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Specially Designed Instruction in Math PDU Session One . Oct 9, 2012 4:306:30. PDU Goal.
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Specially Designed Instruction in Math PDU Session One
Oct 9, 2012
4:306:30
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
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
3x2 – 2y + 8 – 2x2 + 5y
1 unit
1 square unit
1 unit
The greens don’t match up so this means the yellow rod is a variable
1 unit
= X
X
1 unit
=Y
Y
X
= X2
X
=Y2
Y
Y
=XY
X
Y
1 sq unit
Y2
X
Y
x2
XY

+
3+ 2= 1

+

+
Solution is
2x 3

+

+
Solution is
4y +2

+
lets add these polynomials
3x2 – 2y + 8 – 2x2 + 5y

+
Solution is
8 +x2+3y

+
Solution is
8 +x2+3y

+
3x2 2x2=x2
2y + 5y=3y
8
8+x2+3y
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 Nation’s 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.”
Algebra is the most important topic in math
aljebr (Arabic)
“reunion of broken
parts”
study of the rules of operations and relations
All elementary math leads to Algebraic mastery
Automatic
recall of facts
Mastered
standard
algorithms
Robust sense of
number
Estimation Fluency
Positive and negative fractions
Fractions and Decimals
Fluency with
Fractions
Percentages
“ Any approach that continually revisits topics year after year without closure is to be avoided.” NMP
… 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 PreCalculus
…customary sequence of a yearlong courses in Algebra 1, Algebra II, Geometry, and PreCalculus
No research supports one approach over another approach at the secondary level. Spiraling may work at the secondary level. Research is not conclusive .
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
You need all three and
not in a particular order
“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
“Difficulty with learning fractions is pervasive and is an obstacle to further progress in mathematics and other domains dependent on mathematics, including algebra.”
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
“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
Only 8 studies
inconclusive
 rescind recommendation that instruction should be one or the other
“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.”
Number Sense is Innate
Numerosity
to count
perform simple
addition and
subtraction
Number of objects
You don’t’ need to teach these skills. We are born with them and will develop them with out instruction. It is a survival skill.
Babies can count
This is not natural … not a survival skill!
Activation in the brain during arithmetic
Parietal lobe
Motor cortex involved with movement of fingers
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
Subitizing
(latin for instant)
yes
Children who cannot conceptually subitize are likely to have problems learning basic arithmetic processes.
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
40,000 BC
2000 BC
600 AD
Sumerian
Society –
Fertile
Crescent
marking on
clay for counting
Babylonians
base 60 systems
still used today
in telling time and
lat/long
Persian Mathematicians
use “Arabic System”
Notches in bones
30 months
3 years
5 years
witness counting many time
 counting becomes abstract
answer “how many” questions
distinguish various adjectives (separate number from shape, size)
onetoone correspondence
last number in counting sequence is the total number in the collection
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.
7, 5, 9, 11, 8, 3, 7, 2
English speakers get about 45
Native Chinese speakers recall all of the numbers
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
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
3,6
72, 68
…we have no intuition regarding other numbers that modern mathematicians use, such as negative numbers, integers, fractions or irrational numbers…these numbers are not needed for survival, therefore they don’t appear on our internal number line…
How do you explain negative numbers to a 5 year old?
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.
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
Number Module
Number Symbols
The human brain comprehends numerals as quantities, not as words.
Brocca’s Area
Number Words
This reflex action is deeply rooted in our brains and results in an immediate attribution of meaning to numbers.
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
However
We continue to develop number sense for the rest of our lives.
“Our ability to approximate numerical quantities may be embedded in our genes, but dealing with exact symbolic calculations can be an errorprone ordeal.” Sousa
Sharon Griffin Calculation Generalizations
Global Quantity Schema
Initial Counting Schema
more than
1 2 3 4 5
less than
Requires Subitizing
Requires oneonone Correspondence
Internal Number line has been developed
1 10 20 30 40 50
a little
a lot
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
Double internal number line has been loosley developed to allow for two digit operational problem solving
1 10 20 30 40 50
a little
a lot
1 10 20 30 40 50
a little
a lot
Loosely coordinated number line is developed to allow for understanding of place value and solving double digit additional problems.
Double internal number line has been well developed to allow for two digit operational problem solving
1 10 20 30 40 50
a little
a lot
1 10 20 30 40 50
a little
a lot
These two well developed number lines allow for the capability of doing two digit addition calculations mentally.
25 x 30=
Exact
Approximate
The CRA instructional sequence consists of three stages: concrete, representation, and abstract.
In the concrete stage, the teacher begins instruction by modeling each mathematical concept with concrete materials (e.g., red and yellow chips, cubes, baseten blocks, pattern blocks, fraction bars, and geometric figures).
(Harrison & Harrison, 1986: Suydam& Higgins, 1977)
In this stage, the teacher transforms the concrete model into a representational (semiconcrete) level, which may involve drawing pictures; using circles, dots, and tallies; or using stamps to imprint pictures for counting.
or
Concrete  representational using a drawing (semiconcrete)
3 groups of 4 is 12 total or
3 X 4 = 12
representational  abstract using symbols