Secondary Mathematics

630 Pages·2011·6.83 MB·English

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P A R T I C I P A N T ’ S M A N U A L

SECONDARY MATHEMATICS

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© 2010 The New Teacher Project, Inc.

All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted,
in any form or by any means electronic, mechanical, photocopied, recorded, or otherwise, without the prior
written permission of the publisher.

Every effort has been made to contact copyright holders of material reproduced in this book. Any omissions
will be rectifi ed in subsequent printings if notice is given to the publisher.

Bilingual Education

Early Childhood Education

Elementary Education

Elementary Special Education

English as a Second Language

Foreign Language

Generalist Education

Middle School Mathematics

Music

Secondary English Language Arts

Secondary Mathematics

Secondary Science

Secondary Social Studies

Secondary Special Education

Visual Arts

About The New Teacher Project

The New Teacher Project (TNTP) helps school districts and states fulfi ll the promise of public education by ensuring
that all students—especially those from high-need communities—get excellent teachers. A national nonprofi t
organization founded by teachers, TNTP is driven by the knowledge that although great teachers are the best solution
to educational inequality, the nation’s education systems do not suffi ciently prioritize the goal of effective teachers for
all. In response, TNTP develops customized programs and policy interventions that enable education leaders to fi nd,
develop, and keep great teachers and achieve reforms that promote effective teaching in every classroom. Since its
inception in 1997, TNTP has recruited or trained approximately 37,000 teachers—mainly through its highly selective
Teaching Fellows™ programs—benefi ting an estimated 5.9 million students. TNTP has also released a series of
acclaimed studies of the policies and practices that affect the quality of the nation’s teacher workforce, most recently
including The Widget Effect: Our National Failure to Acknowledge and Act on Differences in Teacher Effectiveness
(2009). Today TNTP is active in more than 40 cities, including Baltimore, Chicago, Denver, New Orleans, New York,
and Oakland. For more information, please visit www.tntp.org.

Teaching for Student Achievement, General Education and Special Education

The New Teacher Project’s pre-service training program, Teaching for Student Achievement (TfSA), is currently
in its fi fth edition. The training, which is designed specifi cally with the needs of novice teachers in mind, is based
on the essential knowledge and skills that new teachers need in order to have an immediate impact on the success
of their students. The conceptual framework is informed by the latest pedagogical research as well as TNTP’s own
fi eld-based research of program participants. Over the last several years of development, TNTP has integrated the
best practices of its most successful candidates with those of other successful teachers in high-need schools to
create a training program with two central goals for participants: (1) to make immediate academic progress with
all students; and (2) to close the achievement gap. Teachers who participate in the TfSA curriculum receive
a Guidebook, our central text for the training experience. The Guidebook is available in both general education
and special education versions, and serves as a resource to teachers during the training experience and beyond.

Teaching for Results

Teaching for Results seminars are designed to maximize a new teacher’s effectiveness through a focus on data-
driven instructional planning and delivery that aims to close the achievement gap for all students. Each title in
the series is grounded in an inquiry process through which course participants learn research-based instructional
strategies specifi c to a content area and are taught to constantly evaluate the effi cacy of those strategies against
authentic student performance data. This process encourages new teachers to make real-time adjustments to
their instructional practice with the goal of increasing student achievement.

Titles Offered in the Teaching for Results Series:

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Table of Contents

PARTICIPANT

Orientation Session ............................................................................... 1
Handout OS.1: Personal and Course Goals ..............................................................2
Handout OS.2: Center 1—Content .............................................................................4
Handout OS.3: Center 2—Assessment .................................................................... 17
Handout OS.4: Center 3—Instruction .....................................................................39
Handout OS.5: Strategic Design ...............................................................................48
Handout OS.6: Strategic Design Unit Planning Template ....................................49
Handout OS.7: Unit Plan Self-Assessment ..............................................................55
Handout OS.8: Unit Plan Components ....................................................................56
Handout OS.9: Successes, Goals, and Challenges in My Classroom .................. 61

Session 1: Introduction to Secondary Mathematics ...........................63
Handout 1.1: Free Write Refl ection ..........................................................................64
Handout 1.2: Preparing for College in Secondary Mathematics ..........................65

Handout 1.3: Graphic Organizer for Preparing for College
in Secondary Mathematics ................................................................ 76
Handout 1.4: Secondary Mathematics Content Domain Graphic ........................ 81
Handout 1.5: Applying the Content Domain Graphic ............................................82
Handout 1.6: The Secondary Mathematics Content Domain ...............................83
Handout 1.7: Examining Standards .........................................................................85

Handout 1.8: Sample: How Is the Mathematics Domain Addressed
in Instruction? .....................................................................................86
Handout 1.9: How Is the Mathematics Domain Addressed in Instruction? .......90
Handout 1.10: Instructional Strategies Log ............................................................93

Module 1: Foundations for Teaching Mathematics

Session 2: Setting Up for Success .........................................................97
Handout 2.1: Developing a Vision of Success .........................................................98
Handout 2.2: 40 Developmental Assets for Adolescents Ages 12–18 ..................99
Handout 2.3: Assessment of 40 Developmental Assets ...................................... 102
Handout 2.4: Characteristics of Adolescents ....................................................... 108
Handout 2.5: Mathematics Confi dence Questionnaire ....................................... 110
Handout 2.6: 10 Ways to Increase Mathematics Confi dence .............................. 111
Handout 2.7: Developing Routines for Confi dence .............................................. 113
Handout 2.8: “I Can …” ............................................................................................ 115
Handout 2.9: Student Goal-Tracking Template .....................................................117
Handout 2.10: Four-Point Rubric for Assessing Specifi c Mathematics Skills .. 118
Handout 2.11: Understanding Knowledge for Teaching ......................................120
Handout 2.12: The Content Domain as Knowledge for Teaching ...................... 121

(continued)

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Table of Contents

Table of Contents (continued)

Handout 2.13: Self-Assessment ..............................................................................122
Handout 2.14: Setting Professional Goals .............................................................124

Session 3: Engaging in the Cycle of Inquiry ......................................127
Handout 3.1: Setting Big Goals for Students ........................................................128

Handout 3.2: Assessment Practices That Improve Teacher and
Student Performance .......................................................................129
Handout 3.3: Assessment Inventory ...................................................................... 135
Handout 3.4: What Data Do You Have? .................................................................138
Handout 3.5: The Inquiry Process .........................................................................139
Handout 3.6: Inquiry Process Walk-Through ....................................................... 140
Handout 3.7: Inquiry Process Implementation ..................................................... 143
Handout 3.8: Identifying a Focus Class ................................................................ 145
Handout 3.9: MDTP Diagnostics Overview .......................................................... 146
Handout 3.10: Which Assessment Do I Use? ........................................................ 150
Handout 3.11: How Do I Administer the MDTP Assessment? ............................ 154
Handout 3.12: Sample Student Directions for MDTP .......................................... 155
Handout 3.13: Sample MDTP Test Results Report .............................................. 157
Handout 3.14: Forming Ground Rules ................................................................... 169
Handout 3.15: The Inquiry Process, Step One: Diagnose ....................................171
Handout 3.16: Objectives Checklist ....................................................................... 173

Session 4: Overview of Teaching Standards ......................................175
Handout 4.1: About the National Council of Teachers of Mathematics .............176
Handout 4.2: NCATE/NCTM Standards ................................................................ 179
Handout 4.3: NCTM and the Content Domain ..................................................... 187
Handout 4.4: Sample Geometry Standards........................................................... 188
Handout 4.5: Sample NCTM Standards and State Standards ............................ 189
Handout 4.6: NCTM Standards and State Standards .......................................... 191
Handout 4.7: Sample Unit Plan ............................................................................... 192
Handout 4.8: Unit Plan ............................................................................................ 194

Handout 4.9: Sample: Aligning NCTM Standards to Your
Professional Goals ............................................................................ 196
Handout 4.10: Exit Slip ............................................................................................ 197

Module 2: Problem Solving, Proof, and
Communication in Mathematics

Session 5: Using Data to Drive Instruction I .....................................199
Handout 5.1: Navigating Your Score Report .........................................................200
Handout 5.2: Sample MDTP Test Results Report ................................................206
Handout 5.3: Practice Analyzing Class-Wide Patterns ....................................... 212
Handout 5.4: Analyzing Your Class-Wide Patterns ...............................................214

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Handout 5.5: Common Errors with Basic Skills .................................................. 216
Handout 5.6: Lowest Scoring Item Analysis ......................................................... 219
Handout 5.7: Example Identifying Student Strengths and Gaps ........................ 221
Handout 5.8: Identifying Student Strengths and Gaps ........................................222
Handout 5.9: Student Grouping .............................................................................223
Handout 5.10: Setting Goals ...................................................................................224
Handout 5.11: Teaching Plan for Mathematics .....................................................226
Handout 5.12: Incorporating Peer Feedback into Teaching Plan ......................228
Handout 5.13: Creating Effective Questions for Inquiry Groups .......................230
Handout 5.14: Data Analysis Protocol Notes ........................................................233

Session 6: Mathematical Problem Solving .........................................235
Handout 6.1: Evaluating My Teaching Plan ..........................................................236
Handout 6.2: Article on Problem Solving .............................................................237
Handout 6.3: Problem-Solving Article Discussion ..............................................248
Handout 6.4: How Is Problem Solving Addressed in Instruction? ....................249
Handout 6.5: The RICE Problem-Solving Method ...............................................252
Handout 6.6: Practice with the RICE Method ......................................................253
Handout 6.7: Problem of the Day Routine .............................................................254
Handout 6.8: Problem of the Day Routine Example Problem ............................255
Handout 6.9: Problem-Solving Pitfalls ..................................................................256
Handout 6.10: Critical Foundations for Algebra ..................................................258
Handout 6.11: Key Concepts in Algebra and Number Sense ..............................260
Handout 6.12: The Uses of Variables .....................................................................262
Handout 6.13: The Four Uses of Variables ............................................................263
Handout 6.14: Problem-Solving Skills for College Readiness ............................264
Handout 6.15: Problem-Solving Strategies Overview .........................................265
Handout 6.16: Sample Strategy Implementation PLANNING Sheet ..................269
Handout 6.17: Sample Strategy Implementation REFLECTION Sheet .............. 271

Handout 6.18: Evaluation Criteria for Strategy Implementation
Planning Sheet................................................................................. 272
Handout 6.19: Strategy Implementation PLANNING Sheet................................ 273
Handout 6.20: Strategy Implementation REFLECTION Sheet ........................... 276

Session 7: Mathematical Reasoning and Proof ..................................279
Handout 7.1: Strategy Debrief Group Process ......................................................280
Handout 7.2: Mathematical Reasoning as Equity ................................................ 281
Handout 7.3: Reading Guide for Mathematical Reasoning as Equity ...............288
Handout 7.4: Reasoning and Sense Making .........................................................290
Handout 7.5: Reading Guide for Reasoning and Sense Making .........................296
Handout 7.6: Reframing Mathematical Concepts ................................................297
Handout 7.7: Problem-Solving Practice .................................................................299
Handout 7.8: Problem-Solving Practice—Full Examples ..................................300

Handout 7.9: Aspects of Encouraging Reasoning in the Mathematics
Classroom ..........................................................................................304
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Handout 7.10: Problem-Solving Examples (Resource) .......................................305
Handout 7.11: Types of Reasoning ......................................................................... 315
Handout 7.12: Formal Proofs in Geometry ............................................................316
Handout 7.13: Formal Proofs in Calculus ..............................................................317
Handout 7.14: Practice Using Diagrams to Aid Proofs ....................................... 318
Handout 7.15: Reasoning and Proof Strategies Overview .................................. 319
Handout 7.16: Strategy Implementation PLANNING Sheet ................................322
Handout 7.17: Strategy Implementation REFLECTION Sheet ............................325

Session 8: Literacy and Communication in Mathematics ................. 327
Handout 8.1: Strategy Debrief Group Process ......................................................328

Handout 8.2: Learning Mathematical Language in the Zone
of Proximal Development .................................................................329
Handout 8.3: Learning Mathematical Language ...................................................335

Handout 8.4: The Need for Language Instruction in the Mathematics
Classroom ...........................................................................................336
Handout 8.5: Tools for Socratic Seminars .............................................................338
Handout 8.6: Mathematics Vocabulary to Be Taught Explicitly .........................340
Handout 8.7: Mathematics Literacy Strategies ......................................................342
Handout 8.8: Mathematics Writing Prompts .........................................................348
Handout 8.9: Communicating About Measurement .............................................350
Handout 8.10: Number Sense and Communication ..............................................354
Handout 8.11: Communicating About Algebra ...................................................... 355
Handout 8.12: Mathematical Literacy and Communication
Strategies Overview ........................................................................358
Handout 8.13: Strategy Implementation PLANNING Sheet.................................365
Handout 8.14: Strategy Implementation REFLECTION Sheet ............................368
Handout 8.15: Objectives Checklist ........................................................................ 371

Module 3: Connections, Representation, and
Applications in Mathematics

Session 9: Using Data to Drive Instruction II ...................................373
Handout 9.1: Strategy Debrief Group Process ..................................................... 374
Handout 9.2: Analyzing Your Class-Wide Patterns .............................................. 375
Handout 9.3: Lowest Scoring Item Analysis ......................................................... 376
Handout 9.4: Sample Data Set: First Administration .......................................... 377
Handout 9.5: Sample Data Set: Second Administration ..................................... 378
Handout 9.6: Example Identifying Student Strengths and Gaps ....................... 379
Handout 9.7: Identifying Student Strengths and Gaps ........................................380
Handout 9.8: Findings and Refl ections ................................................................. 381
Handout 9.9: Student Grouping .............................................................................382
Handout 9.10: Setting Goals ...................................................................................383

Table of Contents (continued)

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Handout 9.11: Respond Plan for Mathematics ......................................................385
Handout 9.12: Exit Slip ............................................................................................389

Session 10: Mathematical Connections ..............................................391
Handout 10.1: Revisiting Professional Goals ........................................................392
Handout 10.2: Overview of Mathematical Connections .....................................394
Handout 10.3: Mathematical Connections and the Content Domain ................396
Handout 10.4: Connecting Measurement to the Standards ................................397
Handout 10.5: Connecting Measurement to Other Subjects ..............................399
Handout 10.6: Topics in Discrete Mathematics .................................................... 401
Handout 10.7: Discrete Mathematics .....................................................................403

Handout 10.8: Connecting Discrete Mathematics
to State Standards (Examples) .....................................................405
Handout 10.9: Connecting Discrete Mathematics to My Classroom .................406
Handout 10.10: How Is Discrete Mathematics Addressed in Instruction? ............407
Handout 10.11: Connecting Mathematics to the Real World .............................. 410
Handout 10.12: Mathematical Connections Strategies Overview ...................... 421
Handout 10.13: Strategy Implementation PLANNING Sheet ..............................423
Handout 10.14: Strategy Implementation REFLECTION Sheet..........................426

Session 11: Mathematical Representation .........................................429
Handout 11.1: Strategy Debrief Group Process ....................................................430
Handout 11.2: Dienes on Manipulatives ................................................................ 431
Handout 11.3: Cognitive Scaffold for Dienes on Manipulatives .........................435
Handout 11.4: Representation in Number Sense .................................................437
Handout 11.5: Representations of Integer Mathematics .....................................438
Handout 11.6: Manipulatives and Number Sense ................................................439
Handout 11.7: Manipulatives in Geometry ............................................................443
Handout 11.8: Discovering the Pythagorean Theorem .......................................445

Handout 11.9: Algebra Tiles as Pictorial Representations
of Variable Expressions ................................................................. 447
Handout 11.10: Reproducible Algebra Tiles ..........................................................448
Handout 11.11: Factoring and Foiling with Algebra Tiles ..................................449
Handout 11.12: Online Resources for Mathematical Representations ..............450
Handout 11.13: Written Representations in Mathematics ................................... 451
Handout 11.14: Connecting Verbal and Graphic Representations .....................453
Handout 11.15: Mathematical Representations Strategies Overview ............... 461
Handout 11.16: Strategy Implementation PLANNING Sheet ..............................466
Handout 11.17: Strategy Implementation REFLECTION Sheet ..........................469

Session 12: Technology for Mathematics ...........................................471
Handout 12.1: Strategy Debrief Group Process ................................................... 472
Handout 12.2: General Principles for Evaluating Technology ........................... 473
Handout 12.3: WebQuests: Technology for Students with Disabilities ............. 476

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Handout 12.4: WebQuest: Technology for Exploration and Reinforcement .... 477
Handout 12.5: WebQuest: Technology for Dialogue and Presentation ............. 478
Handout 12.6: Meaningful Incorporation of Calculators .................................... 479
Handout 12.7: Understanding Algebra Through Graphing Calculators ............480
Handout 12.8: Using the Graphing Calculator to Find a Best-Fit Line .............485
Handout 12.9: Using Computer Spreadsheets to Solve Equations ....................489
Handout 12.10: Find the Formula Using a Spreadsheet to Solve a Pattern ......499
Handout 12.11: Refl ecting on Graphing Calculators and Spreadsheets ............505
Handout 12.12: Technology Resources by Content .............................................508
Handout 12.13: How Is Technology Addressed in Instruction? ..........................511
Handout 12.14: Technology Strategies Overview .................................................514
Handout 12.15: Strategy Implementation PLANNING Sheet .............................. 519
Handout 12.16: Strategy Implementation REFLECTION Sheet .........................522

Session 13: Project-Based Learning ...................................................525
Handout 13.1: Strategy Debrief Group Process ...................................................526
Handout 13:2: What Is Project-Based Learning (PBL)?...................................... 527
Handout 13.3a: Overview of Multiple Intelligences in the Classroom ..............529
Handout 13.3b: Multiple Intelligences Self-Assessment......................................533
Handout 13.3c: Understanding Your Intelligences ..............................................537
Handout 13.4: Using Multiple Intelligences for PBL and Mathematics .............538
Handout 13.5: Project Criteria Checklist ..............................................................539
Handout 13.6: Sample Project ................................................................................540
Handout 13.7: Steps for Designing a Project ........................................................ 552
Handout 13.8: Designing a Project ........................................................................555
Handout 13.9: PBL Resources ................................................................................ 557
Handout 13.10: Objectives Checklist .....................................................................559

Module 4: Closing

Session 14: Responding to Student Needs .........................................561
Handout 14.1: Project-Based Learning Debrief ....................................................562
Handout 14.2: Tuning Protocol ..............................................................................563
Handout 14.3: Student Study ..................................................................................565
Handout 14.4: Understanding Mathematics Diffi culties .....................................566
Handout 14.5: Strategies for Addressing Disabilities in Mathematics .............. 570
Handout 14.6: Summary: Diffi culties in Understanding Mathematics.............. 575
Handout 14.7: Case Studies in Mathematics Disabilities .................................... 576
Handout 14.8: Analyzing the Case Studies ........................................................... 577
Handout 14.9: Intervention Plan ............................................................................ 578
Handout 14.10: Exit Slip .......................................................................................... 579

Table of Contents (continued)

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Session 15: Closing Session ................................................................581
Handout 15.1: Sentence and Question Stems for Feedback ...............................582
Handout 15.2: Analyzing Student Progress in the Content Domain .................583

Handout 15.3: Synthesizing Strategy Implementation Planning
and Refl ection Sheets .....................................................................584
Handout 15.4: Managing for Success .....................................................................587
Handout 15.5: Management Strengths and Weaknesses.....................................590
Handout 15.6: Setting Professional Growth Goals ..............................................592

Teaching for Results: Secondary Mathematics

References: Teaching for Results 2010 .......................................593

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Course Overview

Overview

The Secondary Mathematics seminar is designed to help new teachers
increase their effi cacy as content area teachers of secondary mathematics.
This course will systematically guide teachers in gaining new insights on
the way secondary mathematics content is organized to understand better
what students are expected to know and be able to do at each grade level.
A comparison of the organization of the content area—referred to as a
content domain—to the local and state standards and any local curricular
resources such as textbooks or district pacing guides will empower teachers
to examine critically the tools they have been given to ensure student mastery
and to determine if there are particular areas of study that will need to be
supplemented with the teacher’s own repertoire of instructional strategies. To
aid in this process, the course is set up so that participants are exposed to an
extensive bank of content-specifi c instructional strategies. Teachers are also
asked to assess constantly student performance in relation to standards and
expectations of content mastery through a course routine called the inquiry
process. With a clear picture of where students are, and where they need to
be to achieve at high levels, teachers are best equipped to effectively and
effi ciently plan, teach, reteach when warranted, and make changes to their
instructional practice to ensure measurable results in their efforts to close the
achievement gap.

The interaction between the areas of focus mentioned above, which can
be parsed into three broad components of content, assessment, and
instruction, lies at the heart of the course. Each experience in this course
aims to help teachers become more effective at understanding and delivering
the content for which their students must master, administering and utilizing
assessment tools and data to inform instruction to move students toward
standards mastery, and ultimately toward closing the achievement gap. Course
goals are listed below and provide measurable descriptions of the knowledge
and skills participants will be able to demonstrate by the end of the Secondary
Mathematics course.

x
Course Overview

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(continued)

The Secondary Mathematics seminar is organized around the following course
outcomes aligned to the three key course themes of content, assessment,
and instruction. The ultimate goal of this course is to equip teachers with
the skills and habits of mind to help them close the achievement gap. By the
end of the course, participants will

Content

� develop depth of understanding about content and pedagogy to effectively
make mathematics content accessible to students

� critically analyze the unique topics, structures, and vocabulary of
secondary mathematics content to make connections across topics while
planning and delivering instruction

Assessment

� implement various assessments (diagnostic, formative, summative) to
collect data about and monitor progress of student achievement

Instruction

� make instructional choices to modify or supplement curriculum to meet
standards and achieve big goals for a diverse group of learners

� facilitate opportunities for students to develop connections between
mathematics skills, concepts in the domain, and real-world applications

COURSE GOALS:

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Course Overview

We believe that high-quality teachers can and must close the achievement gap. Participants are
responsible for ensuring that all of their students meet or exceed grade-level standards. It is critical that
participants adjust their instruction to increase the academic achievement of students who are performing
below grade level to help them catch up, as well as bolster the academic achievement of students who are
currently meeting minimum grade level expectations.

To help ensure that what is learned in the course is being integrated into the daily work of teaching to
close the achievement gap, participant competencies for each course theme are presented below along
with digestible and accessible reminders of how to achieve the goals on a daily basis. Those reminders
are referred to as key messages throughout the course.

Teaching for Results: Competencies Key Messages

It is your responsibility to understand deeply the content you teach in order to make
learning meaningful to all students.

Each of the core subjects you teach has a unique organizational structure, vocabulary, and requirements.
Understanding these elements will help you make connections for your students between skills and
knowledge, as well as within and between content areas. Most important, you will be able to help your
students generalize and apply knowledge to new and complex academic tasks.

Assessment is a key component to closing the achievement gap. Teachers need to use
ongoing assessments to be highly effective in the classroom.

You will need to monitor student progress carefully, critically analyze results, and make immediate and
careful changes to your own teaching practices to achieve measurable progress in student performance.
You need to understand how to select effective assessments so that you can continuously document student
progress toward meeting or exceeding grade-level standards by the end of the school year.

You will invest students in their academic development by being transparent about their performance, sharing
results, and providing them with the ongoing assessment tools they need to understand and drive their own
academic success.

Teachers must choose instructional strategies carefully to maximize their impact on
student achievement.

Your instructional choices should support student mastery of standards, be informed by student
achievement data, and be differentiated to meet student needs. This will help ensure that your efforts in
the classroom will lead toward academic achievement for every student. Your curriculum is a tool, but
whether your students make gains is up to you. No curriculum on its own will be effective for all students.

CONTENT

ASSESSMENT

INSTRUCTION

Course Overview (continued)

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Course Assessments

For each competency there is a related assignment or series of assignments
that enables participants to demonstrate mastery of the competencies upon
which this course is grounded. These assignments comprise the course
assessments. Some assignments are completed during the sessions, while
others are given as homework. Note that there are multiple assignments
required as part of this course, not all of which are part of the course
assessments. Some of the assignments are formative in nature, while the
assignments that comprise the course assessments are meant to be capstone
assignments related to each competency. A full list of assignments can be
found in the syllabus.

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