Supporting MATH skills
Models
for selecting interventions
Classwide
Instruction
Individual
Instruction
Models
1. Won’t Do or Can’t Do
Can’t do reasons:
Accuracy, Fluency,
Too hard
2. Powerful Effective teaching
3. Morningside Big 5 Math Targets
Tool Skills
Computation
Solving standard
word problems
Generic Problem
solving
Math Vocabulary
Conversation
Tool skills
• Reading numbers quickly
– 1-9 digit numbers
• Writing numbers quickly and legibly
– 1-9 digit numbers
•
•
•
•
Copying and taking dictation
Place value
Quickly identifying, explaining, manipulating
Math facts fluency
– Number family method
– E.g., 5 9 45 = 5x9 = 45, 9x5= 45, 45~9 + 5, 45~5 + 9
• Number sense of number relationships
Computation
• Working with number combinations
• Addition, subtraction, Multiplication and
division
• Whole numbers, fractions, decimals, &
simple percentage calculations
• “Mental” arithmetic
• Direct and explicit instruction of algorithms
Solving standard work problems
• Typical story problems involving quantities
• Between 20 to 70 word lengths
• Students learn to
– Write an algebraic equation with one unknown
– Determine the unknown by manipulating the
equation and doing arithmetic
• Direct instruction for 10 types of word
problems
• Fluency building
How To….
1)Select students for intervention
using screening data
1) Identify Can’t do problems
2) Conduct math level of assessment.
3) Set up progress monitoring.
4) Plan for implementation.
Who will do what and when.
CBM- To identify who
• Screen early for struggling students (K-1)
– Number Identification
– Quantity Discrimination
– Missing Number
• On going Screening for basic skills (1-6)
– Computation
• Screen for Solving and Predict end of year test (2-6)
– Concepts and Applications
Fuchs Manual:
http://www.rti4success.org/index.php?option=com_content&task=blogca
tegory&id=10&Itemid=56
Concepts and Applications
–
–
–
–
18–25 problems
year-long.
Timed
Score: total
number of
correct blanks
Grade
Time limit
2
8 minutes
3-4
6 minutes
5-6
7 minutes
Early Skills
1. orally identify the
number ( 0 thru100).
13
7
2. orally identify the
larger number
4 __ 6
3. orally identify the
missing number
3
3
Why CBM?
To identify level for Intervention
Common computation errors:
1. Inadequate tool skills and mastery of facts
2. Ineffective strategy (error in steps in
procedure) but sound facts
3. Inadequate facts but effective strategies
To assess TOOL Skills
Single CBM Tools skills
Tool Skill
Single probes
Reading numbers
Correct
Errors
ACCURACY PROBLEM
SLOW AND MANY ERRORS
Writing numbers
Place value
Add facts
Subtract facts
Multiply facts
Division facts
FLUENCY PROBLEM
SLOW AND FEW ERRORS
Using End-of-Year
Benchmarks
Grade
Probe
Kindergarten
First
Maximum score
Benchmark
Data not yet available
Computation
First
30
20 digits
Data not yet available
Second
Computation
45
20 digits
Second
Concepts and Applications
32
20 blanks
Third
Computation
45
30 digits
Third
Concepts and Applications
47
30 blanks
Fourth
Computation
70
40 digits
Fourth
Concepts and Applications
42
30 blanks
Fifth
Computation
80
30 digits
Fifth
Concepts and Applications
32
15 blanks
Sixth
Computation
105
35 digits
Sixth
Concepts and Applications
35
15 blanks
How to…Tool skills
Class-wide Math Intervention
Classwide monitoring
of skill acquisition
After classwide instruction
Tool Skill Flashcard Intervention
15 Min per Day
• Model, Guided Practice, Independent Timed
Practice with delayed error correction
MATH FLASHCARD
PARTNERS
Demonstration Math 1: Flashcards
Tear to make 6 flashcards.
Write answers on back.
Your partner is someone sitting next to you.
STEP 1: GET READY
Get out flashcards and a regular pencil and a
colored pen.
Move to math partner seating.
STEP 2: GUIDED PEER PRACTICE for 3 minutes
Partner 1 SAYS flashcard ANSWERS .
Partner 2 SHOWS FLASHCARDS for 3 seconds.
SAY answer if no answer or mistake.
Trade jobs when timer rings
Do new job for another 3 minutes
Demonstration Math 2 Practice Test
STEP 3: GET READY
Turn passed out practice paper over.
Write name on back.
STEP 4: 2 minute TIMED INDEPENDENT PRACTICE
At “GO!” turn paper over and write answers.
Hold paper in the air when timer rings.
STEP 5: GRADE and CORRECT ERRORS
Mark answers right or wrong with colored pen using
overhead answer key.
Write correct answers to missed problems.
Answers on overhead
ANSWER KEY
2
3
x 3
x 2
x
x
x
x
x
x
x
8
3
x
3
6
x
2
3
x
3
3
x
8
3
6
6
24
18
6
9
24
7
3
3
6
9
3
3
7
8
3
5
3
8
3
x
x
x
x
x
x
21
18
27
21
24
15
24
2
3
3
2
5
3
3
7
8
3
3
3
5
3
x
x
x
x
x
x
6
6
15
21
24
9
15
7
3
3
0
1
3
3
3
0
3
0
3
4
3
x
x
x
x
x
x
21
0
3
9
0
0
12
4
3
3
2
3
3
3
6
0
3
3
3
2
3
x
x
x
x
x
x
12
6
9
18
0
9
6
6
3
3
5
1
3
3
8
9
3
5
3
9
3
x
x
x
x
x
x
18
15
3
24
27
15
27
3
3
3
5
2
3
3
1
2
3
0
3
9
3
9
x
15
x
6
x
3
x
6
x
0
x
27
RECORD
Demonstration MATH 3
Number of digits correct per 2 minutes
50
45
40
35
30
Score
Errors
25
20
15
10
5
0
1
2
3
4
5
6
7
8
9 10 11 12 13 14 15
Practice day
MATH FLASHCARD
PARTNERS
STEP 1: GET READY
Get out flashcards and a regular and a colored pen.
Move to math partner seating.
STEP 2: GUIDED PEER PRACTICE for 3 minutes
Partner 1 SAYS flashcard ANSWERS .
Partner 2 SHOWS FLASHCARDS for 3 seconds.
SAY answer if no answer or mistake.
Trade jobs when timer rings
Do new job for another 3 minutes
STEP 3: GET READY
Turn passed out practice paper over.
Write name on back.
STEP 4: 2 minute TIMED INDEPENDENT PRACTICE
At “GO!” turn paper over and write answers.
Hold paper in the air when timer rings.
STEP 5: GRADE and CORRECT ERRORS
Mark answers right or wrong with colored pen using overhead answer key.
Write correct answers to missed problems.
STEP 6: RECORD
Write score on progress graph.
Pass paper to teacher.
To assess if knows strategies
even with inadequate facts
Computation Error Analysis
Test strategies with “easy” facts to target gaps
Fluency: Time 2 to 5 minutes and mark last problem
Accuracy: Allow to finish untimed
1)
21
+ 43
2)
125
+721
3)
26
+ 5
4)
92
+18
5)
215
+495
6)
54
-12
7)
367
-121
8)
51
-22
9)
40
- 11
10)
932
-167
11)
34
x 2
12)
12
x 24
13)
67
x 2
14)
15)
37
x 25
523
x 5
•(other similar probes with decimal, fractions, rounding, place value)
SKILL
ADDITION
Addition Score > 20 digits per minute?
Add two columns no regrouping
Add three columns no regrouping
Add one column regrouping
Add two columns regrouping
Add three columns regrouping
TEST # JOHN TIM
1
2
3
4
5
SUBTRACTION
Subtract > 20 digits per minute?
Subtract two columns no regrouping
Subtract three columns no regrouping
Subtract two columns with regrouping
Subtract three columns with regrouping and zero
Subtract three columns with regrouping
MULTIPLICATION
Multiplication > 20 digits per minute?
Multiply one digit no regrouping
Multiply two digits no regrouping
Multiply one digit with regrouping
Multiply two digits with regrouping
6
7
8
9
10
11
12
13, 14
15
JANE
TINA
Yes





No


No



Yes





Yes
No
Yes
No






No

Yes







Yes



No
Sample computation Sequence
3RD GRADE
1. addition and subtraction facts 0-20
2. fact families addition and subtraction 0-20
3. 3 digit addition without and with regrouping
4. 3 digit subtraction without and with regrouping
5. 2 and 3 digit addition and subtraction
6.
7.
8.
9.
with and without regrouping
multiplication facts 0-9
division facts 0-9
fact families multiplication and division 0-9
add/subtract fractions with like denominators
flash cards
practice set – same as skill
practice set – same as skill
practice set – same as skill
practice set – same as skill
flash cards
flash cards
practice set – same as skill
practice set – same as skill
(3rds, 4ths, 8ths, 10ths, no regrouping)
10. single digit multiplied by double/triple digit
practice set – same as skill
without regrouping
11. single digit multiplied by double/triple digit
practice set – same as skill
with regrouping
12. single digit divided into double/triple digit
without remainders
13. add and subtract decimals to the hundredths
practice set – same as skill
practice set – same as skill
Fuchs Math: 17 Scripted Topics
•
•
•
•
•
•
•
•
Identifying and writing
numbers to 99
Identifying more, less, and
equal with objects
Sequencing numbers
Using <, >, and = symbols
Skip counting by 10s, 5s,
and 2s
Understanding place value
(introduction)
Identifying operations
Place value (0-50)
•
•
•
•
•
•
•
•
•
Writing number sentences
(story problems)
Place value (0-99)
Addition facts (sums to 18)
Subtraction facts (minuends
to 18)
Review of addition and
subtraction facts
Place value review
2-digit addition (no regrouping)
2-digit subtraction (no
regrouping)
Missing addends
30
How to…Computation
Class-wide Math Intervention
Teacher first reviews steps and write SKILL STEPS on board.
1)
21
+ 43
2)
125
+721
3)
26
+ 5
4)
92
+18
5)
215
+495
6)
54
-12
7)
367
-121
8)
51
-22
9)
40
- 11
10)
932
-167
11)
34
x 2
12)
12
x 24
13)
67
x 2
14)
15)
37
x 25
523
x 5
To subtract two numbers that require regrouping need to know:
1)check ones place (place value)
2)know top number should be bigger
3)know how to take a 10 from tens place to ones place
4)Subtraction facts
MATH COMPUTATION STEP BY STEP COACHES
Tens Ones
1. Look at the ones column.
2. Is top number is smaller than bottom?
YES
So need to make top bigger.
53
- 38
53
-38
4
3. Burrow a 10 from the tens column.
Cross out the top number in tens column.
Subtract 1 from top number  5 – 1 = 4
Write the number on top.
4. Add the 10 to the one column.
10 + 4 = 14
Write in the 1 next to the top number.
5. Subtracted ones column correctly.
(14 – 8= 6)
6. Subtracted tens column correctly.
( 4 - 3 = 1)
53
-38
4
5 14
-38
4
1
54
-38
6
4
5 4
-3 8
16
MATH COMPUTATION COACHES
Demonstration Math 4
Your partner is someone sitting next to you.
STEP 1: GET READY
Move to math partner seating with folder, regular
pencil and a colored pen
Write name on passed out worksheet.
STEP 2: GUIDED PEER PRACTICE for 3 minutes
Partner 1 WRITES as many answers as possible
Partner 2 LOOKS at answers as written
SHOW how to redo missed or stuck answers.
Trade jobs when timer rings
Demonstration MATH 5 : Practice test
STEP 3: GET READY
Turn passed out practice paper over.
Write name on back.
STEP 4: 2 minute TIMED INDEPENDENT PRACTICE
At “GO!” turn paper over and write answers.
Hold paper in the air when timer rings.
MATH COMPUTATION COACHES
STEP 1: GET READY
Move to math partner seating with folder, regular and a colored pen
Write name on passed out worksheet.
STEP 2: GUIDED PEER PRACTICE for 3 minutes
Partner 1 WRITES as many answers as possible
Partner 2 LOOKS at answers as written
SHOW how to redo missed or stuck answers.
Trade jobs when timer rings
STEP 3: GET READY
Turn passed out practice paper over.
Write name on back.
STEP 4: 2 minute TIMED INDEPENDENT PRACTICE
At “GO!” turn paper over and write answers.
Hold paper in the air when timer rings.
STEP 5: GRADE and CORRECT ERRORS
Mark answers right or wrong with colored pen.
Write correct answers to missed problems.
STEP 6: RECORD score on progress graph.
Computation Coaches Intervention
15 Min per Day
• Teachers encouraged to
– Provide applied practice using mastery-level basic
tool skills
– Have stronger do it first. That person can then show
his or her process on paper if needed.
– Make sure students tell how to do it and not just give
answer.
– Scan papers for high error rates
– Do 5-min re-teach for those with high-error rates
Measurement Plan
• Weekly probe of Intervention skill
• Weekly probe of Retention of previously
mastered computational skills
• Monthly probe to monitor progress toward
year-end computational goals
General Findings
• Growth rates and trials to criterion varied
dramatically across skills
• Retention probe was strongest predictor of yearend SAT-9 performance
• Mastery level performance on early skills
predicted fewer trials to criterion on future
related complex skills
• Fluency scores higher than “mastery” predicted
retention of skill over time (about +20 dc/min)
VanDerHeyden, in preparation
Tier 1 Outcome 2 in math
Response Cards
Guided practice
PREP
• For each student: a laminate board, a dry erase marker,
and paper towel.
• Train until all students respond to cues :
– “Cards write” write problem and answer
– “Cards up” hold card above head towards teacher
– “Cards clean” wipe clean
Response Cards
Small group guided practice
After lecture and skill modeling, give 1 problem at a time.
•
•
•
•
Say “Cards write!”
Wait sufficient time.
Say “Cards up!”
Scan students’ answers.
– All correct  praise, say “Clean cards!”, and move on.
– Two or three wrong  give correct response, instruct
the students to check their answers, move on
– More than 25% wrong  reteach and ask again.
• Increase Pace as students learn skill.
Cover - Copy - Compare
STEP 1: Give a copy of the answers or another student’s 100% paper.
STEP 2: Prompt to do the following 5 steps when working on problems.
Look at the solution of the problem.
Cover the correct answer.
Do the problems on your own paper.
Uncover the correct answer.
Compare your answer with the correct answer.
STEP 3: After 4 problems correct, try 4 problems without procedure.
STEP 4: Check work. If incorrect, give help and tell the student to redo
the problem 3 times.
STEP 5: Have student complete worksheet, grade and chart.
Website for cover copy and compare worksheets:
http://www.interventioncentral.org/htmdocs/tools/mathprobe/addsing.php
Recommended computer fluency building
program:
Quarter Mile Math
http://www.thequartermile.com/
· Race format to motivate students
· > 70,000 problems for K through prealgebra (K-9th grade)
For Problem solving ideas:
See Interventioncentral.org by Jim Wright
KEEP IT SIMPLE
PROBLEM
Tools skills
Computation
Accurate
Peer tutor
flashcard
1. Mini skill step
lessons
PLUS
2. Large group:
Peer tutoring
or
Cover copy Compare
Small group:
Response cards
PLUS
3. Daily fluency test
Fluency
Timed trials
Time trials with goal
setting
Problem Solving