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Richard Feynman’s Great Physics Lectures are Finally Available Online


Sometimes it takes decades for works that have had an immense impact to become easily and cheaply available, and this is especially true in the sciences. The difficulty of making a science or math text, with all of its special symbols and diagrams, browser-friendly poses a serious technical challenge. Since 1964, the “Feynman Lectures on Physics” have been a physics education classic, but now, the first volume is finally available online.

Richard Feynman is known not only as one of the great physicists of the twentieth century, but also as one of its great communicators who could explain the most complicated topic with rare clarity. His physics lectures, however, have traditionally been an expensive financial investment and used mainly at the undergraduate level. Now, ambitious high school students can try them out for free. We’ll be looking forward to the rest of the volumes becoming available in the near future.

The Stupidity of the “Math Wars”

we love math tshirt

Unless you’re a professor of education, you may not have noticed that for the last several decades a war has been raging between math education reformers and those from the traditional camp. Their disagreement boils down to one simple question: should students learn traditional computational algorithms (like the division algorithm for two numbers)? Reformers believe that achieving computational fluency is secondary to developing thinking, while traditionalists argue that not only are the basic arithmetic algorithms a fundamental part of mathematics, but that their mastery enables students to progress in the subject.

The historical roots of this conflict are irrelevant to us because like many social policy debates, it is heavily politicized. A more pertinent issue is how does all of this affect parents, teachers, and students. In a recent opinion piece in the New York Times, Alice Crary and W. Stephen Wilson offer an interesting analysis of the false dichotomy created by this ideological battle and come to an obvious conclusion: facility with mechanical mathematical procedures is inseparable from mathematical thinking. A student can’t develop abstract reasoning abilities without being able to easily manipulate the numbers and symbols that represent that reasoning.

Crary and Wilson point out that studying mathematics while de-emphasizing computation is like studying history without the actual historical facts. Even if historians have developed a way of thinking about their subject, they cannot do so without reference to specific facts. Similarly, both professional mathematicians and students of mathematics need to be comfortable with a certain set of basic computation techniques and a body of fundamental facts, before they can either prove new theorems or learn new material.

If you’re a parent or teacher, you need to give your kids an opportunity to practice algebraic and arithmetic computation and at the same time, let them work on challenging non-standard problems that develop their thinking. For a thorough treatment of the computational side of K-8 mathematics we suggest starting with James Milgram’s The Mathematics Pre-Service Teachers Need to Know, and if you need a textbook right away, Singapore Math is a solid choice. For developing problem solving skills and mathematical maturity, we’ve already mentioned a few excellent places to start. The most important takeaway from the “math wars”, is to stay away from standard US math textbooks and to steer clear of anything that has the word reform in it or looks like it was written without any input from well-known professional mathematicians.

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A Book on Modern Mathematics for Elementary School Students

modern math for elementary school

Sadly, professional mathematicians play a mostly decorative role in shaping mathematics education. Research is simply a much more attractive activity than the politics of education reform and curriculum development. There are not enough incentives to lure most mathematicians away from their academic responsibilities and to push them into improving the quality of mathematics instruction, unless of course, those mathematicians are parents concerned with the quality of their children’s education. That is the story of Oleg Gleizer, a mathematician and parent who could not find a suitable mathematics program for his five year old son and decided to take matters into his own hands.

The result of his effort is the book Modern Math for Elementary Schoolers [PDF], which bridges the gap between the requirements of school mathematics and a deeper conceptual understanding of the subject. This is not a replacement for a good textbook because it does not cover all of the standard topics, but it is a vital supplement that opens the doors of high level mathematical thinking to elementary school students. For example, the first chapter introduces number partitions, parity, and other basic properties of numbers using Young diagrams, which are important objects in advanced mathematics. This approach actually makes the topic more visual and easier to understand even though advanced ideas lurk in the background. Other topics that are deeply yet playfully explored in the book include straight line geometry (and its connection to physics), straight edge and compass constructions, modular arithmetic, and algorithms.

In effect, Modern Math for Elementary Schoolers [PDF] is a lively guide and collection of problems for parents and teachers who want to weave a non-superficial mathematics, computer science, and physics narrative into their teaching. Contrary to the title of the book, a significant part of the material in the book will be relevant to students of any age. If you’re looking for something similar to Math from Three to Seven, this book fits the bill perfectly.

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Moebius Noodles: A Mathematical Playground for Young and Old

Moebius Noodles book

Contrary to popular belief, mathematics is not an activity that requires textbooks, calculators, and years of training. Because it consists of such fundamental notions as symmetry, classification, counting, and geometric transformations — all concepts that come naturally to even the youngest children — mathematics can truly be studied at any age. If you have picked up a copy of Math From Three to Seven and are wondering whether there is something similar for kids that are younger still, you should take a look at Moebius Noodles.

This book, the work of Yelena McManaman, Maria Droujkova, and Ever Salazar, is a beautifully illustrated collection of activities that engages young kids (even toddlers) in discovering fundamental mathematical principles and abstractions. For example, why wait until middle school or high school to learn about functions when you can think about them in any almost any context? For instance, Moebius Noodles proposes an activity where a child is given the name of a baby animal (like “kitten”) and must identify the corresponding adult animal name (in this case “cat”). The child has just created a baby-to-mother function and there are endless other possible activities that reinforce this idea of mappings between sets. The book covers basic ideas involving numbers, symmetry, functions, and even a little bit of calculus. If you’re a parent or preschool teacher interested in fun activities that involve both playing with and internalizing fundamental mathematical concepts, then Moebius Noodles is worth your time.

Quadratics Done Simply and Properly


A common middle school algebra topic like quadratics is all too often spread across hundreds of confusing textbook pages and several years of tedious instruction. It’s as if the subject is a mix of rocket science and neurosurgery. In reality, if you set aside some of the more beautiful geometric examples, the basic algebra of quadratic equations should not pose much of a problem for a typical math student. James Tanton confirms this with his Guide to Everything Quadratic [PDF], a short 64 page booklet on the fundamentals of quadratic equations. The guide presents the algebra of quadratics in an intuitive and mathematically sound way with plenty of examples. In addition to the algebra section, there is a section on graphing quadratics, and a section on fitting quadratics to data. As an added bonus, Tanton demonstrates a quick way to graph parabolas and includes a set of exercises that walk the reader through the derivation of the cubic formula, something that almost never appears in the standard school curriculum. If you want a more geometric perspective we recommend the book Lines and Curves. For those approaching quadratics for the first time, Tanton’s guide can replace or at least supplement most of the coverage of quadratics in a standard algebra textbook.

A Comprehensive Guide To Teaching K-8 Mathematics

K-8 math terms

One of the effects of a highly decentralized education system in the US is the lack of a single guide to teaching any single subject. In mathematics, especially at the K-8 level, this has been an acute problem with no easy solution. Teachers have to do their own research, rely on the opinion of colleagues, and hope that their Web surfing or professional development classes lead them to good materials and guides. Unfortunately, even if they find useful bits of content scattered in online forums, websites, or books, how to bring all of it together into one cohesive mathematical narrative remains a mystery. Standard school textbooks, because of their low quality, are unfortunately not useful.

To address this problem, James Milgram, a Stanford mathematician and one of the top math education experts in the country, put together The Mathematics Pre-Service Teachers Need to Know [PDF], a 564 page guide to teaching K-8 mathematics. A few key facts about this monumental work stand out. First of all, unlike many good (but less comprehensive) mathematics books, Milgram’s work does not introduce some radical curriculum intended only for elite Chinese and Russian students toiling away in some underground olympiad training camps. The book was funded by the Department of Education and deals primarily with core parts of the K-8 math curriculum. Secondly, because James Milgram, and many of the people who contributed to the book, are serious research mathematicians and not simply educators chasing the latest education fad, the content in the book is grounded in solid mathematics. Thirdly, Milgram includes a large amount of material borrowed from foreign textbooks (from Russia and Singapore) to illustrate the best practices that have been proven effective in teaching various topics.

The Mathematics Pre-Service Teachers Need to Know [PDF] corrects one of the main flaws of the standard mathematics curriculum — that it is a mile wide and an inch deep — by providing in-depth coverage of all of the core topics and not introducing extraneous concepts that cannot be fully and rigorously developed. At the same time, the book does venture into a few extracurricular areas which are important for developing mathematical maturity. While it can certainly be a definitive guide to K-8 mathematics, Milgram’s work is not a textbook, but a teaching guide. Teachers will find a myriad of pedagogical tips, exercises, and problems, but they will still need to do some work in finding additional challenges for their students. These 12 problems are a good place to start.

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Computer Science Unplugged: A Computational Thinking Curriculum Without the Computer

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The chorus calling for teaching computer science to all children seems to be getting louder by the day. Even the White House seems to think that programming is the new literacy. Programming is clearly an important skill, but the rush to teach programming languages and popular web technologies seems to have eclipsed a much more fundamental aspect of computer science: computational thinking. Billions of lines of code may run today’s infrastructure, helping land airliners and processing billions of dollars in commerce, but behind that code are algorithms and deep mathematical ideas. Unfortunately, most of the theory of computer science is left to either AP or college-level courses, which is too late. Computer Science Unplugged, a free computer science curriculum that features activities, games, and problems, seeks to address that problem. The curriculum comes with a free book that contains engaging activities, some of which are kinesthetic, but which cover topics like binary numbers, information theory, and searching algorithms. Computer Science Unplugged is appropriate for children as young as seven and is a good way to incorporate computer science concepts into regular math classes or enrichment programs. In some ways, the best part of the curriculum is that it does not require a computer and lets students move around.

The Fascinating World of Preschool Mathematics Education and Enrichment

math enrichment and math circles for preschoolers

Teaching math to young kids who don’t know how to read, write, or count is a complicated task. Providing these kids with mathematical enrichment seems like an even more daunting task. Unfortunately, the vast majority of math materials for young kids involve colorful pictures, games, and activities without real mathematical substance. Sure, knowing the names of shapes is important and receiving prizes for this knowledge is fun, but it doesn’t require too much thinking. A more sophisticated but still age appropriate activity would require giving a child three pencils and asking her to place them on a table so that none of the erasers touch the table (the pencils cannot be made to stand vertically). Solving this problem requires the application of three dimensional spatial reasoning, an important long-term skill.

This type of activity has been the cornerstone of elite Eastern European preschool math programs, and until recently was not widely available in the English-speaking world. The recent translation and publication of Alexander Zvonkin’s unique book, Math from Three to Seven: The Story of a Mathematical Circle for Preschoolers, changes that. This memoir gives an in-depth view of a two year math circle that Zvonkin, a professional research mathematician, ran for a group of kids ages three to seven. It meticulously describes every session and reveals a world of problems and activities far beyond the confines of the regular preschool curriculum. Perhaps as valuable as the mathematical content of the book, are the observations and insights that Zvonkin shares with the reader. Anyone interested in math education, not just at the preschool level, will learn a great deal from this one-of-a-kind work. Once you read this, you will be prepared to start your own enrichment program.

Street-Fighting Mathematics: Inexact Reasoning Leading to Deeper Understanding

All too often school teaches us to “guess and check” when a simple exact calculation would lead to the right answer. Guessing the answer to a one variable equation may not further our mathematical knowledge, but is it possible that guessing can lead to deep insights? According to Sanjoy Mahajan, physicist and author of Street-Fighting Mathematics: The Art of Educated Guessing and Opportunistic Problem Solving [PDF], the answer is yes. He makes the case that by using certain basic problem solving strategies one can avoid rigorous and complicated calculations while the result will be the same. Moreover, these strategies and the solutions that they yield lead to a deep understanding of the subject matter. The book is full of examples from mathematics, engineering, and physics and although some parts require knowledge of calculus, it should be accessible to motivated high school students. As a bonus, it is freely available from MIT Press. Here is a TEDx talk that the author gave illustrating the street-fighting techniques in his book.

Trigonometry the Nonboring Way

trigonometric delights
Few subjects have such a reputation for being boring as trigonometry. Most students who study it miss both the historical context and the interesting applications because most textbooks are too dry and skip the story that gives the subject meaning. Eli Maor’s book, Trigonometric Delights, addresses these shortcomings by a offering a historical development of trigonometry that will be eye-opening even to professional mathematicians. It’s not a textbook or a tutorial but an in-depth guide to some of the most important and beautiful theorems and applications of the subject. Having past exposure to trigonometry helps as Trigonometric Delights is not popular mathematics literature intended for casual browsing. The book is ideal for teachers who want to spice up their trigonometry classes with exciting material, and as an added bonus it is completely free online.