Designing Practice

Here’s a handout I gave my students this week (link):

I’ve written about variation theory before. The idea is that, from each problem to the next, only one thing changes. This does three things. First, it scaffolds learning for students who are having a hard time. There is less to focus on because most things stay the same from each problem to the next. Second, it helps students focus on important features of the problem. They can make connections based on cause and effect – changing the problem in this way caused the solution to change in that way. Third, it provides opportunities for more confident students to see shortcuts and make connections they might not make otherwise because of the patterns in the problems.

I don’t want to take too much credit here. This handout is far from perfect. I often design practice activities like this because practice is important, and I can throw something like this together in a few minutes that targets an issue I see students having. In this case, many students seemed to understand the meaning of the inequality sign, but not how to “solve” an inequality as they would solve an equation. For instance, for questions like #2, a lot of students would say something like “x=2.” Which is a true statement! But is not what it means to solve an inequality, in the sense of describing the set of all possible solutions. So we spent some time talking about that distinction, and I threw this together to help students practice and make connections between what it means to solve an equation and what it means to solve an inequality.

But helping my students work through these problems, I noticed a benefit I hadn’t before. The two places that a number of students got stuck were problem #2 and problem #6. And in each case, most of those students had successfully solved the problem above. So instead of needing to explain a problem from scratch, I could explain the problem they were stuck on by connecting it directly to a problem they had already solved and knew how to do — building a bridge from what they already knew to something new. It really changed the character of the conversations I was having with students, and I think it helped them to make connections they wouldn’t have made otherwise.

Practice is important, but how teachers structure practice isn’t often a popular conversation. It’s easy to dismiss things like this as just “worksheets” or “drill and kill” or whatever. But I think this was a really effective tool to address an issue I saw my students having, and I learned something new about how to structure practice in a way that helps students who are having a hard time.

Roster Audit

https://twitter.com/ilana_horn/status/1354905280824422400

Everyone should do this. It’s practice seeing lots of strengths in different students. It’s practice seeing challenging students in a different light. It’s practice expanding my definition of what it means to be a successful student.

Last week a teacher on our team left — the second teacher on our team this year. I talked to my students this week about it, mostly just to let them know that I care about them and I value spending time with them and teachers leaving isn’t a reflection on who they are. And I took time to name strengths they all bring to the classroom and that contribute to our learning community. Having done the roster audit made it really easy to name strengths that applied to every student in the room.

I’ve had lots of tough classes this year, where it feels like it’s students vs teacher or like everyone in the room is taking turns getting frustrated with someone else. But all the classes I taught the last two days felt great. I felt relaxed and positive, students responded and participated, and there might even have been some learning.

So if you’re a teacher reading this, carve out time to do a roster audit. And when you’re done find a way, any way, to share with your students what you value about them.

Grading, Not Grading, and New Teachers

I’ve seen more and more writing recently that seems to argue, “grades do nothing for students, it’s obvious we should get rid of them.” I’m not a fan of grades, but I worry those arguments overlook something important.

Grades provide a feedback loop for students. A student works hard on an assignment, they do well, they receive a good grade, they get a little burst of dopamine, and they’re more likely to put in that same effort in the future. Now that feedback loop is far from perfect. Learning is messy, and many students get frustrated when the put a lot of effort in and they don’t do well. Working hard for grades can become a perverse cycle where the grade is the motivator rather than the learning. Students then don’t work hard on things that aren’t graded or aren’t a large part of their grade. There are plenty more issues I could name. But a lot of young people struggle to see the point in school, and grades can act as a short-term way for students to see that their effort pays off. Short-term incentives are especially helpful for students who struggle with executive function and/or have trouble delaying gratification when it comes to learning.

Now this isn’t actually a “grades are good we should keep them” post. I’d love to imagine schools without grades. A skilled teacher can help her students see that their effort pays off without grades. That skilled teacher can do a lot more to support students who struggle academically and don’t get positive feedback from grades. She can find challenges for students who get good grades without trying very hard. She can help her students to see why learning is intrinsically valuable, not something they should do for a letter on an assignment.

A skilled teacher can probably do everything grades do, and avoid the pitfalls. But that takes expertise and experience. A brand new teacher would have a hard time teaching without grades while juggling everything else they’re trying to do. So here’s my question, for the folks out there with a “here’s how I avoid giving grades” system: How would you support a new teacher using your system? How would you make sure that “no grades” helps every teacher and every student in every school, rather than making life easier for veterans while leaving new teachers out to dry?

I’m happy to be persuaded I’m wrong. I see every day the ways that grades distort learning and leave many students behind. But whenever I see an argument for some “ungrading” system it sounds like it would be pretty challenging for anyone new. I work at a school with a little over 30 teachers, and we’ve had 4 leave in the middle of the year this year. Convince me that your ambitious utopian grading system works in a school where a significant fraction of students are learning from long-term subs, teachers covering extra classes on short notice, and mid-year hires.

To broaden my argument a little bit, I’m honestly less interested in the “are grades good or bad” argument than I am in the “how do we support new teachers” argument. I can’t imagine a world with low teacher turnover coming anytime soon. I love to imagine all the ways education could be better. But the path to a better education system can’t be one designed for experienced teachers that leaves new teachers behind.

Would You Remember?

You wake up first thing in the morning and immediately think of a problem you want to work on. You sit down and get to work. You try a few strategies and they give you some information but you can’t quite get there. You feel stumped. You step away, eat some breakfast, then come back to it and spend a few more minutes working on it. The solution comes to you! You’re excited, and you take a moment to review the steps along the way. It’s fun to look at each attempt now that you know the final solution.

You move on to something else. A few hours later you think back to the problem you were working on. Do you think you would remember the solution?

This is a story about me playing Wordle. And most days I cannot for the life of me remember what the word was later in the day. Despite all the thinking I did, the word doesn’t stick. I’m seeing on social media that a lot of other Wordle players have the same experience.

I wonder if this phenomenon has any lessons for math teachers.

On Testing, and Knowing Students

This is mostly an anti-testing post, but it’s also a post about the different ways teachers know their students.

Labeling a student as being at a “fourth grade level” is dumb.

It’s a huge oversimplification, and it doesn’t have any value for me as a teacher.

Here are some things that are much more useful to know in 7th grade math: How are the student’s mental math skills? Arithmetic skills? Fraction skills? How are they with algebraic thinking? How comfortable are they with geometry? How is their mathematical confidence? How independently can they work? How are their English language skills? What mathematical skills do they have in their native language? Can they work with anyone in the class, or do they only work well with a few other students? How willing are they to take a guess when they aren’t sure? How willing are they to share their ideas?

And there’s lots more I learn about students. There’s the student who usually doesn’t feel like math is worth learning and doesn’t want to pay attention but always enjoys Which One Doesn’t Belong questions. There’s the student who feels like she hasn’t learned anything in years so what’s the point of trying. There’s the student who always seems unhappy to be in math then randomly tells me that math is the only class that students should take in school because it’s actually useful. There’s the student who is always excited to try something new but that sometimes means guessing wildly at answers without great intuition for whether they make sense. I learn all of this in the first weeks of the school year — but from spending time with students in the classroom, and not from any test.

Here’s the other thing. All of the tools that we use to measure a student’s grade level have a ton of noise in them. One day they might be at a 6th grade level, but they get a few different questions on the computer-generated test or guess wrong or are having a tough day and all of a sudden they’re at a 4th grade level. And when these tools try to add more information about their skills in number and operations or algebraic thinking there’s even more noise and more variation. The reality is that the tests don’t measure much of what I need to know about a student to support them, and what they do measure they don’t measure accurately.

I support testing at the group level. A short test, averaged across a large group, can be a useful, rough approximation of where the group is at. It can be useful to decide where to send resources, at the level of a school or a grade or a subpopulation. But it would be awesome if we could end the illusion that testing individual students offers any value to educators beyond what we already know.

Assigning Twenty Problems

Q: How did you feel when you saw this tweet?

I don’t mean to undermine the teacher, but I told my son that doing 5 problems is enough. He will not be doing all 20. pic.twitter.com/6CDubKXniL

— Dᴜᴀɴᴇ Hᴀʙᴇᴄᴋᴇʀ (@dhabecker) January 17, 2022

A: My first reaction was that I don’t like the assignment. Too many problems, too repetitive, too prescriptive. Don’t make them write the equation every time! Ridiculous.

Q: That seems hypocritical! You have your students solve 20 problems in a row. Who are you to judge?

A: Ok but my assignments are different I promise.

Q: We’ll come back to that. First, what was the most thought-provoking tweet in the replies?

A: This one, for sure (bottom tweet):

Disappointed that nobody in this thread is offering any grace that possibly this was a sub plan or some other googled resource for a class context we simply don’t know. Everyone calls for grace in this crazy time and it seems too few are actually willing to offer it.

— Jennifer Michaelis (@MichaelisMath) January 17, 2022

Q: Wow. Ok.

A: Yup.

Q: Ok back to you. Defend yourself. Why doesn’t assigning repetitive practice make you a terrible teacher?

A: First, I don’t think an assignment like that is ever appropriate for homework. Most students fall in two groups. Either they mostly know it and a few problems is enough to practice, or they mostly don’t know it and making them do 20 problems is torture. Maybe there are some in the middle for whom that homework is helpful, but they’re a small group. In class I can provide support to help students who are having a hard time, and make a useful learning experience out of 20 problems. Also, math teachers would do well to remember that we were usually the ones who were good at math in school. It can be hard to empathize with a student who needs 20 problems of practice to get a concept down, but those kids exist.

Q: Ok fine fine but haven’t you read the RESEARCH on spaced and interleaved practice? Repetitive practice like this is BAD cognitive scientists say so.

A: Interleaved practice is a great use for homework! I’d totally support a mix of previous topics instead of that assignment above. But a bit of massed practice early in the learning progression can be a critical step to set students up for success with interleaved practice later. Also have you ever written interleaved homework assignments for every day of the year? It’s a ton of work. I’m in a new school and new grade level this year. That’s not something I’m able to do. It’s a goal, and something I’m working to find more room for, but it’s also a huge challenge and so many other pressing day-to-day things bump it off my plate.

Q: Doesn’t that make you a bad teacher? Because RESEARCH?

A: Fuck off. Also, that’s not how research works, stop using RESEARCH as a cudgel.

Q: So you’re defending assigning 20 questions sometimes. Can you say more about when it’s worthwhile?

A: Sure. It actually connects back to that idea of interleaving. Some topics have more problem sub-types than others. I just finished teaching integer addition and subtraction, and I will absolutely defend 20 integer addition and subtraction problems. Because there are so many different permutations that require different thinking, it’s hard to hit on each possibility with less practice. Similarly, solving two-step equations with and without parenthesis is a good example. There’s a lot of math there, and those problems also don’t take forever so it doesn’t feel cruel to assign 20 in a row. But asking students to find the constant of proportionality from a table or calculate the scale factor 20 times in a row, less useful. So it depends a lot on the topic.

Q: Any other situations?

A: One thing I often try to do is to give students a series of problems that lead somewhere, or that ask students to find patterns in their answers. Maybe I’m trying to motivate the value of a shortcut by helping students see that the way they’re solving a specific problem isn’t efficient, or I’m trying to get students to make a conjecture about what will happen next in a given situation, or I’m helping students gain confidence with a topic right before we jump into a more complex task.

Q: Isn’t that different from the original assignment?

A: Probably. But we have no idea what that teacher was going to do the next day. But yea, I’m still not crazy about the original assignment, just defending the idea of assigning 20 problems in a row in other situations.

Q: Any final thoughts?

A: This has been a hard year. I do things I’m not proud of every day. Literally every day, I’m not exaggerating. But I also value exercising in the afternoons, sleeping a full night, and leaving my work at school. I welcome notes from parents with questions about what I do in class. I am always trying to improve my practice, and I try to look at any feedback as a chance to grow. My students never hold back when they feel bored or don’t understand why they have to learn something. But I am also teaching a new grade level in a new school in a challenging year. If your criticism is to tell me I’m a bad teacher based on one assignment, or to tell me that I should work 11 hour days to be better, you can fuck right off.

Solving Equations, A Follow-Up

I wrote a blog post a few days ago and I didn’t say what I meant to say. Here’s a second try.

I stumbled across a Twitter conversation today in which someone was arguing that we should teach complex numbers to 3rd graders. I won’t link to it, I’m not trying to shame anyone. Finding ways to explore “higher math” is an instinct I’ve had many times in my career, and that I’ve seen many other ambitious math teachers have. We have some idea of what the “rich” or “beautiful” math is, and we want to get to it faster! I think lots and lots of math teachers have had the instinct, “this math is so incredible, I want my students to see it!” Here’s the thing. The full sentence is actually, “I find this math incredible, I want my students to see it.” Humans (me, you, everyone else) are irreparably self-centered, and we have a hard time seeing things from someone else’s perspective.

Here’s what I meant to say in my blog post but didn’t: There is incredible richness and complexity in regular old school math that I think math teachers often take for granted. I wrote about these five equations:

I think many math teachers would agree that the best way to solve equation number one is to multiply both sides by 3. Some might appeal to seeing the equation as one-third of x is equal to 12, which one could argue is a different way of saying the same thing. But equation number two! That’s a zinger. You can solve it by multiplying by 3 also — two strategies could be seeing that it’s equivalent to the first equation, or knowing that 3 is the reciprocal of one-third and one can multiply by the reciprocal to make one. But in 7th grade, where we expect students to solve equations like this, the emphasis is on inverse operations. And that can really take students for a ride! The operation here is multiplication by a fraction, so inverse operations means dividing by that fraction. Again, if you have a good understanding of reciprocals this isn’t too hard — but lots of 7th graders don’t! It’s not intuitive for most 7th graders that dividing by one-third is the same as multiplying by 3, and lots of 7th graders would tie themselves in knots trying to get there. There’s a ton of math here, and I’ve only gotten through two “one-step equations.” (If you’d like an interesting intellectual exercise, pick your favorite method for each of the next three equations, then try to “break” it by finding a new equation for which your method is either very inefficient or doesn’t work. Also, for a gripping tour of the underlying math, Ben Blum-Smith’s blog post is a must-read.)

My point isn’t to argue for one method or another or to get in a back-and-forth about equation solving. My point is that there is way more complexity in solving these equations than one might think. Your typical math teacher could solve any of these in seconds without putting a pencil to paper. Lots of folks would argue that this math is “simple” or “rote,” but I’d argue it’s anything but.

Rather than searching for new, sexy math for students to learn — math that adults find beautiful or worthwhile — let’s spend more time plumbing the depths of school mathematics. What I described above is something most middle school math class sprint past without thinking twice, always in a rush to get somewhere else. There’s a lot there, and those of us with too much mathematical knowledge and experience are likely to overlook it.

How Would You Solve It?

Here are five equations:

I would love to get a dozen middle school math teachers in a room and ask them how they teach students to solve equations like these.

There are so many different possible strategies. The left side is equivalent to a fraction times x in each equation, so one approach is to rewrite each equation in that form and multiply by the reciprocal. A different approach is to focus on inverse operations. Where two numbers are being divided, multiply to make 1, and so on. There are also a bunch of strategies well suited for one or two of these equations that don’t transfer well to the others — but those more specific strategies are the ones that I think a lot of experienced math students would choose.

Here’s a dilemma I’m thinking about:

I want students to have the flexibility to solve each of these equations, and to be able to solve each equation in multiple ways. I think flexibility is the essential ingredient that helps them solve unfamiliar problems in the future. But if I focus on flexibility early in the learning process, some (many?) students will flail because they don’t have a single reliable method for solving an equation, and they feel dumb and discouraged. And if I focus too much on a single problem type and a single method, some (many?) students will struggle to transfer what they know.

What is the best way to strike a balance between fluency with one method and flexibility with multiple methods?

What Is Productive Struggle, Really?

Here are three types of struggle:

• A student tries hard at something. They can’t do it themselves and get help to complete it.
• A student tries hard at something. Eventually they figure it out.
• A student tries hard at something. Eventually they figure it out, and then they get a chance to practice it again successfully.

The first I’m skeptical to call productive struggle. Struggle can take a lot of different forms, but if a sequence ends with me or someone else helping a student and not them thinking on their own, it’s hard for me to see that as a success. I mean, sure, you can say that struggle is important, mistakes grow your brain, overcoming challenges whatever, but then we can call anything we want productive struggle. It’s totally ok struggle to be unproductive sometimes. I don’t think every problem I give students needs to be solvable by everyone, and there’s a lot that can be learned from failure. But needing help can’t be the only thing in a student’s mathematical diet.

The second I think a lot of people would call productive struggle. And it often feels productive in the moment. But then I see that same student getting frustrated and stuck on a similar task the next day because they didn’t have the chance to solidify their learning. While it might have seemed productive at first, I’d argue it’s not productive struggle because it doesn’t lead to durable learning. Productive in the moment, sure. But when that student has a hard time with a similar problem the next day all of the good that we hope comes from the struggle can evaporate away.

The third is what I think distinguishes productive struggle from regular struggle. Struggle should lead to success, and practice is a key ingredient in helping student succeed in the future. I want students to see that they are capable of figuring things out and solving hard problems, and I want them to see that the process of figuring things out is a legitimate way of learning. To do that, they need practice to make sure that learning sticks around.

Memorization and Understanding

Memorize these sequences:

PBPPS=P
NBPPS=N
PBMPS=P
NBMPS=N
PBPNS=P
NBPNS=N
PBMNS=P
NBMNS=N
PSPPB=P
NSPPB=P
PSMPB=N
NSMPB=N
PSPNB=N
NSPNB=N
PSMNB=P
NSMNB=P

Any luck? No? Well if you’re reading this, you probably already know them in a different context.

There are sixteen different possible combinations of positive, negative, addition, and subtraction signs in an integer addition or subtraction problem. Sixteen! And they’re above. For instance, PBMNS=P means that a Positive Bigger number Minus a Negative Smaller number equals a Positive number. (And that’s just whether the answer is positive or negative, without including whether one should add or subtract that unsigned integers to find the answer. Including that last element there are 32 different possible permutations!)

I’m probably somewhere in the middle of the road of opinions on memorization in math. I would like my students to know some things by heart — stuff like single-digit addition, skip-counting up to 5s, a decent fraction of their multiplication facts. Some 7th grade-specific things I’d like them to know are working with 10%, 25%, 50%, 75% and the circumference and area formulas working with circles.

But integer operations are a great example of a concept that, when you look closely, is absolutely impossible without understanding. When I say understanding, what I mean is students recognizing when two things that look different are actually the same. 5 – 3, 5 + (-3), and -3 + 5, are all the same problem. That’s one thing to remember, rather than three — and most students already know how to do solve one of those problems. Or that subtracting a negative is the same as adding a positive, and that gets rid of four different possibilities above.

I think recognizing when two different things are actually the same thing is maybe the fundamental skill in K-8 math. Working with integers is a great example. It would be easy for me, walking up to a student who is stuck on -3 + 5, to focus on helping them to solve that specific problem. Some things I say might help the student solve the problem in front of them, but not help them solve similar problems in the future. My goal in those conversations is to help students see when one problem they don’t know how to solve is actually the same as a different problem they do know how to solve — and why they are the same. That’s the understanding that will actually help them in the future.