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Specially Designed Instruction in Math PDU Session One . Oct 9, 2012 4:30-6:30. PDU Goal.

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Specially designed instruction in math pdu session one

Specially Designed Instruction in Math PDU Session One

Oct 9, 2012


Pdu goal

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

Pdu requirements

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

Pdu requirements1

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)

Pdu requirements2

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)

Specially designed instruction in math pdu session one


Outcomes for session one

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

Math basics quiz

Math basics quiz

  • T F The brain comprehends numerals first as words, then as quantities.

  • T F Learning to multiple, like learning spoken language, is a natural ability

  • T F It is easier to tell which is the greater of two larger numbers than of two smaller numbers

  • T F the maximum capacity of seven items in working memory is valid for all cultures

  • T F Gender differences in mathematics are more likely due to genetics that to cultural factors

Math basics quiz1

Math basics quiz

  • T F Practicing mathematics procedures makes perfect

  • T F Using technology for routine calculations leads to greater understanding and achievement in mathematics

  • T F Symbolic number operations are strongly linked to the brain’s language areas

Manipulative make it concrete

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

3x2 – 2y + 8 – 2x2 + 5y

Cra algebra using algeblocks

CRA Algebra- using Algeblocks

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


Cra algebra using algeblocks1

CRA Algebra- using Algeblocks

1 unit




= X2


Cra algebra using algeblocks2

CRA Algebra- using Algeblocks




Cra algebra using algeblocks3

CRA Algebra- using Algeblocks




Algeblocks key

Algeblocks Key

1 sq unit






Basic mat 3 2

Basic Mat: -3+2



Basic mat 3 2 make 0 pairs

Basic Mat: -3+2 (Make 0 pairs)

-3+ 2= -1



Basic mat 3x 5 2 x

Basic Mat: 3x-5 + (2-X)



Basic mat 3x 5 2 x 0 pairs

Basic Mat: 3x-5 + (2-X) (0 pairs)

Solution is

2x -3



Basic mat 3y 5 y 3

Basic Mat: (3y +5) + (y-3)



Basic mat 3y 5 y 3 0 pairs

Basic Mat: (3y +5) + (y-3) (0 Pairs)

Solution is

4y +2



You try

You try

lets add these polynomials

3x2 – 2y + 8 – 2x2 + 5y

Basic mat 3x 2 2y 8 2x 2 5y concrete

Basic Mat: 3x2 – 2y + 8 – 2x2 + 5yconcrete



Basic mat 3x 2 2y 8 2x 2 5y

Basic Mat: 3x2 – 2y + 8 – 2x2 + 5y

Solution is

8 +x2+3y



Basic mat 3x 2 2y 8 2x 2 5y representational

Basic Mat: 3x2 – 2y + 8 – 2x2 + 5yrepresentational

Solution is

8 +x2+3y



Basic mat 3x 2 2y 8 2x 2 5y abstract

Basic Mat: 3x2 – 2y + 8 – 2x2 + 5yabstract

3x2- 2x2=x2

-2y + 5y=3y



2006 national math panel

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 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.”

2006 panel

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

2008 recommendations

2008 Recommendations

Algebra is the most important topic in math

al-jebr (Arabic)

“reunion of broken


-study of the rules of operations and relations

2008 recommendations1

2008 Recommendations

All elementary math leads to Algebraic mastery

Elementary math focus by end of 5 th grade

Elementary Math Focus- by end of 5th grade


recall of facts




Robust sense of


Estimation Fluency

Middle school math focus by end of 8 th grade

Middle School Math Focus- by end of 8th grade

Positive and negative fractions

Fractions and Decimals

Fluency with



A need for coherence

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

Interactive verses single subject approach

Interactive verses Single Subject Approach

… 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 .

Math wars

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

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

Number sense

Number Sense



“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

Developmental appropriateness is challenged

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



Social motivational and affective influences

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

Teacher directed verses student directed

Teacher directed verses Student directed

Only 8 studies


- rescind recommendation that instruction should be one or the other

Formative assessment

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.”

Low achieving and mld

Low Achieving and MLD

Real world math

Real World Math

Everyone can do math

Everyone Can Do Math

Number Sense is Innate


to count

perform simple

addition and


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

Why do children struggle with 23x42

Why do children struggle with 23x42?

This is not natural … not a survival skill!



Activation in the brain during arithmetic

Parietal lobe

Motor cortex involved with movement of fingers

Which has more

Which has more?

Prerequisite to counting

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


(latin for instant)





2 types of subitizing

2 types of subitizing

Is subitizing necessary

Is Subitizing necessary?


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


Society –



marking on

clay for counting


base 60 systems

still used today

in telling time and


Persian Mathematicians

use “Arabic System”

Notches in bones

Cardinal principle

Cardinal Principle

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)

-one-to-one correspondence

-last number in counting sequence is the total number in the collection

Cardinal principle1

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.

Digit span memory

Digit Span Memory

7, 5, 9, 11, 8, 3, 7, 2

English speakers get about 4-5

Native Chinese speakers recall all of the numbers

Digit span

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

English makes counting harder

English makes counting harder

Mental number line

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


72, 68

Negative numbers

Negative Numbers

…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?

Piaget verses what we know

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.

Mental number line1

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

Number symbols verses number words

Number Symbols verses Number Words

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.

Teaching number sense

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.

Operational sense

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

4 year olds operational sense

4 year olds Operational Sense

Global Quantity Schema

Initial Counting Schema

more than

1  2  3  4  5

less than

Requires Subitizing

Requires one-on-one Correspondence

6 year olds operational sense

6 year olds Operational Sense

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

8 year olds operational sense

8 year olds Operational Sense

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.

10 year olds operational sense

10 year olds Operational Sense

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.

Language and multiplication

Language and Multiplication

25 x 30=



Specially designed instruction in math pdu session one


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, base-ten blocks, pattern blocks, fraction bars, and geometric figures).



  • 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)



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)



  • 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

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