Research Based Tools and Insights for Improving

Mathematics and Science Teachers’ Classroom Practices:

Dr. Marilyn P. Carlson Dr. Marilyn P. Carlson Arizona State UniversityPrincipal Investigator: Project Pathways

Professor, Department of Mathematics and StatisticsDirector, 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

CRESMET

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 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

InterventionOne Example

Carlson & Bloom Problem solving framework -Describes effective mathematical practices

Conceptual Frameworks (e.g.,Function Taxonomy) -Characterizes understanding (e.g., reasoning abilities, connections, notational issues)

Theoretical grounding for

-Designing curricular modules

-Determining course structure

-Instrument development

Lens for researching

emerging practices

and understandings

ReviseCognitive

Frameworks

Design (Revise)

Practices & Instruments

C o n je c tu re

EvaluateImagine

P lan n in g

Executing

Checking

O r ie n t in g

Planning

Executing

CheckingCyc

ling

Bac

k

Cyc

ling

For

war

d

Conjecture Cycle

Some Insights About Some Insights About Effective Problem Effective Problem Solving Practices: Solving Practices:

The Problem Solving The Problem Solving CycleCycle

Planning

PhaseBehavior

Resources Heuristics Affect Monitoring

OrientingSense makingOrganizingConstructing

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?”

ConjecturingImaginingEvaluating

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?”

ExecutingComputingConstructing

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 well-connected 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.

The Multidimensional Problem Solving Framework: A The Multidimensional Problem Solving Framework: A Characterization of Effective Problem Solving PracticesCharacterization of Effective Problem Solving Practices

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) Instructional goals (b) The role of homework and exams (c) Attention to the learning process (d) Teaching Constraints (Factors of Resistance) (e) 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 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)

Instructional goals The primary goal 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 questionTeaching Constraints (Factors of Resistance)

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 Expectation of Meaning Making

Persistence in sense making, initiative in stringing together a logical argument

Mathematical Integrity (intellectual honesty) 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 Decision to Establish Classroom Rules of Engagement Individuals are engaged in meaning making through

various instructional modes 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 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

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 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

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.