Oct 9, 2012 4:30-6:30 Specially Designed Instruction in Math
PDU Session One
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PDU Goal To build the capacity of special educators to provide
quality specialized instruction for students with disabilities in
the area of math, by building content knowledge of mathematics,
assessing students using diagnostic tools, creating lesson based on
a scope and sequence and progress monitoring growth
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PDU Requirements Attend ten sessions (20 hours) 11 hours of
professional development using the How the Brain Learns Mathematics
by David A. Sousa and Teaching Learners Who Struggle with
Mathematics by Sherman, Richardson, and Yandl 9 hours of small
group lesson writing and reflection using the Lesson Study protocol
If a session is missed then you will be responsible for doing a
self study of the missing content and complete the corresponding
exit slip and Lesson Study Protocol
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PDU Requirements Complete a Diagnostic Math assessment on the
targeted student (assessment provided in class)(1 hour) Complete
progress monitoring tool after 5-10 hours of instruction (progress
monitoring tool provided in class) (1.5 hours) IEP meeting for the
targeted student sometime during the PDU (annual, eligibility or
special request) where writing is discussed (2 hours including
planning and meeting)
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PDU Requirements Math lesson plans (10+ hours) Direct
instruction in mathematics for the targeted student (15+ hours)
Reflection Essay (1 hour) Complete a portfolio (1 hour) 9 Lesson
Plans with Lesson Study Protocol Copy of Diagnostic Assessment Copy
of IEP with names crossed out Copy of Progress Monitoring with
Interpretation Copy of Reflection Essay Attend Final PDU peer
review process (2 hours)
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Text
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Outcomes for Session One Participants will have a basic
knowledge of the National Math Panel report of 2008 Participants
will have a foundational knowledge of the psychological processes
of mathematics
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Math basics quiz 1. T F The brain comprehends numerals first as
words, then as quantities. 2. T F Learning to multiple, like
learning spoken language, is a natural ability 3. T F It is easier
to tell which is the greater of two larger numbers than of two
smaller numbers 4. T F the maximum capacity of seven items in
working memory is valid for all cultures 5. T F Gender differences
in mathematics are more likely due to genetics that to cultural
factors
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Math basics quiz 6. T F Practicing mathematics procedures makes
perfect 7. T F Using technology for routine calculations leads to
greater understanding and achievement in mathematics 8. T F
Symbolic number operations are strongly linked to the brains
language areas
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Manipulative make it concrete We are going to add polynomials
using Algeblocks After learning how to use the Algeblocks you will
be able to add and subtract these polynomials in less than 10
seconds Before we can use the concrete manipulative we need to
build some background knowledge. You need a set of Algeblocks and
Algeblocks Basic Mat 3x 2 2y + 8 2x 2 + 5y
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CRA Algebra- using Algeblocks 1 unit 1 square unit The greens
dont match up so this means the yellow rod is a variable X 1 unit =
X
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CRA Algebra- using Algeblocks 1 unit Y =Y X X = X 2
2006 National Math Panel President Bush Commissioned the
National Math Panel To help keep America competitive, support
American talent and creativity, encourage innovation throughout the
American economy, and help State, local, territorial and tribal
governments give the Nations children and youth the education they
need to succeed, it shall be the policy of the United States to
foster greater knowledge of and improve performance in mathematics
among American students.
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2006 Panel 30 members 20 independent 10 employees of the
Department of Education Their task is to make recommendations to
the Secretary of Education and the President on the state of math
instruction and best practices based on research Research includes
Scientific Study Comparison study with other countries who have
strong math education programs
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2008 Recommendations Algebra is the most important topic in
math -study of the rules of operations and relations
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2008 Recommendations All elementary math leads to Algebraic
mastery Major Topics of Algebra Must Include Symbols and
Expressions Linear Equations Quadratic Equations Functions Algebra
of Polynomials Combinatorics and Finite Probability
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Elementary Math Focus- by end of 5 th grade Robust sense of
number Automatic recall of facts Mastered standard algorithms
Estimation Fluency
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Middle School Math Focus- by end of 8 th grade Fluency with
Fractions Positive and negative fractions Fractions and Decimals
Percentages
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A need for Coherence High Performing Countries Fewer Topics/
grade level In-depth study Mastery of topics before proceeding
United States Many Topics/ grade level Shallow study Review and
extension of topics (spiral) Any approach that continually revisits
topics year after year without closure is to be avoided. - NMP
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Interactive verses Single Subject Approach Interactive Single
Subject topics of high school mathematics are presented in some
order other than the customary sequence of a yearlong courses in
Algebra 1, Algebra II, Geometry, and Pre- Calculus customary
sequence of a yearlong courses in Algebra 1, Algebra II, Geometry,
and Pre- Calculus No research supports one approach over another
approach at the secondary level. Spiraling may work at the
secondary level. Research is not conclusive.
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Math Wars Conceptual Understanding verses Standard Algorithm
verses Fact Fluency Debates regarding the relative importance of
conceptual knowledge, procedural skills, and the commitment of
.facts to long term memory are misguided. -NMP Few curricula in the
United States provide sufficient practice to ensure fast and
efficient solving of basic fact combinations and execution of the
standard algorithms. -NMP
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Number Sense number value with small quantities basic counting
approximation of magnitude Informal Place value compose and
decompose numbers Whole number operations commutative, associative
and distributive properties Formal
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Fractions Difficulty with learning fractions is pervasive and
is an obstacle to further progress in mathematics and other domains
dependent on mathematics, including algebra. Conceptual knowledge
leads to Procedural Knowledge -Use fraction names the demarcate
parts and wholes -Use bar fractions not circle fractions -Link
common fraction representations to locations on a number line
-Start working on negative numbers early and often
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Developmental Appropriateness is challenged What is
developmentally appropriate is not a simple function of age or
grade, but rather is largely contingent on prior opportunities to
learn. NRP Piaget Vygotsky
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Social, Motivational, and Affective Influences Motivation
improves math grades Teacher attitudes towards math have a direct
correlation to math achievement Math anxiety is real and influences
math performance
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Teacher directed verses Student directed inconclusive - rescind
recommendation that instruction should be one or the other
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Formative Assessment The average gain in learning provided by
teachers use of formative assessments is marginally significant.
Results suggest that use of formative assessments benefited
students at all levels.
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Low Achieving and MLD Visual representations with direction
instruction Very positive effects Explicit systematic instruction
improve the performance of student with MLD positive effects using
direct instruction
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Real World Math Taught using real word math High performance on
test that had similar real world problems Taught using real word
math Low performance on measures of computation, simple word
problem and equations solving
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Everyone Can Do Math Number Sense is Innate Numerosity Number
of objects to count perform simple addition and subtraction You
dont need to teach these skills. We are born with them and will
develop them with out instruction. It is a survival skill.
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Why do children struggle with 23x42? This is not natural not a
survival skill!
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Numerosity Activation in the brain during arithmetic Parietal
lobe Motor cortex involved with movement of fingers
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Which has more?
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Prerequisite to counting Recognizing the number of objects in a
small collection is a part of innate number sense. It requires no
counting because numerosity is identified in an instant. When the
number exceeds the limit of subitizing, counting becomes
necessary
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Subitizing
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Counting
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2 types of subitizing perceptional conceptual
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Is Subitizing necessary? Children who cannot conceptually
subitize are likely to have problems learning basic arithmetic
processes.
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Counting Is it just a coincidence that the region of the brain
we use for counting includes the same part that controls our
fingers? 8000 BC Sumerian Society Fertile Crescent marking on clay
for counting 600 AD 2000 BC Babylonians- base 60 systems still used
today in telling time and lat/long Persian Mathematicians use
Arabic System 40,000 BC Notches in bones
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Cardinal Principle 30 months3 years5 years -witness counting
many time - counting becomes abstract -answer how many questions
-distinguish various adjectives (separate number from shape, size)
-one-to-one correspondence -last number in counting sequence is the
total number in the collection
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Cardinal Principle Recognizing that the last number in a
sequence is the number of objects in the collection. Children who
do not attain the cardinal principle will be delayed in their
ability to add and subtract.
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Digit Span Memory English speakers get about 4-5 Native Chinese
speakers recall all of the numbers
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Digit Span The magical number of seven items, long considered
the fixed span of working memory, is just the standard span for
Western adults. The capacity of working memory appears to be
affected by culture and training.-Sousa
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English makes counting harder English three forms for ten (ten,
-teen and ty) some numbers dont make sense (eleven, twelve) teens
are confusing (nineteen implies 91 not 19) Chinese place value
friendly simple two or three sound words logical system (22 is
called 2 tens 2)
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Mental Number line typical number line -3 -2 -1 0 1 2 3 4 5 6 7
8 9 brains number line 1 10 20 30 40 50
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Negative Numbers we have no intuition regarding other numbers
that modern mathematicians use, such as negative numbers, integers,
fractions or irrational numbersthese numbers are not needed for
survival, therefore they dont appear on our internal number line
How do you explain negative numbers to a 5 year old?
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Piaget verses what we know Remember that what we once knew
about number sense and children influenced by Piagetian theory
Children's knowledge is more influenced by experience than a
developmental stage with regards to number sense.
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Mental Number Line The increasing compression of numbers on our
mental number line makes it more difficult to distinguish the
larger of a pair of numbers as their value gets greater. As a
result, the speed and accuracy with which we carry out calculations
decreases as the numbers get larger.-Sousa
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Number Symbols verses Number Words Number Module Number Symbols
Broccas Area Number Words The human brain comprehends numerals as
quantities, not as words. This reflex action is deeply rooted in
our brains and results in an immediate attribution of meaning to
numbers.
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Teaching Number Sense Just as phonemic awareness is a
prerequisite to learning phonics and becoming a successful reader,
developing number sense is a prerequisite for succeeding in
mathematics. Berch We continue to develop number sense for the rest
of our lives.
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Operational Sense Our ability to approximate numerical
quantities may be embedded in our genes, but dealing with exact
symbolic calculations can be an error-prone ordeal.- Sousa Sharon
Griffin Calculation Generalizations Major reorganization in
childrens thinking occur at age 5 where cognitive structures
created in earlier years are added to hierarchy This reorganization
occurs every two years60% of children progress at this rate; 20%
slower; 20% faster
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4 year olds Operational Sense Global Quantity SchemaInitial
Counting Schema more than less than 1 2 3 4 5 Requires
SubitizingRequires one-on-one Correspondence
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6 year olds Operational Sense Internal Number line has been
developed This developmental stage is a major turning point because
children come to understand that mathematics is not just something
that occurs out in the environment but can also occur inside their
own heads 1 10 20 30 40 50 a little a lot
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8 year olds Operational Sense Double internal number line has
been loosley developed to allow for two digit operational problem
solving Loosely coordinated number line is developed to allow for
understanding of place value and solving double digit additional
problems. 1 10 20 30 40 50 a little a lot 1 10 20 30 40 50 a little
a lot
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10 year olds Operational Sense Double internal number line has
been well developed to allow for two digit operational problem
solving These two well developed number lines allow for the
capability of doing two digit addition calculations mentally. 1 10
20 30 40 50 a little a lot 1 10 20 30 40 50 a little a lot
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Language and Multiplication 25 x 30= Exact Approximate
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CRA The CRA instructional sequence consists of three stages:
concrete, representation, and abstract.
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Concrete In the concrete stage, the teacher begins instruction
by modeling each mathematical concept with concrete materials
(e.g., red and yellow chips, cubes, base-ten blocks, pattern
blocks, fraction bars, and geometric figures).
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Concrete Studies show that students who use concrete materials
Develop more precise and comprehensive mental representations Show
more motivation and on-task behaviors Understand mathematical ideas
Can better apply these ideas to life situations (Harrison &
Harrison, 1986: Suydam& Higgins, 1977)
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Representational In this stage, the teacher transforms the
concrete model into a representational (semi-concrete) level, which
may involve drawing pictures; using circles, dots, and tallies; or
using stamps to imprint pictures for counting. Concrete
------------------ representational using a drawing (semi-concrete)
or
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Abstract At this stage, the teacher models the mathematics
concept at a symbolic level, using only numbers, notation, and
mathematical symbols to represent the number of circles or groups
of circles. The teacher uses operation symbols (+, , ) to indicate
addition, multiplication, or division. 3 groups of 4 is 12 total or
3 X 4 = 12 representational ---------------- abstract using
symbols