We must talk before we write: Fluency as a catalyst for writing in math classrooms

Dr. Angela McIver, President - Math Foundations, LLC
Dr. Jeanine M. Staples, Assistant Professor of Education - Special Education Department - College of Education - University of Maryland College Park

Introduction

Language fluency is the ability to accurately decode print and symbolic representations for comprehension and use. This involves recognizing and appropriately using vocabulary, generating useful questions about what is read, and responding to prompts for action in relationship to written material. This type of fluency is recognized as optimal when students can demonstrate it with relative ease and in relationship to developmentally appropriate texts. For instance, when a student is able to interpret print in grade-level or below grade-level material aloud, and understand the descriptions, directions, concepts, explanations, ideas, and/or questions they encounter, he or she has demonstrated language fluency. Similarly, when a student is able to verbally interpret and act upon a grade-level or below grade-level symbolic representation (like an algorithm, quantity abbreviation, or fraction), he or she has demonstrated mathematic language fluency. Fluency is taken-for-granted as evolutionary in nearly all academic and career technical classrooms. Many teachers assume that their students will learn over time to exercise language fluency by completing worksheets or drill sheets in isolation. However, research shows that all learners require practice with content-related talk in order to grasp other complex literate practices, like writing (Dyson; Gee; Moje, Young, Readance and Moore; Scherff and Piazza).

As a literacy educator, theorists and practitioner, I have always been concerned about the role of talk in Reading/English/Language Arts (RELA) classrooms. My work with older at-promise students (ages middle school through adult) has consistently shown that students with an underdeveloped relationship with school-based learning commonly have an underdeveloped sense of language fluency. That is, at-promise students too often do not know how to:

Talk about what they do or do not understand about what they read
Identify and use vocabulary to assist their learning processes and knowledge acquisition
Verbally respond to comments, directions, and prompts for action
Accurately articulate how to solve problems
Clarify ambiguities or inconsistencies in print or symbolic representations

In short, what RELA teachers know is that discourse-rich classrooms, in which students have myriad opportunities to talk before, during, and after reading, act as a catalyst for, good writing (Bloome, Dyson, Gee). Within the last five years my conversations with colleagues who have expertise in other content areas like History, Music, Art, Science, and various career technical specialties, reveal that language fluency is lacking in nearly every area of study and career preparation. Yet, this is particularly true within the content area of Mathematics. In the last several years a colleague and I, Angela McIver, have begun to formally research the ways language fluency does or does not occur in reading and mathematics. Our intention is to reveal the similarities that language development has across content areas to make teacher collaboration and professional development meaningful and resulting in effective instruction and authentic assessment for all students. Angela and I have found that mathematics, as a language for communication, requires fluency measures that are similar to the ones listed above for RELA classrooms. Her data-driven findings are described in the discussion below.

Talking the Talk: Math as a Language

As a math consultant for schools who are struggling to meet the needs of older students with weak math foundations, I have often been asked to write curricula that teaches students how to succeed on the open-ended math questions that have become ubiquitous on state tests. The motivation from these schools is to increase test scores in order to meet requirements set forth by the No Child Left Behind Act (NCLB). I often turn down these opportunities because I believe that good instruction doesn't come from the curriculum but from the teacher. As a researcher, I've interviewed hundreds of students (formally and informally) from early adolescents to young adults. These interviews take the form of investigative journalism in that my role is to uncover the multiple reasons why students misunderstand the math that they should have mastered in elementary school. I've come to the conclusion that many RELA teachers have. That is, we must talk before we write. Before we can get students to write about math, they must first be able to talk about math. They must first be able to use math as a language.

As students become increasingly required to write about the math they are doing, their teachers must become adept at listening to what students do and do not understand about the language of math. Listening to what students say is a critical component for helping students turn what they say into exceptional writing. If a student is unable to say it, she is also unlikely to use it when expressing herself (orally or in writing). For example, when I entered graduate school, my courses were littered with terms I had never encountered in my everyday life as a teacher or administrator. Uniquely "education" and "social science" terms like "postmodern metanarrative" and "hegemony" were commonplace, and seasoned graduate students used these terms to describe all types of phenomena occurring in schools across the country and the world. It became clear to me that my success in this new world of graduate school hinged on my ability to master the language. The practice of how students master the language of math is not unlike the practice required of me to master the language of graduate school (and equally not unlike the practice of young children learning to speak). In all instances, the learner must first have opportunities to "hear" the language in many contexts. However, hearing the language is not enough to equip students with the tools necessary to write in the language. Writing fluently develops out of speaking fluently.

What do the Students Say?: Examples from the Field

The concept of mathematics as a language is not new. The National Council of Teachers of Mathematics' (NCTM) focus on communication makes language a far more salient issue in math education than in any other time in mathematics education history. Not only must students be able to arrive at solutions to given problems, they must now be able to justify their answers both orally and in writing. Many new curricula have journaling components to their lessons where students are required to reflect on their mathematical learning experience. Standardized tests contain assessment questions that require students to write explanations to their problems. Not only is a command of the English language a prerequisite for success in today's math classrooms and on standardized assessments but - in order for students to master writing in math - they must first develop a command of language specific to mathematics. Consider the following response of a sixth grade student when I asked him to describe what he liked about math.

CC: Math is alright until you have to mix up numbers and match them up like that. I like math where you can just like do it and use if for different things but when you have to match them up and mix them up different I don’t really like it that much when you have to do it like that.
AM: Can you give me an example? What do you mean, when you have to mix up the numbers?
CC: Oh, you know like three over five plus three over nine for example. And then you gotta keep reducing and stuff like that. It's really a big thing I don't like doing.

In this case, the student was unable to access the vocabulary that makes mathematics a language within its own right. In mathematics, language not only requires an understanding of basic grammar found in the English language, but understanding of language specific to mathematics. A student who had command of the language of mathematics may have responded to the question posed above in the following way:

CC: I like math except when we're dealing with fractions. I get confused when I have to find a common denominator to do different operations like addition and subtraction.
Or, the student may have answered in this way:
CC: I like math but rational numbers confuse me. I get frustrated when I have to find the sum or difference of two fractions. I don’t understand how to find a common denominator.

The idea that math should be considered its own language and therefore worthy of scrutiny is put forth by Wakefield (2000) who suggests that because mathematics shares the same attributes as language, education practitioners should consider how acquisition of mathematical language promotes or hinders their mathematical development. Wakefield (2000) suggests that the following eleven characteristics of mathematics qualify it as its own language:

Abstractions (verbal or written symbols representing ideas or images) are used to communicate.
Symbols and rules are uniform and consistent
Expressions are linear and serial
Understanding increases with practice
Success requires memorization of symbols and rules.
Translations and interpretations are required for novice learners.
Meaning is influenced by symbol order.
Communications requires encoding and decoding
Intuition, insightfulness, and "speaking without thinking" accompany fluency. Experiences from childhood supply the foundation for future development.
The possibilities for expressions are infinite (272-273)

Wakefield (2000) argues that although the NCTM has acknowledged the idea of mathematics as a language, the organization inadequately addresses the issue because it fails to approach math learning "from a truly linguistic perspective" (273). Doing so would require teachers to begin with a focus on the structure of the language of math as a foundation for writing. For example, in order for students to be able to write in math classrooms, they must understand and use symbols that represent mathematical ideas or concepts. Consider the following story problem presented to students who were asked to read it out loud and then answer the question.

Meat at Big Save grocery store costs $2.55 per pound. Ms. Rosario bought 0.85 lb. Did she spend more than, less than or exactly $2.55? Explain your answer.

In my interviews, I found that students were confused by the problem because they were unable to interpret the written symbols embedded within the problem. One student read the problem in the following manner

TJ: Meat at Big save grocery store cost two dollars and fifty-five cents per pound. Ms. Smith bought eighty-five lems. Point eighty-five lems. Did she spend more than, less than, or exactly two fifty-five?
AM: What did Ms. Smith buy?
TJ: She bought meat.
AM: And how much did she buy?
TJ: She bought eighty-five ….point eighty-five lems?
AM: What is that? Do you know what that is?
TJ: No

In this example, the student is unable to interpret the symbolic representation of the word "pound": (lb.) and thus unable to interpret the problem. For teachers faced with students learning to write in math a beginning step would be to listen to what students understand and can explain about math. This step (particularly for at-promise students) meets Wakefield's sixth characteristic of the language of math - "translations and interpretations are required for novice learners."

A second consideration for teachers charged with developing students' writing abilities in math is to understand that writing itself is a reflective process. In order for students to use writing to create, discuss, and solve math problems, they must first be able to reflect on what it is they are doing. Consider the following transcript of an eighth grade student when asked to think about a time she felt successful in math.

AM: I want you to think about a time when you felt really good about something you learned in math - when you were very successful. [I want you to think about] something that you really struggled with and you figured it out, or something that you got really easily that made you feel good. Tell me what it was you learned and why it made you feel so good.
SA: I forgot the name, it was when you make the like…when two numbers are on the top and one is on the bottom, and then you have to figure out what the x was and I had asked the man to show me and he wouldn’t show me and I had to figure out myself and I got it right.

In this example, the student was unable to describe what she was doing even though she felt successful in the activity. However, according to Kieran (2001) "building mathematical knowledge and understanding should be an ever growing conscious, intentional activity for the [student]." It should grow from "conscious reflections on activities that allow [students] to make sense of their actions" (224). For students to become reflective writers in math classrooms, they must be reflective thinkers and speakers as well. My observations and interviews reveal that students rarely view classroom mathematical activities as opportunities to reflect on the actual mathematics taking place. As a result, they fail to develop the type of insight and intuition that facilitates written communication.

Conclusion

Having opportunities to think "out loud" and through processes and possibilities helps students organize their thoughts and turn their thoughts into written work. Unfortunately, this step is often missing in the many math classrooms I visit in my work with schools and job training programs. In these settings, students are often working from workbooks doing self-paced lessons of increasing difficulty. In these classrooms, a very big assumption exists that students can move from emergent numeracy to fluency in virtual silence. Teachers can break this cycle by turning their classrooms into places where math talk serves as an essential component of writing. By learning how to ask questions that develop fluent mathematical thinkers and speakers, teachers lay a foundation for writing. The following are some examples of prompts that help teachers begin this process.

The examples provided in this article serve as a call for teachers to be the catalyst for writing across the curriculum. Math teachers can facilitate the writing process by helping students develop a command of the spoken language of math. We have included an emergent list of prompts that promote writing in math. When used by teachers in academic and career technical areas, students can develop discourse-rich classrooms that act as catalysts for writing effectively across disciplines. Based on years of empirical literacy and numeracy research and extensive experience developing professional tools, we whole-heartedly encourage colleagues to facilitate this pedagogy of talk as a critical way to encourage sound writing in all learning areas.

Works Cited

Bloome, David and Talwalkar, Sarah. "Discourse analysis and the study of reading and writing." Reading Research Quarterly 32.1 (1997): 2–11.

Dyson, Anne Haas. The Brothers and Sisters Learn to Write. New York: Teachers College Press, 2003.

Gee, James Paul. "Teenagers in New Times: A New Literacy Studies Perspective." Journal of Adolescent and Adult Literacy 43.5 (2000): 412-420.

Kieran, Carolyn E. Forman and A. Sfard. "The Mathematical Discourse of 13-Year-Old Partnered Problem Solving and Its Relation to the Mathematics That Emerges." Educational Studies in Mathematics 46.1-3 (2003): 187-228.

Moje, Elizabeth Birr, J. P. Young, J. E. Readance and D. W. Moore. "Reinventing Adolescent Literacy for New Times: Perennial and Millennial Issues." Journal of Adolescent and Adult Literacy 43. (2000): 400-410.

Scherff, Lisa and Piazza, Cory. "The More Things Change the More They Stay the Same: A Survey of High School Students' Writing Experiences." Research in the Teaching of English 39.3 2002: 271-304.

Wakefield, Dara. "Math as a second language." The Educational Forum 64. (2000): 272- 279.

Appendix A

Prompts that Promote Writing in Math

The following prompts help encourage the development of critical thinking skills when working with individual students or groups in math. They represent the first step in helping students organize their thoughts in preparation for writing.