Carl's Teaching Blog

A place to talk about teaching and learning

Category: Philosophy (Page 1 of 3)

My Biggest Current Hope For Math Education

Insight struck twice in our living room a couple weeks ago as my daughter was teaching herself the Peppa Pig theme song, with some help from us. We played it for her, and put some stickers on the keys to help her, but it took a long time for her to actually start playing it in sequence. When she was actually learning this she would sort of work away quietly, until she hit a point where she needed a break. Then she would get really silly, blow raspberries or joking around. After about 4 or 5 rounds of tunnel-vision followed by silliness, she eventually had the song mastered. I was left thinking about what her learning style might mean for her future schooling, as it seems to be quite similar to mine.

Julianne’s focus was as intense as her silliness and she shifted back and forth between the two so quickly, it seemed like they were both part of her problem solving process. This could be a headache for her future teachers. As an elementary schooler, she might be the student who tells jokes, distracts their friends, and gets notes home about behavior, but can answer the questions and tear through the worksheets. It’ll be hard to find success in a skill and drill class. Ideally she would be in a class focused on solving rich problems, explaining big ideas, and collaboration may keep her too busy to get in silly mode, or channel that energy towards something productive. Any teacher who can embrace her learning process and not make Julianne feel like an outcast would be good. Most teachers don’t tolerate a lot of silliness, preferring to punish the student or push them out of class until they get in line. Hopefully, Julianne’s future teacher will focus on creating a teaching environment that funnels Julianne’s focus and silliness, along with the energy that the other students bring, through activities that give students room to explore their curiosity.

There are three big things that give me a lot of hope that Julianne will find the right teacher. One is the internet’s growing collection of math activities, instructional routines, and curricula that speak to a variety of learners. Everyday more lesson ideas are posted on the #MTBoS and around the web for teachers to find. Another source of hope is the increased agency that teachers are given from their administrators, curriculum writers, and school districts. In NYC, teachers are expected to have rich discussions, facilitate group work, and customize or supplement lessons using their discretion. All of this would be unthinkable when I was in school. Lastly, because teachers have materials, and agency, they can make decisions that affirm the identities of their students, prevent the marginalization of students, and seek to disrupt systems of oppression and microaggressions at play in their school. Based on ideas happening on Twitter, with curriculum, and in my district, it seems like the focus on how identity affects the classroom will be at the forefront, going forward. This makes me very optimistic for what my daughter’s teacher will be able to accomplish.

My daughter, a biracial girl, will be treated differently in Brooklyn than me, the only black student in my suburban Detroit 5th grade. We learn similarly, so it may be useful to compare one of my school experiences where a teacher writhed my identity. Of course, the teachers of my classes, and her classes would have to make different choices because of the different culture of the school and our different identities.

I HATED social studies from the beginning of 5th grade. Mr Brownbear, along with my other teachers, sent me to the office so often I thought it was my assigned seat. One day, we took turns drawing little slips of paper from an envelope, each with names for our colonial explorers research project. My turn came, and the teacher pulled me aside with the same tone that I hear before I’d get a pink slip. I wasn’t in trouble  He didn’t let me draw from the same envelope as everyone else, instead giving me a handwritten slip he’d been saving for me. It read “Jean Baptiste Point du Sable.” I learned he was the founder of Chicago, and also a Black man.

Mr Brown Bear validated my identity and it made an unforgettable impact on me and probably on his teaching as well. Mr Brownbear must have made some changes too. He must have looked critically at the materials he was going to teach, and used his agency to change the task to reach the dual goals of satisfying the content and validating my identity. I’d had issues around being the only Black student in my grade which were probably well known among the teachers, so it was pretty clear that he was using his discretion as an attempt to expand the discussion around a unit that would otherwise be dominated by Whiteness. The fact that I’m still talking about shows that teachers who can make these kinds of choices can be impactful. I just wish I could do more than hope that my daughter’s teacher will exercise their discretion in similar ways.

My daughter in the kitchen this week and me at age 5 or so.

Looking to validate kids identities, racial and ethnic backgrounds, gender, languages, should be a natural part of planning for teachers today. With a preponderance of activities, technology, and more ways than ever to collaborate, this  work will soon become a necessity for teachers. The right choice could also alter the focus/silliness ratio for kids like me for weeks, even years. My experience with Mr Brownbear was in a social studies, and I can’t recall any similarly affirming experiences from math. Math was very abstract, rarely relevant, and often with timed tests that made me really hate the subject. My math teacher wouldn’t have the palette of options for affirming kids identities, making math culturally relevant, and eliminating the negative effects of status that will be available to my daughter’s teacher.

Julianne’s teacher will have a lot more agency, and a lot more materials, so her education will be great and she will love math…right? Despite all that’s possible, this future teacher’s discretion will determine what my daughter’s math class feels like. Maybe they will discern that Julianne’s silliness is disrupting the others and kick her out of class. Maybe the teacher will find a way to modify a lesson that affirms my daughter’s identity. The teacher’s discretion will have a great impact in a student feeling confident among their peers, and feeling confident as a mathematician. What’s scary is that it’s hard to know how it will affect the class. What’s even scarier is that the teacher may not know either. The use of discretion will be influenced by the implicit biases that the teacher may not even be aware that they hold. Deborah Ball, whose 2018 AERA talk influenced a lot of my thinking. In the talk she dissects a short clip of her teaching to illustrate the way she used her discretion to support the class seeing black girl brilliance. It is a great clip that everyone should find the time to watch. The increased attention that all of this is getting from the research community and on twitter makes me hopeful. I’m hopeful that maybe the people in the research community and on twitter can start to create a conversation about students’ identities and how they can be validated as they are learning math. Especially if they are kids who can be a little silly.

What’s my role in the inevitable conversation about race in math education, and the #MTBoS?

First off, I’ve experienced the #MTBoS as a very welcoming place. Its certainly very white, but I’m used to that. At three I moved to an entirely white neighborhood, was the only black student in my class through 5 years of elementary school, and I went to primarily white MS, HS and Colleges (MSU, Harvard & Columbia). I can count the number of black males in my undergrad math education classes on my thumbs. When I reflect on how I made it this far, luck is the main answer. A slight turn of events could have led me to a trajectory that is out of this profession, or towards incarceration, or left me dead. A busted tail light and a cop with a mean streak could have easily ended my life because those are the values our country espouses, but I was lucky. My parents were also lucky, and wanted to risk giving their kid an education in a good school in the suburbs. Because of all this luck I became a math teacher and an assistant principal, and have become very comfortable with learning in primarily white spaces along the way.

In most of these spaces, conversations about race is avoided, or serves to further existing beliefs. When it comes up it feels like you are simultaneously on trial, the expert witness, and the lawyer presenting the case to a jury that you may need to ask for help later or just tension friends with. One day in middle school Dexter Adams was saying to a rapt audience about how his parents drove through Detroit and heard gun shots. I spend the night at my Grandma’s once a month, and I know that those sounds were probably fireworks or cars backfiring. After some deliberation, I remember deciding it wasn’t worth it to reply. It’s anxiety provoking.

It’s even more anxiety provoking in college where you’re trying to sub-consciously prove you’re not the affirmative-action case that people seem to think you are. Speaking out is a 3-ring circus. If you’re making a point or sharing your experiences, be sure to 1) get the point across, lest they miss it and go out and ruin kids lives, 2) be positive enough to ensure they won’t hate you for challenging their long-held beliefs, only to wind up a partner on a project later, and 3) be entertaining enough that they can’t exercise the privilege of checking out. It’s hard to do even if you have great teachers and classmates, which I’ve been lucky to have. It’s still hard to garner the emotional energy to have these conversations in my primarily white school now, where I swear everyone is on the same page.

I was introduced to the #MTBoS through friends I had met that I wanted to keep in touch with. In posting and sharing the things that I wanted to work on, mainly PBL and PrBL, I was able to quickly find a number of people with great ideas, abundant resources and welcoming positive energy. Despite everyone’s different contexts and perspectives, everyone is willing to chip in because we are all fighting this larger, asynchronous, guerrilla war against traditional math education. Conversations about race, social justice, the whiteness present in math education have come up over time and they are handled well. When they happen, the conversations do a good job of unearthing the experiences facing us #MTBoSers, and the populations we serve, as we all navigate the systemic injustices that emerge in our classrooms. Maybe some of the conversations are successful because they result in people understanding someone else’s perspective, or further one or two people’s thinking about something. Along the way some people get upset, unfollow people, and send tweets to let everyone know they are checking out, “I came here for math, but what you’re posting now scares me…” But the conversation keeps moving forward. Forward towards giving all kids a meaningful math education sanitized of the white male privilege that has been baked in through so many years.

There is nothing wrong about saying the #MTBoS is a community that started with lots of white people. That is a result of the system that we live in. If all the math programs, and all the teacher programs are primarily white, it follows that math education twitter will also be primarily white. The danger is whether we don’t questing the role that whiteness plays in determining the culture of math that students experience. Math has been a vehicle that reproduces societal inequalities while being billed as meritocratic and immune from biases. If the #MTBoS set out on the journey of reshaping our students’ math experiences it was certainly only a matter of time before they must deal with the latent white male privilege. This is now complicated because, the #MTBoS, and math education, is primarily white.

In looking at the #MTBoS’s whiteness the most important question to me is: “Can this primarily white group of math educators effectively practice anti-racism?” I don’t know the answer to that question, but I’d argue a serious consideration of this question would further the conversation than the “Why aren’t there more educators of color?” The second question is kind like asking “Why is the Canary dead?” while the first question is like asking “How do we get out of this deadly coal mine?” White Privilege is a large intractable villain that’s buried deep beneath our society like those tunnels in Stranger Things. Fighting it is going to be forever and take our whole lives, and that of our kids. Focusing on easily measured successes like the number of POC may trick people into thinking that the battle is a finite one, when it’s really an infinite one. It’s a battle that will require lots of persistence and collaboration, much like long battle that currently being waged by the #MTBoS against the traditional math system. Traditional math education is part of what is currently impeding many POC from becoming math teachers, so it is no surprise that there aren’t many who could immediately join the #MTBoS in this fight. “Can this group of mostly white kids take on this big insidious monster that no one even sees, knowing that we might not look like the people who should be facing it?” Just like in Stranger Things, it may not be the perfect team, but it may have to be the perfect team.

As someone who doesn’t participate much in conversations around race on #MTBoS or on the internet, here are three things that I think should be considered by anyone:

1) We are all from really different contexts. Conversations are happening across a number of racial, political, geographical contexts, making it difficult to provide the necessary backstory to go along with what is going on at their school.

2) The work is messy. It’s hard to know whether a social justice lesson that worked in one context won’t get you a visit to the superintendent’s office in another district. And it’s hard to know what to say when the student who beamed during that lesson divulges that they overheard the n-word in the teacher’s lounge yesterday.

3) This is people’s jobs. We are all professionals, most people’s real names and institutions are stated. Most people don’t know how to have conversations and feel like they are assured to be safe when a parent, or a student, or a superintendent might see what you’re posting.

So it’s hard to engage in a conversation that crosses many contexts, about work that is hard in nature, on a platform that poses actual risk. All of those problems get amplified when you are a person of color and talking about race in a primarily white space. Does the person reading this tweet really understand the way I experience race in my context? Am I confident that the solution I tried with my kids will work in other people’s classrooms (and not cause unintentional harm)? Am I being viewed differently than the white parties in this conversation, and could that change the way I am viewed professionally? As someone fully aware of the thin sliver of luck separating my trajectory from so many others, I worry about that luck running out every time I open my mouth.

For me personally to be involved in a conversation about race I’d need 3 things. The number 2 thing I need is what I’ll call a brave space. A teacher I know shared an article with me after her school had some tough conversations about race at their school meeting. The article (which I can email) basically says that the idea of a safe space is a myth, as safety is usually preserved for the majority culture. This is more or less what a grad school program director told me as they defended a white student who offended the people of color in one of my grad classes. Race was born out of violence and it’ll probably take some violent conversations to face it, so a promise of safety could serve to stymie whatever could come out of the conversation. The 3rd thing I’d need is the same thing I’ve always gotten from #MTBoS, and that is constant support and positive energy. The likes, and the retweets of people who you might not be following yet may be the 1st thing that needs to happen. The 1st thing I need to happen is to not have my baby daughter having a 103 fever, and my other daughter not insist that I show her spiderman videos whenever my laptop is open. I’ve wanted to get on #ClearTheAir, or respond to some of the email chains that I’ve been on, but there just isn’t enough time, or at least not yet. Maybe after October.

Problem-Solving, Theory-Buliding and Collaboration: How I stopped sucking

“Let them do it.” This was the constant mantra that Joe a retired NYC math teacher turned math coach repeated often to me during the classroom management struggles of my third year. At the time I was hoping to write material relevant to my South Bronx classroom by pulling from a different textbooks and 2007-era Google searches. Joe followed a similar routine when he made material to teach his East Harlem population using whatever people used in the early 90s. He quickly noticed something off about my tasks that I wouldn’t realize until after a few months working together. My efforts to control student behavior had seeped into a lot of the work that I had them do. Students weren’t doing the work of solving real problems, or making real connections with their prior math knowledge.  Instead they were walking down these narrow pathways of my own thinking. That’s exactly what I thought would help them, but in reality, I was doing them harm.
My teaching moves were as constricting as my curriculum, but I thought that this was necessary. My student population posed a lot of behavior and academic challenges. Changing the focus from my thinking to their thinking allowed me to stop worrying about their math levels or 8th grade scores and instead on how to build off the last lesson so kids really learn. When kids made mistakes, I typically told them what to do as clear and fast as possible. Joe, as politely as he could, shut that down.  Instead he’d ask me to start a conversation up at the front of the class to dissect the student error. He once said: “MAKE SURE THAT YOU NEVER, NEVER, NEVER GIVE AN ANSWER AND DON’T COMMENT ON ANY WRONG ANSWERS. Students will learn more from wrong answers that they discover themselves than they will if you tell them.” This is a quote from one of his emails, with his emphasis by the way. He continues, “They have been trained to look to the teacher to verify their answers and we want them to start depending on their own evidence for confirmation.” It was weird realizing that I was playing into this pyramid scheme of answer-getting, with the kids’ help, and preventing them from building their own critical thinking skills in the process. In trying to keep order, and keep my administrator from seeing chaos spill into the hallway, I played the answer giver that kids seemed to want, not the math teacher they needed.
To get students to do the kind of work they needed to build their thinking skills, I needed new tasks and new practices. One task Joe gave me that I still use today is Tina’s quilt squares. This task was, open, visual and had exactly the kind of thinking my kids needed to do. The old Carl would have totally strangled the joy  out of this. Taking his advice, I “let the kids do it.” The class was as quiet as any of my lectures as they all worked to find pattern. Some kids approached the task by actually drawing out each stage of the pattern. Others probed the tables for patterns in search of a larger explanation. These two types of student thinking were so interesting that I tried to validate both of them at the necessary end of class discussion and it became very clear which people should be presenting, and what I should, and shouldn’t, say about their thinking.
I thought about Joe, and my early years of teaching when I read The Two Cultures of Mathematics that Michael Pershan passed along. The paper’s author describes a tension among mathematicians among people who align with one of two statements:
(i) The point of solving problems is to understand mathematics better.
(ii) The point of understanding mathematics is to become better able to solve problems.
This creates a useful distinction of two kinds of math doers, the theory-builders and the problem-solvers. My first years was a time where I was torn between needing my students to become problem-solvers and wanting them to be theory builders. I was so focused on helping the students solve problems like being on the test, that I didn’t have time to let them really build the theory, so I figured I could do it for them with my awesome worksheets. The two styles seemed to be avenues for the kinds of approaches that students could work on and the kind of thinking that students could show in their student work. Looking back on it now, it was clear that I really didn’t understand what it meant for students to be problem-solving, theory-building, or doing mathematics in the first place.
The theory-building and problem solving avenues are different flavors of “Doing mathematics.” According to the Task Analysis Rubric by Stein et al., tasks are considered “Doing mathematics” if they:
  1. Require complex and non-algorithmic thinking.
  2. Require students to explore and understand the nature of mathematical concepts, processes or relationships.
  3. Demand self-monitoring or self-regulation of one’s own cognitive processes.
  4. Require students to access relevant knowledge and experiences and make appropriate use of them in working through the task.
  5. Require students to analyze the task and actively examine task constraints that may limit possible solution strategies and solutions.
  6. Require considerable cognitive effort and may involve some level of anxiety for the student due to the unpredictable nature of the solution process required.
The kinds of tasks that can fit into “Doing Math” could be either theory-building or problem solving, and possibly both. Tina’s quilt square was my first example of such a task, and seeing it first hand changed the rest of my teaching career. Students were doing complex thinking, analyzing the result, using multiple solutions. The kind of tasks I was doing before Joe would fall under the task guides memorization and procedures without connections, with little focus on making connections or, well, anything in that “Doing Math” category.
Before Joe, I spent hours doing the theory-building, and the problem solving for my kids. I then packaged up my results in a way that would help students easily retrace my steps. They didn’t see how these steps are useful, as they didn’t approach a real problem. They didn’t see how these steps could be built from prior understanding, as we only talked about theory superficially. All my efforts were going towards weren’t going towards making good future mathematicians, but good retracers. It stayed that way because any thoughts about doing things differently would throw my pacing off. It wasn’t until Joe’s frank meetings, and All-Caps emails, that I realized my current style had to change.
The whole point in going to teach on the east coast was to change the world with my teaching. Helping kids learn math seemed like the best path towards that goal because it could improve their test scores and make them eligible for higher education opportunities. Yet, I was lying to myself when I thought that it was enough for my class to do well on tests. My failed first test of calculus was proof that kids who do well on tests can still struggle at the next level. The right goal would be to make kids mathematicians. To help them actually learn how to solve big problems and make the connections needed to build theories. The application of those skills lies beyond just the math classroom, but can be actually used to think about tackling larger social problems and skills. The only way to get to my larger goal would be to not just teach Tina’s Quilt Squares the way Joe did, but to change my own thinking about my profession so I could make my own tasks to give to my future mathematicians.
After Joe, I sought to approach the the task of teaching my classes as my kids approached the task of learning algebra. I approached it like a problem-solver, looking to try and solve lots of little problems while simultaneously looking to connect big ideas and make sense of things like a theory-builder. Just like my students began to learn from each other and work things out in conversations, I began to talk with Joe and later other educators which helped me develop my practice. This was before I found the #MTBoS or joined NCTM or MFA, so this took a lot of time. Luckily Joe, and my Principal allowed to approach the task of improving as a teacher without having to retracing some steps or memorize some procedures, as it is probably the reason I’m spending another fall preparing to teach.
* * * * *
In writing this piece, I read W.T. Gowers 2000 article about The Two Cultures of Mathematics, of which I understood about 65%. I do know that is not about K-12 schools and what goes on there, but is actually a bit of a call two action about two cultures that exist among people in the field of mathematics. The prominent theory-builders belong to a culture that studies the fashionable ideas which are at the center of that field, while problem-solvers work around the periphery. This distinction reminded me a lot of math education. It feels like there is a prominent culture of around that approaches things in the old ways, like in this recent pro-memorization NYT article. Meanwhile the world seems to ignore ideas that come from problem-solvers working around the periphery in classrooms like Joe working with me.
The problem-solvers Gowers describes, who were creating the field of combinatorics, could actually benefit the theory-builders with their unique ways to solve problems. Technological advances would allow for major advances in combinatorics and math as a whole as the two cultures learned to collaborate and move the field forward. Hopefully writing this story of what happened in a cramped South Bronx classroom, might describe a different approach to how to improve teaching and learning. And hopefully the larger conference that it is a part of could promote the kind of collaboration across the world of mathematics education that has been seen among the problem-solvers and theory-builders of mathematics.

Grading, Assessments, and Hot Pockets

So you’re at a PD, a really awesome one at that. Everybody is quietly thinking about the prompt “What is assessment?” Your neighbors are writing things like “Assessment is knowing where kids are, where they need to go, and what you should do next.” These poetic statements allude to many parts of a real-time data gathering and analyzing process . Diagnostic assessment, summative assessment, formative assessment are all critical pieces of information that end up letting the teacher know what they need to maximize student growth and learning. The information gained from assessment become the ingredients that “Chef Teacher” can use to create any number of delicious stews, or salads, or souffles.

The facilitator tells everyone to stop writing and to stand up and share with someone new. After 15 awkward seconds of trying to lock eyes with someone, you find a partner across the room. After shaking hands you read your poetic statement with a serious flourish. Your partner responds with the following:

Assessment is how you give kids grades.

You wonder for a second if their table was given the same task. This statement describes a calculation chore that happens at the middle and the end of each term. Grades are what you show to parents and administrators if they want to know how the kids are doing. Assessment is a process that ensures that you have the information at any given point to be able to make the grade, but also to do so much more. Assessment can help you make decisions in the moment, tweak tomorrow’s lesson, or even alter your unit structure. Your assessments can tease out which students understand what you taught today and which ones are relying on the trick they learned last year. Viewing assessment as only a tool for finding grades is like “Chef Teacher” going to the kitchen, by passing all the groceries, and microwaving a Hot Pocket.

You rack your brain for how to begin a conversation about Grading, Assessments,…and Hot Pockets, when your partner cracks a smile. Turns out he was messing with you. He didn’t really believe that Assessment is solely for producing grades, but lots of teachers out there do. How would you describe all the things that assessment could be to someone who thinks it is only for getting the numbers to put on the report card?

Explaining my lack of “help”-fulness

This year our school is talking about student work in mixed groups. We have been placed into 7 groups of teachers and social workers, each of whom are related to one student. After each session, the teacher bringing the work gets ideas for their teaching, and the group gains insights into the student and how our work affects them. These conversations have only involved essays so far, but this past Friday I was the presenter.

Due to realities of our schedule I provided a student’s partially finished math project for our descriptive inquiry group to look through. It was a project where the student had to create a set of equations that then help her solve a larger problem. The student make a mistake early on in the assignment and continued finishing the work, not being able to see that answers stopped making any sense. The discussion about this did not just allow for us to talk about the student. It allowed the members of the group a chance to step into a math teachers shoes and decide to how to respond to student misconception.

Talking about this student’s work flared up and we ended up having to scrap the rest of the inquiry protocol. The issue that broke our group apart happened after I explained the project and everyone gave their initial impressions. Someone noticed that the student made the a mistake. “The student should have multiplied these answers by x,” the teacher stated, referring to the column with numbers far to small to make sense in the situation, “so the teacher should show them what correct answer should be.” I began to feel a little uncomfortable. My instincts say the first thing to do would be to understand why the student made the mistake. I would need to ask a series of questions before I gave any kind of instruction. Thesequestions would intend to help the student to understand why the need to correct it, not to correct the multiplication, thus preventing the student from making sense of the problem.

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