Monthly Archives: April 2013

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.

Photo Credit: Enokson

A Quick Look at the Biochemistry and History of Modern Frozen Food

Most of us know that freezing food prevents it from spoiling, but the fact that quick freezing is better than slow freezing is a more subtle point that not everyone may know. As always, Henry Reich delivers a to the point video that addresses this issue and illustrates the science behind modern frozen food. As usual in such videos, some details need to be skipped, but there are enough scientific nuggets here (if you pause it) for further exploration. Even if you don’t take the time to learn about the Arrhenius equation, you will be much more appreciative of modern refrigeration.

Instructables: A Place To Learn Do-It-Yourself Engineering


An oft-repeated maxim is that learning by doing is the best type of learning. As easy as it is to believe, it seems that humanity, at least the part living in the developed world, is quickly moving in the direction of complete dependance on complicated gadgets designed by high tech companies that recruit brilliant scientists. Even if one wanted to build something, how can it be possible to compete with the world of highly advanced industry? As it turns out, the point is not to compete but to learn how to hack — hacking, not in the sense of breaking into computer systems, but in the sense of making things. That is the idea behind, Instructables, a site devoted to collecting community-generated do-it-yourself projects. Whether you want to make complex origami sculptures or your own boat, there are numerous visual guides and detailed instructions to help you out. Each project, is effectively a hands-on lesson and can be a starting point for further exploration of physics, math, and engineering principles.

Preschool Computer Science: Kids Programming Adults


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.

How Much Money Is There on Earth? An Entertaining Introduction to the Monetary System

How much money exists on Earth may sound like a silly question, but answering it leads to several important concepts underlying the modern economy. Michael Stevens, in his usual engaging style, answers the question and turns a normally dry subject into an exciting one. His video discusses the creation of money, how it acquires it’s value, and even touches upon some esoteric topics like the dirtiness of physical currency. If you’re an expert on the topic, there may not be many revelations here, but if you’re new to the subject of economics, this is a fun place to start.

One Hour With Richard Feynman: Imagining How Nature Works

Once in a rare while, a genius unlocks a secret of nature, moves humanity forward, and secures a prominent place in the annals of science. Sometimes, more rarely, that same person, also conveys the excitement of discovery and the most complex phenomena in the simplest most beautiful language. Richard Feynman, one of the greatest physicists of the twentieth century, is that extremely rare person. In this one hour documentary he talks about the wondrous ways of nature with the air of someone who is both the keeper of its secrets and who is at the same time as fascinated by it as a child. Feynman talks about the jiggling of atoms to explain heat, surface tension of water, and how fire work. He discusses magnetism and reveals why any “why” question can lead to an infinite rabbit hole of explanations. This video should be mandatory viewing for anyone studying science and should be a powerful reminder about the power of imagination, not just the power of theory.

Google Science Fair 2013: Pre-College Research Recognized

Google Science Fair

Winning science competitions is clearly not a requirement for future successful scientific careers, and the winners of these competitions are not necessarily geniuses, although they are definitely bright. That said, an event like the Google Science Fair can be a good motivator for students. This annual event lets students from all over the world submit their research projects online, which are then evaluated by a panel of experts that includes Nobel laureates. The winners receive various awards and there are several age categories with their own prizes, but of course the future opportunities that come with winning are the greatest reward. Perhaps, the most important lesson from such a competition is that learning should not be limited by school textbooks and that sufficiently motivated students can go far if they follow their intellectual curiosity. If you have a science fair project ready you still have a week to submit it. Even if you don’t take part this year, following the work of the winners is an educational experience in itself. Good luck!

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.

For the Love of Physics: Science as a Performance Art

Inspiring future scientists takes great teachers who are often talented performers. The beauty of physics as an academic subject is that it lends itself well to awe-inspiring demonstrations and performances. Walter Lewin, an MIT physics professor and legendary lecturer, is one of those talented teachers and performers who squeezes out of physics every drop of excitement that can be conveyed to a lay audience. In this lecture, filled with some of his most famous demonstrations, he puts his life on the line to illustrate the principles of classical mechanics, explains why the sky is blue while clouds are white, and leaves the lecture hall on a rocket. The lecture does not require mathematical sophistication, which makes it accessible to middle school students, but it will inspire anyone to pursue physics.

Computer Science Unplugged: A Computational Thinking Curriculum Without the Computer

cs unplugged image

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.