Three Small Elements of Trauma-Informed Teaching

I think the movement toward trauma-informed education is important. However, a lot of the stuff I read about trauma-informed teaching is vague enough that it’s not very helpful. I’m not saying it’s wrong or dangerous, just not very practical for busy teachers. I want to share a few small, specific things I try to think about when teaching students with a history of trauma. I’m not claiming to be an expert — this is one of those posts that I’m writing so I do something more often.

First, a piece of framing. I love this comic by Michael Giangreco and Kevin Ruelle:

The goal of trauma-informed teaching isn’t to figure out who the “trauma kids” are and treat them differently, or to treat anyone differently at all. It’s to identify practices that support everyone, but are particularly important for students who have a history of trauma. Here are three things I try to do:

Don’t threaten. It sounds obvious not to threaten kids, but teachers do it all the time in all sorts of small ways. “If you keep blowing off your work like this, you’re gonna be in 7th grade again next year!” or “If you don’t pay attention you’re going to sit up front for the rest of the month!” These might not sound like threats at first glance, but they are. And for some students, they trigger a power struggle that has nothing to do with learning. I’m not saying students shouldn’t have consequences. But consequences should be shared clearly and dispassionately ahead of time, not when we’re frustrated and trying to regain control.

Walk away. When I’ve got 25 kids in front of me and an issue to deal with, it can be hard to take a step back. I want that kid to get their workbook out, or plug their Chromebook in, or whatever, and move on to teaching. But if I sit there and say “you need to take your workbook out right now” and try to stare or tower over them until they do it, I put that student in a high-pressure situation. Instead, I can say “Hey, it seems like you’ve got something on your mind. I’m going to give you some space for a few minutes. Can you get your workbook out when you’re ready?” Rather than creating a pressure cooker I walk away, and I can circle back and try to have a conversation again without getting into a power struggle over compliance. I’m not saying to ignore negative behavior — that’s why I stop by to acknowledge what’s going on and then circle back later. Giving space and circling back usually works better than trying to force the issue right there. It’s hard to remember in the moment, but I don’t need to solve every problem right away.

Rebuild relationships. When conflict happens in class, there are two things I need to do to fix things: figure out what happened, and let them know I care. Figuring out what happened doesn’t mean saying “what’s going on with you today” before launching into a tirade about why the student is out of line. It means listening, being willing to own something I can do differently, and understanding what triggered a negative interaction. In these conversations I often learn simple things I can avoid in the future. Then, letting student know I care is often as simple as letting them know I’m excited to see them tomorrow, and then giving them a smile and a fist bump in the hall before class. These ideas come from the world of restorative practices. But there’s a misconception that restorative practices need to be big huge events, or only happen after fights or other major conflicts. Rebuilding is equally important after I get frustrated about something small in class or there’s a miscommunication that makes a student mad. And it doesn’t need to be a giant sit-down event. It can be a relatively short conversation — and that short conversation can prevent the need for something bigger down the line.

I’m not an expert on trauma-informed teaching. There’s definitely a lot more to it than these three tips. But I think these are also low-hanging fruit; if more teachers (myself included) can change their practice in some small ways, it can make a big difference.

Backflips and Great Teaching

Watch this video:

This is fantastic teaching. Why can’t school be more like this?

Well ok, some of this is hard to do in schools. But I think I can learn a few things. A few observations of what works well here:

  • Motivation. This kid wants to learn how to do a backflip. Motivated students learn more.
  • Complexity. Learning a backflip is for him, and the teacher breaks it down into small pieces that are simple and accessible. Learning is best in manageable chunks.
  • Forward progress. The kid can move on when he is ready. Students want to feel consistent forward progress.
  • Visible learning. The kid can tell that he is getting closer step by step. Learning is best when it’s easy to see.

Why I Don’t Tell Students “You Are a Math Person”

It’s that time of year when I see the “you are a math person” message. I love the message. I would love to live in a world where more people think they are math people.

But that’s not something I do. I have that same goal — I want my students to see themselves as math people, as people who are capable of doing math and enjoying math. But I am hesitant to start the year by telling students that I’m confident they will be math people by the end of my class, or something along those lines.

My first hesitation is simple: show, don’t tell. There’s nothing wrong with telling each student they are a math person if my pedagogy backs that up. But I’d rather focus on the student’s experience than the teacher’s words. As the year goes on, am I building up students’ confidence? Am I helping them to see the different ways they can be smart in math class? Am I supporting them when they have a hard time, and challenging them when they need it? There’s no easy answer, and it takes a lot of work and a lot of pieces fitting together. I’ve written about how I structure routines, challenge assignments, and more to help students feel successful, but there is always more work to do.

Second, what I say at the beginning of the year matters a lot less than what I say in tough moments when students are frustrated and feel dumb. When a student lashes out because they still don’t get it, how I respond matters a ton — way more than anything I said the first day of class. Those moments require knowing students, being thoughtful with my language, and meeting the student where they are. And those are moments that make a huge difference.

Third, I fall short. There are plenty of students who don’t feel like they’re math people at the end of my class, for lots of reasons. I don’t think that telling them they are math people again is the thing that will make the difference. I don’t want to make promises that I’m not sure I can keep.

I don’t mean this as a criticism of teachers telling students they are math people. My guess is that all of those teachers are thoughtful and skilled in the ways they help students reach that goal. But I’m hesitant about some of the rhetoric I see. Helping students see that they, too, can be a math person is a lot more than a t-shirt or an inspiring speech.

Structuring Spaced Practice

Students need to practice in math class, and I want to provide as much spaced practice as I can to maximize the value of their time practicing.

Here is my quick and dirty summary of spaced practice. Spaced practice refers to practicing the same concept multiple times, spaced out by hours, days, or weeks. There is an enormous body of research behind it as probably the most powerful learning technique from cognitive science research. (Not saying it’s the best learning technique in general, just the best one that’s easy to study.) Without spaced practice, students are much less likely to retain what they know, or be able to apply what they know in new situations.

Ok, now to the nitty gritty. I use DeltaMath as my practice platform. It’s not perfect, but it’s far superior to other platforms I’ve tried and makes my life way easier. The thing I like best is that I can decide exactly which topics, and how many questions, to assign students.

(Important context: my district is on a four-day week, so most weeks I teach Monday-Thursday.)

I’m lucky to teach hour-long blocks, which typically leaves me time for DeltaMath at the end of class after I finish the lesson in the OUR curriculum. Monday through Wednesday I assign students 4-10 problems on one skill, depending on how long each problem takes. These are mostly skills practicing what we are working on in the lesson, but sometimes they are previews of something that’s coming up. For instance, my third unit is on area and circumference of circles, and requires some rounding. During the second unit, I’ll do a few mini-lessons on rounding and assign 2-3 days of DeltaMath skills on rounding to different places. This gives students practice with the skill and helps them focus on the circles and not the rounding in the upcoming unit.

Each Thursday I set aside a bigger chunk of time to practice, and I assign 6-10 skills, 2 problems each. These skills are a mix of the current unit, future unit prep like the rounding example above, and practice from previous units with an emphasis on skills that come up in future years.

The first part of the week I recognize I’m assigning blocked practice. I think this is important to get students some sustained time with a concept to become more confident. I can assign all the spaced practice I want, but for every student who doesn’t know how to solve the problems it’s a waste of their time. The last day of the week is my chance to bring in spaced practice. Only assigning students 2 problems from each skills allows me to assign a larger number of skills and give spaced practice on more skills.

There’s no hard and fast rule for what skills I pick. I’m always trying to figure out which skills are best to prepare for future units, and building in mini-lessons to prepare students for upcoming content. I also choose skills to spiral largely based on what I think is most important. Students will do lots of spaced practice of integer operations and equation solving after I teach those skills, but I won’t assign them scaled drawings very often because I see it as less important.

One final thought. I am a believer in the research on spaced practice. I think it is a compelling and underutilized technique for teachers to use. I’m grateful to have DeltaMath as a way to provide spaced practice without spending all my time writing problems. I’ve written my own spaced problem sets in the past, and it takes forever. But I have no idea what the right balance is — how often to spiral problems, how many problems of each type, how much instruction students need before I start putting something into the rotation, and more. I’d love to hear more examples of how teachers do it. I worry that, despite all the hype about spaced practice, the little details are both important and challenging to figure out, and it’s easy to say it’s too much work and put it to the side.

Challenge Assignments

Math is a big place, and I want my students to see lots of examples of what math can look like. One way I’ve tried to do this is through weekly challenge assignments. They meet a practical need — giving students something mathematical to do when they finish early — and they can also help broaden students’ ideas of what math looks like.

In the past when I thought of challenge assignments I would think of “problems.” The internet has lots of puzzle-like tasks that ask students to experiment, connect ideas, or have a stroke of insight on the way to a solution. Problems are one part of what I do with challenge assignments, but there’s much more out there. Students can explore different digital math tools, learn a bit about a new idea, play with interactives, create art, and more. Below is a list of some resources I’ve used to create these assignments. There are plenty of problems in there — problems are one important part of what math is — but there’s a lot of other stuff as well.

Offering broader options as challenge assignments has an extra benefit: it’s not always the students who have good grades or excel in other parts of math class who enjoy working on them. Plenty of times a student who doesn’t think of themselves as good at math finds a challenge assignment they enjoy working on. I teach 7th grade, but many of these resources can be adapted across grades.

One note before I start: I know that creating another weekly assignment can feel like an extra burden for teachers. Last year when I first designed these, I would keep a running list of potential ideas for challenge assignments. Then, about once a month, I would take 20-30 minutes to pick through my list, play with some ideas, figure out which work best, and mock up some simple assignments in Google Classroom. The assignment is usually just a screenshot of a problem or a link to a website and 1-2 quick reflection question. It is a bit of extra work, but I enjoyed that time exploring new mathematical ideas. I hope this post can act as a bank of resources for teachers to do something similar for their classes.

The Good Stuff

Ok here we go. Some ideas for challenge assignments:

Mathigon. Set students loose with a Penrose Tiles or other tiles for a tesselation challenge in Polypad, play with tangrams, work through a course (I had a good success with the graph theory one), explore the timeline of mathematics or the almanac of interesting numbers, or just grab some problems from the calendar puzzles.

Play With Your Math is my favorite resource for problems, and each problem is easy to play with and explore.

The Hour of Code has a ton of fun options. I haven’t explored most of them, but the Super Mario one was fun for my students.

Puzzles! My students have enjoyed the Fifteen puzzle, Game About Squares, and the Blue Box Game. Naoki Inaba’s puzzles are awesome, and Sarah Carter has a great collection. This puzzle website also has a ton of different online puzzles with different difficulty options (scroll down to see the different puzzles).

Math art! Annie Perkins’ math art challenge is fantastic. Desmos and Mathigon also both run art contests.

The EDC’s SolveMe puzzles are fun, and have an option for students to create their own.

Ben Orlin’s Math Games With Bad Drawings is a great source of mathematically-oriented games. If you search around on the internet you can find some of his games without having to order the book. Ben’s probability fables are also thought-provoking, and while they’re better for older students I enjoyed using “The Wise Monkey.”

Explorables. There are lots of places on the internet where you can explore an interactive simulation and see what happens. Complexity Explorables are really interesting — while many of them are over my students’ heads, it can still be fun to explore and try to make something pretty, and the epidemic and traffic ones are more accessible. Vi Hart and Nicky Case’s Parable of the Polygons is really thought-provoking and well-designed, as is this gerrymandering game.

Euclidea! I find these puzzles so much fun, and they’re a great introduction to constructions for students of different ages.

Finally, some miscellaneous fun: broken calculator puzzles, the nerd search, Vi Hart’s hexaflexagons, the locker problem, the Josephus problem, and just randomly searching the internet for “riddles” or “logic puzzles.”

This isn’t anything close to exhaustive. But I hope this is enough to get a teacher started, and get a glimpse of how much is out there.

Polypad + Activity Builder

Polypad in Desmos Activity Builder!

This is great timing for something I’ve been working on, and I want to use this post to think out loud and hopefully get some feedback on what I’m thinking about.

The short version: I want to use Polypad to help my 7th grade students develop flexibility in their number sense. A huge part of working with fractions and proportions is being able to put numbers together and break them apart, and knowing times tables is much less important than knowing the factors and multiples of different numbers and being able to see how different numbers are connected.

I set out to use Polypad to build some number sense tasks with a few guidelines:

  • Students should be able to solve problems through experimentation if they don’t have a good strategy right away
  • Problems should be solvable in lots of different ways
  • Problems should involve as many different types of arithmetic thinking as possible

A few notes before diving in:

  • These aren’t designed to be done without teacher guidance. I’ll need to show students how to use the tools, what the different manipulatives represent, and more
  • Desmos Activity Builder is a helpful platform for these because I can separate problems by screen, something that’s not really possible in Polypad
  • This isn’t an activity I would give to students as-is, but a bank of different problem types I’d like students to play with and practice
  • I’m imagining putting together activities with a few problems like these and having students explore them once a week or so
  • The last few screens are early finisher slides for students to explore after completing a few number sense screens.

This link is a sample of what I have right now. I’d love feedback! I’ve been playing with Polypad over the summer, and the Desmos integration inspired me to toss my drafts into an activity and share it. What am I missing? Which problem types seem helpful, which don’t? How can these be better?

Finally, a missing piece I hope is fixed soon: you can’t create questions in the Activity Builder version of Polypad. I made all of these in a regular Polypad and cut-and-pasted them in. It’s a bit of extra work, and makes it tougher to borrow and edit someone else’s Polypad from an Activity Builder.

Another year in the books

When I was in high school I had a few classes where I could make a notecard of key terms or formulas or whatever for a final. Whenever I made a notecard I found that it wasn’t very useful; the time and effort I took to record all that information meant I remembered it. It was a helpful way to review, but it was the review that was helpful and not the notecard.

That’s pretty much how I feel about blogging. I often jot notes in a quiet moment during class or quickly after 4th period before I eat lunch. And while I don’t often reread my blog posts from months or years ago, the act of writing and organizing my thinking helps me to learn from my successes and failures. Often when I write a post it takes a turn I don’t expect, and I learn something new by the time I’m finished. At least half the posts I start I don’t finish, but the drafting helps me to explore new ideas.

Anyway, it’s another year in the books. It’s definitely been my least successful as a teacher. My students just didn’t learn much math. Beyond math, I’m not sure they’re any more creative or curious or independent than they were 10 months ago. All things I’m excited to work on next year. Still, I have a better work-life balance than in the past and more things that are fulfilling outside of work. I’m happy doing what I’ll doing and I’ll keep doing it.

Unfortunately teacher blogging is mostly dead. Michael Pershan has some useful thoughts here. I’m sad that so many teachers I admired and learned from are gone. It’s unlikely blogging will make a comeback, and I’m not extroverted enough to replace blogging with Twitter or Instagram or whatever else. So I’ll keep blogging away. But I’d like to say that, even if it won’t lead to professional advancement or fame or riches, blogging can be a great way to grow as a teacher. For me, nothing can replace the process of brainstorming, drafting, and composing a blog post when I want to reflect on and learn from my experiences in the classroom.

So if you’re someone who’s thinking about it, dive in! There isn’t the community there used to be, but there are folks out there willing to read and comment and chat, and you’ll learn a ton along the way.

I’ll take the summer off from writing as I have for a while now. I already have a bunch of ideas and partial drafts of posts for next year, and I’ll look forward to another year of sending my thoughts out into the void. See you in August.

A few wins this year

My biggest goal as a math teacher is to help students see their potential as learners. It’s nice if students use something they learned in the “real world” or pursue a career where their mathematical skills open doors. I’d love for that to happen as well. But math class is often a place where students feel dumb, where they become convinced of things they can’t do rather than things they can do. I want students to see that, with effort and the right support, they can do hard things.

I fell short of this goal for many students most days in the last school year. So I want to take a moment to reflect on a few small wins at the end of the year, and to try and capture some successes that I want to cultivate next next year.

We did the Desmos escape room last week. It was really fun! (The linked version turns off “hard mode,” which I would recommend for middle schoolers limited to a single class period.) One student who is rarely engaged loved it, and was helping a bunch of other students when they get stuck, laughing, and totally focused. I heard her say “I’m so smart” a few times. That’s the opposite of what I usually hear her saying in math class — she’s usually telling me she doesn’t care about something. And that day I got to agree that yes, she is smart.

I offer a weekly challenge assignment that students can work on instead of some of their weekly practice if they want. A student finished with a group quiz and I suggested he try the challenge, which was an Hour of Code Super Mario game that week. He told me that he wasn’t the type of student who did challenges. But I pushed him to give it a try. When I checked in later in class he was having a great time, most of the way through it and excited to finish.

One regular day we finished an activity and I asked students to try 10 practice problems to check their understanding. One student looked at my in mild horror. He was convinced there was no way he could solve 10 problems. I don’t often give that many problems of one type in a row, but we were matching nets and prisms/pyramids, and I assigned more because they were quick and accessible. I gave him some encouragement and moved along. I came back a few minutes later and he was almost finished, and clearly impressed with himself.

Now all three of those moments feel mundane. They’re not my flashiest lessons or life-changing moments, and alone they aren’t something that will change a kid’s identity in school. But if I add up enough of these moments, students can change their perception of what they’re capable of. Those moments aren’t easy to create. I need to find those “just-right” challenges, build up students’ independence, know what work a student is ready for on their own and when they need help, offer that help when they ask, build trust, and cultivate a space where students treat each other well and are willing to take risks. That’s not stuff that’s easy to put into a blog post and it not an applause line you’ll hear at a keynote. But perfecting those little teacher moves, and helping students realize they can do more than they thought, is the real magic of teaching for me.

Fractions, and What I Would Love to See Cut From the Standards

An argument I’m bored of is “students shouldn’t do anything a computer can do” vs “fluency is important as the foundation of other skills, never get rid of anything ever.” Both sides are wrong. It’s a good thing that slide rules aren’t part of the curriculum today, and math class shouldn’t fight technological change. But humans can’t outsource all mathematical thinking to computers. Fluency with foundational skills still supports higher-level thinking, even if that feels inconvenient. The arguments I’m more interested in are about what, exactly, we should get rid of, and why. Students will learn more if we can get rid of some topics so they can focus more on others.

Here’s a great problem from Robert Kaplinsky, and it’s the type of problem I want my students to be able to solve:

How far apart are the freeway exits?

The foolproof algorithm for subtracting with mixed numbers is to first convert mixed numbers to improper fractions, then find common denominators, then subtract, and then simplify and/or convert back to a mixed number. But that’s not how I hope students solve this one. I hope they recognize that they can ignore the 1 for the moment, and focus on 1/2 – 1/4. This strategy doesn’t work as well when a problem requires borrowing, but it’s the best approach to take here. And the context screams “do this efficiently.” If students know that they can do 1/2 – 1/4, and they can compare the size of those fractions in their head, they will know right away the answer is a little bigger than 1. That’s all the information you need in this context.

Then on to common denominators. The foolproof algorithm for finding common denominators is to multiply the two numbers together. In this case, the common denominator would be 2*4 = 8. This is a good strategy if the denominators are 3 and 7, or 8 and 10, or 9 and 12. You won’t always find the best denominator, but when the numbers are messy it’s easier just to multiply and simplify at the end if necessary. But that strategy throws away students’ intuition for quarters and halves. And that’s valuable intuition! I hope most students know off the top of their head that 1/2 = 2/4, and use that quick change to find the answer more efficiently.

So here’s my argument for adding and subtracting fractions:

Students need to understand why common denominators are necessary to add and subtract. They also need to be able to add and subtract with common denominators, and have some fluency finding common denominators when one denominator is a multiple of the other (2/4/8, 2/3/4/6/12, and 5/10 seem most important to me). I think students should know how to find an LCM. It’s a skill I love because it’s good practice with multiples/divisibility, which are always worth practicing. But for a tricky problem like 5/12 + 7/20 we should teach students to use Desmos or WolframAlpha or any of a number of other tools that can do the messy stuff for them. Desmos can solve any problem along these lines easily, and give the answer in either decimal or fraction form.

Similarly, students should be able to add and subtract mixed numbers by hand when the denominator meet the criteria above and there isn’t any carrying or borrowing necessary. I honestly think that if there’s carrying or borrowing they’re best off punching the numbers into a calculator, and using their intuition to ensure the answer makes sense. That’s what I would do.

I know standards seem impossible to mess with because of the still-radioactive politics connected to the Common Core. Still, I would love to see someone pick through the standards with an eye for what we can cut our outsource to computers. Teachers often complain that there’s too much stuffed in the standards — and the complex arithmetic, especially with decimals and fractions, is the content that often takes the most time in elementary and early middle school. So let’s get rid of some things. Not by saying “computers can do that, throw it all away,” but by saying “here are the essential arithmetic skills students need, and they’ll be fine if they let computers do the rest.”

Predictable Practice

Students should get math problems right every class.

More than that, students should have some predictable practice that they can feel confident with every class.

(To be clear, I also think students should see problems they haven’t been explicitly taught to solve each class. If teachers only expose students to problems we’ve shown students before, we teach them that they can’t think for themselves. But that’s not the point of this post.)

For a significant chunk of students, math class feels like a steady diet of failure. Learning math means solving problems, but if every problem a student sees is too hard for them to solve on their own, we send a message that math is just an endless parade of excessive challenges.

So I think each day students should see a few problems within a predictable structure that they know how to solve.

Some people will deride this as “rote practice” or “drill and kill.” First, those terms are often poorly defined. Practice is important, but what makes practice rote rather than useful? I’m not advocating for 30-problem worksheets or 1-39 odd. Instead, I try to keep a few skills handy that a) students have had a chance to build confidence with, and b) are foundational for future learning and worth gaining more fluency with. I start practice with 2-3 of these problems. Nothing crazy, but enough to build a bit of confidence. Ideally, that confidence helps students try problems that are harder, either because they are non-routine or because they are new.

I use DeltaMath as my practice platform, and I use it in large part because it’s easy to assign a few problems from a bunch of different skills. I avoid assigning 20 of one type of problem at once. But spaced practice is essential for long-term retention and understanding, As students feel confident with more types of problems, I can use those problems as a springboard to build confidence toward new content. It becomes a virtuous cycle. Practice helps students gain confidence on a topic, spaced review of that topic helps students feel successful, and that success helps students take risks with more challenging math.