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I will be conducting my action research with sixth grade mathematics students.

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APA-formatted reference page for my literature review sources

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AMLE - Association for Middle Level Education

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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.

Williams, T., Rosin, M., & Kirst, M. W. (2011). Gaining ground in the middle grades. Education outlook. No. 1. American Enterprise Institute for Public Policy Research.

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

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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.

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As science and math are slated for more emphasis in our classrooms, writing should not be sacrificed. Indeed, in the past two decades, neuroscience and cognitive science research have provided increasing evidence that correlates creativity with academic, social, and emotional intelligence. Writing can help the brain to develop the logical functions required for successful math and science learning.

Writing to Develop Executive Function

During the school years, especially from ages 8-18, the most rapid phase of maturation is taking place in the prefrontal cortex. This is a critical time during which the brain is developing the individual's executive functions. These include judgment, critical analysis, induction, deduction, delay of immediate gratification for long-term goals, recognition of relationships (symbolism, conceptualization), prioritizing, risk assessment, organization, creative problem solving. There are also emotional aspects to executive function, including the ability to identify one's emotional state, exert emotional self-control, and reflect about emotional response choices.

When it comes to math and science, writing brings more than literacy and communication advantages. The practice of writing can enhance the brain's intake, processing, retaining, and retrieving of information. Through writing, students can increase their comfort with and success in understanding complex material, unfamiliar concepts, and subject-specific vocabulary. When writing is embedded throughout the curriculum, it promotes the brain's attentive focus to classwork and homework, boosts long-term memory, illuminates patterns, gives the brain time for reflection, and when well-guided, is a source of conceptual development and stimulus of the brain's highest cognition.

How the Brain Stores Information

There is an involuntary information intake filter that determines what sensory input is accepted into the brain. Input must also pass through an emotional filter, the amygdala, where the destination of that information is determined. When stress is high, the intake filter selectively admits information related to perceived threat, virtually ignoring other sensory input. The high stress state also directs the amygdala switching station to conduct information to the lower, reactive brain, where long-term retrievable memories cannot be formed. In addition, the behavioral outputs of the lower brain are limited to fight (act out), flight (self-entertainment sometimes misinterpreted as ADHD), or freeze (zone out).

Use Code Names to Reduce Anxiety

Fear of making mistakes in front of classmates is one of the greatest sources of anxiety for students. This fear impacts learning. With the help of blogs and wikis, a teacher can set up an environment that enables students to post anonymously via a code name that's known only by the teacher. These code names afford individual privacy so students can express themselves without the fear that limits in-class responses and questions.

With these tools, we can introduce a variety of writing activities that reduce the stress that blocks passage through the amygdala to the reflective prefrontal cortex. For instance, students can write descriptive responses to math or science questions, as well as predictions, hypotheses, and questions. Other useful writing exercises include journaling, newspaper editorials to defend a position, and formal research-style formatted reports of student experimentation with data analysis.

These activities provide all students with the opportunity to actively participate in learning, as they receive timely feedback, reflect, revise, and risk making mistakes. In this way, writing can build confidence and reveal gaps in foundational knowledge. Students can share creative insights, and build their capacities to communicate their ideas and defend their opinions.

Through these shared written reflections about content and concepts students have opportunities to express creative hypotheses, alternative perspectives, and concerns about their understanding. Especially with peer anonymity, there is accountability and peer interaction, without the concern about mistakes that is so paralyzing to many students during class time. As students consider and define their opinions, conclusions, and predictions in writing, their brains construct valuable concept networks. (More on these below.)

When learning is examined through shared writing, students are exposed to multiple approaches to solving problems. This is so important in building the flexibility and open-minded approach to other cultures as the science, math, and technology world is indeed global. Furthermore, students have the chance to communicate using their own words. They build communication skills they will surely use in their collaborations now and in the future science and math communities they will enter.

Make Writing Relevant to Help Curb Dropout Rate

Writing can also reduce the neural processing blockades that result from the stress of boredom -- the most frequent reason high school dropouts give for leaving school. We know that students are engaged when material is personally relevant, and connects to real world issues and problems. And when this happens, there is increased information flow through the attention and emotional filters to the higher processing prefrontal cortex.

Writing can increase both personal relevance and confidence. Personal relevance comes when students can write for creativity and personal expression. Even when the facts of the math or science are not debatable, individual responses to the information are appropriate writing topics. When writing is incorporated in learning and assessment, there is increased opportunity to produce the ideal situation for active, attentive learning with collaboration, revision, and metacognition through personalization, and creativity.

Reminding students of previous successes promotes confidence, as does providing them the opportunity to recognize their own progress over time. One way to help them recognize their progress is through their written assessments: For instance, along with the content you're assessing, you can also have students write their responses to both the learning itself and to recognition of their progress. These can be maintained in computer files or portfolios and reviewed as evidence of successful, incremental progress with student opportunities for metacognition about strategies used for success.

What's Going on in There? Writing "Sprouts" Conceptual Brain Networks

The construction of conceptual memory networks builds the most valuable neural architecture a brain owner can have. These networks serve as "nets" to catch and hold new input with similar patterns, and "work" when activated for creative transfer -- use of the information learned in one context for application in a new context.

Concept networks are the valuable tools the brain uses in the highest orders of thinking. When the brain seeks to predict the best response, answer, solution to a problem or make a choice, the executive function control networks in the prefrontal cortex send out messages to the memory association areas, such as the hippocampus and memory storing cortex of each hemisphere. These messages activate stored prior knowledge memories that relate to the new situation. The more extensive the brain's collection of memory networks, the more successful it will be in activating the best prior knowledge to predict the best responses, answers, and choices for any new situation. The greater the links and cross-connections among networks of stored information, the greater a person's access to multiple storage centers of background knowledge to use in response to the new problem or opportunity.

Writing as Memory Cement and Concept Clue

Memory bundles, such as Piaget's schema, contain category-related information linked in circuits based on commonalities (such as similar sounds, visual images). These bundles of neurons are linked together because they have been used together repeatedly.

The many varieties of writing can serve to guide the brain to recognize, construct, and extend its patterns. Writing can illuminate sequential procedures that students need to learn in mathematics and science, from factoring an equation to the photosynthetic chemical process. Prior knowledge can be activated to link with new input through writing and drawing mind maps or graphic organizers, and new learning can be added into visible and mental patterns when students write analogies and other comparisons.

The neural activity or mental manipulation that transforms formulas, procedures, graphs, and statistical analyses into words represents the brain's recognition of patterns. When this is also done in writing, the facts, procedures, and observations are processed symbolically in the writing process -- giving the memory another storage modality and truly illuminating the patterns for the brain to follow as it adds new learning to existing concept networks.

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The Brain-Based Benefits of Writing for Math and Science Learning

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As science and math are slated for more emphasis in our classrooms, writing should not be sacrificed. Indeed, in the past two decades, neuroscience and cognitive science research have provided increasing evidence that correlates creativity with academic, social, and emotional intelligence. Writing can help the brain to develop the logical functions required for successful math and science learning.

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Categorisation and analysis of explanatory writing in mathematics

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I am studying the effects of oral and written communication on sixth grade mathematics understandings. I have begun brainstorming data collection methods and am open to suggestions.

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The main themes that I have identified among my sources are expository writing, reflective writing, and oral communication.

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Sagor, R. (1992). How to conduct collaborative action research. Alexandria, VA: ASCD.

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Mills, G. (2014). Action research: A guide for the teacher researcher. Upper Saddle River, NJ: Pearson Education.

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Edutopia is a free source of information, inspiration, and practical strategies for learning and teaching in preK-12 education. We are published by the George Lucas Educational Foundation, a nonprofit, nonpartisan organization.

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The 2002 and 2007 reports of the National Assessment of Educational Progress (NAEP) writing as ses sment (National Center for Education Statistics, 2008), administered to 8th and 12th graders, show an increase in writing scores. Applebee and Langer's (2006) analysis of NAEP data as well as other sources published during the decade preceding their review revealed that the more frequently students reported writing one or more paragraphs in science and social studies, the higher their writing achievement.

The results, however, were lower in mathematics. "It may be that at the classroom level, the role of writing in mathematics instruction has not been well conceptualized" (Applebee & Langer, 2006, p. 14). Indeed, many teachers find it more natural to integrate writing and science (e.g., Varelas, Pappas, Kokkino, & Ortiz, 2008) or writing and social studies (e.g., Jones & Thomas, 2006). Wolsey (2010) examined the complexity of student writing and vocabulary learning in a cross-disciplinary writing project involving English, science, and social studies. Where was mathematics?

That we see so few examples of the integration of writing and mathematics in educational literature seems surprising, considering that the mathematics education community has affirmed the importance of such integration for many years. As early as 1989, the National Council of Teachers of Mathematics (NCTM) identified learning to communicate mathematically as a major goal for students.

In 1992, Countryman's book, Writing to Learn Mathematics: Strategies That Work, K-12, captured the attention of mathematics educators amid a flurry of interest and ideas (e.g., Quinn & Wilson, 1997; Sipka, 1990). Almost a decade later, NCTM (2000) specifically stressed writing as "an essential part of mathematics and mathematics education" (p. 60) in the landmark document Principles and Standards for School Mathematics.

Still, many teachers struggle to link writing and mathematics and honor the integrity of both disciplines at the same time. Teachers of writing might say that if students are assigned to describe the process they used in solving a problem with no revision or editing, the quality of integration is drawn into question. Teachers of mathematics might say that if students are asked to write a report on a famous mathematician they may not be engaged in developing mathematical reasoning no matter how many drafts they write. Although the appropriate balance may be elusive, the endeavor is nevertheless worthy of being undertaken (e.g., Applebee, 1984; Bossé & Faulconer, 2008; Newell, 2008).

There are two levels of integration that teachers may use as a beginning point. Writing without revision, the first level, can be readily worked into mathematics instruction. Writing with revision, the second level, may take more time but enables teachers to connect the writing process more fully with mathematics instruction. Each level can be appropriate under differing circumstances.

In Carter's (2009) description of writing tasks in her mathematics instruction, both levels are reflected. She had her students write in what she called "mathematical notebooks" (p. 607), which required little or no revision. She also had her students write about mathematics in a process-oriented way during writing workshop. Some wrote stories such as "The Hexagon Adventure" and "The Magical Subtracting Frying Pan." Others wrote autobiographies such as "How I Learned to Do Math in Kindergarten and Beyond."

Although Carter's (2009) reflective mathematical journals and stories about mathematical concepts in writing workshop may not represent an ideal solution, they demonstrate one teacher's exploration of the possibilities. This article presents six additional examples, including student work, in which teachers have attended to the goals of both writing and mathematics. By no means are these ideas new. Teachers may have been introduced to them before, but perhaps not in the context of mathematics. Although the following suggestions are not exhaustive, teachers have found them to be a promising place to begin.

Writing without revision

Learning logs

As students start class, they are given a prompt to which they respond for a few minutes in writing. The task is not designed as a mathematics problem per se, but rather to encourage students to focus on mathematics.

One fifth-grade teacher typically used learning logs to review previously learned material. On one occasion, the prompt she posed was, What did we learn about mean, median, and mode? Most students wrote at least a half page filled with definitions and examples (see Figure 1A). While studying probability, one student wrote the following definition: "The 'probability' of something is how likly your gonna get, pick, or find something." He then gave an example using shapes and colored cards (see Figure 1B).

Figure 1A: Learning log entries by fifth graders - definitions and examples of mean, median, and mode

Figure 1B: Learning log entries by fifth graders - definition and examples of probability

This teacher found that the quality of the learning logs improved as students shared their work: "[When a student shared] a log that showed clear understanding, this provided others with an example to follow." Along with finding learning logs to be an effective way to introduce or close a lesson, the teacher observed that the quality of student discussion of mathematics during lessons was richer when students were expected to write.

Learning logs honor the integrity of writing when students write their own connections and examples. At the same time, they foster mathematical understanding by engaging the minds of students in transforming information from facts to be memorized to the construction of meaning (e.g., Newell, 2008).

Think-write-share

Teachers often ask questions and count on having at least one or two students raise their hands. One 2011fourth-grade teacher tried to involve everyone in the class by allowing think time and expecting students to write before sharing. For one think-write-share, she asked, "What is an equivalent fraction?" She gave the students a few minutes to think and then asked them to write their responses before calling on students to share, using a document camera. In reflecting on the use of this strategy, the teacher commented,

I assumed that everyone understood [what an equivalent fraction is], because we've done it for a while. However, as we shared, I soon realized that some of my students thought that equivalent fractions were [only] fractions that were the same, for example, 1&#8260;4 = 1&#8260;4.

To address this misconception, the teacher strategically selected students to share based on the thinking she had seen demonstrated: "Kids who were getting it wrong, kids who were getting it a little, and those who knew it well." Once several clear responses had been shared and discussed (see Figure 2A), the teacher invited all students to write again, revising their original work to reflect their increased understanding (see Figure 2B).

Figure 2A: Think-write-shares by fourth graders - example showing a clear understanding of equivalent fractions

Figure 2B: Think-write-shares by fourth graders - example of equivalent fractions, before and after revision

The think-write-share strategy heightens student engagement in writing. Concurrently, students are held accountable for their own mathematical understanding.

Note-taking/note-making

Students may be accustomed to taking notes, but now ask them to make notes as well. Along with listing the main points of a lesson, students can write their own reflections and perceptions. One fifth-grade teacher used the strategy of note-taking/note-making in mathematics to extend the notion of whole numbers to integers, including the concept of negative number.

She asked her students to fold their papers in half vertically. On the left side, she directed students to define integers and construct number lines to demonstrate relative size of integer pairs. On the right side, students wrote their own reactions and observations. One student wrote, "I will remember that on the negitive side the bigger it gets the smaller it is. I thought that was cool." Another student wrote, "It is weird that -2 is greater than -5." This same student went on to create his own analogy: "It is like a piano, lower to higher with middle C. Smaller to larger with zero. Zero and C aren't small, big, high, or low" (see Figure 3).

Figure 3: Note-taking/Note-making: A fifth grader's conception of integers

Note-taking/note-making honors both writing and mathematics. It encourages students to make connections between new concepts and previously learned material and their personal experiences.

Writing with revision

Shared writing

A third-grade teacher used shared writing with her students to review what they had learned in their geometry unit. Using chart paper, the teacher recorded a student-generated word bank that included face, edge, vertex, congruent figures, and polygons. On subsequent sheets, the teacher used a black pen to record in sentences what the students shared about their learning. Next she guided them to suggest revisions and additions to their shared writing and recorded those suggestions with a different colored pen.

Some changes related to word choice ("learning about geometry" became "learning and exploring geometry"), while other changes related to geometry concepts. In the shared writing, students had listed 3-D shapes such as spheres, cones, cubes, and cylinders and had given real-life examples. During revision, students added the following sentence that went beyond visualization to include attributes: "Some 3-D figures will have vertexes (corners), edges, and faces." Students then suggested three titles and voted on which title was best (see Figure 4).

Figure 4: A shared writing about geometry composed and revised by third graders and their teacher

This teacher valued shared writing as an engaging strategy to help students internalize mathematics:

I feel that this experience has helped them go beyond recording the information in their math journals and just doing our assignments.... I think it helped to make it more personal and [validate] what was important to them.

As students engaged in revising, they improved the writing while extending their mathematical thinking (see Whitin & Whitin, 2000). Because the mathematical ideas are constructed through group interaction, students help one another learn to communicate mathematically.

Class book

Using a similar process, a teacher can make a class book with students. Once a shared writing is completed on chart paper, the teacher numbers the sentences and assigns each student a numbered part to write as a final draft for a page in the class book. Very little time is required for the teacher to move from student to student and edit sentences. As the teacher edits, students are encouraged to create representations for the idea on their page.

When the students are finished, the teacher gathers the pages in order and adds a cover with a title. When the shared writing has only a few sentences, the teacher assigns several students the same number and creates several covers. In this way, students produce multiple copies of the same book. One could be kept in the classroom and others given to teachers of younger grades for their reading centers.

A fourth-grade teacher used a variation of this strategy with her students. This teacher's prompt was to create and illustrate a word problem using fractional parts of a set of 24 (see Figure 5). Every student's work became a page in the class book.

Figure 5: A class book page about word problems written and illustrated by fourth graders

Class books provide a sense of audience as well as an opportunity for students to revise and edit their writing. Simultaneously, the format invites students to "create and use representations to organize, record, and communicate mathematical ideas" (NCTM, 2000, p. 67).

Alphabet books

One fifth-grade teacher helped her students create alphabet books to develop mathematics vocabulary. She assigned students letters of the alphabet and sent them on a search for new and complex words in their mathematics textbook, notes, thesaurus, and mathematics dictionary (e.g., Monroe, 2006). Students were assigned to use each word in a meaningful sentence, draw an appropriate representation, or write a real-world connection (see Figure 6). The teacher explained,

Because the format of an alphabet book requires brevity, I can readily respond to what students have written individually in short conferences, offering both compliments and suggestions for improvement. I can discuss their mathematical thinking and their writing in the same conference.

Figure 6: A class book page about word problems written and illustrated by fourth graders

Students then revise and edit as they prepare a final draft for publication. This teacher prefers to have the book professionally printed, using an inexpensive online publishing tool (e.g., mightyauthors.com).

Alphabet books not only introduce students to a delightful genre, but also allow students to experience every phase of the writing process. At the same time, they honor the integrity of mathematics by helping students develop vocabulary essential for mathematical thinking. The format invites the use of meaningful communication through words, graphic representations, and symbols.

A call for additional strategies

These six ideas represent only a place to begin. Creative teachers will use these ideas as springboards to additional strategies for integrating their writing and mathematics instruction. Numerous and varied opportunities for this integration support students as they learn to think their way into mathematics and make it their own (Zinsser, 1988) and also become more aware of the range of writing possibilities available to them (Newell, 2008). Such opportunities are not superfluous or simply nice to do if there is enough time. They are too important to ignore.

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The 2002 and 2007 reports of the National Assessment of Educational Progress (NAEP) writing as ses sment (National Center for Education Statistics, 2008), administered to 8th and 12th graders, show an increase in writing scores.

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First Person

I started my teaching career at a large comprehensive high school, teaching math exactly as I had been taught. Each day, I introduced my students to a new type of problem, solved a problem for them, and wrote the procedure in listed steps alongside my solution. Then I assigned them additional problems to solve on their own using the procedure that I provided. My students' progress was slow with this approach, and I felt like I was struggling to reach them.

After four years of teaching, I switched gears and spent my fifth year teaching physics to 9th graders at a new engineering-themed small school in the Bronx borough of New York City. That year was completely different from my past teaching experience, both in terms of the content I was teaching and the type of instruction I was trying to provide my students. I used an inquiry-based curriculum with project-based assessments in this physics course with mixed success.

I finished the year feeling like the course had gone better than the teacher-centered math courses I had taught during the previous four years, but I wasn't able to instantaneously change the learning in my classroom as much as I had hoped. Again, I found myself struggling to accurately assess what had gone wrong and how to improve.

See Also: Guided Reading: How to Make Kids Hate (or Love) to Read

Vocabulary Lessons

My principal contacted me over the summer because she was looking for someone to attend a training on teaching English-language learners. Our district's department of education required someone from each school to attend a weeklong summer professional-development course, and no one else from my school was available or wanted to go. Initially, I felt like the training wouldn't be applicable to my practice, but I decided to attend mostly to "take one for the team." But, in fact, it was this PD experience that opened my eyes as to what had gone wrong the previous year and started to provide me with concrete strategies to improve the teaching and learning in my classroom. I learned, specifically, about the central role of language development in learning, even in math classes.

Through this activity and others like it, I started to think about how much vocabulary my students actually needed to learn in my classroom, especially if I was going to talk to them with appropriate mathematical and scientific words as they learned the new content. Regardless of whether my students were ELLs or former ELLs or neither, all the new content they were learning required them to learn new academic language as well, and I needed to teach this language to them. At this point in my career, I made a commitment that going forward, I would make a conscious effort to help my students develop both spoken and written academic language.

I continued to teach a variety of math and science classes for four years after this eye-opening summer training. I experimented with different ways to get my students talking and writing about math and science. I tried grouping my students purposefully, providing different participation structures like "round robins" and "novel ideas only," and asking my students to write lab reports in science class and essays to identify and explain their mistakes in solving problems in math class. I even had my calculus students read a nonfiction book in literature circles about Newton and Leibniz's independent discoveries of calculus and required them to keep a journal about their reading and the connections they were making to their own learning in calculus.

It was an exciting time for me as I tried to help improve the teaching and learning in my classroom, and I definitely felt more successful as a teacher. My students' written work gave me insight into what they understood and what they didn't in a way I had never experienced when the majority of the assessment in my classroom was summative and focused on correct answers rather than explanations and processes.

Academic-Language Development

After nine years of teaching, I left the classroom to take a position at an amazing nonprofit organization where I planned and facilitated PD for preservice and new math teachers. The Common Core State Standards had just started to be implemented, and they were creating a different culture from the one in which I had learned to teach. Students were now being asked to explain their thinking and the processes they used to solve problems and not just to provide the correct answer.

Additionally, the preservice teachers I was working with were required to complete the edTPA, a performance-based assessment, as a part of their certification process, and I had to learn this assessment in order to support them through the process. Part of the edTPA stresses the development of academic language of students, even in a math classroom. Specifically, it requires preservice teachers to identify one language function essential for students to learn in connection with the math lesson they're teaching. It suggests "compare/contrast," "conjecture," "describe," "explain," and "prove" as language functions to be frequently used in a math classroom. As I tried to support the preservice teachers through this process, I began to look around to find examples of how other math teachers in our community were supporting language development in their classrooms.

When I was specifically looking for ways the teachers in our community were supporting their students' academic-language development, I noticed that it was something that many teachers took seriously and incorporated almost daily. One common method that some teachers utilized came out of work traditionally associated with debate teams. They would ask students to justify their work using a "claim warrant" explanation. For example, while referencing a picture of triangle ABC, a student could say or write, "My claim is that angle A is the biggest angle. My warrant is that it is across from the longest side: BC."

I also saw teachers adopt the idea of a problem without a question. They'd present students with a diagram or a situation and then ask questions like, "What is this situation about? What quantities do you have? What questions could be asked?" This process takes students away from traditional procedural methods of solving problems and gets them to think about different situations mathematically and to describe mathematical scenarios verbally and in writing.

A group of beginning math teachers that I worked with on an inquiry project gave their students examples of problems that were solved incorrectly and asked them to write about the misconception presented in the solution, using sentence starters like "I believe that ...," "I can see that ...," "My evidence for this idea is ...," or "I think this means ..."

A Window on Understanding

All these methods and others were shared among the teachers in our community and were experimented with in different ways, all in hopes of helping develop students' mathematical understanding, generally, through the use of writing. These methods provided for a symbiotic relationship between teaching and learning that previously didn't exist between the two. Through verbal and written strategies, the students gained a deeper understanding of the content.

Meanwhile, students' writing gave teachers a window into students' understanding so that they could really address the content learning of their students. It enabled teachers to assess exactly what their students did and did not understand and adjust their teaching accordingly.

This fall, I return to the classroom, where I will be teaching 7th grade math and science. I re-enter the classroom with a wealth of knowledge from my work as a PD coordinator, knowledge I did not have when I started my teaching career. I hope to be able to synthesize all that I learned, both as a teacher myself and as an educator of educators, to develop my students' mathematical and scientific abilities through writing.

If there's one thing I learned as a math educator, it's that while it's important to have students learn to solve problems and to talk about solutions, it is equally, if not more important, to teach them to write about their strategies and thought processes, or they will always struggle to exhibit their mathematical understandings. And I need to be able to accurately assess exactly where my students are in their content development, so that I can reach them where they are and adjust my teaching accordingly.

I now believe that the key to creating a classroom environment with a true symbiotic relationship between teaching and learning is writing, so next year, my students will be doing a lot of it.
Coverage of the implementation of college- and career-ready standards and the use of personalized learning is supported in part by a grant from the Bill & Melinda Gates Foundation, at www.gatesfoundation.org. Education Week retains sole editorial control over the content of this coverage.
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Students in My Math Classes Next Year Will Do a Lot of Writing. Here's Why

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First Person I started my teaching career at a large comprehensive high school, teaching math exactly as I had been taught. Each day, I introduced my students to a new type of problem, solved a problem for them, and wrote the procedure in listed steps alongside my solution.

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Journal writing can be a valuable technique to further develop and enhance your mathematical thinking and communication skills in mathematics. Journal entries in mathematics provide opportunities for individuals to self-assess what they've learned. When one makes an entry into a math journal, it becomes a record of the experience received from the specific math exercise or problem-solving activity. The individual has to think about what he/she did in order to communicate it in writing; in so doing, one gains some valuable insight and feedback about the mathematical problem-solving process. The math no longer becomes a task whereby the individual simply follows the steps or rules of thumb. When a math journal entry is required as a follow up to the specific learning goal, one actually has to think about what was done and what was required to solve the specific math activity or problem. Math instructors also find that math journaling can be quite effective. When reading through the journal entries, a decision can be made to determine if further review is required. When an individual writes a math journal, they must reflect on what they have learned which becomes a great assessment technique for individuals and instructors.

If math journals are something new, you will want to use the following strategies to assist the implementation of this valuable writing activity.

A journal should be written at the end of a math exercise.

Journal entries should be in a separate book, one used specifically for mathematical thinking.

Math journals should contain specific details about the areas of difficulties and areas of success.

The math journal entries should take no more than 5-7 minutes.

Math journals can be done with children and adults. Younger children will draw pictures of the concrete math problem they have explored.

Math journals should not be done daily, it's more important to do math journals with new concepts in areas specifically related to growth in mathematical problem-solving.

Be patient, math journaling takes time to learn. It is critical to understand that math journaling is an entry of the mathematical thinking processes.

There's no right or wrong way of thinking!

Math Journal Prompts to Get You Started

I knew I was right when......

If I missed____________ I would have to__________________.

The thing you have to remember with this kind of problem is........

Tips I would give a friend to solve this problem are.........

I wish I knew more about......

How many times did you try to solve the problem? How did you finally solve it?

Could you have found the answer by doing something different? What?

What method did you use to solve this problem and why?

Was this hard or easy? Why?

Where else could you use this type of problem-solving?

What would happen if you missed a step? Why?

What other strategies could you use to solve this problem?

Write 4 steps for somebody else that will be solving this problem.

What would you like to do better next time?

Were you frustrated with this problem? Why or why not?

What decisions had to be made when solving this problem?

What do you like about math? What don't you like about math?

Is math your favorite subject? Why or why not?

"When one has to write about problem-solving strategies, it helps to clarify thinking. We will often discover solutions to problems when we write about the problem".

Another strategy that helps to retain math concepts and support understanding is knowing how to take great notes in math.

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Math Journaling in the Classroom and Out

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Journal writing can be a valuable technique to further develop and enhance your mathematical thinking and communication skills in mathematics

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The Importance of Writing in Mathematics

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by Barry Garelick and Katharine Beals

At a middle school in California, the state testing in math was underway via the Smarter Balanced Assessment Consortium (SBAC) exam. A girl pointed to the problem on the computer screen and asked "What do I do?" The proctor read the instructions for the problem and told the student: "You need to explain how you got your answer."

The girl threw her arms up in frustration and said, "Why can't I just do the problem, enter the answer and be done with it?"

The answer to her question comes down to what the education establishment believes "understanding" to be, and how to measure it. K-12 mathematics instruction involves equal parts procedural skills and understanding. What "understanding" in mathematics means, however, has long been a topic of debate. One distinction popular with today's math reform advocates is between "knowing" and "doing." A student, reformers argue, might be able to "do" a problem (i.e., solve it mathematically), without understanding the concepts behind the problem solving procedure. Perhaps he has simply memorized the method without understanding it.

The Common Core math standards, adopted in 45 states and reflected in Common Core-aligned tests like the SBAC and the PARCC, take understanding to a whole new level. "Students who lack understanding of a topic may rely on procedures too heavily," states the Common Core website. "... But what does mathematical understanding look like?" And how can teachers assess it?

"One way ... is to ask the student to justify, in a way that is appropriate to the student's mathematical maturity, why a particular mathematical statement is true or where a mathematical rule comes from." ( http://www.corestandards.org/Math/).

The underlying assumption here is that if a student understands something, she can explain it-and that deficient explanation signals deficient understanding. But this raises yet another question: what constitutes a satisfactory explanation?

While the Common Core leaves this unspecified, current practices are suggestive: Consider a problem that asks how many total pencils are there if 5 people have 3 pencils each. In the eyes of some educators, explaining why the answer is 15 by stating, simply, that 5 x 3 = 15 is not satisfactory. To show they truly understand why 5 x 3 is 15, and why this computation provides the answer to the given word problem, students must do more. For example, they might draw a picture illustrating 5 groups of 3 pencils.

Consider now a problem given in a pre-algebra course that involves percentages: A coat has been reduced by 20% to sell for $160. What was the original price of the coat?"

A student may show their solution as follows:

x = original cost of coat in dollars
100% - 20% = 80%
0.8x = $160
x = $200

Clearly, the student knows the mathematical procedure necessary to solve the problem. In fact, for years students were told not to explain their answers, but to show their work, and if presented in a clear and organized manner, the math contained in this work was considered to be its own explanation. But the above demonstration might, through the prism of the Common Core standards, be considered an inadequate explanation. That is, inspired by what the standards say about understanding, one could ask "Does the student know why the subtraction operation is done to obtain the 80% used in the equation or is he doing it as a mechanical procedure - i.e., without understanding?"
Providing instruction for explanations-the road to "rote understanding"
In a middle school observed by one of us, the school's goal was to increase student proficiency in solving math problems by requiring students to explain how they solved them. This was not required for all problems given; rather, they were expected to do this for two or three problems per week, which took up to 10 percent of total weekly class time. They were instructed on how to write explanations for their math solutions using a model called "Need, Know, Do." In the problem example given above, the "Need" would be "What was the original price of the coat?" The "Know" would be the information provided in the problem statement, here the price of the discounted coat and the discount rate. The "Do" is the process of solving the problem.

Students were instructed to use "flow maps" and diagrams to describe the thinking and steps used to solve the problem, after which they were to write a narrative summary of what was described in the flow maps and elsewhere. They were told that the "Do" (as well as the flow maps) explains what they did to solve the problem and that the narrative summary provides the why. Many students, though, had difficulty differentiating the "Do" section from the final narrative. But in order for their explanation to qualify as "high level," they couldn't simply state "100% - 20% = 80%"; they had to explain what that means. For example, they might say, "The discount rate subtracted from 100% gives the amount that I pay."

An example of a student's written explanation for this problem is shown in Figure 1.

For problems at this level, the amount of work required for explanation turns a straightforward problem into a long managerial task that is concerned more with pedagogy than with content. While drawing diagrams or pictures may help some students learn how to solve problems, for others it is unnecessary and tedious. As the above example shows, the explanations may not offer the "why" of a particular procedure.

Under the rubric used at the middle school where this problem was given, explanations are ranked as "high", "middle" or "low." This particular explanation would probably fall in the "middle" category since it is unlikely that the statement "You need to subtract 100 -20 to get 80" would be deemed a "purposeful, mathematically-grounded written explanation."

The "Need" and "Know" steps in the above process are not new and were advocated by Polya (1957) in his classic book "How to Solve It". The "Need" and "Know" aspect of the explanatory technique at the middle school observed is a sensible one. But Polya's book was about solving problems, not explaining or justifying how they were done. At the middle school, however, problem solving and explanation were intertwined, in the belief that the process of explanation leads to the solving of the problem. This conflation of problem solving and explanation is based on a popular educational theory that being aware of one's thinking process - called "metacognition" - is part and parcel to problem solving (see Mayer, 1998).

Despite the goal of solving a problem and explaining it in one fell swoop, in many cases observed at the middle school, students solved the problem first and then added the explanation in the required format and rubric. It was not evident that the process of explanation enhanced problem solving ability. In fact, in talking with students at the school, many found the process tedious and said they would rather just "do the math" without having to write about it.

In general, there is no more evidence of "understanding" in the explained solution, even with pictures, than there would be in mathematical solutions presented in a clear and organized way. How do we know, for example, that a student isn't simply repeating an explanation provided by the teacher or the textbook, thus exhibiting mere "rote learning" rather than "true understanding" of a problem-solving procedure?
Requiring explanations undoes the conciseness of math
Math learning is a progression from concrete to increasingly abstract. The advantage to the abstract is that the various mathematical operations can be performed without the cumbersome attachments of concrete entities - entities like dollars, percentages, groupings of pencils. Once a particular word problem has been translated into a mathematical representation, the entirety of its mathematically-relevant content is condensed onto abstract symbols, freeing working memory and unleashing the power of pure mathematics. That is, information and procedures handled by available schemas frees up working memory. With working memory less burdened, the student can focus on solving the problem at hand (Aditomo, 2009). Thus, requiring explanations beyond the mathematics itself distracts and diverts students away from the convenience and power of abstraction. Mandatory demonstrations of "mathematical understanding," in other words, impede the "doing" of actual mathematics.
"I can't do this orally, only headily"
The idea that students who do not demonstrate their strategies in words and pictures must not understand the underlying concepts assumes away a significant subpopulation of students whose verbal skills lag far behind their mathematical skills, such as non-native English speakers or students with specific language delays or language disorders. These groups include children who can easily do math in their heads and solve complex problems, but often will be unable to explain - whether orally or in written words - how they arrived at their answers.

Most exemplary are children on the autistic spectrum. As autism researcher Tony Attwood has observed, mathematics has special appeal to individuals with autism: it is, often, the school subject that best matches their cognitive strengths. Indeed, writing about Asperger's Syndrome (a high functioning subtype of autism), Attwood notes that "the personalities of some of the great mathematicians include many of the characteristics of Asperger's syndrome." (Attwood, 2007)

And yet, Attwood adds (ibid.), many children on the autistic spectrum, even those who are mathematically gifted, struggle when asked to explain their answers. "The child can provide the correct answer to a mathematical problem," he observes, "but not easily translate into speech the mental processes used to solve the problem." Back in 1944, Hans Asperger, the Austrian pediatrician who first studied the condition that now bears his name, famously cited one of his patients as saying that, "I can't do this orally, only headily" (Asperger, H., 1991 [1944]).

Writing from Australia decades later, a few years before the Common Core took hold in America, Attwood adds that it can "mystify teachers and lead to problems with tests when the person with Asperger's syndrome is unable to explain his or her methods on the test or exam paper" (Attwood, 2007, p. 241). Here in post-Common Core America, this inability has morphed into an unprecedented liability.

Is it really the case that the non-linguistically inclined student who progresses through math with correct but unexplained answers - from multi-digit arithmetic through to multi-variable calculus - doesn't understand the underlying math? Or that the mathematician with the Asperger's personality, doing things headily but not orally, is advancing the frontiers of his field in a zombie-like stupor?

Or is it possible that the ability to explain one's answers verbally, while sometimes a sufficient criterion for proving understanding, is not, in fact, a necessary one?
What is really being measured?
Measuring understanding, or learning in general, isn't easy. What testing does is measure "markers" or byproducts of learning and understanding. Explaining answers is but one possible marker.

Another, quite simply, are the answers themselves. If a student can consistently solve a variety of problems, that student likely has some level of mathematical understanding. Teachers can assess this more deeply by looking at the solutions and any work shown and asking some spontaneous follow-up questions tailored to the child's verbal abilities. But it's far from clear whether a general requirement to accompany all solutions with verbal explanations provides a more accurate measurement of mathematical understanding than the answers themselves and any work the student has produced along the way. At best, verbal explanations beyond "showing the work" may be superfluous; at worst, they shortchange certain students and encumber the mathematics for everyone.

As Alfred North Whitehead famously put it about a century before the Common Core standards took hold:

It is a profoundly erroneous truism ... that we should cultivate the habit of thinking of what we are doing. The precise opposite is the case. Civilization advances by extending the number of important operations which we can perform without thinking about them.

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Katharine Beals is a lecturer at the University of Pennsylvania Graduate School of Education and an adjunct professor at the Drexel University School of Education. She is the author of Raising a Left-Brain Child in a Right-Brain World: Strategies for Helping Bright, Quirky, Socially Awkward Children to Thrive at Home and at School. Barry Garelick has written extensively about math education in various publications including The Atlantic, Education Next, Educational Leadership, and Education News. He recently retired from the U.S. EPA and is teaching middle and high school math in California. He has written a book about his experiences as a long term substitute in a high school and middle school in California: "Teaching Math in the 21st Century".
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Anindito Aditomo. Cognitive load theory and mathematics learning: A systematic review, Anima, Indonesian Psychological Journal, 2009, Vol. 24, No. 3, 207-217

Hans Asperger. "Problems of infantile autism," Communication: Journal of the National Autistic Society, London 13, 45-32), 1991 [1944]).

Tony Attwood. The Complete Guide to Asperger's Syndrome, Jessica Kingsley Publishers, Philadelphia, PA, 2007. (p. 240)

G. Pólya, How to Solve It: A New Aspect of Mathematical Method, Doubleday, Garden City, NY, 1957

Richard Mayer. Cognitive, metacognitive, and motivational aspects of problem solving, Instructional Science, 1998, March, Vol. 26, Issue 1-2, pp 49-63

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Math Problems: Knowing, Doing, and Explaining Your Answer

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by Barry Garelick and Katharine Beals At a middle school in California, the state testing in math was underway via the Smarter Balanced Assessment Consortium (SBAC) exam. A girl pointed to the problem on the computer screen and asked "What do I do?"