1. How the Brain Learns Math Specially Designed Instruction in Math PDU

2. Old 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 alegblocks using the CRA strategies CLOCLO

3. Manipulative make it concrete We are going to add and subtract 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

4. 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

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

6. CRA Algebra- using Algeblocks Y Y =Y 2

7. CRA Algebra- using Algeblocks X Y =XY

8. Algeblocks Key 1 sq unit X Y x2x2 Y2Y2 XY

9. Basic Mat: -3+2 + -

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

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

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

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

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

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

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

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

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

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

20. 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 CLOCLO

21. 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.”

22. 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

23. 2008 Recommendations Algebra is the most important topic in math -study of the rules of operations and relations

24. 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

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

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

27. 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

28. 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.

29. 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

30. 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

31. 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

32. 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

33. 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

34. Teacher directed verses Student directed inconclusive - rescind recommendation that instruction should be one or the other

35. 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.”

36. 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

37. 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

38. Effective Teaching Modeling CLO Clearly Communicated Academic Language Checks for Understanding of the CLO

39. Group T Chart - Reflection of NMP What was new information? How is this going to change your practice?

40. 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 CLOCLO

41. 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.

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

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

44. Which has more?

45. 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

46. Subitizing

47. Counting

48. 2 types of subitizing perceptional conceptual

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

50. 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

51. 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

52. 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.

53. Digit Span Memory English speakers get about 4-5 Native Chinese speakers recall all of the numbers

54. 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

55. 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)

56. 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

57. 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?

58. 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.

59. 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

60. 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.

61. 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.

62. 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

63. 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

64. 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 50 a little a lot

65. 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 50 a little a lot 1 10 20 30 40 50 a little a lot

66. 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 50 a little a lot 1 10 20 30 40 50 a little a lot

67. Language and Multiplication 25 x 30= Exact Approximate

68. Assessing 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 CLOCLO

69. Does the instructional approach impact the determination of a disability? processing speed reasoningnumber sense visual-spatial

70. Types of Math Disorders Number Sense Counting Skill Deficits Arithmetic Skill Procedural 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

71. Primary Assessments How Children Learn Mathematics by M. Sharma Digit span-student repeats in a string of numbers either forwards or backwards Magnitude comparison-student chooses the largest of visually or verbally presented numbers Missing number-student names a missing number from a sequence of numbers between zero and 20 Number knowledge test-basic measure of number sense Numbers from dictation-student writes numbers from word dictation Number identification-student identifies numbers between zero and 20 from printed numbers Quantity discrimination-student identifies the larger of two printed numbers

72. Screeners that provide this information SkillScreenerWhat does it tell us Missing NumberMissing 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 relationships -Leads 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

73. 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

74. Screeners that provide this information SkillScreener Cognitive AwarenessMath their Way Operations Sequential DirectionsMath their Way Operations; Classroom Impact Questionnaire Recognize PatternMonitoring Basic Skills Progress; Mathematics Navigator Rapid RecallRapid Automatic Naming; Monitoring Basic Skills Progress VisualizeMath their Way Operations Visual Spatial SensePattern Block Design Fluid ReasoningMonitoring Basic Skills Progress; Mathematics Navigator Levels of masteryMath their Way Operations and Place Value

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

76. 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.

77. Quantity Discrimination Early Numeracy Skill Number Sense Administration Time 1 minute Administration Schedule Beginning of Kindergarten to end of First Grade Score1 point for each correct Quantity Discrimination Wait RuleIf the student does not respond within 3 seconds on a quantity pair, mark a (/) through the pair Discontinue RuleIf the student us unable to correctly complete the quantity discrimination in the first 5 pairs.

78. 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”

79. Quantity Discrimination Scoring 1 point for correct discrimination 1 point for a self correction if done within 3 second

80. Quantity Discrimination Scoring 0 point for incorrect discrimination 0 point for correct self correction if done after 3 second 0 point for number other than appears in the QD0 points for pairs skipped

81. Quantity Discrimination Practice ]

82. Quantity Discrimination

83. DPS CBM Benchmark Guidelines for SLD Eligibility Determination The score for fall 1 st grade was 8 According to the score where did the student fall for QD for fall 1 st grade? At or Above Benchmark? Below Benchmark? Well Below Benchmark?

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

85. 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.

86. Missing Number Early Numeracy Skill Number Sense Administration Time 1 minute Administration Schedule Beginning of Kindergarten to end of First Grade Score1 point for each correct Missing Number Wait RuleIf the student does not respond within 3 seconds on a quantity pair, mark a (/) through the number Discontinue RuleIf the student us unable to name the missing number in the first 5 pairs.

87. 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.”

88. Missing Number Scoring 1 point for missing number read correctly 1 point for missing number self corrected, if done within 3 second

89. Missing Number Scoring 0 point for incorrect missing number 0 point for missing number if done after 3 second 0 points for pairs skipped

90. Missing Number Practice ]

91. Missing Number

92. DPS CBM Benchmark Guidelines for SLD Eligibility Determination The score for fall 1 st grade was 8 According to the score where did the student fall for QD for fall 1 st grade? At or Above Benchmark? Below Benchmark? Well Below Benchmark?

93. Math Their Way Screener

95. Pre-number Concepts K-2 nd Grade 2rd-12 grade (select subtest tests based on knowledge of the student’s skills)

96. Pre-number Concepts and Skills Counting by Rote Memory One-to-One Correspondence Instant Recognition Conservation of Number Counting Backwards Estimation of Objects Numeral Recognition Numeral 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.

97. Pre-number Concepts and Skills Counting by Rote Memory One-to-One Correspondence Conservation of Number Instant Recognition Counting Backwards Estimation of Objects Numeral Recognition Numeral 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

98. Pre-number Concepts and Skills Counting by Rote Memory One-to-One Correspondence Instant Recognition Conservation of Number Counting Backwards Estimation of Objects Numeral Recognition Numeral 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

99. Pre-number Concepts and Skills Counting by Rote Memory One-to-One Correspondence Instant Recognition Conservation of Number Counting Backwards Estimation of Objects Numeral Recognition Numeral 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

100. Pre-number Concepts and Skills Counting by Rote Memory One-to-One Correspondence Instant Recognition Conservation of Number Counting Backwards Estimation of Objects Numeral Recognition Numeral 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 wasn’t a guess

101. Pre-number Concepts and Skills Counting by Rote Memory One-to-One Correspondence Instant Recognition Conservation of Number Counting Backwards Estimation of Objects Numeral Recognition Numeral Forms Objective: counting backwards from various starting points Materials: 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

102. Pre-number Concepts and Skills Counting by Rote Memory One-to-One Correspondence Instant Recognition Conservation of Number Counting Backwards Estimation of Objects Numeral Recognition Numeral Forms Objective: counting backwards from various starting points Materials: 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

103. Pre-number Concepts and Skills Counting by Rote Memory One-to-One Correspondence Instant Recognition Conservation of Number Counting Backwards Estimation of Objects Numeral Recognition Numeral Forms Objective: estimation of quantity Materials: 3 jars labeled A, B, C; jar A 25 objects; jar B 50 objects; jar C 100 objects Procedure: 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

104. Pre-number Concepts and Skills Counting by Rote Memory One-to-One Correspondence Instant Recognition Conservation of Number Counting Backwards Estimation of Objects Numeral Recognition Numeral Forms Objective: child names the numeral out of sequence from memory Materials: number cards 0-10 and 11-20 Procedure: randomly place number 0-10 on the table; ask the child to point to a number and tell you the name; repeat with numbers 11-20 Extension: try two and three digit numbers 1837100249 1118131720121419 5 15

105. Pre-number Concepts and Skills Counting by Rote Memory One-to-One Correspondence Instant Recognition Conservation of Number Counting Backwards Estimation of Objects Numeral Recognition Numeral Forms Objective: child names the numeral out of sequence from memory Materials: number cards 0-9; blank paper; pencil Procedure: 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

106. Number Operations K-12 Grade

107. Number Operations Simple Addition/Subtraction Concept level Simple Addition/Subtraction Connecting Level Simple Addition Symbolic Level Simple Addition Visualization Level Objective: child shows knowledge and understanding of combinations within each number from 3-10 Materials: beans Procedure: 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

108. Number Operations Simple Addition/Subtraction Concept level Simple Addition/Subtraction Connecting Level Simple Addition Symbolic Level Simple 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

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

110. Number Operations Simple Addition/Subtraction Concept level Simple Addition/Subtraction Connecting Level Simple Addition Symbolic Level Simple Addition Visualization Level Objective: child shows that he or she can visualize addition and subtraction problems and find the solution without materials Materials: none Procedure: tell the child a number story; ask the child to close their eyes and visualize the story in their head

111. 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

112. Place Value K-12 Grade

113. Place Value Building Large Numbers with Manipulatives Concept level Building Large Numbers with Manipulatives connecting level Building Large Numbers with Manipulatives symbolic level Regrouping Concept Level Regrouping 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 24

114. Place Value Building Large Numbers with Manipulatives Concept level Building Large Numbers with Manipulatives connecting level Building Large Numbers with Manipulatives symbolic level Regrouping Concept Level Regrouping 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 24

115. Place Value Building Large Numbers with Manipulatives Concept level Building Large Numbers with Manipulatives connecting level Building Large Numbers with Manipulatives symbolic level Regrouping Concept Level Regrouping 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

116. Place Value Building Large Numbers with Manipulatives Concept level Building Large Numbers with Manipulatives connecting level Building Large Numbers with Manipulatives symbolic level Regrouping Concept Level Regrouping Connecting Level Regrouping Symbolic Level Objective: Child demonstrates regrouping with manipulatives Materials: equations that require regrouping; base ten blocks; place value chart Procedure: verbalize an equation as the child solves the problem on the mat Extension: build 3 and 4 digit numbers 18 + 7

117. Place Value Building Large Numbers with Manipulatives Concept level Building Large Numbers with Manipulatives connecting level Building Large Numbers with Manipulatives symbolic level Regrouping Concept Level Regrouping Connecting Level Regrouping Symbolic Level Objective: Child demonstrates regrouping with manipulatives Materials: equations that require regrouping; base ten blocks; place value chart; equation cards Procedure: show an equation card; child solves the problem on the mat Extension: build 3 and 4 digit numbers 18 + 7

118. Place Value Building Large Numbers with Manipulatives Concept level Building Large Numbers with Manipulatives connecting level Building Large Numbers with Manipulatives symbolic level Regrouping Concept Level Regrouping Connecting Level Regrouping Symbolic Level Objective: Child demonstrates regrouping with manipulatives Materials: equations that require regrouping; base ten blocks; place value chart; equation cards Procedure: show an equation card; child solves the problem on the mat Extension: build 3 and 4 digit numbers 18=7

119. Correct Digit MBSP

120. Correct 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.

121. Correct Digit Math SkillComputation Administration Time 1 to 6 minutes depending on the grade level (see MBSP Manual) Administration Schedule 1 st to 6 th grade Score1 point for each Correct Digit 1 point for each Correct Problem Wait RuleNo wait rule Discontinue RuleNo discontinue rule

122. Correct Digit Directions SEE PAGE 2 in the MBSP Book for Directions

123. Scoring Math CBM How to score Math CBM-Count the total number of correct digits (CD). 25 +16 41 2 CD 35 +16 50 1 CD 25 x16 150 250 406 8 CD

124. Scoring Math CBM Scored as Correct Scored 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 6’s and 9’s. With 6’s and 9’s, 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.

125. Scoring Math CBM Other Considerations All 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

126. Correct Digit Practice 2 cd 0 cd1 cd 0 cd 1 cd 2 cd

127. Correct Digit

128. DPS CBM Benchmark Guidelines for SLD Eligibility Determination The score for fall 2 nd grade was 15 According to the score where did the student fall for CD for fall 2 nd grade? At or Above Benchmark? Below Benchmark? Well Below Benchmark?

129. Rapid Automatic Naming Processing Speed Ability to recall over learned material Hints at memory related issues with mathematic learning

132. 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 secondsGrade level >111< K 111-95K 94-761 st grade 75-672 nd grade 66-643 rd grade 63-594 th grade 58-525 th grade 51-496 th grade 48-457 th grade 45-408 th grade <409 th grade +

133. MBSP Concepts and Application K-6 grade level assessment Fluid reasoning and grade level skills

134. Concepts and Application: MBSP The student is given a sheet containing concepts and application math problems appropriate for their grade level. There are 18 to 24 problems per sheet. The student simply answers the problems.

135. Concepts and Application Math SkillProblem Solving Administration Time 6 to 8 minutes depending on the grade level (see MBSP Manual) Administration Schedule 2nd to 6 th grade Score1 point for each blank correctly completed with additional points for correct digit Wait RuleNo wait rule Discontinue RuleNo discontinue rule

136. Concepts and Application Directions SEE PAGE 12 in the MBSP Book for Directions

137. Concepts and Application Scoring Student receives one point for each blank that is correct with additional points for certain types of problems. 2 points 3 points

138. Concepts and Application Scoring 2nd-4 th Grade: For word problems and money problems, one point is awarded for each correct digit. Money problems must have the money symbol correct ($, decimal or ¢). 2 points for correct digit with money symbol 3 points for correct digits with money symbol

139. Concepts and Application Scoring 5 th and 6th Grade: For word problems and charts and graphs, one point is awarded for each correct digit. 3 points for correct digits

140. Concepts and Application Practice

141. Concepts and Application

142. DPS CBM Benchmark Guidelines for SLD Eligibility Determination The score for spring 2 nd grade was 19 According to the score where did the student fall for spring 2 nd grade? At or Above Benchmark? Below Benchmark? Well Below Benchmark?

143. Mathematics Navigator

144. Indicate the skill deficit, however knowledge about number sense, fluid reasoning, rapid recall and visual-spatial processing can be observed Performance LevelInterpretation 0-25%Guessing Level; go down one level 25%-50%Student needs skill development 50%-75%Student needs portions of the skills 75% and upStudent should be tested on the next level

145. If incorrect then interview How did you get that answer? Record the response

146. If incorrect then interview How did you get that answer? Record the response

147. If incorrect then interview How did you get that answer? Record the response

148. Evidence of deductive reasoning Evident of inductive reasoning Evidence of proficiency in visual spatial reasoning Evidence of procedural knowledge

149. Pattern Block Design Visual-Spatial Processing

150. Pattern Block Design Objective: To determine a child’s ability to complete a pattern black design with and without guidelines Materials: Pattern blocks and Block Design Cards Directions: Have the student complete the images using the pattern blocks; first have them complete the design with guidelines; next have them complete the design without guidelines.

151. Mathematics Fishbone

152. Number Sense (Math their Way; quantity discrimination; missing number ) Operations ( Math their Way; ) Reasoning and Problem Solving (Monitoring Basic Skills Progress; Pattern Block Design; Mathematics Navigator Screeners ) Fluency (Monitoring Basic Skills Progress; Rapid Automatic Naming ) Counting by Rote Memory: One-to-One Correspondence: Instant Recognition: Conservation of Number: Counting Backwards: Estimation of Objects: Numeral Recognition: Numeral Forms : MBSP Correct Digit CBM (k-6; use 6 th grade for 6-12): Quantity Discrimination CBM: Color naming RAN: MBSP Applications (k-6): Mathematics Navigator Screener (6-12): Missing Number CBM: Pattern Block Design : + & - Concept Level: + & - Connecting Level: + & - Symbolic Level: + * - Visualization Level: Regrouping + & - Concept Level: Regrouping + & Connecting Level: Regrouping + & Symbolic Level: Place Value Concept Level: Place Value Connecting Level: Place Value Symbolic Level : Regrouping + & - Concept Level: Regrouping + & Connecting Level: Regrouping + & Symbolic Level:

153. Number SenseOperations Reasoning and Problem SolvingFluency Counting by Rote Memory: to 100 One-to-One Correspondence: to 10 Instant Recognition: 10/10 Conservation of Number: 1/3 Counting Backwards: from 30 Estimation of Objects: 3/3 Numeral Recognition: 10/10 Numeral Forms : 10/10 Cardinality: 5/5 Fluency Assessment: 1. 14%iles 2. 18%ile 3. 22%ile Color naming RAN: 5 grade level Problem Solving and Applications and Interview : SS 92 Key Math Pattern Block Design or Visual Spatial Reasoning : SS 88 Key Math + Concept Level: 3/3 + Connecting Level: 0/3 + Symbolic Level: 3/3 + Visualization Level: 2/3 Regrouping + Concept Level: 0/3 Regrouping + Connecting Level: 0/3 Regrouping + Symbolic Level: 3/3 Place Value Concept Level: 3/3 Place Value Connecting Level: 3/3 Place Value Symbolic Level : 3/3 Name: _____Greg_____ Grade: 5 th Executive Functioning Skills (structured or unstructured): - Poor focus in Math; excellent focus in Reading and unstructured environments Other: -Older sibling have no issues with math; parents didn’t report any history of math difficulty - Concept Level: 3/3 - Connecting Level: 0/3 - Symbolic Level: 3/3 - Visualization Level: 3/3 Regrouping - Concept Level: 0/3 Regrouping - Connecting Level: 0/3 Regrouping - Symbolic Level: 1/3 Verbal Reasoning: SS 102 Poor performance in Math

154. Is there evidence to suggest difficulty with executive functi oning? Is there evidence to suggest difficulty with number sense? Is there evidence to suggest difficulty with reasoning ? Root Cause of Math Difficulty Student has the ability to sustain focus when basic skills are automatic Is there evidence to suggest difficulty with conceptual understanding of operations ? Student is able to learn through various methods (mastery, inquiry) yes no 1. 2. 3. Is there evidence to suggest difficulty with processing speed Is there evidence to suggest problems with concepts and application? yes no yes no yes no yes no yes no Prioritize the concerns 1.Conceptual Understanding of Operations 2.______________________________ 3.______________________________ 4.______________________________ 5.______________________________ 6.______________________________ Executive Functioning Concerns that impact Procedural Math Reasoning Concerns that impact problem solving Number Sense Concerns Computation Concerns Math Fluency Concerns Concepts and Application Concerns Create treatment plan Name: ___________Greg ___________

155. Treatment Plan for Greg Direct InstructionAccommodations and Modifications of the Core Curriculum Home to School Connections Recommendations: instruction of the conceptual understanding of operations using a CRA approach; continue to build visualization concepts through 10 frame computation exercises Areas of Concern: Lacks operational conceptual understanding Home Engagement:  High X Medium  Low  None Goals : Increase understanding of operational concepts Accommodations: Allow use of computing devices when doing problem solving tasks; make sure CRA approach is used in introducing new concepts Meaningful Homework Tasks : allow use of manipulative to solve basic computation problems; homework that follows a CRA approach. Plan: 15 minutes of supplement during math instruction by the special education teacher; use Origo and Hands on Standards Materials Modifications: No modifications needed at this time

156. Number SenseOperations Reasoning and Problem SolvingFluency Counting by Rote Memory: to 100 One-to-One Correspondence: to 10 Instant Recognition: 3/10 Conservation of Number: 1/3 Counting Backwards: from 30 Estimation of Objects: 0/3 Numeral Recognition: 10/10 Numeral Forms : 10/10 Cardinality: 0/5 Fluency Assessment: 1. 8%iles 2. 8%ile 3. 7%ile Color naming RAN: 6th grade level Problem Solving and Applications and Interview : SS 81 Key Math; evidence of strong reasoning ability Pattern Block Design or Visual Spatial Reasoning : SS 71 Key Math + Concept Level: 2/3 + Connecting Level: 0/3 + Symbolic Level: 2/3 + Visualization Level: 2/3 Regrouping + Concept Level: 0/3 Regrouping + Connecting Level: 0/3 Regrouping + Symbolic Level: 3/3 Place Value Concept Level: 1/3 Place Value Connecting Level:1/3 Place Value Symbolic Level : 0/3 Name: ___Samuel_____ Grade: 6 th Executive Functioning Skills (structured or unstructured): - Poor focus in in structured and unstructured environments Other: -Parents didn’t report any history of math difficulty - Concept Level: 2/3 - Connecting Level: 0/3 - Symbolic Level: 0/3 - Visualization Level: 0/3 Regrouping - Concept Level: 0/3 Regrouping - Connecting Level: 0/3 Regrouping - Symbolic Level: 1/3 Verbal Reasoning: SS 88 Poor performance in Math

157. Is there evidence to suggest difficulty with executive functi oning? Is there evidence to suggest difficulty with number sense? Is there evidence to suggest difficulty with reasoning ? Root Causes of Math Difficulty Student has the ability to sustain focus when basic skills are automatic Is there evidence to suggest difficulty with conceptual understanding of operations ? Student is able to learn through various methods (mastery, inquiry) yes no 1. 2. 3. Is there evidence to suggest difficulty with processing speed Is there evidence to suggest problems with concepts and application? yes no yes no yes no yes no yes no Prioritize the concerns 1.Executive Functioning 2.Number Sense 3.Operations 4.Visual Spatial Reasoning 5.______________________________ 6.______________________________ Executive Functioning Concerns that impact Procedural Math Reasoning Concerns that impact problem solving Number Sense Concerns Computation Concerns Math Fluency Concerns Concepts and Application Concerns Create treatment plan Name: __________Sam _____________

158. Treatment Plan for Sam Direct InstructionAccommodations and Modifications of the Core Curriculum Home to School Connections Recommendations: provide structured environment; teach metacogntiion skills related to executive functioning skills ; develop instant recognition of number at the conceptual level; develop cardinality; instruction on the conceptual level of operations; develop visual spatial skills Areas of Concern: Lacks operational conceptual understanding; poor executive functioning; poor number sense; Home Engagement:  High X Medium  Low  None Goals : Increase instant recognition of number and cardinality; Increase understanding of operational concepts; increase visual spatial reasoning skills Accommodations: Allow use of computing devices when doing problem solving tasks; make sure CRA approach is used in introducing new concepts; allow manipulative; extra time to complete tasks; distraction free environment to complete work Meaningful Homework Tasks : allow use of manipulative to solve basic computation problems; homework that follows a CRA approach; games to develop number sense and computational understanding Plan: 30 min outside the general education classroom; daily subitizing and counting skills and games; Use Origo and Hands on Standards materials; puzzle work Modifications: No modifications needed at this time