The Interaction of Language and Math: The # = (a [x-1]) = Word Connection

1. The Interaction of Language and Math:The # = (a [x-1]) = Word Connection Lori C. Josephson, M.A. 57st Annual International Dyslexia Conference November 2006 lorijosephson@adelphia.net

2. Core Vocabulary Necessary for Concept Teaching: denominator plus eight-eighteen-eighty Multiple Meaning Vocabulary time, times round-rounding hour, are, our The Two Basic Issues:

3. “Dyslexics have difficulty with language. If mathematics is taught through the medium of language, if children are told what to do and expected to remember a sequence of verbal instructions, then dyslexic children are going to find this hard. We are asking them to rely on an area in which we know they are cognitively weak.”Mary Kibel

5. Necessary Solutions: • Link Concept/Vocabulary Learning to Action: Literally ‘tie’ or ‘link’ the language to visual and kinesthetic images • Create a physical representation of the concept: Students manipulate to solve problems Language runs alongside the manipulating Increases opportunities for non-verbal learning and reduces dependence on language alone

6. Orton-Gillingham Multisensory Learning Triangle

7. Mathematical Application of O-G Linkages: • Concrete Representation • Pictorial Representation • Abstract Presentation

8. Oftentimes, students require considerable ‘overlearning’ of language to ensure that abstract terms and symbols are linked to a concrete base!! regrouping reduce to lowest terms % = per cent x * (ab) how many more?

9. Domains to Consider….. • Concept Formation • Numeration • Geometry • Fractions/Decimals • Operations • Addition/Subtraction/Multiplication/Division • Mental Computation Ability • Applications • Measurement • Time/Money • Estimation/Data Interpretation • Problem Solving

10. Numeration Concept Formation • Counting • Rote Memory & Ordinal Counting • The Confusing ‘Teens’ • ‘ty’ means “10” • Temporal Terminology • Before/After • Sequence • First, second, next, last • Directional Terminology • Down/Up, Rounding Up/Down • Left/Right • Proportional Terminology • Greatest/Least/Greater Than or Equal to • Positive/Negative Integers

11. Geometry Concept Formation • Knowledge of Prepositions • Inside, outside, above, below • Compare/Contrast Terminology • Alike/Different • Similar/Congruent • Parallel • Parallel/Intersect • Heavy Language Component • Many multiple meaning words: a ray vs. array, figure, shape, degree, angle vs. angel, area • Unfamiliar terminology for many students that is difficult to learn without concrete representations

12. Fractions/Decimals Concept Formation • Concrete Representations a MUST!! • Confusing Vocabulary • Tens/Tenths • Hundreds/Hundredths • Heavy Language Component • Least Common Factor • Greatest Common Multiple • Reciprocal • Cross-Product • Multi-step Problem Solving Necessary

14. Case Study - Devon • 8th Grader with Language Learning/Organizational Difficulties • Average Cognitive Ability • Working on Subtracting Fractions with ‘Borrowing’: 8 1/4 6 - 2 2/3 - 1 11/12

16. Case Study – Allison • 5th Grader with Language Learning Issues; very well organized; very well developed memory skills; weaker concept formation • Difficulty realizing that fraction = decimal = per cent without concrete presentation

17. Operations: Addition • Concrete Representation • Symbol Confusion: + = ______ • Vocabulary: • in all, all together • plus • addend, missing addend, sum (vs. some) • trading, regrouping, carrying • total • fact family

18. Operations: Subtraction • Concrete Representation • Inverse of addition • Symbol Confusion: - + ______ = • Directional Confusion • Vocabulary: • minus, take away • how many more? fewer? less? how many are left? • subtrahend, minuend, difference • borrowing, regrouping, trading • rapid retrieval of many linguistic facts • The ‘Zero’ Issue • Avoidance of just ‘crossing out the numbers’ • Positive Use of Language: “Bigger Botttom Better Borrow” • “Talking one’s way through the procedure” to ensure understanding of concept underlying the procedure

19. Operations: Multiplication • Concrete Representation • Concept of repeated addition • “Long” multiplication • Well developed counting skills: skip counting • Symbol Confusion: x * (ab) ____ = • Directional Confusion – especially ‘long’ multiplication • Heavy Vocabulary Load • muliplier, multiplicand, product • array (vs. ‘a ray’) • each, time vs. times • Rapid retrieval of many linguistically based facts

21. Operations: Division • Concrete Representation • Concept of repeated subtraction • “Long” division • Inverse of mutiplication • Symbol Confusion: ÷ ⁄ = • Placement of answer depending on horizontal/vertical presentation • Directional Confusions Abound • Heavy Vocabulary Load • Dividend, divisor, quotient • Rapid retrieval of a great deal of linguistic material • Many multi-step verbal directions • Each, group • Reliance upon language to “talk oneself through the procedures” • Important to have a good grasp on all previously taught numerical operations using whole numbers (+ - x)

23. Operations: Mental Computations • Heavy dependence on working memory (computation) • Heavy dependence on long-term memory (fact retrieval) • Heavy dependence on rapid naming ability • Heavy dependence on linguistic/conceptual sequencing ability

25. Case Study - Jonathan • 6th Grader - Gifted Dyslexic, difficulty with sequencing, rapid naming, visual memory

26. Applications: Measurement • Knowledge of ‘spatial’ vocabulary: • tallest/shortest ---hottest/coldest • shortest/longest ---lightest/heaviest • most/least • Working knowledge of various units of measurement in terms of: • overall organization: metric vs. English system length/weight/liquid measure, etc. • equivalencies: 12 inches = 1 foot • Heavy Vocabulary Load: Multiple Meanings/Abstractions • foot/feet; area; square units = units²; abbreviations for units of measure (foot = ft.; pound = lb.; ounces = oz., etc.) • perimeter vs. area • degree(s) 32º Fahrenheit 0ºCelsius • squared ² vs. cubed ³

28. Applications: Time & Money • Knowledge of ‘temporal’ vocabulary: • first, next, last • today, tomorrow, yesterday, date, seasons (spring, summer, fall/autumn, winter) • second, hour, day, week, month, year, decade, century, millenium • quarter to, quarter after • Knowledge of American currency and equivalencies: • penny, nickel, dime, quarter (!), half-dollar, dollar • Heavy Language Load • ‘Making change’ • Multiple Meanings • ‘Account balance’; ‘checking account’ • ‘Interest’

29. Applications: Estimation, Data Interpretation, Verbal Problems • Heavy Language Load • ‘about’; ‘estimate’ /es ti mit/ vs. /es tim āt/ • most likely; least likely; probable vs. probability • Graphs: bar/picto/picture/pie/line • Key Words for solving verbal problems: • Connecting key words to correct operation • Translating words into images • Solving multi-step problems • Sorting relevant/irrelevant information

30. General Recommendations • DON’T ASSUME ANYTHING!!! Include an assessment of students’ math vocabulary knowledge either at the outset of the school year and/or at the beginning of each new concept chapter • DIRECTLY TEACH THE STUDENTS THE VOCABULARY YOU WOULD LIKE THEM TO KNOW AND USE • Use tried and true multisensory structured language methods to teach students mathematically related vocabulary and symbols

32. General Recommendations…. • Assist students with rapid retrieval of mathematically related vocabulary and the basic ‘facts’: • Make vocabulary/symbol flash cards and conduct a ‘quick drill’ to assist in ‘overlearning’ that is necessary • Teach ‘trick of 9’s’, ‘trick of 8’s’, ‘counting on’, ‘fact families’ to assist with +/- fact retrieval • Teach ‘skip counting’ (3’s She’ll Be Comin’ Round the Mountain ♫; 4’s Jingle Bells ♪; counting by 1’s, 2’s, 5’s, 10’s; finger trick for 9’s); ‘fact families’ to assist with ×/÷ fact retrieval

34. General Recommendations…. • Spend A LOT of time on vocabulary words having multiple meanings/similar sounds!! pattern rounding are 'left'? degrees hour/our/are quarter weight/wait time(s) average date 18/80 season area

35. General Recommendations…. • Use MNEUMONICS!! • Subtraction with Regrouping: “Bigger Bottom, Better Borrow” • Order of Operations “Please Excuse My Dear Aunt Sally” Parentheses, Exponents, Multiplication/Division, Addition/Subtraction • Long Division “Daddy , Mommy, Sister, Brother” Divide, Multiply, Subtract, Bring Down

36. General Recommendations….. • Provide opportunities for imagery/picture drawing • Use of Verbal Rehearsal • Weave/review concepts continuously using diagnostic teaching methods

37. Scope & Sequence of Mathematics Vocabulary Grade 1 digit fewer same plus equals sorting match bar graph minus different o’clock fact family total none shapes number line inch more addition dime

38. Scope & Sequence of Mathematics Vocabulary Grade 2 pattern zero triangle calendar compare quarter subtract cone place value shape rounding half hour fractions estimate regroup months pictograph hundreds problems balance scale

39. Scope & Sequence of Mathematics Vocabulary Grade 3 symmetric forms half dollar quart centimeter cylinder cubic units remainder minuend hexagon Roman Numerals octagon product equivalent number pattern software round down missing factor mixed number below zero

40. Scope & Sequence of Mathematics Vocabulary Grade 4 average volume diameter degrees Celsius perimeter end point graphics denominator hardware monitor per cent congruent figure polygon prism quotient even number vertex parallel end point whole numbers

41. Scope & Sequence of Mathematics Vocabulary Grade 5 probability linear diagonals metric units geometry isosceles numerator infinite set flowchart median formula circumference ratio latitude eliminate decimal point reciprocal probability expanded form negative numbers

42. Scope & Sequence of Mathematics Vocabulary Grade 6 exponent kilogram inverse elapsed time symbols variables trillion cross-product pentagon equation algorithm frequency table trapezoid prediction hectometer unit pricing centimeter checking account scientific notation geometric solids

43. Case Study - Tyler • 7th grader with severe language learning issues; retrieval/processing speed issues; highly motivated; average cognitive ability • Results of mathematics vocabulary assessment revealed many gaps in vocabulary knowledge leading to many gaps in concept development and calculation ability • Currently using Transitional Mathematics vs. Everyday Mathematics

44. “We are going to TAKE 25 FROM 61. WRITE DOWN 61 first (I sometimes wrote the first figure I heard before the second one.) WRITE DOWN 25 UNDERNEATH it. Put the 2 UNDER the 6 and the 5 UNDER the 1. Draw a line UNDERNEATH. Start at the bottom on the RIGHT. Take 5 AWAY FROM 1. It won’t go. Start again. Borrow 10 FROM the 6. (Confusion here because you take smaller numbers from bigger ones, and 10 take away 6 is 4.) ‘Where do I put the 4?’ There isn’t a 4 in the sum. Now pay attention….start again. You are borrowing 10 FROM 60. (Confusion again because that seems to leave a 50 somewhere). You borrow the 10 from the 60 and add it to the 1 to make 11. The you take 5 AWAY FROM 11. That leaves 6. Put the 6 DOWN, UNDER the line BELOW the 5. There is no need to take so long. Take the one you have borrowed AWAY FROM the 6. ‘Which 6?’ Then take 2 AWAY FROM the 6. ‘Which 6?’. Then take 2 AWAY FROM 5. That leaves 3. If you like you can pay back

45. the 10 to the 2 and that makes 3. Then you take 3 AWAY FROM 6, and you get the same answer, 3. Put the 3 DOWN, on the LEFT of the 6. Not that 6, the one in your answer. Read the answer from LEFT to RIGHT—36. J. Street, Dyslexic who eventually taught herself to subtract using her own private method, which involved adding, despite the disapproval of her teacher who said that this method was too confusing!! --J. Street ‘Sequencing and Directional Confusion in Arithmetic, Dyslexia Review, 1976 61 - 25 36

46. One of my very favorite dyslexic individuals……

47. Case Studies: Sean Kyle Jaron Kelly