Assessment of Mathematics
Unit 8
NC SIPII MATH
What teachers do = What teachers
think
• True! What we think will effect what we learn.
• Learning occurs instantaneously. T F
• You can only practice what you know.
• Proficiency builds gradually. T F
TF
How do we build motivation?
• If we give students easy tasks, are we really
building success or self – confidence?
• Do learning styles exist?
– LS Inventory assessment:
http://www.engr.ncsu.edu/learningstyles/ilsweb
.html
• Homogeneous Grouping- Is it good or bad? /
Full Inclusion?
• Student overachievement or low expectations
Failure - Who is to Blame
• Failure to learn may stem from
– Poor past experiences
– Lack of prior knowledge
– Lack of good instruction
• What is instruction?
– Good instruction is informed by assessment.
– What about the things we can’t control?
• IQ
• Learning style
• Family home
“If the student has not learned –
the teacher has not taught!”
• Missing or erroneous prior knowledge may
be your biggest barrier
• Task difficulty is directly related to prior
knowledge.
• It is not the task it is the skill!
Let’s Learn about Hacky Sack
• Hacky Sack is the trademarked name of a
type of footbag. The name "hacky sack"
came from the inventors of the footbag,
John Stalberger and Mike Marshall.
Marshall suffered a fatal heart attack in
1975, however Stalberger continued the
business. At a later date, Stalberger sold
the title to Wham-O.
Moves
• Inside Kick
• Outside Kick
• Toe Kick
• Knee Kick
RULES
• 1. No Hands (except when serving), No Arms - Shoulders
are technically not allowed and are widely accepted among
the average hack circle.
• 2. Always serve the bag to someone else, unless of course
you are alone. Footbag is traditionally a game of courtesy,
hence "The Courtesy Toss": a light lob usually toward the
receivers knee.
• 3. Don't bogart that bag. - Don't always hog it 'till you drop it
because that is bad for everybody else. Being able to pass
well is important to almost all footbag games.
• 4. Don't say "sorry". Everyone makes mistakes, especially
when learning, so sorries are unnecessary.
• 5. Try not to give knee passes Passes from the knee tend to
go straight to the ground.
Learning Process
• Learning orientation vs. performance orientation
(completing the task)
• How can you show which is important in your
classroom?
Fundamentals of Assessment
How is all of this connected to
the RTI Model?
AMOUNT OF RESOURCES
REQUIRED TO MEET THE
STUDENT’S NEEDS
Tier IV
IEP
Consideration
Tier III
Student
Study
Team
Tier II
Consultation
With Other
Resources
Tier I
Consultation
 Intensive Interventions 1-7%
Between
Teachers
Parents
-
 Strategic Interventions 5-15%
 Core Curriculum 80-90%
INTENSITY OF NEEDS
Needs -circles -pub
What happens in each tier?
• Tier I- at least 3 times per year for
benchmarking.
• Tier I / Tier II (small group) start to move
into progress monitoring to determine how
your students are responding to your
intervention.
• Tier III (individualized) Progress monitoring
continues.
Modified
PSM
What is the problem?
Assessment
What does the data
show?
Why is this
happening?
Curriculum Issue?
Instructional Issue?
Student Issue?
What is our plan?
What are we going to
do? What interventions
are needed? How will
we measure success?
Did the plan work?
What does the data
show?
Implement the plan Who will do what,
where, when, and how
often? How will fidelity
of implementation be
determined?
80%-15%-5% Rule
Apply the “80-15-5” rule to determine if the
focus of the intervention should be on the
core curriculum (and instruction), on
subgroups of underperforming learners, or
individual students (T. Christ, 2008)
Source: Christ, T. (2008). Best practices in a problem analysis. In A.
Thomas & J. Grimes (Eds.), Best practices in school psychology V
(pp. 159-176).
Application of the 80%-15%-5% Rule
• If less than 80% of students are successfully meeting
academic or behavior goals, the intervention focus is on the
core curriculum, INSTRUCTION, and general student
population.
• If no more than 15% of students are not successful in
meeting academic or behavior goals or have met the goals,
the intervention is on small group ‘treatments’ or
interventions.
• If no more than 5% of students are not successful in
meeting academic or behavioral goals, the intervention
focus is on the individual student.
– Source: Christ, T. (2008). Best practices in a problem analysis. In A. Thomas & J.
Grimes (Eds.), Best practices in school psychology V (pp. 159-176).
Tier I
What is the problem?
Assessment
What does the data
show?
What is our plan?
What are we going to
do? What interventions
are needed? How will
we measure success?
Did the plan work?
What does the data
show?
Implement the plan Who will do what,
where, when, and how
often? How will fidelity
of implementation be
determined?
Tier I
• Universal Screening (both high and low achievers
may need interventions)
• Universal Screening is a general outcome measure
used to identify underperforming students and to
determine the rate of increase for students.
• A Universal Screening will not identify why
students are underperforming, rather it will identify
which student is below / above the expected
performance criteria for a given grade level in
reading, writing, spelling, and math.
Primary Prevention
Fuchs & Fuchs
Tier 1 CBM (Curriculum Based Measurement) is a
tool used to screen all students in the class to
identify those potentially at risk for poor
outcomes at the end of the year.
Number Knowledge
Test
http://clarku.edu/numberworlds
Name__________________________
Score _____
Date________
Preliminary
Let’s see if you can count from 1 to 10. Go ahead.
Number Knowledge Test Level 0 (4-year-old level)
____ 1. Can you count these Counters and tell me how many there are?
(Place 3 Counters in a row in front of the child.)
____ 2a. (Show stacks of counters, 5 vs. 2, same color)
Which pile has more?
____ 2b. (Show stacks of counters, 3 vs. 7, same color)
Which pile has more?
____ 3a. This time, I’m going to ask you which pile has less.
(Show stacks of counters, 2 vs. 6, same color.)
Which pile has less?
____ 3b. (Show stacks of Counters, 8 vs. 3, same color.)
Which pile has less?
____ 4. I’m going to show you some Counters.
(Show a line of 3 blue and 4 white Counters in a row, as follows: BWBWBWW.)
Count just the white Counters and tell me how many there are.
____ 5. (Pick up all the Counters from the previous question. Show a mixed array—not a row—of 7
white and 8 blue Counters)
Here are some more Counters.
Count just the blue counters and tell me how many there are.
*Go to Level 1 if 3 or more correct
In order to receive credit an item that has an a and b, both a and b must be correct to be
counted in the score.
Use data to develop a plan
Grade
Computation Cut-Off for
one minute
Grade 1
< 5 digits
Grade 2
< 10 digits
Grade 3
< 10 digits
Grade 4
< 10 digits
Grade 5
< 15 digits
Grade 6
< 15 digits
Other sources of data:
-Scores on the NKT
-One-on-one interviews
-Analysis of student work
-Skill-based measures
Fuchs & Fuchs Note: These figures may change pending additional RTI research.
Tier I
What is the problem?
Assessment
What does the data
show?
What is our plan?
What are we going to
do? What interventions
are needed? How will
we measure success?
Did the plan work?
What does the data
show?
Implement the plan Who will do what,
where, when, and how
often? How will fidelity
of implementation be
determined?
Concepts and Applications
Student copy of a
Concepts and
Applications test:
– This sample is
from a secondgrade test.
– The actual
Concepts and
Applications test is
3 pages long.
Concepts and Applications
Grade
2
Time
limit
8 minutes
3
6 minutes
4
6 minutes
5
6
7 minutes
7 minutes
Length of test varies by grade.
Tier 1
What is the problem?
Assessment
What does the data
show?
What is our plan?
What are we going to
do? What interventions
are needed? How will
we measure success?
Did the plan work?
What does the data
show?
Implement the plan Who will do what,
where, when, and how
often? How will fidelity
of implementation be
determined?
Tier I
What does instruction in this
classroom look like?
How is this instruction different from
what we see in most classrooms?
Think about the Medical Model.
Tier I
What is the problem?
Assessment
What does the data
show?
What is our plan?
What are we going to
do? What interventions
are needed? How will
we measure success?
Did the plan work?
What does the data
show?
Implement the plan Who will do what,
where, when, and how
often? How will fidelity
of implementation be
determined?
Data to Consider
•Continue Fall, Winter, and Spring benchmarks
• Assess students who did not meet the first
benchmark more frequently
•Continue group progress monitoring
• Analyze additional student work samples
Tier II
What is the problem?
Assessment
What does the data
show?
What is our plan?
What are we going to
do? What interventions
are needed? How will
we measure success?
Did the plan work?
What does the data
show?
Implement the plan Who will do what,
where, when, and how
often? How will fidelity
of implementation be
determined?
• Student interviews
• Conceptual understanding
• Student work
• Skill-based measurements
www.interventioncentral.org
• http://www.interventioncentral.org/index.ph
p/cbm-warehouse#mathComputation
– Here we can find:
– GOMs – such as mixed computation probes
– SBMs- such as specific calculation issues
– MMs – such as quantity discrimination and early
numeracy fluency
– NORMS for CDs
– Instructional support ideas for different tiers
Performance Task
1. Assess the mathematical errors
in the student work.
2. Design a plan for instruction for
your student.
Connecting Forms of a Value
What fractional part of the figures are shaded?
1.
4
4
9
6
2.
Circle ¾ of the group of boxes.
3.
Put a ring around 3/8 of the group of triangles.
4.
5.
Draw an arrow where the ½ should be.
0
1
2
Put a circle on the number line where 2/3 should be.
0
1
2
3
4
Addition of Fractions
1.
2 1 3
 
5 5 10
2.
3 1 4
 
8 4 12
3.
4 1 5
 
9 3 12
4.
5.
1 1 2
 
2 3 5
3
2
5
4 5 9
7
9
16
Tier II
What is the problem?
Assessment
What does the data
show?
What is our plan?
What are we going to
do? What interventions
are needed? How will
we measure success?
Did the plan work?
What does the data
show?
Implement the plan Who will do what,
where, when, and how
often? How will fidelity
of implementation be
determined?
Plan should consider:
• CONTINUING EFFECTIVE TIER I INSTRUCTION
• TARGETED INSTRUCTION
• INTERVENTION GROUPS (1:7)
• SKILL-BASED INSTRUCTION
•
IES Recommendations
Tier II and III
• Instruction during the intervention should be
explicit and systematic.
Provide:
– Models of proficient problem solving
– Verbalization of thought processes
– Guided practice
– Corrective feedback
– Frequent cumulative review
• Interventions should include instruction on how to
solve word problems that is based on common
underlying structures.
IES Recommendations
Tier II and III
• Intervention materials should include
opportunities for students to work with visual
representations of mathematical ideas and
interventions should be proficient in the use of
visual representation of mathematical ideas.
• Interventions at all grade levels devote about 10
minutes in each session to building fluent
retrieval of basic arithmetic facts.
IES Recommendations
Tier II and III
• Instructional materials for students receiving
interventions should focus intensely on in-depth
treatment of whole numbers in grades K-5 and
on rational numbers in grades 4 through 8.
These materials should be selected by a
committee.
• Monitor the progress of students receiving
supplemental instruction and other students
who are at risk.
• Include motivational strategies in Tier II and
Tier III interventions.
Tier II
What is the problem?
Assessment
What does the data
show?
What is our plan?
What are we going to
do? What interventions
are needed? How will
we measure success?
Did the plan work?
What does the data
show?
Implement the plan Who will do what,
where, when, and how
often? How will fidelity
of implementation be
determined?
Structures of Addition
Join and Part-Part Whole How many altogether?
– There is something, and you get more of it?
– There are two kinds, how many all together?
?
Start Unknown What did I start with?
– Some are given away, some are left; how many were there to
start?
?
Taken
Left
Compare--total unknown How many do I have?
- I know one amount and I have some amount more than that.
How many do I have?
?
Addition types Adapted from Carpenter, Fennema, Franke, Levi and Empson, 1999, p. 12 in Adding it Up, NRC 2001.
Structures of Subtraction
Ask yourself if a problem is a subtraction problem—
Does it fit one of these three types?:
You’ve got some amount
and “take away” from it:
What’s left?
?
– The Classic “Take away”
(how many left?)
?
– Comparison
(difference between? who has more?)
You compare to see:
Who has more or less?
?
– Deficit/Missing amount
(what’s missing?)
You need some more to get
where you want to be:
What is the missing amount?
Multiplication Structures
Repeated Addition
Array/row-column
Counting Principle
Division Structures
Measurement/Repeated Subtraction
“How many 2s can I get out of 10?”
2
10
2
2
2
2
If I have 10 cups of beans and I give out 2 cup
portions, how many servings will that provide?
?
?
10
Partitive/Unitizing/Fair Shares
?
“How many would one person get? or “What would that mean in
relation to 1?” If 2 people find $10 how much will each person get ?
2
10
Product/Factor
“If I have an area of 10 and one side is 2, how long is the other side?”
?
Virtual Manipulatives
•
National Library of Virtual Manipulatives
– http://nlvm.usu.edu/en/nav/vlibrary.html
Tier II
What is the problem?
Assessment
What does the data
show?
What is our plan?
What are we going to
do? What interventions
are needed? How will
we measure success?
Did the plan work?
What does the data
show?
Implement the plan Who will do what,
where, when, and how
often? How will fidelity
of implementation be
determined?
MULTIPLICATION:
4-digit number times 2-digit number:
with regrouping
Item 1:
17 CD/17 CD Total
7083
x 57
49581
354150
403,731
On-line resources to help with data
•
http://www.thatquiz.org/
• http://www.aimsweb.com/measures-2/
• http://www.easycbm.com/
• http://enumeracy.com/
Tier III
What is the problem?
Assessment
What does the data
show?
What is our plan?
What are we going to
do? What interventions
are needed? How will
we measure success?
Did the plan work?
What does the data
show?
Implement the plan Who will do what,
where, when, and how
often? How will fidelity
of implementation be
determined?
www.interventioncentral.org
• http://ic.gorgeswebsites.com/index.php?option=com_content&vi
ew=article&id=195
• Here we can find:
– GOMs – such as mixed computation probes
– SBMs- such as specific calculation issues
– MMs – such as quantity discrimination and early numeracy
fluency
• Early numeracy probes
– http://progressmonitoring.org/probes/earlyn
umeracy.html
Tier III
What is the problem?
Assessment
What does the data
show?
What is our plan?
What are we going to
do? What interventions
are needed? How will
we measure success?
Did the plan work?
What does the data
show?
Implement the plan Who will do what,
where, when, and how
often? How will fidelity
of implementation be
determined?
The Tier III plan should be:
• CONTINUING EFFECTIVE TIER II
INSTRUCTION
• MORE INTENSIVE
• MORE INDIVIDUALIZED
• Take into consideration the IES
Recommendations
Implementation should be:
•
•
•
•
•
MORE EXPLICIT
MULTI-SENSORY
MORE SYSTEMATIC
MORE DIRECT
CUMULATIVE
Tier III
What is the problem?
Assessment
What does the data
show?
What is our plan?
What are we going to
do? What interventions
are needed? How will
we measure success?
Did the plan work?
What does the data
show?
Implement the plan Who will do what,
where, when, and how
often? How will fidelity
of implementation be
determined?
On-line resources to help with
• http://www.interventioncentral.org/index.ph
p/academic-resources
• http://enumeracy.com/
Tier III
What is the problem?
Assessment
What does the data
show?
What is our plan?
What are we going to
do? What interventions
are needed? How will
we measure success?
Did the plan work?
What does the data
show?
Implement the plan Who will do what,
where, when, and how
often? How will fidelity
of implementation be
determined?
At Tier III
– PROGRESS MONITORING SHOULD
TAKE PLACE 2-3 Times PER WEEK
– WE SHOULD ONLY USE CBMs THAT
ARE VERYSENSITIVE TO CHANGE AT
THIS TIER
Case Study:
•
•
•
•
•
James-2nd Grade
James can identify the cardinal number of a set
and can relate the numeral to that number.
He can skip count in series of 10, 2, 5, 9, 4
without the use of manipulatives with
automaticity.
He has developed automaticity with basic
addition facts 1-6.
He understands the concepts of one more and
one less.
He continues to use fingers to count up and
count down for addition and subtraction. His
fluency is weak.
James’ Data
Baseline 13 digits per two minutes (2nd point)
Monitored for CDs for approximately 20 weeks
Last 3 data points ranged between 14 and 16- CDs
(target should be about 18 to 20)
Baseline Data
Baseline:
• PM Day
• 1
• 2
• 3
Date
CDs per 2 min.
5 January 2009
12
5 January 2009
13
5 January 2009
15
Tier I
•
•
•
•
•
•
•
Phase Change: Tier I
PM Day Date
CDs per 2 min.
4
15 January 2009
12
5
28 January 2009
12
6
7 February 2009
13
7
21 February 2009
13
8
30 February 2009
14
Tier II
•
•
•
•
•
•
•
•
•
•
•
Phase Change: Tier II
PM Day Date
9
8 March 2009
10
16 March 2009
11
20 March 2009
12
25 March 2009
13
29 March 2009
14
5 April 2009
15
7 April 2009
16
12 April 2009
17
17 April 2009
CDs per 2 min.
13
14
13
14
14
12
13
13
13
Tier III
•
•
•
•
•
Phase Change: Tier III
PM Day
Date
CDs per 2 min.
18
23 April 2009
14
19
28 April 2009
15
20
1 May 2009
16
Learning Task-Case Study
• Using the information you already knew and
what you learned today, work in groups to
formulate a plan for instruction for James.
• How will you evaluate his progress?
Possible interventions …
• Math Review: Promote Mastery of Math Facts
through Incremental Rehearsal
(interventioncentral.org)
• Subitizing
• Number Lines/Hundreds Boards
• Base Ten Instruction
• The Structures of Addition and Subtraction
Tier IV--Intensive
Referral to Special Education
If the student has not shown progress even with
targeted SBI instructional plans.
When do we make the referral?
TIERS I -III PROVIDE DATA SET FOR TIER IV
EVALUATION
Be Careful---• We do not have the same correlations as we
have with reading probes.
• We need to be cautious about applying what
we know about Reading to the world of
Mathematics.
• In Math, we must think about what we are
measuring. We are not in the same place in
Math as we are in Reading.
Web Resources for Assessment and
Intervention
•
•
•
•
•
•
•
•
•
•
•
•
•
•
Yearly Progress Pro
http://www.mhdigitallearning.com/
AIMS Web
http://www.edformation.com/
Intervention Central
http://www.interventioncentral.org/
Number Worlds Home
http://clarku.edu/numberworlds/
John Woodward, Transitional Math
http://www2.ups.edu/faculty/woodward/home.htm
Progress Monitoring
http://www.progressmonitoring.net/
National Center on Student Progress Monitoring
http://www.studentprogress.org/chart/chart.asp
References
“Assisting Students Struggling with Mathematics”: Response to Intervention (Rti) for
Elementary and Middle Schools” IES National Center for Education Evaluation and
Regional Assistance, NCEE 2009-4060, U.S. Department of Education
Ball, Deborah (1992) “Magical Hopes: Manipulatives and the Reform of Math Education”,
American Educator, Summer 1992
Ball’s Website: http://wwwpersonal.umich.edu/~dball/
Fuchs, Lynn “The Prevention and Identification of Math Disability
Using RTI”, September 18, 2008 Presentation
Gersten, Russell, Clark, B, Jordan, N, Center on Instruction, “Screening
for Mathematics Difficulties in K-3 Students” 2007.
Gersten, Russell, Jordan, N, Flojo, J., “Early Identification and Interventions for
Students with Mathematical Difficulties”, Journal of Learning Disabilities, Volume
38, Number 4, July August 2005
Gickling, Edward, PhD, Instructional Assessment in Mathematics, March 2003,
Presentation at Exceptional Children’s Conference
Griffin, Sharon. (2003). Mathematical Cognition, Royer, ed. Greenwich, CT.: Infoage
Publishing.
Ma, Liping (1999) Knowing and Teaching Elementary Mathematics. Edison, NJ,
Lawrence Erlbaum Associates.
Mayer, Richard (2003). Mathematical Cognition, Royer, Ed.. Greenwich, CT.: Infoage
Publishing.