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Summing up Math Language: Frameworks, Activities and Ideas to Empower
Brenda Custodio– ESL Consultant, Ohio State University, [email protected] Luciana de Oliveira– Associate Professor, University of Miami, [email protected] Judith O'Loughlin– Consultant, Language Matters, LLC, [email protected]
Kate Reynolds– Title III Grant Curriculum Designer, Missouri State University, [email protected]
TESOL International Association, Seattle, WA, March 22, 2017.
Session ObjectivesParticipants learn to 1. analyze key mathematic vocabulary, functions and discourse patterns common at
different grade levels, 2. engage ELLs with vocabulary, phrase and sentence construction (i.e., sentence
level to discourse level competencies), 3. acquire a framework to transition from key words to word problems, 4. debate the “universality” of math (e.g., worldwide calendars, different date systems,
math symbols/notation, long division, metric system, military time, etc.), and 5. develop literacy strategies for helping ELs develop better math skills (e.g., math
notebooks, think alouds).
Math Language: Discourse Patterns, Linguistic Functions and Vocabulary.
›A Professor and teacher educator in TESL/TEFL, currently works with in-service educators at Missouri State University.
›Publications: –Reynolds, K.M. (March, 2015). Approaches to inclusive English classrooms: A teacher’s handbook for content based instruction. Bristol: Multilingual Matters.
–Reynolds, K.M. (Ed.) (in press). Vocabulary volume of the ELT Encyclopedia. Alexandria, VA: TESOL and Wiley.
–Reynolds, K.M. (in press). Developing content-based objectives from academic standards. Upcoming chapter in M.A. Snow and D. Brinton (Eds.), Handbook of Content-Based ESL Instruction, 2nd edition. Cengage.
–Reynolds, K. M. and Jiao, J.J. (2012). Aha! Measuring Teachers’ Content-Based Learning. MinneWITESOL Journal, (29): 127-153.
–Reynolds, K. M. (2010). Exploration: One journey of integrating content and language objectives. In J. Nordmeyer & S Barduhn (Eds.), Integrating Language and Content. Alexandria, VA: TESOL.
›
Dr. Kate Mastruserio Reynolds– Title III Grant Curriculum Designer, Missouri State University, [email protected]
Objective
›Participants will be able to analyze key mathematic vocabulary, functions and discourse patterns common at different grade clusters.
http://www.mextesol.net/journal/index.php?page=journal&id_article=471
›Scarcella’s Academic Language Theoretical Framework (2003)
›Emphasizes 3 dimensions:
–Linguistic dimension–Cognitive dimension–Sociocultural/psychological dimension
›Scarcella, 2003.
Linguistic dimension
WORD LEVEL
SENTENCE LEVEL
PARAGRAPH LEVEL
2 + 3 = 5
What are some ways that the teacher, text, or test might phrase
this equation?
Possible answers:
•How many altogether?
•How many in all?
•How much is 3 and 2?
•What is the sum of…?
•What is 2 plus 3?
•Add the two numbers.
•Three squares and two more are…
•Three plus two equals…
Common Types of Mathematics Vocabulary
Slavit, D., & Ernst-Slavit, G. (2007).
Math Vocabulary by Grade Cluster
Math Vocabulary by Grade Cluster
Math Vocabulary by Grade Cluster
Math Vocabulary by Grade Cluster
Math Vocabulary by Grade Cluster
Math Vocabulary by Grade Cluster
Math Vocabulary by Grade Cluster
A Whole Lot of Words! Tiers of Vocabulary
http://www.learningunlimitedllc.com/wp-content/uploads/2013/05/tiered-3.png
Tiers of Vocabulary›Tier 1: Common, everyday words. Most children know when they enter school in their native language(s). Bilingual children would know these words in one or both languages.
›Examples: dog, ball, mom, dad, house, car
Beck, McKeown, and Kucan (2013)
https://images-na.ssl-images-amazon.com/images/G/01/img15/pet-products/small-tiles/23695_pets_vertical_store_dogs_small_tile_8._CB312176604_.jpg
Tiers of Vocabulary›Tier 2: Words that are employed in many content areas. Important for students to know and understand long-term. Essential for academics.
›Academic and cognitive processing words. Student will see repetitively in texts, texts, and oral discourse throughout schooling.
›Examples: describe, justify, contrast, elaborate
Beck, McKeown, and Kucan (2013) http://p2cdn4static.sharpschool.com/UserFiles/Servers/Server_87286/Image/Vridder/Staff/BloomRevisedTaxonomy.jpg
Tiers of Vocabulary›Tier 3: Content-specific vocabulary. The bolded, defined words in textbooks and glossaries. Terminology and jargon.
›Words for learning a specific academic topic. Necessary for learning the academic concepts and building students' background knowledge.
Beck, McKeown, and Kucan (2013) http://static.wixstatic.com/media/5cd6ef_b39fcd12c5a34ea7ae8ca26e54233076.jpg_srz_435_283_85_22_0.50_1.20_0.00_jpg_srz
Task: Analysis of Math Vocabulary›Directions: Using the Math text/pages provided, please identify and categorize the academic vocabulary into three sections using the 3 Tiers that would be new encounters for ELs.
Tier 1 Tier 2 Tier 3
A Whole Lot of Words! Bernier’s Model of Academic Vocabulary›Content terms—routinely occur in lectures and textbooks as “common knowledge” references to course material within the disciple.
–“regular” math terms and jargon, –archaic language (e.g., abacus, trope, addend), –non-history terms or terms borrowed from other fields (e.g., lemma, root, kite, harmonic mean),
–obscure acronyms and symbols (e.g., AAS, e, GCF, GLB, IQR (see more at http://www.mathwords.com/a_to_z.htm), and
–non-English vocabulary (e.g., alpha, chi, gamma, eta, pi, epsilon, quotient).
Bernier, 1997, pp. 96-97.
Bernier’s Model of Academic Vocabulary›Language terms—refers to vocabulary technically outside the boundaries of the content, but that frequently finds its way into course lectures, readings, and assignments.
–metaphors (offering the olive branch), –colloquial usages (fell on deaf ears), –class-based constructions (syllabus, office hours, mortgage, stocks and bonds), and
–cultural idioms (two for one, ballpark figure, my 2 cents, a dime a dozen, a penny saved is a penny earned, “classic” works of fiction allusions).
Bernier, 1997, pp. 96-97.
Bernier’s Model of Academic Vocabulary›Language Masking Content—includes the fluid boundaries between the two previous categories.
›Terms appropriated by teachers and scholars that hide content due to –variant or multiple meanings (e.g., area, continuous, functions, place value, right),
–unfamiliar metaphors (8-hour day/24-hour day; > sign is like a hungry alligator; numerator is like the top cookie in an Oreo) and
–oxymorons (constant variable, bigger half, random order, sharp curve). (See more at http://www.english-for-students.com/Mathematics.html).
Bernier, 1997, pp. 96-97.
Task: Analysis of Math Vocabulary›Directions: Using the Math text/pages provided, please identify and categorize the academic vocabulary into three sections using Bernier’s categories.
Content Terms Language Terms Content Masking Language
How should we teach the words?Evidence-Based Teaching of Academic Vocabulary
What does current research tell us about teaching vocabulary?
1. Use frequency lists– Nation & Waring, 1997
2. Teach vocabulary explicitly –Marzano & Pickering, 2005
3. Actively engage students –Folse, 2004
4. Use repetition and multiple exposures through practice and materials –
McKeown et al., 1985
5. Use rich oral language- Carlo, August & Snow, 2005
6. Employ narrow, extensive reading for incidental learning- Day, Omura,
Hiramatsu, 1991; Schmitt, & Carter, 2000
https://www.google.com/search?q=academic+language+contexts&rlz=1C5CHFA_enUS723US730&source=lnms&tbm=isch&sa=X&ved=0ahUKEwiJ-aXU_o_SAhXjy1QKHYTfDc4Q_AUICCgB&biw=1027&bih=812#tbm=isch&q=academic+and+social+language+overlap&imgrc=V19M13TGIvZMuM:
Ways to Develop Academic Vocabulary› Clarify embed the meaning of new vocabulary
› Contextualize language
› Create real comprehension check opportunities and activities to measure the learner’s degree of understanding; monitor comprehension
› Direct learners’ attention to key ideas and vocabulary through
– intonation, – visuals, – realia, or – writing on blackboard
› Emphasize key vocabulary
› Limit use of idioms, slang, and colloquial expressions
› Use gestures, expressions and other non-verbal ways to make meaning
6 Steps for Teaching Academic Vocabulary1. Provide a description, explanation, or
example of the new term.
2. Ask students to restate the description, explanation, or example in their own words.
3. Ask students to construct a picture, pictograph, or symbolic representation of the term.
4. Ask students to contextualize the term. Where might they see it in their lives?
5. Practice the term often in a short space of time by
1. Vocabulary notebooks2. Graphic organizers (Frayer’s Model)3. Think-pair-share about terms4. Play games with the terms
Adapted from Marzano, R. J. (2004).
http://3.bp.blogspot.com/-VbZV_5I0J2A/UAMIA26DCdI/AAAAAAAAPKw/ADNDLNyIeHU/s1543/IMG_3120.JPG
Pre-Teaching Math Vocabulary?A Unique Feature of Math Vocabulary
›The academic vocabulary of math is different from other content areas, because the math vocabulary and definitions are intertwined.
›It is difficult to define some math terms through pre-teaching.
›The reason for this is that math terms describe a concept/activity.
›One cannot teach the term without teaching the mathematic concept.
Task: Defining Math Vocabulary›Directions:
1. Choose one term from the text book.2. Write the textbook’s definition of the term3. Write a formal definition of the term
Term + class/category + differentiation from other items in category (Spaghetti is a kind of pasta that is long and thin and usually served with sauce.)
4. Write an informal, or learner friendly, definition of the term.5. Choose a visual or set of visuals you would use to help
explain this term. Draw it/them.
Patterns of Text: Discourse Patterns
›Directions:
›Think-Pair-Share
›Questions:
›What are the patterns of language at the sentence level?
›What are the types of sentences present in math texts?
›What have you noticed about the language of math texts?
At the Sentence Level
Grammar in Math Sentences›Incomplete sentences/phrases
–A number minus 6›Simple sentences
–Maddy has two cupcakes.›If/then constructions
–If Tariq has 5 and Juan has 3, then what do they have all together?
›Wh- questions & Inverted subject–How many are there?–Does she have 10 or 12?
›Compounds and Series–6 is 2 greater than 4 and five times as high as
›Reversals–The number a is five less than b. Correct equation: a = b -5
What other grammatical patterns occur in math discourse?
› There is/are- dummy subject
› Present tense; timeless present
› Prepositions are key to comprehension.
› Abbreviations, symbols, labels
› 2nd and 3rd person
Chamot, 2009; Li, 2012
http://resultncutoff.in/wp-content/uploads/2015/03/maths-ftr.jpg
Task: Identifying Patterns at the Sentence Level
›Directions: Using your math book/pages,
–identify 3-4 patterns at the sentence level, and
–2 sentence starters.
http://mathtrailblazers.uic.edu/mtb4-curriculum/components/student-materials/
Simplifying FractionsThere are two major difficulties when simplifying fractions. One is finding a number that is a common factor when it's
not "obvious", and the other is simplifying completely. To deal with either style problem it is helpful to have a plan and approach each problem systematically.
The easiest way is to try dividing both the numerator (top number) and the denominator (bottom number) by each prime number. The rules of divisibility will simplify this process:
Start with 2: EVEN numbers (ones that end with 2, 4, 6, 8, or 0) can be divided by two without a remainder (i.e., they are divisible by 2).
Then go to 3: Find the SUM OF THE DIGITS (Add the digits together). If the sum can be divided by three then the number is divisible by 3. [NOTE: Since you can tell by looking if a number is divisible by two or by five, you may want to use the "eyeball approach" before checking three....]
Next try 5: Numbers that end with 5 or 0 are divisible by by five.Go on to 7, 11, 13, 17 and so on. Unfortunately there is no easy way to determine whether the number will be
divisible by these -- you just have to try dividing by each. But, you can stop trying when the answer is less than the divisor.
The most challenging fractions to simplify are generally the ones that don't look like they can be simplified. For example:
Twenty-six can be divided by two (because it's even), but 65 can't. Sixty-five can be divided by five, but 26 can't... the fraction looks like it can't be simplified --BUT WAIT-- factor either number and then try dividing by the other, less obvious, factor. And always double-check to see if you can simplify ONE MORE TIME.
At the Paragraph Level
Patterns of Text: Functions of Language
Michael Halliday’s Functions of Language (1975)
7 Language Functions
1. Instrumental
2. Regulatory
3. Interactional
4. Personal
5. Imaginative
6. Heuristic (investigate)
7. Representational
›Translate into these classroom interaction patterns:
–Express an opinion–Summarize–Persuade–Question–Inform–Sequence–Disagree, agree, reach consensus–Debate–Evaluate –Justify
Function Examples Classroom Experiences Instrumental- language is used to communicate preferences, choices, wants, or needs
"I want to ..." Problem solving, gathering materials, role playing, persuading
Personal- language is used to express individuality
"Here I am ...." Making feelings public andinteracting with others
Interactional- language is used to interact and plan, develop, or maintain a play or group activity or social relationship
"You and me ....""I'll be the cashier, ...."
Structured play, dialogues and discussions, talking in groups
Regulatory- language is used to control
"Do as I tell you ....""You need ...."
making rules in games, giving instructions, teaching
Representational- language used to explain
"I'll tell you.""I know."
Conveying messages, telling about the real world, expressing a proposition
Heuristic- language is used to find things out, discover, wonder, or hypothesize
"Tell me why ....""Why did you do that?""What for?"
Question and answer, routines,inquiry and research
Imaginative- language is used to create, explore, and entertain
"Let's pretend ....""I went to my grandma's last night."
Stories and dramatizations, rhymes, poems, and riddles, nonsense and word playwww.communityinclusion.org/elm/Professionals/.../Halliday-handout.docx
Functions Common in Math Texts
› Sequential language› Process-oriented› Comparatives› Argument› Cause & effect› Narrative (word problems)] › Commands and directives http://2.bp.blogspot.com/-T93pPfGa0jM/UhrQ-vsZtUI/AAAAAAAAHhU/
wYHNglnd0qQ/s1600/cause+and+effect+Linguistic+patterns.jpg
Task: Find a Function of Language
›Directions:
1. Find a function in your text.
2. Answer these questions: – Which one is it? – Does the text explain how to do
it?– Does the text present a model
of the language needed?
How do you select math academic language?1. Identify functions of language embedded in
or congruent with the math topic and task.
2. Evaluate the enabling skills necessary to complete academic tasks and determine if the students have the academic language to complete them. If not, explicitly teach students the language to perform the tasks.
3. Provide sentence starters and dialogues to teach and model the academic language function.
4. Provide interactional practice to discuss math reasoning.
5. Allow students options in selecting academic language to learn and practice.
6. Adapt content, simplify or elaborate, when the language is too complex and not essential. Be sure to focus on the big picture.
https://s-media-cache-ak0.pinimg.com/736x/c3/ee/1a/c3ee1a56e5314ffb7b3343550acf8b94.jpg
REFERENCES› Beck, I., McKeown, M., & Kucan, L. (2013). Bringing words to life, 2nd ed. NY, NY: Guilford Press.› Bernier, A. (1997). The challenge of language and history: Terminology from the student optic.” In M. A. Snow and D. M. Brinton (eds.) The Content-Based Classroom : Perspectives on Integrating Language and Content (1st Ed.) (pp. 96-97). New York: Longman.
› Carlo, M. S., August, D., & Snow, C.E. (2005). Sustained vocabulary: Strategy instruction for English language learners. In E. F. Heibert and M. L. Kamil (Eds.), Teaching and Learning Vocabulary: Bringing Research to Practice (pp. 173-154). Mahwah, NJ: Lawrence Erlbaum.
› Cazden, C. B. (2001). Classroom discourse: The language of teaching and learning. Portsmouth, NH: Heineman.› Chamot, A. U. (2009). The CALLA handbook: Implementing the cognitive academic language learning approach (2nd ed). NY, NY: Addison-Wesley.› Chapin, S. H., O’Connor, C. & Anderson, N. C. (2009). Classroom discussions: Using math talk to help students learn (2nd Ed.). Sausalito, CA: Math Solutions Publications.
› Day, R. R., Omura, C, Hiramatsu, M. (1991). Incidental EFL vocabulary learning and reading. Reading in a Foreign Language, 7(2): 541-551.› Li, W. E. [李文瑤 ]. (2012). Genre analysis of word problems in junior secondary school mathematics textbooks for ESL learners in Hong Kong. (Thesis). University of Hong Kong, Pokfulam, Hong Kong SAR. Retrieved from http://dx.doi.org/10.5353/th_b5541038
› Folse, K. (2004). Vocabulary Myths: Applying Second Language Research to Classroom Teaching. Ann Arbor: University of Michigan Press› Herbel-Eisenmann, B., Steele, M., & Cirillo, M. (2013). Developing teacher discourse moves: A framework for professional development. Mathematics Teacher Educator, 1(2), 181-196.
› Herbel-Eisenmann, B. A., & Otten, S. (2011). Mapping mathematics in classroom discourse. Journal for Research in Mathematics Education, 42, 451-484.› Marzano, R. J. (2004). Building background knowledge for academic achievement: Research on what works in schools. Alexandria, VA: ASCD.› Marzano, R., & Pickering, D. (2005). Building academic vocabulary: Teacher’s manual. Alexandria, VA: ASCD.› McKeown, M., Beck, I., Onanson, R., & Pople, M. (1985). Some effects of the nature and frequency of vocabulary instruction on the knowledge and use of words. Reading Research Quarterly, 20: 522-535.
› Moschkovich, J. (2008). I went by twos, he went by one: Multiple interpretations of inscriptions as resources for mathematical discussions. The Journal of the Learning Sciences, 17(4), 551–587.
› Nathan, M.J., Long, S.D., & Alibali, M.W. (2002). The symbol precedence view of mathematical development: A corpus analysis of the rhetorical structure of algebra textbooks. Discourse Processes, 33(1), 1-21.
› Nation, I.S.P., & Waring, R. (1997). Vocabulary size, text coverage, and word lists. In N. Schmitt and M. McCarthy (Eds.), Vocabulary: Description, Acquisition and Pedagogy (pp. 6-19). Cambridge: Cambridge University Press.
› Scarcella, R. (2003). Academic English: A conceptual framework. Berkeley, CA: University of California Linguistic Minority Research Institute.› Schmitt, N., & Carter, R. (2000). The lexical advantages of narrow reading for second language learners. TESOL Journal, 9(1), 4-9. › Schleppegrell, M. J. (2007). The linguistic challenges of mathematics teaching and learning: A research review. Reading & Writing Quarterly, 23, 139-159.› Sfard, A. (2008). Thinking as communicating: Human development, the growth of discourses, and mathematizing. Cambridge, UK: Cambridge University Press.
› Slavit, D., & Ernst-Slavit, G. (2007). Teaching mathematics and English to English language learners simultaneously. Middle School Journal (November): 4-11.› Wagner, D., & Herbel-Eisenmann, B. (2009). Re-mythologizing mathematics through attention to classroom positioning. Educational Studies in Mathematics, 72(1), 1-15.
Secondary Students:Math for New Arrivals
*Brenda Custodio is a retired middle and high school ESL teacher and school administrator of a newcomer high school.
*She received her Ph. D. from Ohio State University in TESOL and now works at the university level preparing teachers for a career in TESOL.
*She is the author of two books: › Custodio, B. (2011).How to Design and Implement
a Newcomer Program. Boston: Allyn & Bacon.
- Custodio, B. & O’Loughlin, J. B. (2017) Students with Interrupted Formal Education: Bridging Where They Are and What They Need. Thousand Oaks, CA: Corwin.
Brenda Custodio– ESL Consultant, Ohio State University, [email protected]
Developing Numeracy Skills
› Think of learning math as a learning the language of numerary
› Start with a pretest to determine current numeracy skills
› Offer development classes on basic math skills and math vocabulary
› Allow students to move to more advanced math classes as their skills increase
› Provide sheltered math for students with strong math skills but limited English ability
Keys Tips for Teaching Numeracy Skills
Math is NOT Necessarily a “Universal Language”
•Writing numerals varies with different languages•Many countries reverse the use of the comma and the decimal point ($2.300,25)•Long division varies by country•Some countries use different local or religious calendars•Many countries use a 24-hour day•Most countries put the day before the month•Metric system used in every other country but US
Mathematics has its own VocabularyMultiple Meaning Words
› Table› Line› Plane› Times› Plot› Angle› Average› Combine› Figure› Faces
Phrase Words for Mathematical Relationships› If› Because› Unless› Alike, similar, same› Different from› Probably› Not quite› Always, all the time› Never
Components of a Strong Numeracy Program
1.Math Vocabulary, Number Sense, and Math Symbols2.Operations and Basic Math Skills3.Fractions, Decimals, and Percents4.Measurement5.Data Analysis and Statistics6.Other Math Concepts such as Geometry, Algebra, and
Word Problems
(These components require math vocabulary)
Challenges of Teaching Math to Newcomers
Cummins says that “mathematical concepts and operations are embedded in language” and “must be taught explicitly if students are to make strong academic progress in mathematics.”
He also says that new English learners trying to learn a new language at the same time as new math concepts presents a “heavy cognitive load,” and without the strong anchor of background knowledge, the task may be overwhelming.
Seven Strategies for Numeracy Development
1.Help students look for patterns rather than solutions.2.Stress that there are multiple ways to solve a problem.3.Encourage peer-group collaboration.4.Utilize visuals and manipulatives to teach abstract
concepts.5.Ground new concepts in practical, real-life situations.6.Encourage the use of journals or logs that require
students to explain their actions with written words.7.Focus on math vocabulary and math sentences.
Support with Manipulatives - Fractions
More Work with Manipulatives – Value Lines
Sample Math Vocabulary Organizer
Picture Book Examples for Math Concepts
References
Ciancone, Tom. (1996) “Numeracy in the Adult ESL Classroom. Toronto Board of Education.Cummins, Jim. (2006) “Every Student Learns.” Scott Foresman-Addison Wesley Mathematics Series, p. iv, xviii.Custodio, Brenda (2011) How to Design and Implement A Newcomer Program. Allyn and Bacon.
Challenges for Grades 3-5 English Learners: Sentence and Discourse Level in Word Problems
› Former K-8 ESL/Special Education Adjunct Graduate Professor
› NJ City University, Bilingual/Special Education endorsement candidates
› Ohio State University, Bilingual Elementary Graduate Candidates
She is the author of:› O’Loughlin, J. B. (2010). Academic Language Accelerator.
New York, NY: Oxford University Press.› O’Loughlin, J. B. (2013). “What Time is It?” in Gottlieb, M, &
Ernst-Slavit, G. Academic Language in Diverse Classrooms: Promoting Content and Language Learning, Mathematics, Grades 3-5. Thousand Oaks, CA: Corwin.
› Custodio, B. & O’Louglin, J. B. (2017). Students with Interrupted Formal Education: Bridging Where They Are and What They Need. Thousand Oaks, CA: Corwin.
Judith O'Loughlin– Consultant, Language Matters, LLC, [email protected]
Word Problems in Mathematics
• Present significant comprehension difficulties • particularly challenging for English Language Learners
(ELLs)• These comprehension challenges have to do with
• lexical complexity • sentence complexity
• Permeate mathematics textbooks and standardized tests
• Seen as important measures of mathematical understanding
Word Problems - Discourse Level - Addition›Basic Elementary Word Problems - Addition
› Someone had ____ and ____. How much did she have: – in all,– together, – combined, – in total?
› Example: Sarah earned 58 dollars last week from her paper route. This week she earned 47 dollars. How much money did she earn for both weeks combined?
Word Problems - Discourse Level - Addition ›Elementary Word Problems - Addition
›Adding to an existing amount is challenging.
›Someone has ____ and bought _____ and ____.
›Example: While building the house, Charlie noticed that they were running out of nails, so he told his father he’s going to buy some. If they still have 9 nails left and Charlie bought 2 boxes of nails, one with 55 nails, the other with 31 nails, how many do they have now?
Word Problems - Discourse Level - Subtraction›Basic Elementary Word Problems: Subtraction
›Linear Structure - Whole to Parts
›There are ______ and ______ are _____ . How many are not?
›Looking from the whole to the part.
›Example: Jan has 20 apples and 7 are red. How many are not red?
Word Problems Mixed Addition and Subtraction Word Problems
›Two step problems
›Total amount needed is _______. First person does/makes ________ and a second person does/makes _____. How many should the third person do/make?
›Example: A catering service needs to prepare 500 pieces of fish fillets for the mayor’s 50th birthday party. Team one prepares 189 pieces and team two prepares131 pieces. How many pieces does team three have to prepare?
Word ProblemsEstimating - Addition›Story problem provides specific numbers. However, students are asked to estimate the total.
›First person has______ and second person has _____. Estimate how many all together.
›Example: Thirteen students from Mrs. D’s class want to go camping. Eighteen students from Mrs. C’s class want to go camping. Estimate how many students want to go camping altogether.
Word ProblemsMultiplication›Story problems with cost (money) and number of items.
›Utilize groupings/categories (containers, teams, classes, rows, rolls, foods, etc.) and number of items in each.
›There are ___ (name of container) and _____ in each. How many in all?
›Example: There are seven girls on stage. Each girl is holding nine flowers. How many flowers are there in all?
Word ProblemsMultiplication - Wording Issues› One day Michelle decided to count her savings. She opened her
piggy bank and sorted out different coins and dollar bills. If she counted a total of 20 nickels (a nickel is equivalent to 5 cents) what is the total value of money does she have in nickels?
› She also found that she has 10 pieces of $5 bills. What is the total value of money does she have in $5 bills?
Word ProblemsDivision›Take the whole and determine number of parts.
›There were ____ (total number of items) then determine how many items would each person/object contain.
›Examples: Eighteen fish were caught on a deep-sea fishing boat. If each person on the boat caught 2 fish, how many people were on the boat?
›A total of 8 people paid a total of $24 for admission. If each ticket cost the same amount, how much did each ticket cost?
Word ProblemsMixed Operations›Addition and Division:
–Greg has 91 erasers and Jane gives him 8 more. Greg gives each of his 9 friends an equal number of erasers. How many erasers does each friend get?
›Addition and Subtraction: –Jane bought a large cheese pizza. The pizza was divided into 12 slices. Jane ate 2 slices, Marvin at 3 slices, and Mary ate one slice. How many slices did they eat together? How many slices were left over?
Word ProblemsElapsed Time›Elapsed time problems have a starting time when an action or event started and an ending time. ›Simple calculations, such as subtraction, will not provide an accurate answer.
›Examples: –Alexa went to the bookstore at 5:45 PM. She left the bookstore at 9:10 PM. How long was Alexa at the bookstore?
–Henry leaves arrives at work at 8:15 AM. He eats lunch at 12:00 PM. Then returns to his desk at 1 PM and works until 4:30 PM. How long does Henry work in the morning? In the afternoon after lunch?
Word ProblemsElapsed Time
Solving Elapsed Time Problems: Tools› Manipulatives: Student clocks › Story Board Frames to Sequence Elapsed Time
› 5:45 9:10 PM
Mathematics Word Wall Examples
Literacy Strategies Help ELs Learn Math›Personal Math Dictionaries- key vocabulary words in math problems.
›Math Diaries- Create your own math stories.
›Strategy Flash Cards or Book Marks- Create cards with simple directions and examples
–Crossing Out Distractors–Highlighting Key Operation Words and Numbers–Think Aloud Prompts and Sentence Starters
›Comic Book Templates- Draw the Problem
›Manipulatives- counters, number cubes, dice, tangrams, rods, unifix cubes, etc.
Picture Books: Building Background for Math Instruction› Pastry School in Paris, C. Neuschwandeer› Perimeter, Area, and Volume, David A. Adler› Full House: An Invitation to Fractins, D.A. Dodds› A Very Improbable Story, E. A. Einhorn› Sir Cumference (geometry picture book series), C. Neushwandeer› Equal Shmequal: A Math Adventure, V. Kroll› How Many Seeds in a Pumpkin? M. McNamara
Picture Books: Building Background for Math Instruction› Twizzlers Percentages, J. Pilotta and R. Bolster› The Grapes of Math, G. Tang and H. Bridges› Multiplying Menace: The Revenge of Rumplestiltskin, P. Calvert and
W. Geehan› The Wishing Club: A Story About Fractions, D. J. Napoli and A. Curry› Equal Schmequal, V. Keroll and P. O’Neill› A Place for Zero, A. Sparagna LoPresti and P. Hornung› Perimeter, Area, and Volume, D. A. Adler and E. Miller
References›O’Loughlin, J. B. (2013). Grade 3: What Time is It? in (Gottlieb, M. & Ernst-Slavit, G. (Eds.), Academic Language in Diverse Classrooms: Promoting Content and Language Learning. Thousand Oaks, CA: Corwin.
› Word Problem Examples:– Reading and Math Worksheets. K-5– Learning:www.k5learning.com
› Spectrum Math Series. (2015) Greensboro, NC: Carson-Dellosa Publishing Group.
› Apps for Math Problem Practice– Bedtime Math (www.BedtimeMath.org)– Interactive Telling Time– Khan Academy– Pizza Fractions (Ages 6-8)– Splash Math: Third Grade math, Mutlitplication, Fractions, and More – Telling Time - Little Matchups Game
A Framework for Analyzing Word Problems: Beyond Key Words
Chair and Associate Professor, Dept of Teaching and Learning, University of Miami
Incoming President-Elect, TESOL International Association
Publications – math-relatedde Oliveira, L. C., Sembiante, S., & Ramirez, J. A. (in press). Bilingual academic
language development in mathematics for emergent to advanced bilingual students. In S. Crespo & S. Celedón-Pattichis, & M. Civil (Eds), Access and equity: Promoting high quality mathematics in grades 3-5. National Council of Teachers of Mathematics (NCTM).
de Oliveira, L. C. (Series Ed.) (2014/2016). The Common Core State Standards and English Language Learners. Alexandria, VA: TESOL Press.
- Civil, M., & Turner, E. (Eds.) (2014). The Common Core State Standards in Mathematics for English Language Learners: Grades K–8.
- Bright, A., Hansen-Thomas, H., & de Oliveira, L. C. (Eds.) (2015). The Common Core State Standards in Mathematics and English Language Learners: High School.
de Oliveira, L. C. (2012). The language demands of word problems for English language learners. In S. Celedón-Pattichis & N. Ramirez (Eds.), Beyond good teaching: Advancing mathematics education for ELLs (pp. 195-205). Reston, VA: National Council of Teachers of Mathematics.
de Oliveira, L. C. (2011c). In their shoes: Teachers feel like English language learners through a math simulation. Multicultural Education, 19(1), 59-62.
de Oliveira, L. C., & Cheng, D. (2011). Language and the multisemiotic nature of mathematics. The Reading Matrix, 11(3), 255-268.
Kenney, R., & de Oliveira, L. C. (2015). Building functions from context: A framework for connecting ELLs’ understandings of natural language and symbol sense in algebra. In A. Bright, H. Hansen-Thomas, & L. C. de Oliveira (Eds). The Common Core State Standards in Mathematics and English Language Learners: High School (pp. 57-70). Alexandria, VA: TESOL Press.
Kenney, R., & de Oliveira, L. C. (2015). The role of symbol sense in mathematical semiotic systems for English language learners. Teaching for Excellence and Equity in Mathematics, 6(1), 7-15.
Luciana de Oliveira– Chair and Associate Professor, University of Miami, [email protected]
Presentation
›Demonstrate how teachers can –identify these challenges –focus on developing academic literacy and mathematics content simultaneously
›Based on de Oliveira (2012) and de Oliveira (2016) – a language-based approach to content instruction (LACI)
Framework for Analyzing Word Problems at the Elementary Level
›5 Qs to help teachers –identify the language demands of word problems before they present it to students
–get a better sense of the structure of the word problem and the language demands for ELLs
›simultaneous focus on –the mathematical concepts integrated in the word problem
–aspects of language with which ELLs may have difficulty
Framework for Analyzing Word Problems at the Elementary Level
Guiding Qs to Ask Language Demands to Identify
Tasks for Teachers to Perform
1. What task is the student asked to perform?
Type of questions and their structure – e.g. how many, how much
To analyze the question by identifying what it is asking
2. What relevant information is presented in the word problem?
Overall clause construction – the verbs and who, what, to whom
To break down the clause by finding what information is presented
3. Which mathematical concepts are presented in the information?
Specific clause construction – numerical information presented in different parts of the clause
To connect the mathematical concepts needed by looking for specific numerical information presented in the clause
Guiding Qs to Ask Language Demands to Identify
Tasks for Teachers to Perform
4. What mathematical representations and procedures can students use to solve the problem based on the information presented and the mathematical concepts identified?
Question + overall clause structure + specific clause structure
To connect all previously analyzed pieces to determine a variety of mathematical representations and procedures that can be used to solve the problem
5. What additional language demands exist in this problem?
Language “chunks”: nouns, verbs, prepositional phrases within clauses - not as isolated elementsConnections between clauses to determine how different parts of the word problem are connected.
To identify any aspect of language that seems problematic for ELLs not recognized through the previous guiding questions
Word Problems and Key Language ChallengesWord Problem 1:
• Indiana Statewide Testing for Educational Progress-Plus (ISTEP+) Item Sampler
• 3rd grade (Indiana Department of Education, 2002)• items = examples of the types of problems typically found• academic standards for Grade 2 assessed in Grade 3:
• number sense, computation, algebra and functions, geometry, measurement, and problem solving
• Example: sample item for the problem solving section
Word Problem 1
Denise is buying candy for 3 of her friends. She wants to give
each friend 4 pieces of candy. If each piece of candy costs 5¢,
how much money will Denise spend on candy for her friends?
Clause 1: Denise is buying candy for 3 of her friends.
Clause 2: She wants to give each friend 4 pieces of candy.
Clause 3: If each piece of candy costs 5¢,
Clause 4: how much money will Denise spend on candy for her friends?
How teachers can analyze the language demands of the word problem by focusing on each guiding question:
Guiding Questions
Language Demands of the Word Problem
1. What task is the student asked to perform?
• Look at the WP, find the question mark (?), look closely at what the Q asks students to do
• The -if clause “if each piece of candy costs 5¢”
• The symbol ¢• The construction spend on candy (vs. the
book is on the table)• Human participant: Denise→ the one who will be spending
money; “for her friends” →indicate Denise’s 3
friends
2. What relevant information is presented in the word problem?
• Different important aspects of the text: what is being presented in terms of the
problem (see the following table)
Clauses and Relevant Information Provided in Word Problem 1
Clause Relevant Information ProvidedDenise is buying candy for 3 of her friends.
Who? = DeniseWhat is she doing? = is buyingWhat? = candyFor whom? = 3 of her friends
She wants to give each friend 4 pieces of candy.
Who? = She [Denise]What does she want? = wants to giveTo whom? = each friendWhat? = 4 pieces of candy
If each piece of candy costs 5¢
What? = each piece of candyHow much is each piece of candy? = 5¢
Guiding Questions
Language Demands of the Word Problem
3. Which mathematical concepts are presented in the information?
• For Word Problem 1, our analysis of the question reveals that it is asking about a quantity (how much money)
• The following table helps teachers connect the information and the mathematical concepts presented in the problem (based on Huang & Normandia, 2008)
Information Provided Mathematizing the Problem Situation
Clause 1: Denise is buying candy for 3 of her friends.
Total number of friends = 3
Clause 2: She wants to give each friend 4 pieces of candy.
Number of pieces of candy = 4Each friend = 1Therefore, 4 pieces of candy for 1 friend
Clause 3: If each piece of candy costs 5¢
Price for each (1) piece of candy = 5¢
How teachers can analyze the language demands of the word problem by focusing on each guiding question:
Guiding Questions
Language Demands of the Word Problem
4. What mathematical representations and procedures can be used to solve the problem based on the information presented and the mathematical concepts identified?
• Connect the information, the mathematical concepts presented in the problem, and the mathematical representations and procedures that can be used to solve the problem
• The following table helps teachers draw on the crucial mathematical language and ideas inherent in the construction of the word problem to plan lessons that enhance ELLs’ having access to the problem
Information Provided
Mathematical Concepts
Mathematical Representations and
ProceduresClause 1: Denise is buying candy for 3 of her friends.
Total number of friends = 3
Addition 4 + 4 + 4: 3 groups of 4 because there are 3 friends.4 + 4 + 4 = 12 pieces of candy total that Denise will buyClause 1 RepresentationFriend 1
Friend 2
Friend 3
Clause 2 RepresentationFriend 1
Friend 2
Friend 3
Clause 2: She wants to give each friend 4 pieces of candy.
Number of pieces of candy = 4Each friend = 1Therefore, 4 pieces of candy for 1 friend
12 pieces of candy
1 piece of candy
Information Provided
Mathematical Concepts
Mathematical Representations and
ProceduresClause 3: If each piece of candy costs 5¢
Price for each (1) piece of candy = 5¢
1 piece of candy = 5¢12 pieces of candy total that Denise will buy5 + 5 + 5 + 5 + 5 + 5 + 5 +5 + 5 + 5 + 5 + 5 = 60¢
Clause 3 RepresentationFriend 1
Friend 2
Friend 3
5¢ 5¢ 5¢ 5¢
5¢ 5¢ 5¢ 5¢
5¢ 5¢ 5¢ 5¢
12 groups of 5¢
How teachers can analyze the language demands of the word problem by focusing on each guiding question:
Guiding Questions
Language Demands of the WP
5. What additional language demands exist in this problem?
• Identify language “chunks” such as a combination of nouns, verbs, and prepositional phrases within clauses → relationships of language chunks to other parts of WP
• Reference devices “words that stand for other words in a text” (Schleppegrell & de Oliveira, 2006, p. 263)
For example, the human participant Denise: Denise in Clause 1 She in Clause 2 The noun group 3 of her friends in Clause 1 these participants are picked up in Clause 2
as each friend
these participants are picked up in Clause 4 as her friends
Language-based Approach to Content Instruction LACI
› Refers to a simultaneous focus on language and content
› Content-area instruction: major context for building language and literacy in the education of ELLs
(August & Shanahan, 2006)–Need to fully understand the demands of mathematics
discourse for ELLs–Need to clearly consider the language demands of
mathematics at the elementary school level › discipline-specific academic support in language and literacy development
LACI
›ELLs need to be able to engage with the meanings presented›Content is never separate from the language through which that content is manifested›Description of potential linguistic challenges and framework can help teachers:
–be more proactive in helping ELLs learn the ways language is used to construct mathematical knowledge
–better understand the language demands and address them in their curriculum
“accessibility”
LACI
Accessibility ›making texts more accessible means more than simplifying the language through which content is manifested›ELLs need access to mathematics discourse›A matter of social justice
–If ELLs are not given opportunities to engage and participate in experiences involving the use of appropriate mathematics discourse, they will continue to be at a disadvantage.
Why Use LACI in mathematics?
›Description of the language demands of WP based on guiding questions and framework can help teachers:
–analyze the construction of word problems–make content accessible to ELLs by providing them access to the ways in which knowledge is constructed in mathematics
–not to simplify these word problems but to enhance teachers’ understandings about how mathematical disciplinary discourse is constructed in them
References› de Oliveira, L. C. (under contract). A language-based approach
to content instruction (LACI) for English language learners: Academic language in the content areas. University of Michigan Press.
› de Oliveira, L. C. (2016). A language-based approach to content instruction (LACI) for English language learners: Examples from two elementary teachers. International Multilingual Research Journal, 10(3), 217-231.
› de Oliveira, L. C. (2012). The language demands of word problems for English language learners. In S. Celedón-Pattichis & N. Ramirez (Eds.), Beyond good teaching: Advancing mathematics education for ELLs (pp. 195-205). Reston, VA: National Council of Teachers of Mathematics.
› de Oliveira, L. C., & Cheng, D. (2011). Language and the multisemiotic nature of mathematics. The Reading Matrix, 11(3), 255-268.
Elementary School
›[Mathematics textbook]›Three sisters attended a movie that cost $5 per person. Each sister spent $2 on popcorn. Their mother gave them $30 to spend for all three. How much money was left?
Middle School›Middle School: [Section 3 Sequences and Equivalent Equations. This section explores patterns in mathematics and in nature to help students find patterns and use them to make predictions as a problem-solving strategy]
›At the end of February, Sarah began to save for a $340 mountain bike. At the time she had $173 in her savings account. Her savings increased to $208 in late March and $243 in late April. If her savings pattern continues, when will she be able to buy her bike?
High School Level
›[Section 2-8 Explaining Multiplication and Division Related Facts. Chapter 2 is about using “algebra to explain” by examining algebraic properties, such as distribution and commutativity. This section seems to focus on using commutativity to find “facts.” This problem is the last one in the section before review problems are started]
›Meli went grocery shopping. Her least expensive purchase was a drink. She bought bread which cost twice as much as the drink, salad that was four times as much as the drink, and laundry detergent that was five times as much as the drink. Her bill came to $18. How much did each item cost Meli?
Thank you for your attention and participation!› Feel free to contact us.
– Judith O'Loughlin– Consultant, Language Matters, LLC, [email protected]
– Kate Reynolds– Title III Grant Curriculum Designer, Missouri State University, [email protected]
– Brenda Custodio– ESL Consultant, Ohio State University, [email protected]
– Luciana de Oliveira– Associate Professor, University of Miami, [email protected]
