Good Morning!
Christopher Kaufman, Ph.D.
(207) 878-1777
e-mail: info@kaufmanpsychological.org
web: kaufmanpsychological.org
1
Mind Over Math
The Neuropsychology of
Mathematics and Practical
Applications for Instruction
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
Agenda
Morning
Afternoon
8:30 -
Neuroanatomy 101
(A Quick User’s Guide to the
Brain)
12:30
Practical/Implications
Strategies for Classroom
and Remedial Instruction
9:00 -
The Brain on Math (AKA: The
Neuropsychology of
Mathematics)
2:00
Mini-Break
2:15
More Strategies
2:45
Q&A
3:00
Adjourn
10:30
Break
10:45
When Brains and Math
Collide! The Neuropsychology
of Math Disorders (With a
Side Trip into Math Anxiety)
11:30
Lunch
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 don’t ever want to have
to feel it again.”
David Berg, Educational Therapist
Author of, Making Math Real
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
Neuroanatomy 101: A Quick
User’s Guide to the Brain
DA’ BRAIN: Its two hemispheres
and four lobes
The Hemispheres Fancifully Illustrated . . .
Sequential,
Factual
Processing
Integrative,
‘Big Picture’
Processing
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
Right Hemisphere
• Has greater capacity for handling informational
complexity because of it’s 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
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?
The Four Lobes
FRONTAL
LOBE
PARIETAL
LOBE
OCCIPITAL
LOBE
TEMPORAL
LOBE
The Neuropsychology of Math
(AKA: The Brain on Math)
The Nature of Math
• It’s sequential and cumulative (earlier skills continually
form the basis for newer skills across the grade span)
• It’s conceptual (lots of ideas and
themes must be understood and
‘reasoned’)
• It’s procedural (lots of rules and algorithms must be
mastered to calculate – perform ‘numerical operations’
• It’s highly variable from a skill perspective (math is a
many varied thing!)
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)
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
Math Ability & the Neurodevelopmental Functions
(Portions adapted from the work of Mel Levine)
Spatial-Motor
Visualizing problems/procedures,
comprehending angles (and other elements
of geometry), creating charts, graphs, etc., and
maintaining sufficient grapho-motor accuracy to
solve problems correctly on paper
Attention
Maintaining sufficient cognitive
energy and attention on work
Executive Functioning
Planning, organizing, monitoring
the quality of work (also determining
what is/is not important for problem
solving)
MATH
Temporal-Sequential
Following sequences
and multiple steps
(Levine)
Memory
Recalling facts, procedures,
and rules, recognizing
patterns, and problem solving
Language
Processing written language and
spoken information – in directions,
problems – and understanding/recalling
technical math vocabulary
Attention
Controls
There is no single
math processing center!
Working
Memory
Executive
Functioning
Neuromotor
Functions
Spatial
Comprehension
Language
Memory
(LTM)
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
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
How much language is required to solve this?
1013
- 879
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!
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)
Visual-Spatial Functioning and
Mathematics
• We’re 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/factual processing and the
right related to simultaneous/spatial/holistic
processing
Left Parietal Lobe: Center of
Arithmetic Processing?
Area associated
with arithmetic
processing
15% bigger
In
Einstein’s
Brain!
Side Bar Issue:
Einstein’s Brain
• Actually weighed a bit less than the average for
brains of it’s 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
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 brain’s
‘big picture,’ ‘integration center’)
Novel stimuli are processed in the right
hemisphere
Many aspects of math are visualspatial 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 . .)
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
Your Turn . . .
Which figures to the right match
the ones to the let?
A closer look at the frontal lobe
Central
Sulcus (or
‘Fisure’)
Math strategies
and problemsolving directed
from here!
Frontal Lobe Specifics
(Adapted from Hale & Fiorello, 2004)
Dorsolateral
Prefrontal Cortex
Planning
Strategizing
Sustained Attention
Flexibility
Self-Monitoring
-------------------------------
Orbital Prefrontal
Impulse Control
(behavioral inhibition)
Emotional Modulation
Motor Cortex
Executive Skill and Math
Math’s Changing Face (It’s new again)
Out with the
explicit
teaching of
facts and
standard
algorithms . .
And in with
constructionist math
curricula that
emphasize
discovery learning
and the selfconstruction of
math know-how
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
BREAK TIME!
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)
=?
Attention problems
lead to other careless
errors (i.e., Forgetting to
regroup, etc.)
Organizational/planning
deficits lead to work
poorly organized on the
the page (or work not
shown)
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, it’s STM put to work.
• Long-Term Memory (LTM): Information and
experiences stored in the brain over longer periods of
time (hours to forever)
The Brain’s Memory Systems
Working Memory: Some kids
have got ‘leaky buckets’
• Levine: Some kids
are blessed with
large, ‘leak proof,’
working memories
• Others are born with
small WM’s that leak
out info before it can
be processed
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?
WM capacity tends to predict students’ ability to direct and monitor cognition.
How Large is the Child’s
Working Memory Bucket?
Case 1: Rachel Recallsitall
algorithm
Case 2: Nicky Normal
Case 3: Frankie
Forgetaboutit
fact
directions
fact
algorithm
facts
directions
algorithm
42
42
Working memory: A fundamental
element of math functioning
So much of learning
and academic
performance
requires the
manipulation of
material held in the
mind’s temporary
storage faculties
• 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
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.
Let’s Look at a Classic
Word Problem . .
• Sharon has finished an out-of-town
business meeting. She is leaving Chicago
at 3:00 on a two-hour flight to Boston. Her
husband, Tom, lives in Maine, 150 miles
from Boston. It’s his job to pick up Sharon
at the airport as soon as the flight lands. If
Tom’s 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?
Math Anxiety
Mathematics is the
supreme judge;
from its decisions
there is no appeal.
~Tobias Dantzig
Math Anxiety on a Brain Level
(or, ‘When the amygdala comes along for the ride’)
Amygdala
Temporal
lobe
Hippocampus
Bottom line: It’s crucial to keep kids from getting overly anxious
during math instruction (or they may always be anxious during
math instruction!)
Research (and common sense)
clearly indicates . . .
As anxiety
goes up . .
Working memory
Capacity goes down!
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
When Brains and Math
Collide!
Subtypes of Math Disabilities
and Their Neuropsychological Bases
Can you say, “Dyscalculia?”
Sure you can!!
Occur as often
As RD’s!!
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!
Epidemiology of
Math Disabilities
• Occur in about 1 - 6% of the population (Rourke, et al., 1997; DSMIV-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 LD’s, Math LD occurs more often in boys than girls
• MD’s 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 (‘MD’s’) occur just as
often as reading disabilities (‘RD’s’) – this has big implications for
the RTI process!!
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)
Semantic/Language-Based MD’s
• 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)
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.
• They’ll 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
• They’ll 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.
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)
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 . . .
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
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 visualspatial 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)
Error Patterns Associated with
the 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’
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)
Student Profiling to Inform Instruction
and Learning Plan
Student’s Name: _______________
Attention/EF Language
Memory
Neuromotor
Emotional
Neuro
Profile
Math Fact Skill
Academic
Profile
Strategies
Algorithm Skill
Math Concepts
Problem Solving
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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.
Implications for Instruction
BRINGING THE NEUROPSYCHOLOGY OF
MATH INTO THE CLASSROOM
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 6:1
Conceptual and Procedural
Knowledge
Conceptual knowledge has a
greater influence on procedural
knowledge than the reverse
Strong
Conceptual
Knowledge
Procedural
Knowledge
Weak
Sousa, 2004
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)
Step One: Understand a Child’s
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
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, what’s 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
Example of an Algorithm Error (revealed
via a ‘think aloud’ examination)
(Hale & Fiorello, 2004, p. 211)
64
+13
14
“First I look to see if it’s
addition or subtraction.
Okay, it’s 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.”
A Great Calvin and Hobbs Example
‘John has a problem with
multiplication’
• What kind of problem? How broad is the scope?
• Kids who can’t (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 can’t solve 16 X 7 on paper may have an algorithm
process difficulty (working memory or arithmetic
reasoning deficiency – frontal lobe or parietal lobe)
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/algorithms as kids problem solve
(and after they’ve 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!
Strategies to Build Number Sense
Meet Caleb
Caleb’s a ‘feisty little guy’
(to quote his mother) who’s just
entered kindergarten. He wore
sandals to school, but took them off
somewhere in the classroom and now
can’t seem to find them. He’s 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
1:1 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.
What, exactly, is number sense?
• Definitions abound in the literature . . .
Berch, 1998: Number sense is an emerging
construct that refers to a child’s 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.’
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)
Building Number Sense
• It’s critical that parents, during the preschool years, really
talk to kids about numbers and amounts and magnitude
(“Let’s 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, it’s 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)
10
9
8
7
6
5
4
3
2
1
Building Number Sense: Some
Concrete Strategies (Bley & Thorton, 2001)
1.
More or less than 10?
•
8+4: Is this more than 10 or less than 10? (kids should check with
manipulatives and number line work)
What’s 5+5? 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
Building Number Sense: More
Strategies (Bley & Thorton, 2001)
2. What can’t it be?
Provide computational problems and a choice
of two (or more) possible answers. Ask the children to predict which of the
choices couldn’t be possible and to state why.
Example: A. 28 + 37 = 65
B. 28 + 37 = 515
Verbalized response: The answer can’t be 515. It’s not even 100, because 50 +
50 is 100, and both numbers are less than 50.
3. What’s 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
It’s B. The problem is sort like 100 – 50, and the answer to that is 50, and so 48
is closest.
Digi-Blocks
Strategies Targeting
Semantic/Memory Weakness
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. She’s
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, she’s 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 they’d just show me
what to do and make it clear, I could do it – I
wish they’d just show me what they
mean!”
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
Subtraction
Multiplication Division
Sum
Add
Plus
Combine
Increased by
More than
Total
Take away
Remaining
Less than
Fewer than
Reduced by
Difference of
Product
Multiplied by
Times
Of
3 X 3 = 3(3)
Quotient
Per
A (as in gas is
$3 a gallon)
Percent (divide
by 100)
Operations Language Chart
in a Simpler Form
Add
=
Plus
=
+
Subtract = Take Away
Multiply
Divide
=
=
Times
=
Divided By
=
Minus
=
–
X
=
Per
=
÷
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!)
Illustrating the Pythagorean Theorem
c
a
Teacher: John, can you remind
us what an hypotenuse is?
b
John: Um, nope – I haven’t got
a clue . . .
Teacher: John, we’ve spent the
last two days talking about this
stuff.
John: So?! I don’t remember,
All right?! What’s your problem?!
Geez!!
13
5
12
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
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 doesn’t 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 it’s 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)
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
Meltzer et al., 2006
Gimme the facts, Madam,
just the facts . .
“Terrific Tens” Strategy
+
1
2
3
4
5
6
7
8
9
9
8
7
6
5
4
3
2
1
10
10
10
10
10
10
10
10
10
Meltzer et al., 2006
And then there’s good ‘ol
‘Touch Math’
Developers and it’s 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
Multiplication Fact Strategies
0 Rule: 0 times any number is 0
1’s Rule: 1 times any number is the number itself
2’ Rule: Counting by two’s
5’s Rule: The answer must end in a 5 or 0 (e.g., 35 or 60)
10’s Rule: The answer must end in a 0 (10, 40, 80, etc.)
9’s Rule: Two-hands counting rule
2 hands ‘Rule’ when it comes
To solving the tricky 9’s!
Meltzer et al., 2006
A key developmental asset in teaching
kids division and division facts . . .
Greed (balanced by an insistence on fairness)
“How many do we each get?”
Strategies Targeting Executive
Functioning (Procedural/Algorithmic)
Weakness
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: “He’s just
so all over the place with it!”). Although Andrew is a reasonably wellmotivated youngster who’s 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: “It’s like
I’m always at square one with him on this stuff”). Completing
assignments of all kinds is also a big issue for this kid.
The most important thing to
remember in helping ADHD
(“EFD”) kids with math
It’s all about . . .
Diminishing demands
on working memory
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 child’s instructional
environment, with emphasis given
to presentation of algorithm steps
in easy to follow formats
A key distinction:
Factual Memory vs. Procedural Memory
Factual memory . .
Procedural memory
Refers to an individual’s ability to
recall discrete bits/units of
information
Refers to an individual’s ability to
remembers processes; that is,
procedural steps
(e.g,.7 X 7 = 49, the capital of
France is Paris, my mother’s
middle name is Dorothy, ‘sh’
makes the /sh/ sound)
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:
Working memory demand:
Fairly minimal
Moderate to marked, depending
upon the process being
recalled
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., “Let’s just first circle all the signs on
the page” or “let’s 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.
Thus, in teaching two digit by
one digit multiplication (47 X 6)
•
First ensure the child’s 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.
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
A good multiple digit multiplication ‘box template’
X
+
(Adapted from Bley & Thorton, 2001)
A better multiple digit multiplication ‘box template’
+
7
2
3
X
3
2
1
4
4
6
2
1
6
9
2
3
1
3
6
(Adapted from Bley & Thorton, 2001)
Long Division Algorithm Box Template
7
—
0
4
9
3
4
8
2
8
–
6
8
6
3
R5
5
4
3
=
4
X
4
X
4
=
64
Pneumonics/Heuristics: Excellent Ways to Help
EFD Kids Learn and Retain Arithmetic Algorithms
Does McDonalds Sell Burgers Done Rare?
1.Divide
2.Multiply
3.Subtract
4.Bring Down
5.Repeat (if
necessary)
Improving Error Checking
Top Three Hits
The 3 most common errors
a kid exhibits in math
Example:
Steven’s Top 3 Hits:
1. Misreading directions
2. Misreading signs
3. Arriving at errors that can’t
possibly make sense.
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?
Strategies Targeting Visual/Spatial Weakness
For Kids with NLD: Emphasize the
Verbal
1.
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)
2.
Emphasize the verbal (simple, direct, concrete) over the visual
whenever possible
3.
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.
4.
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.
5.
Complex visuals can really freak out kids with visual/spatial
weakness (avoid busy graphs, maps, and charts)
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
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
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 25
John’s
Division
Card
MAKING THE ABSTRACT CONCRETE
A) The problem: What’s 5/8 of 16?
___ ___ ___ ___ ___ ___ ___ ___
B) Concrete illustration of 5/8:
C) Concrete illustration of 5/8 of 16
___ ___ ___ ___ ___ ___ ___ ___
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
(Adapted from Bley & Thorton, 2001)
D) Answer is 10
“Buy Out”: A great technique for kids
who are ‘motivationally challenged’
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!
34
X 45
56
X 13
89
X 64
92
X 35
76
X 56
83
X 83
69
X 31
78
X 64
27
X 59
39
X 37
71
X 82
90
X 90
Two Effective (Evidence-Based)
Remedial Programs
Case Studies/Student
Profiling