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The Egyptians are known for being ahead of their time in comparison to some civilisations that came after them. This unit looks at how the Egyptians solved mathematical problems in everyday life and the technology they used. An understanding of this area has only been possible following the translation of the Rosetta Stone. This study unit is just one of many that can be found on LearningSpace, part of OpenLearn, a collection of open educational resources from The Open University. Published ...
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Alon Amit, prolific Quora math answerer, argues that an honest representation of mathematical ideas is enough to spark interest in math. It's not necessary to exaggerate the role of math; the golden ratio does not drive the stock market, the solution of the Riemann hypothesis will not kill cryptography, and Grothendieck did not advance robotics. Hi…
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Dave Cole, the author of the Math Kids series of books, talks about introducing kids to math as a fun challenge and puzzle beyond the rote memorization they've come to expect. Kids who like to read are enticed by puzzles and mysteries. Möbius strips, Pascal's triangle, and other concepts that are new to them, make them marvel, "Is this math?" They …
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Neil Epstein, Associate Professor of Mathematics at George Mason University, introduces us to the fractions used by the ancient Egyptians, well before the Greeks and Romans. The Egyptian fractions all had a unit numerator. They could represent any fraction as a sum of unique unit fractions, a fact that was not proved until centuries later. These su…
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Jeanne Lazzarini joins us again to introduce us to the mathematician Luca Pacioli, whose views of numbers and shapes influenced Leonardo da Vinci, leading to a period of art and invention. His book, De Divina Proportione, is the only book ever illustrated by da Vinci. The Renaissance was a period when mathematicians studied art and artists studied …
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Alon Amit, probably the most prolific answerer of math questions on Quora, shares his reasons for his deep involvement. He seeks to share the journey, the exploration and stumbles of solving a problem. He's especially drawn to questions that will teach him things, even if he never completes the answer. He also shares his joy of problem solving with…
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Lee Kraftchick continues his tour of books about math written for the non-mathematician like himself. We also can't let go of Gödel Escher Bach. Lee cites an opinion piece in the Washington Post, titled, "The Problem with Schools Today is Too Much Math," which gives a very narrow view of what math is. He counters it with a response (see theartofmat…
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Lee Kraftchick discusses some of his favorite books for non-mathematicians to explore the breadth of mathematics. These books range from very old to current. Some discuss beautiful proofs, whether math is invented or discovered, and how to think. Lee and Carol agree on the number one greatest book for mathematicians and non-mathematicians alike. Se…
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Ethan Zhao and Edward Yu are the winners in mathematics of the prestigious Davidson Fellow Scholarships, awarded based on projects completed by students under 18. Ethan's project was on learning models and Edward's was on combinatorics. It was math contests and the MIT Primes program that gave them the background to do original research in high sch…
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Lawyer Lee Kraftchick discusses the search for truth and basic principles in the legal community and the surprising parallels and similarities with the same search in the math community. Mathematical and legal arguments follow a similar structure. Even the backwards way an argument is created is the same.…
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Lee Kraftchick, a lawyer with a math degree, discusses some of the surprising parallels between the fields. Math is used directly to make statistical arguments to rule out random chance as a cause. He gives examples from his experience in redistricting and affirmative action. Math is used indirectly in legal reasoning from what is known to justifie…
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Brian Katz, from California State University Long Beach, invites us to explore the various layers of ordinary sounds, informed by a little calculus. The limited frequencies that come out of the wave equation are what separates sounds that communicate (voice, music) from noise. These higher notes are in the sound itself and you can hear them (but al…
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Lawrence Udeigwe, associate professor of mathematics at Manhattan College and an MLK Visiting Associate Professor in Brain and Cognitive Sciences at MIT, is both a mathematician and a musician. We discuss his recent opinion piece in the Notices of the American Mathematical Society calling for "A Case for More Engagement" between the two areas, and …
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Joseph Bennish returns to dig into the math behind the Fourier Analysis we discussed last time. Specifically, it allows us to express any function in terms of sines and cosines. Fourier analysis appears in nature--our eyes and ears do it. It's used to study the distribution of primes, build JPEG files, read the structure of complicated molecules an…
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Joseph Bennish, Professor Emeritus of California State University, Long Beach, joins us for an excursion into physics and some of the mathematics it inspired. Joseph Fourier straddled mathematics and physics. Here we focus on his heat equation, based on partial differential equations. Partial differential equations have broad applications. Fourier …
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Jim Stein, Professor Emeritus of CSULS, returns to complete his (admittedly subjective) list of the ten greatest math theorems of all time, with fascinating insights and anecdotes for each. Last time he did the runners up and numbers 8, 9 and 10. Here he completes numbers 1 through 7, again ranging over geometry, trig, calculus, probability, statis…
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Jim Stein, Professor Emeritus of CSULB, presents his very subjective list of what he believes are the ten most important theorems, with several runners up. These theorems cover a broad range of mathematics--geometry, calculus, foundations, combinatorics and more. Each is accompanied by background on the problems they solve, the mathematicians who d…
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Jim Stein, Professor Emeritus of California State University Long Beach, discusses some bets that appear to be 50-50, but can have better odds with a tiny amount of seemingly useless information. Blackwell's Bet involves two envelopes of money. You can open only one. Which one do you choose? We learn about David Blackwell and his mathematical journ…
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Jeanne Lazzarini joins us again to discuss fractals, a way to investigate the roughness that we see in nature, as opposed to the smoothness of standard mathematics. Fractals are built of iterated patterns with zoom similarity. Examples include the Koch Snowflake, which encloses a finite area but has infinite perimeter, and the Sierpinski Triangle, …
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Joseph Bennish, Prof. Emeritus of CSULB, describes the field of Diophantine approximation, which started in the 19th Century with questions about how well irrational numbers can be approximated by rationals. It took Cantor and Lebesgue to develop new ways to talk about the sizes of infinite sets to give the 20th century new ways to think about it. …
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Jeanne Lazzarini, a math education specialist, returns to discuss tessellations and tiling in the works of Escher, Penrose, ancient artists and nature. We go beyond the familiar square or hexagonal tilings of the bathroom floor. M.C. Escher was an artist who made tessellations with lizards or birds, as well as pictures of very strange stairways. Ro…
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Joseph Bennish returns to take us beyond the rational numbers we usually use to numbers that have been given names that indicate they're crazy or other-worldly. The Greeks were shocked to discover irrational numbers, violating their geometric view of the world. But later it was proved that any irrational number can be approximated remarkably well b…
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Lara Alcock of Loughborough University shares what she learned, by tracking eye movements, about how mathematicians and students differ in the ways they read mathematics. She developed a 10-15 minute exploration training, that increases students' comprehension through self-explanation. We also discuss the transition between procedural math and proo…
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Jon Goga, of Brainy Spinach Math, is using the Roblox gaming platform to bring math learning to kids using something they already enjoy. Along the way, he teaches them some techniques that are useful for mathematicians at any level--breaking down and building up a problem. We also discuss the "inchworm" and "grasshopper" styles of learning.…
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Sunil Singh, the author of Chasing Rabbits and other books, shares fascinating stories that show mathematics as a universal place of exploration and comfort. Stories of mathematical struggle and discovery in the classroom help students connect deeply with the topic, feel the passion, and see math as multi-cultural and class-free.…
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Caron Rivera, a math teacher at a school for elite athletes, shares how she breaks through the myth of the "math person" and teaches athletes to think like mathematicians. Her problem solving technique applies to anything. Through it her students get comfortable with not knowing, with the adventure of seeking the answer. They build their brains in …
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Brian Katz of CSULB joins us once again to discuss mathematical definitions. Students often see them as cast in stone. Prof. Katz helps them see that they're artifacts of human choices. The student has the power to create mathematics through definitions. This is illustrated by the definitions of "sandwich" and "approaching a limit." What makes a go…
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Mark Hendrickson, of Beast Academy Playground, talks about how to bring young kids into the joy, creativity and exploration that mathematicians experience. Kids enjoy art because they are free to try things and shun math for its apparent rigidness. He offers subtly mathematical games that invite even very young children to explore and question.…
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Ian Stewart, prolific author of popular books about math, discusses how math is the best way to think about the natural world. Often math developed for its own sake is later found useful for seemingly unrelated real-world problems. A silly little puzzle about islands and bridges leads eventually to a theory used for epidemics, transportation and ki…
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Joseph Bennish joins us once again to continue his discussion of symmetry, this time venturing into higher dimensions. We explore the complex symmetry groups of the Platonic solids and the sphere and their relationships. We then venture into the 4th dimension, where we see that, with a change to the distance the symmetries are maintaining, we get E…
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We are all born with an intuitive attraction to symmetry, through human faces and heartbeats. Joseph Bennish, of California State University Long Beach, explores the mathematical meaning of symmetry, what it means for one shape to be more symmetric than another, how symmetries form mathematical groups and groups form symmetries, and hints at implic…
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You are a contestant on Let's Make a Deal, hosted by Monty Hall. There are 3 identical doors. Behind only one is the prize car. You make your choice, then Monty Hall opens one of the other doors to reveal a goat and asks whether you want to change your choice. Should you, or does it matter? Paula Sloan talks about the counterintuitive answer, and h…
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There are a lot of jokes that poke fun at mathematicians, how they think and how they fumble around in the real world. Many of them start, "A mathematician, an engineer and a physicist ..." We'll look at what these jokes say about us. The most telling is a little joke that only a mathematician would enjoy, since it gives surprising insight into how…
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