbers through rectangles, and eventually order of operations and the distributive property. I haven’t done the math yet for the distributive property problems in there, but I wonder whether it is introduced better there or in the IM curriculum – I’m going to decide over the summer which way to use to teach it. I also REALLY like the way I’ve done OofO the last two years through the NCTM article about “The Truth about PEMDAS,” but I don’t know if my students will need more of an introduction than the Boss Triangle, or if we can start from there.

I guess my big question is how to incorporate more math-doing to the first three days of school to get my students thinking like mathematicians sooner – but then also wondering whether I need to rush that, or if it’s ok to delay that for a few days… What are your thoughts? Feel free to reply here or to tweet at me @MrKitMath

Goal Setting in Math (part 2)

Tonight, I was talking with Rhonda Bondie about Goal Setting in math classes. I was sharing with her my observations from the previous post that most student goals either focused on their grades, their HW completion, or their participation in class discussions. It rarely focused on their actual math “skills” or strategies or thinking. I acknowledged that it was harder in math class because there was a tension in thinking classrooms between revealing too much information by revealing the topic of study too early (i.e. telling students that we’re going to discover the Pythagorean Theorem might ruin the discovery if a student has been introduced to the formula already in SHSAT prep or Saturday School, whereas telling students we’re looking for a pattern in the areas of the squares doesn’t give it away, but might make it a bit harder to set a goal).

Except, maybe it wouldn’t make it harder! Maybe THAT was the key to setting the goals. I realized that vague “we’re looking for a pattern” connected to what it is mathematicians do (look for, study and analyze patterns), and that maybe the Math Practices were a better way to set up our student goals. And then I made a connection to my absolute favorite diagram from from Amy Lucenta’s and Grace Kelemanik’s book, Routines for Reasoning.


In the book, they argue that the 8 Math Practices are not actually created equal, but instead, there is a hierarchy.

“MP 1: Make Sense of Problems and Persevere in Solving them is an overarching goal.” MP2: reason abstractly and quantitatively, MP7: look for and make use of structure, and MP8: look for and express regularity in repeated reasoning “describe three avenues for mathematical thinking” that allow you to solve problems. The remaining practices, MP3: construct viable arguments and critique the reasoning of others, MP4: model with mathematics, MP5: Use appropriate tools strategically, and MP6: attend to precision, all describe important ways to navigate those three avenues of thinking and play an important role in problem solving, no matter which avenue of thinking you choose. (I’ve paraphrased/summarized/quoted various ideas from pages 3 – 10 in their book in this paragraph)

This made me think about introducing the following ideas to my students at the beginning of the year.

“In this class, our goal is to think like mathematicians. One thing mathematicians do is make sense of problems and persevere in solving them. The way they solve them is by following a particular avenue of thinking. Our goal is to get good at all three avenues, because sometimes we might find a particular type of problem is more easily solved using one type of reasoning than another. We might start using one avenue that we’re typically more comfortable with, and then get stuck as we start to solve the problem. We might have to try a different avenue of thinking to get unstuck. Other times, we might successfully solve a problem using one avenue of thinking, but it doesn’t give us sufficient evidence to convince a skeptic, so we might need to solve a problem using multiple avenues of thinking to convince the skeptics.

As we navigate these three avenues for thinking, there are some important things to keep in mind. We need to be able to construct arguments using our reasoning and critique the reasoning of others, we need to be able to model with mathematics, to use tools strategically, and to attend to precision.

So these eight ideas set our goals for the year. A student who has mastered all 8 has achieved our goal for thinking like mathematicians. They’ve grown in a way that’s more comparable to the previous year as well – for example, maybe in 6th grade, they were struggling with looking for structure and making use of it, so in 7th grade, they’re going to focus on that avenue of thinking. And maybe by the end of 7th grade, they’ve started to master looking for and noticing structure, but they’re still not quite sure how to actually make use of it, so in 8th grade, they’re going to focus on using the structures they notice to solve the problems.

Now, in Grace & Amy’s book, they have an appendix with the following chart:

Screenshot 2018-07-18 02.19.52

You’ll notice that there’s clear questions to “ask yourself” for each avenue and there’s clear “actions” to take when solving problems using each avenue. Let’s focus in on what the rubric might look like now. I’m going to choose just MP7 about Structure to focus on, because I’m most familiar with that practice.

Here’s the CC’s text:

CCSS.MATH.PRACTICE.MP7 Look for and make use of structure.

Mathematically proficient students look closely to discern a pattern or structure. Young students, for example, might notice that three and seven more is the same amount as seven and three more, or they may sort a collection of shapes according to how many sides the shapes have. Later, students will see 7 × 8 equals the well remembered 7 × 5 + 7 × 3, in preparation for learning about the distributive property. In the expression x2 + 9x + 14, older students can see the 14 as 2 × 7 and the 9 as 2 + 7. They recognize the significance of an existing line in a geometric figure and can use the strategy of drawing an auxiliary line for solving problems. They also can step back for an overview and shift perspective. They can see complicated things, such as some algebraic expressions, as single objects or as being composed of several objects. For example, they can see 5 – 3(x – y)2 as 5 minus a positive number times a square and use that to realize that its value cannot be more than 5 for any real numbers x and y.

The Ask Yourself Questions: The Actions:
·       What type of problem is this?

·       Does this remind me of another problem situation?

·       How is this (situation, object, process, etc.) behaving? Can I connect it to something else I know?

·       What are the parts (chunks) of the process?

·       How can I get the answer without doing all the calculations?

·       How can I use properties to uncover structure?

·       How can I change the form of this (number, expression, shape) to surface the underlying structure?

·       Chunk complicated mathematical objects (expressions, shapes, etc.).

·       Connect representations.

·       Change the form of the number, expression, space, e.g., create equivalent expressions.

·       Recall and use properties, rules of operations, and geometric relationships

The Ask Yourself Questions:

  • What type of problem is this?
  • Does this remind me of another problem situation?
  • How is this (situation, object, process, etc.) behaving? Can I connect it to something else I know?
  • What are the parts (chunks) of the process?
  • How can I get the answer without doing all the calculations?
  • How can I use properties to uncover structure?
  • How can I change the form of this (number, expression, shape) to surface the underlying structure?

The Actions:

  • Chunk complicated mathematical objects (expressions, shapes, etc.).
  • Connect representations.
  • Change the form of the number, expression, space, e.g., create equivalent expressions.
  • Recall and use properties, rules of operations, and geometric relationships

Now, I’m currently envisioning a few different ways students could assess themselves about how well they’re using this avenue for thinking.

One, you could put it into the same four-point mastery scale from my last post, though I do find the “above grade level” to be a bit challenging here. One thing we haven’t specified is that deploying this practice looks different at different grade levels (in part because of the types of content the students are exploring), but also because their level of independence with this may vary.

1: Below Mastery 2: Approaching Mastery 3: Proficient(At grade level) 4: Mastery (above grade level)
Rarely uses the ask yourself questions OR cannot make sense of the math problem through asking the questions.

Cannot see chunks or ways to change the form. Does not make connections between representations.

Does not recall or use relevant properties or relationships.

Does not result in successful solution NOR do they switch to a more successful avenue.

Sometimes uses the ask yourself questions to make sense of math problems.

Is sometimes able to successfully chunk, connect, or change the form to solve problems.

Recalls and uses only some of the relevant properties or relationships.

May or may not result in successful solution.

Regularly uses most of the 7 “ask yourself” questions to make sense of math problems.

Chunks, connects, or changes the form to solve problems.

Recalls and uses relevant properties and relationships.

Results in successful solution.

Regularly uses all 7 “ask yourself” questions to make sense of math problems.

Chunks, connects, AND changes the form to solve problems.

Recalls and uses relevant properties and relationships.

Results in successful solution that student can verify using a second avenue for thinking.

Gradeless Gradeless Gradeless Gradeless

Another way I could see assessing it is by using Rhonda Bondie’s “Must Have” and “Amazing” criteria. For example (and I’m not sure I’m in love with this way yet – it’s my first draft still):

Must Have Amazing
·      Ask yourself at least 2 of the “ask yourself” questions to make sense of the problem.

·      Chunks, changes, or connects.

·      Valid mathematical thinking shown that begins the problem.

·      Ask yourself at least 5 of the “ask yourself” questions to make sense of the problem.

·      Does at least two: chunk, change, or connect

·      Can justify why it makes sense to chunk, change, or connect in the way that they did

·      Recalls and uses properties, rules of operations and geometric relationships

·      Can use another avenue for thinking to solve the problem and verify answer.

If you needed to use a 4-point mastery scale, you could do something like:

1: Most of the Must-haves

2: all of the Must-Haves

3: All of the Must-Haves and some of the Amazing

4: All of the Must-Haves and ALL of the Amazing (or MOST, depending)

I’m not sure which of these two ways I like better yet, but I recognize they’re not quite equivalent. Either way, I can now see much more specific goal setting around mathematical thinking if students become comfortable with the math practices as avenues for thinking.

They might say “My goal is to use structure to solve at least two problems in the next unit.” or they might say “My goal is to use two avenues for thinking to solve every problem.” or they might say “My goal is to get better at asking myself these questions when I use this avenue.” or they might focus even more narrowly on something like, “I’m going to improve my ability to chunk problems, and I’m going to use that strategy on at least two problems.”

Now it even makes sense for a teacher to share from one year to the next a student’s goals. Well, Kit was very strong at using MP7: make use of structure, but he really struggled with reasoning abstractly and quantitatively. Encourage him to develop goals around MP2 next year.

I want to do more fleshing these ideas out over the course of the summer. This connect came organically out of a conversation I was having with Rhonda this evening.

What do other people think about these ideas? I’d love to hear from you on here or on twitter/FB!

Goal Setting in Math vs. ELA (Part 1)

A few years ago, I was in a meeting to discuss vertical alignment among the math departments at my middle school. We were looking to trace how the skills and standards built from 6th grade to 8th grade in our math classes. One of the members of the meeting was a special educator who worked with both the math and ELA departments, and she introduced us to a document that the ELA department had been working on where they looked at how the standards built vertically as well. In that moment, I was suddenly even more jealous of ELA teachers than I already was (sometimes, I think I should’ve become an ELA teacher – but that’s a different post!).

If you look at the ELA common core standards across the domains and grade levels, they build on each other very directly. I randomly chose Reading: Literature to look at across the three grades. I chose the first standard within that strand, and I followed it from 4th grade through 10th grade.

Refer to details and examples in a text when explaining what the text says explicitly and when drawing inferences from the text.

Quote accurately from a text when explaining what the text says explicitly and when drawing inferences from the text.

Cite textual evidence to support analysis of what the text says explicitly as well as inferences drawn from the text.

Cite several pieces of textual evidence to support analysis of what the text says explicitly as well as inferences drawn from the text.

Cite the textual evidence that most strongly supports an analysis of what the text says explicitly as well as inferences drawn from the text.

Cite strong and thorough textual evidence to support analysis of what the text says explicitly as well as inferences drawn from the text.

Look at how connected these 6 standards are. In all of them, they reference using textual evidence, but they do so in more and less sophisticated ways across the grade levels. At first, kids are expected to just refer to details, then be able to quote specifics, and eventually they’re citing evidence – first any evidence, then several pieces of evidence, then choosing the strongest evidence, and ultimately, being strong and thorough in their citations. All of the standards mention both what the text says explicitly as well as drawing inferences from the text.

This makes vertical alignment pretty easy and it makes it more obvious when a student is above or below grade level in this standard – for example, if a student is only citing one piece of evidence and they’re in 7th grade, you can see they might be operating at a 6th grade level, whereas if a 7th grade student is citing the strongest piece of evidence, then they might be operating above grade level. I confess there seem to be some degree of subjectiveness on the part of the teacher about assessing whether the evidence cited is the “strongest possible…” But there’s no question in my mind that these 6 standards are linked directly, and that there is a growth in the student.

If I’m a middle school student, I can even do a self-evaluation, rating myself on a four-point mastery scale relatively easily. Let’s say I’m a 7th grade student. I might consider my skills along the following rubric:

1: Below Mastery 2: Approaching Mastery 3: Proficient

(At grade level)

4: Mastery (above grade level)
Quote accurately from a text when explaining what the text says explicitly and when drawing inferences from the text.


Cite textual evidence to support analysis of what the text says explicitly as well as inferences drawn from the text.


Cite several pieces of textual evidence to support analysis of what the text says explicitly as well as inferences drawn from the text.


Cite the textual evidence that most strongly supports an analysis of what the text says explicitly as well as inferences drawn from the text.


Grade 5 Grade 6 Grade 7 Grade 8

Now, admittedly, it’s possible that even as a 7th grade student, if I’m below grade level, I might not even be quoting accurately from the text. But, there’s definitely a progression of the skill, where I can self-assess “Where am I now?” and then “What’s my goal?” And this goal even hints at the above-grade level work for the following year, allowing a student to push themselves. If I’m struggling with the current work, I can even see perhaps where I should have been developing it – if I’m not able to cite several pieces, am I at least citing one? Am I at least quoting something accurately? If not, then I can set a SMART goal: “In the next book club chat, I will cite at least two pieces of textual evidence to support my inference.”

Let’s contrast that with mathematics for a moment. First off, it’s practically impossible to trace a domain by code the same way I did with ELA – from elementary to middle school, the domains change names, and in 8th grade, one domain is replaced with another! I chose to look at Numbers &  Operations in Base ten in Elementary and the Number System in Middle school, as they seemed to flow together. Again, I chose just the first standard in each grade level at this domain. You can see for yourself how much less clear the thread is connecting these.

Recognize that in a multi-digit whole number, a digit in one place represents ten times what it represents in the place to its right. For example, recognize that 700 ÷ 70 = 10 by applying concepts of place value and division.

Recognize that in a multi-digit number, a digit in one place represents 10 times as much as it represents in the place to its right and 1/10 of what it represents in the place to its left.

Interpret and compute quotients of fractions, and solve word problems involving division of fractions by fractions, e.g., by using visual fraction models and equations to represent the problem. For example, create a story context for (2/3) ÷ (3/4) and use a visual fraction model to show the quotient; use the relationship between multiplication and division to explain that (2/3) ÷ (3/4) = 8/9 because 3/4 of 8/9 is 2/3. (In general, (a/b) ÷ (c/d) = ad/bc.) How much chocolate will each person get if 3 people share 1/2 lb of chocolate equally? How many 3/4-cup servings are in 2/3 of a cup of yogurt? How wide is a rectangular strip of land with length 3/4 mi and area 1/2 square mi?.

Apply and extend previous understandings of addition and subtraction to add and subtract rational numbers; represent addition and subtraction on a horizontal or vertical number line diagram.

Know that numbers that are not rational are called irrational. Understand informally that every number has a decimal expansion; for rational numbers show that the decimal expansion repeats eventually, and convert a decimal expansion which repeats eventually into a rational number.

Explain how the definition of the meaning of rational exponents follows from extending the properties of integer exponents to those values, allowing for a notation for radicals in terms of rational exponents. For example, we define 51/3 to be the cube root of 5 because we want (51/3)3 = 5(1/3)3 to hold, so (51/3)3 must equal 5.

What the heck?? It’s practically impossible to see how knowing there are numbers that are not rational is even connected to adding and subtracting integers or how either of those ideas are connected to fraction quotients. Other than the use of the words rational and integer in the RN.A.1 (which comes from the high-school “Number System” standards), I’m not sure that you can see how this standard builds on the middle school ones either.

They do a much better job at connecting the two from elementary school, but maybe that was because of the different domain name? 5.NBT.1 builds clearly on 4.NBT.1, because now in addition to recognizing that a digit in one place of a multi-digit number represents ten times as much as the place to its right, students will also understand that it represents 1/10 of what it represents in the place to its left.

There are other planning documents that show the progression of standards, as you can find here:

And Randall Charles does a great job at summarizing big ideas in mathematics which you can link to the standards:

Cathy Fosnot has place big ideas in the landscape for learning which also includes models and strategies.

All of this tells me there seems to be an extra layer to understanding the way math content is connected to the big picture that doesn’t seem to exist in the ELA standards. It has always made me wonder how helpful information from previous year teachers was in math, especially in middle school – for example, does a student’s understanding of how to add and subtract integers impact their ability to identify a number as rational or irrational? I think not. That’s not to say that the skills for identifying a number as rational or irrational aren’t laid down in earlier grades or carried through to later grades. Quite the contrary – you just have to know which skills are related and how this is all connected. No easy task for a teacher who’s busy mastering their own grade’s standards – hence the progressions, and this other ideas I shared here.

This brings me back to the meeting I was sitting in: vertical alignment within the math department. It’s harder to identify when a student is above or below grade level if you don’t know what your grade-level standard looks like in earlier/later grade levels. It’s harder to differentiate for a student if you don’t know or understand how the math builds over the years. And the arrangement of the standards in math does nothing to shed light on it.

I also couldn’t help but notice the common core’s website is differently arranged for math vs ELA – in the math section, you must click on a grade first (K – 8) and then on the domain, and then on individual standards, whereas in ELA, you needed to choose a domain FIRST, then a grade! That speaks to the idea of vertical alignment more than the discrete topics and standards arrangement that pervade the math standards structure – which also seems to indicate the way that many people seem to consider ELA skills to build on each other whereas many people often view mathematics as discrete topics or skills that are to be memorized (but are seemingly disconnected from each other).

Recognizing this challenge of the math standards made the idea of goal setting challenging. How can you set a goal and see where you were, where you are, and where you’re going without giving away some of the story in mathematics? If I want to develop a conceptual understanding of the Pythagorean Theorem using the areas of tilted squares on grid paper (thanks Betina Zolkower and CMP3 and MAP), then I need to know if the student understands how to find the area of rectangles, triangles, and tilted squares. I need to know if they have already heard of the formula a^2 + b^2 = c^2. I need to know what they know about triangles, specifically right triangles. Understanding the triangle inequality theorem may be helpful, though it’s not necessary. Once they know what it is, they may learn the converse of it, how to use it to find missing side lengths, and how to expand it to work in three dimensions. This connects to their work with exponents (what does that symbol mean? How do you “undo” it? etc.) as well as with solving equations (if you push them to work algebraically vs. working arithmetically).  Eventually, they should understand that the Pythagorean Theorem is generalized for non-right triangles as the law of cosines. They will learn about trig for right triangles eventually as well, and they might study special cases of right triangles. The Pythagorean Theorem is also connected to similar triangles and what they may know about angles of a triangle (including that the sum of the interior angles is 180 degrees). This theorem might also help them as they find the areas and perimeters of composite shapes involving triangles and/or inscribed and circumscribed triangles. They will also eventually learn how to prove the theorem – and they may be introduced to any of the 2000+ proofs that exist.

I bring all of these ideas up to illustrate that it’s much more difficult for a student to self assess where they are in terms of their prior knowledge, their current understanding, and what their goals should be in mathematics class. If I’m a student who is in middle school, just learning about the Pythagorean Theorem, how do I assess which aspect I’m struggling with, or how to go above grade level, or how to set a goal? I don’t have nearly as clear of an idea just based on looking at the standard or even if I looked at a rubric. I’m not sure I could even think about what a rubric would look like for this topic in the same way.

The standards about this theorem exist in 8th grade:

Explain a proof of the Pythagorean Theorem and its converse.
Apply the Pythagorean Theorem to determine unknown side lengths in right triangles in real-world and mathematical problems in two and three dimensions.
Apply the Pythagorean Theorem to find the distance between two points in a coordinate system.
It’s talked about explicitly again here:
Prove theorems about triangles. Theorems include: a line parallel to one side of a triangle divides the other two proportionally, and conversely; the Pythagorean Theorem proved using triangle similarity.
Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems.*
It’s the underpinnings for these two standards, but you need to already know the law of cosines is connected to PT to know that:
(+) Prove the Laws of Sines and Cosines and use them to solve problems.
(+) Understand and apply the Law of Sines and the Law of Cosines to find unknown measurements in right and non-right triangles (e.g., surveying problems, resultant forces).
So let’s see: if I was an 8th grader, learning about the Pythagorean theorem, would I necessarily be able to use a rubric to understand where I was in my understanding? I’m not sure. Here’s one attempt at making a rubric for it…
1: Below Mastery 2: Approaching Mastery 3: Proficient (At grade level) 4: Mastery (above grade level)
Apply the Pythagorean Theorem to determine unknown side lengths in right triangles in mathematical problems in two and dimensions. Apply the Pythagorean Theorem to determine unknown side lengths in right triangles in real-world and mathematical problems in two dimensions. Apply the Pythagorean Theorem to determine unknown side lengths in right triangles in real-world and mathematical problems in two and three dimensions. Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems.*

(NOTE: Big jump to understand what trigonometric ratios are!)

Grade 8 modified Grade 8 modified Grade 8 HS Geometry
When you look at this rubric, I tried to modify the 8th grade standard a bit to show partial understanding. I figure if grade level requires using it in both 2D and 3D contexts, then only being able to use it in the 2D context would be one example of approaching grade level. For level 1, I struggled with what to write, ultimately eliminating the real world problems, thinking this is basically a student who is just plugging and chugging numbers, but doesn’t necessarily understand what they’re doing or how to apply it to a context. (Though, I fully admit that a student who is below grade level in this standard actually might have even bigger challenges – they might not know how to square numbers or how to follow the order of operations or how to solve an equation with exponents – any of those skills could be preventing them from being able to apply this formula). For the above grade level, I tried to pull in the related HS geometry standard – but it mentions trig ratios, and those aren’t something that a student is going to be able to just “push themselves to do” based on what they know so far – in 8th grade, they may never have even heard of trig yet! (depends on their teacher, their school, etc.).  An alternate possibility in the above grade level section could be applying the formula on composite figures, inscribed/circumscribed triangles, or finding patterns in special right triangles.
This means it’s pretty hard to set goals and self-assess. How do I recognize if my difficulty is that I’m forgetting to square the numbers first, and I’m adding a + b and then squaring it? How do I notice that needs to be my fix? If I can use it in two dimensions but not three, I feel like that might be easier to recognize and set a goal around, but I (as a student) might not be sure HOW to figure out how to use it in three dimensions without outside support, whereas I feel like a student who already knows how to cite one piece of evidence can more easily push themselves to cite at least two (or even to choose the BEST evidence) relatively individually.
Today, in my conversation with Rhonda Bondie about goal setting in math, I had an epiphany. All this time, I’ve been trying to wrap my mind around having students set goals related to the math content standards. But they don’t flow well from grade level to grade level, and it’s not always super obvious how they’re connected. It’s hard to set a goal when you don’t have a clear idea about where you are, what your struggles or, or what your goals are. Feedback (from peers and teachers) also plays a role in goal-setting, as the feedback may cue you in on your strengths and struggles. But how do you give each student specific feedback every day in every content standard during the unit? So often, it feels like student goals (when I’ve had them set them before) are either focused on the “wrong” numbers – their grades (I’m going to earn a 90 on all of my quizzes, or on my exams, or on my report card), their HW completion (I’m going to do my HW every day), or their participation in class (I’m going to raise my hand more, or participate in class discussions more). I rarely hear/see my students set goals like “I’m going to learn how to solve problems with the Pythagorean Theorem in 3 dimensions,” – though occasionally, especially at the start of the year, I’ll hear a student set a goal related to math content from the previous year, like my student who said “I want to master long division because I don’t really feel comfortable dividing.” Sometimes, these content goals from the previous year are good goals (in this case, it was a sixth grader, so it was relevant!), but other times, those goals aren’t directly related to the new skills of the grade level (i.e. if an 8th grader said “I’m going to master how to do integer operations” – it’s an important skill for 8th grade standards, but it’s not in-and-of-itself related to any 8th grade standards!).
And then, in the course of my conversation with Rhonda, I had an epiphany about goal setting in math class. I’ll share that in my next post, so stay tuned!