The Root of the Problem:

1. The Root of the Problem: Mathematics And Language Marcia Thompson Lincoln High School

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

3. Purpose • Examine some of the problems created by semantics in acquiring mathematical language • Examine syntax problems as they relate to comprehension • Name some explicit strategies teachers can use to address some of these language issues

4. Mathematics as a Language • 1. Abstractions (verbal or written symbols representing ideas or images) are used to communicate. • 2. Symbols and rules are uniform and consistent. • 3. Expressions are linear and serial.

5. Mathematics as Language • 4. Understanding increases with practice. • 5. Success requires memorization of symbols and rules. • 6. Translations and interpretations are required for novice learners.

6. Mathematics as Language • 7. Meaning is influenced by symbol order. • 8. Communication requires encoding and decoding.

7. Mathematics as Language • 9. Intuition, insightfulness, and “speaking without thinking” accompany fluency. • 10. Experiences from childhood supply the foundation for future development. • 11. The possibilities for expressions are infinite. (Wakefield, 2000, p. 272-273)

8. Semantics Defined: The study of meaning Relationship between signs and symbols Meaning or interpretation of a word, sentence or other language form Semantic Problems: Decoding and symbolic language Polysemy Homophones and Near homophones Synonyms Dense words or phrases Semantics and Mathematics

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

10. Decoding and Symbolic Language • Double decoding to multiple decoding • Look alike symbols • Multiple representations • Shorthand

11. Decoding and Symbolic Language • Cultural Differences • Graphic Representations

12. Teacher Response to Decoding and Symbolic Language Problems • Visuals in the classroom • Explicit teaching of symbolic language during content instruction • Writing about symbolic language and graphic representations

13. Polysemy • Multiple meanings • Vocabulary key indicator of comprehension • More aware of non-mathematical meaning • Example: range

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

15. Teacher Response to Polysemy • Marzano’s Building Background Knowledge for Academic Achievement • Learn from descriptions, not formal definitions through linguistic and non-linguistic ways within the context of a particular concept or topic • Discuss words • Play games with words • Evolves over time which is needed for ELLs • Writing • Drawings • Comparing • Classifying • Metaphors • Analogies

16. Teacher Response to Polysemy • Cognitive Academic Language Learning Approach (Tuttle, 2005) • Write the process in collaboration with another student • Read the problem • Name vocabulary not known • Speak with partner • Supported by teacher

17. Teacher Response to Polysemy • Project Challenge • “Revoicing” • Student speaks their understanding • Classmates “listen” so they can paraphrase in their own words if asked • Teacher models fuller explanation, clarifies

18. Homophones and Near Homophones • Students confuse words with the same sound. • Examples: • Plane-plain • Hour-our • Whole-hole • Weight-wait

19. Teacher Response to Homophones and Near Homophones • Make students aware of homophones • Create a bulletin board • Journal writing within the context of study HOMOPHONES AND NEAR HOMOPHONES

20. Synonyms • Words with the same or nearly the same meaning • Examples • Mathematical operations and context • Everyday language found in word problems

21. Teacher Response to Synonyms • Awareness • “Operation” Word Wall • Journal • Games • Operation match • Win the Deck

22. Dense Words and Phrases • Complex meanings: • Exponent • Coefficient • Variable • Combined concepts: • Common denominator • Least common multiple

23. Teacher Response to Dense Words and Phrases • Explicit instruction • Strategies used with words with multiple meanings could be used • Awareness of the cognitive load place on English Language learners

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

25. Syntax Problems for ELL • Pronouns and Prepositions • Symbolic Representations • Conditional Clauses and Logical Connectors • Comparative Constructs and Complex Sentences

26. Pronouns • Sandy’s family does its laundry at a coin-operated Laundromat. It costs $1.25 per load to use the washing machines and 25¢ per load to use the dryers for 10 minutes. Sandy’s family has 5 loads of laundry to do and each load will need to be in a dryer for 30 minutes. Which expression will give Sandy’s family the total cost of doing these loads of laundry? • Example from Campbell el al., 2007 p. 7

27. Teacher Response to Pronoun problems • Be aware pronouns can cause problems. • Rephrase a problem to help understanding especially if the pronoun depends on inferential understanding. • Talk about what noun the pronoun refers to in the problem.

28. Prepositions • Prepositions link nouns, pronouns, and other phrases to other words in a sentence. • Prepositions can obscure meaning when encountered in mathematical reading. • Examples: • Divided by… • Find the number between…

29. Teacher Response to Preposition Problems • Teacher awareness of other words that are not necessarily mathematical words but effect comprehension • Explicit teaching of grammatical elements within the context of teaching mathematics • Example • Create a preposition wall or journal page • Writing mathematical explanations using prepositions

30. ELLs struggle translate words to symbolic language. Written and symbolic language of math does not always correspond in a one-to-one manner. Examples: 20 divided by 5 is not 6 times as many children (c) as adults (a) 6a=c instead of 6c=a Symbolic Representations

31. Teacher Response • Teach students strategies to help them be successful (Irujo, 2005) • Read the problem multiple times • Draw models • Teach students to ignore irrelevant words • “Think-alouds”

32. Conditional Clauses If…then If and only if… Because… For example… Such that… But… Logical Connectors Show similarity Contradiction Cause and effect Reason and result Chronological sequence or logical sequence Often absent from language Problems of Conditional Clauses and Logical Connectors

33. Problems of Conditional Clauses and Logical Connectors • Lager (nd) example: • Each time the figure number increases by one, the number of blue squares changes by how many? The comma is the sole indicator of the “given-then” construction of the entire interrogative. (p.3)

34. Teacher Response to Conditional Clauses and Logical Connectors • Be aware of the difficulty of these language constructs and make students aware of their linguistic misunderstandings • They will ask the right questions • They will seek correction • Collaboration • Paraphrasing • Rewrite the problem

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

36. Teacher Response to Comparative Constructs and Complex Sentences • Teach comparison words which express relationship of time, space, quantity, direction, order, size, and age. Examples of these: most, many, less, longer, least, greatest, before, after, between, some, many, few • Break sentences into smaller parts that are more understandable. • Write in the other order to help students write more complex mathematical statements.

37. Conclusions • There is a need for mathematics educators to work collaboratively with English teachers in order to find effective strategies to help ELLs deal with this language demand. • There is also a need for action research in this area to make sure appropriate strategies are implemented.

38. Conclusions • There is a need for cultural sensitivity and awareness of differences within the realm of mathematical written notation and contextual situations posed in mathematical problems. • Most importantly, there is a need for teachers to add language objectives for the ELL mathematics classroom in order for students to unlock the language of mathematics for academic success.

39. References • Adams, T. (2003, May). Reading Mathematics: More than words can say. Reading Teacher, 56, 786. Retrieved November 20, 2008 from Ebscohost MasterFILE Premier. • Barton, M. L. & Heidema, C. (2000) Teaching reading in mathematics. Aurora, CO: Mid-Continent Research for Education and Learning. • Bielenberg, B. & Fillmore, L. (2004-2005, December-January). The Englishthey need for the test. Educational Leadership, 62, 45-49. Retrieved November 20, 2008 from Ebscohost Academic Search Premier. • Brown, C. (2005) Equity of literacy-based math performance assessments for English language learners. Bilingual Research Journal, 29(2) 337-363. • Campbell, A., Adams, V. & Davis, G. (2007) Cognitive demands and second-language Learners: A framework for analyzing mathematics instructional contexts. • Mathematical Thinking and Learning, 9(1) 3-30.Retrieved November 30, 2008 from Ebscohost Academic Search Premier. • Irujo, S. (2007) Teaching Math to English language learners: Can research help? The ELL Outlook. Retrieved from http://www.coursecrafters.com/ELL- Outlook/2007/mar_apr/ELLOutlookITIArticle1.htm December 1, 2008. • Freeman D. & Freeman, Y. Essential linguistics: What you need to know to teach. Portsmouth, NH: Heinemann. • Kennedy, J., Hancewicz, E., Heuer, L., Metsisto, D., & Tuttle, C. (2005). Literacy strategies for improving Mathematics instruction. Alexandria, VA: Association for Supervision and Curriculum Development.

40. Lager, C. (2006) Types of mathematical-language reading interactions that unnecessarily hinder Algebra learning and assessment. Reading Psychology, 27, 165-204. Retrieved November 30, 2008 from Ebscohost Academic Search Premier. Lager, C. (n.d.) Unlocking the language of Mathematics to ensure Our English Learners acquire Algebra. Retrieved November 30, 2008 from http://repositories.cdlib.org/ucaccord/pb/pb-006-1004/ Marzano, Robert J. (2004) Building background knowledge for academic achievement: Research on what works in schools. Alexandria, VA: Association for Supervision and Curriculum Development. Scarcella, R. (2008) Developing academic language: Facilitating connections across grades and content areas. Keynote address Dakota TESL Catch the Dream Conference. Retrieved from http://www.bismarckschools.org/uploads/ resources/7932/dakotakeynote1.pdf November 30, 2008. Shields, D. & Findlan, C. & Portman, C. (2005, March). Word meanings. Mathematics Teaching, 190,37-39. Retrieved November 20, 2008 from Ebscohost Academic Search Premier. Wagner, R., Muse, A. & Tannenbaum, K. (2007) Vocabulary acquisition: Implications for reading comprehension. New York: Guilford Press. Wakefield, D.V. (2000). Math as a second language. The Educational Forum, 64, 272– 279. Wheeler, L. & McNutt, G. (1983) The effect of syntax on low-achieving students’ abilities to solve mathematical word problems. The Journal of Special Education, 17(3), 309-315. Retrieved November 30, 2008 from Ebscohost Academic Search Premier. Wong-Filmore, Lily. (2004) The Role of Language in Academic Development. California Department of Education: Regional Support for High Schools.