What Successful Math Teachers Do, Grades 6–12: 80 Research-Based Strategies for the Common Core–Aligned Classroom
Publication Year: 2013
The math teacher's go-to resource—now updated for the Common Core!
What works in math and why has never been the issue; the research is all out there. Where teachers struggle is the “how“—something the research rarely manages to tackle. That's the big service What Successful Math Teachers Do provides. It's a powerful portal to what the best research looks like in practice, strategy by strategy—aligned in this new edition to both the Common Core and the NCTM Standards.
How exactly does What Successful Math Teachers Do work? It couldn't be easier to navigate. The book's eleven chapters organize clusters of strategies around a single aspect of a typical instructional program. For each of the 80 strategies, the authors present: A brief description of that strategy; A summary ...
- Front Matter
- Back Matter
- Subject Index
- Chapter 1: Make Sense of Problems and Persevere in Solving Them
- Aligning Chapter 1 to the Common Core State Standards
- Strategy 1: Help Students Develop Self-Control to Enhance Their Thinking and Independence as Well as to Ease Your Own Workload.
- Strategy 2: Encourage Students to Be Mentally Active While Reading Their Textbooks.
- Strategy 3: Praise Mistakes!
- Strategy 4: Make a Lesson More Stimulating and Interesting by Varying the Types of Questions You Ask Students.
- Strategy 5: Use a Variety of Strategies to Encourage Students to Ask Questions About Difficult Assignments.
- Strategy 6: Use a Question-Asking Checklist and an Evaluation Notebook to Help Students Become Better Learners.
- Strategy 7: Find Out Why Students Rate a Mathematical Task as Difficult So You Can Increase the Difficulty of Exercises and Tests More Effectively.
- Strategy 8: Teach Students to Ask Themselves Questions About What They Already Know About a Problem or Task They Are Working on.
- Strategy 9: Structure Teaching of Mathematical Concepts and Skills Around Problems to Be Solved, Using a Problem-Centered or Problem-Based Approach to Learning.
- Chapter 2: Reason Abstractly and Quantitatively
- Aligning Chapter 2 to the Common Core State Standards
- Strategy 10: Teach Students to Ask Themselves Questions About the Problems/Tasks They Are Working on.
- Strategy 11: Help Students Understand Their Own Thought Processes, and Guide Them in Learning to Think Like Mathematicians.
- Strategy 12: Select and Carefully Structure Homework Assignments So That They Require the Development of Mathematical Thinking and Reasoning. Anticipate Changes That Might Occur While Students Are Working at Home.
- Strategy 13: Emphasize Higher-Level Thinking Objectives in Regular Mathematics Classes So That All Students Incorporate the Features of Enriched Academic and Honors Classes.
- Chapter 3: Construct Viable Arguments and Critique the Reasoning of Others
- Aligning Chapter 3 to the Common Core State Standards
- Strategy 14: Use Classwide Peer Tutoring to Help Your Students Learn Whether or Not They Have Learning Disabilities.
- Strategy 15: Carefully Select Problems for Use in Cooperative Learning Groups.
- Strategy 16: Encourage Students to Work Cooperatively with Other Students.
- Strategy 17: Use Group Problem Solving to Stimulate Students to Apply Mathematical Thinking Skills.
- Strategy 18: Don't Give Students Feedback on Their Performance Too Early.
- Strategy 19: Promptly Give Students Information or Feedback About Their Performance.
- Strategy 20: Increase Your Understanding of Factors That Affect Students' Attitudes Before and After Testing. You May Be Surprised!
- Strategy 21: Get Students to “Think Out Loud” When Solving Problems.
- Chapter 4: Model with Mathematics
- Aligning Chapter 4 to the Common Core State Standards
- Strategy 22: Find Out About Your Students' Motivation Regarding Mathematics, and Use That Knowledge to Refine Your Instruction.
- Strategy 23: When Trying to Determine How to Motivate Students' Interest in Mathematics, Teachers Should Differentiate Between Personal and Situational Interest and Use Both Forms to Increase Students' Motivation to Learn Mathematics. Teachers Also Need to Both Stimulate and Maintain Their Students' Interest.
- Strategy 24: Use School Fund-Raising Projects, Such as Students' Selling Candy or Organizing a Walkathon, as the Basis of Mathematics Lessons.
- Strategy 25: When Doing Inquiry Lessons, Give Students Clearly Written Materials to Guide the Inquiry Process.
- Strategy 26: Use Graphic Representations or Illustrations to Enhance Students' Memory While They Are Listening to You. Abstract Representations Such as Flowcharts Are More Effective Than Colorful Pictures.
- Strategy 27: Playing Makes Understanding Mathematics Easier and More Fun.
- Strategy 28: Assign Homework and Other Projects Requiring Students to Write About Connections Between Mathematics and Other Subjects.
- Strategy 29: Encourage Students to Make Mental Pictures While Applying Rules to Solve Problems.
- Chapter 5: Use Appropriate Tools Strategically
- Aligning Chapter 5 to the Common Core State Standards
- Strategy 30: Use the Jigsaw Technique of Cooperative Learning as an Interesting and Effective Way for Students to Learn.
- Strategy 31: Use Homework as a Way of Delving More Deeply into Important Mathematical Concepts and Skills.
- Strategy 32: Help Students Learn without Relying on Teacher-Centered Approaches. Give Them Carefully Chosen Sequences of Worked-Out Examples and Problems to Solve.
- Chapter 6: Attend to Precision
- Aligning Chapter 6 to the Common Core State Standards
- Strategy 33: Treat Students in Ways That Reflect the Belief That You Have High Expectations for Their Performance.
- Strategy 34: Call on Students More Frequently to Promote Their Achievement.
- Strategy 35: Make Sure to Pause for at Least Four Seconds After Listening to a Student's Communication Before Responding.
- Strategy 36: Emphasize the General Principles That Underlie Solving Specific Types of Problems.
- Chapter 7: Look for and Make Use of Structure
- Aligning Chapter 7 to the Common Core State Standards
- Strategy 37: Teachers Should Be Tactical in Their Use of Questions.
- Strategy 38: Use a Variety of Sequences to Ask Questions.
- Strategy 39: Adolescents Need Extended Support to Acquire the Ability to Visualize.
- Strategy 40: Give Students the Kind of Feedback That Will Most Help Them Improve Their Future Performance.
- Strategy 41: Complex Exercises That Give Students Freedom Tend to Fit the Way Older Students Learn.
- Strategy 42: Provide Hints or Clues or Ask Leading Questions When Students Need Help Solving Problems Instead of Giving Them the Answers. Gradually Phase Out This Support So as to Foster Independent Problem Solving.
- Strategy 43: Examine Your Students' Knowledge of Mathematics, and Use This Information to Write Challenging Word Problems That They Will Enjoy Solving.
- Strategy 44: Students Need Time to Practice Planning Their Solutions to Problems.
- Chapter 8: Look for and Express Regularity in Repeated Reasoning
- Aligning Chapter 8 to the Common Core State Standards
- Strategy 45: Use Questions for Different and Versatile Functions in the Classroom.
- Strategy 46: Use Inquiry-Based Learning in Addition to Problem-Based Learning.
- Strategy 47: Teachers Can Help Students Learn to Ask Better Questions.
- Strategy 48: Use Homework Assignments as Opportunities for Students to Get Practice and Feedback on Applying Their Mathematical Knowledge and Skills.
- Strategy 49: Use Analogies to Help Students Develop More Valid Conceptions.
- Strategy 50: Have Students Study Written Model Solutions to Problems While Learning and Practicing Problem Solving.
- Chapter 9: Manage Your Classroom
- Aligning Chapter 9 to the Common Core State Standards
- Strategy 51: Create Your Own Support Network as Soon as You Begin Your First Teaching Job.
- Strategy 52: Before Beginning a Lesson, Put an Outline of What You Are Going to Cover on the Blackboard.
- Strategy 53: Make Realistic Time Estimates When Planning Your Lessons.
- Strategy 54: Make Classroom Activities Flow Smoothly.
- Strategy 55: Have “Eyes in the Back of Your Head” So You Notice Misbehavior at an Early Stage.
- Strategy 56: Do More Than One Thing at a Time.
- Strategy 57: Work Directly with Individual Students as Often as Possible.
- Strategy 58: Avoid Reacting Emotionally When Evaluating Problematic Situations in the Classroom.
- Strategy 59: To Reduce Math Anxiety, Focus on Both the Thoughts and the Emotions of the Students.
- Strategy 60: Consider Whether a Student's Learning Weakness Might Involve a Deficiency in Auditory Perception.
- Chapter 10: Assess Student Progress
- Aligning Chapter 10 to the Common Core State Standards
- Strategy 61: Feedback on Practice Is Essential for Improving Student Performance.
- Strategy 62: Make Sure Students Pay Attention to the Feedback You Give Them.
- Strategy 63: Systematically Incorporate Review into Your Instructional Plans, Especially Before Beginning a New Topic.
- Strategy 64: Provide All Students, Especially Students Lacking Confidence, with “Formative Assessments” to Allow Them Additional Opportunities to Succeed in Mathematics.
- Strategy 65: Be Aware of Students' Different Levels of Text Anxiety as It Relates to Different Subject Areas, and Use a Variety of Techniques to Help Them Overcome Their Test Anxiety.
- Strategy 66: Do Not Assume That Students Accept Responsibility for or Agree with Their Bad Grades on Tests.
- Strategy 67: If Students Do Not Follow Your Instructions and/or If Their Achievements Do Not Fulfill Your Expectations, the Cause May Not Be Students' Incompetence. It Could Be a Result of Your Self-Overestimation.
- Chapter 11: Consider Social Aspects in Teaching Mathematics
- Aligning Chapter 11 to the Common Core State Standards
- Strategy 68: Make Multicultural Connections in Mathematics.
- Strategy 69: Find Out About Your Students' Families and How Their Values and Practices Might Affect Students' Attitudes and Performance in Mathematics.
- Strategy 70: Reach Out to Parents to Form a Partnership for Educating Elementary and High School Students.
- Strategy 71: Inform Parents That They Should Not Let Media Reports About Studies of Other Children Change Their Views of Their Own Children's Abilities to Be Successful in Mathematics.
- Strategy 72: Some Students Do Not Think They Have Control Over Their Academic Successes and Failures. Help These Students Recognize That They Do Have Some Control.
- Strategy 73: Teach Students, Especially Girls, to Believe That Success in Mathematics Results from Their Efforts.
- Strategy 74: Give Girls the Same Quantity and Quality of Teacher Attention as Boys.
- Strategy 75: Make Special Efforts to Encourage Girls to Study Mathematics.
- Strategy 76: Use Different Motivational Strategies for Girls and Boys.
- Strategy 77: Take into Consideration How Students View Successful Teachers and How This Differs for Girls and Boys.
- Strategy 78: Praise, Encourage, and Help Your Older Students.
- Strategy 79: Does Grade Skipping Hurt Mathematically Talented Students Socially and Emotionally? Don't Worry About Accelerating Talented Students!
- Strategy 80: Use Technology Such as Dynamic Geometry Software to Enhance Student Understanding and Analysis.
To Barbara for her support, patience, and inspiration.
To my children and grandchildren, David, Lauren, Lisa, Danny, Max, Sam, and Jack, whose futures are unbounded.
And in memory of my dear parents, Alice and Ernest, who never lost faith in me.—Alfred S. Posamentier
To my Mom and Dad, whose unceasing love and support, and examples of integrity and hard work have converged to allow me a life limited only by my imagination.—Terri L. Germain-Williams
To my wife, Tae Jin, who has made every sacrifice to ensure my happiness and success.
To my children, Jennifer and Rebecca, who are an eternal source of pride and joy, and in memory of my parents, Stanley and Beatrice, who were always there for me.—Daniel Jaye
Copyright © 2013 by Corwin
All rights reserved. When forms and sample documents are included, their use is authorized only by educators, local school sites, and/or noncommercial or nonprofit entities that have purchased the book. Except for that usage, no part of this book may be reproduced or utilized in any form or by any means, electronic or mechanical, including photocopying, recording, or by any information storage and retrieval system, without permission in writing from the publisher.
National Council of Teachers of Mathematics (NCTM) standards throughout the book are reprinted with permission by NCTM. They are quoted from Curriculum and Evaluation Standards for School Mathematics (1989), Professional Standards for Teaching Mathematics (1991), Handbook of Research on Mathematics Teaching and Learning (1992), Assessment Standards for School Mathematics (1995), Principles and Standards for School Mathematics (2002), and Research Companion to Principles and Standards for School Mathematics (2003). All rights reserved. Standards are listed with the permission of NCTM. NCTM does not endorse the content or validity of these alignments.
All trade names and trademarks recited, referenced, or reflected herein are the property of their respective owners who retain all rights thereto.
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Library of Congress Cataloging-in-Publication Data
Posamentier, Alfred S.
What successful math teachers do, grades 6–12 : 80 research-based strategies for the common core–aligned classroom. — Second edition / Alfred S. Posamentier, Terri L. Germain-Williams, Daniel Jaye.
Includes bibliographical references and index.
ISBN 978-1-4522-5913-0 (pbk.)
1. Mathematics—Study and teaching (Secondary) —Standards—United States. I. Germain-Williams, Terri L. II. Jaye, Daniel. III. Title.
This book is printed on acid-free paper.
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Acquisitions Editor: Robin Najar
Associate Editor: Julie Nemer
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Copy Editor: Amy Rosenstein
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As a direct result of federal pressure on the states to continuously improve their instructional program and ensure that all students are being reached in the teaching process, teachers are being called on to meet professional standards and base their work on research-proven methods of teaching. Educational research, often conducted at universities or on educational sites by university researchers, is reported in educational journals and is most often read by other researchers. All too often, the style in which research reports or articles on research findings are reported is not friendly or appealing to the classroom teacher. The very community—classroom teachers—that could benefit enormously from the findings of many of these educational initiatives rarely learns about these endeavors. It is the objective of this book to bring some of the more useful research findings to the classroom teacher. In our quest for the most salient research findings, we were guided by the Common Core State Standards and the National Council of Teachers of Mathematics standards. Rather than merely presenting the research findings that support these standards, we have attempted to convert them into useful classroom strategies, thus capturing the essence of the findings and at the same time putting them into a meaningful context for the practicing mathematics teacher.
This book is to serve as a resource for mathematics teachers. It should provide a portal to access the many worthwhile findings resulting from educational, psychological, and sociological research studies done in Europe and in the United States. Heretofore, teachers have had very few proper vehicles for getting this information, short of combing through the tomes of research reports in the various disciplines. This book is designed to provide an easy way for the classroom teacher to benefit from the many ideas embedded in these academic exercises.
[Page xiii]The book is designed to be an easy and ready reference for the mathematics teacher—both preservice and inservice. It consists of 11 chapters, each with a theme representing one aspect of the typical instructional program and each guided by the Common Core Standards. Each chapter presents a collection of teaching strategies, concisely presented in a friendly format:The Strategy
This is a simple and crisp statement of the teaching strategy we recommend.What the Research Says
This offers a discussion of the research project that led to the strategy. This section should give the teacher some confidence in, and a deeper understanding of, the principle being discussed as a “teaching strategy.”Teaching to the National Council of Teachers of Mathematics Standards
Here we present the salient National Council of Teachers of Mathematics standard that we are supporting with the strategy.Aligning to the Common Core State Standards
This section not only helps explain the standards but also shows how they can be met.Classroom Applications
This section tells the teacher how the teaching strategy can be used in the mathematics instructional program. Where appropriate, some illustrative examples of the teaching strategy in the mathematics classroom are provided.Precautions and Possible Pitfalls
This is the concluding section for each strategy and mentions some of the cautions that should be considered when using this teaching strategy so that the teacher can avoid common difficulties [Page xiv]before they occur, thereby achieving a reasonably flawless implementation of the teaching strategy.Sources
These are provided so that the reader may refer to the complete research study to discover the process and findings in detail.
We see this book as a first step in bringing educational research findings to the practitioners: the classroom teachers. Perhaps teachers will see that there is much to be gained to enhance teaching by reviewing educational research with an eye toward implementing the findings in their instructional program. Furthermore, it would be highly desirable for researchers to make more of an effort to extend their publications/findings to the classroom teacher. To do otherwise would make the entire activity of educational research irrelevant.
As you read the many instructional suggestions offered in this book, we hope you will continuously think of yourself as the teacher who might implement them. Remember, your personality plays a large role in mapping out an overall instructional strategy. Each teacher brings to the classroom various strengths, and therefore, the research we bring to the reader should be viewed in that context. Nevertheless, the specific research-based tips and strategies offered here will help you focus on certain aspects of your teaching. Teachers who continuously self-evaluate their instructional performance will, undoubtedly, become master teachers.
Amy Rosenstein is to be commended for her fine copy editing of the second edition.Publisher's Acknowledgments
Corwin thanks the following individuals for their contributions to this book:
- Kim Cottini, Math Learning Consultant
- Living Sky School Division
- North Battleford, Saskatchewan, Canada
- Diane S. Coupal, Adjunct Lecturer in Mathematics
- State University of New York at Plattsburgh
- Mathematics Department
- Plattsburgh, NY
- Nathan Herzog, Associate Professor
- William Jessup University
- Rocklin, CA
- Jami Stone, Assistant Professor of Education
- Black Hills State University
- College of Education
- Spearfish, SD
- Steven Willott, National Board Certified Mathematics Teacher
- Francis Howell North High School
- St. Charles, MO
About the Authors
Now that you have read through the many research-based suggestions to improve your instruction, we want you to reflect on the larger picture of mathematics education today. Consider the following situation. A recent visit to a picture-framing shop highlighted a mathematical deficiency that seems to be common in our society. An inspection of the bill for framing two pictures, one 4 inches by 20 inches and the other 12 inches by 12 inches, revealed that they cost the same. When questioned, the proprietor responded that the same amount of framing was used for the two pictures and that the glass was figured on the basis of “united inches.”
He was immediately asked what this unit of measure meant. He indicated that it was the sum of the length and the width; in this case each had 24 united inches, and so the cost was the same for the two pieces of glass. The merchant was asked if he believed the two frames required the same amount of glass. He wasn't sure, but assumed they did, since the two had the same number of united inches. A math teacher listening to this discussion chimed in to give him a quick lesson on rectangle area. The proprietor was amazed to discover that he had been charging the same amount for the two pieces of glass, when, in fact one's area (144 sq. in.) was almost twice that of the other (80 sq. in.). This mathematical illiteracy is particularly alarming, especially in the context of our country's poor showing on a recent Organisation for Economic Co-operation and Development Programme for International Student Assessment study, where the United States came in 28th out of 40 countries being compared on the mathematical achievement of 15-year-olds.
We have become complacent about achievement in mathematics. Adults more often than not take pride in their inability to have mastered school mathematics. Furthermore, when they are told that their children will need to master mathematics in school, [Page 240]they begin to question the reasons for such claims, especially when their children come home with math homework that looks unfamiliar to their parents. Over the years, we have tried to convince others that there is power and beauty in mathematics. This is no easy task. We are often confronted with responses like “I don't need to know arithmetic since I use the [ubiquitous] calculator.” Or, “I don't even have to calculate the ‘best buy’ in the supermarket, since every item has its unit price indicated.” Or “Even ‘miles per gallon’ need not be calculated, since my car's odometer does that for me.” Some even ask, “Why teach mathematics at all?” Why don't they ever ask, “Why teach poetry, literature, music, art, or even science if one is not planning to pursue a career in those fields?” When was the last time an adult needed to use any of these subjects in everyday life?
We need to convince the general populace of the importance of mathematics. Simply saying, as many do, that today's students are involved with real-world applications in the classroom just doesn't cut it. Unfortunately, there are at least two problems with the real-world-math claims: First, the real world of students is often not what adults have chosen as the real world, and to be truly of the real world—rather than artificial models—is generally far too complicated for a school audience. There are times when parents do a “project” at home that involves mathematics or reasoning skills. Often these skills were developed as a result of school math instruction. Parents should involve their children in these projects, which might include setting up a birthday party, buying flooring or carpeting, or calculating expenses (i.e., budget). These would be actual real-world activities.
So where does that leave us? Do we merely stop teaching mathematics for the above-mentioned reasons, or do we try to demonstrate its purpose in other ways? Mathematics has manifested itself in the school curriculum in different ways at various times in our history. In the 18th century, students learned to “reckon,” or do arithmetic, so that they would be able to do the necessary calculations required in trade or farming, for which they also needed some geometry. As time went on, the time available for math instruction increased, as did the material taught: algebra and trigonometry. More advanced instruction began to include some probability and statistics, although this was limited until the “number crunching” machines (i.e., calculators and computers) appeared. Today, we find ourselves with the dilemma of deciding exactly what and how mathematics should be taught. The advent of the computer has had [Page 241]a marked effect on the curriculum. Topics that used to be commonly taught are no longer needed, such as extracting the square root of a number or using logarithms to simplify complex calculations. In short, learning mathematics entails much more than obtaining tools to use in other fields. It is the subconscious acquisition of thinking and reasoning skills coupled with the more sophisticated way we view the physical world that leads the list of the many life enhancements that come with learning mathematics.
We, as members of the select New York State Math Standards Committee, were charged with preparing standards that would provide the necessary understandings and applications of arithmetic and geometry so that a proper transition could be made to a solid understanding and use of algebra, trigonometry, probability, and statistics. Today's youth need a different facility with, and understanding of, numerical concepts than previous generations. The calculator and computer have reduced the need for one to be a lightning-fast calculator, yet despite today's technology, the need to understand number concepts and relationships has not diminished. Calculating a tip in a restaurant or balancing a checkbook still ought to be done easily.
For us to compete favorably in mathematics achievement in the world arena, we need to spend more time doing those things that school mathematics has done well for decades: provide appropriate reckoning skills, enable a reasonably sophisticated view of geometry for use in academics and beyond, and fortify students with the necessary tools of mathematics to pursue whatever academic endeavor they choose to study. The by-product of all of this will be a well-reasoning and able problem-solver. Only if we do this in the context of motivating students with the beauty of mathematics (rather than telling them that what they are doing is for their “real-world” experiences, when they have difficulty accepting this) will our efforts bear fruit.
Resource: What the Authors Say: Enriching Instruction[Page 242]Strategy: Enrich Your Instruction Before You Look to Accelerate the CurriculumWhat the Authors Say
Recently, there has been a surge to move students along in the standard high school mathematics curriculum at a faster pace. In the course of this acceleration, a number of topics have been relegated to the back burner. In short, they were cut. Years ago, there were many useful techniques in algebra that were regularly taught but are not even mentioned today. One example is the teaching of factoring. Students were shown how to factor the sum and difference of two cubes, but today the practice is not mentioned. Every teacher owes it to his or her students to enrich their instruction. This can be done by expanding on a topic being taught, by extending the students' knowledge by showing how the topic being taught can relate to another topic out of the curriculum, or by bringing in historical aspects related to the topic.Teaching to the National Council of Teachers of Mathematics Standards
The National Council of Teachers of Mathematics says enrichment is part of any good teaching performance. The Principles and Standards for School Mathematics state that “mathematics teachers generally are responsible for what happens in their own classrooms [Page 243]and can try to ensure that their classrooms support learning by all students … teachers must challenge and hold high expectations for all their students, not just those they believe are gifted.”1 It is a good professional practice to provide opportunities for all students to enrich their learning of mathematics by encouraging them to explore topics beyond the scope of the syllabus. The following classroom application contains some suggestions for student projects in mathematics.Aligning to the Common Core State Standards
The Common Core State Standards have released documents discussing the application of the standards (both literacy and mathematics) for students with disabilities as well as English language learners. However, the website and published materials lack explicit discussion or documentation regarding enrichment. This may be a result of the “higher” expectations of the standards, as claimed in their introduction to the mathematics standards. This may be a result of the novelty of the movement as a whole. In either case, mathematics educators must consider how to meet the needs of their students. Consider the habits of athletes. They push themselves beyond the expectations of a meet or game during practices so that the requirements of the contest seem easier when compared to their preparation. In mathematics, we can challenge students with solving more intricate or difficult problems (including problems with multiple answers and those with non-integer solutions), as well as the use of proof to enhance understanding. The CCSS provide high expectations that are internationally benchmarked, and we can use them as a guide to help our students become even more skillful mathematicians through meeting and exceeding their standards.Classroom Applications
There are a multitude of topics that can be used to enrich instruction. You can assign individual students to do a small report on a topic that relates to the material being taught. Listed next are some possible topics that students might use for such a project. This list is merely intended as a guide for generating additional topics.
- Advanced Euclidean Geometry
- Algebraic Fallacies
- Algebraic Models
- Algebraic Recreations
- Analog Computer
- Ancient Number Systems and Algorithms
- Arithmetic Fallacies
- Arithmetic Recreations [Page 244]
- Bases Other Than Ten
- Binary Computer
- Boolean Algebra
- Brocard Points
- Calculating Shortcuts
- Cavalieri's Theorem
- Checking Arithmetic Operations
- Conic Sections
- Continued Fractions
- Curves of Constant Breadth
- Cylindrical Projections
- Desargues' Theorem
- Diophantine Equations
- Divisibility of Numbers
- Dynamic Symmetry
- Elementary Number Theory Applications
- The Euler Line
- Extension of Euler's Formula to N Dimensions
- Extension of Pappus's Theorem
- Fermat's Last Theorem
- Fibonacci Numbers
- Finite Differences
- Finite Geometry
- The Five Regular Polyhedra
- The Four-Color Problem
- The Fourth Dimension
- Game Theory
- Gaussian Primes
- Geometric Dissections: Tangrams
- Geometric Fallacies
- Geometric Models
- Geometric Stereograms
- Geometric Transformations
- Geometry of Bubbles and Liquid Film
- Geometry of Catenary
- Geometry Constructions (Euclid)
- Gergonne Problem
- The Golden Section
- Graphical Representation of Complex Roots of Quadratic and Cubic
- Higher Algebra
- Higher-Order Curves
- Hyperbolic Functions
- The Hyperbolic Paraboloid
- Hypercomplex Numbers
- Intuitive Geometric
- Recreations [Page 245]
- Investigating the Cycloid
- The Law of Growth
- Linear Programming
- Lissajous Figures
- Lobachevskian Geometry
- Logarithms of Negative and Complex Numbers
- Magic Square Construction
- Map Projections
- Mascheroni Constructions
- Mathematics and Art
- Mathematics and Music
- Mathematics of Life Insurance
- Maximum-Minimum in Geometry
- Methods of Least Squares
- The Metric System
- Minimal Surfaces
- Modulo Arithmetic in Algebra
- Monte Carlo Method of Number Approximation
- Multinomial Theorem
- Napier's Rods
- The Nine-Point Circle
- The Number Pi, Phi, or e
- Number Theory Proofs
- Paper Folding
- Partial Fractions
- Pascal's Theorem
- Perfect Numbers
- Polygonal Numbers
- Prime Numbers
- Problem Solving in Algebra
- Projective Geometry
- Proofs of Algebraic Theorems
- Properties of Pascal's Triangle
- Pythagorean Theorem: Triples
- Regular Polygons
- The Regular Seventeen-Sided Polygon
- Relativity and Mathematics
- Riemannian Geometry
- Solving Cubics and Quartics
- Special Factoring
- Spherical Triangles
- The Spiral
- Steiner Constructions
- Theory of Braids
- Theory of Equations
- Theory of Perspectives
- Three-Dimensional Curves [Page 246]
- The Three Famous
- Problems of Antiquity
- Unsolved Problems
You might also merely digress from a topic being taught. For example, when teaching probability, you might mention the famous “birthday problem.” When discussing concurrency in geometry, you might want to introduce Ceva's theorem, which makes many difficult theorems almost trivial to prove. We suggest you consider any of the following books, each of which will give you lots of ideas for enrichment:2003). Math wonders to inspire students and teachers. Alexandria, VA: ASCD.(2006). 101+great ideas for introducing key concepts in mathematics. Thousand Oaks, CA: Corwin., & (2010). Teaching secondary mathematics: Techniques and enrichment units. Columbus, OH: Allyn & Bacon/Prentice Hall., , & (Precautions and Possible Pitfalls
Whenever you embark on material that is not part of the curriculum and where students will essentially not be held responsible for learning the material, there is a tendency on the part of the students to stretch this activity out as long as possible. By keeping the teacher from moving along, figuring that if the teacher runs out of time and cannot cover a required topic, the students hope to have less material to study for when the next test comes along. So we urge you to do the enrichment with an eye toward a properly balanced time schedule. Don't let the enrichment material dominate the class activities. Remember that you are still responsible for covering the prescribed course work. Experience will help you appropriately estimate the proper time allotment for such enrichment activities.Note
1. The National Council of Teachers of Mathematics. (2000). Principles and standards for school mathematics (p. 375). Reston, VA: Author.
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