Specially Designed Instruction in Math PDU Session Two

Specially Designed Instruction in Math PDU Session Two
November 26, 2012 4:30-6:30

Text Chapter Seven: Recognizing and Addressing Mathematics Difficulties

Outcomes for Session Three
Participants will understand the different categories of math learning disabilities Participants will deepen their understanding of dyscalculia Participants will be able to conduct and interpret math diagnostic screeners to determine the area for intervention

conceptual based math procedural based math
Does the instructional approach impact the determination of a disability? conceptual based math procedural based math reasoning number sense processing speed visual-spatial

Types of Math Disorders
Difficulty understanding the concept associated with fluid reasoning Counting Skill Deficits Associated with Number Module dysfunction Arithmetic Skill Number Sense Visual-Spatial Deficit Procedural Disorders Memory Deficit Associated with Executive Functioning Non-verbal reasoning Rapid Recall of over learned material

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

Screeners that provide this information
Skill Screener What does it tell us Missing Number Missing Number CBM These 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 Number Identification 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 Pattern Monitoring Basic Skills Progress; Mathematics Navigator Rapid Recall Rapid Automatic Naming; Monitoring Basic Skills Progress Visualize Visual Spatial Sense Pattern Block Design Fluid Reasoning 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 above Highly predictive of dyscalculia

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.

Early Numeracy Skill Number Sense Administration Time 1 minute Administration Schedule Beginning of Kindergarten to end of First Grade Score 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 pair Discontinue 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
1 point for correct discrimination 1 point for a self correction if done within 3 second

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 QD 0 points for pairs skipped

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.

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 above Highly 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 Skill Number Sense Administration Time
1 minute Administration Schedule Beginning of Kindergarten to end of First Grade Score 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 number Discontinue 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
1 point for missing number read correctly 1 point for missing number self corrected, if done within 3 second

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

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.

Missing Number

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 student’s skills)

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. 4 2 … 1 5 3

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

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

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

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

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

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

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

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 1 8 3 7 10 2 4 9 5 11 18 13 17 20 12 14 19 15

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 1 8 3 7 2 4 9 5

Number Operations K-12 Grade

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

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

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

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

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 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

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

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

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

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

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

Correct Digit MBSP

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. 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.

Correct Digit Math Skill Computation 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

Scoring Math CBM 25 x16 150 250 406 8 CD 25 +16 41 2 CD 35 +16 50 1 CD
How to score Math CBM-Count the total number of correct digits (CD). 25 x16 150 250 406 8 CD 25 +16 41 2 CD 35 +16 50 1 CD Counting the correct digits is more sensitive than just if the answer is correct.

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.

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

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

Correct Digit

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 < K 111-95 K 94-76 1st grade 75-67 2nd grade 66-64 3rd grade 63-59 4th grade 58-52 5th grade 51-49 6th grade 48-45 7th grade 45-40 8th grade <40 9th grade +

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

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. 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.

Concepts and Application
Math Skill Problem Solving Administration Time 6 to 8 minutes depending on the grade level (see MBSP Manual) Administration Schedule 2nd to 6th grade Score 1 point for each blank correctly completed with additional points for correct digit Wait Rule No wait rule Discontinue Rule No discontinue rule

Concepts and Application Directions
SEE PAGE 12 in the MBSP Book for Directions Go through the directions in MBSP

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

2nd-4th 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 ¢). 3 points for correct digits with money symbol 2 points for correct digit with money symbol

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

Concepts and Application 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.

Concepts and Application

The score for spring 2nd grade was 19 According to the score where did the student fall for spring 2nd grade? At or Above Benchmark? Below Benchmark? Well Below Benchmark?

Mathematics Navigator

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

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

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

Pattern Block Design Visual-Spatial Processing

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.

Mathematics Fishbone

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

Next time we meet… Chapter 3 Teaching Students who struggle with Math: Place Value

Specially Designed Instruction in Math PDU Session Two

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