Here is a lovely, simple explanation of how limits and infinitesimals work (and when they don’t).
Well worth a read, because it isn’t afraid to tell the whole story in non-technical language.
Here‘s maths teacher Dan Meyer talking about how to get people buying into maths problems by asking a simple question and removing all the spoon-feeding steps (including plugging numbers into formulae).
This man is super-enthusiastic, as well as a very clear thinker.
I’ve used the full version of Mathematica and it’s a tour de force. It’s also massively more sophisticated than most people need (as well as hugely overpriced).
So try this cut-down online version.
Try typing in Plot sin (x^2) / e^x or something…
There are very few books which explain things with the degree of clarity I seem to need. Even the very best authors sometimes use imprecise English and expect the reader to fill in the gaps using ‘context.’ I seem not to do context. 😉
I’m providing a list of publications below which are worth reading. If you want to really understand the subject matter, I suggest working through at least two on each subject. Some of this stuff is University level, so expect to come out with some new questions of your own!
Melanie Mitchell –Complexity
John Gribbin –Chaos, Complexity…
Peter Atkins –Galileo’s Finger
Trigonometry gets confusing at angles bigger than a right angle because you can’t draw a little triangle and say ‘opposite over hypotenuse’ or whatever.
Rather than adopt some kind of handle-turning approach (eg C.A.S.T.), that may get you the correct result but which doesn’t add to understanding, try thinking in terms of the graphs of the trig. functions eg Sin x. This allows you to use the symmetry in these plots to see why you get the answers (rather than just remembering some opaque formula).
Sometimes, if you aren’t getting a new concept, it’s really useful to hear someone else’s version.
Here is a video about Khan Academy. This represents the future of education and is also a great place to get an additional perspective on a wide variety of interesting topics.
In case you were in any doubt, the scientific literature is not error free. Text books are still frequently written which contain misconceptions (as distinct from poor explanations).
This site offers a pretty strident criticism of many such publications, together with a clearer interpretation of much of the underlying thinking.