Oct 9, :30-6:30 Specially Designed Instruction in Math PDU Session One

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 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 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 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)

Text

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 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

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 brain’s language areas

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

CRA Algebra- using Algeblocks 1 unit 1 square unit The greens don’t match up so this means the yellow rod is a variable X 1 unit = X

CRA Algebra- using Algeblocks 1 unit Y =Y X X = X 2

CRA Algebra- using Algeblocks Y Y =Y 2

CRA Algebra- using Algeblocks X Y =XY

Algeblocks Key 1 sq unit X Y x2x2 Y2Y2 XY

Basic Mat:

Basic Mat: -3+2 (Make 0 pairs) = -1

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

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

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

Basic Mat: (3y +5) + (y-3) (0 Pairs) + - Solution is 4y +2

You try lets add these polynomials 3x 2 – 2y + 8 – 2x 2 + 5y

Basic Mat: 3x 2 – 2y + 8 – 2x 2 + 5y concrete + -

Basic Mat: 3x 2 – 2y + 8 – 2x 2 + 5y + - Solution is 8 +x 2 +3y

Basic Mat: 3x 2 – 2y + 8 – 2x 2 + 5y representational + - Solution is 8 +x 2 +3y

Basic Mat: 3x 2 – 2y + 8 – 2x 2 + 5y abstract 3x 2 - 2x 2 =x 2 -2y + 5y=3y 8 8+x 2 +3y

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 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 Algebra is the most important topic in math -study of the rules of operations and relations

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

Elementary Math Focus- by end of 5 th grade Robust sense of number Automatic recall of facts Mastered standard algorithms Estimation Fluency

Middle School Math Focus- by end of 8 th grade Fluency with Fractions Positive and negative fractions Fractions and Decimals Percentages

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 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.

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

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

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

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

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 inconclusive - rescind recommendation that instruction should be one or the other

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 Visual representations with direction instruction Very positive effects Explicit systematic instruction improve the performance of student with MLD positive effects using direct instruction

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

Everyone Can Do Math Number Sense is Innate Numerosity Number of objects to count perform simple addition and subtraction 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.

Why do children struggle with 23x42? This is not natural … not a survival skill!

Numerosity Activation in the brain during arithmetic Parietal lobe Motor cortex involved with movement of fingers

Which has more?

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

Subitizing

Counting

2 types of subitizing perceptional conceptual

Is Subitizing necessary? Children who cannot conceptually subitize are likely to have problems learning basic arithmetic processes.

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

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

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 English speakers get about 4-5 Native Chinese speakers recall all of the numbers

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 three forms for ten (ten, -teen and –ty) some numbers don’t 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)

Mental Number line typical number line brains number line

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… 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 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 Module Number Symbols Brocca’s 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.

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 “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 children’s 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

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

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 a little a lot

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 a little a lot a little a lot

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 a little a lot a little a lot

Language and Multiplication 25 x 30= Exact Approximate

CRA The CRA instructional sequence consists of three stages: concrete, representation, and abstract.

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).

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)

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

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