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A Free Stanford Online Course to Develop Your Mathematical Thinking

Most math classes try to teach computation skills and not much more. This is true not only in school where students memorize mechanical procedures and how to plug numbers into formulas, but also in university courses targeting scientists and engineers. Computational fluency is important but it is only a stepping stone to mathematical maturity.

Higher level mathematics requires the ability to prove mathematical statements, which in turn, requires the ability to think logically and create convincing and rigorous arguments. If this sounds more like something taught in law school, it’s because much of math education has been divorced from actual math. We’ve addressed the topic of mathematical thinking before, but now there is an online course that teaches the most foundational concepts in non-computational math.

That course, Introduction to Mathematical Thinking is offered by Stanford University via Coursera and is taught by Keith Devlin, who is a well known and charismatic math popularizer, educator, and researcher. The main purpose of this course is to serve as a transition between computation-fixated school math classes and undergraduate math major courses. In some ways, this is a traditional course that many university math departments require, but as a MOOC it is now accessible to anyone from high school students to math teachers. As Professor Devlin says in the introductory video (below), the course does not teach students new mathematics; instead it teaches them how to think mathematically and work with the standard mathematical language that involves notions like equivalence relations and logical quantifiers.

The course has been offered before with tens of thousands of students and has received excellent reviews. If you have never been exposed to anything beyond plug and chug math, this course is for you. Once you acquaint yourself with the basics of mathematical thinking, you will gain a deeper appreciation of some important topics that are usually left out of the regular school math curriculum.

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.

Photo Credit: faungg

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

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.

How to Start Your Own Math Circle or Enrichment Program

math circle session

Traditionally, math enrichment programs are run by professional mathematicians with an interest in education or by teachers with an interest in math competitions, but for most other people the idea of starting their own program seems like a daunting task. Fortunately, a few years ago, Sam Vandervelde and the Mathematical Sciences Research Institute put together Circle in a Box, a definitive guide on starting your own math enrichment program. It includes almost two hundred pages of advice on everything from the logistics of setting up an enrichment program to a fairly large set of suggested math topics and problems. There is even a section on how to apply for funding. Circle in a Box focuses primarily on setting up a math circle as opposed to any other type of enrichment program. Math circles are informal problem solving and discussion groups that were extremely popular for decades in Eastern Europe and which have played a crucial role in the development of several generations of mathematicians. Unlike school math clubs which usually focus on preparing students for specific math competitions, math circles are more flexible and their aim is to introduce a greater range of mathematical ideas (not simply problem solving tricks) and to explore even nontraditional topics in depth.

In our experience, the approach outlined in the book is similar to the one used by The Math Circle, one of the oldest math circles in the United States and by the Gentle Knowledge Math Circle, one of the first free out of school math enrichment programs. The author is the founder of the Stanford Math Circle and is well-known in the world of math outreach. If you’re a teacher, a parent, or simply a math enthusiast who is interested in starting your own program, this book along with Mathematical Circles (Russian Experience) will be an invaluable guide.

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)

Art of Problem Solving Classes

aops_logoWe prefer to review high quality online courses that are free, but for the Art of Problem Solving (AOPS) classes we need to make an exception. The Internet is flooded with free courses taught by first-rate instructors, but with scale comes an often overlooked problem. A popular free online course that has an enrollment in the thousands cannot provide the kind of student-teacher interaction that is vital for learning. Art of Problem Solving math classes are relatively inexpensive, but because of limited enrollment, students can communicate with their instructors in real-time. Unlike massive open online courses, Art of Problem Solving classes are built on fairly basic technology that does not include audio or video, but this does not take away from the learning experience because students receive individual attention and are required to do in-class work that is immediately available for their instructors to review. The other defining aspect of these classes is the quality of the mathematical content. Unlike various other online resources, AOPS focuses on sometimes difficult yet engaging problems (after all, problem solving is in their name) instead of simple textbook exercises. For advanced or motivated students in grades 5-12 who do not have access to local high quality math instruction and for homeschoolers, these classes are worth looking into.