Writing Across the Curriculum:

Thinking Made Visible

When students are tasked with writing about their process for determining a solution, they make their thinking visible. Teachers can gain key insights into the depth of student understanding, and they can use that knowledge to deliver specific, actionable feedback and additional instruction as necessary.

Many math standards and assessments now ask that students explain their reasoning when it comes to solving math problems. Common Core math standards, for instance, state that students should be able to “construct viable arguments and critique the reasoning of others.”

The National Council of Teachers of Mathematics (NCTM) states that math instruction should enable all students to:

• “Organize and consolidate their mathematical thinking through communication”
• “Communicate their mathematical thinking coherently and clearly to peers, teachers, and others”
• “Analyze and evaluate the mathematical thinking and strategies of others”
• “Use the language of mathematics to express mathematical ideas precisely”

In “Principles and Standards for School Mathematics,” NCTM writes that:

“Through communication, ideas become objects of reflection, refinement, discussion, and amendment. When students are challenged to communicate the results of their thinking to others orally or in writing, they learn to be clear, convincing, and precise in their use of mathematical language.”

Writing is a metacognitive task. As such, the process of writing can help students work through complex concepts. Math education expert Marilyn Burns notes that when students write in math class,

“It requires students to organize, clarify, and reflect on their ideas… Writing in math class isn’t meant to produce a product suitable for publication, but rather to provide a way for students to reflect on their own learning and to explore, extend, and cement their ideas.”

When students defend their mathematical solutions in writing, they provide clear evidence of their understanding. Writing forces them to slow down and explain a solution in their own words. As William Zinsser states in Writing to Learn,  “Writing is how we think our way into a subject and make it our own.”

Math teachers shouldn’t worry about the intricacies of grammar when they review writing. Rather, you should encourage students to demonstrate their understanding of a subject. If students can explain how they obtained an answer, they can likely replicate that work. When teachers review writing about math, they can see whether students fully grasp a concept – or whether they’re just regurgitating formulas.

Making Math Relevant

While written answers make an appearance every so often in traditional math classes, it usually happens with an abstract question – “Train A goes west at 50 km/h and train B goes east at 40 km/h. Both trains depart at the same time, 1000 km apart. When will they cross?”

When it comes to writing in math, you can do more to engage your students. You can help them make real-world connections to math.

Dr. Mona Kiani, a ThinkCERCA School Success Manager, was teaching an 8th-grade class in 2017 when she posed an essential question to her students:

“How do the financial costs and benefits of different opportunities after high school compare?”

Mona’s class read a ThinkCERCA text leveled for 8th grade, which told them about Javier, a student evaluating three degree options and their impact of cost versus pay over time. Mona’s class calculated the economic benefits of each degree, and wrote arguments for which degree Javier should pursue.

Then, Mona extended the lesson. She asked her students to evaluate their choices for life after high school. Every student calculated the benefits and drawbacks of certain options – like attending a community college for two years and then transferring to a state university; going straight to a four-year college; going straight into the workforce.

“Students were immediately excited about the task – understanding finances and how to budget most effectively is inherently interesting to most middle and high school students,” Mona says.

“Students asked thoughtful questions about post-high school life, like what age people retire, what loans are, and what the drawbacks of getting credit cards are. They refined their internet research skills, hopped on the phone to get car insurance quotes, consulted with their parents about budgeting for food, talked to their advisors about dorms versus apartments, and some even swore they’d stop asking their parents for fashion items after creating a budget and seeing how expensive life can be.”

Through the writing activity, students were able to build strong personal connections to the concepts they learned about in math class. They not only learned how to evaluate career options – they understood the real-world importance of their math instruction as well.

Ideas and Activities to Practice Writing in Math

• Use CERCA to show students how to justify an answer in writing. Like Mona’s class, you can use the CERCA Framework to guide students through writing a formal argument in math. They may argue for the correct answer to a problem, or they could argue for the best way to solve a problem. By writing out a formal argument, they explain their processes and justify their thinking.
• State the CLAIM – What solution will students prove?
• Cite EVIDENCE – What evidence (student work and equations) support the claim?
• Explain REASONING – Why does the evidence (work) support the claim (solution)?
• Address COUNTERARGUMENTS – If students are evaluating multiple ways to solve a problem, they can address a second argument for how a person could go about finding a solution.
• Use AUDIENCE-appropriate language – Who is the audience of the assignment? What symbols or equations need to be explained?

• Ask students to write a one-page journal entry explaining a mathematical concept. Students can spend the first ten minutes of a class period working on the entry, and can complete the informal writing sample at home. Two high school mathematics teachers described this process for NCTM. The teachers wrote that the students wanted to keep writing journal entries after an initial trial period because “they learned to explain themselves better mathematically… our comments on the journals gave them immediate feedback on their understanding of the material.”

• Use graphic organizers to help students clarify their thoughts. By writing down information and structuring it on a page, students can analyze the relationship among mathematical concepts. After nine middle school teachers asked students to use the “four corners and a diamond” graphic organizer (in which students write down what they know; what they need to know; possible methods to solve the problem; attempts at solving the problem; and their response to the question) scores before and after students used the graphic organizer suggested that it “may significantly help students coordinate their mathematical ideas, methods, thinking, and writing,” according to Indiana University Southeast Associate Professor Alan Zollman.

• Have students write and reflect with the “Climbing and Diving” technique. This process, described by ASCD author Vicki Urquhart, has students take ten minutes to write down all that they have learned about a lesson or concept. Then, they read their writing and choose one concept to write about for an additional ten minutes, giving students a chance to reflect and organize their thoughts on the concept.

• Give students a few minutes to write down their thoughts before sharing them in a class discussion. The “think-write-share” method ensures that all students are involved in answering a group question, rather than just a few students jumping to the answer and raising their hands to share. It can help students articulate a problem, beyond just solving for it. This strategy also “heightens student engagement in writing,” Brigham Young University Professors Brad Wilcox and Eula Ewing Monroe note. “Concurrently, students are held accountable for their own mathematical understanding.”

• Provide a space for students to discuss a concept in small groups. Burns notes that “having students talk in small groups enables more students to express their ideas. After a discussion, remind students that they can write about any idea they heard, as long as it makes sense to them and they can explain it.”

Math Writing Prompts

• How did adding the shot clock change NBA scoring? [Assign in ThinkCERCA]
• When should health officials quarantine infected individuals to prevent diseases from becoming epidemics? [Assign in ThinkCERCA]
• What actions should be taken to protect communities along the Atlantic Coast from the effects of sea level rise? [Assign in ThinkCERCA]
• What conclusion can you draw about the relationship between the right to vote, voter registration, and voter turnout? [Assign in ThinkCERCA]

Math Discussion and Writing in Action

What does literacy in math look like? This video shows how one elementary school makes writing and discussion a core element of math class.