Current educational reform calls for middle grades students who have the foundational skills to be "high school" ready (Williams, Rosin & Kirst, 2011). The Common Core State Standards Initiative (2010) echoes this call for students who are college and career ready. The Common Core Mathematics Standards describe outcomes for middle grades students where they reason abstractly and quantitatively and construct viable arguments and critique the reasoning of others. This parallels emphasis in the language arts standards, which specifies that students engage with complex texts and academic language, and develop the ability to read, write, and speak grounded in evidence from texts. Writing in mathematics provides opportunities for these standards to be realized. Though writing in mathematics has experienced increased interest over the last decade, there is still a paucity of research to inform educators and policymakers. According to the National Institute for Literacy (2007), the improvement of students' writing skills is related to students' capacity to learn. Given the strong connections between writing and learning, mathematics students who have experiences with written communication can experience the rich, deep conceptual learning envisioned by the Common Core State Standards for Mathematics.

**Types of Writing in the Mathematics Classroom**

**
Tenets of
***This We Believe*
addressed:

- Students and teachers engaged in active learning
- Curriculum is challenging, exploratory, integrative, and relevant
- Educators use multiple learning and teaching approaches

Writing in the mathematics classroom can take many forms (Pugalee, 2005). One of the most common forms is expository writing in which students are asked to explain or describe their mathematical process when solving a rich task or problem. Another form is journal writing in which students could be prompted to write about a variety of topics such as a time they helped explain mathematics to a classmate, to write as many examples of a certain concept as they can conjure up, or to write about a recently discussed concept or procedure with which they are struggling. Writing also can be part of a long-term project such as a research paper about a history of mathematics topic or an exploration involving significant mathematics. Writing in mathematics can take place in the classroom or at home, individually or in a pair or bigger group.

**Theoretical Frameworks for Writing in Mathematics**

Mathematics education research supports the claim that writing in the mathematics classroom promotes a deeper understanding of concepts and procedures. Writing helps students extend their critical thinking abilities as well as the ability to link a new idea to relevant prior knowledge (e.g., Bicer, Capraro, & Capraro, 2013; Bradley, 1990; Craig, 2011; Powell, 1997; Pugalee, 2004, 2001). Writing promotes an individualized and constructive approach to the learning of mathematics, encouraging students to create their own problem solving knowledge (Carr & Biddlecomb, 1998; Steele, 2007). In general, the act of writing is viewed as requiring a deliberate analysis that encourages an explicit association between current and new knowledge that becomes part of a deliberate web of meaning (Vygotsky, 1987) promoting metacognitive frameworks that extend students' reflection and analysis (Pugalee, 2004; Pugalee, 2001). Boscolo and Mason (2001) contend that "writing can improve students' learning by promoting active knowledge construction that requires them to be involved in transforming rather than a process of reproducing" (p. 85). Writing helps students focus on their thinking about the mathematical processes in which they are engaged. It is through this reflection and analysis that critical and metacognitive thought processes become visible through the written word, thus building a strong foundation for continued growth of procedural and conceptual understanding.

**Writing and Learning Mathematical Content**

Writing in mathematics supports students in learning mathematical concepts and procedures. The National Council of Teachers of Mathematics' *Principles and Standards for School Mathematics* (NCTM, 2000, p. 60) highlights that, "Students who have opportunities, encouragement, and support for speaking, writing, reading and listening in mathematics classes reap dual benefits: they communicate to learn mathematics, and they learn to communicate mathematically." A mixed-methods study involving 293 middle grades students found that students' mathematical knowledge was extended as a result of their writing in mathematics (Reilly, 2007). The students increased the detail of explanations related to their approaches to learning mathematics. Their writing showed how their understanding of mathematical concepts changed over time.

Learning mathematics content and problem solving are intricately linked. Research indicates that writing supports problem solving in mathematics. In one study (Hensberry & Jacobbe, 2012) students wrote their thought processes through a structured diary designed to mirror Polya's (1985) heuristic for problem solving: understanding the problem, devising and carrying out a plan, and looking back. The problem-solving diaries were found to be an effective tool in improving students' solution strategies. As students wrote about their thinking they also demonstrated development of problem-solving skills. In a similar study, Arslan and Altun (2007) involved students in writing that involved reflecting on their use of six heuristic strategies: simplify the problem, guess and check, look for a pattern, make a drawing, make a systematic list, and work backwards. The experimental group significantly outperformed the control on the problem-solving post-test and retention test. Research on writing and problem solving shows that students apply better problem-solving skills when compared to students in nonwriting mathematics experiences (Bicer, Capraro & Capraro, 2013). Other research involving writing in the form of journal entries found that writing supports learning and makes mathematics more applicable and relevant to students (Clarke, Waywood & Stephens, 1993). Research supports writing as a learning tool for mathematics promoting students' conceptual understanding of mathematical concepts, problems solving, and metacognitive thinking.

**Writing and Developing Mathematical Thinking**

The Common Core notes that students should monitor and evaluate their problem-solving progress and develop alternatives as necessary (CCSSI, 2010). Fello and Paquette (2009) report that writing requiring students to describe why a procedure or strategy supports their formulation of connections between ideas and concepts and promotes deepening mathematical understanding. Akkus and Hand (2005) extended their work with writing in science in developing a mathematics reasoning heuristic consisting of a framework where students respond to key reflective questions: What is my question/problem?, What can I claim about the solution?, What did I do?, What are my reasons?, What do others say?, and Reflection. Their research found that classrooms where the heuristic was applied performed significantly better than control classrooms. This work supports the use of writing to promote mathematical reasoning and communication through dialogue. Research also supports the use of writing with academically low-achieving students (Baxter, Woodward & Olson, 2005). Four seventh grade low-achieving students in a classroom emphasizing communication demonstrated marginal participation in group and whole-class discussions; however, the students' math journals included entries where they demonstrated their abilities to explain their mathematical reasoning showing their conceptual understanding and skills in representing problems. This study was part of a series of studies focusing on low-achieving students in reform mathematics classrooms. The study looked closely at the writing assignments of a small number of particular students in order to focus closely on the performance of low-achieving, often marginalized students, in a targeted classroom of 28 students.

**Research-Based Instructional Practices**

As Common-Core aligned assessments—such as those by the Smarter Balanced Consortium— include many opportunities for students to communicate their mathematical process in writing, teachers are now charged with readying students for this new level of language demand. However, simply incorporating writing into mathematics instruction does not automatically result in well elaborated explanations nor immediate deepening of the writer's mathematical understanding (Shield & Galbraith, 1998; O'Connell, Beamon, Beyea, Denvir, Dowdall, Friedland, & Ward, 2005). Researchers agree that writing in mathematics requires the development of precision in the use of mathematical language; therefore, teachers should provide direct instruction to students and model appropriate processes (e.g., Adu-Gyamfi, Bossé & Faulconer, 2010). For instance, Shield and Galbraith (1998) proposed that a well elaborated written presentation of a mathematical procedure should include a generalized statement of the procedure, a demonstration of the procedure, a link from the procedure to prior knowledge, and a justification of the use of the procedure. The authors argue that teachers need to explicitly model the above mentioned components of mathematical expository writing in order for the writing to be effective in extending students' understanding. Similarly, Steele (2007) states that it is important for teachers to not only ask students to write about their problem solving but to prompt further by asking students to give their reasons for solving the problem in a certain way, to describe their procedure and why it works, and to justify why their solution is correct. In a teaching experiment in fourth through sixth grade classrooms, O'Connell et al., (2005) found that as participating teachers began providing students with support for writing about mathematics such as specific lessons on ways to more effectively describe ideas in writing, graphic organizers, and modeling, the more rewarding the writing tasks were for the students. Based on a review of research, Bossé and Faulconer (2008) provide many useful techniques for increasing both reading and writing in the mathematics classroom such as creating a "print rich environment" for students, highlighting text structures and vocabulary present in textbooks, requiring and supporting note taking, and modeling integrating and connecting multiple representations. The authors also include helpful guidelines for "avoiding negative consequences in the classroom" as a backlash from students not used to having a focus on writing in the mathematics classroom.

Lynch-Davis (2011) reports on research focused on teachers' comments to students' responses in mathematics journals. The research provides insights on assessment constraints, which is often an ignored area in studying instructional practices. She reports that teacher feedback to written work in mathematics is short and unelaborated with most of the comments being evaluative and focused on quality of the written response. Such practices are ineffective in eliciting better responses from students. Interpretative listening approaches focused on student thinking are more effective in promoting the type of communication that builds students' mathematical reasoning. In a study of four middle school teacher-researchers who engage in action-research for the first time on their instructional practice of incorporating writing in their classes, Ishii (2003) reported that the participating teachers found the main benefit of writing was to increase student discourse in the classroom. A few of the teachers posited that writing, if paired with an in-depth discussion and debate, could dramatically impact students' problem-solving abilities.

**Writing in Mathematics and Student Assessment**

Steele (2007) argues that when students only present procedures and algorithms in response to a problem, it is difficult for teachers to assess the depth of their knowledge. Asking students to write provides a valuable window into students' problem-solving strategies and their understanding of the problem at hand. Brown (2005) describes the use of a three-step process in which students engage in reflection and writing after completing a formative assessment. Students review their test and describe strengths and weaknesses. Students are guided to become better writers and extend their mathematical knowledge and language. Writing in mathematics provides unique opportunities for students to engage in self-assessment and become critical of the mathematical writing of others, and it serves as a window for teachers to better understand the mathematical thinking of students. Though writing is typically seen as a tool for student learning, the power of student writing as a vehicle for teachers to assess students' mathematical understanding cannot be underestimated.

**Conclusion**

Research supports that writing in mathematics promotes the learning of mathematical concepts at a conceptual level as well as builds proficiency with skills and procedures. In order for this vision for mathematically literate students as described in The Common Core State Standards Initiative (2010) and other standards documents to be fully realized, students must have many opportunities to develop and elaborate their mathematical thinking through writing. However, for students to find value in the writing component of a mathematics classroom, teachers need to do more than incorporate writing tasks in the mathematics classroom and provide explicit and detailed instruction to students about appropriate writing practices.

#### References

Adu-Gyamfi, K., Bossé, M. J., & Faulconer, J. (2010). Assessing understanding through reading and writing in mathematics. *International Journal for Mathematics Teaching & Learning*.

Akkus, R., & Hand, B. (2005, October). Mathematics reasoning heuristic (MRH): Writing-to-learn. In *27th annual meeting of the North American Chapter of the International Group for the Psychology of Mathematics Education, Roanoke, Virginia*.

Arslan, Ç., & Altun, M. (2007). Learning to solve non-routine mathematical problems. *Elementary Education Online*, *6*(1), 50-61.

Baxter, J. A., Woodward, J., & Olson, D. (2005). Writing in mathematics: an alternative form of communication for academically low‐achieving students. *Learning Disabilities Research & Practice*, *20*(2), 119-135.

Bicer, A., Capraro, R. M., & Capraro, M. M. (2013). Integrating writing into mathematics classroom to increase students' problem solving skills. *International Online Journal of Educational Sciences*, *5*(2).

Boscolo, P., & Mason, L. (2001). Writing to learn, writing to transfer. In P. Tynjala, L. Mason, & K. Lonka (Eds.), *Writing as a learning tool *(pp.83-104)*. *Dordrecht, The Netherlands: Kluwer Academic Publishers.

Bossé, M. J., & Faulconer, J. (2008). Learning and assessing mathematics through reading and writing. *School Science and Mathematics*, *108*(1), 8-19.

Bradley, C.A. (1990). The relationship between mathematics language facility and mathematics achievement among junior high school students. *Focus on Learning Problems in Mathematics 12*(2), 15-31.

Brown, S. A. (2005). You made it through the test; what about the aftermath?. *Mathematics Teaching in the Middle School, 11*(2), 68-73.

Carr, M., & Biddlecombe, B. (1998). Metacognition in mathematics: From a constructivist perspective. In D. J. Hacker, J. Dunlosky, and A. C. Graesser (Eds.), *Metacognition in educational theory and practice *(pp.69-91)*. *Mahweh, NJ: Lawrence Erlbaum Associates.

Clarke, D. J., Waywood, A. & Stephens, M. (1993). Probing the structure of mathematical writing,* Educational Studies in Mathematics, 25*(3), 235-250.

Common Core State Standards Initiative. (2010). Common Core State Standards for Mathematics. Washington, DC: National Governors Association Center for Best Practices and the Council of Chief State School Officers.

Craig, T. S. (2011). Categorization and analysis of explanatory writing in mathematics. *International Journal of Mathematical Education in Science and Technology*, *42*(7), 867-878.

Fello, S. E., & Paquette, K. R. (2009). Talking & Writing in the Classroom. *Mathematics Teaching in the Middle School*, *14*(7), 410-414.

Hensberry, K. K., & Jacobbe, T. (2012). The effects of Polya's heuristic and diary writing on children's problem solving. *Mathematics Education Research Journal*, *24*(1), 59-85.

Ishii, D. K. (2003). First-time teacher-researchers use writing in middle school mathematics instruction. *The Mathematics Educator, 13*(2), 38-46.

Lynch-Davis, K. (2011). Responding to journal writing in the middle grades mathematics classroom. *National Teacher Education Journal*, *4*(2).

National Institute for Literacy. (2007). *What content-area teachers should know about adolescent literacy. *Washington, DC: Author.

National Council of Teachers of Mathematics (NCTM). (2000).* Principles and standards for school mathematics*. Reston, VA.

O'Connell, S., Beamon, C., Beyea, J., Denvir, S., Dowdall, L., Friedland, N., & Ward, J. (2005). Aiming for understanding: lessons learned about writing in mathematics: reflect and discuss. *Teaching Children Mathematics. 12*(4), 192-199.

Polya, G. (1985). How to solve it. Princeton: Princeton University Press.

Powell, A. B. (1997). Capturing, examining, and responding to mathematical thinking through writing. *The Clearing House, **71*(1), 21-25.

Pugalee, D. K. (2005). *Writing to develop mathematical undertanding. *Norwood, MA: Christopher Gordon Publishers.

Pugalee, D. K. (2004). A comparison of verbal and written descriptions of students' problem-solving processes. *Educational Studies in Mathematics, 55, *27-47.

Pugalee, D. K. (2001). Writing, mathematics, and metacognition: Looking for connections through students' work in mathematical problem solving. *School Science and Mathematics, 101*(5), 236-245.

Reilly, E. M. (2007). *Writing to learn mathematics: A mixed method study*. ProQuest.

Shield, M., & Galbraith, P. (1998). The analysis of student expository writing in mathematics, *Educational Studies in Mathematics, 36,* 29-52.

Steele, D. (2007). Understanding students' problem-solving knowledge through their writing. *Mathematics Teaching in the Middle School, 13*(2), 102-109.

Vygotsky, L. S. (1987). Thinking and speech. In R. W. Rieber & A. S. Carton (Eds.), *The* *collected works of L. S. Vygotsky *(pp. 39-243). New York: Plenum Press.

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#### Annotated References

Akkus, R., & Hand, B. (2005, October). Mathematics reasoning heuristic (MRH): Writing-to-learn. In *27th annual meeting of the North American Chapter of the International Group for the Psychology of Mathematics Education, Roanoke, Virginia*.

The researchers tested a mathematics reasoning heuristic with a student template engaging students in reflective writing about key questions: 1. What is my question (problem)?; 2. What can I claim about the solution?; 3. What did I do?; 4. What are my reasons?; 5. What do others say?; and 6. Reflection. A one-way analysis of variance (ANOVA) showed a non-significant result on pre-test scores between control and treatment classrooms, and a significant difference between the control group and the treatment groups in favor of the treatment groups for the post-test. Writing also helped in identifying student misconceptions. The writing tasks assisted students in performing on related assessments of mathematics but also helped the teacher better understand the levels of students' understanding related to the mathematics topics.

Clarke, D. J., Waywood, A. & Stephens, M. (1993). Probing the structure of mathematical writing,* Educational Studies in Mathematics, 25*(3), 235-250.

This study focused on the effectiveness of journal writing and its relationship to mathematics learning. Benefits that grow from the use of writing in a classroom are described. The study found that journaling can enhance the connection between learning and writing and make mathematics more applicable and relevant to students. The findings are based on data from 500 students in grades seven through eleven. The writing also demonstrated a powerful link between language and mathematics. The authors also describe a relationship between the students' writings and perceptions of mathematics and mathematical activity.

Pugalee, D. K. (2004). A comparison of verbal and written descriptions of students' problem solving processes. *Educational Studies in Mathematics*, *55*(1-3), 27-47.

This study investigated the relationship between writing and problem solving. The study compared data of students who wrote descriptions of their mathematical problem solving processes to those who provided only verbal descriptions while solving a mathematics problem. Performance on the written problem-solving tasks was significantly better than performance on the tasks with verbalized descriptions (*p < *0.05). An analysis of the written data showed the presence of a metacognitive framework: understanding the problem and what the problem is asking (orientation), planning and selecting strategies (organization), monitoring progress (execution), and evaluating the outcomes (verification).

#### Recommended Resources

Lyman, F. (2003). *Think-pair-share smart card*. San Clemente, CA: Kagan Cooperative Learning Resources for Teachers.

O'Connell, S. (2001). *Math, the write way: Thinking and writing about math*. Grand Rapids, MI: Frank Schaffer Publications.

Pugalee. (2005). *Writing to develop mathematical undertanding. *Norwood, MA: Christopher Gordon Publishers.

Writing in Mathematics. http://www.mathwire.com/writing/writing1.html

#### Author Information

**Amélie Schinck-Mikel** is an assistant professor of mathematics education in the department of mathematics at Cal Poly San Luis Obispo. She is interested in socio-cultural issues in mathematics education, pre-service teacher education, language and mathematics learning, and in-service professional development. Her email address is aschinck@calpoly.edu.

**David K. Pugalee** is professor of education at the University of North Carolina at Charlotte where he is the director of the Center for Science, Technology, Engineering and Mathematics Education. He is interested in language and communication and how they influence STEM teaching and learning. His email is david.pugalee@uncc.edu.

*Published June 2014*