Expanding Mathematical

Thinkingfor Teachers and

StudentsCalifornia Math and Science

PartnershipPajaro Valley Unified School District

Santa Cruz County

Pajaro Valley Unified School District

• 16 elementary schools

• 6 middle or junior high schools

• 3 high schools

• 6 charter schools

PVUSD Student Demographics for Targeted

Schools

• 77% free or reduced lunch

• 57% English language learners

• 90% Latino

• 54% migrant students

CaMSP in PVUSD

Mathematics in Grades 5 through Algebra I

Two Primary Goals:1. Improve teachers’ content and

pedagogical content knowledge, particularly in the core areas of algebra and functions

2. Improve student achievement

A Multi-Faceted Approach to Professional Development

• Algebraic Thinking Institute

• Lesson Study

• MARS Performance Assessments

• Individualized Coaching

“The new vision of mathematics teaching and learning… is one in which all students investigate complex problems in mathematics in supportive, collegial communities…

…To achieve this vision teachers need new knowledge, skills, behaviors, and dispositions…

…Because it is difficult if not impossible to teach in ways that one has not learned…

…teachers also need opportunities to learn challenging mathematics and reflect on their own learning and teaching in supportive, collegial communities.”

Loucks-Horsley, Love, Stiles, 2009

Algebraic Thinking Institute

• Week-long summer institute presented by Dr. Joanne Rossi Becker of San Jose State University

• Active learning with an emphasis on problem solving and deepening understanding of content around big ideas in algebra

Big Ideas in Algebra

• Variable• Equality• Function• Representation• Structure• Modeling• Generalization

Algebraic Thinking Institute

Throughout the institute, teachers:

• Grapple with and collaboratively solve difficult mathematical problems

• Discuss and present a variety of possible solutions and learn from each other

• Reflect on instructional strategies used and how to implement them in their own classrooms, especially making them accessible to English language learners

Professional Development through

Lesson Study

The chief value of curriculum revision lies not in the product made, but in the process of the making, for it rekindles the intellectual life of all those who participate in the enterprise.

Lesson Study Institute

• Week-long summer institute presented by Dr. Joanne Lieberman and Dr. Don Pierce of California State University at Monterey Bay

• History and overview of the Lesson Study process. Guidance provided in goal setting and planning.

Guiding Principals of Lesson Study

• Student learning is primary.

• Decisions are data driven.

• Student data provides the basis for continuous improvement in schools.

Lesson Study Goals

• Investigate and implement effective mathematics instructional strategies.

• Promote the value of student discourse and what lesson characteristics enhance it.

• Create shared expectations for student thinking and learning.

• Increase collaboration and respect for each other.

Professional Development

TRADITIONAL• Begins with an answer• Driven by an expert• Is workshop-based• Communication flows

from trainer to participants

• Reinforces hierarchical relationships

• Research informs practice

LESSON STUDY• Begins with a question• Driven by participants• Is classroom-based• Communication flows

in conversation among participants

• Reinforces reciprocal relationships

• Research informs practice and practice becomes new research

What is Lesson Study?

Lesson Study is…

• Teacher-lead, ongoing professional learning

• Directed by a common overarching goal

• Focused on subject content in the context of student thinking

• Informed by outside expertise

What is Lesson Study?

Lesson Study is not…

• Teacher training

• Centered on creating a perfect lesson

• Conducted in isolation

• Limited to a single lesson study cycle

Lesson Study Cycle1. RESEARCH

Study curriculum and standards.

Consider long-term goals for student learning and

development.

Research best practices.2. PLAN

Select research lesson.

Anticipate student thinking.

Plan data collection and lesson.

3. CONDUCT RESEARCH LESSON

One team member teaches, others collect data.

4. REFLECTShare data.

What was learned about student learning?

What are implications for this unit and more broadly?

What learnings and new questions do we want to carry forward in our

work?

Lesson Study Cycle - Revision

1. RESEARCHStudy curriculum and

standards.

Consider long-term goals for student learning and

development.

Research best practices.2. PLAN/REVISE

Revise research lesson.

Anticipate student thinking.

Plan data collection and revised lesson.

3. CONDUCT RESEARCH LESSON

One team member teaches, others collect data.

4. REFLECTShare data.

What was learned about student learning?

What are implications for this unit and more broadly?

What learnings and new questions do we want to carry forward in our

work?

Lesson Study Culmination

• Teams come together for a day to celebrate and share experiences.

• Teams create presentations to showcase their lessons and reflect on their learnings from the lesson study process.

Lesson Study Presentations

• Overview of where lesson fits in the course and unit

• Overview of goals of lesson

• Sequence of activities in lesson

• Analysis of what the team learned and revisions made

SURFACE AREA AND VOLUME OF RECTANGULAR PRISMS

Harendra, Emily, Kelli, Tomas, Erin

Introduction

What is the difference between surface area and volume of a rectangular prism?

How do I find the surface area of this net?

How do I find the volume of this prism?

Goals

Mathematical objectives: Students calculate surface area of nets

in square units

Students develop a conceptual understanding of volume as layers of cubes

Students distinguish between square and cubic units

Student Engagement

Students counted squares on each face of the net to calculate surface area

Students cut and folded nets, creating rectangular prisms

Students used cubes to fill prisms, calculating volume

Students strategized methods to find volume without enough cubes

Student Thinking

Surface Area The number of

squares on a face is that face’s surface area

There are three pairs of faces

All six faces are added together to get total surface area

Volume The number of

cubes that filled the prism equals the volume

Students counted how many cubes it took to fill one layer and then counted how many layers they needed to fill the prism

Volume could be calculated as layers or slices

Revisions

Units changed Listed the six sides for each figure on the

recording sheet for surface area Broke the lesson down into two days,

didn’t talk about dimensions in the first lesson

Reduced the number of nets Lesson quiz revised

Lesson Study video

Benefits of Lesson Study

• Provides valuable collaboration opportunities to solve common problems.

• Fosters deeper understanding of mathematics and student learning across grade levels.

• Reveals important insights about instructional practices.

Benefits of Lesson Study continued

• Allows teachers to focus on helping all students learn.

• Promotes shared reflection based on classroom evidence.

• Makes concrete what reflection means, what problem solving looks like, and what thinking entails.

MARS Performance Assessments

“We must ensure that tests measure what is of value, not just what is easy to test. If we want students to investigate, explore, and discover, assessment must not measure just mimicry mathematics.”

Everybody Counts, 1989

MARS Performance Assessments

• The Mathematics Assessment Resource Service (MARS) is an NSF-funded collaboration between U.C. Berkeley, Michigan State and the Shell Centre in Nottingham England.

• The MARS Assessments target second grade through Geometry, aligned with the California and NCTM National Mathematics Standards.

Anatomy of a MARS task

Entry level – access into the problem

Core mathematics – meeting standards

Ramp up – conceptually deeper, beyond

Entry Core Ramp Up

Implementing MARS

• Formative assessments

• Instructional tool to deepen conceptual understanding

• Re-engagement lessons to address student misconceptions

MARS as Professional Development

• Scoring MARS

• Analyzing student strengths and misconceptions

• Creating opportunities for re-engagement

MARS to CST Correlation

(insert 4-quadrant Correlation to CST)

Individualized Content Coaching

“Content-Focused Coaching centers on students’ learning in the lessons but is also about teachers’ learning from the process. In the short term, teachers refine how they teach particular lessons to specific groups of students. In the long term, they develop professional habits of mind and general teaching expertise.”

West and Staub, 2003

Acquiring New Skills/Behaviors

Training vs.

Impact

Concept Understand

ing

Skill Attainment

Application

Presentation 85% 15% 10%

Modeling 85% 18% 15%

Practice 85% 80% 15%

Coaching 90% 90% 80%

Joyce and Showers, 1982

Guiding Principles of Content

Coaching• Develops teacher capacity to

increase the number of students learning at high levels

• Focuses on guiding the teachers to examine:– the student work,– the mathematics,– and the pedagogy related to

mathematics

Math Coaches…

• Help teachers extend their understanding of mathematical knowledge and instructional strategies.

• Work collaboratively with teachers to assess student thinking and develop effective lessons for all students.

• Foster professional habits of mind.

Teacher Support through Coaching

• During class: observing lessons, team teaching, and providing demo lessons using the coaching cycle*

• Outside of class: lesson planning, examining student work, content training, formative assessment evaluation

*Coaching cycle: pre-conference, lesson, post-conference

Pajaro Valley Unified School DistrictCaMSP cohort 4

tamra_mullin@pvusd.netsuzannah_young@pvusd.netfrances_whitney@pvusd.net