Mind Over Math

1
Good Morning!
Christopher Kaufman, Ph.D. [phone] 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
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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
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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
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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
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DA BRAIN Its two hemispheres and four lobes
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The Hemispheres Fancifully Illustrated . . .
Sequential, Factual Processing
Integrative, Big Picture Processing
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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

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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?
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The Four Lobes
FRONTAL LOBE
PARIETAL LOBE
OCCIPITAL LOBE
TEMPORAL LOBE
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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)

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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

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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
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There is no single math processing center!
Neuromotor Functions
Attention Controls
Working Memory
Spatial Comprehension
Executive Functioning
Memory (LTM)
Language
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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

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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

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How much language is required to solve this?
1013 - 879
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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)

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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

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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

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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

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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 . .)

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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

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Your Turn . . .
Which figures to the right match the ones to the
let?
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A closer look at the frontal lobe
Central Sulcus (or Fisure)
Math strategies and problem-solving directed from
here!
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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
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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 . .
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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
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BREAK TIME!
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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.)
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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)

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The Brains Memory Systems
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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

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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?
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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
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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

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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.

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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?

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Math Anxiety
Mathematics is the supreme judge from its
decisions there is no appeal.  Tobias Dantzig
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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!)
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Research (and common sense) clearly indicates . .
.
As anxiety goes up . .
Working memory Capacity goes down!
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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
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When Brains and Math Collide!
Subtypes of Math Disabilities and Their
Neuropsychological Bases
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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!
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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)

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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)
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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)

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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
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Learning to Remember
December 7, 2010 Augusta Civic Center

• Essential Brain-Based Strategies for Improving
  Students Memory Learning

Christopher Kaufman, Ph.D.
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Implications for Instruction
BRINGING THE NEUROPSYCHOLOGY OF MATH INTO THE
CLASSROOM
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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
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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)

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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

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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