The Root of the Problem:The Root of the Problem:

MathematicsMathematics

And And

LanguageLanguageMarcia

ThompsonLincoln High

School

Math RegisterMath Register ……many mathematics teachers have often many mathematics teachers have often

believed that because mathematical believed that because mathematical concepts (e.g. π, function, limit) are concepts (e.g. π, function, limit) are universal and often explicated with universal and often explicated with numeric examples, mathematics learning numeric examples, mathematics learning is less language-dependent than other is less language-dependent than other core content areas. However, this core content areas. However, this misconception has slowly begun to change misconception has slowly begun to change as language continues to be shown crucial as language continues to be shown crucial to learning mathematics because it serves to learning mathematics because it serves both as a means of representation and as both as a means of representation and as a means of communication. (Lager, p. 167)a means of communication. (Lager, p. 167)

PurposePurpose Examine some of the problems Examine some of the problems

created by semantics in acquiring created by semantics in acquiring mathematical languagemathematical language

Examine syntax problems as they Examine syntax problems as they relate to comprehensionrelate to comprehension

Name some explicit strategies Name some explicit strategies teachers can use to address some of teachers can use to address some of these language issuesthese language issues

Mathematics as a LanguageMathematics as a Language

1. Abstractions 1. Abstractions (verbal or written (verbal or written symbols representing symbols representing ideas or images) are ideas or images) are used to used to communicate.communicate.

2. Symbols and rules 2. Symbols and rules are uniform and are uniform and consistent.consistent.

3. Expressions are 3. Expressions are linear and serial.linear and serial.

Mathematics as LanguageMathematics as Language

4. Understanding 4. Understanding increases with increases with practice.practice.

5. Success requires 5. Success requires memorization of memorization of symbols and rules.symbols and rules.

6. Translations and 6. Translations and interpretations are interpretations are required for novice required for novice learners.learners.

Mathematics as LanguageMathematics as Language

7. Meaning is 7. Meaning is influenced by influenced by symbol order.symbol order.

8. Communication 8. Communication requires encoding requires encoding and decoding.and decoding.

Mathematics as LanguageMathematics as Language

9. Intuition, 9. Intuition, insightfulness, and insightfulness, and “speaking without “speaking without thinking” accompany thinking” accompany fluency.fluency.

10. Experiences from 10. Experiences from childhood supply the childhood supply the foundation for future foundation for future development.development.

11. The possibilities 11. The possibilities for expressions are for expressions are infinite. (Wakefield, infinite. (Wakefield, 2000, p. 272-273)2000, p. 272-273)

Semantics and MathematicsSemantics and Mathematics

Semantics Defined:Semantics Defined: The study of The study of

meaning meaning Relationship Relationship

between signs and between signs and symbols symbols

Meaning or Meaning or interpretation of a interpretation of a word, sentence or word, sentence or other language formother language form

Semantic Semantic Problems:Problems: Decoding and Decoding and

symbolic languagesymbolic language PolysemyPolysemy Homophones and Homophones and

Near homophonesNear homophones SynonymsSynonyms Dense words or Dense words or

phrasesphrases

Decoding and Symbolic Decoding and Symbolic LanguageLanguage

In reading mathematics text one must decode and In reading mathematics text one must decode and comprehend not only words, but also signs and comprehend not only words, but also signs and symbols, which involve different skills. Decoding symbols, which involve different skills. Decoding words entails connecting sounds to the alphabetic words entails connecting sounds to the alphabetic symbols, or letters…In contrast, mathematics signs symbols, or letters…In contrast, mathematics signs and symbols may be pictorial, or they may refer to and symbols may be pictorial, or they may refer to an operation, or to an expression. Consequently, an operation, or to an expression. Consequently, students need to learn the meaning of each symbol students need to learn the meaning of each symbol much like they learn “sight” words in the English much like they learn “sight” words in the English language. In addition they need to connect each language. In addition they need to connect each symbol, the idea it represents, and the written or symbol, the idea it represents, and the written or spoken term that corresponds to the idea.spoken term that corresponds to the idea. (Barton (Barton and Heidema 2002, p.15)and Heidema 2002, p.15)

Decoding and Symbolic Decoding and Symbolic LanguageLanguage

Double decoding to multiple decodingDouble decoding to multiple decoding

Look alike symbolsLook alike symbols

Multiple representationsMultiple representations

ShorthandShorthand

and

x2

Decoding and Symbolic Decoding and Symbolic LanguageLanguage

Cultural Cultural DifferencesDifferences

Graphic Graphic RepresentationsRepresentations

Teacher Response to Decoding Teacher Response to Decoding and Symbolic Language and Symbolic Language

ProblemsProblems Visuals in the classroomVisuals in the classroom Explicit teaching of symbolic language during Explicit teaching of symbolic language during

content instructioncontent instruction Writing about symbolic language and graphic Writing about symbolic language and graphic

representationsrepresentations

PolysemyPolysemy

Multiple meaningsMultiple meanings Vocabulary key Vocabulary key

indicator of indicator of comprehensioncomprehension

More aware of non-More aware of non-mathematical mathematical meaningmeaning

Example: rangeExample: range

PolysemyPolysemy

Activity: With your group, write as Activity: With your group, write as many mathematical words you can many mathematical words you can think of with multiple meanings.think of with multiple meanings.

Teacher Response to Teacher Response to PolysemyPolysemy Marzano’s Marzano’s Building Background Knowledge Building Background Knowledge

for Academic Achievementfor Academic Achievement Learn from descriptions, not formal definitions Learn from descriptions, not formal definitions

through linguistic and non-linguistic ways within through linguistic and non-linguistic ways within the context of a particular concept or topicthe context of a particular concept or topic

Discuss wordsDiscuss words Play games with wordsPlay games with words Evolves over time which is needed for ELLsEvolves over time which is needed for ELLs

WritingWriting DrawingsDrawings ComparingComparing ClassifyingClassifying Metaphors Metaphors AnalogiesAnalogies

Teacher Response to Teacher Response to PolysemyPolysemy

Cognitive Academic Cognitive Academic Language Learning Language Learning Approach (Tuttle, Approach (Tuttle, 2005)2005) Write the process in Write the process in

collaboration with collaboration with another studentanother student

Read the problemRead the problem Name vocabulary not Name vocabulary not

knownknown Speak with partnerSpeak with partner Supported by teacherSupported by teacher

Teacher Response to Teacher Response to PolysemyPolysemy

Project ChallengeProject Challenge ““Revoicing”Revoicing”

Student speaks their Student speaks their understandingunderstanding

Classmates “listen” so they Classmates “listen” so they can paraphrase in their own can paraphrase in their own words if askedwords if asked

Teacher models fuller Teacher models fuller explanation, clarifiesexplanation, clarifies

Homophones and Near Homophones and Near HomophonesHomophones

Students confuse words with the Students confuse words with the same sound.same sound.

Examples: Examples: Plane-plainPlane-plain Hour-ourHour-our Whole-holeWhole-hole Weight-waitWeight-wait

Teacher Response to Teacher Response to Homophones and Near Homophones and Near

HomophonesHomophones Make students Make students

aware of aware of homophoneshomophones

Create a bulletin Create a bulletin boardboard

Journal writing Journal writing within the context within the context of studyof study

HOMOPHONES ANDNEAR

HOMOPHONES

SynonymsSynonyms

Words with the same or nearly the same meaningWords with the same or nearly the same meaning ExamplesExamples

Mathematical operations and contextMathematical operations and context Everyday language found in word problemsEveryday language found in word problems

Teacher Response to Teacher Response to SynonymsSynonyms

AwarenessAwareness ““Operation” Word WallOperation” Word Wall JournalJournal GamesGames

Operation matchOperation match Win the Deck Win the Deck

Dense Words and PhrasesDense Words and Phrases

Complex meanings:Complex meanings: ExponentExponent CoefficientCoefficient VariableVariable

Combined concepts:Combined concepts: Common denominatorCommon denominator Least common multipleLeast common multiple

Teacher Response to Dense Teacher Response to Dense Words and PhrasesWords and Phrases

Explicit instructionExplicit instruction

Strategies used with words with Strategies used with words with multiple meanings could be usedmultiple meanings could be used

Awareness of the cognitive load Awareness of the cognitive load place on English Language learnersplace on English Language learners

Syntax Problems for ELLSyntax Problems for ELL

““Even when students Even when students know the vocabulary know the vocabulary and the computations and the computations required, it is possible required, it is possible that the organization of that the organization of the words (i.e., the the words (i.e., the syntax) prevents some syntax) prevents some students from students from understanding the understanding the problem.” (Wheeler and problem.” (Wheeler and McNutt, 2001, p.309) McNutt, 2001, p.309)

Syntax Problems for ELLSyntax Problems for ELL

Pronouns and PrepositionsPronouns and Prepositions Symbolic RepresentationsSymbolic Representations Conditional Clauses and Logical Conditional Clauses and Logical

ConnectorsConnectors Comparative Constructs and Comparative Constructs and

Complex SentencesComplex Sentences

Pronouns Pronouns

Sandy’s family does its laundry at a coin-Sandy’s family does its laundry at a coin-operated Laundromat. operated Laundromat. It costs $1.25 per It costs $1.25 per loadload to use the washing machines and to use the washing machines and 25¢ per load to use the dryers for 10 25¢ per load to use the dryers for 10 minutes. Sandy’s family has 5 loads of minutes. Sandy’s family has 5 loads of laundry to do and each load will need to laundry to do and each load will need to be in a dryer for 30 minutes. Which be in a dryer for 30 minutes. Which expression will give Sandy’s family the expression will give Sandy’s family the total cost of doing these loads of laundry? total cost of doing these loads of laundry?

Example from Campbell el al., 2007 p. 7Example from Campbell el al., 2007 p. 7

Teacher Response to Pronoun Teacher Response to Pronoun problemsproblems

Be aware pronouns can cause Be aware pronouns can cause problems.problems.

Rephrase a problem to help Rephrase a problem to help understanding especially if the understanding especially if the pronoun depends on inferential pronoun depends on inferential understanding.understanding.

Talk about what noun the pronoun Talk about what noun the pronoun refers to in the problem.refers to in the problem.

PrepositionsPrepositions

Prepositions link nouns, pronouns, and Prepositions link nouns, pronouns, and other phrases to other words in a other phrases to other words in a sentence.sentence.

Prepositions can obscure meaning Prepositions can obscure meaning when encountered in mathematical when encountered in mathematical reading.reading. Examples:Examples:

Divided Divided byby…… Find the number Find the number betweenbetween……

Teacher Response to Teacher Response to Preposition ProblemsPreposition Problems

Teacher awareness of other words that are Teacher awareness of other words that are not necessarily mathematical words but not necessarily mathematical words but effect comprehensioneffect comprehension

Explicit teaching of grammatical elements Explicit teaching of grammatical elements within the context of teaching within the context of teaching mathematicsmathematics ExampleExample

Create a preposition wall or journal page Create a preposition wall or journal page Writing mathematical explanations using prepositionsWriting mathematical explanations using prepositions

Symbolic RepresentationsSymbolic Representations

ELLs struggle ELLs struggle translate words to translate words to symbolic language.symbolic language.

Written and Written and symbolic language symbolic language of math does not of math does not always correspond always correspond in a one-to-one in a one-to-one manner.manner.

Examples:Examples: 20 divided by 5 is not 20 divided by 5 is not

6 times as many 6 times as many children (c) as adults children (c) as adults (a) (a)

6a=c instead of 6a=c instead of 6c=a6c=a

520

Teacher ResponseTeacher Response

Teach students strategies to help Teach students strategies to help them be successful (Irujo, 2005)them be successful (Irujo, 2005) Read the problem multiple timesRead the problem multiple times Draw modelsDraw models Teach students to ignore irrelevant Teach students to ignore irrelevant

wordswords ““Think-alouds” Think-alouds”

Problems of Conditional Problems of Conditional Clauses and Logical Clauses and Logical

ConnectorsConnectors Conditional ClausesConditional Clauses

If…thenIf…then If and only if…If and only if… Because…Because… For example…For example… Such that…Such that… But…But…

Logical ConnectorsLogical Connectors Show similarityShow similarity ContradictionContradiction Cause and effectCause and effect Reason and resultReason and result Chronological Chronological

sequence or logical sequence or logical sequencesequence

Often absent from Often absent from languagelanguage

Problems of Conditional Problems of Conditional Clauses and Logical Clauses and Logical

ConnectorsConnectors Lager (nd) example:Lager (nd) example:

Each time the figure number increases Each time the figure number increases by one, the number of blue squares by one, the number of blue squares changes by how many? changes by how many? The comma is The comma is the sole indicator of the “given-then” the sole indicator of the “given-then” construction of the entire interrogative. construction of the entire interrogative. (p.3)(p.3)

Teacher Response to Teacher Response to Conditional Clauses and Conditional Clauses and

Logical ConnectorsLogical Connectors Be aware of the difficulty of these Be aware of the difficulty of these

language constructs and make language constructs and make students aware of their linguistic students aware of their linguistic misunderstandingsmisunderstandings They will ask the right questionsThey will ask the right questions They will seek correctionThey will seek correction

Collaboration Collaboration ParaphrasingParaphrasing Rewrite the problemRewrite the problem

Comparative Constructs and Comparative Constructs and Complex SentencesComplex Sentences

Brown (2005):Brown (2005):For example, the use of For example, the use of comparatives (e.g., higher than, comparatives (e.g., higher than, greater than, as much as), passive greater than, as much as), passive voice (e.g., X is added to Y), reversed voice (e.g., X is added to Y), reversed ways of stating the known and ways of stating the known and unknown variables (e.g., X is 2 less unknown variables (e.g., X is 2 less than Y; the correct equation is X = Y – than Y; the correct equation is X = Y – 2, not X – 2 = Y) can exacerbate 2, not X – 2 = Y) can exacerbate confusion.confusion. (p. 340-341) (p. 340-341)

Teacher Response to Teacher Response to Comparative Constructs and Comparative Constructs and

Complex SentencesComplex Sentences Teach comparison words which express Teach comparison words which express

relationship of time, space, quantity, direction, relationship of time, space, quantity, direction, order, size, and age. Examples of these:order, size, and age. Examples of these: most, most, many, less, longer, least, greatest, before, many, less, longer, least, greatest, before, after, between, some, many, fewafter, between, some, many, few

Break sentences into smaller parts that are Break sentences into smaller parts that are more understandable.more understandable.

Write in the other order to help students write Write in the other order to help students write more complex mathematical statements.more complex mathematical statements.

ConclusionsConclusions

There is a need for mathematics There is a need for mathematics educators to work collaboratively with educators to work collaboratively with English teachers in order to find English teachers in order to find effective strategies to help ELLs deal effective strategies to help ELLs deal with this language demand. with this language demand.

There is also a need for action research There is also a need for action research in this area to make sure appropriate in this area to make sure appropriate strategies are implemented. strategies are implemented.

ConclusionsConclusions

There is a need for cultural sensitivity and There is a need for cultural sensitivity and awareness of differences within the realm of awareness of differences within the realm of mathematical written notation and mathematical written notation and contextual situations posed in mathematical contextual situations posed in mathematical problems.problems.

Most importantly, there is a need for Most importantly, there is a need for teachers to add language objectives for the teachers to add language objectives for the ELL mathematics classroom in order for ELL mathematics classroom in order for students to unlock the language of students to unlock the language of mathematics for academic success.mathematics for academic success.

ReferencesReferences Adams, T. (2003, May). Reading Mathematics: More than words can say. Adams, T. (2003, May). Reading Mathematics: More than words can say.

Reading Teacher, 56, Reading Teacher, 56, 786. Retrieved November 20, 2008 from Ebscohost 786. Retrieved November 20, 2008 from Ebscohost MasterFILE Premier.MasterFILE Premier.

Barton, M. L. & Heidema, C. (2000) Teaching reading in mathematics. Aurora, CO:Barton, M. L. & Heidema, C. (2000) Teaching reading in mathematics. Aurora, CO:Mid-Continent Research for Education and Learning.Mid-Continent Research for Education and Learning.

Bielenberg, B. & Fillmore, L. (2004-2005, December-January). The Englishthey need Bielenberg, B. & Fillmore, L. (2004-2005, December-January). The Englishthey need for the test. for the test. Educational Leadership, 62,Educational Leadership, 62, 45-49. Retrieved November 20, 2008 45-49. Retrieved November 20, 2008 from Ebscohost from Ebscohost Academic Search Premier.Academic Search Premier.

Brown, C. (2005) Equity of literacy-based math performance assessments for EnglishBrown, C. (2005) Equity of literacy-based math performance assessments for Englishlanguage learners. language learners. Bilingual Research JournalBilingual Research Journal, 29(2) 337-363., 29(2) 337-363.

Campbell, A., Adams, V. & Davis, G. (2007) Cognitive demands and second-language Campbell, A., Adams, V. & Davis, G. (2007) Cognitive demands and second-language Learners: A framework for analyzing mathematics instructional contexts.Learners: A framework for analyzing mathematics instructional contexts.

Mathematical Thinking and Learning, 9Mathematical Thinking and Learning, 9(1) 3-30.Retrieved November 30, 2008 (1) 3-30.Retrieved November 30, 2008 from Ebscohost Academic Search Premier.from Ebscohost Academic Search Premier.

Irujo, S. (2007) Teaching Math to English language learners: Can research help? Irujo, S. (2007) Teaching Math to English language learners: Can research help? The ELL OutlookThe ELL Outlook. Retrieved from http://www.coursecrafters.com/ELL-. Retrieved from http://www.coursecrafters.com/ELL-Outlook/2007/mar_apr/ELLOutlookITIArticle1.htm December 1, 2008.Outlook/2007/mar_apr/ELLOutlookITIArticle1.htm December 1, 2008.

Freeman D. & Freeman, Y. Freeman D. & Freeman, Y. Essential linguistics: What you need to know to teach.Essential linguistics: What you need to know to teach.Portsmouth, NH: Heinemann.Portsmouth, NH: Heinemann.

Kennedy, J., Hancewicz, E., Heuer, L., Metsisto, D., & Tuttle, C. (2005). Kennedy, J., Hancewicz, E., Heuer, L., Metsisto, D., & Tuttle, C. (2005). LiteracyLiteracystrategies for improving Mathematics instruction. strategies for improving Mathematics instruction. Alexandria, VA: Alexandria, VA:

Association Association for Supervision and Curriculum Development.for Supervision and Curriculum Development.

Lager, C. (2006) Types of mathematical-language reading interactions that Lager, C. (2006) Types of mathematical-language reading interactions that unnecessarily unnecessarily

hinder Algebra learning and assessment. hinder Algebra learning and assessment. Reading Psychology, 27Reading Psychology, 27, 165-204. , 165-204. Retrieved November 30, 2008 from Ebscohost Academic Search Premier.Retrieved November 30, 2008 from Ebscohost Academic Search Premier.

Lager, C. (n.d.) Unlocking the language of Mathematics to ensure Our English Lager, C. (n.d.) Unlocking the language of Mathematics to ensure Our English Learners acquire Algebra. Retrieved November 30, 2008 from Learners acquire Algebra. Retrieved November 30, 2008 from http://repositories.cdlib.org/ucaccord/pb/pb-006-1004/http://repositories.cdlib.org/ucaccord/pb/pb-006-1004/

Marzano, Robert J. (2004) Marzano, Robert J. (2004) Building background knowledge for academic achievement:Building background knowledge for academic achievement:Research on what works in schools. Research on what works in schools. Alexandria, VA: Association for Supervision and Curriculum Alexandria, VA: Association for Supervision and Curriculum Development.Development.

Scarcella, R. (2008) Developing academic language: Facilitating connections across Scarcella, R. (2008) Developing academic language: Facilitating connections across grades and content areas. Keynote address Dakota TESL Catch the Dreamgrades and content areas. Keynote address Dakota TESL Catch the DreamConference. Retrieved from http://www.bismarckschools.org/uploads/ Conference. Retrieved from http://www.bismarckschools.org/uploads/ resources/7932/dakotakeynote1.pdf November 30, 2008.resources/7932/dakotakeynote1.pdf November 30, 2008.

Shields, D. & Findlan, C. & Portman, C. (2005, March). Word meanings. Shields, D. & Findlan, C. & Portman, C. (2005, March). Word meanings. MathematicsMathematicsTeaching, 190,Teaching, 190,37-39. Retrieved November 20, 2008 from Ebscohost 37-39. Retrieved November 20, 2008 from Ebscohost

AcademicAcademicSearch Premier.Search Premier.

Wagner, R., Muse, A. & Tannenbaum, K. (2007) Wagner, R., Muse, A. & Tannenbaum, K. (2007) Vocabulary acquisition: Vocabulary acquisition: Implications for reading comprehension. Implications for reading comprehension. New York: Guilford Press.New York: Guilford Press.

Wakefield, D.V. (2000). Math as a second language. Wakefield, D.V. (2000). Math as a second language. The Educational ForumThe Educational Forum, , 6464, 272–, 272–279.279.

Wheeler, L. & McNutt, G. (1983) The effect of syntax on low-achieving students’ Wheeler, L. & McNutt, G. (1983) The effect of syntax on low-achieving students’ abilities to solve mathematical word problems. abilities to solve mathematical word problems. The Journal of SpecialThe Journal of Special Education, 17Education, 17(3), 309-315. Retrieved November 30, 2008 from Ebscohost (3), 309-315. Retrieved November 30, 2008 from Ebscohost Academic Search Premier.Academic Search Premier.

Wong-Filmore, Lily. (2004) The Role of Language in Academic Development. Wong-Filmore, Lily. (2004) The Role of Language in Academic Development. California California Department of Education: Regional Support for High Schools.Department of Education: Regional Support for High Schools.