## Reading & Writing in the Disciplines

# Writing in Mathematics

### WHAT ARE THE WRITING DEMANDS OF MATHEMATICS CLASSROOMS?

In mathematics classrooms, there are many times when students are called upon to write, especially if we consider writing to include mathematical expressions or equations; representations such as tables, graphs, or other images; and narrative text.

Take a few minutes to look at the curriculum materials you are currently using, the additional resources you draw on, and any assessments you are using with your students. Find examples of where students are expected to write, such as the following:

- Problems that students are expected to solve with an expectation that they “show their work”
- Explanations that students are expected to provide about their thinking
- Justifications that students are expected to offer regarding their solutions and their solution strategies
- Proofs that students are expected to construct using a format or template that you have provided
- Posters or other group projects through which students are expected to present their solutions and explain their thinking

Also consider the extent to which you have communicated your expectations regarding student note taking during class or the extent to which you engage students in reflecting on what they are learning and what questions they have about that content.

**Reflect: **What literacy demands are associated with writing in your mathematics classroom? When, how often, and for what purpose are students expected to write?

### WHY IS MATHEMATICAL WRITING IMPORTANT?

Writing provides students with opportunities to communicate their mathematical thinking in an organized and coherent way. Students need to consider what mathematical terminology to use in their communication, what expressions or equations might be helpful as they lay out their solutions to problems they are solving, and what models or representations might provide additional insights into those solutions. Essentially, students need to be able to consider how someone else might make sense of their writing; this often involves being explicit about aspects of their thinking that they might otherwise take for granted but which often can contribute to even greater mathematical insight.

While the Standards for Mathematical Practice do not specifically identify and discuss mathematical writing, there are many that focus on the importance of being able to communicate one’s mathematical thinking, and thus have implications for writing. These include being able to construct viable arguments, critique the reasoning of others, model everyday situations with expressions and equations, draw conclusions and interpret results, and communicate precisely to others. All of these practices are consistent with what it means to write within the discipline of mathematics.

Countryman (1992), in her book *Writing to Learn Mathematics: Strategies That Work, *discusses her own efforts to support mathematical writing in her high school mathematics classroom. She offers the following reasons why she thinks writing is important:

Writing can provide opportunities for students to construct their own knowledge of mathematics. . . . Writing has given them a chance to practice inferring, . . . symbolizing, organizing, interpreting, linking, explaining, planning, reflecting, and activating. Writing helps students make sense of mathematics (pp. vi–vii).

Many others also make the argument that opportunities to write mathematically help students deepen their understanding of the mathematics content itself (e.g., Bosse & Faulconer, 2010; Burns, 2004).

Finally, it should be noted that on some occasions, mathematical writing may serve as a personal log of one’s own mathematical thinking, including questions and confusions as well as new mathematical insights, and may not be intended as a communication to others. This kind of writing may be less precise and analytical, but it plays an important role as students work to construct their mathematical understandings.

The use of this kind of informal writing in mathematics classrooms can help support the full engagement of students (Countryman, 1992). For instance, during whole-class discussions, only one person is speaking at a time. The other students should be actively listening, but some may become disengaged. Even during small-group discussions, where more students have opportunities to speak, others may sit back and just listen passively. When students have opportunities to reflect and write their own thoughts before discussions begin, there is more likely to be full engagement in the discussion.

### MATHEMATICAL WRITING AS FORMATIVE ASSESSMENT

Mathematical writing can also provide teachers with insights into the mathematical thinking of their students, including their level of proficiency with mathematical vocabulary and notation and their fluency with important models and representations. When doing formative assessments, teachers might ask themselves, What do students understand about the mathematics associated with the problems and tasks they are being asked to solve? How well do they articulate justifications for their solutions or identify their reasons for each step in a proof? How are students making connections across important mathematical ideas as they write reflectively about a mathematics topic? How do students communicate their analyses of the range of solution strategies that might arise as they examine and respond to the mathematical thinking of others? Having students engage in mathematical writing also allows confusions or misconceptions to surface so they can be discussed and addressed.

Feedback to students on their writing, either from the teacher or from other students, can often help students become more powerful mathematical writers. This feedback can consist of questions about anything that is not clear in the mathematical writing, suggestions for improving the writing, or comments on what is powerful in the writing. All these different kinds of feedback provide students with opportunities to revisit and reengage with their mathematical thinking and how effectively they are communicating that thinking.

An additional strategy that can help students strengthen their writing is to show them examples of strong mathematical writing and have discussions about the features that make it strong as well as to share anonymous examples of writing that needs to be strengthened and have discussion about how it might be strengthened. The use of rubrics that identify the features of strong mathematical writing, particularly with regard to solutions for “open response” items or rich tasks, can also be extremely helpful.

When mathematical writing is intended to serve as a personal log of students’ own mathematical thinking, including questions and confusions as well as new mathematical insights, and not a communication to others, expressing a genuine interest in their writing and providing encouragement may be more powerful and appropriate than giving feedback on the writing itself.

### THE IMPORTANCE OF SMALL-GROUP AND WHOLE-CLASS DISCUSSION

Just as small-group work and whole-class discussions can be powerful supports for student reading and sense making in mathematics classrooms, they also serve as an important support for student writing in mathematics classrooms. For instance, the third Standard for Mathematical Practice (MP3) calls for students to “construct viable arguments and critique the reasoning of others.” Students who learn to do this through small-group and whole-group discussion may find it easier to lay out their “viable arguments” and “critique the reasoning of others” in writing. Similarly, the sixth Standard for Mathematical Practice is “attend to precision.” Students who have opportunities to practice communicating precisely to others during small-group and whole-class discussion are better poised to convey this communication more precisely in writing.

For example, Burns (2004), in her list of strategies for incorporating writing during mathematics class, offers the following: “Have students discuss their ideas before writing” (p. 32). She explains that most students find it easier to talk than to write, and opportunities to talk can help students organize their ideas, and get feedback on those ideas. In general, many teachers feel it is easier for students to get their thoughts down on paper if they have a chance to practice articulating those thoughts verbally before they write (e.g., Anderson & Little, 2004; Baxter et al., 2002; Chapin et al., 2013; Lynch & Bolyard, 2012).

At the same time, some teachers have also found that asking students to write about their thoughts is a good way to prepare for a small-group or whole-class discussion, where students can get productive feedback on their thinking and their writing, and where both that thinking and that writing can be strengthened (e.g., Countryman, 1992; Fernstein, 2007). There are also benefits that come to the readers of the mathematical writing of others: it provides them with opportunities to think about and discuss different mathematical points of view and broaden their own perspectives (e.g., Burns, 2007).

### Instructional Practices That Support and Strengthen Mathematical Writing

In the next set of videos, you will see teachers engaging their students in some form of writing that communicates the students’ mathematical thinking. While the particular contexts vary and the particular audiences differ, these videos highlight the range of opportunities that are present to support and strengthen the mathematical writing of students.

In all of the videos, you will see that small-group and whole-class discussions are often used to support and strengthen mathematical writing. Students may be given opportunities to talk about their mathematical thinking in small groups before trying to get their mathematical ideas down on paper. A whole-class discussion where ideas are generated and discussed may precede a mathematical writing assignment. There may be times when students move back and forth between discussing their thinking and writing about their thinking. The work of trying to verbalize one’s thinking can be a helpful support for trying to write about one’s thinking.

These videos also provide opportunities to see how feedback serves to strengthen student writing. Some of this feedback may be provided informally, as teachers look over what students are writing and make comments, or when students look over each other’s written work and ask questions. Some of this feedback may be provided more formally, in writing, with time for students to read and reflect on that feedback and make revisions to their written work. In all of these kinds of situations, student writing is being strengthened.

As you watch these videos, consider the instructional strategies you see at play and reflect on which of these you might want to try in your own classrooms. Also consider what the teachers say and what the students say about what it looks like and feels like to write mathematically. Finally, keep in mind that strengthening and supporting student writing in mathematics class is a process that takes place over time. The students you see in these videos may be accustomed to writing on a regular basis, but this may not be what the process looked like at the beginning of this effort.

### WHAT DOES THE RESEARCH LITERATURE SAY?

While there are many recommendations regarding the importance of writing in mathematics classrooms, the research on the impact of mathematical writing on student learning is more limited. A summary of the research identifies a number of studies reporting that students demonstrate greater mathematical understanding and learning through “writing to learn” approaches; however, researchers making this claim explain that writing has this impact because it requires students to “investigate and consider mathematical concepts and connections and practice communicating such to others” (Bosse & Faulconer, 2010, p. 10). This claim is consistent with the many recommendations regarding the role of mathematical writing in classrooms by mathematics education professional organizations (e.g., NCTM, NCSM) and is also at the heart of many of the Standards for Mathematical Practice.

There is little research to date on the impact of strategies designed to strengthen and support student writing on the actual quality of writing that students produce. However, anecdotal data suggest that disciplinary literacy practices are more successfully incorporated into mathematics classrooms when teachers demonstrate an interest in reading and writing in mathematics themselves in order to learn more mathematics, share these learning experiences with their students, and provide structured opportunities for students to engage in these disciplinary literacy practices themselves (Bosse & Faulconer, 2010).

Another aspect of the question of the impact of mathematical writing on student learning has to do with the extent to which teachers use what they learn from student writing to reflect on and strengthen their instruction. Unfortunately, there does not currently appear to be any research on this question; teachers learn a good deal about their students’ mathematical thinking from a variety of sources, and it would be difficult to attribute the impact to student writing specifically. However, research on the impact of *effective formative* assessment strategies on student learning suggests that this can have an impact on student learning of mathematics.

The NCTM research brief* Five “Key Strategies” for Effective Formative Assessment* (2007) identifies the following five strategies:

- Clarifying, sharing, and understanding goals for learning and criteria for success with learners
- Engineering effective classroom discussions, questions, activities, and tasks that elicit evidence of students’ learning
- Providing feedback that moves learning forward
- Activating students as owners of their own learning
- Activating students as learning resources for one other

[Reprinted with permission from the National Council of Teachers of Mathematics, the public voice of mathematics education, supporting teachers to ensure equitable mathematics learning of the highest quality for all students. Membership information can be found at www.nctm.org.]

The author of the research brief provides solid research evidence for the impact of each of these strategies and concludes the following:

The available research evidence suggests that considerable enhancements in student achievement are possible when teachers use assessment, minute-by-minute and day-by-day, to adjust their instruction to meet their students’ learning needs. (p. 4)

The mathematical writing of students certainly provides at least some of that critically important formative assessment data.

Because providing feedback to students on their mathematical writing is an important strategy for strengthening student writing, it is useful to consider what this research brief suggests about what kind of feedback has been found to be helpful. There is evidence that feedback consisting of comments that force students to “engage cognitively” (p. 2) in the work provides greater support for student performance than feedback consisting of grades or scores or even comments in combination with grades or scores. In other words, feedback that communicated what needed to be improved and how it might be improved was more effective than feedback that focused on the students themselves and how they compared to each other.

Some examples of feedback that is “cognitively engaging” include the following:

- There are 5 answers here that are incorrect. Find them and fix them.
- You’ve used substitution to solve all these simultaneous equations. Can you use elimination?
- You seem to be confusing sine and cosine. Talk to Katie about how to work out the difference.
- Compare your work with Ali and write some advice to another student tackling this topic for the first time.

As the author of the brief points out, these examples all serve to put “the ball back in the student’s court” (p. 3), adding that students will need time set aside to read, respond to, and act on this feedback.

### RECONNECTING WITH OUR MATHEMATICIANS

As you think back on the videos you’ve reviewed and reflected on, consider how the students in these videos are engaged in communicating and writing mathematically in ways that reflect the authentic work of mathematicians.

Mathematicians regularly write about their mathematical thinking, including the notes they take and the thoughts they have as they tackle a difficult problem or attempt a new proof. They need to be able to use this writing to consolidate and communicate their thoughts to others. Sometimes these communications are informal, including exchanging drafts of solutions or thoughts about obstacles and insights with colleagues and collaborators. Sometimes they are more formal, such as final papers that are presented to colleagues or submitted for publication.

This is true for applied mathematicians, who use mathematics to find solutions for problems in the real world, often as part of a team, where a variety of perspectives are shared and discussed and where solutions are finally laid out in the form of a report. This is also true for theoretical mathematicians, who work on rich mathematical problems or proofs that advance the field, are responsible for documenting their thinking processes as they work toward a solution or a proof, and need to be able to clearly and compellingly present these solutions or proofs to others.

As you reflect on what you saw in these videos, an important question to consider is how teachers provided structured opportunities for students to engage in mathematical writing. How did they organize and structure these opportunities? How were these opportunities situated in the larger context of small-group and whole-class discussion? What did the teachers seem to learn from the mathematical writing of their students? How was feedback provided on that writing? As you contemplate these questions, it is important to keep in mind that there are times when students may need to engage in mathematical writing on their own to document their thoughts and questions or take notes on what they are learning during class. There are also times when students engage in mathematical writing as part of a small-group effort to document and explain their thinking. Finally, there are times when there are more structured opportunities for students to engage in mathematical writing as they solve particular problems or work on particular proofs.

As you consider what you have seen in these videos, consider how you can use what you are exploring here to strengthen and support the engagement of your own students in the disciplinary literacy practice of writing mathematically on an ongoing basis.