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I disagree with the notion that the unit circle approach preceding the triangle approach should be, if it is contextualized historically, "baffling." To this end, I suggest two pieces if you are wondering about how the unit circle gets itself into the pedagogical broaching of a subject that etymologically appears to be the study of triangles/three sided ...


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I find it easier to think about a single number (x or y of unit circle) than a ratio. Also helpful with angles greater than 180 or negative. I learned it this way back in the early 80s...so it's not like some totally new fangled approach. Just because it seems strange to you or you get some agreement on this message board, doesn't validate your opinion....


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I've noticed this trend as well, and it's baffling. The only justification I can see for it is that one of the main topics in precalculus is "functions," so they introduce the sine and cosine functions first. Then they say "Hey, guess what, these apply to triangles." However, this strikes me as being completely backward: they should introduce the sine ...


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"Solving (routine) problems fast" is a useful skill, but very far from what you really need. You have to be proficient in reading, understanding and criticizing proofs, come up with your own proofs. Be able to plan (and execute the plan) to attack ill-defined problems. Use tools, like a computer algebra system, Google or math.SE to ease over routine stuff/...


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I then progressed to the point that I could manage to do 85% of all the questions in a minute. Rarely did I take 1.5-2 minutes (on the really difficult ones). It sounds like you were doing very simple, plug-and-chug problems. For example, evaluate the integral from x = -3 to x = 4 of (4x³+3x²)dx. Problems that require detailed proofs, or examination of ...


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Another example is string theory as a candidate for the theory for quantum gravity. After numerous consistency checks (the so-called "theoretical experiments") it still fails to be validated by the experimental data.


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My answer here to a different question has a link to some "college-level contest" problems that are arranged by topic as you request. The issue is that these contest problems are not really "Algebra I" exactly. So this may be helpful only to future readers, not to you personally, because they are sorted by topic just well enough that you will find that "...


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"Why it is said that Finland has a particularly good education system, but Finland's performance on international mathematics competitions is quite often at relatively intermediate level?" Because the first part is a comparison of population averages and the latter part is a comparison of population extremes. If the distributions vary in shape, than you ...


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My general idea of math results is that it only shows how good one is at that thing, and it's not very easily comparable. For example, you could be good at calculation and go extremely well in tests (that could be checked with a calculator) but be very weak on concepts. Likewise, you could be very good at concepts but not be good at, say, answering multiple ...


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You seem to be seeking research to show there are better options than note-taking. I can't answer the question you posed, but you will need to overcome the evidence that note-taking is effective. For example: Jennifer Gonzalez. "Note-taking: A Research Roundup" September 9, 2018. Link. "Whether it’s taking notes from lectures (Kiewra, 2002) or ...


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Judea Pearl & Dana Mackenzie, in their new book The Book of Why (p.190ff), explain the paradox in a way I hadn't seen before. Pearl imagines changing the rules to "Let's Fake a Deal," where "Monty opens one of the two doors you didn't choose, but his choice is completely at random." Of course he could open the door containing the car/prize, ruining ...


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Also, realize that many authors write "complete" (self-contained) texts, and by their own recommendations, may suggest various ways an instructor may want to approach the subject. For example, an author of a text with 13 chapters, may suggest, in the event that an instructor is focusing on foo, to focus on chapters 1 - 3, chapter 4 (sections 1-3), and ...


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Yes they do so in many places, I think the point is that the students are forced to finish a certain curriculum (depends on the country ofcourse), for example when students finish the $9^{th}$ grade they must know this and this and that. So in the years before the $9^{th}$ grade the teachers focus on what have to be finished and leave the other topics that ...


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My sense is that often authors are encouraged (required?) by publishers to make books very 'complete'. This means the book has all the topics that any instructor would be likely to want for a subject. However, it also means that books are often too big to entirely cover in a semester (or year). Thus an instructor must pick and choose chapters/sections ...


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I tend to see this in upper level college courses a lot. My impression is that the schools are trying to appear to have a solid course by using an iconic textbook (not often the best pedagogically). Often the course is too short to really cover the content properly. I don't like the practice. Prefer to spend more time or more realistically, use a book ...


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