Considerations around Math Instruction for Students with Disabilities

Oct 9, 20124:30-6:30Specially Designed Instruction in Math PDU Session One PDU GoalTo 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 growthPDU 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 InterpretationCopy of Reflection Essay Attend Final PDU peer review process (2 hours)

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Outcomes for Session One Participants will have a basic knowledge of the National Math Panel report of 2008Participants will have a foundational knowledge of the psychological processes of mathematics Math basics quizT 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 quizT F Practicing mathematics procedures makes perfect T F Using technology for routine calculations leads to greater understanding and achievement in mathematicsT F Symbolic number operations are strongly linked to the brains language areas

Manipulative make it concreteWe are going to add polynomials using AlgeblocksAfter 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 + 5yCRA Algebra- using Algeblocks 1 unit 1 unit 1 square unit The greens dont 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 = X2 CRA Algebra- using Algeblocks Y Y=Y2 CRA Algebra- using Algeblocks X Y=XY Algeblocks Key 1 sq unit XYx2Y2XYBasic Mat: -3+2 +-Basic Mat: -3+2 (Make 0 pairs) +--3+ 2= -1Basic Mat: 3x-5 + (2-X)+-Basic Mat: 3x-5 + (2-X) (0 pairs)+-Solution is 2x -3Basic Mat: (3y +5) + (y-3)+-Basic Mat: (3y +5) + (y-3) (0 Pairs)+-Solution is 4y +2You try lets add these polynomials

3x2 2y + 8 2x2 + 5yBasic Mat: 3x2 2y + 8 2x2 + 5yconcrete +-Basic Mat: 3x2 2y + 8 2x2 + 5y+-Solution is 8 +x2+3yBasic Mat: 3x2 2y + 8 2x2 + 5yrepresentational+-Solution is 8 +x2+3yBasic Mat: 3x2 2y + 8 2x2 + 5yabstract 3x2- 2x2=x2 -2y + 5y=3y88+x2+3y

2006 National Math Panel President Bush Commissioned the National Math PanelTo 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.

2006 Panel30 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 programs2008 Recommendations Algebra is the most important topic in math al-jebr (Arabic)reunion of broken parts-study of the rules of operations and relations 2008 Recommendations All elementary math leads to Algebraic masteryMajor 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 5th grade

Robust sense of number Automaticrecall of facts Masteredstandard algorithmsEstimation Fluency Middle School Math Focus- by end of 8th grade Fluency withFractions Positive and negative fractionsFractions and Decimals

Percentages A need for Coherence

High Performing Countries Fewer Topics/ grade levelIn-depth study Mastery of topics before proceeding United States Many Topics/ grade levelShallow study Review and extension of topics (spiral) Any approach that continually revisits topics year after year without closure is to be avoided. -NMPInteractive 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 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. -NMPFew curricula in the United States provide sufficient practice to ensure fast and efficient solving of basic fact combinations and execution of the standard algorithms. -NMPYou need all three and not in a particular order Number Sense Too often we assume that the informal skill were already developed. Students with MLD actually need direct instruction in these skills 36Fractions 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 often37Developmental 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 VygotskyPiaget notions of developmental appropriateness is challenges and proven to not be correct for math learning. Children are capable of understanding magnitude of number as young as 2 months. Doesnt mean the whole theory is wrong, just the notions of what we once thought was developmentally appropriate.Vygotsky approaches have never been scientifically tested in classrooms. 38Social, 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 Only 8 studiesinconclusive - 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 Real World Math Everyone Can Do Math

Number Sense is InnateNumerosity 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. Babies can count Why do children struggle with 23x42?

This is not natural not a survival skill!Numerosity

Activation in the brain during arithmeticParietal lobe Motor cortex involved with movement of fingers

Number sense and finger movement are both located in essentially the same part of the brain. Counting on finger dates back to early number sense development needed for survival. e.g. while hunting are you going to have more luck going after one deer or five deer?

Numberosity is number sense; an understanding of magnitude. It activates soon after birth and is hardwired as an adaptation for survival in the human brain.

This is number sense. a persons ability to recognize that something has changed in a small collection

We have number sense because numbers have meaning to us

the ability to compare the sizes of two collections shown simultaneously and the ability to remember number of objects presented successively in time46Which has more?This is an example of number sense. Quickly look at the quantities on the right and left. Which one has more? Your numerocity portion of the brain is activated. No language is needed to complete this task.47Prerequisite 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 (latin for instant)Subitizing the numbers 1-4 can be instantly recognized. Why? It is likely that subitizing is a primitive cerebral process while counting involves more sofisticated. Note that het last one was difficult to subitize. This is because the brain is wired for up to 4 objects. After four objects we cannot subitize random order of objects.

Brain scans indicate that when subitizing one to four items, areas in the visual cortex were activated while areas involving attention were quite. 49Counting After the number of four items we no longer subitize and we count. When this happens we now require attention (executive functioning) and visual attention.502 types of subitizing perceptual subitizing involves recognizing a number without using other mathematical processes. You instantly recognizes the number (used by babies and animals); helps children separate collections of objects into single units and connects each unit with only one number word, thus developing the concept of counting.

conceptual subitizing allows one to know the number of a collection by recognizing a familiar pattern (like on dice) 51Is Subitizing necessary?yesChildren 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 AD2000 BCBabylonians- base 60 systems still used todayin telling time and lat/longPersian Mathematicians use Arabic System40,000 BC

Notches in bones

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

English speakers get about 4-5Native Chinese speakers recall all of the numbers read the following list of numbers allowed. Now I am going to cover them up. You have 20 seconds to try to memorize this list of numbers. Now write down the list of numbers on your exit slip. How many did you remember?

When you are trying to remember these numbers you are using a verbal memory loop. Your short term memory is very short so you must rehearse the numbers to remember them but you only have 2 seconds to rehearse these numbers. Most English speakers can only remember up to 7 digits. The English words are too long and many have multiple syllables. Numbers in Chinese are very short. Because of this they can retain up to 12 numbers at one time.

Chinese speakers process numbers in a different part of the brain than Englsih speakers. 56Digit 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.-SousaEnglish makes counting harder Mental Number line typical number line -3 -2 -1 0 1 2 3 4 5 6 7 8 9brains number line 1 10 20 30 40 503,672, 68First the brains number lines doesnt go into negative numbers Second, the smaller numbers are equally spaced out but the larger numbers on the line are very close to each other. As a result when given two numbers to compare, if the number are small, the brain can quickly determine the larger number. If the numbers are larger it takes the brain longer to determine the larger number. This has an impact on mathematical learning 59Negative 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 lineHow do you explain negative numbers to a 5 year old?Typically teacher use metaphors such as money barrowed from a bank, temperatures below zero, or simply an extension of the number line to the left of zero. 60Piaget verses what we knowRemember that what we once knew about number sense and children influenced by Piagetian theoryChildren's knowledge is more influenced by experience than a developmental stage with regards to number sense.

These are two types of number lines you find in a typical kindergarten classroom. The American number line starts at 0 but the Canadian number line starts at -10. Why would Canadian kindergarten children be able to understand negative numbers in kindergarten? If they can understand them then why can American children understand them? What type of experience would Canadian children with negative numbers?61Mental 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.-SousaNumber Symbols verses Number Words

Number Module Number Symbols Broccas AreaNumber 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 HoweverWe 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.- SousaSharon Griffin Calculation Generalizations 4 year olds Operational Sense Global Quantity Schema Initial Counting Schema

more than less than 1 2 3 4 5Requires Subitizing Requires 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 1 10 20 30 40 50a 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. 1 10 20 30 40 50a little a lot 1 10 20 30 40 50a 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. 1 10 20 30 40 50a little a lot 1 10 20 30 40 50a little a lot Language and Multiplication 25 x 30=

Exact Approximate I want you to recite the alphabet and solve this two digit multiplication problem at the same time.

It is impossible because of where the brain stored multiplication facts in the same part of the brain used for language tasks like reciting the alphabet. We memorize facts using language.

This also explains why second language learns always revert to the first language to solve mathematical problems. It is less taxing on cognitive reserves. They have to take in the problem in the second language, translate to the first language, calculate the problem and then translate to the second language, thus taking a longer period of time.

When doing estimation, making approximations or determining the larger of two numbers the parietal lobes are activated on both right and left hemispheres. These are the number sense centers in the brain. They are naturally developed centers.

When doing exact calculations, the language processing areas in the frontal lobes are activated. As we became more sophisticated in our concept of number we needed to tie language to numbers. When we are problem solving we are using language. It is very difficult to do both calculations and language activates at the same time.

This doesnt mean that children with language processing issues have number issue as it is different parts of the brain. It just may mean they have difficulty expressing the numbers. 70CRAThe CRA instructional sequence consists of three stages: concrete, representation, and abstract.

71Concrete 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).

ConcreteStudies show that students who use concrete materials Develop more precise and comprehensive mental representationsShow more motivation and on-task behaviorsUnderstand mathematical ideasCan better apply these ideas to life situations

(Harrison & Harrison, 1986: Suydam& Higgins, 1977)

RepresentationalIn 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)orAbstract 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