
Tools to make hard problems easier to solve. In this book, Sanjoy Mahajan shows us that the way to master complexity is through insight rather than precision. Precision can overwhelm us with information, whereas insight connects seemingly disparate pieces of information into a simple picture. Unlike computers, humans depend on insight. Based on the author's fifteen years of teaching at MIT, Cambridge University, and Olin College, The Art of Insight in Science and Engineering shows us how to build insight and find understanding, giving readers tools to help them solve any problem in science and engineering. To master complexity, we can organize it or discard it. The Art of Insight in Science and Engineering first teaches the tools for organizing complexity, then distinguishes the two paths for discarding complexity: with and without loss of information. Questions and problems throughout the text help readers master and apply these groups of tools. Armed with this three-part toolchest, and without complicated mathematics, readers can estimate the flight range of birds and planes and the strength of chemical bonds, understand the physics of pianos and xylophones, and explain why skies are blue and sunsets are red. The Art of Insight in Science and Engineering will appear in print and online under a Creative Commons Noncommercial Share Alike license.
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It beautifully treats estimation and problem solving techniques, illustrated by examples from science and engineering. Instead of aiming for a complete, thorough and accurate treatment of problems, its goal is to teach shortcuts to sacrifice some accuracy for much reduced effort. This is a refreshing change to academia where rigor is often pursued at all cost. But in the real world rigor rarely matter, and simplifications are almost always worthwhile, especially initially since we can always refine models if required.
I first read it as an undergraduate and use the estimation and problem solving techniques from it almost daily. Though well hidden, the pdf is available for free on the website of the publisher.
Tim Gowers' Mathematics: A Very Short Introduction is a popular book on doing mathematics. Not a textbook that teaches you mathematics, so wouldn't give this as the only book, but popular "what's the field like" books could be very interesting to a high schooler.
Also, not suitable for the only book, Penrose's The Road to Reality. It gets very advanced and probably can't be fully tackled without additional mathematical education, but it tries to be an honest exposition of the math needed for modern physics from the ground up without explicitly resorting to external knowledge. I would have loved a "this will teach you all of the math if you can get through it" book like this even if I never did manage to get through it.
For stuff like the bad social science you mentioned, often you can catch it if you know basic statistics along with the common tricks people use to p-hack. See, for example, "The garden of forking paths" by Gelman.
Even if the statistical analysis is not given explicitly, or the issue is more subtle, for social science that touches on everyday life, you can usually tell if it'll hold up just with common sense. Try yourself using this quiz: https://80000hours.org/psychology-replication-quiz/