Developing Diagnostic
Interviews in Mathematics
Karen Karp
Amy Lingo
Project ABRI
University of Louisville
Topics

Overview of RtI Model, a multi-tiered intervention approach

Diagnostic interview: a component of formative
assessment
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Designing and implementing diagnostic interviews
RtI:
3-Tiered Model
Tertiary Prevention:
specialized & individualized
strategies for students with
continued failure
~5%
~15%
Secondary Prevention:
supplementary strategies
for students who
do not respond to primary
Primary Prevention:
school-wide or class-wide
systems for all
students and staff
~80% of Students
Components of A Strong RtI Model
Incorporates a
regular screening
process
Includes
evidenced based
practices
Integrates
progress
monitoring
Instructs with
preventative
methodology
Uses diagnostic
assessment to
align intervention
Newman-Gonchar, R., Clarke, B., & Gersten, R. (2009). A summary of nine key studies:Multi-tier intervention and response to interventions for
students struggling in mathematics. Portsmouth, NH: RMC Research Corporation, Center on Instruction.
Tier 1 - Universal Mathematics Instruction

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Implementation of core
mathematics instruction
Instruction with
methodology addressing
both conceptual and
procedural understanding.
Implementation of
instruction with fidelity
Identification of Students Through Screening

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Universal Screener
Building level team to facilitate the implementation of the
screening and progress monitoring
Use benchmarks or growth rates to identify students at
low, moderate, or high risk for developing mathematics
difficulties.
Student Body
is given a Universal Screening
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Universal Screenings
(What we hear schools are using)
DIBELS
AIMS Web
Think Link
GRADE
MAP
Universal Screening: Determines students with
possible mathematics difficulties.
This is not an endorsement of the products,
but a listing of those described by schools.
Students Identified by the Universal Screening are
given more in-depth mathematics assessment.
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In-Depth Assessment determines:
Tier of Support
Specific Areas of Need
Plan of Intervention
In-Depth
Assessment
CCSS- Math Domains
Counting and Cardinality
Operations and Algebraic Thinking
Number and Operations in Base 10
Number and Operations- Fractions
Geometry
Data and Measurement
Standards of Mathematical
Practice
1.
2.
3.
4.
5.
6.
7.
8.
Make sense of problems and persevere in solving
them.
Reason abstractly and quantitatively.
Construct viable arguments and critique the
reasoning of others.
Model with mathematics.
Use appropriate tools strategically.
Attend to precision
Look for and make use of structure.
Look for and express regularity in repeated
reasoning.
C
R
U
A
E
What Does It Mean
to Understand Mathematics?
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Understanding is the measure of quality and quantity
of connections between new ideas and existing ideas
Knowing  Understanding (students may know
something about fractions, for example, but not
understand them)
Richard Skemp named the ends of the continuum of
understanding
Instrumental
Relational
understanding
understanding
What to do and why
Just doing it
Implications for Teaching

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The need to
replace the
question “Does the
student know it?”
with the question
“How does the
student understand
it?”
Early number
concepts
Copyright © Allyn and Bacon 2010
Computation
I thought they had it!
Petit, M., Laird, R., & Marsden, E. (2010). A focus on fractions: Bringing Research to the Classroom. New York: Routledge. .
Diagnostic Interview
•
•
•
•
•
•
Gathers in-depth information about an individual
student’s knowledge and mental strategies.
Provides evidence of prior knowledge, naïve
understandings and students’ ways of thinking about
concepts.
Focuses on a task or problem where students are asked
to either verbalize their thinking or demonstrate ideas
through models or drawings
Emphasizes the collection of evidence
Is not a teaching opportunity
Uses errors to identify barriers to understanding, to
inform instructional decisions
Diagnostic Interview - Example
Philip, R. & Cabral, C. (2005). IMAP: Integrating mathematics and pedagogy to illustrate children’s reasoning. San Diego State University Foundation: Pearson.
Student Teacher Interaction
Philip, R. & Cabral, C. (2005). IMAP: Integrating mathematics and pedagogy to illustrate children’s reasoning. San Diego State University Foundation: Pearson.
3 3/8
Philip, R. & Cabral, C. (2005). IMAP: Integrating mathematics and pedagogy to illustrate children’s reasoning. San Diego State University Foundation: Pearson.
Procedural Error
Philip, R. & Cabral, C. (2005). IMAP: Integrating mathematics and pedagogy to illustrate children’s reasoning. San Diego State University Foundation: Pearson.
Talk Aloud-Verbalization
Philip, R. & Cabral, C. (2005). IMAP: Integrating mathematics and pedagogy to illustrate children’s reasoning. San Diego State University Foundation: Pearson.
Drawing-Visual Representation
Philip, R. & Cabral, C. (2005). IMAP: Integrating mathematics and pedagogy to illustrate children’s reasoning. San Diego State University Foundation: Pearson.
Difference in Solutions
Philip, R. & Cabral, C. (2005). IMAP: Integrating mathematics and pedagogy to illustrate children’s reasoning. San Diego State University Foundation: Pearson.
Student Decision-making
Philip, R. & Cabral, C. (2005). IMAP: Integrating mathematics and pedagogy to illustrate children’s reasoning. San Diego State University Foundation: Pearson.
How to design a diagnostic assessment
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Look at what students need to know common core
Give a task
Record and/or make notes
Have materials available for students to use
The difference between a regular task and a diagnostic
interview is that you are sitting and listening to the
student talk about their thinking and how they are solving
the task. Questions are used to probe and materials and
other representations are used to see the depth of
understanding.
Interventions Following Diagnostics
 Use
of screening
information
 Use of diagnostic
interview
 Create a plan for
intervention
Recommendations for identifying and
supporting students struggling in mathematics

Recommendations are based on strong,
moderate and low levels of evidence resulting
from comprehensive reviews of current
research literature.
Gersten, R., Beckmann, S., Clarke, B., Foegen, A., Marsh, L., Star, J. R., & Witzel, B. (2009). Assisting students struggling with mathematics:
Response to Intervention (RtI) for elementary and middle schools (NCEE 2009-4060). Washington, DC: National Center for
Education Evaluation and Regional Assistance, Institute of Education Sciences, U.S. Department of Education. Retrieved from
http://ies. ed.gov/ncee/wwc/publications/practiceguides/.
General Screening and Intervention
Recommendations
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Screen all students
Choose appropriate instructional materials
Intervention with explicit instruction
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Modeling
Talk aloud (verbalization)
Guided practice
Feedback (correction of errors)
Frequent review of progress
Strong
Moderate
Low
Problem solving instruction based on common underlying
structures
Visual representations
Screening and Intervention
Recommendations

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10 minutes per session devoted to fluency building of
basic mathematics facts
Progress monitoring
Strong
Integration of motivational strategies
Moderate
Low
Recommendations for students identified as
low-achieving

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On a regular basis; and
For the purpose of building computation and problem
solving proficiency;
 Explicit instruction including opportunities for asking and
answering questions
 Think aloud opportunities regarding decisions during
problem solving
 Dedicated time to foundational skills necessary for grade
level mathematics learning
National Mathematics Advisory Panel. Foundations for Success: The Final Report of the National Mathematics Advisory Panel, U.S. Department of Education: Washington, DC, 2008.
Recommendations from research involving
small group interventions
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Explicit instruction
Concrete--Semi-concrete--Abstract approach
Modeling
Underlying mathematical structures
Examples (consideration of range and sequence)
Independent work with immediate corrective feedback
Visuals (drawings & diagrams)
Note: The interventions may be effective for other student groupings. This listing specifically targets small groups.
Newman-Gonchar, R., Clarke, B., & Gersten, R. (2009). A summary of nine key studies: Multi-tier intervention and response to interventions for students
struggling in mathematics. Portsmouth, NH: RMC Research Corporation, Center on Instruction.
Hedden’s Continuum - CSA
Effective Practices for Teachers
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Explicit Instruction
A range of instructional examples, a sequence from
concrete-representational-abstract
Verbalization by the students and the teacher
Use of visual representation
Multiple heuristic strategies
Formative assessment information provided to teachers
Peer-assisted learning (1:1 tutoring)
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Cross age (more effective)
Within classroom same grade, role exchange
Performance based
Jayanthi, M., Gersten, R., Baker, S. (2008). Mathematics instruction for students with learning disabilities or difficulty learning mathematics: A guide for
teachers. Portsmouth, NH: RMC Research Corporation, Center on Instruction.
Thank you

Contact information.

Karen Karp
Amy Lingo

karen@louisville.edu
amy.lingo@louisville.edu