The Power of Mathematical Visualization by James S. Tanton (The Great Courses)
We are loving our Great Courses Plus subscription. We leapt right in with The Power of Mathematical Visualization, and I recommend it enthusiastically.
The great physicist, Richard Feynman, used visualization extensively to understand problems. Temple Grandin, author of Thinking in Pictures, suggests we may be too hasty in trying to prevent and cure autism, since mild forms ot Autism Spectrum Disorder, at least, lead people to different and possibly important ways of thinking. But what about the rest of us, who are neither geniuses nor autistic (nor autistic geniuses)? Visualization is still a powerful and fascinating way to think about math, from basic arithmetic to esoteric and high-powered concepts.
Back when The Great Courses was called The Teaching Company, the course format was primarily a talking-head lecture, perfectly suited for audio-only listening in the car while commuting to work. Many of the courses still work well for that, but the company has grown and expanded considerably, not just in content but also in format. The Power of Mathematical Visualization includes plenty of diagrams, images, and physical demonstrations.
James Tanton is an engaging teacher. He does spent a bit too much time complaining about the unhelpful ways he was taught math in school, but I soon learned to forgive and ignore that. His enthusiasm for math is infectious. You can watch the introductory lecture on YouTube, if you want to check it out.
Here's a picture of the course contents. (Click for a larger image.)
I'd recommend this course to anyone, including all our grandchildren. Certainly the oldest three could get a lot out of most of the lectures. The six-year-olds, being especially interested in math, are also good candidates. Even the younger ones would benefit from at least being in the same room while others are watching. You never know what they are absorbing from their surroundings.
James Tanton has another series on The Great Courses Plus, called Geometry: An Interactive Journey to Mastery. Would I ever have picked that one out of their long list of topics that are at first glance much more interesting? Not likely. But we're watching it now, because Tanton is such a good presenter, and so far it's as intriguing as Visualization.
I took a course on Argumentative Fallacies years ago when the company was called The Teaching Company. It was very good. However, I no longer have any disposable income for purchasing courses. :(
It may still be out of reach for you, Diane, but the streaming (Great Courses Plus) is a great deal, especially if you're used to buying the physical courses, which even at 70% off (the only way to buy) can seem daunting.
For streaming, the regular price is $180/year, far cheaper than cable TV (and as far as I'm concerned, much more worthwhile). Last year, I got a Black Friday deal that gave me $20 off and a free Roku device that enables us to watch on our TV in the comfort of our family room, instead of sitting around the computer. But we still can watch on our computers, and our phones, if we want to.
Best of all, you can share the Great Courses Plus with family and friends—up to 10 devices. Thus, out of one Black Friday deal I got access to the GCP, plus terrific Christmas presents for Porter and all our kids and grandkids. Unbeatable.
One thing I find particularly interesting about this course is my conviction that much of what it shows has probably made its way into the schools at one time or another as "new math" (as far back as my own school days) or reviled implementations of the Common Core. As such, some people might find it discouraging. But there is a universe of difference between offering a fun alternative way of understanding math concepts, and the unhelpful drudgery that results when educational theorists decide that they've now found the one best way to teach math, and everyone had better learn it that way.
One of my Oswego colleagues was known for telling a student who was stuck on a problem "Visualize it!" Sometimes that was enough.