Considerations around Math Instruction for Students with Disabilities

How the Brain Learns Math Specially Designed Instruction in Math PDUOld Dogs Learn New Tricks Participants will be able to orally explain how to add and subtract polynomials using algeblocks using academic vocabulary after guided modeling of the algeblocks guided practice of the alegblocksusing the CRA strategies CLO

Manipulative make it concreteWe are going to add and subtract 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

The National Math Panel Report Participants will be able to orally summarize and apply the key findings on the meta-analysis of the National Math Panel report using academic vocabulary after a review of the findings using visual supports CLO

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 30Fractions 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 often31Developmental 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. 32Social, 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 Effective Teaching Modeling

CLO Clearly Communicated Academic LanguageChecks for Understanding of the CLO See who remembers the CLO for this section?Explain that this is a rigorous task (complex and in-depth)We are going to be checking for understanding of the CLO 38Group T Chart - Reflection of NMPWhat was new information?How is this going to change your practice?How the Brain Learns Math Participants will be able to orally summarize and apply the recent scientific brain based finding using academic vocabulary after a review of chapter 1 in How the Brain Learns Math using visual supports CLO

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 time43Which 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.44Prerequisite 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. 46Counting 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.472 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) 48Is 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. 53Digit 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 56Negative 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. 57Piaget 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?58Mental 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. 67Assessing for a Math SLD Participants will be able to orally explain; give and diagnosis a battery of math screeners to diagnosis math learning disabilities using academic vocabulary after oral explanation of the math screeners guided practice of giving the screeners CLO

conceptual based mathprocedural based mathDoes the instructional approach impact the determination of a disability?

processing speed reasoning number sense visual-spatial Types of Math Disorders Number Sense Counting Skill Deficits Arithmetic SkillProcedural Disorders Memory Deficit Visual-Spatial Deficit Associated with Number Module dysfunction Difficulty understanding the concept associated with fluid reasoning Associated with Executive Functioning Rapid Recall of over learned material Non-verbal reasoning Primary Assessments How Children Learn Mathematics by M. Sharma Digit span-student repeats in a string of numbers either forwards or backwardsMagnitude comparison-student chooses the largest of visually or verbally presented numbersMissing number-student names a missing number from a sequence of numbers between zero and 20Number knowledge test-basic measure of number senseNumbers from dictation-student writes numbers from word dictationNumber identification-student identifies numbers between zero and 20 from printed numbersQuantity discrimination-student identifies the larger of two printed numbersScreeners that provide this information Skill Screener What does it tell usMissing Number Missing Number CBMThese are all screeners that hint at basic number sense dysfunction due to developmental dyscalculia or acquired dyscalculia Number Module in Left Parietal Lobe Cannot conceptualize numbers Unable to understand number relationshipsLeads to difficulty with developing operational sense Difficulty with estimation Number Knowledge Math their Way Pre-Number Concepts and Skills Number Dictation Math their Way Pre-Number Concepts and Skills Number Identification Math their Way Pre-Number Concepts and Skills Quantity Discrimination Quantity Discrimination CBM Post- Primary Assessments How Children Learn Mathematics by M. Sharma Psychological Perspectives in assessing math learning needs; Journal of Instructional Psychology(2005) K. Augustyniak, J. Murphy and D.K. Phillips Levels of cognitive awareness (Is the child thinking while doing math?)Follows Sequential Directions Recognized patterns Estimate quantities Rapid recall of over learned facts Visualize and manipulate mental pictures Sense of Spatial Orientation and Organization Ability to do deductive reasoning Ability to do inductive reasoning Levels of mastery (connect to existing knowledge, uses concrete to build a model, draws representations of the model, translates into mathematical notation; applies to real world; teaches the concepts Screeners that provide this information Skill Screener Cognitive Awareness Math their Way Operations Sequential Directions Math their Way Operations; Classroom Impact Questionnaire Recognize PatternMonitoring Basic Skills Progress; Mathematics Navigator Rapid Recall Rapid Automatic Naming; Monitoring Basic Skills Progress Visualize Math their Way Operations Visual Spatial Sense Pattern Block Design Fluid Reasoning Monitoring Basic Skills Progress; Mathematics Navigator Levels of mastery Math their Way Operations and Place Value Quantity Discrimination Number Sense Typically K-1st grade skill; consider using with older students if you suspect a number sense issue; use first grade norms for all grades 2 and aboveHighly predictive of dyscalculia Quantity Discrimination The student is given a sheet containing pairs of numbers. In each number pair, one number is larger than the other. The student identifies the larger number in each pair.

Quantity Discrimination Early Numeracy SkillNumber Sense Administration Time 1 minute Administration Schedule Beginning of Kindergarten to end of First GradeScore 1 point for each correct Quantity Discrimination Wait Rule If the student does not respond within 3 seconds on a quantity pair, mark a (/) through the pairDiscontinue Rule If the student us unable to correctly complete the quantity discrimination in the first 5 pairs. Quantity Discrimination Directions The sheet on your desk has pairs of numbers. In each set, one number is bigger than the other. When I say, 'start,' tell me the name of the number that is larger in each pair. Start at the top of this page and work across the page [demonstrate by pointing]. Try to figure out the larger number for each example.. When you come to the end of a row, go to the next row. Are there any questions? [Pause] Start. NOTE: If the student has difficulties with speech production, the examiner can use this alternate wording for directions: When I say, 'start,' point to the number that is larger in each pair Quantity Discrimination Scoring Quantity Discrimination Scoring Quantity Discrimination Practice

]Turn to the appropriate page in your practice book. I am going to be the student. You will grade the sheet in your practice booklet. 81Quantity Discrimination

DPS CBM Benchmark Guidelines for SLD Eligibility Determination The score for fall 1st grade was 8

According to the score where did the student fall for QD for fall 1st grade?

At or Above Benchmark?Below Benchmark?Well Below Benchmark?

Missing Number Number Sense Typically K-1st grade skill; consider using with older students if you suspect a number sense issue; use first grade norms for all grades 2 and aboveHighly predictive of dyscalculia

Missing Number The student is given a sheet containing multiple number series. Each series consists of 3-4 numbers that appear in sequential order. The student states aloud the missing number.

Missing Number Early Numeracy SkillNumber Sense Administration Time 1 minute Administration Schedule Beginning of Kindergarten to end of First GradeScore 1 point for each correct Missing Number Wait Rule If the student does not respond within 3 seconds on a quantity pair, mark a (/) through the numberDiscontinue Rule If the student us unable to name the missing number in the first 5 pairs. Missing Number Directions The sheet on your desk has sets of numbers. In each set, a number is missing. When I say, 'start,' tell me the name of the number that is missing from each set of numbers. Start at the top of this page and work across the page [demonstrate by pointing]. Try to figure out the missing number for each example.. When you come to the end of a row, go to the next row. Are there any questions? [Pause] Start. NOTE: If the student has difficulties with speech production, the examiner can give the student a pencil and use this alternate wording for directions: When I say, 'start, write in the number that is missing from each set of numbers.Missing Number Scoring Missing Number Scoring Missing Number Practice

]Turn to the appropriate page in your practice book. I am going to be the student. You will grade the sheet in your practice booklet. 90Missing Number

DPS CBM Benchmark Guidelines for SLD Eligibility Determination The score for fall 1st grade was 8

According to the score where did the student fall for QD for fall 1st grade?

At or Above Benchmark?Below Benchmark?Well Below Benchmark?

Math Their Way Screener

Pre-number Concepts

K-2nd Grade 2rd-12 grade (select subtest tests based on knowledge of the students skills)Pre-number Concepts and Skills Counting by Rote MemoryOne-to-One Correspondence Instant RecognitionConservation of Number Counting Backwards Estimation of Objects Numeral RecognitionNumeral Forms How far can you count?Have the child count as far as possible up to 100.Have the. child count by 2s, 5s, and 10s.12345Pre-number Concepts and Skills Counting by Rote MemoryOne-to-One Correspondence Conservation of Number Instant RecognitionCounting Backwards Estimation of Objects Numeral RecognitionNumeral Forms Objective: verbally counting while physically or mentally touching the object once Materials: 24 counters, 5 blocks each a different color Procedure: group counters by 4, 8, and 12; child chooses which to count out loud; count other groups; count all.5 blocks of a different color; count the cubes; begin with the blue cube and count all of them; count the cubes but make the green one the last cube; count the cubes but make the yellow cube five

Pre-number Concepts and Skills Counting by Rote MemoryOne-to-One Correspondence Instant RecognitionConservation of Number Counting Backwards Estimation of Objects Numeral RecognitionNumeral Forms Objective: verbally counting while physically or mentally touching the object once Materials: 24 counters, 5 blocks each a different color Procedure: group counters by 4, 8, and 12; child chooses which to count out loud; count other groups; count all.5 blocks of a different color; count the cubes; begin with the blue cube and count all of them; count the cubes but make the green one the last cube; count the cubes but make the yellow cube five

Pre-number Concepts and Skills Counting by Rote MemoryOne-to-One Correspondence Instant RecognitionConservation of Number Counting Backwards Estimation of Objects Numeral RecognitionNumeral Forms Objective: recognized groups of 2,3,4, and 5 w/out counting Materials: 14 counters Procedure: group counters 2, 3, 4, and 5 randomly; ask to the child to point to the group of three, four, two, five, three, etc. Do not allow the child time to verbally or physically count the objects

Pre-number Concepts and Skills Counting by Rote MemoryOne-to-One Correspondence Instant RecognitionConservation of Number Counting Backwards Estimation of Objects Numeral RecognitionNumeral Forms Objective: a quantity remains consistent Materials: 20 blocks Procedure: align two sets of 10 blocks; ask to the child if there are the same number of blocks in each set; if the child says yes then spread out one set of blocks and then ask if there are the same number of blocks in each set; if the child says yes then they have conservation of number; ask the child to explain their answer to make sure it wasnt a guess

Pre-number Concepts and Skills Counting by Rote MemoryOne-to-One Correspondence Instant RecognitionConservation of Number Counting Backwards Estimation of Objects Numeral RecognitionNumeral Forms Objective: counting backwards from various starting pointsMaterials: 20 counters Procedure: ask child to place 7 counters in a row; cover one counter and ask the child how many are there now; continue covering one counter at a time and asking how many; repeat with larger amounts; add counters and see if the child can count on from the original amount or do they need to count all objects

Pre-number Concepts and Skills Counting by Rote MemoryOne-to-One Correspondence Instant RecognitionConservation of Number Counting Backwards Estimation of Objects Numeral RecognitionNumeral Forms Objective: counting backwards from various starting pointsMaterials: 8-10 stacks of Unifix cubes with different number of cubes in each stack Procedure: place the stacks in a row; point to one stack at a time and ask the child to tell you how many cubes are in each stack; if they count silently ask them to explain to you how they got that number; as you point to the next stack see if they are counting on or backwards or starting from the first cube each time

Pre-number Concepts and Skills Counting by Rote MemoryOne-to-One Correspondence Instant RecognitionConservation of Number Counting Backwards Estimation of Objects Numeral RecognitionNumeral Forms Objective: estimation of quantity Materials: 3 jars labeled A, B, C; jar A 25 objects; jar B 50 objects; jar C 100 objectsProcedure: Ask the child toe estimate how many beans are in jar A; then say If there are ___ beans in jar A, then how many beans do you think there are in Jar B? ; repeat with Jar C; you are not looking for a accuracy in the estimation but how they compare one jar to another jar; ask the child to explain their estimation

Pre-number Concepts and Skills Counting by Rote MemoryOne-to-One Correspondence Instant RecognitionConservation of Number Counting Backwards Estimation of Objects Numeral RecognitionNumeral Forms Objective: child names the numeral out of sequence from memory Materials: number cards 0-10 and 11-20Procedure: randomly place number 0-10 on the table; ask the child to point to a number and tell you the name; repeat with numbers 11-20Extension: try two and three digit numbers

18371002491118131720121419515Pre-number Concepts and Skills Counting by Rote MemoryOne-to-One Correspondence Instant RecognitionConservation of Number Counting Backwards Estimation of Objects Numeral RecognitionNumeral Forms Objective: child names the numeral out of sequence from memory Materials: number cards 0-9; blank paper; pencilProcedure: show the child a number and have them write it on the paper; look for reversals, initial position, ease or fluency of writing; pencil grip and position;Extension: write the numbers from memory

183702495

Number Operations K-12 Grade

Number Operations Simple Addition/Subtraction Concept levelSimple Addition/Subtraction Connecting Level Simple Addition Symbolic LevelSimple Addition Visualization Level Objective: child shows knowledge and understanding of combinations within each number from 3-10Materials: beansProcedure: have the child place five beans in your hand; place some in the open hand and others in the closed hand; child must determine how many beans are in the closed hand; repeat through all possible number combinations up to 10

Number Operations Simple Addition/Subtraction Concept levelSimple Addition/Subtraction Connecting Level Simple Addition Symbolic LevelSimple Addition Visualization Level Objective: child reads an equation and solves using objects Materials: beans; equation cards Procedure: show the child an equation card; ask the child to use the beans to show what the card means; have them do both horizontal and vertical

Number Operations Simple Addition/Subtraction Concept levelSimple Addition/Subtraction Connecting Level Simple Addition Symbolic LevelSimple Addition Visualization Level Objective: child shows that they can record an addition and subtraction problem and solve with manipulativesMaterials: beans; pencil and paper Procedure: Verbally tell the child an addition equation; ask them to record the equation and solve with manipulatives

Number Operations Simple Addition/Subtraction Concept levelSimple Addition/Subtraction Connecting Level Simple Addition Symbolic LevelSimple Addition Visualization Level Objective: child shows that he or she can visualize addition and subtraction problems and find the solution without materials Materials: noneProcedure: tell the child a number story; ask the child to close their eyes and visualize the story in their head

Number Operations Multiplication For students who are ready, you might want to consider doing a few multiplication problems at the concept, connecting and symbolic level Place Value K-12 Grade

Place Value Building Large Numbers with Manipulatives Concept levelBuilding Large Numbers with Manipulatives connecting levelBuilding Large Numbers with Manipulatives symbolic levelRegrouping Concept LevelRegrouping Connecting Level Regrouping Symbolic Level

Objective: Child demonstrates understanding of large numbers with manipulatives Materials: base ten blocks 10s and 1s; place value mat Procedure: ask the child to build two digit numbers on the place value mat

24Place Value Building Large Numbers with Manipulatives Concept levelBuilding Large Numbers with Manipulatives connecting levelBuilding Large Numbers with Manipulatives symbolic levelRegrouping Concept LevelRegrouping Connecting Level Regrouping Symbolic Level

Objective: Child demonstrates understanding of written numbers with manipulatives Materials: base ten blocks 10s and 1s; place value mat; two digit number cards Procedure: ask the child to build two digit numbers on the place value mat

24Place Value Building Large Numbers with Manipulatives Concept levelBuilding Large Numbers with Manipulatives connecting levelBuilding Large Numbers with Manipulatives symbolic levelRegrouping Concept LevelRegrouping Connecting Level Regrouping Symbolic Level

Objective: Child demonstrates understanding of written numbers with manipulatives Materials: base ten blocks 10s and 1s; place value mat; two digit number cards Procedure: tell the child a two digit number; child records the number then demonstrates the number with the base ten blocks Extension: build 3 and 4 digit numbers

24

Place Value Building Large Numbers with Manipulatives Concept levelBuilding Large Numbers with Manipulatives connecting levelBuilding Large Numbers with Manipulatives symbolic levelRegrouping Concept LevelRegrouping Connecting Level Regrouping Symbolic Level

Objective: Child demonstrates regrouping with manipulativesMaterials: equations that require regrouping; base ten blocks; place value chartProcedure: verbalize an equation as the child solves the problem on the mat Extension: build 3 and 4 digit numbers

18 + 7Place Value Building Large Numbers with Manipulatives Concept levelBuilding Large Numbers with Manipulatives connecting levelBuilding Large Numbers with Manipulatives symbolic levelRegrouping Concept LevelRegrouping Connecting Level Regrouping Symbolic Level

Objective: Child demonstrates regrouping with manipulativesMaterials: equations that require regrouping; base ten blocks; place value chart; equation cardsProcedure: show an equation card; child solves the problem on the mat Extension: build 3 and 4 digit numbers

18 + 7Place Value Building Large Numbers with Manipulatives Concept levelBuilding Large Numbers with Manipulatives connecting levelBuilding Large Numbers with Manipulatives symbolic levelRegrouping Concept LevelRegrouping Connecting Level Regrouping Symbolic Level

Objective: Child demonstrates regrouping with manipulativesMaterials: equations that require regrouping; base ten blocks; place value chart; equation cardsProcedure: show an equation card; child solves the problem on the mat Extension: build 3 and 4 digit numbers

18=7

Correct Digit MBSPCorrect Digit: MBSP The student is given a sheet containing computation problems appropriate for their grade level. There are 25 problems per sheet. The student simply answers the problems.

These are the directions for completing the correct digit probes using the Monitoring Basic Skills Porgress (MBSP) probes. The scoring is very similar in Aimsweb. 120Correct Digit Math SkillComputation Administration Time 1 to 6 minutes depending on the grade level (see MBSP Manual)Administration Schedule 1st to 6th grade Score 1 point for each Correct Digit 1 point for each Correct Problem Wait Rule No wait rule Discontinue Rule No discontinue rule Correct Digit Directions SEE PAGE 2 in the MBSP Book for Directions

Go through the directions in MBSP 122Scoring Math CBMHow to score Math CBM-Count the total number of correct digits (CD). 25+16 41

2 CD 35+16 50

1 CD 25x16 150 250 4068 CDCounting the correct digits is more sensitive than just if the answer is correct.123Scoring Math CBMScored as CorrectScored as correct: If the student has the correct answer, credit is given for the longest method used to solve the problem, even if work is not shown.If a problem has been crossed out or started, but not completed, the student receives credit for any correct digits.Reversed or rotated digits are scored as correct with the exception of 6s and 9s. With 6s and 9s, it is not possible to tell which one the student meant to write.In multiplication problems, any symbol used as a place holder is counted as a correct digit as long as it is holding a place that needs to be held.

124Scoring Math CBMOther ConsiderationsAll errors are marked with a slash.Parts of the answer above the line, such as carries or borrows, are not counted as correct digits.In division, a basic fact is when both the divisor and the quotient are 9 or less. The total CD is always 1. Remainders of 0 are not counted as correct digits.If a student finishes in less than 2 minutes, note the number of seconds it took to complete and prorate the score

125Correct Digit Practice

2 cd2 cd2 cd0 cd1 cd0 cd0 cd1 cd1 cd2 cd2 cd2 cdTurn to the appropriate page in your practice book. I am going to be the student. You will grade the sheet in your practice booklet. 126

Correct Digit 127DPS CBM Benchmark Guidelines for SLD Eligibility Determination The score for fall 2nd grade was 15

According to the score where did the student fall for CD for fall 2nd grade?

At or Above Benchmark?Below Benchmark?Well Below Benchmark?

Rapid Automatic Naming Processing Speed Ability to recall over learned material Hints at memory related issues with mathematic learning RAN Norms Have the child name the colors on each page. Use a stopwatch to calculate the time it takes for them to name the colors. Add the RAN 1 and RAN 2 to determine a score. Total number of seconds Grade level >111< K111-95K94-761st grade 75-672nd grade 66-643rd grade 63-594th grade 58-525th grade 51-496th grade 48-457th grade 45-408th grade