Vocabulary understanding is a major contributor to overall comprehension in many content areas including mathematics. Effective methods for teaching vocabulary in all content areas are diverse and long standing. The importance of teaching and learning the language of mathematics is vital for the development of mathematical proficiency. Students' mathematical vocabulary learning is a very important part of their language development and ultimately mathematical proficiency. This article draws upon currently available research-based evidence to provide a: (a) rationale for teaching vocabulary, (b) review of research that supports the importance of teaching mathematics vocabulary, and (c) description of specific strategies to teach mathematics vocabulary. Implications and the need for future research are also addressed.

Figures - uploaded by Elizabeth Hughes

Author content

All figure content in this area was uploaded by Elizabeth Hughes

Content may be subject to copyright.

ResearchGate Logo

Discover the world's research

  • 20+ million members
  • 135+ million publications
  • 700k+ research projects

Join for free

Full Terms & Conditions of access and use can be found at

http://www.tandfonline.com/action/journalInformation?journalCode=urwl20

Download by: [Duquesne University] Date: 29 October 2015, At: 12:44

Reading & Writing Quarterly

Overcoming Learning Difficulties

ISSN: 1057-3569 (Print) 1521-0693 (Online) Journal homepage: http://www.tandfonline.com/loi/urwl20

The Language of Mathematics: The Importance of

Teaching and Learning Mathematical Vocabulary

Paul J. Riccomini, Gregory W. Smith, Elizabeth M. Hughes & Karen M. Fries

To cite this article: Paul J. Riccomini, Gregory W. Smith, Elizabeth M. Hughes & Karen M. Fries

(2015) The Language of Mathematics: The Importance of Teaching and Learning Mathematical

Vocabulary, Reading & Writing Quarterly, 31:3, 235-252, DOI: 10.1080/10573569.2015.1030995

To link to this article: http://dx.doi.org/10.1080/10573569.2015.1030995

Published online: 08 May 2015.

Submit your article to this journal

Article views: 355

View related articles

View Crossmark data

The Language of Mathematics: The

Importance of Teaching and Learning

Mathematical Vocabulary

PAUL J. RICCOMINI

The Pennsylvania State University, University Park, Pennsylvania, USA

GREGORY W. SMITH

Hartwick College, Oneonta, New York, USA

ELIZABETH M. HUGHES

Duquesne University, Pittsburgh, Pennsylvania, USA

KAREN M. FRIES

Francis Marion University, Florence, South Carolina, USA

Vocabulary understanding is a major contributor to overall

comprehension in many content areas, including mathematics. Effec-

tive methods for teaching vocabulary in all content areas are diverse

and long standing. Teaching and learning the language of mathemat-

ics is vital for the development of mathematical proficiency. Students'

mathematical vocabulary learning is a very important part of their lan-

guage development and ultimately mathematical proficiency. This arti-

cle draws on current research-based evidence to (a) provide a rationale

for teaching vocabulary, (b) offer a review of research that supports the

importance of teaching mathematics vocabulary, and (c) describe

specific strategies for teaching mathematics vocabulary. It also

addresses implications and the need for future research.

Developing the language of mathematics is an essential aspect of teaching

mathematics to young children; this process continues throughout an

individual's mathematics education. Because the understanding of

Gregory W. Smith is now at The University of Southern Mississippi in Hattiesburg, MS.

Address correspondence to Paul J. Riccomini, Department of Education Psychology,

Counseling, and Special Education, The Pennsylvania State University, 214 CEDAR, University

Park, PA 16802, USA. E-mail: pjr146@psu.edu

Color versions of one or more of the figures in the article can be found online at

www.tandfonline.com/urwl.

Reading & Writing Quarterly, 31: 235–252, 2015

Copyright # Taylor & Francis Group, LLC

ISSN: 1057-3569 print=1521-0693 online

DOI: 10.1080/10573569.2015.1030995

235

Downloaded by [Duquesne University] at 12:44 29 October 2015

mathematical vocabulary affords access to concepts, mathematical instruction in

the areas of language is imperative (Monroe, 1998 ). The term language is

defined as ''the words, their pronunciation, and the methods of combining them

used and understood by a community'' (''Language,'' 2013 ). Although this defi-

nition simplifies a rather complex idea, it highlights the importance of vocabu-

lary development within language. Specifically, in relationship to the language

of mathematics, the ability to use words (i.e., vocabulary) to explain, justify, and

otherwise communicate mathematically is important to the overall development

of mathematical proficiency. In addition, research shows that language is a

pivotal component of mathematics success (Seethaler, Fuchs, Star, & Bryant,

2011), and a student's general knowledge of mathematical vocabulary can

predict mathematical performance (van der Walt, 2009 ).

Proficiency in mathematics depends on a continuous growth and blend of

intricate combinations of critical component skills such as concepts,

procedures, algorithms, computation, problem solving, and language

(Riccomini, Sanders, & Jones, 2008 ). The National Research Council (2001 )

further described proficiency through five interconnected strands: (a) under-

standing mathematics, (b) computing fluently, (c) applying concepts to solve

problems, (d) reasoning logically, and (e) engaging and communicating with

mathematics. Clearly, the importance of students learning the language of

mathematics is highlighted in both descriptions. In addition, the National

Council of Teachers of Mathematics (2006) placed an emphasis on language

development through the adaptive reasoning strand, which the National

Research Council (2001 ) described as the ''capacity for logical thought, reflec-

tion, explanation, and justification'' (p. 116). Mathematical proficiency includes

the ability to communicate and reason through written and spoken language.

Further emphasis on the importance of language in the development

of mathematical proficiency is evident in the newly formed Common Core

State Standards in Mathematics (CCSSM; National Governors Association

Center for Best Practices & Council of Chief State School Officers, 2010).

In addition to the comprehensive set of grade-level standards described

in the CCSSM, there are eight Standards for Mathematical Practice that have

clearly embedded the importance of language to mathematical proficiency.

The language focus is described in the sixth Standard for Mathematical

Practice, ''Attend to precision,'' which includes the following description:

''In the elementary grades, students give carefully formulated explanations

to each other. By the time they reach high school they have learned to

examine claims and make explicit use of definitions'' (p. 7). Undoubtedly,

language development and specifically vocabulary are now new points of

emphasis and important aspects for teachers to begin to address as per

the newly adopted CCSSM.

Although the language of mathematics can be confusing (Rubenstein

& Thompson, 2002 ), it is necessary for the communication of higher

order mathematics reasoning (Sloyer, 2003 ). Goals requiring the use of

236 P. J. Riccomini et al.

Downloaded by [Duquesne University] at 12:44 29 October 2015

higher order mathematics reasoning are unmistakably present in national

mathematics organizations such as the National Research Council and

National Council of Teachers of Mathematics and in the CCSSM. In order

to meet these goals, students must effortlessly use, understand, and apply

mathematical words, symbols, and diagrams routinely during mathematics

activities. If students' language development is weak or underdeveloped,

their overall mathematics learning will become slowed (van der Walt,

Maree, & Ellis, 2008 ). Schwartz and Kenney (1995 ) organized the lan-

guage of mathematics into more commonly utilized language terms; for

example, mathematical nouns or objects were classified as numbers, mea-

surements, and functions, whereas verbs were actions associated with

problem solving and reasoning. This organizational framework not only

represents the process that individuals go through when they problem

solve but also provides a way to assess mathematical development

(Kenney, 2005 ).

In an effort to improve students' overall mathematical performance,

educators need to recognize the importance of, and use research-validated

instructional methods to teach, important mathematical vocabulary. The

purpose of article is to provide teachers with an overall understanding of

the impact of mathematical vocabulary on proficiency and specific

evidence-based instructional strategies to promote the learning of essential

vocabulary in mathematics.

DIFFICULTIES STUDENTS EXPERIENCE WITH

MATHEMATICAL VOCABULARY

There are many challenges for students in their learning of the language

of mathematics. Communicating mathematically is a complex task for

even the most mathematically advanced student. The ability to effectively

communicate (expressively and receptively) through the language of

mathematics requires mathematical understanding; a robust vocabulary

knowledge base; flexibility; fluency and proficiency with numbers, sym-

bols, words, and diagrams; and comprehension skills. Many students

strugglewithsomeoralloftheimp ortant mathematical concepts,

especially students with learning disabilities (Morin & Franks, 2010 ).

Every day in their mathematics classrooms, students encounter a

text-centered instructional setting that creates unintended barriers to their

learning (Anderson-Inman & Horney, 1998 ). It is important to recognize

the many and varied difficulties that present challenges for students; find-

ing instructional strategies and activities to help students overcome these

difficulties is imperative.

According to the research of Rubenstein and Thompson (2002 ), there

are at least 11 categories of difficulties associated with learning the language

Mathematical Vocabulary 237

Downloaded by [Duquesne University] at 12:44 29 October 2015

of mathematics. The categories are defined in the following manner:

(a) meanings are context dependent (e.g., foot as in 12 inches vs. the foot

of the bed), (b) mathematical meanings are more precise (e.g., product as

the solution to a multiplication problem vs. the product of a company),

(c) terms specific to mathematical contexts (e.g., polygon, parallelogram,

imaginary number), (d) multiple meanings (e.g., side of a triangle vs. side

of a cube), (e) discipline-specific technical meanings (e.g., cone as in the

shape vs. cone as in what one eats), (f) homonyms with everyday words

(e.g., pi vs. pie ), (g) related but different words (e.g., circumference vs. per-

imeter), (h) specific challenges with translated words (e.g., mesa vs. table),

(i) irregularities in spelling (e.g., obelus [ ] vs. obeli ), (j) concepts may be

verbalized in more than one way (e.g., 15 minutes past vs. quarter after),

and (k) students and teachers adopt informal terms instead of mathematical

terms (e.g., diamond vs. rhombus, orin the house vs. in the division

bracket). Undoubtedly, the many difficulties that students face when learn-

ing the language of mathematics are complex and can negatively impact

their language development.

An important first step in helping students to learn and use the language

of mathematics is for teachers to understand the many difficulties that

vocabulary presents students (Monroe & Orme, 2002 ). It is only with this rec-

ognition and understanding of the specific difficulties that teachers can then

begin to address the instructional needs of their students from a language

perspective. Effectively designing and delivering vocabulary instruction is a

needed course of action. Although a common belief with many teachers is

that simply exposing students to new vocabulary words through rich

context-specific interactions is the best way to teach vocabulary, many stu-

dents will require more systematic and explicit instructional techniques

and purposeful instructional activities to facilitate their learning (Marzano,

2004).

Providing appropriate academic language support is important for all

learners, especially in the mathematics classroom, where the ongoing devel-

opment of explicit mathematical vocabulary is essential (Bay-Williams &

Livers, 2009 ). There are three main purposes to teaching essential vocabulary

in mathematics class to increase students' effective use of mathematical lan-

guage. First and most obvious is to provide initial instruction to promote the

understanding and storage of word meanings in long-term memory. Second,

and only after students have developed an understanding, the goal of instruc-

tion becomes to help students become fluent and maintain the word mean-

ing over time. Third, the end result of achieving the first two goals is that

students are able to easily and accurately use the language of mathematics

to explain and justify mathematical concepts and relationships. Without the

instructor first teaching basic understanding and facilitating fluency with

vocabulary words, the purposeful and effective use of the language of

mathematics will likely not occur.

238 P. J. Riccomini et al.

Downloaded by [Duquesne University] at 12:44 29 October 2015

GENERAL APPROACHES AND TECHNIQUES FOR TEACHING

MATHEMATICAL VOCABULARY

One would assume that mathematical vocabulary is taught at some level

during mathematics classes; however, language development is often over-

looked by math teachers (Adams, 2003 ; Riccomini & Witzel, 2010 ). It is

important that teachers apply general vocabulary instructional techniques

to mathematical vocabulary on a regular basis. Developing and then using

a systematic plan for teaching vocabulary throughout the year will maximize

and facilitate improved understanding of essential vocabulary for students

(Manzo, Manzo, & Thomas, 2006).

Marzano's (2004) six steps for educators to maximize student learning of

essential vocabulary incorporates components of the following evidence-

based instructional strategies that aid in achieving positive academic out-

comes across content areas: (a) explicit instruction (Bottge, Heinrichs, Mehta,

& Hung, 2002 ; Test & Ellis, 2005 ), (b) stimulating prior knowledge (Stroud &

Schwartz, 2010 ; Yeh et al., 2012 ), (c) repetition (Joseph, Eveleigh, Konrad,

Neef, & Volpe, 2012 ; Kluge, Ritzman, Burkolter, & Sauer, 2011 ), (d) differen-

tiating instruction (Geisler, Hessler, Gardner, & Lovelace, 2009 ; Jones, Yssel,

& Grant, 2012), and (e) cooperative learning (Ryve, Nilsson, & Patterson,

2013; Wang, 2012). Although his recommendations cut across content areas,

they form the basis for the specific strategies and techniques that are

described. First, teachers should begin vocabulary instruction by providing

students with an informal description, explanation, or example of the new

term or phrase either directly or through indirect means. This will help stu-

dents begin the process of connecting the new meaning to their prior knowl-

edge. Second, it is important to provide students with opportunities to restate

the teacher-provided descriptions, explanation, or examples in their own

words. This opportunity reinforces the connections to their prior knowledge.

Third, to help strengthen the linkage to prior knowledge, students are asked

to construct a picture, symbol, or graphic representation of the term or

phrase. This is especially important for younger children who have less prior

knowledge specific to mathematics.

Fourth, as students become more familiar and comfortable with the

language (i.e., learning and using the terms), it is vital for teachers to provide

students with periodic opportunities to reengage in a variety of activities to

help them further develop and enrich their knowledge. Students often only

develop surface-level understanding of the material, and without opportu-

nities for further engagement students will not gain the desired deep under-

standing necessary for mathematical reasoning and communication. Fifth,

involving small-group and= or peer-to-peer discussions on specific terms

further develops a deeper understanding and reduces misconceptions that

may have formed. Sixth, to facilitate long-term retention, teachers must

Mathematical Vocabulary 239

Downloaded by [Duquesne University] at 12:44 29 October 2015

provide opportunities for the students to revisit these essential and

already-learned terms through such things as game-like activities that

students will find enjoyable (see Figure 1).

By grounding their vocabulary instruction in Marzano's six steps, tea-

chers are likely to see an improvement in mathematics language develop.

These six steps articulated are neither new nor innovative but frequently

get pushed to the side during mathematics instruction for a host of reasons

(e.g., time constraints, not valued, lack of teacher training). Learning math-

ematical vocabulary through daily mathematics instruction that emphasizes

the six general recommendations is important and essential for many stu-

dents, especially struggling students and students with disabilities. Because

mathematics naturally progresses from less complex to more complex skills,

mastery of vocabulary is essential for long-term success in mathematics

(Monroe & Panchyshyn, 2005 ); hence, the use of specific instructional strate-

gies supported by research is necessary.

ACTIVITIES FOR TEACHING MATHEMATICS VOCABULARY

When students with poor language skills struggle with learning important

mathematical vocabulary terms, educators should consider using strategies

specifically developed for learning content vocabulary. Although there are

many methods of facilitating the learning of vocabulary, five specific techni-

ques for helping students learn and remember essential mathematical

FIGURE 1 Concept map based on six recommendations by Marzano (2004 ) for effective

vocabulary instruction.

240 P. J. Riccomini et al.

Downloaded by [Duquesne University] at 12:44 29 October 2015

vocabulary are described in this article: (a) explicit vocabulary instruction,

(b) mnemonic strategies, (c) fluency building through multiple exposures,

(d) game-like activities, and (e) technology applications.

Explicit Vocabulary Instruction

Educators recognize that children may naturally learn vocabulary through

incidental or embedded learning experiences; however, for many students

these types of mathematics learning encounters are not sufficient. Instead

of simply exposing students to mathematics vocabulary, it is necessary to

directly teach vocabulary (sometimes in isolation) and provide opportunities

for numerous and meaningful practices across contexts. The language of

mathematics consists not only of words and text but also of symbols and

diagrams; explicit instruction can help build the connections between these

elements of mathematics language (Van de Walle, 2001).

Explicit articulation of vocabulary terms, definitions, and uses takes the

guesswork out of making meaning of unfamiliar terms and focuses the stu-

dents' learning on correct use and application. Research suggests that explicit

instruction of new vocabulary with opportunities for use through incidental

learning is more effective than incidental learning in isolation (Sonbul &

Schmitt, 2010 ) and across ages and grade levels (Taylor, Mraz, Nichols,

Rickelman, & Wood, 2009). Explicit instruction is an established, highly effec-

tive instructional approach that can be used independently or in conjunction

with other teaching strategies and techniques (Archer & Hughes, 2011).

Explicit vocabulary instruction requires teachers to introduce and teach a

new word and its meaning through a systematic and purposeful presentation.

This direct presentation highlights the importance of the new word, connects

to prior knowledge, and allows students to engage with the multiple uses of

the word (Lee & Jung, 2004 ). Common elements of explicit instruction

include logically sequencing key skills, reviewing prior skills and knowledge,

providing step-by-step teacher models of new skills along with opportunities

for guided and independent practice, and assisting students with connections

between new and existing knowledge (Archer & Hughes, 2011 ). There is a

strong literature base supporting explicit instruction for teaching vocabulary

in content areas such as reading, science, and social studies (e.g., Harmon,

Hedrick, & Wood, 2005 ; Hong & Diamond, 2012 ; Jitendra, Edwards, Sacks,

& Jacobson, 2004 ; McKeown & Beck, 2002 ; Stahl & Fairbanks, 1986 ; White,

Graves, & Slater, 1990).

Concurrent with instruction, the teacher is checking for student under-

standing and encouraging active learning through frequent questioning and

guided activities to promote student independence. After explicit instruction

in new vocabulary terms, students could also (a) create concept maps;

(b) keep individual math dictionaries of terms, illustrations, and examples;

and (c) develop word walls with new terminology (Van de Walle, 2001 ). This

Mathematical Vocabulary 241

Downloaded by [Duquesne University] at 12:44 29 October 2015

type of instruction is neither incidental nor accidental; the teacher plans and

carefully directs all aspects of the lesson.

Mnemonic Strategies

Mnemonic instruction refers to strategies and techniques used to improve

learning in memorable and motivating formats. Mnemonic strategies help

students learn new information by connecting it to their prior knowledge

(Mastropieri & Scruggs, 2007 ). Mnemonic instructional practices have 30

years of research support including a diverse set of learners and across mul-

tiple content areas supporting their use as an evidence-based technique

(Forness, Kavale, Blum, & Lloyd, 1997 ; Jitendra et al., 2004 ; Mastropieri &

Scruggs, 1989 ). In addition to enhancing the academic performance of

low-performing, as well as average- and above-average-achieving, students,

mnemonic instruction benefits students with disabilities (Kavale & Forness,

1999).

One specific mnemonic instructional practice, the keyword strategy, has

the greatest application to teaching mathematical vocabulary. Overwhelm-

ingly positive evidence exists for the use of the keyword mnemonic tech-

nique to teach content vocabulary to students with disabilities (e.g.,

Mastropieri, Scruggs, & Fulk, 1990 ; Scruggs & Mastropieri, 2000 ). Unfortu-

nately, few mathematics-specific examples of the keyword mnemonic

strategies are available (see Sanders, 2007).

Educators using the keyword strategy teach students meanings of new

vocabulary terms by selecting a similar-sounding word and a picture, draw-

ing, or computer graphic that represents the essential information to learn

(Atkinson, 1975 ; Kavale & Forness, 1999 ; Mastropieri & Scruggs, 2007 ). By

providing students with a tool to anchor a new term with a similar-sounding

word already known by the student, teachers enable students to better recall

the meaning of the new term. Further strengthening the effectiveness of the

keyword strategy is the use of a picture representation that highlights the

critical attributes of the new term. Either this illustration can be created by

the student, or to save time the teacher can create the illustration. The last

part of the keyword technique is to create a sentence that connects the

keyword and the desired definition. This is a powerful memory-aiding device

to help students learn and remember essential mathematical vocabulary.

The example keyword mnemonics for the terms parallel lines and ray

highlight the three critical aspects for teachers using this strategy (see

Figures 2 and 3 ). First, the unfamiliar terms are anchored to a familiar

keyword: Parallel lines is anchored to pair of elves, and ray is anchored to

run away. Second, a visual image is created that accentuates the key features

of the new term and captures the keyword. In the examples provided, the

visual images clearly depict the key features of the definitions and the

keywords. Third, a sentence is developed to connect the information in a

242 P. J. Riccomini et al.

Downloaded by [Duquesne University] at 12:44 29 October 2015

meaningful and memorable fashion. For parallel line: '' The pair of elves are

the same distance apart and will never intersect. The pair of elves are on

parallel lines.'' For ray: ''Start here!!! Run away and never stop running ray!''

The key to maximizing the effectiveness of the keyword mnemonic is to

incorporate the developed keywords mnemonics into the regular classroom

instructional routine. Simply presenting students with a keyword mnemonic

will not likely result in the desired learning and remembering on the part of

the students. Combining the keyword mnemonic strategy with other instruc-

tional activities typical in mathematics classes can maximize its effectiveness.

The keyword mnemonic strategy is easily incorporated into bulletin

boards, warm-ups, game activities, SmartBoard presentations, teacher-

directed or student-centered instructional time, and even peer tutoring. Com-

bining the keyword mnemonic strategy with other instructional activities

typical in mathematics classes can maximize its effectiveness.

Fluency Building Through Multiple Exposures

Fluency in mathematics is often associated with basic arithmetic facts (e.g.,

55¼ 25) and other computational-type problems (e.g., long division, per-

fect squares), at times overshadowing the vital role of vocabulary recognition

and understanding. The National Reading Panel (National Institute of Child

Health and Human Development, 2000 ) highlighted the importance of

repeated and multiple exposures to new vocabulary to build fluency. Being

fluent with mathematics vocabulary may allow learners to more readily

recognize what is required to solve a problem, therefore having more cogni-

tive energy to dedicate to more laborious tasks, such as calculating solutions

that require multiple steps. As with explicit instruction, fluency is achieved

through planned, purposeful, and targeted practice of specific content.

FIGURE 2 Example keyword mnemonics for the term parallel lines.

Mathematical Vocabulary 243

Downloaded by [Duquesne University] at 12:44 29 October 2015

Traditional ways to practice fluency include the use of flashcards, in

which one side of an index card has the vocabulary term and the other has

the definition and a visual. The creation of the cards also acts as a rehearsal

activity and can help with learning and remembering vocabulary or can be

paired with other activities to teach sight words (Kaufman, McLaughlin, Derby,

& Waco, 2011 ). Students can rehearse the vocabulary and practice recalling the

word or definition. Because the cards contain both the word and the

definition, students receive immediate feedback, which has been linked to

improved learning (Epstein et al., 2002). Previously mastered vocabulary

can be set aside, thus maximizing time spent on learning new material.

Flashcards can be used independently (e.g., in reciting) or with a peer or

parent. They can be used at home, at school, or in other settings and integrated

with other practices (e.g., games, metacognitive strategies). Strengths to using

cards include their ease of use and the opportunity for students to practice

through repetitive exposures of the vocabulary word; however, this type of

practice isolates the word from the context in which it is used.

A variation of the traditional approach was described by Taylor and

colleagues (2009 ), in which one side of the index card is divided into quad-

rants; the new vocabulary word is listed in the top right quadrant with the

definition in the bottom right quadrant. The left two quadrants are used to

draw a picture supporting the definition of the word. On the back of the

index card, the students describe the relationship between the picture and

the new term. This approach, like traditional flashcards, is easy to create

and use; but unlike traditional flashcards, this approach includes an example

of how the vocabulary word is used in context and incorporates elements

highlighted by Marzano (2004).

Although some vocabulary-building activities require dedicated allot-

ments of time, building fluency through multiple exposures to vocabulary

FIGURE 3 Example keyword mnemonic for the term ray.

244 P. J. Riccomini et al.

Downloaded by [Duquesne University] at 12:44 29 October 2015

can often be accomplished through frequent but brief 5- to 10-min activities

(Stump et al., 1992 ). This versatility allows teachers to incorporate

fluency-building activities during brief opportunities of time (e.g., transitions

between exchanging classes, at the end of a lesson, while passing out class-

room materials) and maximize instructional time. An engaging activity might

include passing out mathematics vocabulary cards so that each student has

one card. Students then circulate around the classroom to form clusters of

related words (students form clusters based on how the words are related).

Once the clusters are formed, the teacher can then lead a discussion about

how certain terms may fit into more than one category. Another idea is to

have students play vocabulary line frog while waiting in line for specials

or lunch. The person at the end of the line has an opportunity to jump to

the front of the line (or second place, if there is a designated line leader) if

the student provides a correct definition or uses the term correctly in context.

To increase opportunities to respond, this can be done as a lightning round

that the teacher leads at a rapid pace.

Game-Like Activities

Teachers should use a variety of different techniques when teaching

vocabulary, which may include game-like activities (Covington, 1992 ;

Johnson, von Hoff Johnson, & Schlichting, 2004 ). Educational games are

ideal for engaging students in motivating activities (Charlton, Williams, &

McLaughlin, 2005). Games may be used to improve sight recognition

(Berne & Blachowicz, 2008 ) or to improve and maintain understanding

of essential vocabulary (Wells & Narkon, 2011). In addition, using

game-like activities is an excellent way to make learning mathematical

vocabulary fun and more appealing to students.

In general, teachers have established game activities designed to serve

various learning objectives in their classrooms. A common game format used

by many teachers is based on the popular television program Jeopardy! This

game format is used in many classrooms, not just mathematics classrooms,

because of its easy-to-learn format and applicability across many different

content areas. Because the game is organized into categories (e.g., Geometry,

Algebra), it is simple to add an additional category devoted exclusively to

vocabulary. There are templates available online to aid in creating and cus-

tomizing this type of quiz game (e.g., http://www.edtechnetwork.com/

powerpoint.html).

Wells and Narkon (2011) explained three games (i.e., Mystery Word,

Word-O, and Word Sorts) that can be used to motivate student learning. In

Mystery Word, a vocabulary word is selected from a list, and the leader

provides clues about the mystery word until the class is able to surmise what

the word is. Word-O is an adapted form of Bingo, and Word Sorts allows

students to work with a list of words to compare and contrast words in an

Mathematical Vocabulary 245

Downloaded by [Duquesne University] at 12:44 29 October 2015

effort to form categories of words (see Wells & Narkon, 2011). More chal-

lenging rounds may include words that are not overtly similar.

Using game-like activities throughout the course of the academic year

affords students opportunities to attend to continued vocabulary develop-

ment in mathematics in a fun, recreational manner. The playful learning

opportunities may be both interesting and motivational for students

(Charlton et al., 2005; Wells & Narkon, 2011).

Technology Applications

Students with disabilities often struggle with mathematics content in mid-

dle school and high school. They are faced with a text-centered world and

often lack the skills to read and write at sufficient levels to meet the chal-

lenges of secondary education (Anderson-Inman & Horney, 1998 ). How-

ever, teachers can use various instructional techniques and strategies to

help their students overcome many of the barriers to learning the language

of mathematics. Furthermore, technology applications may become an

effective aid for students in the future. As Anderson-Inman and Horney

(1998 ) stated, ''Computer-based solutions represent the future in educa-

tors' effort to help students with learning disabilities achieve in school

up to their potential'' (p. 248).

Instructional technology can enhance and support mathematics

instruction by offering teachers and their students visual and auditory stim-

uli and interactive simulations that make mathematics real for students (e.g.,

demonstrating how data collection can be utilized to find solutions to

everyday problems). Although very few studies have specifically addressed

vocabulary development with instructional technology, there is evidence

that suggests that improved learning outcomes are possible (Hebert &

Murdock, 1994 ;Koury,1996 ). Instructional technology can include a range

of applications, such as apps, streaming audio and video, software

programs, computer simulations, video and audio demonstrations, and

graphics programs (e.g., graphing calculators). The Internet now allows

studentstoaccessrealdatathatcanthenbeusedtosolveauthenticmean-

ingful problems and provide visual representations not easily created or

accessible in past mathematics classrooms. Students can learn through

interactive computer games that can be highly motivating and challenge

students at their optimal learning levels (Gee, 2004). As the technology con-

tinues to improve in both access and learner effectiveness, instructional

technology has great potential to be a powerful teaching tool for educators

and learning aid for students.

Empirical research supports the use of instructional technology (e.g.,

calculators, graphing calculators, video discs, software applications) by

educators in the areas of basic facts as well as problem solving, telling time,

ratios and proportions, fractions, and decimals (e.g., Bouck, 2010 ; Cawthon,

246 P. J. Riccomini et al.

Downloaded by [Duquesne University] at 12:44 29 October 2015

Beretvas, Kaye, & Lockhart, 2012 ; Hofmeister, 1989 ); however, a minimal

amount of empirical research is available regarding the efficacy of instruc-

tional technology for teaching vocabulary specific to mathematics for

low-achieving students. It is logical that technology applications can and

should be developed and applied to enhance instruction of essential vocabu-

lary in mathematics.

Learning technical mathematical vocabulary may require much more

than the status quo for mathematics instruction. With ever-increasing

advancements in instructional technology, the possibility of significantly

impacting the overall mathematical performance of low-achieving students

and students with disabilities through the application of technological

advances is substantial. Unfortunately, very few researchers have examined

the effectiveness of using instructional technological applications to teach

mathematical vocabulary specifically, an obvious gap in the knowledge base

on evidence-based vocabulary instruction. This is an area that should be

explored and further developed.

LIMITATIONS AND THE NEED FOR FUTURE RESEARCH

Although the need for vocabulary instruction in mathematics is great, there is

limited published research that focuses specifically on interventions for

developing vocabulary in mathematics. Therefore, scholars are tapping into

the rich vocabulary research available through literacy research and extend-

ing it across content areas such as mathematics, as seen in the framework

described by Marzano (2004 ). Building content-specific vocabulary research

from literacy research, as sometimes done here, is a natural extension;

however, there are some limitations within this bridge, including the way

mathematics vocabulary is often presented with limited context clues (e.g.,

''Find the slope '') and the 11 caveats described by Rubenstein and Thompson

(2002 ). Research is needed to identify and analyze instruction with particular

attention to these characteristics that distinguish mathematics vocabulary

from other expressions of vocabulary. Equally important to how mathematics

vocabulary is taught is the question of when mathematics vocabulary should

be taught and how it should be assessed. Mathematics is a content area that

builds from prerequisite skills to more advanced skills, calling teachers' atten-

tion to when students should be expected to master vocabulary and how to

distinguish between limited skills and limited vocabulary, and thereby access

to the skills, when the two are intertwined.

Given the limited availability of intervention research specific to math-

ematics vocabulary, generalization of these suggestions should be made with

caution. Like all classroom instruction, instructional decisions should be

made based on data supporting students' response to instruction. Therefore,

teachers should collect data on the effectiveness of mathematics vocabulary

Mathematical Vocabulary 247

Downloaded by [Duquesne University] at 12:44 29 October 2015

interventions being implemented and make continued educational decisions

based on those data.

IMPLICATIONS FOR PRACTICE

Given the large number of terms encountered throughout the course of a

year and the varying ability and readiness of students to learn new

vocabulary, teachers must judiciously select words to teach and help stu-

dents not only to learn the new terms as they are encountered but also to

continue to remember previously learned terms from year to year.

Although using and encountering terms in naturalistic contexts facilitates

vocabulary development, for many students, especially struggling stu-

dents, this development may be fragmented and disjointed; therefore,

the consistent and purposeful use of vocabulary building can greatly assist

students. Although there is not one right way to build vocabulary skills, a

theme that ran through all supports described is clear: purposeful word

instruction with multiple opportunities for students to respond and prac-

tice vocabulary in multiple contexts. From explicitly introducing a vocabu-

lary word to playing a word game while waiting in line to go to lunch, the

instruction should be methodically planned and executed with purpose

and precision. Capitalizing on instructional time and providing multiple

opportunities for students to successfully learn, use, and practice new

and critical vocabulary is important.

CONCLUSION

As the language of mathematics continues to become an emphasis in the

development of mathematical proficiency, there is no question about the

importance of spending instructional time to teach mathematics vocabu-

lary. van der Walt (2009 ) emphasized that vocabulary within the language

of mathematics is an aspect of instruction that requires specific attention.

While vocabulary continues to emerge as an essential aspect of language

development in mathematics, resources supporting mathematics vocabu-

lary need to become more prevalent in mathematics literacy. This article

has presented an overview of the impact of mathematical vocabulary on

proficiency and evidence-supported instructional strategies for incorporat-

ing mathematics vocabulary instruction into classroom learning. Rich

development and understanding of mathematics vocabulary is essential

for students to become actively engaged in mathematics past mundane

computational requirements to thorough understanding and meaning

making. Educators have the responsibility to provide students with instruc-

tion that best supports learning, academic success, and lifelong success.

The strategies and techniques described in this article can help teachers

248 P. J. Riccomini et al.

Downloaded by [Duquesne University] at 12:44 29 October 2015

accomplish this responsibility once they recognize the importance of the

language of mathematics.

REFERENCES

Adams, T. L. (2003). Reading mathematics: More than words can say. The Reading

Teacher,56, 786–795.

Anderson-Inman, L., & Horney, M. (1998). Transforming text for at-risk readers. In D.

Reinking, L. D. Labbo, M. C. McKenna, & R. D. Kieffer (Eds.), Handbook of

literacy and technology: Transformations in a post-typographic world

(pp. 15–43). Mahwah, NJ: Erlbaum.

Archer, A. L., & Hughes, C. A. (2011). Explicit instruction: Effective and efficient

teaching. New York, NY: Guilford Press.

Atkinson, R. C. (1975). Mnemotechnics in second-language learning. American

Psychologist,30, 821–828.

Bay-Williams, J. M., & Livers, S. (2009). Supporting math vocabulary acquisition.

Teaching Children Mathematics,16, 238–245.

Berne, J. I., & Blachowicz, C. L. Z. (2008). What reading teachers say about

vocabulary instruction: Voices from the classroom. The Reading Teacher , 62,

452–477.

Bottge, B. A., Heinrichs, M., Mehta, Z. D., & Hung, Y. H. (2002). Weighing the

benefits of anchored math instruction for students with disabilities in general

education classes. Journal of Special Education , 35 , 186–200.

Bouck, E. C. (2010). The impact of calculator type and instructional exposure

for students with a disability: A pilot study. Learning Disabilities ,16 ,

141–148.

Cawthon, S. W., Beretvas, S., Kaye, A. D., & Lockhart, L. (2012). Factor structure of

opportunity to learn for students with and without disabilities. Education Policy

Analysis Archives,20(41), 1–30.

Charlton, B., Williams, R. L., & McLaughlin, T. F. (2005). Educational games: A tech-

nique to accelerate the acquisition of reading skills of children with learning dis-

abilities. The International Journal of Special Education , 20 , 66–72.

Covington, M. V. (1992). Making the grade: A self-worth perspective on motivation

and school reform. New York, NY: Cambridge University Press.

Epstein, M. L., Lazarus, A. D., Calvano, T. B., Matthews, K. A., Hendel, R. A., Epstein,

B. B., & Brosvic, G. M. (2002). Immediate feedback assessment technique pro-

motes learning and corrects inaccurate first responses. Psychological Record , 52,

187–201.

Forness, S. R., Kavale, K. A., Blum, I. M., & Lloyd, J. W. (1997). Mega-analysis of

meta analysis: What works in special education and related services. Teaching

Exceptional Children, 29(6), 4–9.

Gee, J. P. (2004). Learning by design: Games as learning machines. Interactive

Educational Multimedia,8, 15–23.

Geisler, J. H., Hessler, T., Gardner, I., & Lovelace, T. S. (2009). Differentiating writing

interventions for high-achieving urban African American elementary students.

Journal of Advanced Academics,20, 214–247.

Mathematical Vocabulary 249

Downloaded by [Duquesne University] at 12:44 29 October 2015

Harmon, J. M., Hedrick, W. B., & Wood, K. D. (2005). Research on vocabulary

instruction in the content areas: Implications for struggling readers. Reading

& Writing Quarterly,21, 261–280.

Hebert, B. M., & Murdock, J. Y. (1994). Comparing three computer-aided instruction

output modes to teach vocabulary words to students with learning disabilities.

Learning Disabilities Research & Practice,9, 136–141.

Hofmeister, A. M. (1989). Mainstreaming students with learning disabilities for

videodisc math instruction. Teaching Exceptional Children , 21 (3), 52–60.

Hong, S. Y., & Diamond, K. E. (2012). Two approaches to teaching young children

science concepts, vocabulary, and scientific problem-solving skills. Early

Childhood Research Quarterly,27, 295–305.

Jitendra, A. K., Edwards, L. L., Sacks, G., & Jacobson, L. A. (2004). What research says

about vocabulary instruction for students with learning disabilities. Exceptional

Children,70, 299–321.

Johnson, D. D., von Hoff Johnson, B., & Schlichting, K. (2004). Logology: Word and

language play. In J. F. Baumann & E. J. Kame'enui (Eds.), Vocabulary instruc-

tion: Research to practice gap (pp. 179–200). New York, NY: Guilford Press.

Jones, R. E., Yssel, N., & Grant, C. (2012). Reading instruction in Tier 1: Bridging the

gaps by nesting evidenced-based interventions within differentiated instruction.

Psychology in the Schools,49, 210–218.

Joseph, L., Eveleigh, E., Konrad, M., Neef, N., & Volpe, R. (2012). Comparison of the

efficiency of two flashcard drill methods on children's reading performance.

Journal of Applied School Psychology,28, 317–337.

Kaufman, L., McLaughlin, T. F., Derby, K. M., & Waco, T. (2011). Employing reading

racetracks and DI flashcards with and without cover, copy, and compare and

rewards to teach site words to three students with learning disabilities in read-

ing. Educational Research Quarterly , 43 (4), 27–50.

Kavale, K. A., & Forness, S. R. (1999). Efficacy of special education and related

service. Washington, DC: American Association of Mental Retardation.

Kenney, J. M. (2005). Mathematics as language. In Literacy strategies for improving

mathematics instruction (pp. 1–6). Alexandria, VA: Association for Supervision

and Curriculum Development.

Kluge, A., Ritzman, S., Burkolter, D., & Sauer, J. (2011). The interaction of drill and

practice and error training with individual differences. Cognition, Technology &

Work,13, 103–120.

Koury, K. A. (1996). The impact of preteaching science content vocabulary using

integrated media for knowledge acquisition in a collaborative classroom. Jour-

nal of Computing in Childhood Education,7, 179–197.

Language. (2013). In Merriam-Webster's online dictionary . Retrieved from http://

www.merriam-webster.com/dictionary/language?show=0&t=1370883898

Lee, H., & Jung, W. S. (2004). Limited English-proficient (LEP) students and mathemat-

ical understanding. Mathematics Teaching in the Middle School, 9, 269–272.

Manzo, A. V., Manzo, U. C., & Thomas, M. M. (2006). Rationale for systematic

vocabulary development: Antidote for state mandates. Journal of Adolescent

& Adult Literacy, 49, 610–619.

Marzano, R. J. (2004). Building background knowledge for academic achievement.

Alexandria, VA: Association for Supervision and Curriculum Development.

250 P. J. Riccomini et al.

Downloaded by [Duquesne University] at 12:44 29 October 2015

Mastropieri, M. A., & Scruggs, T. E. (1989). Constructing more meaningful relation-

ships: Mnemonic instruction for special populations. Educational Psychology

Review,1, 83–111.

Mastropieri, M. A., & Scruggs, T. E. (2007). The inclusive classroom: Strategies for

effective instruction. Columbus, OH: Merrill Prentice Hall.

Mastropieri, M. A., Scruggs, T. E., & Fulk, B. J. M. (1990). Teaching abstract vocabu-

lary with the keyword method: Effects on recall and comprehension. Journal of

Learning Disabilities, 23, 69–74.

McKeown, M. G., & Beck, I. L. (2002). Direct and rich vocabulary instruction. In J. F.

Baumann & E. J. Kame'enui (Eds.), Vocabulary instruction: Research to practice

(pp. 13–27). New York, NY: Guilford Press.

Monroe, E. E. (1998). Using graphic organizers to teach vocabulary: Does available

research inform mathematics instruction. Education , 118 , 538–542.

Monroe, E. E., & Orme, M. P. (2002). Developing mathematical vocabulary. Prevent-

ing School Failure, 46, 139–142.

Monroe, E., & Panchyshyn, R. (2005). Helping children with words in word

problems. Australian Primary Mathematics Classroom , 10 (4), 27–29.

Morin, J. E., & Franks, D. J. (2010). Why do some children have difficulty learning

mathematics? Looking at language for answers. Preventing School Failure , 54,

111–118.

National Council of Teachers of Mathematics. (2006). Principles and standards for

school mathematics. Reston, VA: Author.

National Governors Association Center for Best Practices & Council of Chief State

School Officers. (2010). Common Core State Standards for Mathematics.

Washington, DC: Authors.

National Institute of Child Health and Human Development. (2000). Report of the

National Reading Panel: Teaching children to read: An evidence-based assess-

ment of the scientific research literature on reading and its implications for

reading instruction. Retrieved from http://www.nichd.nih.gov/publications/

nrp/smallbook.htm

National Research Council. (2001). Adding it up: Helping children learn mathemat-

ics. Mathematics Learning Study Committee; J. Kilpatrick, J. Swafford, & B.

Findell (Eds.); Center for Education, Division of Behavioral and Social Sciences

and Education.Washington, DC: National Academies Press.

Riccomini, P. J., Sanders, S., & Jones, J. (2008). The key to enhancing students' math-

ematical vocabulary knowledge. Journal on School Educational Technology,

4(1), 1–7.

Riccomini, P. J., & Witzel, B. S. (2010). Response to intervention in mathematics.

Thousand Oaks, CA: Corwin Press.

Rubenstein, R., & Thompson, D. (2002). Understanding and supportingchildren's math-

ematical vocabulary development. Teaching Children Mathematics, 9, 107–112.

Ryve, A., Nilsson, P., & Patterson, K. (2013). Analyzing effective communication

in mathematics group work: The role of visual mediators and technical terms.

Educational Studies in Mathematics,82, 497–514.

Sanders, P. S. (2007). Embedded strategies in mathematics vocabulary instruction: A

quasi-experimental study. Retrieved from http://tigerprints.clemson.edu/cgi/

viewcontent.cgi?article=1163&context=all_dissertations

Mathematical Vocabulary 251

Downloaded by [Duquesne University] at 12:44 29 October 2015

Schwartz, J. L., & Kenney, J. M. (1995). Assessing mathematical understanding

and skills effectively (Interim report of the Balanced Assessment Program).

Cambridge, MA: Harvard Graduate School of Education.

Scruggs, T. E., & Mastropieri, M. A. (2000). The effectiveness of mnemonic instruc-

tion for students with learning and behavior problems: An update and research

synthesis. Journal of Behavioral Education , 10 , 163–173.

Seethaler, P. M., Fuchs, L. S., Star, J. R., & Bryant, J. (2011). The cognitive predictors

of computational skill with whole versus rational numbers: An exploratory

study. Learning and Individual Differences , 21 , 536–542.

Sloyer, C. W. (2003). Mathematical insight: Changing perspective. Mathematics

Teacher,96, 238–242.

Sonbul, S., & Schmitt, N. (2010). Direct teaching of vocabulary after reading: Is it

worth the effort? English Language Teachers Journal , 64 , 253–260.

Stahl, S. A., & Fairbanks, M. M. (1986). The effects of vocabulary instruction: A model

based meta-analysis. Review of Educational Research , 56 (1), 72–110.

Stroud, M. J., & Schwartz, N. H. (2010). Summoning prior knowledge through

metaphorical graphics: An example chemistry instruction. Journal of Edu-

cational Research, 103, 351–366.

Stump, C. S., Lovitt, T. C., Fister, S., Kemp, K., Moore, R., & Schroeder, B. (1992).

Vocabulary intervention for secondary level youth. Learning Disabilities

Quarterly,15, 207–222.

Taylor, D. B., Mraz, M., Nichols, W. D., Rickelman, R. J., & Wood, K. (2009). Using

explicit instruction to promote vocabulary learning for struggling readers.

Reading & Writing Quarterly,25, 205–220.

Test, D. W., & Ellis, M. F. (2005). The effects of LAP fractions on addition and

subtraction of fractions with students with mild disabilities. Education & Treat-

ment of Children, 28(1), 11–24.

Van de Walle, J. A. (2001). Elementary and middle school mathematics: Teaching

developmentally (4th ed.). New York, NY: Addison Wesley Longman.

van der Walt, M. (2009). Study orientation and basic vocabulary in mathematics in

primary school. South African Journal of Science and Technology , 28 , 378–392.

van der Walt, M., Maree, K., & Ellis, S. (2008). A mathematics vocabulary question-

naire for immediate use in the intermediate phase. South African Journal of

Education,28, 489–504.

Wang, M. (2012). Effects of cooperative learning on achievement motivation of

female university students. Asian Social Science ,8 (15), 108–114.

Wells, J. C., & Narkon, D. E. (2011). Motivate students to engage in word study using

vocabulary games. Intervention in School and Clinic , 47 , 45–49.

White, T. G., Graves, M. F., & Slater, W. H. (1990). Growth of reading vocabulary

in diverse elementary schools: Decoding and word meaning. Journal of

Educational Psychology, 82, 281–290.

Yeh, T., Tseng, K., Cho, C., Barufaldi, J., Lin, M., & Chang, C. (2012). Exploring the

impact of prior knowledge and appropriate feedback on students' perceived

cognitive load and learning outcomes: Animation-based earthquakes instruc-

tion. International Journal of Science Education ,34 , 1555–1570.

252 P. J. Riccomini et al.

Downloaded by [Duquesne University] at 12:44 29 October 2015

... Learning of the language of mathematics is part of learning mathematics [1,2]. The language of mathematics may be considered a precise, global language with no ambiguities. ...

... Sometimes words are omitted, phrases are shortened, and gestures and pronouns are used instead of the mathematical terms. Teachers try to help students understand the new mathematical concepts and move from informal, everyday language to formal language of mathematics [1,2]. ...

... Mathematics is considered as a global language, which students must master to be able to understand the concepts and communicate their ideas fluently. Mathematics language consists of words, numbers, symbols, and diagrams and is the key to access mathematical concepts [1]. In mathematics education, it involves the ability to use words to explain concepts, justify procedures and communicate mathematically [1]. ...

Language is an essential aspect of teaching and learning mathematics. It is necessary for communicating, transmission of concepts and ideas, and formation of meaning of mathematical concepts. In mathematics, besides symbols, which are usually common in different languages, words and expressions are used, which may invoke different concept images to students in various languages. Some words are used in mathematics and in everyday language with different meanings, while others are used only in mathematics or in mathematics and other disciplines in similar but non-identical ways. In Mathematical Analysis, the used vocabulary is gradually enhanced, and the concepts are defined in a more formal way. In the current study, the language used regarding mathematics of change is examined, focusing on rate of change and in relation to misconceptions of students.

... These results further highlight the importance of general language skills in mathematical learning for second-language learners. For firstlanguage learners, the present results are consistent with other research showing that students develop differentiated mathematical vocabulary which is related to mathematical proficiency (e.g., Kung et al., 2019;Powell et al., 2017;Purpura et al., 2017;Riccomini et al., 2015). For second-language learners, the present findings suggest that students' proficiency in the language of instruction is critical in the early grades, presumably until students have developed more skill in both the instructional language and in mathematics. ...

... language. Individual differences in mathematical vocabulary skills may be more closely tied to school experiences than are general vocabulary skills (Riccomini et al., 2015). ...

Language skills play an important role in mathematics development. Students (7 to 10 years of age) learning school mathematics either in the same language used at home (first-language learners; n = 103) or in a different language (second-language learners; n = 57) participated in the study. Relations among cognitive skills (i.e., receptive vocabulary, working memory, quantitative skills), domain-specific language skills (i.e., mathematical vocabulary, mathematical orthography), word-problem solving, arithmetic fluency, and word reading were investigated. Second-language learners had lower scores on measures with strong language components (i.e., receptive vocabulary, subitizing, and word-problem solving) than first-language learners, whereas they performed equally well on other tasks. Mathematical vocabulary and receptive vocabulary contributed to word-problem solving success for first-language learners, whereas only receptive vocabulary in the language of instruction related to mathematical outcomes for second-language learners. Mathematical vocabulary was related to arithmetic fluency for both groups, but mathematical orthography was not. For both groups, students' word reading was predicted by receptive vocabulary but not by quantitative skills, highlighting the domain-specific nature of these skills. These findings have implications for supporting mathematical learning in second-language students.

... Furthermore, endowing mathematical meaning to parts of a shape is linked to the development of mathematical language [31][32][33][34]. Language development starts from what Gee [31] called primary discourse, mathematical language informally used in the family and social contexts whereas the language linked to a secondary discourse is used in formal practices in institutions such as schools. ...

... Introducing standardized geometry words during the instruction helped the students' speech to evolve from primary speech to secondary speech [31]. In this way, we can consider that the use of standardized geometry terminology is linked to the development of forms of reasoning about figure attributes [32][33][34]. ...

This paper reports sophistication levels in third grade children's understanding of polygon concept and polygon classes. We consider how children endow mathematical meaning to parts of figures and reason to identify relationships between polygons. We describe four levels of sophistication in children's thinking as they consider a figure as an example of a polygon class through spatial structuring (the mental operation of building an organization for a set of figures). These levels are: (i) partial structuring of polygon concept; (ii) global structuring of polygon concept; (iii) partial structuring of polygon classes; and (iv) global structuring of polygon classes. These levels detail how cognitive apprehensions, dimensional deconstruction, and the use of mathematical language intervene in the mental process of spatial structuring in the understanding of the classes of polygons.

... Trotzdem hat sich in der Allgemeinen Didaktik, insbesondere aufgrund der empirischen Lehr-Lernforschung im Mathematikunterricht (u. a. Kunter, Baumert, Blum, Klusmann, Krauss & Neubrand, 2011), die einheitliche Meinung durchgesetzt, dass Lernprozesse mit einer kognitiv aktivierenden Aufgabenkultur und intelligentes Üben besonders effektiv sind (Riccomini, Smith, Hughes & Fries, 2015). Völlig andere Ansprüche an den Fachunterricht hinsichtlich der Skala adaptives Üben von Regelhaftem benennen die Expertinnen und Experten der gesellschaftswissenschaftlichen Unterrichtsfächer, in deren Mitte der Biologieunterricht auftaucht. ...

Was ist das Transversale von qualitätsvollem Fachunterricht in unterschiedlichen Schulfächern? Finden sich unterschiedliche oder gemeinsame Merkmale (Basisdimensionen) für die Lernwirksamkeit in den untersuchten Fächern? Zur Beantwortung dieser Fragen wurden Interviewartikel von rund 300 Fachdidaktikerinnen und Fachdidaktikern bzw. Lehrpersonen aus 17 Schulfächern beigezogen, die in der vorliegenden Buchreihe „Unterrichtsqualität: Perspektiven von Expertinnen und Experten" erschienen sind. Die Texte wurden mittels eines lexikometrischen Verfahrens ausgewertet; es wurde eine Frequenzanalyse durchgeführt und mit einer explorativen Faktorenanalyse nach einer korrelativen Struktur gesucht, um abschließend hierarchische Cluster der Fächer im Hinblick auf ausgewählte Merkmale wirksamen Fachunterrichts zu erstellen. Es zeigt sich, dass den untersuchten Basisdimensionen lernwirksamen Unterrichts je nach Schulfach unterschiedliche Bedeutung beigemessen wird. Ergebnisse empirischer Studien, denen häufig nur ein Unterrichtsfach zu Grunde liegt, können deshalb möglicherweise nur mit Vorsicht verallgemeinert werden. Dieses Ergebnis wird als Hinweis gedeutet, dass der Ansatz einer Metawissenschaft Allgemeine Fachdidaktik vertieft werden sollte.

... (Teacher 4) The teachers indicated the importance of clarifying the mathematical vocabulary since it carries meaning in the given problems. According to Teacher 1, these mathematical key terms are important to teach (Riccomini, Smith, Hughes & Fries, 2015) and to clarify because they carry meaning and directive in terms of what should be solved. As for Teacher 4, clarification of these terms is important especially when it is linked to application of such similar terms in other 'subjects' as opposed to their application in mathematics in order to draw a clear distinction (Owens, 2006;Widdows, 2003). ...

Flexible teaching of mathematics word problems is essential to improve learning. Flexible teaching is vital in terms of providing meaningful learning, creating inclusive learning spaces and making content accessible. As such, teachers need to strive to provide flexible teaching of mathematics word problems in order to optimise and maximise learning. In line with this notion, therefore, the qualitative case study reported in this article aimed to explore the implementation of one aspect of universal design for learning (UDL), namely multiple means of representation (MMR), to guide flexible teaching of mathematics word problems. Data were collected using focus group discussions, reflection and observation sessions in which five high school mathematics teachers and a Head of Department were involved. The teachers participated in a mini-workshop on the application of the UDL principles which was organised to introduce and induct them to the approach. The study showed that MMR can be used to help guide flexible teaching of mathematics word problems by providing varied options for comprehension: options for language, mathematical expressions and symbols, as well as options for perception. The findings of the study recommend the need for teachers to adapt their teaching by considering the application of the MMR principle to guide and promote flexible teaching of mathematics word problems.

Research rarely focuses on how deaf and hard of hearing (DHH) students address mathematical ideas. Complexities involved in using sign language (SL) in mathematics classrooms include not just challenges, but opportunities that accompany mathematics learning in this gestural-somatic medium. The authors consider DHH students primarily as learners of mathematics, and their SL use as a special case of language in the mathematics classroom. More specifically, using SL in teaching and learning mathematics is explored within semiotic and embodiment perspectives to gain a better understanding of how using SL affects the development, conceptualization, and representation of mathematical meaning. The theoretical discussion employs examples from the authors' work and research on geometry, arithmetic, and fraction concepts with Deaf German and Austrian learners and experts. The examples inform the context of mathematics teaching and learning more generally by illuminating SL features that distinguish mathematics learning for DHH learners.

  • Rheta N. Rubenstein
  • Denisse R. Thompson

Imagine a teacher running her hands across her desk as she tells her students, "A plane is a perfectly flat surface." The students listen quietly, but one of them is thinking, "I thought a plane was something that flies."

Learn when and how teachers can use rich mathematical vocabulary to develop and maximize students' learning, particularly English Language Learners and struggling readers.

  • B. Charlton
  • R.L. Williams
  • T.F. McLaughlin

This study evaluated the effects of educational games on the performance of eight elementary school students with learning disabilities. The effects of educational games were evaluated in a multiple baseline design across students. The results indicated that each student improved their performance on reading when educational games were in effect. These differences were also educationally significant. Practical considerations and implications of educational games for adoption in the classroom were discussed.

  • D.W. Test
  • M.F. Ellis

One of the hardest math skills for students to learn is fractions. This study used a multiple probe across participants to evaluate the effectiveness of a mnemonic strategy called LAP Fractions on six middle school students' ability to add and subtract fractions. Five of six participants were able to achieve mastery, and all six students maintained gains over a period of 6 weeks. Results are discussed in terms of implications for research and practice.

  • Hea-Jin Lee
  • Woo Sik Jung

The number of Limited-English-Proficient (LEP) learners in the United States is dramatically and steadily increasing every year. In 1993–1994, U.S. public schools enrolled more than 2.1 million LEP students, with more than 90 percent of them coming from non-English-speaking countries (McCandless, Rossi, and Daugherty 1996). A study by the National Center for Education Statistics estimates a current enrollment of 3.4 million LEP students in grades K–8 (Buck 2000). This change in student demographics and the importance of language proficiency in mathematics require increasing awareness of instructional practices.