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Mind Over Math

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Title: Mind Over Math


1
Good Morning!
Christopher Kaufman, Ph.D. (207) 878-1777 e-mail
info_at_kaufmanpsychological.org web
kaufmanpsychological.org
2
Mind Over Math
  • The Neuropsychology of Mathematics and Practical
    Applications for Instruction

3
I never did very well in math - I could never
seem to persuade the teacher that I hadn't meant
my answers literally.  Calvin Trillin
4
Agenda
Morning
Afternoon
  • 830 - Neuroanatomy 101
  • (A Quick Users Guide to the Brain)
  • 900 - The Brain on Math (AKA The
    Neuropsychology of Mathematics)
  • 1030 Break
  • 1045 When Brains and Math Collide! The
    Neuropsychology of Math Disorders (With a Side
    Trip into Math Anxiety)
  • 1130 Lunch

1230 Practical/Implications Strategies for
Classroom and Remedial Instruction 200 Mini-Bre
ak 215 More Strategies 245 Q
A 300 Adjourn
5
Math refusal from an FBA perspective . . .
The student who hides his head under his hood or
exclaims, This is BORING! is usually saying, I
hate this repeated feeling of not being
successful, and I dont ever want to have to feel
it again.
David Berg, Educational Therapist Author of,
Making Math Real
6
Your Turn . .
  1. Choose a kid from your caseload who struggles
    significantly with math.
  2. Take a few moments to complete the first part of
    the Personal Case Study Form

7
Neuroanatomy 101 A Quick Users Guide to the
Brain
8
DA BRAIN Its two hemispheres and four lobes
9
The Hemispheres Fancifully Illustrated . . .
Sequential, Factual Processing
Integrative, Big Picture Processing
10
Left Hemisphere
  • Where spoken and written language are primarily
    processed (greater hemispheric specialization in
    boys)
  • Where language originates (language-based
    thoughts develop in the left hemisphere)
  • Where phonemes, graphemes, grammar, punctuation,
    syntax, and math facts are processed
  • Where routine, overlearned information is
    processed

11
Right Hemisphere
  • Has greater capacity for handling informational
    complexity because of its interregional
    connections
  • Has greater capacity for processing novel
    information
  • Tends to be more dominant for processing
    creative, imaginative, flexible thinking
  • Tends to be more dominant for emotional aspects
    of writing
  • More common source of spatial/visual-motor
    deficits

12
Your Turn . . .
  • Take a moment to consider . .
  • Which elements of math functioning would be more
    likely processed in the left hemisphere?
  • Which elements of math functioning would be more
    likely processed in the right hemisphere?

Why?
13
The Four Lobes
FRONTAL LOBE
PARIETAL LOBE
OCCIPITAL LOBE
TEMPORAL LOBE
14
The Neuropsychology of Math (AKA The Brain on
Math)
15
The Nature of Math
  • Its sequential and cumulative (earlier skills
    continually form the basis for newer skills
    across the grade span)
  • Its conceptual (lots of ideas and
  • themes must be understood and
  • reasoned)
  • Its procedural (lots of rules and algorithms
    must be mastered to calculate perform
    numerical operations
  • Its highly variable from a skill perspective
    (math is a many varied thing!)

16
Arithmetic Skill An Intrinsic Capacity?
  • Research suggests . .
  • Infants demonstrate number sense early in
    development (Sousa, 2005)
  • 8-month olds can reliably distinguish individual
    objects from collections (Chiang and Wynn, 2000)

17
Has math sense been selected for by evolution?
(Sousa, 2004)
  • Our most ancient ancestors were best able to pass
    on their genes if . . .
  • They could quickly determine the number of
    predators in a pack
  • They could determine how much to plant to feed
    the clan

18
Math Ability the Neurodevelopmental Functions
(Portions adapted from the work of Mel Levine)
Temporal-Sequential Following sequences and
multiple steps (Levine)
Spatial-Motor Visualizing problems/procedures, com
prehending angles (and other elements of
geometry), creating charts, graphs, etc., and
maintaining sufficient grapho-motor accuracy to
solve problems correctly on paper
Memory Recalling facts, procedures, and rules,
recognizing patterns, and problem solving
Attention Maintaining sufficient cognitive energy
and attention on work
MATH
Executive Functioning Planning, organizing,
monitoring the quality of work (also
determining what is/is not important for
problem solving)
Language Processing written language and spoken
information in directions, problems and
understanding/recalling technical math vocabulary
19
There is no single math processing center!
Neuromotor Functions
Attention Controls
Working Memory
Spatial Comprehension
Executive Functioning
Memory (LTM)
Language
20
Left vs. Right Brain Math Skill
  • In general terms . .
  • Left Hemisphere More responsible for processing
    of arithmetic (tasked to determine exact answers
    using language processes)
  • Right Hemisphere Responsible for estimating
    approximate magnitude using visual-spatial
    reasoning skills

21
Verbal Functioning and Math Ability
  • Related to the language centers of the temporal
    lobe and posterior frontal lobe
  • The ability to store and fluidly retrieval digit
    names and math facts is mediated by the temporal
    lobe
  • Frontal and temporal language systems are used
    for exact computations because we tend to talk
    our way through calculations

22
How much language is required to solve this?
1013 - 879
23
Side Bar Issue Vocabulary Deficits and Math
  • Math is replete with technical terms, phrases,
    and concepts (i.e., sum, factor,
    hypotenuse, perimeter, remainder)
  • Math also requires the following of often
    detailed verbal instructions
  • Students with limited language comprehension
    skills can struggle greatly with math, even if
    they have no difficulty recalling math facts and
    the specific terms related to them!

24
Visual/Verbal Connections Related to Math
Functions
  • Also temporal lobe areas related to language
    functioning
  • Occipital-Temporal Convergence links the visual
    element of digits to their verbal counterparts
  • This area allows for the attaching of fixed
    symbols to numerical constructs (Feifer Defina,
    2005)

25
Visual-Spatial Functioning and Mathematics
  • Were talking primarily about processing in the
    parietal lobe (site of spatial processing) and
    occipital lobe (the site of visual processing)
  • Left and right hemispheres are involved, with the
    left being associated with arithmetic/sequential/f
    actual processing and the right related to
    simultaneous/spatial/holistic processing

26
Left Parietal Lobe Center of Arithmetic
Processing?
Area associated with arithmetic processing
15 bigger In Einsteins Brain!
27
Side Bar IssueEinsteins Brain
  • Actually weighed a bit less than the average for
    brains of its time/age
  • But, had greater neuronal density than most
    brains and was about 15 wider in the parietal
    lobe region (and had fewer sulci in this area)
  • Thus, he had somewhat greater brain capacity in
    the areas associated with arithmetic and spatial
    reasoning ability

28
More on Right Hemisphere Functioning and Math
Skills
A (not the) visual-spatial processing center
(left parietal also processes visual-spatial
information) Approximations of magnitude are
largely made in the right parietal lobe Mental
rotation and similar spatial reasoning tasks tend
to be processed in the right hemisphere Math
concepts are reasoned in the right hemisphere
(the brains big picture, integration
center) Novel stimuli are processed in the
right hemisphere

29
Many aspects of math are visual-spatial in nature
  • Visualization and construction of numbers
  • Visualizing of the internal number line
  • Visualizing of word problems (easier to determine
    the needed operations if one can picture the
    nature of the problem)
  • Geometry (duh . .)

30
Are boys intrinsically better at math than girls?
  • NO (pure and simple)
  • Boys do have better mental rotation skills
  • This may give them greater confidence in
    attacking certain kinds of math problems (Feifer
    DeFina, 2005)
  • Overall, though, there is growing consensus in
    the field that any advantage boys have over girls
    in math is a product of cultural/societal
    convention

31
Your Turn . . .
Which figures to the right match the ones to the
let?
32
A closer look at the frontal lobe
Central Sulcus (or Fisure)
Math strategies and problem-solving directed from
here!
33
Frontal Lobe Specifics (Adapted from Hale
Fiorello, 2004)
Motor Cortex
Dorsolateral Prefrontal Cortex Planning Strategi
zing Sustained Attention Flexibility Self-Monitori
ng ------------------------------- Orbital
Prefrontal Impulse Control (behavioral
inhibition) Emotional Modulation
34
Executive Skill and Math
Maths Changing Face (Its new again)
And in with constructionist math curricula that
emphasize discovery learning and the
self-construction of math know-how
Out with the explicit teaching of facts and
standard algorithms . .
35
Executive dysfunction impacts
  • Self-directed learning
  • Discovery-based learning
  • Self-initiated strategy application
  • Collaborative learning

This is why so many kids with EFD have struggled
with constructionist math curriculums
36
BREAK TIME!
37
Impact of Executive Dysfunction on Math
Working memory problems lead to poorly executed
word problems
Impulse control problems lead to careless
errors (e.g., misread signs)
?
Organizational/planning deficits lead to work
poorly organized on the the page (or work
not shown)
Attention problems lead to other careless errors
(i.e., Forgetting to regroup, etc.)
38
The Three Primary Levels of Memory
  • Sensory Memory (STM) The briefest of memories
    information is held for a few seconds before
    being discarded
  • Working Memory (WM) The ability to hold
    several facts or thoughts in memory temporarily
    while solving a problem or task in a sense,
    its STM put to work.
  • Long-Term Memory (LTM) Information and
    experiences stored in the brain over longer
    periods of time (hours to forever)

39
The Brains Memory Systems
40
Working Memory Some kids have got leaky buckets
  • Levine Some kids are blessed with large, leak
    proof, working memories
  • Others are born with small WMs that leak out
    info before it can be processed

41
Your Turn . . .
A Working Memory Brain Teaser!
I am a small parasite. Add one letter and I am a
thin piece of wood. Change one letter and I am a
vertical heap. Change another letter and I am a
roughly built hut. Change one final letter and I
am a large fish. What was I and what did I become?
42
How Large is the Childs Working Memory Bucket?
WM capacity tends to predict students ability to
direct and monitor cognition.
Case 3 Frankie Forgetaboutit
Case 1 Rachel Recallsitall
Case 2 Nicky Normal
algorithm
fact
facts
algorithm
directions
fact
directions
algorithm
42
43
Working memory A fundamental element of math
functioning
  • Mental math (classic measure of working memory
    skill)
  • Word Problems
  • Recalling the elements of algorithms and
    procedures while calculating on paper
  • Interpreting and constructing charts/graphs
  • So much of learning and academic performance
    requires the manipulation of material held in the
    minds temporary storage faculties

44
The majority of studies on math disabilities
suggest that many children with a math disability
have memory deficits (Swanson 2006) Memory
deficits affect mathematical performance in
several ways
  • Performance on simple arithmetic depends on
    speedy and efficient retrieval from long-term
    memory.
  • Temporary storage of numbers when attempting to
    find the answer to a mathematical problem is
    crucial. If the ability to use working memory
    resources is compromised, then problem solving is
    extremely difficult.
  • Poor recall of facts leads to difficulties
    executing calculation procedures and immature
    problem-solving strategies.
  • Research also shows that math disabilities are
    frequently co-morbid with reading disabilities
    (Swanson, 2006). Students with co-occurring math
    and reading disabilities fall further behind in
    math achievement than those with only a math
    disability. However, research shows that the most
    common deficit among all students with a math
    disability, with or without a co-occurring
    reading disability, is their difficulty in
    performing on working memory tasks.

45
Lets Look at a Classic Word Problem . .
  • Sharon has finished an out-of-town business
    meeting. She is leaving Chicago at 300 on a
    two-hour flight to Boston. Her husband, Tom,
    lives in Maine, 150 miles from Boston. Its his
    job to pick up Sharon at the airport as soon as
    the flight lands. If Toms average speed while
    driving is 60 miles per hour, at what time (EST)
    must he leave his house to arrive at the airport
    on time?

46
Math Anxiety
Mathematics is the supreme judge from its
decisions there is no appeal.  Tobias Dantzig
47
Math Anxiety on a Brain Level (or, When the
amygdala comes along for the ride)
Bottom line Its crucial to keep kids from
getting overly anxious during math instruction
(or they may always be anxious during math
instruction!)
48
Research (and common sense) clearly indicates . .
.
As anxiety goes up . .
Working memory Capacity goes down!
49
The best math anxiety limerick ever?
There was a young man from Trinity,Who solved
the square root of infinity.While counting the
digits, He was seized by the fidgets,Dropped
science, and took up divinity. Author Unknown
50
When Brains and Math Collide!
Subtypes of Math Disabilities and Their
Neuropsychological Bases
51
Can you say, Dyscalculia? Sure you can!!
Occur as often As RDs!!
Developmental Dyscalculia defined DD is a
structural disorder of mathematical abilities
which has its origin in a genetic code or
congenital disorder of those parts of the brain
that are the direct anatomico-physiological
substrate of the maturation of the mathematical
abilities adequate to age, without a
simultaneous disorder of general mental functions
(Kosc, 1974, as cited by Rourke et al., 2005)
Huh?!
Said more simply! Dyscalculia refers to any
brain-based math disability!
52
Epidemiology of Math Disabilities
  • Occur in about 1 - 6 of the population (Rourke,
    et al., 1997 DSM-IV-TR) Geary (2004) says 5
    8. A recent Mayo Clinic study suggested the
    incidence in the general population could be as
    high as 14 (depending upon which definition of
    math LD is used . .)
  • Like all LDs, Math LD occurs more often in boys
    than girls
  • MDs definitely run in families (kids with
    parents/siblings with MD are 10 times more likely
    to be identified with an MD than kids in the
    general population)
  • Important take home point Math disabilities
    (MDs) occur just as often as reading
    disabilities (RDs) this has big implications
    for the RTI process!!

53
Types of Math Disability (MD)
  1. Verbal/Semantic Memory (language based,
    substantial co-occurrence with reading
    disabilties)
  2. Procedural (AKA anarithmetria substantial
    overlap with executive functioning and memory
    deficits)
  3. Visual-Spatial (substantial overlap with NLD)

54
Semantic/Language-Based MDs
  • Characterized by poor number-symbol association
    and slow retrieval of math facts (Hale
    Fiorello, 2004)
  • Commonly co-occur with language and reading
    disorders (Geary, 2004)
  • Are thought to relate to deficits in the areas of
    phonological processing and rapid
    retrieval/processing of facts from long-term
    memory
  • Math reasoning skills (i.e., number sense and
    ability to detect size/magnitude) are generally
    preserved (Feifer DeFina, 2005)

55
Error Patterns Associated with the
Verbal/Semantic Subtype
  • These kids tend to struggle recalling and
    processing at the what (as opposed to the
    how) level.
  • Theyll forget (or will have great trouble
    learning) the names of numbers, how to make
    numbers, the names/processes of signs (i.e.,might
    often confuse X with ), and multiplication
    facts
  • Theyll make counting errors and other errors
    related to the exact nature of math (always
    have to rediscover the answer to problems such
    as 8 4 and 7 X 3).
  • May arrive at the right answer, but have trouble
    explaining how they got there.

56
The Procedural Subtype of MD(Feifer DeFina,
2005 Hale Fiorello, 2004)
  • Disrupts the ability to use strategic algorithms
    when attempting to solve math problems
  • That is, kids with this subtype of MD tend to
    struggle with the syntax of arithmetic, and have
    difficulty recalling the sequence of steps
    necessary to perform numerical operations (leads
    to lots of calculation errors!)
  • Often seen in conjunction with ADHD/EFD subtypes,
    because the core deficit is thought to relate to
    a frontal lobe/executive functioning weakness
    (particularly working memory difficulties and
    slow processing speed)
  • These kids tend to rely fairly heavily on
    immature counting strategies (counting on fingers
    and through the use of hash marks on paper)

57
Working Memory and the Procedural Type of MD
  • How much working capacity and sequential
    processing skill is needed to solve the
    following?
  • An elementary school has 24 students in each
    classroom. If there are 504 students in the whole
    school, how many classrooms are there?

I forget how you do . . .
58
(No Transcript)
59
Error Patterns Associated With the Procedural
Subtype of MD
  • Like kids with verbal/semantic MD, kids with the
    procedural subtype make errors related to
    exactness (as opposed to estimating magnitude
    or comprehending concepts)
  • Errors are not related to the what, but are
    instead related to the how (e.g., How do you
    subtract 17 from 32? How do you calculate the
    radius of a circle?)
  • These kids know their facts (e.g., might easily
    recall addition multiplication facts), but
    struggle greatly with recalling the
    steps/procedures involved in subtraction with
    regrouping and multiple digit multiplication.
  • Often do better on quizzes of isolated basic
    facts, but struggle with retrieval of the same
    facts to solve word problems or longer
    computations

60
The Visual-Spatial Subtype of MD
  • Heavily researched by Byron Rourke (leading
    researcher in the field of nonverbal learning
    disabilities NLD)
  • This subtype relates to deficits in the areas of
    visual-spatial organization, reasoning, and
    integration
  • Difficulties with novel problem solving generally
    compound math reasoning struggles
  • At a brain level, the deficits are thought to
    relate to processing deficiencies in the right
    (and, to some extent, left) parietal lobe (were
    visual-spatial-holistic processing occurs)

61
Error Patterns Associated withthe Visual-Spatial
Subtype
  • Fine-motor problems incorrectly formed/poorly
    aligned numbers
  • Strong fact acquisition, but struggles with
    comprehending concepts
  • New concepts and procedures are acquired slowly
    and with struggle (must first understand visual
    concepts on a very concrete level before they can
    grasp the abstraction)
  • May invert numbers, or have difficulty grouping
    numbers accurately into columns
  • Tend to have marked difficulties grasping the
    visual form of mathematical concepts (i.e., may
    be better able to describe a parallelogram than
    to draw one)
  • Often have difficulty seeing/grasping big
    picture ideas (get stuck on details and struggle
    with seeing the forest for the trees

62
Key Facts Related to Math Disabilities Across the
Grade Span
  • The verbal/semantic subtype is usually most
    obvious in the early primary grades, given the
    emphasis on math fact acquisition (many kids with
    NLD do fine in math through third grade or so).
  • The procedural and visual/spatial subtypes become
    more obvious as algorithmic and conceptual
    complexity increases!
  • Bottom line As procedural and conceptual
    complexity increase, the demands on the frontal
    and parietal lobes increase (Hale Fiorello,
    2004)

63
Student Profiling to Inform Instruction and
Learning Plan
Students Name _______________
Neuromotor
Attention/EF
Language
Memory
Emotional
Neuro Profile
Math Fact Skill
Math Concepts
Problem Solving
Algorithm Skill
Academic Profile
Strategies
64
LUNCH TIME!!!
65
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Operators Standing By!
Shameless self-promotion slide!!!!
Brookes Publishing Company
34.95
66
Learning to Remember
December 7, 2010 Augusta Civic Center
  • Essential Brain-Based Strategies for Improving
    Students Memory Learning

Christopher Kaufman, Ph.D.
67
Implications for Instruction
BRINGING THE NEUROPSYCHOLOGY OF MATH INTO THE
CLASSROOM
68
Firstly The state of affairs . . .
(An empty glass)
There has been relatively little in the way of
high quality math instruction research! Reading
studies outnumber math studies at a ratio of 61
69
Conceptual and Procedural Knowledge
Conceptual knowledge has a greater influence on
procedural knowledge than the reverse
Strong
Conceptual Knowledge
Procedural Knowledge
Weak
Sousa, 2004
70
Key Research Finding
  • Adults often underestimate the time it takes a
    child to use a newly learned mathematical
    strategy consistently (Shrager Siegler, 1998,
    as cited by Gersten et al., 2005)

71
Step One Understand a Childs Specific Problem(s)
  • Look for deviations for normal development (re
    the acquisition of counting and early arithmetic
    skills)
  • Look for error patterns that are suggestive of
    weakness in the semantic/memory,
    procedural/algorithmic, and visual-spatial domains

72
An Important First Intervention Step Look for
Error Patterns (Hale Fiorello, 2004, p. 211)
  • Math fact error (FE) Child has not learned math
    fact, or does not automatically retrieve it from
    LTM (Teacher Michael, whats 4 X 4? Michael
    Um, 44?)
  • Operand error (OE) Child performs one operation
    instead of another (e.g., 6 3 for a 6 X 3
    problem)
  • Algorithm error (AE) Child performs steps out
    of sequence, or follows idiosyncratic algorithm
    (i.e., attempts to subtract larger from smaller
    number)
  • Place value error (PE) Child carries out the
    steps in order, but makes a place value error
    (common among kids with executive functioning and
    visual/spatial deficits)
  • Regrouping errors (RE) Child regroups when not
    required, forgets to subtract from regrouped
    column during subtraction, or adds regrouped
    number before multiplication

73
Example of an Algorithm Error (revealed via a
think aloud examination)(Hale Fiorello,
2004, p. 211)
  • 64
  • 13

First I look to see if its addition or
subtraction. Okay, its addition, so you always
go top to bottom and left to right. So I add 6
4, and that equals 10, and then 1 3 equals 4.
And then I add them together, top to bottom, and
so 10 4 equals 14.
14
74
A Great Calvin and Hobbs Example
75
John has a problem with multiplication
  • What kind of problem? How broad is the scope?
  • Kids who cant (despite adequate instruction and
    chances to practice) seem to recall the product
    of 8 X 7 have a fact recall difficulty (LTM
    deficiency temporal lobe)
  • Kids who have no difficulty recalling the product
    of 8 X 7, but cant solve 16 X 7 on paper may
    have an algorithm process difficulty (working
    memory or arithmetic reasoning deficiency
    frontal lobe or parietal lobe)

76
THE CORE STRATEGIES
  1. Emphasize the development of an internal number
    line (in grades K and 1) to build number sense
  2. Teach the concept and the algorithm (not just the
    algorithm in isolation), and keep teaching the
    algorithm until mastery
  3. Distributed practice works better than massed
    practice (smaller doses of practice over time is
    better than a lot all at once)
  4. Emphasize the verbalization of strategies/algorith
    ms as kids problem solve (and after theyve
    arrived at a solution)
  5. Build automaticity of fact retrieval
  6. Minimize demands on working memory/simultaneous
    processing (encourage kids to download info from
    working memory to paper by encouraging thinking
    on paper)
  7. Enhance the explicit structure of math problems
    (using multiple colors, graph paper, boxing
    techniques, etc.)
  8. Body-involved, kinesthetic learning is good!

77
Strategies to Build Number Sense
78
Meet Caleb
Calebs a feisty little guy (to quote his
mother) whos just entered kindergarten. He wore
sandals to school, but took them off somewhere
in the classroom and now cant seem to find
them. Hes knows his primary colors and all
basic shapes, but his letter/number ID and
formation skills seem low. He can count to 20 in
a rote manner, but seems unsure as to what the
numbers mean (e.g., yesterday said that 4 was
more than 6). Also, his ability to count
with 11 correspondence is still shaky (can only
do it with direct adult support). He gets
frustrated very easily in task contexts and is
apt to cry and throw things when stressed.
79
What, exactly, is number sense?
  • Definitions abound in the literature . . .
  • Berch, 1998 Number sense is an emerging
    construct that refers to a childs fluidity and
    flexibility with numbers, sense of what numbers
    mean, ability to perform mental mathematics, and
    ability (in real life contexts) to look at the
    world and make magnitude comparisons.

80
Number Sense and Environmental Factors
  • Most kids acquire number sense informally through
    interactions with parents and sibs before they
    enter kindergarten
  • Well-replicated research finding Kids of
    moderate to high SES enter kindergarten with much
    greater number sense than kids of low SES status
  • Griffin (1994) found that 96 of high SES kids
    knew the correct answer to the question, Which
    is bigger, 5 or 4? entering K. Only 18 of low
    SES kids could answer the question correctly
    (this study controlled for IQ level)
  • Number sense skill in K and 1st grade is
    critical, as it leads to automatic use/retrieval
    of math info and is necessary to the solution of
    even the most basic arithmetic problems (Gersten,
    2001)

81
Building Number Sense
  • Its critical that parents, during the preschool
    years, really talk to kids about numbers and
    amounts and magnitude (Lets count these stairs
    as we climb them!)
  • Head Start and other preschool programs for low
    SES kids should really push number concept games
    and related activities (just as they should push
    phonological awareness activities as a precursor
    reading skill)
  • During the K and 1st grade years, its essential
    for children to develop a mental (internalized)
    number line and to play with this line in
    various ways
  • Without strong number sense, kids often are
    unable to determine when a numeric response makes
    no sense (i.e., 5 12 512)

82

10

9

8

7

6

5

4

3

2

1
83
Building Number Sense Some Concrete Strategies
(Bley Thorton, 2001)
  • More or less than 10?
  • 84 Is this more than 10 or less than 10? (kids
    should check with manipulatives and number line
    work)
  • Whats 55? Is 5 9 more or less than that? How
    do you know?
  • Variations for older grades
  • More or less than ½? Ask students to circle in
    green all fractions on a sheet that are more than
    ½.
  • Closer to 50 or 100? Have students circle in
    green those numbers that are closer to 50 than
    100, using both visual and mental number lines
  • Over or under? Provide repeated instance in which
    students are asked to decide which of two given
    estimates is better and explain their reasoning.
  • E.g., 652 298 ? A. Over 400 B. Under 400

84
Building Number Sense More Strategies (Bley
Thorton, 2001)
  • 2. What cant it be? Provide computational
    problems and a choice of two (or more) possible
    answers. Ask the children to predict which of the
    choices couldnt be possible and to state why.
  • Example A. 28 37 65 B. 28 37 515
  • Verbalized response The answer cant be 515.
    Its not even 100, because 50 50 is 100, and
    both numbers are less than 50.
  • 3. Whats closest? Ask the children to predict
    which of the answer choices is closest to the
    exact answer? How do you know?
  • Example 92 49 ? A. 28 B. 48 C. 88
  • Its B. The problem is sort like 100 50, and
    the answer to that is 50, and so 48 is closest.

85
Digi-Blocks
86
Strategies Targeting Semantic/Memory Weakness
87
Meet Katie . . .
Katie is a generally shy and sweet-natured 7th
grader with a longstanding speech/language
impairment. Although her once profound
articulation difficulties have abated in response
to years of SL therapy, she continues to have a
hard time with receptive language tasks of all
sorts. Shes of basically average intelligence,
but has gotten numerous accommodations over the
years related to literacy tasks. Although math
computation had been her area of relative
strength, shes had a much harder time in middle
school now that the technical math vocabulary
demands have really increased. Her father reports
that she now hates math and says things like,
If theyd just show me what to do and make it
clear, I could do it I wish theyd just show me
what they mean!
88
When language comprehension is the problem
Carefully teach math vocabulary, with all the
possible forms related to the different
operations posted clearly in the classroom
Addition Sum Add Plus Combine Increased by More than Total Subtraction Take away Remaining Less than Fewer than Reduced by Difference of Multiplication Product Multiplied by Times Of 3 X 3 3(3) Division Quotient Per A (as in gas is 3 a gallon) Percent (divide by 100)
89
Operations Language Chart in a Simpler Form
Add Plus Subtract Take
Away Minus Multiply
Times X Divide Divided By
Per
90
When language comprehension is the problem
  • Link language to the concrete (have a clear
    visual and kinesthetic examples of all concepts
    readily available)
  • Teach math facts and basic vocabulary in a
    variety of ways (brains love multi-modal
    instruction!)
  • Use lots of manipulatives to clearly demonstrate
    taking away, total, divisor.
  • Make liberal use of kinesthetic/multisensory
    demonstrations
  • Have kids put math vocabulary into their own
    words (and then check for the accuracy of these
    words!)

91
Illustrating the Pythagorean Theorem
c
a
Teacher John, can you remind us what an
hypotenuse is? John Um, nope I havent got a
clue . . . Teacher John, weve spent the last
two days talking about this stuff. John So?! I
dont remember, All right?! Whats your
problem?! Geez!!
b
13
5
12
92
Other language targeted strategies
  • Trying to always present a concrete visual (draw
    it out) whenever you present the oral/verbal
    form of math concept (kids who have significant
    language deficiencies should have quick cheat
    sheets available)
  • Keep verbal instructions short and to the point
  • Having kids read instructions into a tape
    recorder and then play them back

93
When factual (declarative) memory is the problem
  • Ensure that the child clearly grasps the concept
    (i.e., that 3 X 4 mean 3 four times)
  • If the child doesnt grasp the concept, then
    teach the concept in multiple ways until he does
    (kids grasp/recall math facts much better when
    they get the concepts behind them)
  • Drills (i.e., flashcards) really work (kids
    retain rote information best when its
    acquired/practice right before sleep)
  • Fact family sorts (e.g,. Sorting flash cards by
    into families)
  • Use games (e.g,. Multiplication War - see
    supplemental handout)
  • Graph progress with the kid (kids often love to
    see their improvement, and the graphing, by
    itself, is a worthwhile math activity)

94
Three Kinds of Math Facts
Autofacts Math facts a student knows
automatically Stratofacts Math facts a student
can figure out using an an idiosyncratic strategy
(i.e,. counting on fingers and using
hashmarks) Clueless Facts Math facts a
student cannot recall or access at all
Gimme the facts, Madam, just the facts . .
Meltzer et al., 2006
95
Terrific Tens Strategy
9
1
2
3
4
5
6
7
8
9
8
7
6
5
4
3
2
1

10
10
10
10
10
10
10
10
10
Meltzer et al., 2006
96
And then theres good olTouch Math
Developers and its proponents claim that it
bridges manipulation and memorization Also
often called a mental manipulative
technique Multi-sensory, in that kids
simultaneously see, say, hear, and (most
importantly) touch numbers As they learn to count
and perform an array Of computational
algorithms Published by Innovative Learning
Concepts Curriculum now extends into secondary
grades
97
Multiplication Fact Strategies
0 Rule 0 times any number is 0 1s Rule 1
times any number is the number itself 2 Rule
Counting by twos 5s Rule The answer must end
in a 5 or 0 (e.g., 35 or 60) 10s Rule The
answer must end in a 0 (10, 40, 80, etc.) 9s
Rule Two-hands counting rule
2 hands Rule when it comes To solving the
tricky 9s!
Meltzer et al., 2006
98
A key developmental asset in teaching kids
division and division facts . . .
Greed (balanced by an insistence on fairness)
How many do we each get?
99
Strategies Targeting Executive Functioning
(Procedural/Algorithmic) Weakness
100
Meet Andrew . .
Andrew, a fourth grader, knows his multiplication
and division facts cold, but has had gobs of
difficulty getting double/multiple digit
multiplication and has had even more difficulty
performing even the most basic aspects of long
division (to quote his teacher Hes just so all
over the place with it!). Although Andrew is a
reasonably well-motivated youngster whos
attended some extra help sessions with his
teacher (and will seemingly get the
multiplication and division algorithms in these
sessions), he seemingly forgets the procedures
by the time he gets home or to school the next
day (Mom Its like Im always at square one
with him on this stuff). Completing assignments
of all kinds is also a big issue for this kid.
101
The most important thing to remember in helping
ADHD (EFD) kids with math
Its all about . . . Diminishing demands on
working memory
102
Mastery of algorithms is important in the end,
but . .
Go slowly, in a very stepwise manner, and
scaffold, scaffold, scaffold!!
Download as much as possible into the childs
instructional environment, with emphasis given
to presentation of algorithm steps in easy to
follow formats
103
A key distinction Factual Memory vs. Procedural
Memory
  • Factual memory . .
  • Refers to an individuals ability to recall
    discrete bits/units of information
  • (e.g,.7 X 7 49, the capital of France is Paris,
    my mothers middle name is Dorothy, sh makes
    the /sh/ sound)
  • Working memory demand
  • Fairly minimal
  • Procedural memory
  • Refers to an individuals ability to remembers
    processes that is, procedural steps
  • e.g., How to bisect an angle, how to swing a golf
    club, how to bake blueberry muffins, how to
    divide 495 by 15
  • Working memory demand
  • Moderate to marked, depending upon the process
    being recalled

104
Helping EFD (ADHD) Kids with Math First Steps
  • To the extent possible, avoid multiple step
    directions (and good luck with that . . .)
  • Have the kids do one thing (and only thing) at a
    time (e.g., Lets just first circle all the
    signs on the page or lets just highlight the
    key words in this word problem)
  • Mel Levine Break algorithms down into their most
    basic sub-steps and carefully, slowly teach each
    sub-step.

105
Thus, in teaching two digit by one digit
multiplication (47 X 6)
  • First ensure the childs single digit
    multiplication facts are solid (or that he is at
    least facile in the use of the chart/grid)
  • Second, achieve mastery of single by double digit
    multiplication without regrouping (24 X 2) (will
    likely need lots of massed practice at this
    stage)
  • Third, introduce the concept of carrying in
    double digit multiplication, but do so in a
    manner that makes use of the parts of the times
    tables a kid has mastered (e.g., 24 X 5) (again,
    lots of massed practice here)
  • Fourth, bring in more challenging multiplication
    elements from the higher, scarier end of the
    times table (e.g., 87 X 9)
  • Than move, after mastery, by adding a third digit
    to the top number, and then a fourth, always
    building in plenty of time for massed practice,
    and distributed practice in the form of reviews
    of earlier, easier stuff.

106
Helpful Strategies to Aid Algorithm Acquisition
and Practice
  • Graph paper rocks!
  • Box templates are even better
  • Box templates that include written reminders are
    even better
  • Box templates that include written reminders and
    include color coordination are even better

107
A good multiple digit multiplication box
template
X

(Adapted from Bley Thorton, 2001)
108
A better multiple digit multiplication box
template
3
2
7
2
3
X
6
4
4
1
2
1
9
6

3
2
1
3
6
(Adapted from Bley Thorton, 2001)
109
Long Division Algorithm Box Template
9
4
0
R 5
7
8
4
3
2
8

8
6
3
6

5
3
4
4
4
4

X
X
64
110
Pneumonics/Heuristics Excellent Ways to Help EFD
Kids Learn and Retain Arithmetic Algorithms
Does McDonalds Sell Burgers Done Rare?
  • Divide
  • Multiply
  • Subtract
  • Bring Down 
  • Repeat (if necessary)

111
Improving Error Checking
P.O.U.N.C.E P Change to a different color pen
or pencil to change your mindset from that of a
student to a teacher O Check Operations (Order
right?) U Underline the question (in a word
problem) or the directions. Did you check the
question and follow the directions? N Check
the numbers. Did you copy them down correctly. In
the right order? Columns straight? C Check you
calculations. Check for the types of calculation
errors you tend to make. E Does your answer
agree with your estimate? Does your answer make
sense?
  • Top Three Hits
  • The 3 most common errors
  • a kid exhibits in math
  • Example
  • Stevens Top 3 Hits
  • Misreading directions
  • Misreading signs
  • Arriving at errors that cant possibly make sense.

112
Strategies Targeting Visual/Spatial Weakness
113
For Kids with NLD Emphasize the Verbal
  • Kids with pronounced visuo-spatial
    comprehension/integration deficits often struggle
    with forming in LTM visual images of objects and
    particularly struggle with visual representations
    of concepts (i.e., an isosceles triangle)
  • Emphasize the verbal (simple, direct, concrete)
    over the visual whenever possible
  • The goal for these students is to construct a
    strong verbal model for quantities and their
    relationships in place of the visual-spatial
    mental representation that most people develop.
  • Descriptive verbalizations also need to become
    firmly established in regard to when to apply
    math procedures and how to carry out the steps of
    written computation.
  • Complex visuals can really freak out kids with
    visual/spatial weakness (avoid busy graphs, maps,
    and charts)

114
Other Strategies Targeting Visual-Spatial Weakness
  • Fewer items on a page
  • Avoid flashcards (too visual better to do rote
    learning via auditory exercises e.g., via
    rhymes)
  • Use blocks to isolate problems on the page (see
    next slide)
  • Emphasize the use of concrete manipulatives in
    the teaching of abstract concepts (being able
    pick up, feel, and talk about manipulatives helps
    these kids)
  • Encourage these kids to think on paper (help
    them draw very simple pictures stick figures --
    to represent what is going on in a math problem
    (Levine)
  • Kinesthetic learning experiences may be
    particularly helpful for this population,
    providing clear verbal explanations accompany the
    demonstrations

115
Addition (plus) Do these first
Subtraction (minus) Do these next
47 56
88 -45
83 31
45 -24
29 93
62 -39
68 55
96 -48
116
Division Cards A Great Device for NLD Kids
Problem 5 255
Question 1 Is there a number which can be
multiplied by 5, and be equal to or less than
2? Answer No, and so zero is placed above the 2
and the card is shifted to the right to get a
bigger number. Question 2 Is there a number
which can be multiplied by 5, and be equal to or
less than 25? Answer Yes, and the number is 5,
so a 5 is placed above the dividend. Etc.
05
5 2 5
Johns Division Card
117
MAKING THE ABSTRACT CONCRETE
A) The problem Whats 5/8 of 16?
___ ___ ___ ___ ___ ___ ___ ___
B) Concrete illustration of 5/8
C) Concrete illustration of 5/8 of 16
___ ___ ___ ___ ___ ___ ___ ___








D) Answer is 10
(Adapted from Bley Thorton, 2001)
118
Buy Out A great technique for kids who are
motivationally challenged
89 X 64
56 X 13
34 X 45
Operates from the perspective That few things are
as motivating As the chance to get out of
work Thus, kids are motivated to work By the
opportunity to work their way Out of
work E.g. For every two problems you do, you
get to cross out one!
83 X 83
76 X 56
92 X 35
27 X 59
78 X 64
69 X 31
39 X 37
90 X 90
71 X 82
119
Two Effective (Evidence-Based) Remedial Programs
120
Case Studies/Student Profiling
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