Late Penalties

“Students need late penalties to teach them responsibility, life has deadlines.”

That teacher preaching about how life has deadlines was once a student whose teachers gave late penalties. They always turned everything in on time. Then they became the teacher who takes three weeks to grade an assignment, and says “I’ll get back to you about that” and never does, and shows up late to department meetings.

And that kid who can never get assignments in on time will turn in their application for the summer job they want three weeks early, show up every day on time, and get asked back for always going above and beyond.

We often aren’t teaching what we think we’re teaching.

NB: I’m not saying that every teacher who has deadlines takes forever to grade assignments. I am saying that there are tens of thousands of teachers who fit the description above. Their existence casts doubt on the claim that giving late penalties teaches people to meet deadlines.

NNB: I’m not even saying late penalties are evil or whatever. Have them if you want them. Just be honest that you have them for your convenience, and recognize that you’re not teaching what you might wish you’re teaching.

Anthropomophizing

Here are some things I try to avoid saying to myself:

“Oh yeah, that class moves a bit faster than my other two classes, we always get through the lesson early.”

“My first period class understands that.”

“This group is having a tough time with independent work right now.”

I think most teachers naturally anthropomorphize their classes. It’s an intellectual shortcut, seeing the class as one entity rather than 24 different students. But when I do this I’m usually talking as if the whole class is the average student. What I’m actually saying is, “the average student can move a bit faster” or “the average student understands that.” This thinking makes students who struggle or students who are ready to move on invisible. Sure, on average the class is doing well. But it’s easy to go from there to “the students in that class don’t need as much support” or “they all understand this, we can move on” and to leave the one kid who’s struggling behind.

If I teach one class faster than another, do I actually do justice to every kid in that class? It might feel ok because the average student is doing fine — but by moving faster, am I ignoring the students who need the most help by focusing on the average? I bet most of the time, the answer is yes. I try to avoid these generalizations, and try to hold in my head the complexity of all the students in the class and not the average.

Hinge Questions

I’ve loved the idea of hinge questions for a while. The idea: I design a question that elicits evidence of whether students understand the day’s lesson. I get answers from everyone in the class, and I use those answers to inform what I do next. (I use finger voting to see student answers, but lots of other strategies work as well.)

But whenever I’ve tried to pull this off it felt hollow. I usually don’t get to the hinge question until the lesson was almost over. At that point there’s not much I can do to adjust my teaching beyond informing my next day’s lesson. Also, I often see students nail the hinge question but clueless later in the week about what we learned.

So this year I’m trying something new. I’m saving the hinge question for the next day, and asking it after the warmup and before we launch into the day’s lesson. This has three big advantages:

First, this assesses whether students can recall and remember something, not whether they can do what we were doing together a few minutes before. When I tried hinge questions before I got lots of “false positives,” where students seemed to understand in the moment but couldn’t recall the concept later. Delaying the question reduces false positives so I get a more accurate sense of what students know.

Second, this can do more to inform my teaching. If I learn students struggled with the previous day’s topic, I know to slow down and maybe provide an extra problem or two of practice anywhere that idea comes up in the current lesson, or add more scaffolding for that concept, or even do an impromptu mini-lesson on scratch paper before jumping in.

Third, it serves as a reminder of what we recently learned. We typically do a brief turn-and-talk explaining why the answer to the hinge question is what it is, or debating between two common answers. That can refresh students’ memories and remind them of something they might otherwise not be able to retrieve that day, setting them up for success in the day’s lesson.

I’m two months into the school year and I’m still forgetting to do these sometimes, but I’m a fan. I love the idea of hinge questions, but in the past they just didn’t feel actionable enough. Asking a hinge question at the start of the next class has made a huge difference in making that information actionable.

More Effective Grading

I’ve seen a lot of chatter about grading recently, especially about ungrading and other large changes to grading systems. Right now I am a happy user of a pretty traditional, percentages/categories/points grading system. I’m not opposed to different grading systems, and I’m happy there are teachers experimenting and trying new things. I’ve used different forms of standards-based grading in the past, and I’m happy having moved back to a more traditional system. The reason: the benefits of alternative grading didn’t seem worth the effort. I do think it made a positive difference for student learning, but that difference was pretty small, and it was a ton of work. My time is precious, and I think it’s better spent on other things. Again, not saying standards-based grading or other grading systems are bad, only that I’m skeptical they’re the best bang for my buck. I’m not saying traditional grading is great. I agree that it’s statistical nonsense, it incentivizes a lot of behaviors that aren’t learning, and more. But I can overcome some of those issues without spending tons of time building a new system.

Now this post isn’t to dissuade anyone from trying something new. But I think there are two traps a lot of teachers fall into with assessment and grading systems, and I have two pieces of advice that are helpful whether you’re going gradeless, staying traditional, or anything in between.

Be clear. Too often, no matter the system, students don’t know how I determine grades or what exactly they’re I’m grading students on. A few years ago when the school I was at switched to standards-based grading, I made a list of the skills I hoped students would learn (drawn from the Common Core Standards for Mathematical Practice) and wrote up a rubric. Two months into the year I asked students to describe in their own words what it looked like to demonstrate each skill. Many of their answers were totally different from what I was looking for. I bet this is equally true in my traditional grading system now — I’m going to survey students soon on the basic elements of my current grading system. So consider this a challenge: no matter what type of system you use, ask students to describe what they are graded on in their own words. Is it what you expect? If not, what pieces can you make more clear?

Make sure students know how to be successful. The second part of my survey asks students, “What do you need to do to be successful in Mr. Kane’s class?” No matter how you grade, my guess is you’ll get some surprising answers. Do students understand what they should do to make sure they’re learning, to address something they don’t understand, and to improve the quality of their work? I admit that this is often more of a problem for traditional graders like me. Grades tell students what to pay attention to. The classic problem is that students focus too much on what’s graded and not enough on everything else. But I think many teachers with a fancy alternative grading system would benefit from asking this question to their students as well. Do students know how to be successful? And if not, how can you show them?

I’ll say it again. I’m happy with points-based grading, and I’m also happy that lots of teachers are pushing the boundaries of what grades are and how they’re communicated. I’m not trying to convince anyone to go one way or the other with this post. But everyone would benefit by making sure their system is clear, and that their students know how to be successful. The best first step is simple: ask your students.

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!

Ready for our first announcement as Desmos Classroom @Amplify? Polypad is now available in Activity Builder✨.

Polypad is an open canvas with virtual manipulatives that can be used to create, discover, and play with math.

What will YOU build? Tag us in your creations. pic.twitter.com/PNydbr5Yr6

— Desmos Classroom (@desmosclassroom) August 16, 2022

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.