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:


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.

“Work Hard, Be Nice”

There was a big hullabaloo a while back about how the former KIPP slogan of “work hard be nice” is racist. I agree with much of it. But this year I’m having a tough time with our 7th graders. There’s a lot of meanness, a lot of disengagement, and a lot of negative feelings about being in school. I find a lot of my conversations with students involve asking them to be nice or to work hard. Which seems like a contradiction, and has helped me understand more nuance in the problem with that slogan.

First, “work hard be nice” is probably necessary for a functioning school community, but it’s not sufficient. In other words, I don’t want students to hear the message that “if you work hard and be nice you will be successful.” There are lots of other obstacles that make it hard for students to learn, and some students experience more obstacles than others. If “work hard be nice” is the be-all end-all we are being dishonest and setting students up for frustration and failure. There’s a lot of support schools need to offer students so that kindness and hard work can lead to success.

Second, it’s important that I dig beneath the surface to understand the why. Why are students being mean to each other, or lashing out at adults? Why are students feeling disconnected from school? In a lot of cases there are good reasons for the behaviors I’m seeing, and in the rest I probably don’t understand the full situation. If the message is “work hard be nice” but school isn’t interested in understanding why that isn’t always easy, we’re talking to a wall. And when we understand the reasons behind the challenges students experience, we can do a much better job building a community that supports every student.

I’m not arguing that teachers should start telling students to “work hard be nice” every day. I’m noticing that a lot of my language echoes those two ideas, and trying to understand how to message those ideas to students in ways that support everyone’s learning and well-being.

Constant Forward Motion, Perpetual Review

Constant forward motion, perpetual review.

I can’t find where I first heard this phrase, but it’s been the answer to a challenge I’m having this year. (If anyone reading this knows where it came from, I’d love to give credit.)

I often see teachers complain about pacing guides that plan the entire year to the day, or argue that teachers should spend as much time on a topic as it takes for students to learn it. In theory it sounds great to adapt my teaching day by day to meet my students’ needs. But my reality right now is that even if I spend an extra week or two on a topic, some students are going to struggle. There’s no clear line I can draw that tells me I’m done. Is it 80% of students understanding a topic? 90%? 100% isn’t even close to realistic right now. And what exactly is my goal – is it that students can solve the hardest problem on a concept, or only something procedural and straightforward? I could go down this rabbit hole forever, and I could spend months on topics that might otherwise take three weeks.

My solution this year is constant forward motion and perpetual review. We’re using Illustrative Math’s OER curriculum, and I follow their scope and sequence with some modifications. We make steady progress through the curriculum, and at the same time I am constantly supplementing. We do warmups practicing foundational skills students will need for upcoming topics, mini-lessons on topics I anticipate students need a refresher on, extra practice when it’s clear students will benefit, and lots more little pieces that fit before and after the activities from the curriculum to support students. But I don’t say, “we’re going to spend an extra week on this because it’s not going well,” — if I did we would never get anywhere. Instead, I try to get better at supplementing the curriculuum, I set goals that I hope are attainable for every student, and we keep moving forward.

It’s easy to criticize teachers who move on when not all students understand something. It’s a lot harder to articulate a clear benchmark for making those decisions when the reality is much less clear-cut. None of the options are good. Moving on while some students feel confused isn’t great for them. But spending tons of time on some topics and giving others short shrift is setting students up for failure next year. That extra time often doesn’t solve the problems it’s intended to solve, and feels frustrating for everyone. I also think there can be a lot of value in moving on from a topic and revisiting it later, rather than banging our collective heads against a wall and expecting something to change.

I’m not arguing that I have the perfect answer. I’m just saying that these questions are harder than Twitter sound bites can make them seem, and the idea of constant forward motion and perpetual review has been valuable for me this year.

Against Taking Hands

Why do teachers have students raise hands to answer a question?

There are lots of reasons not to. When I take hands I’m typically reinforcing participation from a subset of students while normalizing some students not sharing their ideas. I want all students to do math every day in math class. I don’t think that taking hands is a good way to encourage participation from every student.

I’m also getting a biased sample of what the class knows and potentially deceiving myself about how well a lesson is going. I know intellectually that a few students answering my questions correctly doesn’t represent the whole class, but that positive momentum feels good and fuels my confirmation bias.

Finally, taking hands leads to all sorts of rabbit holes – some valuable, but some not. There are plenty of great times to explore rabbit holes, and I often love when students ask about something they’re curious about and I try to create times for exactly that. But there are other moments when we’re working through an idea and the logic of the new concept is hanging by a thread. In those moments I think it’s better for me as the teacher to have more control over where we’re going to support the learning of every student.

I’ve mostly stopped taking hands this year, and the biggest reason is just how stark the divide is between students who participate and students who don’t. 75% of the class almost never raises their hand, and that underscores for me the value of trying something different. I want students to participate, but participation looks like solving problems, working with a partner or small group, and opportunities to ask me questions individually rather than in front of the class. I want to value student thinking — but I can do that by sharing great ideas I see, rather than only sharing the ideas of students who raise their hand.

I do see value in raised hands. It encourages students to share ideas, take risks, and listen to each others’ thinking. But that only functions if there is a critical mass of kids who are interested in participating. There are lots of other ways to get participation – individual and partner work, turn and talks, group conversations, and more. And I can leverage all of those in different ways to get all of the good stuff that another teacher might get from raised hands, without reinforcing the inequities of who participates and who doesn’t.

The Five Practices

It’s interesting what pieces of my teaching are portable from one school to another. I’ve taught at three very different schools and lots of things that worked in one just don’t work in the next. Becoming a better teacher means understanding my students and my context and that means knowing when to let go of a practice that worked somewhere else. The last eight weeks have involved lots of trial and error as I make adjustments and figure out what works for these students and this context, especially during this weird year.

But as I’ve gotten deeper into this school year one teaching tool feels like a clear winner across all the schools I’ve worked at. The Five Practices have become such a core teaching tool in my pedagogy that I no longer even really think about planning it, it happens so automatically. Several times each day I ask students to try a piece of math, and use student thinking to create some learning for the whole class. Every day I anticipate how students might approach a problem, monitor their approaches, select and sequence student thinking to share with the class, and support students in making connections between ideas. Doing anything else just feels wrong. Teaching without students doing math for more than a few minutes feels like flying blind, and ending a task without sharing some student thinking feels like a waste of their time and energy. It looks different on the surface with different students, but my core thinking as a teacher is the same.

Why Times Tables?

Lots of teachers disagree about whether students need to know their times tables. I’m teaching 7th grade for the first time and I’m thinking a lot about times tables, because some of my students aren’t fluent and I want to understand whether or not that matters.

Context: I’m teaching proportions right now. Here’s a problem I recently gave my students:

I think a reasonable person does this with a calculator. Sure you could do it by hand, but why? Real humans who want to avoid making a mistake definitely use calculators.

Here’s another problem:

Sure, this one is totally reasonable with mental math. But if it’s ok to allow a calculator for the problem above, why not for this one?

Here’s my take:

Times tables aren’t useful for solving problems, they’re useful for learning to solve problems.

Let’s look at the two problems again. The first problem I wouldn’t want to use when introducing a new concept. There’s too much going on. But the second problem is a great introduction to proportions. If students can see that each number on the right is three times the number on the left, they will have a better understanding of the patterns in the table and the idea of a proportional relationship. They could use a calculator to see that pattern — but all the effort involved in using a calculator saps valuable working memory that could focus on making other connections.

Ok so takeaway one: times tables maybe aren’t useful for solving problems, they’re useful for learning to solve problems. Here’s takeaway two: the traditional 12×12 times table is silly and arbitrary. What’s so special about 12? After teaching 7th grade for the last six weeks, here are the times tables I wish my students knew:

Ok the tens look a little awkward, but the idea is simple. I can’t use the same numbers in every problem, so I would love students to be fluent in a decent variety of multiplication facts. But I don’t care about 9×7, or 12×8, or 6×11. It’s been rare in the last few weeks of my class that those facts come up.

Final point. It’s not only about multiplication. If a students knows that 4×6=24, that’s awesome. But it is just as important that they know that 24 / 6 = 4. And one piece of knowledge that keeps coming up, and that I think is another distinct skill, is knowing that if 4 x something = 24, that something is 6. Each of these facts creates opportunities to make connections and better understand new ideas because of the working memory they make available while looking at a new problem.

To summarize: times tables are important, but it’s all about quality over quantity. I don’t need students to know every fact, I need them to know a smaller subset forwards, backwards, and sideways to free up space in working memory. And the point of freeing up that space isn’t to solve problems, it’s to notice other patterns while solving problems that help students learn new ideas.

I Think “Meet Students Where They Are” Is Sometimes Bad Advice

I think “meet students where they are” is sometimes bad advice.

My goal this year is to teach my 7th graders 7th grade math. Plenty of my students are “behind.” It would be easy to say, “well they aren’t ready for 7th grade math so we’ll start with something they are ready for.”

I’m trying all sorts of things to provide support. Adding scaffolding to help students access what we’re learning. Taking time each week to practice foundational skills that students will need soon. Giving students chances for retakes and lots of time on assessments to make success attainable. I’m constantly trying to find new ways to give more students access. What I won’t do is go backwards and say, “ok I have some students on a 3rd grade level so we’ll start by teaching 3rd grade standards.” That’s demoralizing, it lowers the bar for everyone, and I can fit in mini-lessons and chances for extra practice while working on grade-level math.

I don’t mean that I’m just going full steam ahead with a normal 7th grade math curriculum. I’ll have to cut a few less essential things from some units to have the time to do the important stuff well and give students support they need. Part of providing scaffolding is creating a bridge from where students are to where I want us to get. But I worry that the advice “meet them where they are” can lower expectations and lead to a never-ending spiral of remediation.

When I’m at my best we are making steady progress while also putting a few minutes each day toward review of essential foundational skills. I try to simultaneously scaffold students to access the next topic while identifying things they need to work on from previous years and building in time for mini-lessons and practice. I’m not perfect and I often fall short. But I do what I can to keep us moving forward. My goal isn’t to figure out all the 3rd grade standards students don’t know. It’s to figure out what I need to teach them to access 7th grade standards.