CRESMET
Research Based Tools and Insights for
Improving Mathematics and Science
Teachers’ Classroom Practices:
Dr. Marilyn P. Carlson
Arizona State University
Principal Investigator: Project Pathways
Professor, Department of Mathematics and Statistics
Director, CRESMET
Center for Research on Education in Science,
Mathematics, Engineering, and Technology
This work was supported, in part, by grant no. 9876127 from the National
Science Foundation
Project Pathways

Partnership of ASU and five school districts
(ASU: Mathematicians, scientists, engineers, math and science educators,
professional development experts)
Primary Goal:
• To produce a research-developed, refined & tested
model of inservice professional development for
secondary mathematics and science teachers.
Core Strategies:
•
Four integrated math/science graduate courses + linked teacher
professional learning communities (lesson study approach)
Supporting Strategies:
•
•
•
Motivational activities for students
Tools for ELL students
Information & networking for guidance counselors
Initial Assumptions:
•
•
•
•
•
Many teachers have weak understanding of
fundamental concepts they are expected to teach
Teachers hold naïve beliefs about teaching and
learning
Teachers have little time to reflect on and modify
their teaching practices
Math and science teachers need support in learning
to connect STEM concepts across disciplinary
boundaries
Short term interventions have little impact on
teaching practice
Pathways Objectives for Teachers
•
Deepen teachers’ understanding of foundational mathematics &
science concepts and their connections (function, rate-of-change,
covariation, force, pressure)
•
Improve teachers’ reasoning abilities and STEM habits of mind
(problem solving, scientific inquiry, engineering design)
•
Support teachers in adopting “expert” beliefs about STEM
learning, STEM teaching, and STEM methods (See STEM BILT
Taxonomy)
•
Support teachers in reflecting on and modifying their classroom
instruction
Pathways Objectives for Students
•
Increase student persistence in challenging course-taking
•
Improve problem solving, scientific inquiry, engineering
design practices
•
Improve students’ understanding of foundational
concepts and reasoning abilities that have been
documented to be essential for continued STEM course
taking
•
Improve confidence and interest in math & science
Role of Frameworks--Linking
Intervention and Assessment

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Drive curriculum development for all four courses
and learning communities
Guide classroom structure and practices
Provides a lens for analyzing qualitative data
Frames assessment tool development
Analysis of qualitative data has resulted in
development of new frameworks and adaptations of
existing frameworks
The Reflexive Relationship Between
Frameworks and Intervention
Design (Revise)
One Example
Practices &
Instruments
Carlson & Bloom Problem solving
framework
-Describes effective
mathematical practices
Conceptual Frameworks
(e.g.,Function Taxonomy)
-Characterizes understanding
(e.g., reasoning abilities,
connections, notational issues)
Revise
Cognitive
Frameworks
Theoretical grounding for
-Designing curricular modules
-Determining course structure
-Instrument development
Lens for researching
emerging practices
and understandings
Some Insights About
Effective Problem
Solving Practices:
The Problem Solving
Cycle
Conjecture
Imagine
Conjecture Cycle
Orienting
Executing
Checking
Planning
Cycling Forward
Cycling Back
Planning
Executing
Checking
Evaluate
The Multidimensional Problem Solving Framework: A
Characterization of Effective Problem Solving Practices
Phase
Resources
Heuristics
Affect
Monitoring
Mathematical concepts, facts
and algorithms were
accessed when attempting to
make sense of the problem.
The solver also scanned
her/his knowledge base to
categorize the problem.
The solver often drew
pictures, labeled unknowns,
and classified the problem.
(Solvers were sometimes
observed saying, “this is an
X kind of problem.”)
Motivation to make sense of
the problem was influenced
by their strong curiosity and
high interest. High
confidence was consistently
exhibited, as was strong
mathematical integrity.
Self-talk and reflective
behaviors helped to keep
their minds engaged. The
solvers were observed
asking “What does this
mean?”; “How should I
represent this?”; “What does
that look like?”
Conceptual knowledge and
facts were accessed to
construct conjectures and
make informed decisions
about strategies and
approaches.
Specific computational
heuristics and geometric
relationships were accessed
and considered when
determining a solution
approach.
Beliefs about the methods
of mathematics and one’s
abilities influenced the
conjectures and decisions.
Signs of intimacy, anxiety,
and frustration were also
displayed.
Solvers reflected on the
effectiveness of their
strategies and plans. They
frequently asked themselves
questions such as, “Will this
take me where I want to go?”
and “How efficient will
Approach X be?”
Executing
Computing
Constructing
Conceptual knowledge,
facts, and algorithms were
accessed when executing,
computing, and constructing.
Without conceptual
knowledge, monitoring of
constructions was
misguided.
Fluency with a wide
repertoire of heuristics,
algorithms, and
computational approaches
were needed for the efficient
execution of a solution.
Intimacy with the problem,
integrity in constructions,
frustration, joy, defense
mechanisms, and concern
for aesthetic solutions
emerged in the context of
constructing and
computing.
Conceptual understandings
and numerical intuitions were
employed to reflect on the
reasonableness of the
solution progress and
products when constructing
solution statements.
Checking
 Verifying
 Decision
making
Resources, including wellconnected conceptual
knowledge, informed the
solver as to the
reasonableness or
correctness of the solution
attained.
Computational and
algorithmic shortcuts were
used to verify the
correctness of the answers
and to ascertain the
reasonableness of the
computations.
As with the other phases,
many affective behaviors
were displayed. It is at this
phase that frustration
sometimes overwhelmed
the solver.
Reflections on the efficiency,
correctness, and aesthetic
quality of the solution
provided useful feedback to
the solver.
Behavior
Planning
Orienting
Sense making
Organizing
Constructing

Conjecturing
Imagining
Evaluating
Assist teachers in learning to promote these practices
in their students
 Respect the process of change
STEM Beliefs Instrument about Learning and
Teaching Mathematics (Instrument Taxonomy)
Confidence
(a) Teachers Confidence in their mathematical ability
(b) Teachers confidence in their scientific reasoning ability
(c) Teachers confidence in their pedagogical ability
Methods of Mathematics and Science
(a) Process of learning mathematics and science
(b) Learnability of mathematics and science
(c) Authority in learning mathematics and science

III.
Methods of Teaching Mathematics and Science
(a)
(b)
(c)
(d)
(e)
Instructional goals
The role of homework and exams
Attention to the learning process
Teaching Constraints (Factors of Resistance)
Instructional Approaches/Teacher Classroom Practices
Development of tools to assess teachers’ cognitive development,
beliefs, values and classroom practices (e.g., Beliefs About Learning
and Teaching Mathematics Taxonomy
Understanding (Learning Dimension)
 Understanding mathematics requires special abilities that only some
people have.
 Understanding mathematics requires the individual to engage in sense
making to construct meaning.
Authority (Methods of Learning Mathematics)
Authority in learning mathematics and science
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I try to get my students to rely on their own reasoning and logic when
verifying the correctness of their solutions
It is important that students look to their teacher verify the correctness of
their solutions.
(Teaching Dimension)

The primary
Instructional
goalsgoal of my instruction is to teach facts, skills/procedures that my
students may need to score well on an exam.

The primary goal of my instruction is to promote deep and coherent understanding of
central concepts
The role of homework and exams

The primary goal of my exams is to assess if my students can memorize facts and
carry out procedures.
Attention to the learning process

It is important to understand what a student is thinking when s/he asks a question
Teaching Constraints (Factors of Resistance)
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The pace at which I must cover material does not allow me to teach ideas deeply

My school administrators value my efforts to get students to understand ideas deeply.

I am able to adapt my pacing so I can spend instructional time on central concepts of
my courses
Discuss (with members of your
partnership) the three major beliefs
that your project is trying to change in
either teachers or students.
Write one likert style item for each
belief that you identified.
Other Findings that are guiding
Pathways Interventions

Dispositions that supported effective cognitive and
metacognitive behaviors
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Expectation of Meaning Making
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Persistence in sense making, initiative in
stringing together a logical argument
Mathematical Integrity (intellectual honesty)
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All conjectures were based on a logical
foundation
They did not pretend to know they didn’t
How these findings are informing the
Pathways Project

Informing the nature of our classroom and PLC
curriculum
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Decision to Establish Classroom Rules of Engagement
Individuals are engaged in meaning making through
various instructional modes
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Allow time for individual construction through various
instructional modes
All individuals are held accountable for expressing meaning.
Classroom Rules of Engagement are
Negotiated and Enforced by All Members of
the Classroom Community
Individuals are expected to:
 Persist in sense making
 Speak meaningfully
 Attempt to make logical connections
 Exhibit Mathematical Integrity
 Don’t pretend to understand when you don’t
 Base conjectures on a logical foundation
 Support peers in making their own constructions
Special Role of Instructor for Graduate
Course and Facilitator of Professional
Learning Community
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Intervene as appropriate to reinforce classroom rules
of engagement
Model effective mathematical practices
Promote opportunities for teachers to develop and
value effective classroom practices
Assist teachers in learning to promote these practices
in their students
 Respect the process of change
Other Instruments for Assessing
Pathways Progress and Effectiveness

PCA (Precalculus Concept Assessment) Instrument
 Previously developed and validated
 Modified late in year 1

VAMS (Views About Mathematics Survey; previously developed)
 STEM BILT (STEM Beliefs Instrument About Learning and
Teaching); developed--early in the validation process

RTOP (Reformed Teacher Observation Protocol)
 Already validated and published

Learning Community Observation Protocol (LCOP)
 Early draft is available; still collecting qualitative data to identify
critical variables
The Role of Concept Assessment
Instruments in Evaluation
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Provides coherent assessment of central ideas and what
is involved in understanding those ideas
Provides valid and reliable assessment of student
understanding
Easy to administer to large populations
Can assess course and instructional effectiveness
May serve as a tool to assess readiness for a course
Limitations of Concept Assessment
Instruments
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Do not reveal thinking of individual students
Do not reveal insights into the process of
learning
Are not tied to specific instruction or course
materials
Limited in assessing creative abilities
Closing Remarks
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Use of individual cognitive models of learning
or knowing, such as the Multidimensional
Problem Solving Framework increase the
purposefulness of instructional materials,
instructional actions, and serve as guiding
frameworks for instrument development.