For young mathematicians who are looking for interesting and challenging mathematical topics to supplement the usual school curriculum, there are a variety of options available, many of them free of charge. Here’s a short list of recommended resources.
YouTube channel – Numberphile
Brady Haran’s popular channel has been running since 2011. It covers all manner of interesting mathematical topics, in entertaining 10-15 minute videos, at a level accessible to a clever 8th-10th grader. The videos sometimes feature interviews with famous mathematicians. Calculations are done with marker pen on brown wrapping paper. The channel is very good as a launching point for more detailed investigations. Be sure to check out some of the spinoff channels – e.g., Matt Parker’s StandUpMaths and James Grimes’s SingingBanana.
YouTube channel – 3Blue1Brown
Grant Sanderson’s channel is well known for its clear presentation, smooth delivery, and standout graphics. It tackles subjects at a deeper level than Numberphile- but the superb visualizations help to make those subjects accessible.
YouTube channel – Mathologer
Burkard Polster’s channel covers a variety of interesting topics. Polster has a PhD in mathematics, and covers math at a deeper level than Numberphile or 3Blue1Brown. It’s aimed at an older high school or early college student level. He sometimes leaves questions partially unsolved as exercises for the viewer.
YouTube channel – Insights into Mathematics
Canadian Professor Norman Wildberger teaches mathematics at UNSW in Sydney, Australia. Most of his videos are full length 30-50 minute lectures, either recorded in the classroom or presented on a whiteboard. He excels at presenting complicated topics clearly, and showing links to the history and philosophy of mathematics. Most topics are presented at an early college undergraduate level. As a Finitist, Prof. Wildberger brings some very unconventional yet thought provoking ideas to mathematics.
YouTube channel – Michael Penn Math
Micheal Penn is a Professor of Mathematics at Randolph College, a small liberal arts institution. He delivers lectures on mathematics at the advanced high school or college level, as well as tackling problems from various mathematics competitions. His videos are characterized by neat blackboard work and clear explanations.
YouTube channel – MindYourDecisions
Presh Talwalkar delivers mathematical puzzles and riddles – they make good activities for a rainy mathematical day.
Wikipedia
Although Wikipedia is not a textbook, it is a reliable, accessible reference for all areas of mathematics. Over the last ten years, the quality of the mathematical content has improved tremendously. There are more examples to illustrate complex concepts, and all content is well linked to related ideas and footnoted with references. Wikipedia is a great place to start investigation of a new topic.
Wolfram Math World
Steve Wolfram is a theoretical physicist and creator of Mathematica, the mathematical computer language. The online version Wolfram Alpha is no doubt known to every math student who ever struggled with a homework problem. Wolfram Math World is a very comprehensive but somewhat terse mathematical reference. It’s a resource for refreshing your memory or finding a relation but perhaps a not ideal for learning a topic for the first time.
Cut the Knot
Alexander Bogomolny was a mathematician who started Cut The Knot in 1996 and continued developing it up until the day before he passed in 2016. The site contains a wealth of very interesting topics, presented in a friendly narrative style, and full of personal insights. Highly recommended.
The Art of Problem Solving
In addition to providing very high quality online instruction for talented students of mathematics(for a fee), The Art of Problem Solving has an active online forum where you ask and get answers to mathematical questions.
Math Stack Exchange
Math Stack Exchange is an internet rarity – a forum where people ask intelligent mathematical questions and get thoughtful, robust mathematical answers. If you’re stuck on a difficult problem, there’s a good chance that someone else has had the same problem and posted it to the Stack Exchange forum. It covers topics from high school through graduate school math.
Martin Gardner
No discussion of amateur or recreational mathematics would be complete without the mention of Martin Gardner (1914-2010). As the editor of the Mathematical Games column in Scientific American from 1956 until 1986, he inspired a generation of young mathematicians. He introduced new areas of mathematics to the general public in an accessible, engaging style, often looking at the connections between math and games, magic, and puzzles. He wrote wrote a number entertaining math books, and his old Scientific American columns can be found at the public library or can be purchased online.
Khan Academy
Rather than being a source for novel mathematics, Khan Academy is primarily a resource for learning the standard curriculum in an effective and sometimes novel way. The range of material covered is extensive, from Kindergarten to Calculus, and the presentation includes text, video instruction, and exercises.
If you have a resource you’d recommend that’s not listed here, please let us know!