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.

The World We Live In

I learned this week that I’m moving classrooms next year. My first thought was that I’ll be closer to the staff bathroom. My second thought was that, while the rooms are almost identical, the new entryway is set back about 12 inches, and if I prop a table up it will create a larger space that isn’t visible from the window next to the door. Good thing, next year’s 7th grade class is a lot bigger.

We’ll be farther from the back stairwell, tougher to run if that seems like the right move. The window is at ground level rather than 20 feet up, though it’s not designed as an emergency exit.

Should I cover up the window next to the door? Propping a table will block most of their view, but a tall assailant will be able to see over it. There’s enough room, but in lockdown drills our legs start to cramp and if we stick them out too far they’ll be visible. But what if I cover up the window, and we’re doing a loud activity, and Frank comes back from the bathroom and knocks but no one hears him and he’s a shy kid and doesn’t want to knock too loudly so he sits there wondering if we’ll ever let him in.

I’ll have to explain at the start of the year to knock loudly because we live in a world where it’s more important to hide from potential active shooters than to see there’s a student at the door.

Word Problems Are Terrible

Word problems are terrible. Exhibit 1:

Exhibit 2:

(I realize that the second problem isn’t real, but it feels real to me because I’ve seen so many problems in a similar vein, problems that don’t quite make sense but we use anyway.)

These are two different ways we teach students to ignore the context, which is (hypothetically) the point of word problems. First by teaching them that their intuition isn’t valuable because they have to do it my way even if they have a better idea, and second by teaching them that the information in the context isn’t relevant to begin with.

I’m not a fan of the many key words/circle the numbers/underline the question strategies out there that have students mindlessly circling stuff or ignoring everything except the phrase “all together.” But the issue isn’t that they’re unhelpful in the real world. The issue is that they are helpful in the fake world of word problems.

There are great word problems out there. I’m teaching right now with the Open Up Resources curriculum, and the word problems are far better than standard math class fare. But better is far from perfect, and there’s still lots of pseudocontext. The go-to word problem strategy in the Open Up curriculum is Three Reads, which is a great routine. But Three Reads assumes that there’s context worth understanding. The whole thing feels like a bit of a charade working with pseudocontext.

I don’t have a great solution. My first takeaway is that, while I’m not a fan of the key words/circle the numbers/underline the question strategies out there, I understand why they exist and why they’re a rational response to irrational problems. Second, I don’t want to let word problems dictate the way I teach. I still teach them, and give them to students to get them thinking, but I don’t see word problems as an effective metric of success in teaching.

Ambitious Instruction and Falling Short

100% agree. Read the post, watch the video, read the comments. This is the type of learning I wish I got more of when I was a young teacher.

Leading discussions is hard! The teaching here falls under an umbrella that some folks call “ambitious instruction.” I think that’s a great name because it captures the idea that ambitious instruction flops sometimes. Getting students to share tentative ideas and engage with each other’s thinking is fantastic at its best, but pretty cringey at its worst. Dealing with wrong answers is hard! It’s hard for teachers, it’s hard for the student who puts themself on the spot, and it’s hard for other students in the room trying to make sense of the moment. And Marilyn Burns, one of my educational heroes, is in the comments sharing that she made a similar mistake recently in a classroom.

Here’s an idea I couldn’t shake after watching and reading.

Plenty of times I’ve tried something like this and I feel like I’ve moved backwards in my classroom culture without any learning to show for it. Maybe it’s a comment one student makes to another. Maybe it’s the expression on my face giving the game away and shutting a student down. Maybe it’s a long, tortuous discussion that doesn’t really reach the finish line and feels wasted. Maybe I dive deep into one student’s thinking but I don’t do a good job of bringing the rest of the class along for the ride, and they’re bored and confused.

My question is, do I try to facilitate these moments more because they’re so hard? The more at-bats I get the more balls I’ll put into play. Do I try more often, knowing that I’ll often come up short, but reveling in the classes when I get it right? Or do I try less, and go for quality over quantity? Do I save discussions for the activities I know are richest and most valuable, hoping that it’s a higher-percentage shot and I get more bang for my buck, with a bit less risk? I really don’t know the answer. When I started writing this post I was leaning in the direction of less, quality over quantity. But the reality is, I don’t have much of an idea most days what is going to get students’ wheels turning, and it might be less quantity without more quality.

I really have no idea here. I love these moments of ambitious teaching, and at their best they can get students thinking in a way most other teaching can’t. But I’ve also taught lots of lessons where the discussion seems to do more harm than good. No easy answers. But thanks again to Dan and his commenters for giving me a lot to chew on, and some great tools to do a little better next time.

Routines and Consistency

I love routines. I’m always working to find new ones that are valuable for my students. I see two major advantages in routines. First, as students get better at the routine over time, it frees up their mind to focus on the math, rather than thinking about what happens next or listening to my directions. Second, as students have a chance to practice a routine, it provides an opportunity for students who often don’t feel smart in other parts of math class to get better at elements of the routine and see themselves getting better at math. Ideally this is a self-sustaining cycle that empowers students. Learning math can feel like a moving target — just as you’re getting good at one thing we’re off to something else. Routines keep that target in place and give students a chance to dig in to a specific skill.

In the past I’ve experimented with different routine structures, often doing a different routine for each day of the week, or mixing them into my teaching as they seem to fit. But the last few months I’ve tried sticking with routines for extended periods of time instead. I’ve done a few weeks of Visual Patterns, a few weeks of Which One Doesn’t Belong, a few weeks of Estimation 180, and a few weeks of Contemplate then Calculate. I think this approach amplifies the benefits of routines, while allowing me to be efficient and focused in the skills I’m teaching. My goal is to see students getting better at the routine over time. For some students, that wasn’t happening without consistent practice. This is extra helpful with students who are often absent. All the routines provide valuable practice with number sense, and I’m not very picky about which routines fit where in my curriculum. If I stick with a routine for a few weeks, it gives students a chance to get good at that routine, see themselves getting good at that routine, and recognize their ability to become a stronger and more flexible mathematical thinker.

The other thing I love about this is the opportunity to use more routines throughout the year than I have days in the week. I wish I started it sooner, but I’m excited to bring in Decide and Defend, Number Talks, Fraction Talks, Would You Rather, and more.

Mockingbirds and Kestrels

I’m trying to work through the book “To Mock a Mockingbird (and Other Logic Puzzles)” by Raymond Smullyan. I’ve spent the last few weeks working through a couple of chapters on a thing called “combinatorial logic.” Here is a sample:

A certain enchanted forest is inhabited by talking birds. Given any birds A and B, if you call out the name of B to A, then A will respond by calling out the name of some bird to you; this bird we designate by AB. Thus AB is the bird named by A upon hearing the name of B. Instead of constantly using the cumbersome phrase “A’s response to hearing the name of B,” we shall more simply say: “A’s response to B.” Thus AB is A’s response to B. In general, A’s response to B is not necessarily the same as B’s response to A — in symbols, AB is not necessarily the same bird as BA. Also, given three birds A, B, and C, the bird A(BC) is not necessarily the same bird as (AB)C.

A bird is called a kestrel if for any birds x and y, (Kx)y = x. In general it is not true that if Ax=Ay then x=y. However, it is true if A happens to be a kestrel K. Prove that if Kx=Ky then x=y.

(I chose this example not because it’s the first or the easiest, but because it requires a bit less exposition than most others.)

Long story short, I had an incredibly hard time my first few tries learning this concept. I couldn’t solve a single problem. I would read a solution, then go back and try to recreate it without success. I spent hours on a handful of problems, trying them, giving up, coming back to them again, looking at the solution, trying to move forward, realizing I didn’t understand enough, and circling back again. I’m happy I stuck with it because I finally had a breakthrough and I’m now making steady progress, but it was frustrating.

That learning experience has me thinking about how I structure the first time students encounter a topic. It took me a while to get my head around the notation of combinatorial logic, and the way of thinking to solve problems like these. It’s easy for me to forget the size of some of the conceptual shifts students make in 7th grade math. From thinking proportionally to solving problems with negative numbers to working with complicated equations, there are lots of things that are obvious to me but challenging for students. One thing I love about trying to teach myself new math is how humbling it is. Math is hard, and teaching things I’ve known for years can fool me into thinking it’s easy. Struggling through a concept is an important dose of empathy for my students.

Class Size

I have no interest in listening to people sound off with their hot take on “would you rather have your child in a large class with an amazing teacher or a small class with a mediocre teacher,” or any version of that question. I guess it’s fine to play with hypotheticals but schools are making real decisions and those aren’t the choices. I find the whole discourse about class size a frustrating mess of people talking past each other. Because the problem is full of contradictions! Stop pretending you have the answer.

My hot takes:

  • The research does not show the giant positive effects that many people seem to think we’d see from smaller class sizes.
  • But who cares, even if the differences in test scores are marginal, smaller class sizes are a huge factor in teacher job satisfaction and are worth it to make schools humane places to work and learn.
  • But large reductions in class sizes are hugely expensive because you need so many more teachers. Does your school have a line of teachers at the door wanting to work there? Mine doesn’t, and to shift from that dumb hypothetical above to a real one tens of thousands of schools are dealing with right now, I bet most parents would rather have a large class size than a rotating cast of long-term subs.
  • But do we design our education system to scrape by in the sorry reality we have or for the ideals we should strive toward?
  • But ok these conversations often leave out the numbers. Reducing class sizes when there are >40 kids in a room is urgent. I think anything over 30 is unsustainable beyond the occasional exception. I’ve floated at or just above 20 this year in 7th grade math and it’s great, no one needs to advocate for me to have smaller class sizes. I teach in a small rural district so we’re at the mercy of chance. Looks like next year I’ll have around 24 and that’s also fine by me but I wouldn’t want to go much higher. Let’s recognize that resource are finite, let’s direct them toward the schools with the most pressing need.
  • But recognize that, even if folks win class size reductions, you’re not likely to see a magical rise in test scores. It’s not about test scores, it’s about helping students feel seen and valued, and keeping teachers from drowning.
  • But it’s not only about teachers. I’d happily take a bunch of extra kids if it meant we could hire more mental health professionals, special education paraprofessionals, and other student support folks.

Smaller class size? Yes, but recognize it’s complicated, be realistic about the benefits, and prioritize where they’re needed most.