Writing about Math: Solving Equations

Not long ago, I read an article by Marilyn Burns in which she explained how she used to view math and writing as “oil and water.” She thought that the two subjects had nothing to do with one another, and “writing played no role in [her] math classroom.” But now, she says, she “can no longer imagine teaching math without making writing an integral aspect of students’ learning.” In this article, Burns goes on to offer a number of suggestions for how teachers can incorporate writing into their math classes. This article really stuck with me, and so now, at least once each week, I like to ask my students to write about the math that we have been doing in class.

Their writing can take a number of forms. Here are just a few of them, all of which are mentioned by Burns in her article:

  • Students reflect on the process of learning, or what they learned that day/week.
  • Students write about their thinking and process in solving a nonroutine math problem.
  • Students write about what they know about a particular mathematical idea.

At some point during the last few class cycles, I realized that students were struggling to internalize, talk about, and deeply understand the process that goes into solving a one-variable equation. I thought this would be a good time to do some writing, and I asked students to write about their processes. I have done this activity several times now, and I like it because I have learned so much each time. The first time I asked students to write about solving an equation, not a single student was able to arrive at a correct answer or write about the process clearly. This was a big eye-opener for me. It meant that something was missing from my teaching of this key algebraic process, and it helped me to reevaluate. That’s one of the things I like the most about asking students to write about math: It helps me to assess whether or not my students have deep understanding of key concepts.

How I Solved It

First, I subtracted 2x from both sides so that all of the x‘s would be on one side of the equation. This left me with 3x + 7 = 28.

5x + 7 = 2x + 28
–2x        –2x
3x + 7 = 28

Next, I want to get all of my constants on the other side of the equations. Since I know that what I do to one side of the equation, I also need to do to the other, I chose to subtract 7 from both sides.

3x + 7 = 28
      –7     –7
3x = 21

Now I see that three times some number is equal to 21. I know by now that the answer is 7, but I still need to cancel out the coefficient 3 that is in front of the x. I can do this by dividing both sides of the equation by 3.

3x = 21
3       3
x = 7

This leaves me with x = 7.

A note on my solution: I don’t expect students to be quite as specific as I was here, but I do want to see explanations that go beyond “cancel the 2x” or “bring down the 3,” which is what I saw a lot the first time that I did the activity. I want to see descriptions of specific operations and rationales for why the student chose those operations.

Anticipating Student Solutions

In terms of process, I don’t anticipate that all students will follow the same sequence of steps that I did. Some students might choose to subtract 7 from both sides as their first step, or they might collect the variables on the right side of the equation rather than I did. This, however, would result in a negative coefficient of x, which I think will throw my students off, as they still aren’t completely comfortable with negative numbers, especially in an unfamiliar context like this one.

What I’m really anticipating is the use of imprecise language. I think that students will make references to “canceling” or “bringing down” certain numbers, which doesn’t really have much mathematical meaning. When I did this activity for the very first time, I get some responses that I really didn’t expect. Here are a few, exactly as they were written by students:

  • “The first thing you do is write your equation. After you put your numbers in place you need to add. Your next step is to subtract and last divide.”
  • “Add 2 to both sides made 2x negative bring down 7 = bring positive 28. Add 28 to both sides make 28 negative take 28 from positive 7 then bring down 3x. Divide 21 by 3 = x = 7.”
  • “First bring the 2x over to the 5x then you add 5x and 2x that gives you 7x. Then you bring the add sign down and the seven and the equal sign plus 28. Next you take the subtract 7 to both sides you get 21 then you divide 7x by 7x and 21 by 7x then you divide 21 by 7x and you get x equals 3.”

With the exception (maybe) of the first one, each of these evidenced some understanding of how equations work. But it was clear that not everything was in place yet. So I’m expecting similar struggles this time around. However, the group I’m teaching now is generally stronger and has spent a lot more time working on algebra than the first group had.

Supporting Productive Struggle

This time around, I am going to ask students to work on the problem individually, and then get together in groups of three. In their groups, they will share what they wrote and try to come up with a version that they can all agree upon and will share with the class.

At the stage when students are solving the problem individually, I think that there will be a lot of struggle. Students might fixate on the fact that I’m asking them to write something, and so I anticipate some students freezing up because they don’t know the answer to the problem yet. If this is the case, I’ll suggest that they take a few minutes to struggle with the problem before writing about it. I also think that once students do start writing, they won’t be very specific. I’ll try to check in with everyone and ask questions. For example, if something is unclear to me, I might say: “Here’s how I’m understanding this. Is that what you meant?” Or: “This seems interesting. Can you talk to me a little more about it?” I will also ask questions that push back agains words like “cancel” or “bring down.” I might ask students to tell me what they mean when they use those words.

When students are working in groups, I will focus my questioning on refining their responses. I will help them to see how transition words would be important in this context. I will ask similar questions about the vocabulary that they use, but this time I will ask each group member about it to make sure that every students understands what their group wrote, not just the person who wrote it.

Extension Questions

If any students/groups finish early, I will first recommend that they check their answer. I’m assuming that some of the students would have done this already, but if they haven’t, they should take the time to do it. They will need to plug their value of back into the equation and make sure that both sides are equal. This is something that we’ve practiced in class before, so they should be familiar with the way it’s done.

What they haven’t done, though, is written about the process of checking the answer. So if a student or group finishes early, I would ask them to write about the steps they would take to check their answer, and I would ask them to use a few key words, like “substitute” and “balance.” I think that this will help reinforce some key vocabulary that they will need to call upon time and time again in our work with algebra and functions.

Depending on the groups, I might also give out another equation for them to solve and think about. The one in the handout requires three steps. Maybe I will give out another one that uses this distributive property, or requires that students combine like terms on the same side of the equation.

Student Work

 Azia, Travis, and Feliciano

This group did a great job with the activity. What I really liked about their approach is the use of transition words. You can see that I circled them after the group shared out their process. I also really liked how well they used words describing specific operations. This is what I was really looking for during this activity.

This group also included something that I had really hoped to see. They articulated the idea of keeping things balanced in the short phrase “on both sides.” Other groups did this as well, but because Azia’s group was the first to present, I really wanted to point it out to everyone.

Iman, Luis, and Krystal

This group’s presentation was a little bit more problematic, but it brought up some great points for discussion. I liked how this group mentioned the idea of “getting the x‘s together,” but in the same sentence they mention that “what you do to one side you do the opposite to the other side.”

I knew what the group meant by this, but I asked them to talk about it a little more. When they started talking about step two, Krystal caught their mistake and pointed out that you actually need to do the same thing to both sides of the equation. I also had to ask them what they meant when they wrote that they “brought down 3x + 7 = 28.” In step 3, I really liked how the group said that they “did the opposite of multiplication…which is division.” The class talked about why we knew to divide in order to cancel out the coefficient of the variable. This was a good way for me to make sure that every student left the class knowing that a coefficient is multiplied by the variable.

Pedro and Jesse

I’ve included Pedro and Jesse’s work here because I was so impressed by it. Even more impressive is the fact that Pedro wrote this by himself in just a few minutes, and then he used what he wrote to teach the process to Jesse, who really struggles in class.

Pedro’s work also correctly articulates the idea that “what we do to one side we do to the other. This is what Iman’s group wrote about incorrectly, and so it was good that we got to talk about this idea again.

Jaime, Jason, and Jhoana

This group is from a different class than the other three. This class is composed of nonnative English speakers who really needed to work on their writing. But because they are seeking their diplomas, they still need to work on math. So this activity worked out well with this new group.

What I thought was interesting here is how the group solved the problem perfectly on the first sheet of poster paper, but their description of the process had some small mistakes. First, the group says that we need to subtract negative 3 on both sides of the equation, even though that isn’t what they did on the first sheet of poster paper. I knew what they meant, though, and so we were able to clarify this when we talked about it. We also talked about how the phrase “take away 4 dividing both sides by 4” could be confusing to someone learning this for the first time. Other students pointed out that having “take away” and “divide” describe the same step made it sound like they were using both subtraction and division.

Jose, Salvador, and Gabriela

I really loved this group’s approach to the task. They tried to fit in all of the important phrases, and they described every step in great detail. There are some obvious issues with spelling/grammar, but all of the other students recognized how clear and thorough this was. This brought up a good discussion about how, in life and on the TASC, clarity in writing is much more important than spelling every word correctly.

Final Thoughts

Before each group talked about their work, we did a gallery walk. I wrote three questions on the board to help students focus their thinking as they looked at their classmates’ work.

  • Did the other groups solve the problem differently than you did? What was different?
  • What did you like about the other groups’ descriptions of how they solved the problem?
  • Were there any mathematical words or phrases that showed up a lot?

The last question in particular really resonated with students, and I heard them talking about it as they walked around the room and looked at all the posters/boardwork. Most of the groups used the same sequence of steps, and so we really got to dig into the language of what each group wrote. The students were quick to point out that certain things showed up in several of the successful descriptions:

  • “what you do to one side you do to the other”
  • “on both sides”
  • “opposite operation”
  • “canceling out”

I was really struck by how much better these two groups did with the activity than the first group that I did it with. I also ended class feeling confident that my students had a generally solid understanding of how to solve one-variable equations.

What I Might Change

I wouldn’t change anything about the structure of the activity, and I frequently ask my students to journal about their thinking or their process in solving a problem. The next time I do this as a whole-class activity, though, I might change the problem that I ask them to write about. I was hoping that students would take different steps in solving the problem, but that didn’t come out this time, which was a little surprising. I suppose that’s because the process of solving one-variable equations doesn’t invite much creative thinking. It does require strategic thinking, though, and so I’m interested in seeing what my students would produce if I gave them an equation that was a little more challenging.

Unexpected Challenges

When students were working individually, they had a really hard time writing about their process, even if they were able to solve the problem correctly on their own. The most interesting and productive work happened after the groups came together and started talking. Because so many people had trouble writing, though, I noticed that in a few groups one person emerged as the leader and simply copied what they wrote onto the poster paper. In these cases, I tried to make sure to check in with everyone else in the group to see if they were following what the group “leader” had written.

Student Takeaways

I think that my students got a lot out of this activity. It’s always good for them to collaborate on math problems where the process, not the answer, is focus of the group’s work. I think that this activity met my goal of helping us to establish a shared vocabulary that we can all use when solving one-variable equations, and I think it helped struggling students to really slow down and think about process. But in addition to learning the math, my students had to think hard about how they wanted to communicate it. They had to make sure that they were writing clearly and using transition words to guide their reader through the process. Taken as a whole, I think that this went a long way in terms of building my students’ confidence in their own ability to solve and talk about algebraic equations.

Advice for Teachers

Use writing activities in math class! Many of our students want to pass the HSE exam and go on to college. To achieve both of these goals, students need to be strong in both math and writing. Activities like this one help students to develop a deeper conceptual understanding of big mathematical ideas while also helping them to become better, more audience-conscious writers. Initially, students will struggle to write about math, but if you make it a routine in your classroom, they will quickly get more comfortable doing it. These activities are also fun, and they can offer a nice change of pace in your math class!