Carl's Teaching Blog

A place to talk about teaching and learning

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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.

Clog: Launching the #ThinkingClassroom again

So my thinking classroom may have just given the best math class interactions I’ve ever had. More on that later. First let me get up to speed with this experiment.

Last cycle I had a good experience with the Thinking Classroom. I’ve written a few blog posts about it, including one about my plans, another about my awesome and excessive randomizer, and another about kids taking notes. The experiment ended in January because the marking period closed, but I decided to reboot it with a new kids on a different schedule. It’s really great having a second chance to go over the norms with a fresh group and get more practice on-boarding them. Having a bank of problems to use is an added bonus. So far, we have been doing  visual patterns, including my new favorite visual pattern lesson for VNPS, which I could write a whole other post about. (Actually, I’ll just write it now. This lesson has lots of patterns, you can cut them up into strips and give different linear and quadratic patterns to different kids. Give them to different groups and then have them walk around and check out the patterns that the other kids worked on). My classes on this schedule are an hour and a half long, and the VNPS seemed like it would be too long for that, so I have been doing Desmos activities in the remaining time. As the class comes to an end I’ll have kids work on using Desmos to make models of data that is important to them.

Today we used a different shell center problem and it just kept on giving. I showed the kids this prompt and asked them to think about how many would be int 30cm x 30cm and the 40cm x 40 cm and an equation.

The kids ended up doing a lot of thinking, including two students who weren’t here for the first cycle. Two of the three groups got to an equation after 45 minutes, but one group defined the 20cm case as n=1, while the other made an equation where n=20. This was led to a great conversation that we have been having about equivalence, but also about this thread of what I guess I would call “Mathematician’s License,” (like “poetic license”) One of the groups was trying to convince themselves that they had done it wrong, and I had to stop them and say “You made an equation the describes this situation, and that group did as well, and they aren’t the same. That’s ok. In fact, that’s what mathematicians do!”

All the groups seemed only 80 percent clear on the problems, so I made an extension question. I asked the kids about the 20cm x 30cm rectangle and they were all able to think about an equation, and no one’s equations were the same. The two groups from above actually tried solving it using the other groups value for n! It was great to see ideas spreading around the classroom. All the kids were supposed to sit down to work on the Desmos activity, but those two groups kept lingering around their boards. One group had a functional equation and they kept working on the ways to ‘simplify’ it. The other group was struggling a little with their equation, because they wanted to figure out how to write it so that it could apply to rectangles of any size!!!! I was shocked that a kid would even ask that. I told him that we would probably need some multivariable equation to try and figure that out and he was like “are we going to do that this cycle.” What!?!!? I had to restrain myself from not nerding out because we needed to get to work on the Desmos activity. I’ll say this is the first time I’ve ever had a student linger around and do more math, let alone ask me to do more. I think it shows that the kids have built up a lot of confidence and ownership in the first 4 days of the class.

Clog: Why won’t the kids take notes???

My kids are awful note takers. It’s not just me, this is what they are telling me. Each Friday I ask kids to do a 6-question weekly reflection with questions like “This week I took useful notes.” For that item, students all consistently report the lowest scores over the first 4 weeks of the class. It’s clearly the biggest takeaway from doing all of these weekly reflections, so I should probably address it.

Notes were an important takeaway from what I learned when I began looking into the thinking classroom. In the past I didn’t do notes, but I had really well laid out tasks which included everything I want students to take notes on. These tasks would be organized in such away that they could review the important things whenever they studied. The act of deciding what things are important seems like an important academic skill, and by not expecting them to do this meant I was setting them up for failure in higher ed. The meaningful notes that Peter Liljedahl described echoed this:

…teachers can highlight particular parts of the work that is on the boards, but it is important that the students select themselves, and synthesize and reorganize notes on their own.

Unless of course they don’t. I’m not sure why the students aren’t reporting that they are taking notes, but I can assume that my teaching has something to do with it. I usually I rush through the notes at the end of the class, perhaps because I don’t yet stop early to give them to time to write things down. I could write a whole post around my issues with pacing and the thinking classroom (tl;dr version: “how is class over already!!!”), but maybe it would help to give kids some more guidance around the note taking process. Perhaps the reasons they are saying they struggle taking useful notes isn’t because they don’t have time, but that they aren’t used to doing it. I have been giving them some structure so far which you can see in this google doc. On the first class I gave them the just the first page, a note sheet with a few blank lines and said “Write a letter to your future dumber self about what we learned today.” I quickly pivoted to something with more structure. They got a two-page document with a number of important things that the we would cover over the whole course. It had lots of places for kids to put in things that were relevant, and they could do so over the course of the class. What I found was that kids treated it like a worksheet, looking to show me one that was totally filled out (one kid even went and googled some of the new stuff), while others filled out nothing while planning to sort of wait until later when they would “know what to write.” The last page on this is what I made for today.

Download (PDF, 23KB)

Today I decided to make it a point for kids to take notes on an important equation, the vertex form of the quadratic equation, since they will need for the project. I made a note sheet and I had the kids do it after we worked through a bunch of different quadratic equations and graphs on whiteboards. The activity they worked on encouraged them to make connections between tables and equations and we stopped with 15 minutes to allow time for note-taking. Then I handed out the note sheet above and basically told kids “We’re going to take notes now, these are going to be important later, and you can use what you see on the boards to write down the important parts”. This was definitely the best notes they have taken so far, even though some kids kept their paper blank and others wrote things they will not be able to refer to later.

So it wasn’t a complete fix to our problem with note taking, but I’m not taking it personally. It is a new routine, and it is one that contradicts a lot of norms that in the school altogether, as well as messages that the kids are being sent about school and learning. Our school is about experiential learning, so students who choose to come here probably aren’t kids who instinctively look to document things with the future mind. Kids also have some pretty damaging thoughts about how learning and knowledge happen. Ronny, who spent the whole time with a blank paper in front of him, “I already know this, so I don’t need to take notes…the right answer will just come to me later.” What a difficult set of beliefs to set for oneself. Leonardo DaVinci, Martin Luther King, Abaraham Lincoln and all these other genius needed volumes of papers to write their thinking down, but somehow when you get done learning you can skip the notes? We talked some more and he has a lot of reason to believe what he does. He is basically saying that the Regents is the only place where it is important to use what you’ve learned, and in his 11 years of experience, he was able to be successful. This way of thinking doesn’t set up Ronny for success in the future and places an impossible burden on him for when he gets older. If you struggle in college, wouldn’t it be nice to say “my notes are bad” instead of having to say “I guess my brain stopped working”? Seems like some kids don’t think of my class a chance to practice using the effort to become smarter, but as a chance to reinforce the smartness that they already have.

Clog: Bringing along the stragglers

The Thinking Classroom experiment may be a success. The white board contact paper is starting to gather dust in the corners, the kids trust the fancy randomizer, and there are consistently good days. I’m all ready to write a post about successes or a sort of “growth” post about my what I’ve learned the most (spoiler alert: it’s about managing flow). It’s just, well…there are these two kids who I’m worried about.

On Wednesday we started on an open middle problem and two of the four pairing had a student sitting down, facing away from the board, and looking at their phone. Paul likes to mind his own business, and isn’t too interested in talking with me or any student. Denisha has flirted with pursuing a GED, with support from her parents, but decided to give this year one last shot, but so far hasn’t earned any credits in any class that didn’t her performing music.  They both didn’t hear the full explanation about the class during registration, so maybe this isn’t what they expected. I am aware with both having behavior issues in previous math classes which have escalated to administrative interventions. While both of their attendance is around 50%, it surprised me that neither have had an even moderately successful day working with their groups.

On Wednesday, Paul was struggling in a grouping that made me question throwing out the randomizer. He was paired with a student who has had excellent attendance, and is pretty bright, but isn’t really a big “includer.” This student has a tendency to stare off into space, deep in thought, before emerging with an answer that he can’t seem to communicate without writing it on the board. Paul had been paired with this student before and didn’t seem to be contributing anything, even though I asked. It seemed then that Paul wasn’t being vocal then, but his body language showed he wasn’t even participating. I had to try even harder to get the marker into Paul’s hand, but he was said he “don’t learn like this.” He seemed sad that he wasn’t able to pull the answers out of the air like his groupmate and wanted me to sit next to him and tell him what to do.

The randomizer also placed Denisha in a pairing that wasn’t ideal. Her partner was probably the third least talkative kid in the class so I knew I would spend a lot of time over there getting the conversation going. The two were both quiet, Denisha seemed detached. Denisha’s partner seemed quiet because they weren’t clear about the next step in the problem, while Denisha seemed like she was trying to hide and not be seen. If I visited, she would look up and confidently assert that she knew what they were doing, but wouldn’t be able to answer any questions about the actual problem. After some questioning it seemed like she wasn’t aware of what the problem was or what her partner was doing to figure it out. When I wasn’t nearby she would zone off or get sucked back in to her phone as if she was waiting for us to go over the assignment.

Both of these students seem to struggle asserting themselves in math and are also uncomfortable with collaborating around a task. The tasks that I am giving out seem like they are pretty “low-floor,” but Paul and Denisha are not accessing or engaging with them in up until Wednesday. They don’t seem to feel safe in this environment. One of my big goals in this experiment is to create a culture where students spend more of their time feeling safe, but it seems like these two students are spending most of their time avoiding feeling “dumb.” It’s tempting to want to blame the victim here. To imagine the kids could have been more successful if they attended regularly, or to assume that these kids would “get it” if they “just got on board.” I think I should do more to meet their needs (and am open for ideas here). I’ve tried using more manipulatives and am going to work more on having students interact socially before they start working on math, but it’s hard to see if it is working or not.

On Friday I had a pretty successful day with Paul. First I explained my new grading system for the class in very clear terms. I then re-explained the whole philosophy of the class again to connect their grade to the class. Then we had the weekly quiz, which Paul would was unprepared for. Paul, and another student instead worked on quizzes from earlier weeks that they missed. Then I had those students work on a problem maximizing the area of a rectangle given a  perimeter together using manipulatives. Paul was much more confident in working with this new partner, and was more vocal when I asked them to explain why they thought a square isn’t a rectangle. We ended with a conversation of what he needs to do to earn credit. I think it was a big step forward for Paul. Denisha wasn’t there, but maybe I have an approach that I can try with her. Not changing the expectations, but being as clear as possible about them, providing more ways to access the problem, and keeping to work to build confidence.

CLOG: Good Day Today! …And My Somewhat Excessive Randomizer

Yesterday I posted about how I began my Thinking Classroom experiment. The experiment actually started in the 2nd week of november, so at this point I’m actually 3 weeks in to it. Today I actually had the best class that I’ve had so far. It wasn’t a perfect class, but it was good enough to make me forget some of those rough days that happened along the way. There is a lot to write about, and I may not get around to it, but I will start with a digression to talk about my randomizer.

The Randomizer

Before my first class I made a randomizer that basically assigned seats to random numbers, and assigned those randomized seats to students. It was basically one that Joel Bezaire described. My school has really unpredictable attendance, however, and also has kids show up late. I needed a randomizer that wouldn’t assume that we would have the whole roster there and be smart enough to take any number of kids each day. Based on the kids in class the sheet should figure out how many pairs are needed, and then assign the random seats. This thing does all that automatically whenever a new kid is inputted into this google form that I have kids fill out (and I can also decide how many kids I want to have in each group).

It works like this. Kids fill out a google form with their names and click sign-in. This also helps me keep track of attendance and lateness. Then as the names come in, the spreadsheet puts today’s group on the right. It also automatically adds more groups as needed, and some blank seats in group “N” in case we need to form a new group. Check it out in the video below.

Bonus: If you use google slides, you can paste the two columns from the spreadsheet into your google slides, then select that you want to “link to the spreadsheet” so it will automatically fetch the new results. This way you can automatically pull up the random seatings by clicking “Update” in the slides and avoid having to even open up the spreadsheet.

This is what my random groupings table looks like in google slides. You can click the ‘update’ button at the top and the table will pull in any new kids that weren’t added before

If you want to try this out without doing the spreadsheet wrangling, I made a version that should be pretty easy to plug your classes into. All you basically need to do is:

  1. Make your own sign-in sheet with 3 questions: Name, Period, and “Sign-In” or “Sign-Out”. It should look like this one: Randomizer Sign-In Sheet.
  2. When you’ve made your sheet, click on the responses tab and click on the green square that allows you to view a spreadsheet. Make a new spreadsheet and call it whatever your heart desires.
  3. Go to the Randomizer Spreadsheet (Here you can see the results of the sign-in sheet, as they come in from the form above, and you can see the results come in.
  4. Go to the “Randomizer” tab of the spreadsheet above, and click on little menu on the right, then select the option “Copy to” and choose the sheet that you made in step 2.
  5. It should almost be working, There is this one cell on the randomizer that is highlighted in orange that will have an error message. Click inside it, and confirm that it says “=arrayformula(‘Form Responses 1’!A:E)”. [Note: you shouldn’t have to alter anything, you should just have to double click and then press enter.]

So that’s my randomizer, let me know if you think it is useful. I can go more in depth about how it works but I want to say a little bit about why I think the class was better.

Today’s Firsts

Today’s class was a highlighted by a few small firsts. It was the first time that I tried to do a mid-work-time consolidate. This was an idea that I talked to someone about at NCTM Seattle and it was a move that would make a lot of sense. I wanted the kids to work on two sets of visual patterns one linear, and one quadratic, so I wanted to stop after the linear one and make sure everyone felt confident thinking about how we go about describing linear patterns before moving on. Once students got through with the linear set I could consolidate what we’ve all done as a class before moving on to the quadratic set of patterns. I was just about to do this mid lesson consolidation when I decided to roll out another first.

Almost all of the groups were at a linear equation for the first set of visual patterns except for one. This pairing was a girl who had only been in the class once before and another kid with better attendance whose work usually gets presented first and is kind of shy. Both students could use the confidence boost of figuring it out. They explained the equation in words, but somehow got stuck when it came time to write it on the dry erase contact paper that was attached to the storage closet in the back of class. While they were going back and forth about what to say I just sort of drifted away. My attention turned towards the front of the class where the two other groups were sitting idly with their linear equations locked and loaded. They would need to do something while we waited for the big consolidation, so I asked them to do a different problem as they waited, which was a first. For one group, a visual pattern that I had on the board got them immediately engaged. The other group was asked to look around the room at a problem they hadn’t done but others had. (In retrospect, I should have just given that second group the visual pattern as well because I think that group wasn’t as engaged in the consolidation later). In a few minutes we were able to do the halfway consolidation, with all groups feeling confident that they can handle any visual pattern I can throw at them.

All the groups had their confidence tested when they were faced with a non-linear visual pattern, and they were all able to use a recursive strategy and explicit strategy to find further patterns. It was actually pretty impressive seeing the group from the back of class with the most intuitive strategy to figure out what the equation was.

It was a pretty good day, I am not sure how I am going to get the kids who weren’t here (about 1/3 of the class) up to speed on what they missed. Any ideas?


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