Blog Archives

An Online Problem Solving Course for Young Kids

logo for young mooc math class

We’ve mentioned the work of James Tanton, Maria Droujkova, and Yelena McManaman before, and now they have teamed up to offer a one month long online math problem solving course. mpsMOOC13: Problem Solving for the Young, the Very Young, and the Young at Heart revolves around a small set of accessible nonstandard math problems that kids and parents solve together. The solutions and discussions are recorded and reported on the course website resulting in a community-generated math education research project.

The course has already started, but you can still do all of the problems and follow the discussions. If you’re homeschooling this course will be especially useful, and you should stay tuned for similar future courses from this team.

NRICH: An Organized Collection of Math Enrichment Problems and Activities


If you liked the expository writing in Plus Magazine, you may enjoy Nrich, a sister project from Cambridge University. The site features hundreds of math problems and activities organized by grade and ability level, as well as by topic. Although Nrich has a section for students, teachers who need to prepare lesson plans may find it more useful. The site content is closely aligned with US and British math curriculum standards, which should make it particularly appealing to educators.

An outstanding feature of the site is it’s emphasis on math enrichment topics that are usually outside a standard school curriculum, yet close enough to it to be relevant in a regular math class that needs to follow strict education guidelines. Another welcome aspect of Nrich is that professional mathematicians, not just math educators oversee the project, making sure that it is both mathematically sound and relevant. There is even a forum for those who need math help. Nrich may not be the easiest site to navigate, but it does contain a convenient topic directory that organizes all of the content. This project is worth exploring and should contain something useful for anyone teaching or learning math.

Plus Magazine: A Collaboration Between Mathematicians and Educators

plus magazine

Unfortunately, too many of the English language math textbooks that students see on a daily basis are written exclusively by professional educators without any serious input from mathematicians. As a result, these books are too much about teaching procedures and not enough about inspiring future mathematicians and scientists. At the other extreme, textbooks (usually at the college level) written by mathematicians tend to be dry and extremely dense. They may present all of the necessary definitions, lemmas, and theorems, but there is not enough room left for applications.

Plus Magazine, a University of Cambridge project is an attempt to correct this situation. It is an online publication that features articles, podcasts, book reviews, and news stories that makes mathematics relevant to those who are don’t grasp its importance and it is a collaboration between full-time educators and full-time researchers and practitioners of mathematics. You can find an article on why the violin is so hard to play and learn about the research of a recent Abel Prize winner (one of the top awards for mathematical research). The site also includes a collection of interesting nonstandard math problems and quite a bit of the content is related to physics. This should be a useful resource both for teachers and high school students.

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

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.

Problem Solving Russian Style

math circles book cover
You’ve read about the lack of proper problem solving in schools and you’ve even started thinking about a few recommended nonstandard elementary problems, now what? You could browse the Web looking for puzzles or math olympiad problems, but a much better approach would be to find a source of problems that is structured by topic, ability level, and that has been tested on tens of thousands of students of all backgrounds. Mathematical Circles: Russian Experience is exactly what you need. Most of the topics in this book, including parity, combinatorics, basic number theory, the pigeon hole principle, proof by induction, invariants, and inequalities often appear in math competitions, but the goal of this book is not narrowly focused on competition preparation. In some ways that would be just as bad as “teaching to the test.” The ultimate aim of Mathematical Circles (Russian Experience) is to start with quite simple problems that anyone can solve and then, in bite-size increments, increase the difficulty of the problems until a whole branch of mathematics has been introduced. The selection of the problems, the detailed guide for teachers, and the depth of coverage makes this book stand out among other great problem solving books. It is geared towards middle school and high school teachers who would like to enrich the standard school curriculum, but even regular students who don’t attend math clubs and competitions would benefit. In fact, we would recommend this book as the best form of standardized test preparation. Anyone who can solve at least a few of the problems in each of the sections of the book is, in our experience, ready to tackle the hardest SAT problem. As is typical for Russian math literature there are a few extra fun topics included, such as strategy games, that one rarely encounters in English-language books. If you’re looking for one book that contains a complete problem solving curriculum that has stood the test of time, this is a good place to start.

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)