Mathematical Encounters: For the inquisitive mind
After a while you will begin to catch the spirit of offbeat, nonlinear thinking, and you may be surprised to find your aha! Only an elementary knowledge of math is needed to enjoy this entertaining compilation of brain-teasers. It includes a mixture of old and new riddles covering a variety of mathematical topics: Carefully explained solutions follow each problem. Over a period of 25 years as author of the Mathematical Games column for Scientific American, Martin Gardner devoted a column every six months or so to short math problems or puzzles.
This volume contains a rich selection of 70 of the best of these brain teasers, in some cases including references to new developments related to the puzzle. Brooklyn, Touching Cigarettes, and 64 other problems involving logic and basic math. In this book Sydney Padua transforms one of the most compelling scientific collaborations into a hilarious set of adventures, starring Ada Lovelace and Charles Babbage. Extremely funny and utterly unusual, this book comes complete with historical curiosities, extensive footnotes and never-before-seen diagrams of Babbage's mechanical, steam-powered computer.
Uncle Albert and his intrepid niece, Gedanken, enter the dangerous and unknown world of a thought bubble. Discover why you can't break the ultimate speed barrier, how to become older than your mother, how to put on weight without getting fat, and how to live forever without even knowing it. Other books in the series include: A selection of mathematical puzzles, stories, tricks and short articles - great to read all in one go, or to dip into.
The content varies between simple logic puzzles to introductions to more advanced topics such as the Four Colour Theorem, which tells us that we can colour in any map using only four colours, so that no bordering countries have the same colour. A collection of strange mathematical facts and stories. This anthology covers a whole range of ages, maths and mathematicians, and includes probability paradoxes, jumbled Shakespearean sonnets, record-breaking monkeys and typewriters, and theories of big game hunting. Also featured are stories of people who looked for logical loopholes in the American Constitution or calmed their nerves with algebra.
This collection by best-selling author David Wells, a Cambridge math scholar and teacher, includes more than puzzles, from the "mind sharpeners" of a medieval monk to the eitheenth-century Ladies' Diary, the Hindu Bhakshali manuscript, and riddles and popular rhymes. None require any mathematics beyond the most elementary algebra and geometry - and few require even that.
Complete answers appear at the end. Famed puzzle expert Martin Gardner explains the mathematics behind a multitude of mystifying tricks: Each of these are actually demonstrations of probability, sets, number theory, topology and other braches of mathematics. No skill at sleight of hand is needed to perform the more than tricks described in this book because mathematics guarantees their success. Can maths be creative? This book sets out to prove that it can, through a selection of short articles on surprising maths in everyday life.
Through lots of intriguing problems, involving card tricks, polar bears and, of course, socks, Rob Eastaway shows shows how maths can demonstrate its secret beauties in even the most mundane of everyday objects. With a foreword by Tim Rice, this book will change the way you see the world.
Why is it better to buy a lottery ticket on a Friday? Why are showers always too hot or too cold? And what's the connection between a rugby player taking a conversion and a tourist trying to get the best photograph of Nelson's Column? These and many other fascinating questions are answered in this entertaining and highly informative book, which is ideal for anyone wanting to remind themselves - or discover for the first time - that maths is relevant to almost everything we do.
Dating, cooking, travelling by car, gambling and even life-saving techniques have links with intriguing mathematical problems, as you will find explained here. Whether you have a PhD in astrophysics or haven't touched a maths problem since your school days, this book will give you a fresh understanding of the world around you. In , Edwin A. Now, Ian Stewart has written a fascinating, modern sequel to Abbott's book. Through larger-than-life characters and an inspired story line, "Flatterland" explores our present understanding of the shape and origins of the universe, the nature of space, time, and matter, as well as modern geometries and their applications.
Ever since the Sphinx asked his legendary riddle of Oedipus, riddles, conundrums, and puzzles of all sizes have kept humankind perplexed and amused. Each chapter introduces the basic puzzle, discusses the mathematics behind it, and includes exercises and answers plus additional puzzles similar to the one under discussion.
Here is a veritable kaleidoscope of puzzling labyrinths, maps, bridges, and optical illusions that will keep aficionados entertained for hours. An exploration of surprising ways maths occurs in our everyday lives, centred around five famous unsolved problems in mathematics. Topics include how to detect an art forgery, winning strategies in Monopoly, and how to crack a code. Sprinkled with games and links to interactive online content so you can try out some of the ideas for yourself!
This is the complete guide to exploring the fascinating world of maths you were never told about at school.
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Stand-up comedian and mathematician Matt Parker uses bizarre Klein Bottles, unimaginably small pizza slices, knots no one can untie and computers built from dominoes to reveal some of the most exotic and fascinating ideas in mathematics. Starting with simple numbers and algebra, this book goes on to deal with inconceivably big numbers in more dimensions than you ever knew existed. And always with something for you to make or do along the way.
In this book are twenty more curious puzzles and fantastical mathematical stories from one of the world's most popular and accessible writers on mathematics. This is a strange world of never-ending chess games, empires on the moon, furious fireflies, and, of course, disputes over how best to cut a cake.
Each chapter - with titles such as, "How to Play Poker By Post" and "Repealing the Law of Averages" - presents a fascinating mathematical puzzle that is challenging, fun, and introduces the reader to a significant mathematical problem in an engaging and witty way. Illustrated with clever and quirky cartoons, each tale will delight those who love puzzles and mathematical conundrums. The appeal of games and puzzles is timeless and universal. In this book, David Wells explores the fascinating connections between games and mathematics, proving that mathematics is not just about tedious calculation but imagination, insight and intuition.
The first part of the book introduces games, puzzles and mathematical recreations, including knight tours on a chessboard. The second part explains how thinking about playing games can mirror the thinking of a mathematician, using scientific investigation, tactics and strategy, and sharp observation. Finally the author considers game-like features found in a wide range of human behaviours, illuminating the role of mathematics and helping to explain why it exists at all. This thought-provoking book is perfect for anyone with a thirst for mathematics and its hidden beauty; a good high school grounding in mathematics is all the background that is required, and the puzzles and games will suit pupils from 14 years.
Collected over several years by Peter Winkler, dozens of elegant, intriguing challenges are presented in this book. The answers are easy to explain, but without this book, devilishly hard to find. Creative reasoning is the key to these puzzles. No involved computation or higher mathematics is necessary, but your ability to construct a mathematical proof will be severly tested - even if you are a professional mathematician. For the truly adventurous, there is even a chapter on unsolved puzzles.
Beautifully crafted and immensely enjoyable, the problems in this book require minimal technical knowledge, being accessible to young secondary school pupils. However, there is an astonishing range in difficulty; while some of the problems are fairly straightforward, others are significantly tougher, with a great deal of ingenuity and clarity of thought needed to make progress. Whether you are a student preparing for a maths competition, an educational establishment seeking to supplement your problem solving resources, or an individual looking for a different sort of challenge, Elastic Numbers is a unique collection, and will push you to the very edge of your abilities.
Thinking Mathematically is perfect for anyone who wants to develop their powers to think mathematically, whether at school, at university or just out of interest. This book is invaluable for anyone who wishes to promote mathematical thinking in others or for anyone who has always wondered what lies at the core of mathematics. Thinking Mathematically reveals the processes at the heart of mathematics and demonstrates how to encourage and develop them. Extremely practical, it involves the reader in questions so that subsequent discussions speak to immediate experience.
This book tells the story of one of the biggest adventures in mathematics: This is the story of how humankind has come to its understanding of the bizarre world of symmetry - a subject of fundamental significance to the way we interpret the world around us. Our eyes and minds are drawn to symmetrical objects, from the sphere to the swastika, from the pyramid to the pentagon. When do the hands of a clock coincide? How likely is it that two children in the same class will share a birthday? How do we calculate the volume of a doughnut?
Mathematics for the Curious provides anyone interested in mathematics with a simple and entertaining account of what it can do. Author Peter Higgins gives clear explanations of the more mysterious features of childhood mathematics as well as novelties and connections that prove that mathematics can be enjoyable and full of surprises.
Higgins poses entertaining puzzles and questions tempting the reader to ponder math problems with imagination instead of dread. Mathematics for the Curious is an accessible introduction to basic mathematics for beginning students and a lively refresher for adults.
Mathematics for the Imagination provides an accessible and entertaining investigation into mathematical problems in the world around us. From world navigation, family trees, and calendars to patterns, tessellations, and number tricks, this informative and fun book helps you to understand the maths behind real-life questions and rediscover your arithmetical mind. This is a highly involving book which encourages the reader to enter into the spirit of mathematical exploration. A stimulating account of development of basic mathematics from arithmetic, algebra, geometry and trigonometry, to calculus, differential equations and non-Euclidean geometries.
Also describes how maths is used in optics, astronomy, motion under the law of gravitation, acoustics, electromagnetism, and other aspects of physics. In this book, Professor Ogilvy demonstrates the mathematical challenege and satisfaction to be had from geometry, the only requirement being two simple implements straight-edge and compass and a little thought. Topics including harmonic division and Apollonian circles, inversive geometry, the hexlet, conic sections, projective geometry, the Golden Section and angle trisection are addressed in a way that brings out the true intellectual excitement inherent in each.
Also included are some unsolved problems of modern geometry. This book offers a fascinating glimpse into the world of mathematics and mathematicians. It is designed for the reader who has no advanced mathematical background of special aptitude, but who wants to acquaint him or herself with the intellectually stimulating and aesthetically satisfying aspects of the subject. After illuminating the role of the mathematician and dispelling several popular misconceptions about the nature of mathematics, Professor Ogilvy takes you on a lively tour of the four basic branches of the subject: Focusing on the interesting, and even amusing, aspects of mathematics, he points out the interconnections between the branches and presents mathematics as a vital subject whose frontiers are continually expanding.
Promoting Traditional Religion in a - download pdf or read online. New church buildings are arising and lots of older church buildings are redefining themselves to outlive.
Math Maturity - IAE-Pedia
Download e-book for iPad: What is the most modern mystery weapon for those who wish hearty, slow-cooked foodstuff yet wouldn't have hours to spend within the kitchen? You guessed it the strain cooker! However, there is much more to math maturity. For example, a student needs to learn how to learn math, how to self-assess his or her level of math content knowledge, skills, and math maturity, how to make use of aids to doing math, how to relearn math that has been forgotten or partially forgotten, how to make effective use of online sources of math information and instruction, how to make effective use of technological aids to both learning and doing math, and so on.
This section begins with a brief discussion of Pedagogical Content Knowledge, followed by a brief discussion about teaching math.
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Together, the two topics provide insight into roles of math maturity in math education. All teachers learn about content knowledge, pedagogical knowledge, and pedagogical content. Quoting from the article:. Pedagogical content knowledge identifies the distinctive bodies of knowledge for teaching. It represents the blending of content and pedagogy into an understanding of how particular topics, problems or issues are organized, represented, and adapted to the diverse interests and abilities of learners, and presented for instruction. Pedagogical content knowledge is the category most likely to distinguish the understanding of the content specialist from that of the pedagogue.
An article on technological pedagogical content knowledge is available in http: A good teacher in any discipline such as math needs to have an appropriate balance of knowledge and skills in content, pedagogy, and pedagogical content knowledge. The teacher also needs to understand the concept of maturity in the disciplines he or she teaches, and how to help students gain in maturity within these disciplines.
The content of math has been growing steadily for many thousands of years. Math is a broad, deep, discipline of study that is important in its own right and important in representing and helping to solve the problems in many other disciplines.
See the Web document, What Is mathematics? Our pedagogical knowledge—the sciences of teaching, learning, and cognitive neuroscience—has been improving since formal schools were started more than 5, years ago, shortly after reading and writing were developed. Advances in technology, such as the development of the printing press, Information and Communications Technology, and cognitive neuroscience have have greatly changed the processes of teaching and learning. One of the largest challenges our math education system faces comes from the progress occurring in developing artificially intelligent computers that can solve or greatly help in solving the types of math problems that traditionally students have learned to solve using pencil, paper, math tables, and "simple" tools such as a slide rule or four-function calculator.
Increasingly, we live in a world where we all routinely use tools that solve quite complex math problems. GPS, telephones, and the graphics and action in computer games provide excellent examples. High school math and science classes routinely teach students to use a "high end" calculator to perform calculations, graph functions, and solve equations. Part of today's math maturity is knowing capabilities and limitations of math tools that are now readily available. Note that the general idea that such tools are available and relatively easy to learn how to use is a component of math maturity.
Another aspect of math maturity is the ability to communicate to a computer details of a math problem that one wants solved. The Web makes it possible for a person to have a computer solve or greatly help in solving a very wide range of math problems. A "modern" math education includes instruction in how to make effective use of such computer capabilities, and modern math maturity includes insights into the capabilities, limitations, and implications of the steadily growing availability and power of such tools.
The number line is one of the big ideas in math. What do you remember about your early encounters with the number line? Perhaps you can still remember being in an elementary school classroom with a segment of the number line on a poster that extended across the front of the room. The number line is a complex component of math. In your early math education you learned that there are positive and negative numbers.
You know that, for any pair of numbers, either they are the same or one is larger than the other. Introspect as your mind mulls over the fact that -5 is larger than Think about how a young student's mind deals with this situation. This IAE-pedia document contains a number of short sections titled Math maturity food for thought.
You can increase your level of math maturity by spending time reflecting on these "exercises" and discussing them with your fellow students. Some are appropriate for use in teaching elementary school students. As you were reading the previous two paragraphs, you encountered the statement, "It is 'impossible' to divide by zero. Can you provide arguments that convince you and that might convince others that this is a correct statement?
A more mathematically mature mind questions assertions such as, "It is impossible to divide by zero. This approach to learning with questioning and understanding is applied both in math and in other disciplines of study. At any grade level, a teacher might encounter a student who raises such questions and who searches for answers. Math inquisitiveness is one aspect of math maturity. However, this is a tricky situation. We can readily have students memorize the statement that it is impossible to divide by zero. We can have students memorize that zero is the only number that one cannot divide by.
But, we want more than just this rote memory. We want some level of understanding. We can help each student to develop a mental picture of the answers that one gets by dividing a number such as 8 by smaller and smaller positive numbers. This provides the student with an explanation that it is impossible to divide 8 by zero because "the answer is larger than any positive number.
What does it mean when one says the answer is larger than any positive number? Is "infinity" a number? It also begins to provide some insight into "infinity" and the mathematics of infinite series. Of course, the more mathematically mature students will likely ask further questions, such as what does it mean to say the number line extends "forever" in each direction and that there is no largest positive number? Why is it that zero divided by 8 is the same as zero divided by 12?
Is there any other number that has this same peculiarity as zero? And, suppose that it were possible to divide by zero. Would eight divided by zero be the same as 12 divided by zero? At the earliest levels of learning math, students can encounter or pose very challenging and perplexing math questions. There are many opportunities for the more mathematically gifted and talented, the more mathematically cognitively developed, the more mathematically inquisitive or creative, and so on, to demonstrate an above average level of math maturity relative to his or her peers.
You may be interested in Jo Boaler's video about teaching math for understanding.
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Boaler is a math education Professor at Stanford University. The video includes examples of challenging problems that can be used with a wide range of students. A variety of organizations have created math standards. The following is quoted from the linked document:. At every grade level from elementary school on up, teachers of math will encounter students with widely varying knowledge, skills, and interests in math, and with varying levels of math maturity.
This situation suggests an important question: What do teachers themselves need to know to be effective math teachers? The following paper provides a discussion of some research on this question:. Here is an example from the article. Answer the following questions. The authors want preservice and inservice teachers to have an understanding of the depth complexity of the math they are teaching. If teachers lack this depth, what can they expect of their students except rote learning with little understanding?
Try to put yourself into the mind of a young student. As a third grader, you know how to count and how to add integers. Both activities seem relevant to your life. Then the teacher starts explaining that there are two kinds of integers—odd and even. Both "odd" and "even" are part of your third grade working vocabulary. Perhaps you laugh as you think about what it might mean for an integer to be labeled as "odd. Now, back to thinking as a teacher. Are the math words odd and even just words to be memorized and a bunch of drill and practice activities to carry out?
Or, do they have deeper math meaning to you? Why would anyone care whether 0 is odd or even? Suppose that a student asks you the question: Math Department faculty and Teacher Education faculty have long argued the merits of these two approaches. In the preparation of elementary school teachers, the standard compromise that has been worked out consists of a certain amount of college coursework in each of the two areas listed.
The nature and amount of the coursework varies in different teacher education programs throughout the country. One of the main themes in the Math Maturity document you are currently reading is that both approaches are important, but that both need to place considerably more emphasis on math maturity. The teacher needs a reasonably high level of both math content maturity and math pedagogical knowledge maturity. Determining what constitutes an appropriate balance between these two areas will vary with the teacher and with the students that the teacher is teaching.
Math maturity provides a useful framework for addressing some of the math education challenges. Very roughly speaking, math maturity focuses on the long-term understanding, retention, and ability to make use of the math that one has studied. What lasts as one forgets the finer details of what one has studied? What lasts as rote memories fade? This section is a work in progress. The general purpose of this section is to examine components of and measures of adult numeracy, and see how they relate to math maturity.
There is a parallel between the general ideas of literacy and numeracy. Groups working to define and measure adult numeracy tend to move beyond measure of basic math content knowledge and skills. These include a variety of self-assessment instruments and online instruction. As one example, see Ginsburg, L.
Council for Advancement of Adult Literacy. While individual writers and researchers have emphasized different aspects of adult numeracy and different priorities, one common theme is the need to recognize that the context and ability to apply mathematical knowledge to and reason about the numerical aspects of situations is important. Mathematicians use mathematical maturity to mean, loosely, a mixture of mathematical experience and insight that is not taught directly, but which grows and ripens from substantial exposure to complex mathematical concepts and processes.
I have to chuckle when I read this statement.
History of Mathematics
It lacks the precision of communication that mathematicians prize, and it doesn't provide much help to students working to improve their own level of math maturity or that of students they are preparing to teach. Much of the power of math lies in its relatively high level of abstractness. Think about a young child learning the number words one, two, three, etc.
The child eventually learns that by saying the words and making a one-to-one correspondence with a set of objects, the final number said is the quantity of objects in the set. That is a major math-learning step. Later the child encounters the symbols 1, 2, 3, etc. These are shorthand symbols for the words one, two, three and likely they are learned memorized before the child encounters and learns the alphabetic representations one, two, three, etc.
Do you find it interesting that we have children learn the abstract shorthand representations for the natural language words one, two, three, etc. Quoting David Tall from http: One thing implied here is that, as the symbols and the manipulations become more and more abstract, and more difficult for the student to relate to what is known, then the student "learns" with less and less understanding. In many cases, a student would face a daunting task trying to work out a referent that has meaning to the student.
Click here to learn more about communicating in the language of mathematics. Effective communication using the language of mathematics is an important component of the content of math and is an important indicator of a growing level of math maturity. Math maturity is not just some that one has or does not have. Nor is it a specific component of math content that is taught in schools. Rather, one's level of math maturity grows through the study and use of math. Notice that the list does not contain any specific math content.
Rather, it is a list of what mathematicians do. The last item in the bulleted list is steadily growing in importance. The overall discipline of math is now divided into the three categories: Computational math involves a combination of math and artificial intelligence, and is now becoming a routine way in which people solve problems. As an example, think of a GPS. Some mathematicians are highly skilled in the use of computer tools and, indeed, may use them in their research and math problem solving.
The disciplines of Mathematics and of Computer and Information Science strongly overlap. Many other mathematicians have a more modest but still quite personally useful level of knowledge of the overall field of computers and information science and its underlying mathematics. Larry Denenberg has a Ph. He approaches Mathematical Maturity from a different perspective. One can evaluate a math lesson plan or unit of study in terms of how it contributes to a student gaining math maturity. See Good Math Lesson Plans. The general notion of "maturity" in a discipline applies to every discipline—indeed to every voluntary adult activity.
You know that Mozart composed music when he was quite young. However, it is obvious to music critics that his early compositions were quite immature. Similar statements are often made about the work of other "young" artists and writers. One way to describe increasing math maturity is to talk about a person making progress toward "being" a mathematician. That is, increasing math maturity is progress toward learning to think like a mathematician and to function effectively in the culture of mathematicians.
This mathematical thinking is applied over both a wide range of components of the discipline of mathematics and also in areas outside of mathematics. Being logical and thinking logically are applicable in many disciplines. Think of the field of law, for example. However, the reasoning and logical arguments used by most people in most disciplines are different from the precise mathematical logic that is inherent to the discipline of mathematics and to the education of mathematicians.
Richard Feynman captures part of this idea in his statement:. This section explores various aspects of working to improve our math education system. It places particular emphasis on various aspects of math maturity. A math teacher needs maturity in both math content knowledge and in math pedagogical content knowledge. Quite a bit of this comes through on-the-job learning. Most teachers become considerably more effective as they proceed through their first half-dozen years of teaching.
The adage, "You have to teach it do it in order to really understand it" rings true. Many people believe that the math education system in the U. Over the years, there have been considerable efforts to improve the effectiveness of our math education system. Many of these efforts have focused on developing better curriculum and books, providing better preservice and inservice education for math teachers, requiring more years of math courses for precollege students, and setting more rigorous standards.
There has also been a strong emphasis on encouraging women and minorities to take more math coursework. Dissatisfaction with our math education system persists, with reason. Current efforts to improve our math education system tend to be mostly focused on the same approaches that have not proven very successful in the past.
The general opinion seems to be that if we can just do more and do it better with these approaches, our math education system will improve. A mathematical comment to this approach might be: If one is walking to get to a particular place, but the place seems always equally far away, perhaps one is walking in a circle. Brain science research is progressing more rapidly than is our implementation of the results in our educational systems.
Here are five important areas for improvement that have received much less attention than I feel they deserve. Brain science is currently one of the fastest areas of growth in human knowledge. The IAE-pedia site Brain Science provides brief introductions to more than 40 topics in educational cognitive neuroscience that I believe all teachers should know about. In recent years, math educators have also had to deal with the steadily increasing capability and availability of calculators and computers.
In essence, we now need an educational system with a focus on helping human brains and computer brains to work together at posing and solving problems. We need to prepare students to work effectively in environments where the computer capabilities increase significantly year by year.
One important component of the history of math focuses on the development of aids to "doing" math. The abacus provides a good example. The calculator and computer are more modern examples. The combination of computers and artificial intelligence has made possible computer algebra systems that can solve a huge range of math problems. Such systems raise the question, "If a computer can solve a particular category of math problems, what do we want students to learn about solving this category of problems?
Our current math education system is still rather weak in teaching and learning in a manner that appropriately deals with forgetting. See Learning, Forgetting, and Relearning. We know that students in math classes eventually or quite quickly forget much of what they supposedly have learned. Although we spend quite a bit of time on review, we still face the constructivist problem that we are expecting students to build construct new knowledge on top of knowledge that they either never had or have forgotten.
The nature and extent of forgetting what one has learned or supposedly learned varies from discipline to discipline. Think of your math education as consisting of a combination of some big ideas with a large number of smaller little ideas. For example, think about the idea of square root versus various paper-and-pencil methods for calculating square root. The idea of square root is bigger than that of a specific paper-and-pencil computational procedure. Or, to expand on this example, the idea of an n-th root of a number is a bigger idea than the idea of square root.
The idea of an n-th degree polynomial having n solutions is a bigger idea than the idea of n-th root.