Math + Comics = Learning That’s Fun! Help students build essential math skills and meet math standards with 80 laugh-out-loud comic strips and companion mini-story problems. Each reproducible comic and problem set reinforces a key math skill: multiplication, division, fractions, decimals, measurement, geometry, and more. Great to use for small-group or independent class work and for homework! For use with Grades 3-6.
Correction (Thanks to Prof. Leong, see comments below):
“Manjul Bhargava’s PhD advisor is Andrew Wiles of Princeton University, not Benedict Gross. However, Bhargava is an undergraduate student of Gross in Harvard University.”
Parents, students and teachers often argue, with little evidence, about whether U.S. high schools begin too early in the morning. In the past three years, however, scientific studies have piled up, and they all lead to the same conclusion: a later start time improves learning. And the later the start, the better.Biological research shows that circadian rhythms shift during the teen years, pushing boys and girls to stay up later at night and sleep later into the morning. The phase shift, driven by a change in melatonin in the brain, begins around age 13, gets stronger by ages 15 and 16, and peaks at ages 17, 18 or 19.
Does that affect learning? It does, according to Kyla Wahlstrom, director of the Center for Applied Research and Educational Improvement at the University of Minnesota. She published a large study in February that tracked more than 9,000 students in eight public high schools in Minnesota, Colorado and Wyoming. After one semester, when school began at 8:35 a.m. or later, grades earned in math, English, science and social studies typically rose a quarter step—for example, up halfway from B to B+.
Read more at: http://www.scientificamerican.com/article/school-starts-too-early/?&WT.mc_id=SA_WR_20140827
When teaching Quadratic equation in Algebra class using the conventional math pedagogy, this is what you get …
Teaching Math without word, especially for autistic and dyslexic kids, using pictures, video games… look how easy is to explain difficult concepts – even for adults – why (-2)x (-3) = + 6 ?
Today all school maths subjects are taught separately as: Arithmetic, Geometry, Trigonometry and Algebra, influenced by Euler in 1727.
Since 18th century Maths has evolved rapidly with the biggest revolution of Modern (Abstract) Maths in 19th century from the French prodigy Galois in Group Theory.
The 20th and 21st centuries Maths continues to expand from Galois Abstract Maths to a chaotic state where no single mathematician can claim to know all aspects of Maths like Newton, Euler, Gauss…did.
It is time to re-look at Euler’s outdated Maths pedagogy of 4 distinct disciplines… Can these 4 subjects be taught as ONE combined ‘Math’ (americans spell as singular) subject ?
Euler was invited by Peter I of Russia in 1727 to work in the
Petersburg Academy of Sciences. He introduced the fundamental math
disciplines in school math education:
1. Arithmetic
2. Geometry
3. Trigonometry
4. Algebra
these 4 are taught as separate and specific subjects, versus 19 duplicated disciplines in Europe.
Euler influenced not only in Russia schools, but in schools worldwide today.
Source: Russian Mathematics Education
Vol 1: History and world significance
Vol 2: Programs and practices
(Publisher: World Scientific)
This is a revolutionary approach (2005) to teach Secondary / High school Trigonometry by using purely algebra, no geometry and no picture, no sine, cosine, tangent, etc.
New concepts:
Vector as an order pair (x, y)
Quadrance = magnitude of vector
Perpendicular of 2 vectors
Parallel
Spread (angle between 2 vectors)
Australia: University of Sydney
Canada: University of Toronto
China: 北京大学
Japan: Tokyo University
Hong Kong: 香港大学
France: Ecole Normale Supérieure, Paris
India: Indian Institute of Technology, Delhi
Taiwan: 国立台湾大学
UK: Cambridge University
USA: Harvard University
Clouds are not spheres, mountains are not cones, and lightening does not travel in a straight line. The complexity of nature’s shapes differs in kind, not merely degree, from that of the shapes of ordinary geometry, the geometry of fractal shapes.
Now that the field has expanded greatly with many active researchers, Mandelbrot presents the definitive overview of the origins of his ideas and their new applications. The Fractal Geometry of Nature is based on his highly acclaimed earlier work, but has much broader and deeper coverage and more extensive illustrations.
Excellent and educational post by famous Mathematician Timothy Gowers on how to solve Math (Olympiad) problems.
(Post is at the bottom of this article)
Many students often give up immediately when facing a difficult maths problem. However, if students persist on for some time, usually they can come up with a solution or at least an idea on how to solve the problem. That is a great achievement already!
Never give up, even when your Maths question looks like this!
Quote: What I wrote gives some kind of illustration of the twists and turns, many of them fruitless, that people typically take when solving a problem. If I were to draw a moral from it, it would be this: when trying to solve a problem, it is a mistake to expect to take a direct route to the solution. Instead, one formulates subquestions and gradually builds up a useful bank of observations until the direct route becomes clear. Given that we’ve just had the football world cup, I’ll draw an analogy that I find not too bad (though not perfect either): a team plays better if it patiently builds up to an attack on goal than if it hoofs the ball up the pitch or takes shots from a distance. Germany gave an extraordinary illustration of this in their 7-1 defeat of Brazil.
This is a one-of-a-kind reference for anyone with a serious interest in mathematics. Edited by Timothy Gowers, a recipient of the Fields Medal, it presents nearly two hundred entries, written especially for this book by some of the world’s leading mathematicians, that introduce basic mathematical tools and vocabulary; trace the development of modern mathematics; explain essential terms and concepts; examine core ideas in major areas of mathematics; describe the achievements of scores of famous mathematicians; explore the impact of mathematics on other disciplines such as biology, finance, and music–and much, much more.
Unparalleled in its depth of coverage, The Princeton Companion to Mathematics surveys the most active and exciting branches of pure mathematics, providing the context and broad perspective that are vital at a time of increasing specialization in the field. Packed with information and presented in an accessible style, this is an indispensable resource for undergraduate and graduate students in mathematics as well as for researchers and scholars seeking to understand areas outside their specialties.
Features nearly 200 entries, organized thematically and written by an international team of distinguished contributors
Presents major ideas and branches of pure mathematics in a clear, accessible style
Defines and explains important mathematical concepts, methods, theorems, and open problems
Introduces the language of mathematics and the goals of mathematical research
Covers number theory, algebra, analysis, geometry, logic, probability, and more
Traces the history and development of modern mathematics
Profiles more than ninety-five mathematicians who influenced those working today
Explores the influence of mathematics on other disciplines
Includes bibliographies, cross-references, and a comprehensive index
Contributors incude:
Graham Allan, Noga Alon, George Andrews, Tom Archibald, Sir Michael Atiyah, David Aubin, Joan Bagaria, Keith Ball, June Barrow-Green, Alan Beardon, David D. Ben-Zvi, Vitaly Bergelson, Nicholas Bingham, Béla Bollobás, Henk Bos, Bodil Branner, Martin R. Bridson, John P. Burgess, Kevin Buzzard, Peter J. Cameron, Jean-Luc Chabert, Eugenia Cheng, Clifford C. Cocks, Alain Connes, Leo Corry, Wolfgang Coy, Tony Crilly, Serafina Cuomo, Mihalis Dafermos, Partha Dasgupta, Ingrid Daubechies, Joseph W. Dauben, John W. Dawson Jr., Francois de Gandt, Persi Diaconis, Jordan S. Ellenberg, Lawrence C. Evans, Florence Fasanelli, Anita Burdman Feferman, Solomon Feferman, Charles Fefferman, Della Fenster, José Ferreirós, David Fisher, Terry Gannon, A. Gardiner, Charles C. Gillispie, Oded Goldreich, Catherine Goldstein, Fernando Q. Gouvêa, Timothy Gowers, Andrew Granville, Ivor Grattan-Guinness, Jeremy Gray, Ben Green, Ian Grojnowski, Niccolò Guicciardini, Michael Harris, Ulf Hashagen, Nigel Higson, Andrew Hodges, F. E. A. Johnson, Mark Joshi, Kiran S. Kedlaya, Frank Kelly, Sergiu Klainerman, Jon Kleinberg, Israel Kleiner, Jacek Klinowski, Eberhard Knobloch, János Kollár, T. W. Körner, Michael Krivelevich, Peter D. Lax, Imre Leader, Jean-François Le Gall, W. B. R. Lickorish, Martin W. Liebeck, Jesper Lützen, Des MacHale, Alan L. Mackay, Shahn Majid, Lech Maligranda, David Marker, Jean Mawhin, Barry Mazur, Dusa McDuff, Colin McLarty, Bojan Mohar, Peter M. Neumann, Catherine Nolan, James Norris, Brian Osserman, Richard S. Palais, Marco Panza, Karen Hunger Parshall, Gabriel P. Paternain, Jeanne Peiffer, Carl Pomerance, Helmut Pulte, Bruce Reed, Michael C. Reed, Adrian Rice, Eleanor Robson, Igor Rodnianski, John Roe, Mark Ronan, Edward Sandifer, Tilman Sauer, Norbert Schappacher, Andrzej Schinzel, Erhard Scholz, Reinhard Siegmund-Schultze, Gordon Slade, David J. Spiegelhalter, Jacqueline Stedall, Arild Stubhaug, Madhu Sudan, Terence Tao, Jamie Tappenden, C. H. Taubes, Rüdiger Thiele, Burt Totaro, Lloyd N. Trefethen, Dirk van Dalen, Richard Weber, Dominic Welsh, Avi Wigderson, Herbert Wilf, David Wilkins, B. Yandell, Eric Zaslow, Doron Zeilberger
The title of this post is a nod to Terry Tao’s four mini-polymath discussions, in which IMO questions were solved collaboratively online. As the beginning of what I hope will be a long exercise in gathering data about how humans solve these kinds of problems, I decided to have a go at one of this year’s IMO problems, with the idea of writing down my thoughts as I went along. Because I was doing that (and doing it directly into a LaTeX file rather than using paper and pen), I took quite a long time to solve the problem: it was the first question, and therefore intended to be one of the easier ones, so in a competition one would hope to solve it quickly and move on to the more challenging questions 2 and 3 (particularly 3). You get an average of an hour and a half per…
If you want your kids to grow up with Math talent, start young in Music, be it playing simple drum or flute, later at age 4 or 5 progressing to piano or violin, along the way pick up musical theory…
Notice that great mathematicians (or Physicists the close cousins of Math) are often music talents, but the converse not true! Einstein performed violin with an orchestra formed by a group of Nobel Prize Physicists; never heard Mozart or any great musicians proofed any Math Conjectures.
Coming across You’re Getting Old, it struck me that the numbers generated by the site would be perfect for standard form exercises; put in a student’s date of birth and even the young ones will have some big numbers reported! For example, for a 12 year old, the following figures are generated, some updated while you watch.
1. Galois’s mother home-schooling him Latin & other languages before entering Lycée Louis-Le-Grand.
2. William Hamilton: knew 15 languages include Chinese before discovered Quarternions (1,i,j,k) on Monday 16 Oct 1843 walking along Brougham Bridge, Ireland.
3. Pascal, Descartes are philosopher good in writing.
4. Gauss learnt even at old age Russian to read Lobatschefsky’s Non-Euclidean Geometry
5. Cauchy’s father heeded the advice of his neighbour Laplace to teach young Cauchy language before mathematics.
A Southern Song dynasty (南宋) officer. During his 3-yr leaves when his mother died, he generalised 孙子算经 (4th century)’s “Chinese Remainder Theorem” in ‘大衍求一术’. After leaves, he went back to chase money & women, produced no more Maths.
G. Polya / Paul Halmos advocate getting math students to construct not just one but classes of examples to:
1. Extend & enrich own Example Spaces;
2. Develop full appreciation of concepts, definitions, techniques that they are taught.
[Polya, Halmos, Feynman]: they collect and build a personal ‘repertoire’ of “Examples Space” (include counter-examples) for each abstract math idea, which they can relate to a concrete object.
Examples:
Group abelian = (Z,+)
Ring = Z
Principal Ideal = nZ
Equivalence Relation = mod (n)
Cosets = {3Z, 1+3Z, 2+3Z}
…
These are the top 10 tough Mathematics:
1. Motivic cohomology or cohomology Theory 上同调理论
2. Langlands Functoriality Conjecture
3. Advanced Number Theory (eg. Fermat’s Last Theorem) 高等数论
4. Quantum Group 量子群
5. Infinite Dimensional Banach Space 无穷维度巴拿哈空间
6. Local and Micro-local Analysis of Large Finite Group 大有限群之局部与微局分析
7. Large and Inaccessible Cardinals 大与不可达基数
8. Algebraic Topology 代数拓扑学
9. Super-String Theory 超弦论
10. Langlands Theory 非阿贝尔互反性，自守性表现和模数变化
Do we really live in 10-dimensional Space ? Harvard Prof S.T. Yau (1st Chinese Fields Medalist) talked on the inner space of Geometry and String Theory in Physics:
4) GEP Books are an excellent source of DSA questions, since the scope of GAT testing overlaps with the Logic portion of the GEP test. Check out the myriad of GEP Books that can be used to prepare for DSA questions equally effectively.
The Logic portion of GEP test / DSA test is not taught anywhere in the MOE syllabus, and hence the most challenging to prepare for. Your child would need to solve DSA questions like the one below, which is quite obviously not taught anywhere from Primary 1 to Primary 6. However, like all skills, these kind of logic puzzles can be taught, trained, and practiced, in the Mensa book listed below (Scroll down)!
Children can be trained to solve this type of DSA GAT questions easily
If you are looking for more DSA GAT pattern/logic questions, this is the Complete Quiz Book by Mensa. Highly rated on Amazon. These book will be helpful for those seeking for a boost in their DSA GAT scores, since GAT (General Ability Test) is just a politically correct name for IQ Test.
Furthermore, the IQ of a person is not static, it can be changed. The way to change IQ is via reading books and acquiring more knowledge.
Another good book for DSA/GAT/HAST is Ultimate IQ Tests: 1000 Practice Test Questions to Boost Your Brain Power. This book is like the “Ten Year Series” of GAT DSA tests, it will be a good and trusted book for Singaporeans who are used to studying using the practice “Ten Year Series” method, which has undoubtedly worked for generations of Singaporeans (including myself). The 1000 Practice questions (!!!) (similar to GAT) would definitely go a long way in your DSA preparation.
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Many people think that the infamous Cheryl Birthday puzzle is very difficult. However, to a well trained Math Olympian, the Cheryl Birthday question is actually considered comparatively easy! This shows that IQ of a person can be increased by reading, learning, and practicing the relevant books.
P.S. These kind of books are rarely found in Singapore bookstores, not to mention that most decent Singapore bookstores like Borders/Page One have closed down. I have compiled the most helpful books for DSA Score-Boosting in the above link. Hope it helps!
As Singapore is a very high-tech society, there are many children who are addicted to handphones /computer games and as a result have no motivation to learn. Needless to say, this would result in rather severe consequences in exam results if not corrected early. Even for gifted children, the consequence of computer/cellphone addiction is really harmful, not to mention students who already have a weak academic foundation. Hence, motivational books like those listed here are actually of great importance. Only if a child sees the value of learning, will he be interested and self-motivated in learning. Related book:Cyber Junkie: Escape the Gaming and Internet Trap.
NUS High DSA
If you are looking for information regarding NUS High DSA, please click here.
Finally, all the best and good luck for your DSA test!
The problem with the iPad is that there are too many games! Children (and even adults) will find it hard to resist the games. The Kindle would be better for education, since it is primarily a reading device, and there are many educational books available at low cost or even free.
For example, this course CK-12 Algebra I – Second Edition, Volume 1 Of 2 is totally free and costs $0.00 if you have the Kindle. Hence, the Kindle is a much better alternative to iPad for students.
Hua Luogeng (华罗庚) urged using the daily 10-20 mins intervals while waiting for buses, queues, idle times, make it at least 1 hour a day to read Math books which you carry along with you.
Hua advised on speedy self-learning Math :
1) Choose the Best book on the Topic written by the Master (say, Abstract Algebra), read completely and do the exercises.
2) Read other reference books. Read only those new topics not covered in 1).
If not much new things, return them to bookshelf. This way speed up reading many books in short time.
3) Then read International renown Math Journals.
Beware 90% are copy-cats or rubbish by University lecturers to meet their yearly publishing quota. Only < 10% are masterpieces.
4) Pick one topic to do your independent research.
5) Discuss with friends with better knowledge in the field. This way you can be a Master in…
华罗庚 《数论导引 》序言
Preface on “Introduction to Number Theory” by Hua Luogeng (1950).
“Math evolved from concrete to abstract, the former is the source of inspiration of the latter. One cannot just study the abstract definitions and theorems without going back to the source of concrete examples, which prove valuable applications in Physics and other sciences.”
“Mathematics, in essence, is about the study of Shapes and Numbers. From Shapes give rise to the Geometrical Intuition, from Numbers give the Relationship and Concepts ”
“Only by taking an infinitesimal small unit for observation (the differential of history, that is, the individual tendencies of men) and attaining to the art of integrating them (that is, finding the sum of these infinitesimals) can we hope to arrive at the laws of history.”
You’re getting old! 
Mathtuition88 