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Math Wonders to Inspire Teachers and Students
Association for Supervision and Curriculum Development Alexandria, Virginia USA
_________________________
Contents
Foreword
Preface
Chapter 1 : The Beauty in Numbers
1.1 Surprising Number Patterns I
1.2 Surprising Number Patterns II
1.3 Surprising Number Patterns III
1.4 Surprising Number Patterns IV
1.5 Surprising Number Patterns V
1.6 Surprising Number Patterns VI
1.7 Amazing Power Relationships
1.8 Beautiful Number Relationships
1.9 Unusual Number Relationships
1.10 Strange Equalities
1.11 The Amazing Number 1,089
1.12 The Irrepressible Number 1
1.13 Perfect Numbers
1.14 Friendly Numbers
1.15 Another Friendly Pair of Numbers
1.16 Palindromic Numbers
1.17 Fun with Figurate Numbers
1.18 The Fabulous Fibonacci Numbers
1.19 Getting into an Endless Loop
1.20 A Power Loop
1.21 A Factorial Loop
1.22 The Irrationality of v2
1.23 Sums of Consecutive Integers
Chapter 2 Some Arithmetic Marvels
2.1 Multiplying by 11
2.2 When Is a Number Divisible by 11?
2.3 When Is a Number Divisible by 3 or 9?
2.4 Divisibility by Prime Numbers
2.5 The Russian Peasant’s Method of Multiplication
2.6 Speed Multiplying by 21, 31, and 41
2.7 Clever Addition
2.8 Alphametics
2.9 Howlers
2.10 The Unusual Number 9
2.11 Successive Percentages
2.12 Are Averages Averages?
2.13 The Rule of 72
2.14 Extracting a Square Root
Chapter 3 Problems with Surprising Solutions
3.1 Thoughtful Reasoning
3.2 Surprising Solution
3.3 A Juicy Problem
3.4 Working Backward
3.5 Logical Thinking
3.6 It’s Just How You Organize the Data
3.7 Focusing on the Right Information
3.8 The Pigeonhole Principle
3.9 The Flight of the Bumblebee
3.10 Relating Concentric Circles
3.11 Don’t Overlook the Obvious
3.12 Deceptively Difficult (Easy)
3.13 The Worst Case Scenario
Chapter 4 Algebraic Entertainments
4.1 Using Algebra to Establish Arithmetic Shortcuts
4.2 The Mysterious Number 22
4.3 Justifying an Oddity
4.4 Using Algebra for Number Theory
4.5 Finding Patterns Among Figurate Numbers
4.6 Using a Pattern to Find the Sum of a Series
4.7 Geometric View of Algebra
4.8 Some Algebra of the Golden Section
4.9 When Algebra Is Not Helpful
4.10 Rationalizing a Denominator
4.11 Pythagorean Triples
Chapter 5 Geometric Wonders
5.1 Angle Sum of a Triangle
5.2 Pentagram Angles
5.3 Some Mind-Bogglers on p
5.4 The Ever-Present Parallelogram
5.5 Comparing Areas and Perimeters
5.6 How Eratosthenes Measured the Earth
5.7 Surprising Rope Around the Earth
5.8 Lunes and Triangles
5.9 The Ever-Present Equilateral Triangle
5.10 Napoleon’s Theorem
5.11 The Golden Rectangle
5.12 The Golden Section Constructed by Paper Folding
5.13 The Regular Pentagon That Isn’t
5.14 Pappus’s Invariant
5.15 Pascal’s Invariant
5.16 Brianchon’s Ingenius Extension of Pascal’s Idea
5.17 A Simple Proof of the Pythagorean Theorem
5.18 Folding the Pythagorean Theorem
5.19 President Garfield’s Contribution to Mathematics
5.20 What Is the Area of a Circle?
5.21 A Unique Placement of Two Triangles
5.22 A Point of Invariant Distance in an Equilateral Triangle
5.23 The Nine-Point Circle
5.24 Simson’s Invariant
5.25 Ceva’s Very Helpful Relationship
5.26 An Obvious Concurrency?
5.27 Euler’s Polyhedra
Chapter 6 Mathematical Paradoxes
6.1 Are All Numbers Equal?
6.2 -1 Is Not Equal to +1
6.3 Thou Shalt Not Divide by 0
6.4 All Triangles Are Isosceles
6.5 An Infinite-Series Fallacy
6.6 The Deceptive Border
6.7 Puzzling Paradoxes
6.8 A Trigonometric Fallacy
6.9 Limits with Understanding
Chapter 7 Counting and Probability
7.1 Friday the 13th!
7.2 Think Before Counting
7.3 The Worthless Increase
7.4 Birthday Matches
7.5 Calendar Peculiarities
7.6 The Monty Hall Problem
7.7 Anticipating Heads and Tails
Chapter 8 Mathematical Potpourri
8.1 Perfection in Mathematics
8.2 The Beautiful Magic Square
8.3 Unsolved Problems
8.4 An Unexpected Result
8.5 Mathematics in Nature
8.6 The Hands of a Clock
8.7 Where in the World Are You?
8.8 Crossing the Bridges
8.9 The Most Misunderstood Average
8.10 The Pascal Triangle
8.11 It’s All Relative
8.12 Generalizations Require Proof
8.13 A Beautiful Curve
Epilogue
Acknowledgments
Index
About the Author
__________________________________
This book was inspired by the extraordinary response to an Op-Ed article I wrote for The New York Times.* In that article, I called for the need to convince people of the beauty of mathematics and not necessarily its usefulness, as is most often the case when trying to motivate youngsters to the subject. I used the year number, 2,002,** to motivate the reader by mentioning that it is a palindrome and then proceeded to show some entertaining aspects of a palindromic number. I could have taken it even further by having the reader take products of the number 2,002, for that, too, reveals some beautiful relationships (or quirks) of our number system. For example, look at some selected products of 2,002:
2,002 * 4 = 8,008
2,002 * 37 = 74,074
2,002 * 98 = 196,196
2,002 * 123 = 246,246
2,002 * 444 = 888,888
2,002 * 555 = 1,111,110
Following the publication of the article, I received more than 500 letters and e-mail messages supporting this view and asking for ways and materials to have people see and appreciate the beauty of mathematics. I hope to be able to respond to the vast outcry for ways to demonstrate the beauty of mathematics with this book. Teachers are the best ambassadors to the beautiful realm of mathematics. Therefore, it is my desire to merely open the door to this aspect of mathematics with this book. Remember, this is only the door opener. Once you begin to see the many possibilities for enticing our youth toward a love for this magnificent and time-tested subject, you will begin to build an arsenal of books with many more ideas to use when appropriate.
This brings me to another thought. Not only is it obvious that the topic and level must be appropriate for the intended audience, but the teacher’s enthusiasm for the topic and the manner in which it is presented are equally important. In most cases, the units will be sufficient for your students. However, there will be some students who will require a more in-depth treatment of a topic. To facilitate this, references for further information on many of the units are provided (usually as footnotes).
When I meet someone socially and they discover that my field of interest is mathematics, I am usually confronted with the proud exclamation: “Oh, I was always terrible in math!” For no other subject in the curriculum would an adult be so proud of failure. Having been weak in mathematics is a badge of honor. Why is this so? Are people embarrassed to admit competence in this area? And why are so many people really weak in mathematics? What can be done to change this trend? Were anyone to have the definitive answer to this question, he or she would be the nation’s education superstar. We can only conjecture where the problem lies and then from that perspective, hope to repair it. It is my strong belief that the root of the problem lies in the inherent unpopularity of mathematics. But why is it so unpopular? Those who use mathematics are fine with it, but those who do not generally find it an area of study that may have caused them hardship. We must finally demonstrate the inherent beauty of mathematics, so that those students who do not have a daily need for it can be led to appreciate it for its beauty and not only for its usefulness. This, then, is the objective of this book: to provide sufficient evidence of the beauty of mathematics through many examples in a variety of its branches. To make these examples attractive and effective, they were selected on the basis of the ease with which they can be understood at first reading and their inherent unusualness.
Where are the societal shortcomings that lead us to such an overwhelming “fear” of mathematics, resulting in a general avoidance of the subject? From earliest times, we are told that mathematics is important to almost any endeavor we choose to pursue. When a young child is encouraged to do well in school in mathematics, it is usually accompanied with, “You’ll need mathematics if you want to be a _______________.” For the young child, this is a useless justification since his career goals are not yet of any concern to him. Thus, this is an empty statement. Sometimes a child is told to do better in mathematics or else_________________.” This, too, does not have a lasting effect on the child, who does just enough to avoid punishment. He will give mathematics attention only to avoid further difficulty from his parents. Now with the material in this book, we can attack the problem of enticing youngsters to love mathematics.
To compound this lack of popularity of mathematics among the populace, the child who may not be doing as well in mathematics as in other subject areas is consoled by his parents by being told that they, too, were not too good in mathematics in their school days. This negative role model can have a most deleterious effect on a youngster’s motivation toward mathematics. It is, therefore, your responsibility to counterbalance these mathematics slurs that seem to come from all directions. Again, with the material in this book, you can demonstrate the beauty, not just tell the kids this mathematics stuff is great.* Show them!
For school administrators, performance in mathematics will typically be the bellwether for their schools’ success or weakness. When their schools perform well either in comparison to normed data or in comparison to neighboring school districts, then they breathe a sigh of relief. On the other hand, when their schools do not perform well, there is immediate pressure to fix the situation. More often than not, these schools place the blame on the teachers. Usually, a “quick-fix” in-service program is initiated for the math teachers in the schools. Unless the in-service program is carefully tailored to the particular teachers, little can be expected in the way of improved student performance. Very often, a school or district will blame the curriculum (or textbook) and then alter it in the hope of bringing about immediate change. This can be dangerous, since a sudden change in curriculum can leave the teachers ill prepared for this new material and thereby cause further difficulty. When an in-service program purports to have the “magic formula” to improve teacher performance, one ought to be a bit suspicious. Making teachers more effective requires a considerable amount of effort spread over a long time. Then it is an extraordinarily difficult task for a number of reasons. First, one must clearly determine where the weaknesses lie. Is it a general weakness in content? Are the pedagogical skills lacking? Are the teachers simply lacking motivation? Or is it a combination of these factors? Whatever the problem, it is generally not shared by every math teacher in the school. This, then, implies that a variety of in-service programs would need to be instituted for meeting the overall weakness of instruction. This is rarely, if ever, done because of organizational and financial considerations of providing in-service training on an individual basis. The problem of making mathematics instruction more successful by changing the teachers’ performance is clearly not the entire solution. Teachers need ideas to motivate their students through content that is appropriate and fun.
International comparative studies constantly place our country’s schools at a relatively low ranking. Thus, politicians take up the cause of raising mathematics performance. They wear the hat of “education president,” “education governor,” or “education mayor” and authorize new funds to meet educational weaknesses. These funds are usually spent to initiate professional development in the form of the in-service programs we just discussed. Their effectiveness is questionable at best for the reasons outlined above.
What, then, remains for us to do to improve the mathematics performance of youngsters in the schools? Society as a whole must embrace mathematics as an area of beauty (and fun) and not merely as a useful subject, without which further study in many areas would not be possible (although this latter statement may be true). We must begin with the parents, who as adults already have their minds made up on their feelings about mathematics. Although it is a difficult task to turn on an adult to mathematics when he or she already is negatively disposed to the subject, this is another use for this book—provide some parent “workshops” where the beauty of mathematics is presented in the context of changing their attitude to the subject matter. The question that still remains is how best to achieve this goal.
Someone not particularly interested in mathematics, or someone fearful of the subject, must be presented with illustrations that are extremely easy to comprehend. He or she needs to be presented with examples that do not require much explanation, ones that sort of “bounce off the page” in their attractiveness. It is also helpful if the examples are largely visual. They can be recreational in nature, but need not necessarily be so. Above all, they should elicit the “Wow!” response, that feeling that there really is something special about the nature of mathematics. This specialness can manifest itself in a number of ways. It can be a simple problem, where mathematical reasoning leads to an unexpectedly simple (or elegant) solution. It may be an illustration of the nature of numbers that leads to a “gee whiz” reaction. It may be a geometrical relationship that intuitively seems implausible. Probability also has some such entertaining phenomena that can evoke such responses. Whatever the illustration, the result must be quickly and efficiently obtained. With enough of the illustrations presented in this book, you could go out and proselytize to parents so that they can be supportive in the home with a more positive feeling about mathematics.
At the point that such a turnaround of feelings occurs, the parents usually ask, “Why wasn’t I shown these lovely things when I was in school?” We can’t answer that and we can’t change that. We can, however, make more adults goodwill ambassadors for mathematics and make teachers more resourceful so that they bring these mathematics motivators into their classrooms. Teaching time isn’t lost by bringing some of these motivational devices into the classroom; rather, teaching time is more effective since the students will be more motivated and therefore more receptive to new material. So parent and teacher alike should use these mathematics motivators to change the societal perception of mathematics, both in the classroom and outside it. Only then will we bring about meaningful change in mathematics achievement, as well as an appreciation of mathematics’ beauty.
__________________________________
For Example:
1.1 Surprising Number Patterns I
There are times when the charm of mathematics lies in the surprising nature of its number system. There are not many words needed to demonstrate this charm. It is obvious from the patterns attained. Look, enjoy, and spread these amazing properties to your students. Let them appreciate the patterns and, if possible, try to look for an “explanation” for this. Most important is that the students can get an appreciation for the beauty in these number patterns.
_________________________
If you wanna readmore the book, you can download the book in digital version from link below :
Link 1
by Alfred S. Posamentier
Association for Supervision and Curriculum Development Alexandria, Virginia USA
_________________________
Contents
Foreword
Preface
Chapter 1 : The Beauty in Numbers
1.1 Surprising Number Patterns I
1.2 Surprising Number Patterns II
1.3 Surprising Number Patterns III
1.4 Surprising Number Patterns IV
1.5 Surprising Number Patterns V
1.6 Surprising Number Patterns VI
1.7 Amazing Power Relationships
1.8 Beautiful Number Relationships
1.9 Unusual Number Relationships
1.10 Strange Equalities
1.11 The Amazing Number 1,089
1.12 The Irrepressible Number 1
1.13 Perfect Numbers
1.14 Friendly Numbers
1.15 Another Friendly Pair of Numbers
1.16 Palindromic Numbers
1.17 Fun with Figurate Numbers
1.18 The Fabulous Fibonacci Numbers
1.19 Getting into an Endless Loop
1.20 A Power Loop
1.21 A Factorial Loop
1.22 The Irrationality of v2
1.23 Sums of Consecutive Integers
Chapter 2 Some Arithmetic Marvels
2.1 Multiplying by 11
2.2 When Is a Number Divisible by 11?
2.3 When Is a Number Divisible by 3 or 9?
2.4 Divisibility by Prime Numbers
2.5 The Russian Peasant’s Method of Multiplication
2.6 Speed Multiplying by 21, 31, and 41
2.7 Clever Addition
2.8 Alphametics
2.9 Howlers
2.10 The Unusual Number 9
2.11 Successive Percentages
2.12 Are Averages Averages?
2.13 The Rule of 72
2.14 Extracting a Square Root
Chapter 3 Problems with Surprising Solutions
3.1 Thoughtful Reasoning
3.2 Surprising Solution
3.3 A Juicy Problem
3.4 Working Backward
3.5 Logical Thinking
3.6 It’s Just How You Organize the Data
3.7 Focusing on the Right Information
3.8 The Pigeonhole Principle
3.9 The Flight of the Bumblebee
3.10 Relating Concentric Circles
3.11 Don’t Overlook the Obvious
3.12 Deceptively Difficult (Easy)
3.13 The Worst Case Scenario
Chapter 4 Algebraic Entertainments
4.1 Using Algebra to Establish Arithmetic Shortcuts
4.2 The Mysterious Number 22
4.3 Justifying an Oddity
4.4 Using Algebra for Number Theory
4.5 Finding Patterns Among Figurate Numbers
4.6 Using a Pattern to Find the Sum of a Series
4.7 Geometric View of Algebra
4.8 Some Algebra of the Golden Section
4.9 When Algebra Is Not Helpful
4.10 Rationalizing a Denominator
4.11 Pythagorean Triples
Chapter 5 Geometric Wonders
5.1 Angle Sum of a Triangle
5.2 Pentagram Angles
5.3 Some Mind-Bogglers on p
5.4 The Ever-Present Parallelogram
5.5 Comparing Areas and Perimeters
5.6 How Eratosthenes Measured the Earth
5.7 Surprising Rope Around the Earth
5.8 Lunes and Triangles
5.9 The Ever-Present Equilateral Triangle
5.10 Napoleon’s Theorem
5.11 The Golden Rectangle
5.12 The Golden Section Constructed by Paper Folding
5.13 The Regular Pentagon That Isn’t
5.14 Pappus’s Invariant
5.15 Pascal’s Invariant
5.16 Brianchon’s Ingenius Extension of Pascal’s Idea
5.17 A Simple Proof of the Pythagorean Theorem
5.18 Folding the Pythagorean Theorem
5.19 President Garfield’s Contribution to Mathematics
5.20 What Is the Area of a Circle?
5.21 A Unique Placement of Two Triangles
5.22 A Point of Invariant Distance in an Equilateral Triangle
5.23 The Nine-Point Circle
5.24 Simson’s Invariant
5.25 Ceva’s Very Helpful Relationship
5.26 An Obvious Concurrency?
5.27 Euler’s Polyhedra
Chapter 6 Mathematical Paradoxes
6.1 Are All Numbers Equal?
6.2 -1 Is Not Equal to +1
6.3 Thou Shalt Not Divide by 0
6.4 All Triangles Are Isosceles
6.5 An Infinite-Series Fallacy
6.6 The Deceptive Border
6.7 Puzzling Paradoxes
6.8 A Trigonometric Fallacy
6.9 Limits with Understanding
Chapter 7 Counting and Probability
7.1 Friday the 13th!
7.2 Think Before Counting
7.3 The Worthless Increase
7.4 Birthday Matches
7.5 Calendar Peculiarities
7.6 The Monty Hall Problem
7.7 Anticipating Heads and Tails
Chapter 8 Mathematical Potpourri
8.1 Perfection in Mathematics
8.2 The Beautiful Magic Square
8.3 Unsolved Problems
8.4 An Unexpected Result
8.5 Mathematics in Nature
8.6 The Hands of a Clock
8.7 Where in the World Are You?
8.8 Crossing the Bridges
8.9 The Most Misunderstood Average
8.10 The Pascal Triangle
8.11 It’s All Relative
8.12 Generalizations Require Proof
8.13 A Beautiful Curve
Epilogue
Acknowledgments
Index
About the Author
__________________________________
This book was inspired by the extraordinary response to an Op-Ed article I wrote for The New York Times.* In that article, I called for the need to convince people of the beauty of mathematics and not necessarily its usefulness, as is most often the case when trying to motivate youngsters to the subject. I used the year number, 2,002,** to motivate the reader by mentioning that it is a palindrome and then proceeded to show some entertaining aspects of a palindromic number. I could have taken it even further by having the reader take products of the number 2,002, for that, too, reveals some beautiful relationships (or quirks) of our number system. For example, look at some selected products of 2,002:
2,002 * 4 = 8,008
2,002 * 37 = 74,074
2,002 * 98 = 196,196
2,002 * 123 = 246,246
2,002 * 444 = 888,888
2,002 * 555 = 1,111,110
Following the publication of the article, I received more than 500 letters and e-mail messages supporting this view and asking for ways and materials to have people see and appreciate the beauty of mathematics. I hope to be able to respond to the vast outcry for ways to demonstrate the beauty of mathematics with this book. Teachers are the best ambassadors to the beautiful realm of mathematics. Therefore, it is my desire to merely open the door to this aspect of mathematics with this book. Remember, this is only the door opener. Once you begin to see the many possibilities for enticing our youth toward a love for this magnificent and time-tested subject, you will begin to build an arsenal of books with many more ideas to use when appropriate.
This brings me to another thought. Not only is it obvious that the topic and level must be appropriate for the intended audience, but the teacher’s enthusiasm for the topic and the manner in which it is presented are equally important. In most cases, the units will be sufficient for your students. However, there will be some students who will require a more in-depth treatment of a topic. To facilitate this, references for further information on many of the units are provided (usually as footnotes).
When I meet someone socially and they discover that my field of interest is mathematics, I am usually confronted with the proud exclamation: “Oh, I was always terrible in math!” For no other subject in the curriculum would an adult be so proud of failure. Having been weak in mathematics is a badge of honor. Why is this so? Are people embarrassed to admit competence in this area? And why are so many people really weak in mathematics? What can be done to change this trend? Were anyone to have the definitive answer to this question, he or she would be the nation’s education superstar. We can only conjecture where the problem lies and then from that perspective, hope to repair it. It is my strong belief that the root of the problem lies in the inherent unpopularity of mathematics. But why is it so unpopular? Those who use mathematics are fine with it, but those who do not generally find it an area of study that may have caused them hardship. We must finally demonstrate the inherent beauty of mathematics, so that those students who do not have a daily need for it can be led to appreciate it for its beauty and not only for its usefulness. This, then, is the objective of this book: to provide sufficient evidence of the beauty of mathematics through many examples in a variety of its branches. To make these examples attractive and effective, they were selected on the basis of the ease with which they can be understood at first reading and their inherent unusualness.
Where are the societal shortcomings that lead us to such an overwhelming “fear” of mathematics, resulting in a general avoidance of the subject? From earliest times, we are told that mathematics is important to almost any endeavor we choose to pursue. When a young child is encouraged to do well in school in mathematics, it is usually accompanied with, “You’ll need mathematics if you want to be a _______________.” For the young child, this is a useless justification since his career goals are not yet of any concern to him. Thus, this is an empty statement. Sometimes a child is told to do better in mathematics or else_________________.” This, too, does not have a lasting effect on the child, who does just enough to avoid punishment. He will give mathematics attention only to avoid further difficulty from his parents. Now with the material in this book, we can attack the problem of enticing youngsters to love mathematics.
To compound this lack of popularity of mathematics among the populace, the child who may not be doing as well in mathematics as in other subject areas is consoled by his parents by being told that they, too, were not too good in mathematics in their school days. This negative role model can have a most deleterious effect on a youngster’s motivation toward mathematics. It is, therefore, your responsibility to counterbalance these mathematics slurs that seem to come from all directions. Again, with the material in this book, you can demonstrate the beauty, not just tell the kids this mathematics stuff is great.* Show them!
For school administrators, performance in mathematics will typically be the bellwether for their schools’ success or weakness. When their schools perform well either in comparison to normed data or in comparison to neighboring school districts, then they breathe a sigh of relief. On the other hand, when their schools do not perform well, there is immediate pressure to fix the situation. More often than not, these schools place the blame on the teachers. Usually, a “quick-fix” in-service program is initiated for the math teachers in the schools. Unless the in-service program is carefully tailored to the particular teachers, little can be expected in the way of improved student performance. Very often, a school or district will blame the curriculum (or textbook) and then alter it in the hope of bringing about immediate change. This can be dangerous, since a sudden change in curriculum can leave the teachers ill prepared for this new material and thereby cause further difficulty. When an in-service program purports to have the “magic formula” to improve teacher performance, one ought to be a bit suspicious. Making teachers more effective requires a considerable amount of effort spread over a long time. Then it is an extraordinarily difficult task for a number of reasons. First, one must clearly determine where the weaknesses lie. Is it a general weakness in content? Are the pedagogical skills lacking? Are the teachers simply lacking motivation? Or is it a combination of these factors? Whatever the problem, it is generally not shared by every math teacher in the school. This, then, implies that a variety of in-service programs would need to be instituted for meeting the overall weakness of instruction. This is rarely, if ever, done because of organizational and financial considerations of providing in-service training on an individual basis. The problem of making mathematics instruction more successful by changing the teachers’ performance is clearly not the entire solution. Teachers need ideas to motivate their students through content that is appropriate and fun.
International comparative studies constantly place our country’s schools at a relatively low ranking. Thus, politicians take up the cause of raising mathematics performance. They wear the hat of “education president,” “education governor,” or “education mayor” and authorize new funds to meet educational weaknesses. These funds are usually spent to initiate professional development in the form of the in-service programs we just discussed. Their effectiveness is questionable at best for the reasons outlined above.
What, then, remains for us to do to improve the mathematics performance of youngsters in the schools? Society as a whole must embrace mathematics as an area of beauty (and fun) and not merely as a useful subject, without which further study in many areas would not be possible (although this latter statement may be true). We must begin with the parents, who as adults already have their minds made up on their feelings about mathematics. Although it is a difficult task to turn on an adult to mathematics when he or she already is negatively disposed to the subject, this is another use for this book—provide some parent “workshops” where the beauty of mathematics is presented in the context of changing their attitude to the subject matter. The question that still remains is how best to achieve this goal.
Someone not particularly interested in mathematics, or someone fearful of the subject, must be presented with illustrations that are extremely easy to comprehend. He or she needs to be presented with examples that do not require much explanation, ones that sort of “bounce off the page” in their attractiveness. It is also helpful if the examples are largely visual. They can be recreational in nature, but need not necessarily be so. Above all, they should elicit the “Wow!” response, that feeling that there really is something special about the nature of mathematics. This specialness can manifest itself in a number of ways. It can be a simple problem, where mathematical reasoning leads to an unexpectedly simple (or elegant) solution. It may be an illustration of the nature of numbers that leads to a “gee whiz” reaction. It may be a geometrical relationship that intuitively seems implausible. Probability also has some such entertaining phenomena that can evoke such responses. Whatever the illustration, the result must be quickly and efficiently obtained. With enough of the illustrations presented in this book, you could go out and proselytize to parents so that they can be supportive in the home with a more positive feeling about mathematics.
At the point that such a turnaround of feelings occurs, the parents usually ask, “Why wasn’t I shown these lovely things when I was in school?” We can’t answer that and we can’t change that. We can, however, make more adults goodwill ambassadors for mathematics and make teachers more resourceful so that they bring these mathematics motivators into their classrooms. Teaching time isn’t lost by bringing some of these motivational devices into the classroom; rather, teaching time is more effective since the students will be more motivated and therefore more receptive to new material. So parent and teacher alike should use these mathematics motivators to change the societal perception of mathematics, both in the classroom and outside it. Only then will we bring about meaningful change in mathematics achievement, as well as an appreciation of mathematics’ beauty.
__________________________________
For Example:
1.1 Surprising Number Patterns I
There are times when the charm of mathematics lies in the surprising nature of its number system. There are not many words needed to demonstrate this charm. It is obvious from the patterns attained. Look, enjoy, and spread these amazing properties to your students. Let them appreciate the patterns and, if possible, try to look for an “explanation” for this. Most important is that the students can get an appreciation for the beauty in these number patterns.
_________________________
If you wanna readmore the book, you can download the book in digital version from link below :
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