I had an idea to improve education in my country with the help of textbooks by popular authors. I thought to publish videos on YouTube with comments and practical explanations about what the author wanted to say. However, the problem is that any textbook is copyrighted and cannot be quoted. This annoys me wildly, these rules just limit the creative ways of teaching the subject, especially mathematics. What do you think about that? Is it wise to put up such barriers?
Under the laws in most countries, a work such as a book is automatically copyrighted as soon as it is written down. Putting the copyright notice in the book isn't what makes it copyrighted. The notice just makes it easier to sue for copyright violation, because it's harder for the person who copied to claim they didn't know the work was copyrighted.
For this reason, it's pretty much true that all textbooks published within the last century are still in copyright. However, many people do write textbooks and intentionally make them free to access. I run a web site, http://theassayer.org/ , that catalogs such books. People refer to these as open educational resources, OERs.
You can probably produce a commentary or work of criticism on a copyrighted work without violating copyright, and likewise for quoting short sections. This varies from country to country, e.g., US fair use is very permissive.
If you want to make a derivative work from an OER, you can do so. However, you should read the license carefully. Some licenses, such as the CC-By-SA license used by wikipedia, require that your derivative work be released under the same license.
Copyright is related to works, not ideas or facts.
To quote the Oxford Learners Dictionaries:
if a person or an organization holds the copyright on a piece of writing, music, etc., they are the only people who have the legal right to publish, broadcast, perform it, etc., and other people must ask their permission to use it or any part of it
The important part is that copyright is always related to "works"; which means anything tangible in some form or fashion. The details on this vary between different legislatures, but copyright never covers ideas, only ever the expression of those. The source of the term is literally the right to copy something. Ideas cannot be copied, hence the category simply does not apply.
As an example: if you look at copyrights on cooking recipes, it is fine to copy and re-use the actual recipe content (i.e., the "idea" - the list of ingredients and the instructions of how to combine them) as long as you do not copy that verbatim. If there is accompanying text, copyright regulations certainly apply to that. The itemized list ("5 apples", "3 eggs") is usually not copyrighted because it's just too trivial - there are only so many ways to represent this. Of course, if you wrap your recipe in stories about the country where the recipe is from, or add history lessons or some kind of story, then those bits are definitely copyright-worthy material.
The same applies to any other field. Nobody can copyright a mathematical proof, a computer algorithm, an instruction of how to build a house, or any other "content". You can only ever copyright the "work" which explains the ideas (in the case of the computer algorithm, that would be the actual program implementing it, for example). While there are plenty of discussions about where the exact borders lie, you can always take the information content and wrap it in your own words / videos / audio representation.
Finally, be sure not to mix this up with patents - patents are basically the opposite and can be used to protect ideas; the requirements to receive a patent are therefore also much more strict. The copyright is owned by default, in many jurisdictions, or with trivial effort (e.g., just by writing it somewhere near your work).
So to wrap it up: you can use all information from the textbooks freely (as long as you create your own material to express it), but you usually can never just copy stuff verbatim. Exceptions exist ("fair use") but I think that's not what you're asking about.