The Focus of America's Choice Mathematics Professional Development:
Mathematics Offerings:
Conceptual Understanding
The America’s Choice Mathematics model focuses on mathematics as three interrelated elements—concepts, problem solving, and skills. The relationship of these elements is essential for understanding mathematics. Without a conceptual understanding of the mathematics, skills (algorithms, procedures, etc.) and problem-solving strategies, procedures can and will be misapplied. Since the development of the NCTM Principles and Standards for Mathematics in 1989, and the introduction of the New York City Standards in 1996, there has been a push to bring standards-based curricula into New York City Public Schools. In 2003, all elementary and middle schools, except those exempt through either the Empowerment Zone or student performance, have been using Everyday Mathematics in grades K–5 and Impact Mathematics in grades 6–8. Both these curricula focus on building a conceptual understanding of mathematics through the lens of algebra. However, it is common to see teachers introducing procedures not contained within these curricula in their classrooms, simply because a particular procedure is how they were taught and, therefore, learned mathematics and how they understand the mathematics they are teaching. The QUASAR study at the University of Pittsburgh in 1996, demonstrated the importance of procedures in mathematics when students learned these procedures with connections. The connections are the conceptual foundation of the mathematics. In Navigator University, the focus of mathematics is on conceptual understanding and using this understanding to make connections to skills and problem-solving applications.
Lens on Algebra
Mathematics is learned in a continuum–one that spirals and builds over the course of a student’s educational experience. However, mathematics is often taught in a vacuum with teachers isolated from knowing what is in the curricula from year-to-year. It is not uncommon to hear teachers in one grade lament about what students should have learned in a previous grade, finger-pointing backwards: high school to middle school, middle school to elementary school, and even grade to grade. Students are caught in the middle of this lack of communication and piecemeal approach to learning and become frustrated and reluctant learners of mathematics. Teaching mathematics with an algebraic lens means building conceptual understanding of core mathematical concepts in ways that consider application of the mathematics across number systems and procedures that translate into algebra. For example, understanding that addition with whole numbers is about numbers getting larger, whereas addition with integers is not always; or that the vertical (column) algorithm for multiplication does not translate into algebra. For elementary teachers it is essential to know how the mathematics they teach changes as students near algebra, and for secondary teachers it is essential that they know how students have learned mathematics previously in order to support them in application to algebra. In America’s Choice course offerings, all mathematical concepts are explored through the lens of algebra—how does this concept look in algebra and what is the algebraic thinking within this concept.
Pedagogy
Opportunities for learning are when teachers and students interact with content. The pedagogical practices that support learning are based on research on how people learn:
• Engage preconceptions (begin with what is known)
• Organize learning around factual knowledge
and conceptual frameworks
• Meta-cognitive approaches to understand how one understands and learns
In mathematics, learning is through discovery, when students have opportunities to use their prior knowledge to explore and discover new concepts and applications and then have time to reflect on what they have learned and how they learned it. When students struggle with mathematics, often their initial teaching has not provided them with opportunities to discover the mathematics. Giving students a formula and asking them to plug in numbers is not learning. When students come into mathematics without the concepts necessary to support exploration, or with misconceptions about the mathematics, their learning is impeded. Navigator University will explore the pedagogical practices that teachers can use to help students enter into cognitive conflict about the mathematics they are learning, how they have understood that mathematics, and then revise their thinking to revise their misconceptions. Each course focuses on three key components of mathematics:
• The conceptual foundation for the mathematics they are teaching, with an emphasis on learning procedures and algorithms with understanding;
• A focus on how the mathematics supports and builds to algebra;
• The pedagogical practices that support revision of misconceptions that impede
learning.
The Offerings
Coaching Institute
Mathematics Coaches
Coaching is an art; it entails developing expertise in the content area and the ability to work with peers to deepen practice. In this session participants will explore the heart of coaching including:
• The language of coaching
• Understanding the process of change in instructional practice
• Adult learning styles
• Strategies for working with teachers
• Feedback to teachers
• Analyzing data on student progress and using data to inform instructional planning
A combination of seminar, workshop, and practicum, this course is spaced over 3 one-day sessions to provide time for application between sessions.
Session 1: The coaching framework
Session 2: Changing practice
Session 3: Using data and feedback processes
Materials: Coaching Handbook
Price: $750
Vendor Number: AME141000
Contract Number: QC4390I
Dates: September 4, 2009; September 25, 2009; December 1, 2009
Mathematics and Misconceptions—Basic Math Concepts: Addition and Subtraction
Teachers of mathematics: Elementary
The relationship of addition and subtraction is a given in mathematics, but understanding addition as the process of combining two quantities and subtraction as the separating of two quantities is also dependent on recognizing, understanding, and interpreting addition and subtraction in contextual situations. In this course, participants will learn how to represent addition and subtraction situations, how to distinguish between these representations, and how to justify the mathematical reasons for the choices they make. We will explore the inverse nature of addition and subtraction and how the properties of addition can be used to solve problems.
A combination of seminar, workshop and practicum, this course is spaced over 2 one-day sessions to provide time for application between sessions.
Session 1: The conceptual framework for addition and subtraction
Session 2: Using multiple representations and justifications
Materials: Excerpts from Mathematics Navigator Knowing Addition and Subtraction
Price: $500
Vendor Number: AME141000
Contract Number: QR328BJ
Dates: August 14, 2009; September 15, 2009; October 26, 2009
Mathematics and Misconceptions—Basic Math Concepts: Multiplication and Division
Teachers of mathematics: Elementary
Mastering multiplication and division facts is essential for success in upper elementary, middle, and high school mathematics, but teachers are continually frustrated by students who do not know their multiplication and division facts or understand when to multiply or divide. The purpose of this course is to help teachers understand the concepts that support learning multiplication and division facts, beyond rote memorization. Participants will explore strategies to support students to recognize, produce, and use whole number patterns that support the development of these facts; explore factors and multiples within the relationships between multiplication and division facts; use the properties of multiplication to solve multiplication and division problems; and explore the misconceptions that impede student learning of multiplication and division.
Participants will develop knowledge and skills related to
• The cognitive research base regarding conceptual learning and the effect of misconceptions on mathematics learning • Identifying common misconceptions among students at varying levels of mathematics learning
• Analyzing student work to diagnose misconceptions
• Strategies for uncovering students’ misconceptions in math and helping them to learn underlying concepts so that their math learning can progress.
• Analyzing samples of student work that illustrate common kinds of misconceptions
• Analyzing the sources of misconceptions
• Studying lessons designed to help students overcome misconceptions
• Assessing their own students to identify evidence of misconceptions and build a plan to
address them
A combination of seminar, workshop and practicum, this course is spaced over 2 one-day sessions to provide time for application between sessions.
Session 1: The conceptual framework of multiplication and division
Session 2: Multiplication properties and the inverse relationship of multiplication and division
Materials: Excerpts from Mathematics Navigator Knowing Multiplication and Division Facts
Price: $500
Vendor Number: AME141000
Contract Number: QR328BJ
Dates: August 12, 2009; September 16 2009; October 27, 2009
Mathematics and Misconceptions—Division and Fractions
The relationship of division to fractions and an understanding of division models impacts students’ understanding and ability to work with fractions and all forms of rational numbers. Fractions present challenges to teachers and students alike because they differ in notation from whole numbers and can have multiple interpretations—representing part of a whole as well as representing a division problem. Understanding fractions requires examining different representations of fractions, for example number lines and area models, and being able to compare fractions and find equivalent fractions. The ability to represent fractions in more than one way, to connect fractions with decimals, and to use these representations in contextual situations is the focus of this course.
A combination of seminar, workshop and practicum, this course is spaced over 3 one-day sessions to provide time for application between sessions.
Session 1: Division and division models—a conceptual framework
Session 2: The conceptual foundation of fractions, representation of fraction
Session 3: Fractions and division—implications for rational numbers
Materials: Participants guide on types of division, excerpts from Mathematics Navigator Knowing Fractions
Price: $750
Vendor Number: AME141000
Contract Number: QR328BJ
Dates:
Option 1: August 13, 2009; September 18, 2009; October 13, 2009; November 17, 2009
Option 2: March 25, 2010; April 21, 2010; May 19, 2010
Conceptual Learning in Mathematics—Planning for Conceptual Understanding
Mathematics Teachers
Learning mathematics is dependent on developing a strong conceptual basis upon which students can build. Too often, teachers feel pressured to cover the content, leading them to teach procedures that students can use to solve problems. For many students, these procedures are practiced without understanding and, therefore, they are unable to build on what they know. Teaching for conceptual understanding means developing an understanding of the underlying mathematics behind the procedures we use. In this session, participants will:
• Explore the relationship between procedures and concepts
• Discover how the workshop model helps students build their conceptual understanding
• Work with the process strands of the New York State standards to deepen understanding of instructional planning
• Use an annotated planning template to help facilitate the planning process
• Look at student work from planned lessons
A combination of seminar, workshop, and practicum, this course is spaced over 3 one-day sessions to provide time for application between sessions. Participants should bring a student edition of their grade level mathematics resource to the session.
Session 1: Procedure versus conceptual
Session 2: Conceptual lesson planning
Session 3: Revising lesson plans based on student work
Materials: Professional Book—Implementing Standards Based Mathematics in the Classrooms and handouts
Price: $750
Vendor Number: AME141000
Contract Number: QR328BJ
Dates: August 28, 2009; September 17, 2009; October 16, 2009; November 19, 2009
Conceptual Learning in Mathematics—Rational Numbers
Teachers of mathematics (grades 6–8 level)
Participants will develop knowledge and skills related to:
• Strategies for helping students build their mathematical understanding
• Planning conceptual lessons that align with Impact or another middle school mathematics text, as appropriate
• Deepening their understanding of ratio, rational number, and proportional reasoning
• Analyzing the importance of these concepts in students’ mathematics learning
• Learning and practicing strategies for building students’ conceptual learning
• Planning lessons to build conceptual understanding that align with the mathematics text, implement the lessons, reflect on the effectiveness of the strategies, and plan next steps.
A combination of seminar, workshop, and practicum, this course is spaced over 6 one-day sessions to provide time for application between sessions.
Session 1: Division representations and Fractions I
Session 2: Fractions II and Ratio Tables
Session 3: Strip Diagrams and Scale Factors I
Session 4 Strip Diagrams and Scale Factor II
Session 5: Rate
Session 6: Percent
Materials: Excerpts from Mathematics Navigator
Knowing Fractions, Handouts
Price: $1,500
Vendor Number: AME141000
Contract Number: QR328BJ
Dates: September 21, 2009; October 28, 2009; November 18, 2009; January 19, 2010; February 25, 2010; March 24, 2010
Analyzing Student Work, Kindergarten to Grade 3
Teachers and school leaders
This workshop guides participants through a process of studying student work with reference to standards. It models approaches for using student work samples to scaffold students’ learning and develops participants’ facility with the use of different protocols to support collaborative study of student work.
Participants will develop knowledge and skills related to:
• Analyzing student work with reference to standards
• Using student work exemplars and anchors to develop a shared understanding of the characteristics of student work that meets standards
• Identifying patterns of learning in collections of student work and analyzing instructional needs
• Using of a range of protocols for analysis of student work and their differing uses
• Using student work samples as models for learning in a standards-based classroom
A combination of seminar, workshop, and practicum, this course is spaced over 4 one-day sessions to provide time for application between sessions.
Session 1: Standards in mathematics—how good is good enough
Session 2: Identifying evidence in student work
Session 3: Feedback structures
Session 4: Using student work to inform instruction
Materials: Professional Book—Protocols for Looking at Student Work
Price: $1,000
Vendor Number: AME141000
Contract Number: QR328BJ
Dates: October 29, 2009; December 17, 2009; January 20, 2010; February 22, 2010
Analyzing Student Work, Grades 4 to 8
Teachers and school leaders
This workshop guides participants through a process of studying student work with reference to standards. It models approaches for using student work samples to scaffold students’ learning and develops participants’ facility with the use of different protocols to support collaborative study of student work. Participants will develop knowledge and skills related to:
• Analyzing student work with reference to standards
• Using student work exemplars and anchors to develop a shared understanding of the characteristics of student work that meets standards
• Identifying patterns of learning in collections of student work and analyzing instructional needs
• Using of a range of protocols for analysis of student work and their differing uses
• Using student work samples as models for learning in a standards-based classroom
A combination of seminar, workshop, and practicum, this course is spaced over 4 one-day sessions to provide time for application between sessions.
Session 1: Standards in mathematics—how good is good enough
Session 2: Identifying evidence in student work
Session 3: Feedback structures
Session 4: Using student work to inform instruction
Materials: Professional Book—Protocols for Looking at Student Work
Price: $1,000
Vendor Number: AME141000
Contract Number: QR328BJ
Dates: October 30, 2009; December 16, 2009; January 21, 2010; February 23, 2010
Developing Rubrics to Support Student Learning
Mathematics teachers and coaches
Outcomes
This workshop guides participants through a study of different kinds of rubrics and the characteristics that affect their value for teaching and assessment purposes. The workshop also models the process of co-constructing rubrics with students. Participants try out these practices in their own classrooms and reflect on the experience. Finally, faculty co-construct guidelines for effective rubrics to provide for consistency in their use of rubrics with students. Participants will develop knowledge and skills related to:
• Purposes of rubrics, for teaching and for assessment
• Developing teaching and assessment rubrics
• Co-constructing teaching and assessment rubrics with students
A combination of seminar, workshop, and practicum, this course is spaced over 3 one-day sessions to provide time for application between sessions.
Session 1: Criteria and Evidence—the basis of rubrics
Session 2: What does it mean to meet standards— using anchors
Session 3: Using rubrics to inform instructional planning
Materials: Handouts
Price: $750
Vendor Number: AME141000
Contract Number: QR328BJ
Dates: November 16, 2009; December 15, 2009; January 22, 2010
Developing Quality Classroom Discussion
Teachers
Participants will develop knowledge and skills related to:
• The research base supporting the cognitive value of carefully constructed classroom conversations
• Structures to scaffold meaningful discussions
• Verbal and nonverbal management tools to focus and enrich classroom conversations
• Reading and discussion of cognitive research related to the role of conversation in learning
• Analyzing videotaped examples of classroom discussions
• Practice building scaffolding activities to support classroom discussions
• Using classroom conversations in their instruction, analyze their experience and reflect on the implications for ongoing instruction.
A combination of seminar, workshop, and practicum, this course is spaced over 4 one-day sessions to provide time for application between sessions.
Session 1: Accountable Talk in the Mathematics Classrooms
Session 2: Pedagogical strategies to build talk
Session 3: Planning to talk
Session 4: Deepening classroom discussions
Materials: Classroom Discussions-Using Math Talk to Help Students Learn, handouts
Price: $1,000
Vendor Number: AME141000
Contract Number: QR328BJ
Dates: August 27, September 11, September 22, November 20, 2009; February 24, 2010
Reading in Mathematics—Comprehension and Language
Teachers of mathematics
Participants will develop knowledge and skills related to:
• The cognitive research base of the Construction-Integration Model of Comprehension
• The challenges for comprehension presented by math tasks and texts
• Strategies for applying the model of comprehension in helping students learn from informational and non-narrative texts
• Addressing the unique challenges mathematics presents for readers
• Planning lessons that incorporate comprehension instruction into learning from their existing textbooks.
• Reading and discussion of cognitive research related to reading comprehension, focusing on the Construction-Integration Model of Comprehension and its components of textbase and mental model
• Analyzing their textbooks from the perspective of the research base on comprehension and identify the challenges the textbooks present for students’ comprehension
• Experience modeling in a workshop setting of deliberate instruction in comprehension of math tasks based on the model and reflection on the construction of the learning experience
• Implementation of lessons designed to provide instruction on comprehension and reflect on their effectiveness
• Planning comprehension instruction for incorporation into existing curriculum
A combination of seminar, workshop, and practicum, this course is spaced over 3 one-day sessions to provide time for application between sessions.
Session 1: Reading in Mathematics—the challenge
Session 2: Language in Mathematics—making sense
Session 3: Planning for Reading, Writing,
Speaking, and Listening in Mathematics
Materials: Professional Book—Literacy Strategies for Improving Mathematics Instruction, Handouts
Price: $750
Vendor Number: AME141000
Contract Number: QR328BJ
Dates:
Option 1: September 10, 2009; September 23, 2009; December 14, 2009
Option 2: March 23, 2010; April 20, 2010; May 18, 2010
