I find that the impressiveness of a Khan Academy video for me is negatively related to how much I know of the subject. As a math graduate student and calculus teacher, I find Khan's math/calculus videos the least impressive of the lot, his physics/chemistry/biology videos mildly impressive, and his history videos the most impressive.
What this suggests to me is that the Khan Academy is lacking in depth and clarity of presentation, as well as in addressing the subtleties and key issues that would be necessary to impress a person with some knowledge of the subject. Watching the Khan Academy is roughly akin to having a smart kid in your class (who is learning the subject along with you) explain to you what he/she has understood of the subject. It is not really comparable to how an expert teacher would convey the material.
This may not be completely a minus, because the lack of polish and the chumminess of the videos might itself be an endearing factor that makes people more comfortable with the videos. It also makes it easier to scale up and make a larger quantity of videos. Also, the low intensity of the videos makes it easy for a person to watch them when tired and distracted without missing out on too much.
Here are some examples of sloppiness:
(i) In the calculus videos, when I viewed them, the graphs were drawn very shakily, extremely hard to understand, and not well labeled.
(ii) In a video on classical mechanics, there were some inaccurate statements about normal force, describing it as a reaction to gravitational force in the action-reaction sense (this was fixed later, I think). These weren't merely careless errors in speaking, but reflected a deeper lack of understanding.
(iii) The examples and symbol choices are often confusing.
If you are recommending watching Khan Academy videos, I suggest you add the caveat that they should not expect a lot more (in terms of accuracy and quality of explanation) than they would expect learning from their colleagues.
Chris Ferrie has a new series out that includes:
- Quantum Physics for Babies
- Newtonian Physics for Babies
- Optical Physics for Babies
- Quantum Entanglement for Babies
Although not specifically concerning mathematical content, they do include in passing some mathematical ontology. The Newtonian book, for instance, does mention force being the product of mass and acceleration and describes acceleration. Not the best answer to my question, but I still am not finding as much material as I know is out there.
These are a bit like Basher books for toddlers. Big letters with colorful minimalistic pictures. Very simple sentences that stress the big new words the pictures are showing. The same pedagogical goal of just making these words feel safe to the young ones so that they feel comfortable "seeing the map" without needing to understand on any deep level what any of the "places" are really about. Early connections and all.
In looking, I've also found work like "ε-Red Riding Hood and the Big Bad Bolzano-Weierstrass Theorem" by Sunshine DuBois and Colin MacDonald, which although is very cute and clever, is not of the level or content that my child can enjoy. I think there is a clear distinction between a simple board-book style story with a simple core that mentions the big words in meaningful but not essential ways versus stories that have the appearance of children's stories but deliver highly mathematical content that the story depends upon for meaning. I know of many examples of the latter, but they clearly don't have the same pedagogical value.
Best Answer
If you are going to study computer science, you want to read Graham, Knuth, Patashnik: Concrete Mathematics.
Look into solving recursion, combinatorics and maybe a little stochastics. Getting the hang of infinite sequences and series is also advisable. Above stated book covers that all because it is made for computer scientists. You should have math courses covering all that, though.