105

CONTENT AREA LITERACIES

Journal of Adolescent & Adult Literacy 58(2) October 2014 doi: 10.1002/jaal.338 © 2014 International Reading Association (pp. 105–108)

Introduction to column for the volume I value the opportunity to edit the Content Area Literacies department for JAAL . As a former secondary science teacher, it was content area reading that first convinced me of the value of discipline appropriate lit-eracy instruction in my classroom. Over the years, I de-pended on the Journal of Reading , later the Journal of Adolescent & Adult Literacy, for research I could use in my classroom. Voices published in literacy journals such as JAAL are most often those of literacy profes-sionals. In this column, I plan to feature discipline fo-cused researchers from a variety of content areas. These authentic voices do not typically publish in JAAL , but they have valuable perspectives for literacy profession-als. This issue, I have invited two mathematics edu-cators, whose work I have followed during the years, to discuss important issues in mathematical literacy. I welcome your feedback on this and future columns.

Since the latter part of the 20th century, math-ematics classrooms have been undergoing major change

in terms of curricu-lum and instruction (National Council of Teachers of Mathe-matics, 1989 , 1991 , 2000 ). No longer are mathematics class-rooms places where students sit silently at their desks just

listening to the teacher, watching the teacher work some examples, and then replicating that work on similar problems. Current goals (e.g., Common Core State Standards, 2010) demand that students do mathematics meaningfully, with reasoning, and justi-fications. For this to occur,

… classrooms will not be silent places where each learner is privately engaged with ideas. If students are to engage in mathematical argumentation and produce mathematical evidence, they will need to talk or write in ways that expose their rea-soning to one another and to their teacher. These activities are about communication and the use of language. (Lampert & Cobb, 2003 , p. 237)

This focus on communication in the math-ematics classroom is part of what is called math-ematical literacy , namely the ability to read, write, speak, and listen to mathematics with understand-ing (Thompson, Kersaint, Richards, Hunsader, & Rubenstein, 2008 ). In addition to reading and writ-ing words, mathematical literacy requires building meaning with symbols, contexts, graphs, diagrams, and other models as well as the ability to connect and translate among these and other mathematical modes of communication.

As part of this communication focus, we view all students as mathematics language learners (Thompson et al., 2008 ). From this perspective, mathematics in-struction shares many similarities with language arts in-struction. In this article, we highlight similarities in the two disciplines with the goal of helping both literacy and mathematics educators in supporting students’ learning.

Language Development Has Many Levels Like all robust learning, mathematical literacy devel-ops over time as students work on challenging tasks

Literacy in Language and MathematicsMore in Common Than You Think

DENISSE R. THOMPSON & RHETA N. RUBENSTEIN

The department editor welcomes reader comments. Victoria Gillis is Professor and Wyoming Excellence in Higher Education Endowed Chair in Literacy Education at the University of Wyoming, Laramie, WY, USA; e-mail vgillis@uwyo.edu.

Denisse R. Thompson is at the University of South Florida, Tampa, FL, USA; e-mail denisse@usf.edu.

Rheta N. Rubenstein is at the University of Michigan-Dearborn, Dearborn, MI, USA; e-mail rrubenst@umd.umich.edu.

Authors (left to right)

106

JOU

RN

AL

OF

AD

OLE

SC

ENT

& A

DU

LT L

ITER

AC

Y

58(

2)

OC

TOB

ER 2

014

CONTENT AREA LITERACIES

to build networks of meanings around critical con-cepts. As one example, we cite Herbel-Eisenmann, Steele, and Cirillo ( 2013 ) who propose the idea of a Language Spectrum (see Table 1 ) as a way to consider the formality and nature of the language students may use or encounter in different teaching and learning contexts. Like the stages of the writing process typi-cal in language arts—rough draft, peer editing, revi-sion, teacher editing, and final publication—students’ mathematical communication can also be nurtured through their participation in these different contexts. This table helps us all understand better why reading mathematics books is challenging for students. As just one note, if the activities of students’ reporting to the class and writing solutions are omitted or given little attention, then the work of reading the voices of ob-jectified and removed textbook authors is even more daunting.

There are several researched discourse moves teachers may use to facilitate students’ mathemati-cal literacy development (Chapin, O ’ Conner, & Anderson, 2009 ; Herbel-Eisenmann et al., 2013 ). These are likely better known and more commonly practiced by language arts teachers, but are valu-able to mathematics teachers, particularly related to standards that expect students to reason and critique arguments of others as in the Common Core State Standards (National Governors’ Association and Council of Chief State School Officers, 2010 ).

• Waiting for students to respond after asking a question, or waiting after a given student re-sponse for other students to engage with the given response.

• Revoicing a student ’ s response by the teacher to provide clarification or expansion.

• Inviting students to participate by sharing varied solutions, thereby changing the locus of authority from the teacher or the textbook to students and privileging them as individuals who know math-ematics. Students may be cued to listen closely to peers’ responses, to strive to understand, to re-voice a response, to be ready to ask questions, or to evaluate what the speaker is saying.

• Probing a student ’ s thinking from either the teacher or another student to make the think-ing visible and clearer.

• Creating opportunities to engage with another ’ s reasoning by applying someone else ’ s solution approach to a particular problem.

On-going activities using these practices help transform the mathematics classroom to a math-talk learning community in which students and the teach-er co-construct mathematical knowledge. Hufferd-Ackles, Fuson, and Sherin ( 2004 ) provide detailed analyses of how these communities may become more student- and less teacher-centered.

TABLE 1 Characteristics of Mathematical Language in Different Contexts

Context Description Audience Characteristics

Talking in a small group

Language of interaction Others in small group May use pointing and pronouns (this, that), informal language, and reference to physical actions

Reporting to the whole class

Language ofrecounting experience

Class peers not in original group

May use somewhat more formal language, with solutions often retold chronologically and in first person (“and then we…”)

Writinga solution

Languageof generalizing

A reader such as a peer or the teacher

May use “you” as a general actor, more mathematical language, a timeless present tense, and reasoning signals like “because” and “since”

Reading a written description in atextbook

Mathematics register Unknown general reader May use passive voice, no person or action, a focus on relationships, more symbols, actions transformed into nouns (nominalization; we don ’ t bisect but are given bisectors ), dense noun phrases, formal language

Note. Adapted from Herbel-Eisenmann, Steele, and Cirillo (2013), pp. 186–187.

107

Lite

racy

in L

angu

age

and

Mat

hem

atic

s: M

ore

in C

omm

on T

han

You

Thi

nk

A Definition Tells Only Part of a Word ’ s Meaning As in other literacy contexts, meanings rely on much more than definitions and contain embedded rela-tionships. Many of these relationships involve words with few syllables that pack a punch in meaning. For instance, the word is can be used in mathemat-ics with different senses (Herbel-Eisenmann et al., 2013 ). When is is used for classification, as a square is a rectangle , a non-reversible descriptive relation-ship is given. (All squares are rectangles because they are quadrilaterals with four right angles, but not all rectangles are squares.) When is is used for a defini-tion, such as an even number is a number that has a factor of 2 , a reversible identity relationship is given. (Every number with a factor of 2 is even AND every even number has a factor of 2.) These two uses of is illustrate that recognizing the causal relationships inherent in little words ( is , if , then , any , all ) is often complex. This is one example that understanding mathematical language is more complex than just studying vocabulary.

Further, in mathematics, relationships are often implicit, but are expected to be known. For instance, one can define a rectangle as a four-sided plane fig-ure whose angles are all 90°. However, a secondary student who knows a figure is a rectangle should au-tomatically know the following relationships without them being explicitly stated:

• Opposite sides are parallel.

• Opposite sides are congruent.

• Diagonals are congruent.

• Each diagonal divides the figure into two congruent triangles.

• The figure has two lines of symmetry, both parallel to the sides, but not on diagonals.

In many contexts, students are expected to gener-ate and use these relationships on their own to solve a problem, just from the fact that a figure is a rectangle.

As another complexity, in mathematics multiple definitions of a concept are often possible. For in-stance, a rectangle could be defined as a parallelo-gram with one right angle , or as a quadrilateral with four right angles. Depending on what definition is used, different attributes need to be proven.

Important Ideas May Be Implied Just as in reading where there are implicit under-standings, many aspects of mathematical literacy

involve implied meanings, relations, or structures. For example, there is an implied structure in some mathematical notations, such as when two sym-bols are next to each other. For instance, the sym-bol 3 x represents an implied multiplication while the symbol 42 3 represents an implied addition (Rubenstein & Thompson, 2001 ). With vocabulary, as well, meanings shift depending on context. Some words have different meanings in different parts of mathematics (e.g., cube in geometry vs. algebra or round in number vs. geometry) (Thompson & Chappell, 2007 ; Thompson & Rubenstein, 2000 ). Similarly with graphical representations, much of what needs to be gleaned is within or beyond what is explicitly represented. Reading strategies, such as Question-Answer-Response (QAR) (Rafael, 1982), can be adapted by mathematics teachers to aid stu-dents in reading these graphical representations (Rubenstein & Thompson, 2012 ). Overall, teachers need to be alert to implicit as well as explicit mean-ings that words, symbols, graphs, and diagrams convey.

Justification Is an Essential Ingredient in Learning Justification, providing reasons why a solution makes sense, has elevated importance in recent recommendations of the Common Core Standards (2010). When learners can provide the reasoning why procedures work, they are better able to under-stand, retain, use, and transfer that thinking to nov-el situations. As one example to support learners’ development of argumentation, Bénéteau, Bleiler, and Thompson ( 2014 ) used samples of student work based on known misconceptions. Learners critiqued the sample work, paying attention to whether the response was correct, clear, and com-plete. As learners worked in small groups on such critiques, the instructors found themselves becom-ing much more facilitators of instruction. Similarly, learners grew as they dialogued with each other to identify strengths and weaknesses of arguments which resulted in improvements in arguments they produced later.

Conclusion Although there are certainly unique challenges re-lated to developing students’ mathematical literacy as detailed in our references, there are broad literacy principles that transcend mathematics and language learning. We hope this article sheds light on a few

108

JOU

RN

AL

OF

AD

OLE

SC

ENT

& A

DU

LT L

ITER

AC

Y

58(

2)

OC

TOB

ER 2

014

CONTENT AREA LITERACIES

critical areas where language arts and mathematics educators can find common ground, discussing how to engage and support students in reading, writing, and talking as critical components to develop mean-ingful learning.

References Bénéteau , C. , Bleiler , S. K. , & Thompson , D. R. ( 2014 ).

Promoting mathematical reasoning through critiquing stu-dent work . In K. Karp & A. Roth McDuffie (Eds.), Annual perspectives in mathematics education 2014: Using research to improve instruction (pp. 151 – 160 ). Reston, VA : National Council of Teachers of Mathematics .

Chapin , S. H. , O ’ Conner , C. , & Anderson , N. C. ( 2009 ). Classroom discussions: Using math talk to help students learn, Grades K–6 . Sausalito, CA : Math Solutions .

Herbel-Eisenmann , B.A. , Steele , M.D. , & Cirillo , M. ( 2013 ). (Developing) teacher discourse moves: A framework for pro-fessional development . Mathematics Teacher Educator , 1 ( 2 ), 181 – 196 .

Hufferd-Ackles , K. , Fuson , K.C. , & Sherin , M.G. ( 2004 ). Describing levels and components of a math-talk learning community . Journal for Research in Mathematics Education , 35 ( 2 ), 81 – 116 .

Lampert , M. , & Cobb , P. ( 2003 ). Communication and lan-guage . In J. Kilpatrick , W. G. Martin & D. Schifter (Eds.), A research companion to Principles and Standards for School Mathematics (pp. 237 – 249 ). Reston, VA : National Council of Teachers of Mathematics .

National Council of Teachers of Mathematics . ( 1989 ). Curriculum and evaluation standards for school mathematics . Reston, VA : Author .

National Council of Teachers of Mathematics . ( 1991 ). Professional standards for teaching mathematics . Reston, VA : Author .

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

National Governors Association Center for Best Practices and Council of Chief State School Officers (NGA Center and CCSSO) . ( 2010 ). Common Core State Standards for Mathematics . Washington, D.C. : NGA Center and CCSSO . ( http://www.corestandards.org )

Raphael , T. ( 1982 ). Question-answering strategies for children . The Reading Teacher , 36 ( 2 ), 186 – 190 .

Rubenstein , R.N. , & Thompson , D.R. ( 2001 ). Learning math-ematical symbolism: Challenges and instructional strategies . Mathematics Teacher , 94 , 265 – 271 .

Rubenstein , R.N. , & Thompson , D.R. ( 2012 ). Reading visual rep-resentations . Mathematics Teaching in the Middle School , 17 , 545 – 550 .

Thompson , D.R. , & Chappell , M.F. ( 2007 ). Communication and representation as elements in mathematical literacy . Reading and Writing Quarterly: Overcoming Learning Difficulties , 23 ( 2 ), 179 – 196 .

Thompson , D.R. , Kersaint , G. , Richards , J.C. , Hunsader , P.D. , & Rubenstein , R.N. ( 2008 ). Mathematical literacy: Helping students make meaning in the middle grades . Portsmouth, NH : Heinemann .

Thompson , D.R. , & Rubenstein , R.N. ( 2000 ). Learning math-ematics vocabulary: Potential pitfalls and instructional strate-gies . Mathematics Teacher , 93 , 568 – 573 .

The Common Core:Teaching Students in Grades 6–12to Meet the Reading StandardsMaureen McLaughlin and Brenda J. OverturfThis essential resource explains the key points of the CommonCore State Standards reading standards, then aligns each Standardwith appropriate research-based strategies, and shows you how touse those strategies to teach your middle school and high schoolstudents. Classroom applications and student examples will makethis your go-to CCSS resource.

Don’t miss Maureen and Brenda’s K–5 and graphic organizers books!Bundle pricing avai lable at www.reading.org/cc-bundles

Preview a sample chapter and order online at www.reading.org/CC612For priority processing, enter promotion code CC612Call toll free 800-336-7323 (Outside the U.S. and Canada, call 302-731-1600)Join IRA and pay the member price—a 20% savings!

GET YOUR COPY TODAY!

© 2013 | 248 pp. | ISBN 978-0-87207-706-5IRA Members $26.35 | Nonmembers $32.95