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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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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.

Lively Chemistry Crash Course for Those in a Hurry

Although chemical experiments can yield exciting results, the theoretical part of chemistry may appear overly dry to students who are not already interested in it. Those who are studying the subject and need to review it may be overwhelmed by the sheer volume of details that they need to memorize. In both of these cases, it is helpful to have a highly condensed and lively summary of the key concepts. That is exactly what Hank Green accomplishes in his video series, Chemistry Crash Course. The videos in this series are short and entertaining, but they still highlight fundamental concepts. You can think of the collection as an extended trailer for the much broader and deeper subject or as a fun way to review for a chemistry test. The videos do not replace a textbook or a good teacher but they pack enough content into a few minutes that we recommend pausing them to process all of the information. If you have encountered any chemistry at all, these videos will be a bit more useful than if you have never heard about the existence of atoms.

The Theoretical Minimum: An Introduction to Modern Physics for the Curious Amateur

photo of Susskind

After Walter Lewin wows you with his theater of physics and you become intrigued by the possibility of parallel universes, you may be interested in some of the details behind modern physics. Unfortunately, at that point, you will most likely run into a serious roadblock. Contemporary theoretical physics is steeped in advanced theoretical mathematics, and most textbooks are geared towards future researchers, not intellectually curious individuals with limited backgrounds in either subject.

Luckily, Leonard Susskind, a Stanford Physicist and one of the fathers of string theory, comes to the rescue with The Theoretical Minimum, his unique series of courses on modern physics. The outstanding feature of Susskind’s lectures is that they do not shy away from mathematical derivations; the concepts are introduced in a completely rigorous way, yet they are made accessible to people who have never studied much math or science beyond advanced high school courses. In effect, these lectures offer both a physics and mathematics education for the price of one (figuratively speaking — the courses are free). Susskind develops the material from first principles and introduces all of the math that the physics requires. His target audience is adult continuous learners who want more detail than can be found in popular lectures, but bright high school students will benefit from seeing what life as a physics major entails. It doesn’t look too scary at all.

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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Curriculum Notes: Teaching Logarithms and Their History

John Napier

Most textbooks present logarithms as just one more set of mechanical procedures to be memorized and repeatedly applied. Their history of how and why they were invented, however, is rarely presented. In addition, all too often the common sense proofs of the basic properties of logarithms are not emphasized at all. James Tanton, a research mathematician-turned-teacher, presents a quick, rigorous, yet interesting way of teaching logarithms in his take on logarithms essay [PDF]. Like many of his essays, this one contains links to videos on the subject for those who prefer to watch rather than read. If you’re a teacher this will make your lesson preparation easier.