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Good Morning!

Christopher Kaufman, Ph.D. (207) 878-1777 e-mail

info_at_kaufmanpsychological.org web

kaufmanpsychological.org

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

- 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

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

Your Turn . .

- Choose a kid from your caseload who struggles

significantly with math. - Take a few moments to complete the first part of

the Personal Case Study Form

Neuroanatomy 101 A Quick Users 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 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

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

- 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!)

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)

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

There is no single math processing center!

Neuromotor Functions

Attention Controls

Working Memory

Spatial Comprehension

Executive Functioning

Memory (LTM)

Language

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

- 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

Left Parietal Lobe Center of Arithmetic

Processing?

Area associated with arithmetic processing

15 bigger In Einsteins Brain!

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

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

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 . .)

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 problem-solving directed from

here!

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

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 . .

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)

?

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.)

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)

The Brains 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 WMs 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?

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

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

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.

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?

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)

Bottom line Its 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 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!

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!!

Types of Math Disability (MD)

- Verbal/Semantic Memory (language based,

substantial co-occurrence with reading

disabilties) - Procedural (AKA anarithmetria substantial

overlap with executive functioning and memory

deficits) - Visual-Spatial (substantial overlap with NLD)

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)

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.

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 . . .

(No Transcript)

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

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)

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

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

Students Name _______________

Neuromotor

Attention/EF

Language

Memory

Emotional

Neuro Profile

Math Fact Skill

Math Concepts

Problem Solving

Algorithm Skill

Academic Profile

Strategies

LUNCH TIME!!!

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Operators Standing By!

Shameless self-promotion slide!!!!

Brookes Publishing Company

34.95

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 61

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 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

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

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

A Great Calvin and Hobbs Example

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)

THE CORE STRATEGIES

- Emphasize the development of an internal number

line (in grades K and 1) to build number sense - Teach the concept and the algorithm (not just the

algorithm in isolation), and keep teaching the

algorithm until mastery - Distributed practice works better than massed

practice (smaller doses of practice over time is

better than a lot all at once) - Emphasize the verbalization of strategies/algorith

ms as kids problem solve (and after theyve

arrived at a solution) - Build automaticity of fact retrieval
- Minimize demands on working memory/simultaneous

processing (encourage kids to download info from

working memory to paper by encouraging thinking

on paper) - Enhance the explicit structure of math problems

(using multiple colors, graph paper, boxing

techniques, etc.) - Body-involved, kinesthetic learning is good!

Strategies to Build Number Sense

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.

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.

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

- 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)

10

9

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2

1

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

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.

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. 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!

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)

Operations Language Chart in a Simpler Form

Add Plus Subtract Take

Away Minus Multiply

Times X Divide Divided By

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? 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

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 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)

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

Terrific Tens Strategy

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3

2

1

10

10

10

10

10

10

10

10

10

Meltzer et al., 2006

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

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

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 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.

The most important thing to remember in helping

ADHD (EFD) kids with math

Its 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 childs

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 . .
- 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

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.

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.

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

3

2

7

2

3

X

6

4

4

1

2

1

9

6

3

2

1

3

6

(Adapted from Bley Thorton, 2001)

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

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)

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.

Strategies Targeting Visual/Spatial Weakness

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)

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 2 5

Johns Division Card

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)

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

Two Effective (Evidence-Based) Remedial Programs

Case Studies/Student Profiling