The Interaction of Language and Math: The

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The Interaction of Language and MathThe
(a x-1) Word Connection

• Lori C. Josephson, M.A.
• 57st Annual International Dyslexia Conference
• November 2006
• lorijosephson_at_adelphia.net

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The Two Basic Issues

• Core Vocabulary Necessary for Concept Teaching
• denominator
• plus
• eight-eighteen-eighty

• Multiple Meaning Vocabulary
• time, times
• round-rounding
• hour, are, our

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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
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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

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Orton-Gillingham Multisensory Learning Triangle
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Mathematical Application of O-G Linkages

• Concrete Representation
• Pictorial Representation
• Abstract Presentation

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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
how many more?
x (ab)
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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

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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

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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

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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

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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

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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

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Operations Addition

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

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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 ones way through the procedure to
  ensure understanding of concept underlying the
  procedure

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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

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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)

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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

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Case Study - Jonathan

• 6th Grader - Gifted Dyslexic, difficulty with
  sequencing, rapid naming, visual memory

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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

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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

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Applications Estimation, Data Interpretation,
Verbal Problems

• Heavy Language Load
• about estimate /es ti mit/ vs. /es tim at/
• 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

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General Recommendations

• DONT 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

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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 9s, trick of 8s, counting
  on, fact families to assist with /- fact
  retrieval
• Teach skip counting (3s Shell Be Comin Round
  the Mountain ? 4s Jingle Bells ? counting by
  1s, 2s, 5s, 10s finger trick for 9s) fact
  families to assist with / fact retrieval

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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
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General Recommendations.

• Use MNEUMONICS!!
• Subtraction with Regrouping
• Bigger Bottom, Better Borrow
• Order of Operations
• Please Excuse My Dear Aunt Sally
• Parentheses, Exponents, Multiplication/Divisio
  n,
• Addition/Subtraction
• Long Division
• Daddy , Mommy, Sister, Brother
• Divide, Multiply, Subtract, Bring Down

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General Recommendations..

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

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Scope Sequence of Mathematics Vocabulary

• Grade 1
• digit fewer same plus
• equals sorting match bar graph
• minus different oclock fact family
• total none shapes number line
• inch more addition dime

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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

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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

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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

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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

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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

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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

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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 wont 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 isnt 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
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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 RIGHT36.

• 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

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One of my very favorite dyslexic individuals
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Case Studies

• Sean
• Kyle
• Jaron
• Kelly