"Writing in mathematics gives me a window into my students' thoughts that I don't normally get when they just compute problems. It shows me their roadblocks, and it also gives me, as a teacher, a roadmap," says Maggie Johnston, a ninth grade mathematics teacher in Denver, Colorado.
Some mathematics teachers, like Maggie, use writing to its full potential. Do you? As well-crafted as your mathematics lessons might be, without the element of writing, they are not as engaging and cognitively demanding as they could be. Each time you ask students to write, you set the stage for active problem solving, invention and discovery, and improved content learning.
In Maggie's algebra class, students are paired at tables, calculators within easy reach. She begins, "Here is a problem that my Algebra II class did yesterday: Condense 2log4 x – 5logy. Does anyone know how to do this?"
Students mumble, but no one speaks up. Maggie allows a few minutes of wait time, but it's what she does next that might surprise you. Instead of turning to the board and showing students how to solve the problem, she tells her students to write their reactions to being asked to solve it. Maggie is teaching students how to tap into their own thinking through the act of writing—thinking on paper.
Modeling Metacognitive Processes
In Classroom Instruction That Works: Research-Based Strategies for Increasing Student Achievement, authors Robert Marzano, Debra Pickering, and Jane Pollock encourage teachers to emphasize the metacognitive control of processes. They write that research suggests that metacognition, or thinking about one's own thinking, is integral to problem solving and can lead to more self-directed learning.
As problem solvers, when we encounter something new, we first activate prior knowledge, and then our metacognitive processes help us consider possible strategies for solving it. Throughout this problem-solving process, we monitor and evaluate our thinking. Maggie frequently uses writing prompts to get to this metacognitive piece of math learning, and she models the process so that everyone is clear about what to do.
To effectively model metacognition, Maggie has prepared ahead of time one writing prompt and three samples of possible written responses. Today, she uses the prompt: When I see this problem, my first reaction is . . .
Before students begin to write, Maggie models (both orally and in writing) different possible answers. For example, she says, "Here is option one: I realize why I hate math. Option two: I'm a little afraid, but I know that condense means to make smaller. Option three: I'm not exactly sure how to do this problem, but we've been learning about logs, and I think I can do it."
Maggie intentionally gives examples that vary from an unimpressive response to an impressive one. She wants her students to have an idea of how to approach writing, and she wants to help settle any fears they might have about letting her into their thinking.
Using Non-threatening Prompts
Maggie starts at the beginning of the year to develop a system for writing in her math classroom by using non-threatening writing prompts (see below). When students are comfortable with the "how to's" of writing in math class (e.g., explain how you would solve this equation), she moves on, and once students are comfortable at that level, she moves on again, using deeper understanding prompts.
Prompts That Help Build Students' Comfort with Writing
Reflect on your participation in class today and complete the following statements:
- I learned that I . . .
- I was surprised that I . . .
- I noticed that I . . .
- I discovered that I . . .
- I was pleased that I . . .
- My best kept secret about math is . . .
- When I hear someone say "Math is fun!" I . . .
- My three personal goals in math this term are . . .
- I still want to learn about . . .
- After I solved the problem about ________________, I felt . . .
Since she began using writing in her classroom, Maggie has been getting better responses, and she feels better prepared to help all of the students in her classes because she has a window into their thinking.
"By integrating writing into mathematics, students can't bypass orientation and organization before beginning execution. They have to organize themselves, do the problem, and rethink it," she says. "I try to lead them through the metacognitive piece and actually design my writing prompts to do that. That's the most valuable part."
Pairing Prompts and Rubrics
Maggie and her students regularly use a time-saving, three-point rubric. At the beginning of the year, she models the rubric as a tool for them to think about their effort.
In response to the answer, "I realize why I hate math," Maggie asks, "Why is this a one-point response?" A student replies that it doesn't reflect the writer knows anything and adds that the second response, "I'm a little afraid, but I know that condense means to make smaller," shows the student knows a little bit. Maggie agrees, saying, "It tells me they are starting to connect to what they know, but the third one tells me they really are thinking about it."
With the class ready to write, Maggie provides half-sheets of paper that she can quickly collect and review. After skimming the responses, she surmises that most of the students are willing to tackle the problem, although some are less confident about it than others. So she instructs them to try it alone, then to share their answers with a neighbor and come up with a common solution. She reminds them to aim for a three-point response and gives them a couple of minutes to work on it alone before she allows them to work in pairs.
Once students begin to work together, a student comments to his partner, "I'm still confused." The partner replies, "Let's figure it out." Together, they begin to talk their way through the problem.
Maggie uses writing prompts as an informal strategy to help students focus their learning, organize their thinking, and communicate their understanding about a mathematical concept. She uses rubrics as instructional tools to clearly communicate the goals for learning and to provide feedback based on specific criteria.
Rubrics also focus students on the knowledge and skills they are supposed to learn. For students to communicate mathematical understanding, they need to know where they stand in relationship to learning the knowledge or skill. In this way, she uses writing and rubrics to inform learning.
Improving Math Literacy
Like standards, writing can be a roadmap for learning, and perhaps a richer, more detailed one. When most of us reflect on our own middle school experiences, we recall writing in English classes—but in mathematics? Those classes were jam-packed with skill-building and conceptual understanding activities.
For today's students, writing during a math lesson is a way to deepen learning and to get a new perspective. And, for students whose strengths are language-based, writing can be the key to understanding mathematics.
Let's review the steps Maggie took to use writing to improve students' math learning.
- Identify the current mathematics content and the learning goal (e.g., activate metacognitive process).
- Develop or select a question or prompt that will help students access prior content knowledge and focus their learning.
- Introduce the writing prompt to students through a variety of methods.
- Model for students examples of responses of varying quality and discuss each.
- Allow a few minutes for students to respond individually to the prompt.
- Collect and quickly scan responses to get insight about students' readiness to learn.
- Adjust your lesson accordingly. (If many of the students are not ready to try something new, you might need to review earlier content, or you might want students to work in pairs or triads to collaboratively solve a new problem.)
- Asked to express themselves in writing, students must organize their thinking and learning about the content. During the process, there are also opportunities to share their ideas, experience a creative side of mathematics, and learn to value the act of writing. Writing can be as much at home in a middle school mathematics classroom as in an English class.
Sample Three-Point Rubric to Improve Writing
||Levels of Performance
||The response addresses the entire prompt and communicates the appropriate mathematical understanding. The strategy used by students is effective in answering the prompt (including graphs, models, patterns, or pictures).
||The response addresses most of the prompt and demonstrates adequate knowledge and understanding. There is evidence that, with slight feedback, the student could reach the desired response. There may be a few overlooked areas and slight errors in understanding.
||The response demonstrates slight knowledge of the content necessary to complete the prompt. While there was effort made to complete the task, little was successful. Direct instruction would be necessary for the student to be able to accomplish the task.
Adapted from Maggie Johnston's rubric.
Previously published in Middle Ground magazine, April 2009
Vicki Urquhart is a lead consultant at Mid-continent Research for Education and Learning (McREL), where she writes, edits, and produces research-based publications and products. E-mail: firstname.lastname@example.org