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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.

Photo Credit: _Untitled-1

A Possible Mathematical Theory Behind The Coming Cicada Infestation

The eastern United States is about to be overrun by billions of cicadas who will crawl out of the ground and create a deafening commotion. The interesting thing about their emergence is that they only come out every 17 years. Some scientists think that this is a coincidence, but the late Stephen Jay Gould, one of the major figures of evolutionary biology, postulated that the fact that this number is prime might not be an accident. He reasoned that if these periodical cicadas were to come out every, say, 12 years they would coincide with the emergence of predators whose life cycles are 1, 2, 3, 4, 6, and 12 years. Because their life cycle is 17 years, only predators with life cycles of 1 and 17 years coincide with the cicadas and it is easier for them to survive. In other words, periodical cicadas evolved to minimize their exposure to predators. You can learn more about this possible connection between number theory and biology in this Nature article and in a more detailed math paper [PDF] from the Courant Institute at New York University. Even if questions remain about the validity of this particular theory, it is an important reminder that purely mathematical ideas can provide fertile ground for scientific theories in any discipline.

Preschool Computer Science: Kids Programming Adults

drtechniko

If you’re now convinced that preschoolers can learn advanced mathematics, you should not be too surprised to learn that preschoolers can learn computer science equally well. That is the idea behind How to Train Your Robot, a lesson designed to help kids ages 5-7 learn basic programming and computer science. Nikos Michalakis, the man behind the idea has a simple proposition: kids use a starter set of simple commands that consist of primitive symbols to program adults who must obey the instructions exactly as they are written. Once kids are comfortable with the basic commands, they can create their own additional commands, and this is what makes the possibilities endless. For those who are ready to put his ideas to use, Nikos offers a post on how to teach one of his classes based on his own experience. One of the advantages of his approach is that it does not require computers or knowledge of any programming languages and that makes it easy for adults with no programming backgrounds to try it out. If you like this computer-free, running around type of learning, Computer Science Unplugged is a good followup to “How to Train Your Robot” and it introduces more computer science theory to older students.

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.

12 Elementary Math Problems that Capture the Essence of Mathematical Thinking

girl solving problem

One of the most abused terms in mathematics education is problem solving. The term has been hijacked to mean anything from plugging numbers into the quadratic formula to repeating the same steps over and over again when calculating a derivative in calculus class. Neither of these activities could be further from the work of real mathematics, but what kind of problem solving constitutes true mathematical thinking? Alexandre Borovik and Tony Gardiner, both practicing mathematicians, provide a compelling answer in their paper: A Dozen Problems. These twelve problems are accessible even to elementary school students, yet they convey the archetypal paradigms of genuine mathematical thinking. The problems don’t require much mathematical background, certainly nothing beyond the regular school curriculum, but some of them require a good deal of mathematical sophistication. Most of these problems are part of the classical canon of math problems in Russian math literature and have been used in thousands of extracurricular math programs in Russia and the former Soviet Union. This paper is a good starting point if you’re interested in expanding your mathematical horizons beyond the regular school curriculum but are intimidated by difficult olympiad problems that require extensive extracurricular math knowledge.

(Photo credit: Kathy Cassidy)

A Mathematician’s Lament

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There is such breathtaking depth and heartbreaking beauty in this ancient art form. How ironic that people dismiss mathematics as the antithesis of creativity. They are missing out on an art form older than any book, more profound than any poem, and more abstract than any abstract

Much ink has been spilled on the subject of why K-12 mathematics education needs improving, but rarely has anyone made the point in such an eloquent manner as to make it mandatory reading for teachers, parents, and students of mathematics education. This is exactly what former research mathematician and current math teacher Paul Lockhart has done in his impassioned essay “A Mathematician’s Lament” [PDF].

This is not another dry statistics-filled analysis comparing competing education reforms, but a powerful cry for teaching thinking over mindless procedure following. As an added bonus, Lockhart includes two beautiful geometry problems that capture the essence of mathematical reasoning and that you can start playing with right away. Yes, playing — we need more of that in our math classes.

(Photo by Mikey Angels)