Teaching Mathematics in Secondary and Middle School An Interactive Approach

by Cangelosi, James S.

• ISBN13: 9780130950185
• ISBN10: 0130950181
• Edition: 3rd
• Format: Paperback
• Copyright: 2002-05-15
• Publisher: Pearson

Summary

Interactive in its approach, this book focuses on all the complex aspects of teaching mathematics in today's classroom and the most current NCTM standards. It illustrates how to creatively incorporate the standards into teaching along with inquiry-based instructional strategies. The book illustrates how to lead pupils toward meaningful mathematics and strategies for developing mathematics skills.Includes an abundance of illustrative examples, mini case studies, one expansive case study that follows a mathematics teacher through his first year in the profession, cooperative learning activities, field-based activities, and transitional activities. Reviews applying for faculty positions as a mathematics teacher, teaching math from a historical perspective, communication with math, working with students as individuals, working with ESL/EFL and integrating math with other content areas. Includes updated information with respect to the research literature, the publication of PSSM, and advances in technology.For educators teaching mathematics in secondary and middle school.

Table of Contents

Beginning a Career as a Professional Mathematics Teacher	1	(49)
Goal and Objectives for Chapter 1	1	(1)
The Goal	1	(1)
A Note	1	(1)
The Objectives	1	(1)
Preservice Preparation and Professional Portfolios	2	(1)
Selecting a Teaching Position	3	(11)
Applications for Positions	3	(5)
Interviews and Decisions	8	(6)
Teaching Loads and Other Responsibilities	14	(4)
Organizing for the Year	18	(12)
Weeks Before the Opening of the School Year	18	(1)
Help From Colleagues	18	(2)
Planning and Organizing Courses by Writing Syllabi	20	(1)
Arranging and Organizing the Classroom	21	(5)
Organizing a Computerized Management System	26	(3)
Arrangements and Acquisitions	29	(1)
The Beginning of an Eventful School Year	30	(11)
Learning From Experiences	41	(6)
Learning How to Apply a Complex Art	41	(1)
Becoming More Assertive	42	(4)
Benefitting From Instructional Supervision	46	(1)
Preparing for Administrative Supervision	46	(1)
Synthesis Activities for Chapter 1	47	(2)
Transitional Activity From Chapter 1 to Chapter 2	49	(1)
Gaining Students' Cooperation in an Environment Conducive to Doing Mathematics	50	(38)
Goal and Objectives for Chapter 2	50	(1)
The Goal	50	(1)
The Objectives	50	(1)
Well-Planned Lessons Gone Awry	50	(3)
Allocated and Transition Times	53	(3)
Student Behaviors	56	(6)
On-Task Behavior	56	(1)
Teaching Students to Be On-Task	56	(3)
Punishment	59	(1)
Engaged Behavior	60	(1)
Off-Task Behavior	60	(2)
Establishing a Favorable Climate for Learning Mathematics	62	(1)
Communicating and Interacting With Students	63	(7)
Descriptive Instead of Judgmental Language	63	(1)
True Dialogues	64	(3)
Assertiveness, Body Language, and Voice Control	67	(1)
Active Listening and Supportive Replies	68	(1)
Responsibility for One's Own Conduct	69	(1)
Communicating With Students' Parents	70	(2)
A Cooperative Partnership	70	(1)
Teacher-Parent-Student Conferences	70	(1)
Written Communiques	71	(1)
Establishing Standards for Conduct and Routine Classroom Procedures	72	(3)
Necessary Standards for Conduct	72	(2)
Procedures for Smoothly Operating Classrooms	74	(1)
Teaching Standards for Conduct and Procedures to Students	74	(1)
Working With Students as Individuals	75	(9)
The Key: Relating to Students as Individuals	75	(3)
PSSM and Equity	78	(1)
Accommodating the Special-Education Needs of Students	78	(1)
Including Students for Whom English Is Not a First Language	79	(3)
Benefitting From Cultural Diversity	82	(2)
Systematically Dealing With Off-Task Behaviors	84	(1)
Synthesis Activities for Chapter 2	85	(2)
Transitional Activity From Chapter 2 to Chapter 3	87	(1)
Motivating Students to Engage in Mathematical Learning Activities	88	(42)
Goal and Objectives for Chapter 3	88	(1)
The Goal	88	(1)
The Objectives	88	(1)
Motivating Students to Do Mathematics	88	(2)
Intrinsic and Extrinsic Motivation	88	(1)
Connecting Mathematics to Students' Interests: The Problem-Based Approach	89	(1)
Directing Students Into Learning Activities	90	(7)
Directness, Explicitness, and Specificity	90	(4)
Eight Points About Directions	94	(3)
Variety of Learning Activities	97	(1)
Engaging Students in Large-Group Presentations	97	(11)
Student Engagement During Large-Group Presentations	97	(4)
Ten Points About Large-Group Presentations	101	(7)
Responding to Students' Questions	108	(5)
Memory-Level and Reasoning-Level Questions	108	(3)
Student-Initiated Reasoning-Level Questions	111	(1)
Student-Initiated Memory-Level Questions	112	(1)
Asking Students Questions	113	(5)
Student Engagement During Questioning Sessions	113	(1)
Recitation Sessions for Memory-Level Questions	113	(1)
Question-Discussion Sessions for Reasoning-Level Questions	114	(3)
Six Points About Question-Discussion Sessions	117	(1)
Engaging Students in Cooperative-Learning Sessions	118	(3)
Students Learning From One Another	118	(1)
Peer Instruction Groups	118	(1)
Practice Groups	119	(1)
Interest or Achievement Groups	119	(1)
Guidance and Structure for Maintaining Engagement	119	(1)
Nine Points About Cooperative-Learning Sessions	120	(1)
Engaging Students in Independent-Work Sessions	121	(2)
Student Engagement During Independent-Work Sessions	121	(1)
Six Points About Independent-Work Sessions	122	(1)
Motivating Students to Complete Homework Assignments	123	(5)
Appropriate Use of Homework	123	(1)
Use of Homework as Preparation for In-Class Activities	124	(1)
Use of Homework as Extensions of In-Class Activities	125	(1)
Use of Homework as a Follow-Up to In-Class Activities	125	(1)
Four Suggestions About Homework Assignments	126	(2)
Synthesis Activities for Chapter 3	128	(1)
Transitional Activity From Chapter 3 to Chapter 4	129	(1)
Developing Mathematics Curricula	130	(42)
Goal and Objectives for Chapter 4	130	(1)
The Goal	130	(1)
The Objectives	130	(1)
A Curriculum	130	(3)
The Gap Between Research-Based Mathematics Curricula and Typical Practice	133	(10)
The Mystification of Mathematics	133	(5)
Demystifying Mathematics	138	(5)
PSSM's Equity Principle	143	(1)
PSSM's Curriculum Principle	144	(1)
PSSM's Teaching Principle	145	(1)
PSSM's Learning Principle	146	(1)
PSSM's Assessment Principle	147	(1)
PSSM's Technology Principle	147	(1)
PSSM's Content Standards and Expectations	148	(1)
PSSM's Process Standards	149	(15)
Cognitive Processes	149	(7)
Problem Solving	156	(5)
Reasoning and Proof	161	(1)
Communication	161	(1)
Connections	162	(1)
Representation	163	(1)
Designing Mathematics Courses	164	(1)
Teaching Units	164	(5)
Components of a Teaching Unit	164	(1)
The Learning Goal	164	(1)
The Objectives	164	(2)
The Planned Sequence of Lessons	166	(3)
A Summative Evaluation of Student Achievement of the Learning Goal	169	(1)
Daily Planning	169	(1)
Synthesis Activities for Chapter 4	169	(2)
Transitional Activity From Chapter 4 to Chapter 5	171	(1)
Leading Students to Construct Concepts and Discover Relationships	172	(35)
Goal and Objectives for Chapter 5	172	(1)
The Goal	172	(1)
The Objectives	172	(1)
Mathematical Concepts	173	(4)
Conceptualizing	173	(3)
Concept Attributes	176	(1)
Example Noise	176	(1)
Concepts in Mathematics Curriculum	177	(1)
Inductive Reasoning	177	(1)
Construct-a-Concept Objectives	178	(1)
The Construct-a-Concept Learning Level	178	(1)
Stating Construct-a-Concept Objectives	178	(1)
Lessons for Construct-a-Concept Objectives	178	(11)
Challenging but Critical to Teach	178	(1)
Designing Lessons for Construct-a-Concept Objectives	179	(1)
Selecting Examples and Nonexamples	179	(1)
Incorporating the Four Stages Into Construct-a-Concept Lessons	180	(9)
Miniexperiments Relevant to Students' Achievement of Construct-a-Concept Objectives	189	(3)
Discoverable Mathematical Relationships	192	(2)
Discoverable Relationships in Mathematics Curricula	192	(1)
Discovering Relationships for Oneself	192	(2)
Discover-a-Relationship Objectives	194	(1)
The Discover-a-Relationship Learning Level	194	(1)
The Importance of Discover-a-Relationship Learning Level	194	(1)
Lessons for Discover-a-Relationship Objectives	194	(8)
Designing Discover-a-Relationship Lessons	194	(1)
Incorporating the Four Stages Into Discover-a-Relationship Lessons	195	(7)
Miniexperiments Relevant to Students' Achievement of Discover-a-Relationship Objectives	202	(2)
Synthesis Activities for Chapter 5	204	(2)
Transitional Activity From Chapter 5 to Chapter 6	206	(1)
Leading Students to Develop Knowledge and Algorithmic Skills	207	(22)
Goal and Objectives for Chapter 6	207	(1)
The Goal	207	(1)
The Objectives	207	(1)
Mathematical Information to Be Remembered	207	(2)
The Acquisition and Retention of Information	209	(2)
Simple-Knowledge Objectives	211	(1)
Lessons for Simple-Knowledge Objectives	211	(3)
Facilitating Reception and Retention Through Direct Instruction	211	(2)
A Five-Stage Lesson for a Simple-Knowledge Objective	213	(1)
Miniexperiments Relevant to Students' Achievement of Simple-Knowledge Objectives	214	(2)
Stimulus Response	214	(2)
Avoiding Responses Beyond Simple Knowledge	216	(1)
Algorithms	216	(1)
Algorithmic-Skill Objectives	217	(1)
Lessons for Algorithmic-Skill Objectives	217	(6)
Facilitating Algorithmic Skills Through Direct Instruction	217	(1)
Analyzing the Algorithm	217	(1)
Designing Algorithmic-Skill Lessons	218	(3)
Seven-Stage Lesson for an Algorithmic-Skill Objective	221	(2)
Miniexperiments Relevant to Students' Achievement of Algorithmic-Skill Objectives	223	(4)
Emphasis on Process, Not Outcome	223	(1)
Error-Pattern Analysis	224	(3)
Synthesis Activities for Chapter 6	227	(1)
Transitional Activity From Chapter 6 to Chapter 7	228	(1)
Leading Students to Communicate With Mathematics	229	(25)
Goal and Objectives for Chapter 7	229	(1)
The Goal	229	(1)
The Objectives	229	(1)
The Power of Mathematical Language	229	(4)
What Is the Question?	233	(2)
Negotiating the Language of Mathematics	235	(4)
Mathematical Messages	235	(1)
Technical Expressions	236	(3)
Comprehension-and-Communication Objectives	239	(1)
Lessons for Comprehension-and-Communication Objectives	239	(6)
Language Arts Lessons	239	(1)
Comprehension and Communication of Mathematical Messages	239	(4)
Comprehension of and Communicating With Technical Expressions	243	(2)
Speaking, Listening to, and Writing Mathematics	245	(5)
Miniexperiments Relevant to Students' Achievement of Comprehension-and-Communication Objectives	250	(2)
Miniexperiments for Comprehension and Communication of a Message	250	(1)
Miniexperiments for Comprehension and Communication of a Technical Expression	251	(1)
Novelty	251	(1)
Synthesis Activities for Chapter 7	252	(1)
Transitional Activity From Chapter 7 to Chapter 8	253	(1)
Leading Students to Creatively Use Mathematics	254	(23)
Goal and Objectives for Chapter 8	254	(1)
The Goal	254	(1)
The Objectives	254	(1)
Deductive Reasoning for Problem Solving	255	(1)
Application Objectives	256	(1)
Lessons for Application Objectives	256	(2)
Deductive-Learning Activities	256	(1)
Four Stages	256	(2)
A Four-Stage Lesson for an Application Objective	258	(2)
Miniexperiments Relevant to Students' Achievement of Application Objectives	260	(3)
Deciding Flow to Solve Problems	260	(1)
Avoiding ``Giveaway Words''	260	(2)
Extraneous Data	262	(1)
Missing Data	262	(1)
Mixing Example and Nonexample Problems	262	(1)
Non-mathematical Topics for Problems	262	(1)
Doing Mathematics Creatively	263	(1)
Some Thoughts on Creativity	263	(1)
Preserving Creativity	264	(1)
Fostering Creativity	264	(1)
Creative-Thinking Objectives	264	(1)
Lessons for Creative-Thinking Objectives	265	(1)
Synectics	265	(1)
Incorporating Synectics Into Creative-Thinking Lessons	265	(1)
Miniexperiments Relevant to Students' Achievement of Creative-Thinking Objectives	266	(3)
Influencing Students' Attitudes About Mathematics	269	(4)
Affective Objectives	269	(1)
Lessons for Appreciation Objectives	269	(4)
Lessons for Willingness-to-Try Objectives	273	(1)
Miniexperiments Relevant to Students' Achievement of Affective Objectives	273	(2)
Choice, Not Ability or Skill	273	(1)
The Self-Report Approach	274	(1)
The Observational Approach	274	(1)
Synthesis Activities for Chapter 8	275	(1)
Transitional Activity From Chapter 8 to Chapter 9	276	(1)
Assessing and Reporting Students' Progress With Mathematics	277	(55)
Goal and Objectives for Chapter 9	277	(1)
The Goal	277	(1)
The Objectives	277	(1)
Complex Decisions	278	(23)
A Student's Mind: A Complex of Continua	278	(1)
Measurements for Formative Judgments	279	(3)
Measurements for Summative Evaluations	282	(19)
Measurement Usefulness	301	(7)
Both Valid and Usable	301	(1)
Measurement Validity	302	(1)
Measurement Relevance	302	(2)
Measurement Reliability	304	(4)
Common Malpractice	308	(1)
Designing Measurements of Students' Achievement of Unit Goals	309	(6)
A Haphazard Approach	309	(1)
A Systematic Approach	309	(1)
A Test Blueprint	310	(4)
Synthesizing the Measurement	314	(1)
Using Test Results for Formative Feedback	315	(1)
Assigning Grades to Measurement Scores	315	(3)
Grades	315	(1)
Traditional Percentage Grading	315	(1)
Visual-Inspection Grading	316	(1)
Compromise Grading	317	(1)
Portfolios for Communicating Summative Evaluations of Student Achievement	318	(2)
The Complexity of Students' Mathematical Achievements	318	(1)
Individualized Student Portfolios	319	(1)
Tests You Did Not Design	320	(5)
High-Stakes Tests	325	(4)
Government Mandates	325	(1)
Core-Curriculum Tests	325	(1)
Standardized Tests	326	(3)
AP Calculus and Statistics Tests	329	(1)
PSSM-Based Practice and High-Stakes Testing	329	(1)
Synthesis Activities for Chapter 9	329	(2)
Transitional Activity From Chapter 9 to Chapter 10	331	(1)
Technology and Resources for Teaching and Learning Mathematics	332	(34)
Goal and Objectives for Chapter 10	332	(1)
The Goal	332	(1)
The Objectives	332	(1)
Selection and Use of Textbooks	332	(2)
Sources of Ideas for Teaching Mathematics	334	(2)
Colleagues	334	(1)
NCTM and Other Professional Societies	334	(1)
Resource Centers and Inservice Providers	335	(1)
Electronic and Literary Sources on Mathematics Teaching	336	(1)
Hands-On Manipulatives and Concrete Models	336	(10)
Computer-Assisted Mathematics Teaching	346	(11)
Electronic Manipulatives and Computer Simulations	346	(1)
Using Computers to Mathematics	347	(9)
Computer-Based Learning Activities	356	(1)
Calculator-Assisted Mathematics Teaching	357	(4)
Technology for Presentations	361	(3)
Overhead Projector	361	(2)
Visual Presenter	363	(1)
Computer-Based Multimedia Presentations	364	(1)
Extending Your Mathematics Classroom Beyond the Walls of the School	364	(1)
Synthesis Activities for Chapter 10	365	(1)
Transitional Activity From Chapter 10 to Chapter 11	365	(1)
Analyzing Examples of Mathematics Curricula and Instructional Practice	366	(51)
Goal and Objectives for Chapter 11	366	(1)
The Goal	366	(1)
The Objectives	366	(1)
Building From Experiences	366	(1)
A Sixth-Grade Mathematics Course	367	(5)
A Prealgebra Course	372	(3)
An Algebra I Course	375	(26)
A Geometry Course	401	(5)
Algebra II, Trigonometry, and Precalculus	406	(6)
A Life-Skills Mathematics Course	412	(4)
AP and Concurrent-Enrollment Mathematics Courses	416	(1)
Synthesis Activities for Chapter 11	416	(1)
Glossary	417	(6)
References	423	(8)
Index	431	(6)
About the Author	437

Excerpts

Contrary to popular belief, mathematics is an everyday human endeavor by which ordinary people construct concepts, discover relationships, invent algorithms and models, organize and communicate their thoughts in the language of mathematics, execute algorithms, and address their real-world problems. Likewise, by capitalizing on the use of common and personally relevant mathematics problems, mathematics teachers can help students learn and creatively apply mathematics to their everyday lives. Using research-based strategies, effective mathematics teachers guide students to invent and discover new mathematics perspectives and in so doing help students acquire confident attitudes and abilities in mathematics. Case studies in this text, the third edition of Teaching Mathematics an Secondary and Muddle School,demonstrate how. According to consistent findings in research studies cited throughout this book, students develop confident attitudes and abilities in mathematics by engaging in: inquiry lessons that lead them to reason inductively to construct mathematical concepts and discover mathematical relationships direct-instructional lessons that lead them to gain knowledge of conventions and develop algorithmic skills comprehension-and-communication lessons that lead them to take advantage of the special features of mathematical language inquiry lessons that lead them to reason deductively to devise solutions to real-life problems However, most students do not have these experiences. Rather, they acquire a considerably different view of mathematics, perceiving it as a boring string of terms, symbols, facts, and algorithms--truly understood only by rare geniuses. Too many students are asked only to memorize mathematical content without ever discovering, inventing, or creatively applying it. The unhealthy attitudes and inabilities to extend mathematics beyond what is memorized are perpetuated by the most dominant method of teaching mathematics in our schools. Many teachers'' lessons follow a tiresome sequence--lessons void of experiences whereby meaningful mathematics is discovered, invented, or applied: Students are told about a fact or the steps in an algorithm, walked through textbook examples, assigned exercises from the textbook, and given feedback on their work with the exercises. For over a century, mathematics education specialists have encouraged teachers to deviate from common practice by applying research-based strategies. Promise for bringing typically practiced mathematics teaching in line with research-based strategies springs from the widespread dissemination and support for the National Council of Teachers of Mathematics'' (NCTM) plan for school mathematics curriculum reform as articulated in Principles and Standards for School Mathematics(PSSM) (NCTM, 2000b). Teaching Mathematics in Secondary and Middle School: An Interactive Approachis designed to lead you to develop your talent for teaching in accordance with PSSMso that your students eagerly construct mathematical concepts for themselves, discover mathematical relationships, develop and maintain algorithmic skills, communicate in the language of mathematics, and use mathematics to devise solutions to real-life problems. WHERE ARE RESEARCH-BASED STRATEGIES INTEGRATED INTO THIS TEXT? Introducing topics in a spiral fashion, this book actively involves you in researched-based learning activities throughout its 11 chapters. Chapter 1,"Beginning a Career as a Professional Mathematics Teacher," will lead you to vicariously experience the professional activities of Casey Rudd, a beginning mathematics teacher, and expose you to some preliminary ideas for developing mathematics curricula and engaging students in meaningful lessons--ideas that you will further develop in depth as you work with chapters 2-11. Chapter 2,"Gaining Students'' Cooperation in an Environment Conducive to Doing Mathematics," will help you develop strategies that lead students to cooperate in the business of learning mathematics. From your work with other chapters you will learn how to design courses, plan lessons, conduct learning activities, and monitor students'' progress toward meaningful mathematical goals. However, the success of even the best designed curricula depends on how well you establish a classroom environment in which students willingly work on-task and engage in the business of learning--an environment in which students feel free to experiment, make mistakes, raise questions, interact with you and one another, contribute ideas, and expose their thought processes without fear that they are risking embarrassment, harassment, or judgment of their self-worth. Chapter 3,"Motivating Students to Engage in Mathematical Learning Activities," is an extension of chapter 2 that focuses on using problem-based lessons to motivate students to do mathematics, strategies for responding to students'' questions, and strategies for keeping students engaged in the following types of learning activities: large-group presentations, question-discussion sessions, cooperative-learning, independent-work sessions, and homework. Chapter 4,"Developing Mathematics Curricula," will familiarize you with PSSM. You will understand why PSSM-based curricula lead students to do meaningful mathematics whereas typical textbook-driven curricula lead students to perceive mathematics as a linear sequence of meaningless definitions, symbols, rules, and algorithms. From your work with chapter 4, you will develop an advanced organizer for designing courses that are consistent with PSSM-based curricula-an advanced organizer that interrelates the work you will be doing with chapters 5-11. Chapter 5,"Leading Students to Construct Concepts and Discover Relationships," explains and illustrates how to design and conduct inquiry lessons that lead students to construct mathematical concepts and discover mathematical relationships. You will also develop strategies for conducting miniexperiments to monitor students'' progress during these types of lessons and assess how well the objectives were achieved. Chapter 6,"Leading Students to Develop Knowledge and Algorithmic Skills," is organized similarly to chapter 5, but the focus is on using direct-instructional strategies for lessons designed to lead students to acquire and remember mathematical information and develop algorithmic skills. The mini-experiments you will design for this learning level will emphasize the identification and correction of error patterns in students'' execution of algorithms. Chapter 7,"Leading Students to Communicate With Mathematics," is organized similarly to chapters 5 and 6, but the focus is on using a combination of inquiry and direct-instructional strategies for lessons designed to lead students to use mathematics for organizing and communicating ideas and to comprehend the language of mathematics. You will learn how to incorporate conversation, speaking, listening, writing, and reading in your lessons so that students do meaningful mathematics. Chapter 8,"Leading Students to Creatively Use Mathematics," is organized similarly to chapters 5-7, but the focus is on inquiry lessons that lead students to apply mathematics to real-life situations, foster their creativity with mathematics, and develop an appreciation for and willingness to do mathematics. As with chapters 5-7, you will engage in activities that prompt you to design and field-test lessons and miniexperiments. Such activities provide critical experiences upon which you build your teaching talents. You will also collect artifacts (e.g., lesson plans) for the professional portfolio you will have organized from your work with chapter 1. Chapter 9,"Assessing and Reportin