2 Math–Related Areas of SLD
 Math Calculation
 Math Problem Solving
Determining Progress
Measuring Fidelity
Case Studies and Examples
Jason Harlacher, PhD, NCSP
Colorado Department of Education
Colorado Department of Education
I.
II.
Big Ideas in Math
Assessments for math
◦ Academic Skill Deficit (low level)
◦ Insufficient Progress (low growth)
III.
IV.
V.
Determining Progress
Measuring Fidelity
Case Studies and Examples
2


32 million adults (14%) are
identified as illiterate, 50
million can’t read above
4th grade level (education-portal.com)
32% of fourth-graders,
33% of eighth-graders,
scored at Proficient level
(NAEP, 2007)

4th
No change in
grade
scores from 2007 to 2009
Jayanthi et al., 2008

39% of fourth-graders,
34% of eighth-graders
scored at Proficient level
(NAEP, 2009)

No change in 4th grade
scores from 2007 to 2009
(NAEP, 2009)
Ratio of studies on
reading disabilities to
math disabilities was 5:1
from 1996-2005
(compared to 16:1 in
prior decade)
3
Think of at least 2 ways can you add, subtract, multiply, or
divide to get to the number 20.
 Number Sense- “a child’s fluidity and flexibility with
numbers, the sense of what numbers mean, and an
ability to perform mental mathematics...and make
comparisons”
(Gersten & Chard, 1999; p. 20)

Not just knowing math facts, but also flexibility
◦ 8 + 5 = 13, so does 8 + 2 + 3
4
Tell me the sounds in “CAT”

Just as phonemic awareness is a critical early skill for
reading, number sense is viewed in the same manner
5
Developmental Progression
For early mathematics:
Counting
Nonverbal and verbal enumeration of smaller
to larger sets.
Comparing and ordering
Able to compare quantities and sets. Prereq
for idea of movement and change. Need to
envision a mental number line.
Equal partitioning
Ability to equally partition sets (6 into 3 and
3).
Composing and decomposing
Finding subsets within numbers, such as 7
has 3 and 4. Able to deal with larger
numbers more abstractly.
Grouping and place value (unitizing)
Progression in abstractly grouping objects
into sets, particularly in 10s (24 has two 10s
and 4).
Adding to/taking away
Ability to notice and create increases and
decreases in sets of items.
(Clements et al, 2004; Gersten & Chard, 1999; Methe & Riley-Tillman, 2008)
6

National Council for Teacher Mathematics (nctm.org)
Content Strands
(the what)
Math Processes
(the doing)
Number and Operations
Problem Solving
Algebra
Reasoning and Proof
Geometry
Connections
Measurement
Communication
Data Analysis
Representation
http://standards.nctm.org/document/appendix/numb.htm
I.
II.
Big ideas in Math
Assessments for math
◦ Academic Skill Deficit (low level)
◦ Insufficient Progress (low growth)
III.
IV.
V.
Determining Progress
Measuring Fidelity
Case Studies and Examples
8

Computation (Mathematical Calculation)
◦ Completing math problems where students must know
concepts, strategies, and facts (includes decimals, fractions,
percents)

Applications (Mathematical Problem-Solving)
◦ Use and understanding of math concepts to solve problems
(e.g., word problems, measurement, temperature, volume)
(Thurber et al, 2002)
9

What is a deficit? Common guidelines are:
◦ 10th percentile and below
◦ 6 consecutive data points on CBM monitoring graph below
12th percentile
◦ Less than 50% on Criterion-referenced tests
◦ Gap ratio greater than 2.0
Overall convergence of evidence should indicate a deficit

Gap Analysis is a method in which the gap between a
student’s expected level of performance and current
level of performance is quantified in a ratio.
Expected Level
= Gap Ratio
Current Level

Students who are more than 2.0 times below the
expected level of performance are considered to have
a significant gap.
78 Words
per Minute
60 Words
per Minute
78/60
1.3
No
35 Sounds
12 Sounds
35/12
2.91
Yes
35 Correct Digits
40 Correct
Digits
35/40
0.88
No
85% on District
Benchmark
63% on
District
Benchmark
85/63
1.40
No
3.4 Words
per Week
(growth)
1.6 Words
per Week
(growth)
3.4/1.6
2.13
Yes
Anything less than 2.0 is okay, whether they
score above or below the expected level.

Where the Expected Level of Performance comes from
and the context of the student’s school is critical.
Expected Level
Current
Level
Expression
Gap Ratio
Sign. Gap?
78 WPM
(National Norm)
32 WPM
78/32
2.43
Yes
60 WPM
(Local Norm)
32 WPM
60/32
1.88
No
Compared to national norms, the student has a significant gap.
Compared to local norms, student does NOT have a significant gap.
Where would you intervene? With the student or elsewhere?
Remember the theory behind problem-solving model and RTI.

Let’s make a list.

As a table/group, what are 2 assessments you
generally use to show a student’s academic level in:
◦ math calculation?
◦ applications?
14
Computation
Applications
How would you quantify each source of
information to determine if it’s a “skill
deficit”? (Percentile? Gap ratio? Level?)
15






Achievement tests (WIAT, WJ, KTEA, KeyMath,
TEMA,…)
State testing scores (CSAP, Other state scores)
Diagnostic assessments (G-MADE, MAP, KeyMath,
STAR)
Observations (illustrate comparison between peers
and student)
Curriculum based assessment (CBA, CBM, CBE)
Other data indicating relation to peers/grade-level
(district assessments, gradebook, student work,
quizzes, end of unit tests, inventories, )
16

What is insufficient progress?
◦ Let’s talk about this, as it’s not as straight forward as
“academic skill deficit”

Requires a progress monitoring tool
Reliable (consistency)
Valid (measure what’s intended & key indicator)
Sensitive to growth (pick up changes in skills)
Efficient to administer (quick and easy)
Standardized (rule out bias or administrator effect)
Alternate forms (avoid practice effects)
18


CBM fits all of the characteristics of an effective
progress monitoring tool
CBM is a standardized method of assessment that
measures basic skills
◦ Brief, reliable, valid indicators
◦ A thermometer for academic health

Tells you child’s level of risk (screener) and
if instruction is working (progress monitor)
◦ Can monitor anywhere from twice/week
to 3 times/year
Sidenote- CBM is a type of assessment. AIMSweb,
DIBELS, EdCheckUp, easyCBM are companies that have
created passages/probes to use with CBM.
19

Curriculum-Based Measurement (AIMSweb):
◦ Computation
 Math Curriculum-based Measurement (M-CBM)
 Math Computation (M-COMP)
 Fact Probes
◦ Early Numeracy
 Test of Early Numeracy (TEN)
◦ Applications
 Math Concepts and Applied Problems (M-CAP)
20



Older version of math computation (AIMSweb)
2 or 4 minute, mixed-computation probe
Score number of correct digits
◦ Can score answer process with grades 5 & 6

3 Benchmark, 40 Monitoring probes, grades 1 to 6
◦ Will be discontinued in Fall 2011
21
Grade 1
Grade 6
22

Similar to M-CBM, for grades 7 and 8
◦ 4 minutes, group or individual administration

Score number of correct digits or answer process

3 Benchmark, 40 Monitoring Probes
◦ Includes decimals, percents, and fractions
24
25
26




8 minute, mixed computation probes for grades 1 to 8
3 Benchmark and 30 Monitoring probes (AIMSweb)
Group or individual administration
Ease of scoring (right or not)
27
Grade 1,
Benchmark 2
Grade 1,
Benchmark 1
28
Grade 7
29
30

Take 1 minute and complete the M-COMP in your
handouts.

To score: Circle the point value if correct; circle zero if
incorrect.
◦ Alternate answers are correct if shows conceptual
understanding (e.g., ½ = .50).
31
For each correct answer,
write the number of points
each problem is worth on
your answered problems.



Single-skill or multi-skill probes focusing on basic math facts
2 minutes, group or individual admin; 40 probes
Basic fact = 0 to 12
33
34






Monitoring Basic Skills Progress (MBSP)
InterventionCentral.org
Mathfactcafe.com
STAR Math
easyCBM
EdCheckUp



Measure early numeracy skills (number sense) for
grades K-1
3 benchmark, 30 monitoring probes
4 assessments:
38
39
40
41
42
What is the purpose of a screener?
◦ Identify students’ level of performance
(Are they at-risk?)
A diagnostic tool?
◦ To identify specific skill deficits and strengths
(Why are they at-risk?)
A progress monitoring tool?
◦ To monitor a change in students’ level of performance
(Is the instruction effective?)

What varying types of assessment do you have for
building a Body of Evidence related to
MATH COMPUTATION in order to demonstrate…

academic skill deficit and insufficient progress?

Applied math probes for grades 1 to 8
◦ Eight minutes for grades 1 to 6, Ten minutes for 7 and 8



3 Benchmark and 30 Monitoring probes (AIMSweb)
Group or individual administration
Ease of scoring (right or not)
45
counting, number concepts, number naming, money, grid, measurement, time,
charts and graphs, fractions, decimals, applied computation, word problems
46
Grade 2
47
Grade 7
49

Take 1 minute and
complete the M-CAP.
◦ Score in similar manner as
MCOMP.
50



Monitoring Basic Skills Progress
(MBSP)
InterventionCentral.org
easyCBM

What varying types of assessment do you have for
building a Body of Evidence related to
MATH APPLICATIONS in order to demonstrate…

academic skill deficit and insufficient progress?

To measure academic skill deficits/level, use same old
basics. Think of way to quantify it (percentile, gap?)

To measure progress, need CBM scores:
◦ M-COMP, M-CBM, Fact Probes for math computation
◦ M-CAP for applied math
◦ TEN for early numeracy
◦ Other CBM probes available from easyCBM,
EdCheckUp, DIBELS, Monitoring Basic Skills (Fuchs)
54
I.
II.
Big ideas in Math
Assessments for math
◦ Academic Skill Deficit (low level)
◦ Insufficient Progress (low growth)
III.
Determining Progress
◦ Graphing Basics
◦ Determining Response
IV.
V.
Measuring Fidelity
Case Studies and Examples
55

Learning is an interaction between the learner,
curriculum, and environment (instruction).
(Howell & Nolet, 2000)


Have the plan with you.
If you only have the
data, may only focus on
the learner.
The question you are
asking is “Is the
instructional plan
working?”
Positive
Outcomes
40
30
20
10
0
1
2
3
4
5
2nd grader and Words Read
Correct Per Minute score
57
Use a graph with a scale that matches average
range of performance. Otherwise, progress
(or lack thereof) can be misrepresented.
140
120
(Average range, Spring)
100
80
60
40
20
0
1
2
3
4
5
2nd grader and Words
Correct Per Minute score
58
140
Use a graph with a scale that matches average
range of performance. Otherwise, progress
(or lack thereof) can be misrepresented.
120
100
80
60
40
(Average range, Spring)
20
0
1
2
3
4
5
2nd grader and Correct
Digits/ 2 Minutes score
59
140
120
100
80
60
40
20
0
1
2
3
4
5
6
7
8
2/2/2011
9
10
60
140
120
Make sure x-axis is chronological
and that data points represent day
they were administered.
100
80
60
40
20
0
1/1
2/1
3/1
4/1
5/1
2/2/2011
6/1
61
Graphing Basics:
Every graph needs:
- A goal
- An aimline
- A trendline
Trendline
Goal
Aimline/Goal line
2/2/2011
62
1. Establish “Pattern of Performance”
First and foremost, you need a pattern of performance
established.
63
90
80
70
60
50
40
30
20
10
0
Can you predict with
reasonable confidence
the next data point?
1
2
3
4
5
6
7
8
2/2/2011
9
10
64
90
80
70
60
50
40
30
20
10
0
As a general guide, it takes 810 data points to establish a
Can
predict with
reliable pattern
of you
growth.
reasonable confidence
the next data point?
Can you predict with
reasonable confidence
the next data point?
1
2
3
4
5
6
7
8
9
10
65
140
Words Correct/Minute
120
Goal of 110
Pattern of performance?
Can you predict the next
data point with reasonable
confidence?
100
80
60
40
20
0
12/24
1/7
1/21
2/4
2/18
3/3
3/17
3/31
4/14
4/28
5/12
5/26
6/9
140
Words Correct/Minute
Pattern of performance?
Can you predict the next data point
with reasonable confidence?
120
100
Goal of 110
80
60
40
20
0
12/24
1/7
1/21
2/4
2/18
3/3
3/17
3/31
4/14
4/28
5/12
5/26
6/9
140
Words Correct/Minute
120
100
Pattern of performance?
Can you predict the next data point
with reasonable confidence?
Goal of 110
80
60
40
20
0
12/24
1/7
1/21
2/4
2/18
3/3
3/17
3/31
4/14
4/28
5/12
5/26
6/9
140
Words Correct/Minute
Pattern of performance?
Can you predict the next
data point with reasonable
confidence?
120
100
Goal of 110
80
60
40
20
0
12/24
1/7
1/21
2/4
2/18
3/3
3/17
3/31
4/14
4/28
5/12
5/26
6/9
1. Establish “Pattern of Performance”
◦ If no pattern, don’t judge, collect more data.
◦ When in doubt, collect more data.
◦ 8-10 data points within one intervention to have a
technically adequate and reliable growth pattern
 doesn’t mean you can’t make changes before then, but need 8-10
for a reliable slope.

2. What does the trendline look like compared to the
aimline?
2/2/2011
70
120
100
Actual Growth to Expected Growth
Trendline
After 8-10 data points, compare
trendline to aimline.
80
Aimline
60
40
20
0
12/24
1/23
2/22
3/23
4/22
5/22
71


How does the trendline look compared to the aimline?
Positive response:
◦ Trendline is steeper than the aimline. Gap is closing/closed.
2/2/2011
72
120
100
Trendline is steeper than the
aimline. Gap is closing/closed.
Continue “as is”, but consider
ending intervention if goal is met or
above aimline.
Trendline
80
Aimline
60
40
20
0
12/24
1/23
2/22
3/23
4/22
5/22


How does the trendline look compared to the aimline?
Positive response:
◦ Trendline is steeper than the aimline. Gap is closing/closed.

Poor response:
◦ Trendline is flatter than the aimline. Gap is widening.
2/2/2011
74
120
Trendline is flatter than the
aimline. Gap is widening.
100
80
Check fidelity. If poor, improve it
and continue. If good, make an
instructional change!
Trendline
60
Aimline
40
20
12/24
1/23
2/22
3/23
4/22
5/22


How does the trendline look compared to the aimline?
Positive response:
◦ Trendline is steeper than the aimline. Gap is closing/closed.

Poor response:
◦ Trendline is flatter than the aimline. Gap is widening.

Questionable response:
◦ Trendline is parallel with aimline.
◦ Can treat as a “Poor Response”, but intervention may just
need fidelity check or intensified.
2/2/2011
76
120
100
80
Trendline is parallel with aimline. Gap is
staying consistent.
Trendline
Check fidelity. If poor, improve it and
continue. If good, intensify intervention or
make an instructional change!
60
40
Aimline
20
0
12/24
1/23
2/22
3/23
4/22
5/22
Response to Intervention
Baseline
Intervention
Positive
Questionable
Aimline
Performance
Poor
Trendline
Time
140
120
How’s the trendline look
compared to the aimline?
100
80
60
40
20
0
12/24
1/7
1/21
2/4
2/18
3/3
3/17
3/31
4/14
4/28
5/12
5/26
6/9
140
120
How’s the trendline look
compared to the aimline?
100
80
60
40
20
0
12/24
1/7
1/21
2/4
2/18
3/3
3/17
3/31
4/14
4/28
5/12
5/26
6/9
140
How’s the trendline look
compared to the aimline?
120
100
80
60
40
20
0
12/24
1/7
1/21
2/4
2/18
3/3
3/17
3/31
4/14
4/28
5/12
5/26
Would you trust this trendline?!
Don’t forget pattern of performance before
you judge a trendline!
Most data entry programs will calculate a
trendline regardless of how the data looks.
6/9
Is there a pattern of performance?
1.
◦
Positive or Poor Response?
2.
◦
◦
If Positive, continue or consider ending intervention if goal
is met.
If Poor or Questionable, then…
Check fidelity.
3.
◦
◦
4.
If no, gather more data.
If fidelity is poor, fix it and continue with plan.
If good,…
Make changes to alter rate of growth (major changes
if poor response; perhaps minor changes if
questionable response).
1) Pattern of performance?
2) Trendline compared to aimline?
140
120
100
80
60
40
20
0
12/24
1/7
1/21
2/4
2/18
3/3
3/17
3/31
4/14
4/28
5/12
5/26
6/9
1) Pattern of performance?
2) Trendline compared to aimline?
140
120
100
80
60
40
20
0
12/24
1/7
1/21
2/4
2/18
3/3
3/17
3/31
4/14
4/28
5/12
5/26
6/9
“Smallest, yet most powerful factor”
Doesn’t always have to mean “change the program”









Improve fidelity
Opportunities to respond
Accuracy of responses
Pacing
Person facilitating group
Error correction format
Level of explicitness
Amount of review
Time to practice skills








Size of group
Skill focus of intervention
Pre-correction strategies
Format of student response
Incentives during group
Time
Setting
Regroup students
(not exhaustive)
85

For students who are low-performing, suspected of ,or
have a disability
Make more explicit
More opportunities to respond
 For small-group, 8-12 OTRs/minute
 For larger groups it varies based on task, but about 6-8/minute
More immediate and clearer corrective feedback
 Should be accurate with at least 95% of responses and provided
correction 100% of errors
More time to practice skills
Jayanthi et al, 2008; Best Practices in School Psychology V
1) Pattern of performance?
2) Trendline compared to aimline?
140
Interv Change
120
100
80
60
40
20
0
12/24
1/7
1/21
2/4
2/18
3/3
3/17
3/31
4/14
4/28
5/12
5/26
6/9
1) Pattern of performance?
2) Trendline compared to aimline?
140
Tier 2
Tier 3
120
100
80
60
40
20
0
12/24
1/7
1/21
2/4
2/18
3/3
3/17
3/31
4/14
4/28
5/12
Insufficient Progress: Possible Referral?
5/26
6/9
Your district/school will ultimately define “insufficient progress”,
but generally speaking, it is:
8-10 data points within one intervention phase that
indicate a poor or questionable response
But…there are two things to consider when you see a
poor response

Growth Rate and Realistic Progress

Intensity of Instruction
◦ Time
◦ Size of Group
•
What is the growth rate?
Current Score – Beginning Score
# of weeks
= GR
Growth Rate
ROI (Rate of Improvement)
•
How long will it take student to reach goal?
Goal is 110 Words in 10 Weeks
95 Words

65 Words
8 weeks
95-65/8 = 3.75
Words/Week
Will student meet goal in time?
◦ Multiply Growth Rate by weeks left in intervention. Add to current
level.
3.75 Growth Rate X 2 weeks = 7.5 (growth left in intervention)
+ 95 (current level) = 102.5 at end
95 Words

65 Words
8 weeks
95-65/8 = 3.75
Words/Week
When will student reach goal?
Take Goal (110) – Beginning level (65) / Growth rate (3.75) =
45 (gap) /3.75 (growth rate) = 12 weeks to reach goal
(from start of intervention)
25 points
17 points
?
10 weeks
?
What growth rate is needed for the student to reach her
goal in 10 weeks?
25 – 17 = 8 (Gap)
8 ÷ 10 weeks = 0.80 (Growth rate needed)
Fuchs et al (1993)
Deno et al (2001)
Grade
Realistic
Ambitious
Realistic
1
2
3
1.80
2
1.5
2
1.66
3
1
1.5
1.18
4
0.85
1.1
1.01
5
0.5
0.8
0.58
6
0.3
0.65
0.66
Typical
practices
Effective
practices
95
Grade
Realistic
Ambitious
1
0.30
0.50
2
0.30
0.50
3
0.30
0.50
4
0.70
1.5
5
0.75
1.2
6
0.45
1.0
96
Grade
M-CAP,
AIMSweb
2
0.4
3
0.2
4
0.1
5
0.1
6
0.1
7
0.2
8
0.1
Caution:
Realistic growth rates are based on
normative samples and students
performing on grade-level.
Students who are below grade level
and provided targeted instruction
could have more growth relative to
the rates listed here.
97

How much growth can you expect relative to the
intensity of instruction?
◦ Two factors are time and size of the group
◦ More time and smaller group makes it more intense.

When making educational decisions (instructional
change or eligibility), intensity of instruction is
important to consider.

Daily half hour lesson for 12-20 weeks
(D’Agostino & Murphy, 2004)

Daily half hour lessons for 10-20 weeks
(Vaughn et al in press)

30-50 minute sessions three times each week
(O’Connor et al., 2005)

30 minute sessions four times each week
(Burns et al., 2005)

20 minute sessions four times each week for 2.5 yrs
(Torgesen et al, 1999)

50 minute sessions 4 times each week Oct-May
(Torgesen at al, 2003)
RTI Innovations 2010

10-minute daily sessions for 13 weeks
(Van der Hayden et al., 2007)



Tier 2 appears to be:
20-minute sessions three times/week
• average of 10 weeks
(Fuchs et al., 2008)
• 20-30 for
minutes/day
15-minute sessions three times/week
18 weeks
(Bryant et al., 2008)
outside of core
20-40-minute sessions, four to•five
3-5 times/week
times/week
(Gersten et l., 2009)

40-minute sessions
(Fuchs et al., 2005)
Tier 3 appears to be:
• 45-60 minutes/day
outside of core
• 3-5 times/week

Size of group
◦ Tier Two = 3 to 5: 1
◦ Tier Three = 1 to 3: 1
1) Pattern of performance?
2) Trendline compared to aimline?
140
Tier 2
Tier 2:
Tier
Phase
3 2
120
100
80
60
40
20
0
12/24
1/7
1/21
2/4
2/18
3/3
3/17
3/31
4/14
4/28
5/12
5/26
6/9
The intensity of instruction and phase will play into decisions to evaluate.
Regardless of going into Tier 2, 3, or eval, problem-solving never stops.

Ensure basics with the graph (X- & Y-axis, goal, aimline,
trendline)
◦ Always have the instructional plan with you when you examine a
graph


Examine Pattern of Performance first
Then consider Trendline compared to Aimline
Consider growth rates and intensity of instruction when
making decisions.
Never stop problem-solving, whether you evaluate or not.

10
3
I.
II.
Big ideas in Math
Assessments for math
◦ Academic Skill Deficit (low level)
◦ Insufficient Progress (low growth)
III.
IV.
V.
Determining Progress
Measuring Fidelity
Case Studies and Examples
10
4
Fidelity: The extent to which the intervention/instruction
was delivered as intended.
-Did it occur? (Integrity and Sufficiency)
-Did it match the student’s academic skill deficit?
http://www.cde.state.co.us/cdesped/download/pdf/SLDColoradoRulesFedRegs.pdf
10
5
Exercise
Food
Time Lapse
Weight Loss
3 times/week
No dessert!
2 weeks
Goal- 4 pounds
2 times/week
Had dessert
1 day/week.
2 weeks
Loss 1 pound
3 times/week
No dessert
2 weeks
Loss 2 pounds
10
6
Goal Not Met
“not due to
lack of
instruction...”
Fidelity was good
Fidelity was
unclear, not good,
or don’t know
Diet didn’t work
Not sure if diet
would have worked
Ensuring fidelity avoids
misattributing poor response to
diet when poor implementation is
the culprit
Improve fidelity
and redo diet

Direct Assessment
◦ Components of an intervention are clearly outlined in
operational terms and directly observed.
◦ Often calculate a total percentage of number of components
implemented

Indirect Assessment
◦ Assess degree of implementation without directly observing
instruction/intervention. Typically based on self-reports,
interviews, and review of permanent products.
National Research Center on LD- http://www.ldaofky.org/RTI/RTI%20Manual%20Section%204%20-%20Fidelity%20of%20Implementation.pdf
108
From Heartland
Agency in Iowa
10
9
From St. Croix River
Education District
11
0
From Pitt County
Schools, NC
11
1
11
2

Indirect Assessments: Look at…
◦ self reports (rating scales, checklists) on implementation
◦ lessons plans to ensure instruction was planned for and
provided
◦ attendance to ensure student was present to receive
instruction
◦ student products to see that students participated in the
instruction
11
3
Math
10/12
35
45
10/13
45
45
Fire drill ended early, no
fluency practice
From Washoe County
Schools; Reno, NV
11
4
11
5

What are some intervention programs you use that
you could create (or already have) direct assessment
fidelity tools for?

How could (or does) your school indirectly measure
fidelity?

Appropriate instruction implies more than just ‘Did it
occur?’ but also asks if the instruction targeted the
student’s deficits.
 If the student has a deficit in time,
would an intervention targeting
math facts be appropriate?
 Is a fluency intervention
appropriate for a student with
decoding deficits?
11
7
“STAY ON TARGET!”
-We’re all well intentioned, but ensure it’s on
target by doing good problem analysis and
using research-based instruction.
1) Examine skills being targeted during instruction.
2) Compare to skills identified as deficits in student.
Ensure that what
you’re doing is going
to be effective. Starts
with identifying the
problem well.



Fidelity and term “appropriate instruction” ask:
- Did instruction/intervention occur?
- Did instruction/intervention match the student’s
academic skill deficit?
Variety of direct and indirect ways to measure fidelity.
Creating a culture in which fidelity is expected and part of
routine makes it easier to collect data on it.
12
1
Old Model (Discrepancy)
New Model (Problem-Solving)
-Cognitive/Aptitude
-Academic Level (Achievement)
-Social/emotional
-Observation
-etc?
-Academic Level
-Academic Progress
-Previous interventions
-Fidelity information
-Social/emotional
-Observation
-etc?

One person cannot and should not do problem-solving
and RTI on his/her own. The system is not built that
way.

Truly takes a village.
School-Level Teams
Grade-Level Teams
Classroom Teacher
I.
II.
Big ideas in Math
Assessments for math
◦ Academic Skill Deficit (low level)
◦ Insufficient Progress (low growth)
III.
IV.
V.
Determining Progress
Measuring Fidelity
Case Studies and Examples
12
4
6th grade male up for re-evaluation in spring. Currently
classified under LD in reading. Considering math.
I.
Academic Skill Deficit?
12
5
6th grade male up for re-evaluation in spring. Currently
classified under LD in reading. Considering math.
Grade 3, 4, 5; State Math
Assessments
Approaching Standards (next level is
Proficient)
4th grade; Nevada NRT
21st percentile (grade 4)
6th grade (Winter) Benchmark
65% (on-track is 75%)
MCBM (Winter 6th grade)
15 Correct Digits (26 is criterion)
MCAP (Winter 6th grade)
6 Correct Problems (11 is criterion)
KTEA-II (Win 6th)
83 (13th perc) in Computation
80 (9th perc) in Concepts and Application
II. Insufficient Progress?
Tier I (Aug to present)
Tier II (Sept - Jan)
Tier III (Jan to present)
Core block of 60 minutes,
5 days/week using
Everyday Math.
• 20 minutes wholegroup instruction
• 20 minutes of small
group
• 15 minutes of
independent practice
• 5 minutes for
transition, cleanup
30 additional minutes of
instruction, 2 days/week in
group of 6 students.
30 minutes of small
group instruction with 3
students 2 days/week.
Targeting fact fluency using
math games from
curriculum. Also focusing on
money, time, and word
problems with supplemental
core materials.
Targeting math facts,
word problems, and
analyzing graphs to solve
problems. Using direct
instruction math
program.
12
7
New goal line created after intervention
change because a new expected rate of
growth is needed to reach goal
II. Insufficient Progress - number of correct digits on M-CBM
35
Tier 2
Tier 3
30
Goal of 26 CD
25
20
15
10
5
0
9/1
10/1
11/1
12/1
1/1
2/1
3/1
4/1
5/1
6/1
12
8
II. Insufficient Progress number of correct digits on M-CAP
20
18
16
14
12
10
8
6
4
2
0
Tier 2
Tier 3
Goal of 11 CP
9/1
10/1
11/1
12/1
1/1
2/1
3/1
4/1
5/1
6/1
12
9
III. Other factors
 Fidelity◦ Student present 96% of school days. Missed 1 day of math
intervention all year.
◦ Teacher reported 95% of intervention components
implemented (self-report sheet)
◦ Two observations for each tier: 100% of components
implemented for Tier 2 and Tier 3

Observation◦ Student on-task with rest of peers. Completing activities with
accuracy.
13
0
Academic Skill Deficit?
Skill deficit on math CBM and KTEA-II
Insufficient Progress?
Insufficient
progress
computation,
Qualified
under
LD inin Math
but responding in applied math
Computation, but not for
Fidelity? (Occurred?
Scores
above 95% on fidelity measures.
Applied
Math.
Appropriate?)
Targeting computation and applied
math deficits
Response to instruction
Data available. data
Received
Tier 2 and Tier
Will re-examine
for
data?
3.
applied math in 3 weeks
Observation?
On-task and successful during
intervention time. Differentiated
instruction during core.
As a group/table, would you consider qualifying? Why or why not?
13
1
3rd grade male evaluated for reading difficulties in spring.
Informal Reading Inventory, Sept
3rd grade
Level HI (late 1st grade)
District Reading Benchmarks
40%- Off Track (75% On track)
R-CBM, Winter 3rd grade
Read 60 Words Correct per Minute
(criterion is 78); read with 92% accuracy
(target accuracy is 95%)
Survey-level assessment, Winter 3rd Met criterion at late 1st grade level (57
grade
words with 95% accuracy)
STAR Reading Assessment, Fall 3rd
grade
Grade equivalent score of 1.4
WIAT (Achievement Test)
Basic Reading Skills- 86 (18th perc)
Reading Comprehension- 89 (23rd perc)
Curriculum Based Evaluation
Revealed decoding as a deficit. Poor &
inconsistent accuracy rate across
reading passages from grades 1 to 3.
13
2

Insufficient Progress?
SECOND GRADE
Tier I: Oct –Nov
Core block- 90 minutes
of whole and small
group instruction.
THIRD GRADE
Tier I: Sep to present
Core block- 90 minutes of
whole and small group
instruction.
Tier II: Nov –Feb
30 mins/2 times/week
of small group reading.
Group of 6-8 kids.
Targeting fluency and
comprehension.
Tier III: Feb –June
Reduced group size
from 5 students to 3
student. Targeting
fluency and
comprehension.
Tier III: Feb to present
30 minus/twice/week in a
group of 4. Targeting fluency
and decoding.
133
SECOND GRADE: R-CBM (words correct/minute)
Tier 2
Tier 3
New goal line after
intervention change
13
4
Tier 3 (continued from grade 2)
140
Progress in Reading, Grade 3, Words Correct/Minute
120
100
80
60
40
20
0
10/6
11/6
12/6
1/6
2/6
3/6
4/6
5/6
13
5
III. Other factors
 Fidelity◦
◦
◦
◦
Student present 92% of school days in grades 2 and 3.
Student was in different schools and states for grades K and 1.
No fidelity data collected on intervention.
Principal observation noted lack of implementation of high
quality instruction in core (few opportunities to respond,
limited differentiated instruction).
13
6
III. Other factors
 Observation
◦ Observed during whole-class core instruction for 45 minutes.
◦ Was receiving vocabulary instruction and answered correctly
2/2 times.
◦ Attention waivered. Limited interaction with teacher.
◦ Conclusions: Few opportunities to respond, long time on
vocabulary for student (no differentiation), no additional
strategies to facilitate learning for student
137
Did not qualify for services because:
Skill Deficit?
About 1.5-2 grades below current
grade placement. Scoring low, but skill
saydeficit?
there was high fidelity
1. Could not
Insufficient
Progress?
progressintervention
in tier 2. Seems to
(missing
data, Insufficient
second grade
be responding to Tier 3
not matched to need)
Fidelity? (Occurred?
Appropriate?)
2.
Received additional instruction, but not
matched to his need (see 2nd grade
Intensity of instruction
wasLack
limited
intervention focus).
of clear (only
focus
of instruction
in grade
3 core.
60 mins/week of
additional
instruction)
Response to instruction
data?
Available data.
3. Slight response to Tier 3 after targeting
Observation?
Limited effective practices used and
decoding skills
more.
perhaps
mismatch between student
need and instruction.
As a group/table, would you consider qualifying? Why or why not?
13
8

Look over the case study. Take 10-15 minutes to
discuss and make a conclusion.
1) Would you lean more toward qualifying or not qualifying this
student?
2) What is one factor that affected your answer to #1?
3) What would you do next for problem-solving?
13
9

Given your role and your school’s current situation,
what is one piece of information from today that you
will share with other staff in your building?
Given your role and your school’s
current situation, what is one piece of
information from today that you will
share with other staff in your building?
1) The body of evidence for SLD identification under a
problem-solving model is different than one under a
discrepancy model.
2) Consideration of focus and actual delivery of the
instruction is much more pronounced.
3) Fidelity is not an evaluative tool for teachers. It’s about
ensuring the “diet” was followed and if it was appropriate
for the need of the student.
14
1

Thank you so much for your attention and
participation. Please contact me with any questions
you may have.

Jason Harlacher, PhD, NCSP
◦ Principal Consultant, Colorado Dept of Educ
◦ Harlacher_j@cde.state.co.us
Effective teaching practices for
students with SLD in math or low
performance in math.

A few syntheses of research and meta-analyses have
identified key methods for teaching students who are
low in math and students with LD.
◦
◦
◦
◦
Jayanthi et al, 2008
Baker, Gersten, & Lee, 2002
Gersten, Baker, & Chard, 2006
National Math Advisory Panel’s Report
14
4
Use explicit instruction on a regular basis.
1.
◦
◦
◦
◦
Clear modeling with examples
Practice time
Frequent opportunities to respond
Immediate corrective feedback
I do, We do, You do it together, You do it alone
14
5
Use multiple instructional examples.
2.
◦
◦
◦
Careful selection and sequencing of examples
Show variations, but highlight common but critical features
Concrete to abstract, easy to hard, simple to complex
14
6
Have students verbalize decisions and solutions to a
math problem.
3.
◦
Verbalize or think aloud their decisions and solutions




Specific steps (I divide my two to get half)
Generic heuristics (Now I need to check my answer)
Solution format (First add the numbers in the unit column)
Self-questioning format (What should I do first?)
14
7
Teach students to visually represent the information
in the math problem.
4.
◦
◦
More powerful with explicit instruction
More powerful when students use a visual given by teacher
vs. a self-generated one
14
8
Solve problems using multiple/heuristic strategies.
5.
◦
◦
Discuss/show different ways to solve problems.
What is 9 X 5?

Can be nine fives or five nines. Also ten fives minus one five.
14
9
Provide ongoing formative assessment data and
feedback to teachers.
6.
◦
Greater outcomes if teachers are given tips and suggestions
to help decide:



what to teach
when to introduce the next skill
how to group/pair students.
15
0
Provide peer-assisted instruction to students.
7.
◦
Good outcomes if low- and high-performing students are
paired because:



◦
Low-performing student sees better strategies and methods
used by high-performing student
High-performing student solidifies knowledge by
verbalizing/explaining it to other student
Cross-age tutoring more powerful for students with LD (perhaps
of gap of skill deficit?)
PALS, Classwide Peer Tutoring, Team Assisted Individuation
15
1
Baker, Gersten, & Lee (2002). A synthesis of empirical research on teaching mathematics to lowachieving students. The Elementary School Journal, 103 (1), 51-73.
Clements, W. et al (2004). Engaging young children in mathematics: Standards for early
childhood mathematics in education. Mahwah, NJ: Erlbaum.
Education-portal.com
Gersten, R. and Chard, D. Number sense: Rethinking arithmetic instruction for students with
mathematical disabilities. Journal of Special Education (1999), 3, 18–29.
Howell, K., & Nolet, V. (2000). Curriculum-based evaluation.
Jayanthi, M., Gersten, R., & Baker, S. (2008). Mathematics instruction for students with learning
disabilities or difficulty learning mathematics.
http://www.centeroninstruction.org/files/Teaching%20Math%20to%20SLD%20LD%20Guide
.pdf
Methe & Riley-Tillman (2008). An informed approach to selecting and designing early
mathematics interventions. School Psychology Forum, 2, 29-41.
Mathematics Learning Study & National Research Council (2001). Adding it up: Helping children
learn mathematics. Washington, DC: National Academy Press.
National Council of Teachers of Mathematics.
http://standards.nctm.org/document/appendix/numb.htm
Nation’s Report Card. http://nces.ed.gov/nationsreportcard/
Thurber, Shinn, Smolkowski (2002). What is measured in mathematics tests? Construct validity
15
of curriculum-based mathematics measures. School Psychology Review, 31 (4), 498-513. 2
Best Practices in School Psychology (2009). Fifth Edition. NASP,
Burns, M., Appleton, J., & Stehouwer, J. (2005). Meta-Analytic Review of Responsiveness-ToIntervention Research: Examining Field-Based and Research-Implemented Models, Journal of
Psychoeducational Assessment, 23, 381-394
D'Agostino, J. V., & Murphy, J. A. (2004). A meta-analysis of Reading Recovery in United States
schools. Educational Evaluation and Policy Analysis, 26(1), 23-38.
Deno, S. L., Fuchs, L. S., Marston, D.B & Shin, J., (2001). Using curriculum-based measurement to
establish growth standards for students with learning disabilities. School Psychology
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Fuchs, L. S., Fuchs, D., Hamlett, C. L., Walz, L., & Germann, G. (1993). Formative evaluation of
academic progress: How much growth can we expect? School Psychology Review, 22, 27-48.
Gersten, R. et al. (2009). Assisting Students Struggling with Mathematics: Response to
Intervention (RtI) for Elementary and Middle Schools. US Dept of Education. National Center
for Education Evaluation and Regional Assistance.
Newman-Gonchar, R., Clarke, B., & Gersten, R. (2009). Multi-tier intervention and response to
interventions for students struggling in mathematics. Center on Instruction.
O’Connor, R.E., Harty, K., & Fulmer, D. (2005). Tiers of Intervention in Kindergarten Through
Third Grade. Journal of Learning Disabilities, 38, 532-538.
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