Reddit reviews Infinite Powers: How Calculus Reveals the Secrets of the Universe

I think @omeow gives a good answer. Not less calculus as Calculus is the bedrock of so many different areas of maths and science. If you want a good book on this Steve Strogatz's lastes "infinite powers" is awesome: https://www.amazon.co.uk/Infinite-Powers-Calculus-Reveals-Universe/dp/1328879984

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That said, statistics is becoming increasingly important. We need to train everyone, not just Maths grads in more stats. I think if you want to guaruntee a job coming out of an undergraduate degree then Stats is a pretty good bet. Also if you're looking for a primer on stats then David Spiegelhalter's book "The art of statistics" iss a great one: https://www.amazon.co.uk/Art-Statistics-Learning-Pelican-Books/dp/0241398630/ref=sr_1_1?keywords=the+art+of+statistics&qid=1569697929&s=books&sr=1-1

If you're interested in Calculus, Steven Strogatz just wrote a book called Infinite Powers which goes through the historical development of the concepts starting from the Greeks. He gives detailed explanations of methods used by Archimedes to "prove" concepts, or at least to gain intuition about certain questions before formalizing them, and then walks through their development over the years. It's written to be accessible for non-mathematicians but it certainly not one of those pop-science books that just deals with over-simplified explanations of very high-level ideas.

I think your students will be lucky to have a teacher who is so excited to teach them! There's a lot to say here, so I'll just add a couple of points to the discussion.

TIP 1: A BOOK

Steven Strogatz (Cornell math professor and renowned mathematical expositor) has recently come out with a new book called Infinite Powers: How Calculus Reveals the Secrets of the Universe. I haven't read it yet, but based on what I know about it, I suspect you'd find a ton of inspiration from this.

TIP 2: A CONCEPTUAL FRAMEWORK

General framework

Students can easily be overwhelmed by the technical aspects of the subject, but everything we do in calculus can be contextualized via a simple (but brilliant) framework.

(This is something I emphasize in my tutoring, but you might find it helpful when planning classroom lessons as well. One option might be to open the course with a brief overview of calculus based around this framework, perhaps in the first class. Then, each time a new idea is introduced, it can be placed within the framework that you established at the outset.)

1. Approximations (approximate difficult nonlinear problems by easy linear ones)
   
2. Limits (refine your approximations until infinity turns them into exact values)
   
3. Shortcuts (develop systematic shortcuts for calculating important limits)
   
   Approximations, and something close to the idea of a limit, were put to use in ancient Greece (see the work of Archimedes). Thousands of years passed before the third stage was developed and calculus came to fruition - for that, we needed the analytic geometry of Descartes and Fermat.
   
   Altogether, this framework enables us to turn difficult problems about changing quantities into easy problems about geometric quantities. Let's see how this plays out in the two main branches of the subject.
   
   Differential Calculus
   
   The central problem is to find the rate at which a given quantity is changing (with endless applications). We can reframe this as a question about slope. How can we find the slope of a nonlinear curve? For example, how could we find the slope of the parabola y=x\^2 at (3, 9)? This is not obvious at all, but calculus makes it easy, as follows.
   
   
4. We only know how to find the slopes of lines, so let's draw a line that appears to have the same slope as the parabola (the tangent line). Can we find its slope? We'd need two points, but the only point on the line that we know for sure is (3, 9). It seems we're stuck, but we won't give up! Instead, we'll approximate by a secant line.
   
5. We can improve our approximations and watch to see which value they approach... They're approaching 6. We call this the limit, and it must be the answer!
   
6. That was a lot of work. Can we find a shortcut? Whether we use (3, 9) or (4, 16), the process should be the same. Instead of repeating it every time we use a different point, is there a way we could represent multiple values at the same time? Algebra to the rescue. We can use (x, x\^2) as a placeholder. After a little algebra, we get that the slope is 2x. So, what's the slope at (4, 16)? This problem is now as easy as multiplying by 2: 2*4 = 8. From here, we can do something similar for other basic functions (power functions, exponential functions and logarithms, trig. functions and inverse trig. functions) as well as combinations of those functions (sums, products, compositions), and then we'll have shortcuts for all the functions of precalculus.
   
   Integral Calculus
   
   The central problem is to find the accumulated change in a continuously changing quantity. We can reframe this as a question about area! (This can be motivated by considering speed vs. distance.) How can we find the area of a curved (nonlinear) shape? For example, how could we find the area underneath the parabola y=x\^2 between x=0 and x=3?
   
   
7. Apart from the circle (whose area was determined by methods similar to the methods of calculus), we only know how to find the areas of shapes whose sides are straight line segments (like triangles, rectangles...). When it comes to finding areas, the simplest of these shapes is the rectangle, so let's approximate using rectangles.
   
8. We can improve our approximations and watch to see which value they approach... They're approaching 9. We call this the limit, and it must be the answer!
   
9. That was a lot of work. Can we find a shortcut? Here, the fundamental theorem of calculus is the shortcut we're looking for. We can apply it once we build up an inventory of antiderivative formulas for important functions.
   
   Applications
   
   As an example, consider solids of revolution.
   
   
10. Choose an approximating element (e.g. a disk or a shell). Approximate by summing the volumes of these elements.
    
11. Take the limit, so the sum becomes an integral.
    
12. Evaluate the integral using a shortcut (the fundamental theorem of calculus).
    
    That's the idea. I hope it helps!
    
    Edit: Included extra language to clarify the bit about approximating by rectangles.

Is the question "Why is calculus considered so important?"

If so, I would say the reason comes down to what Calculus is capable of. It is, at its core, a language for describing how things change. It also provides a set of intuitions about how things change which is critical for almost any area of applied science and mathematics.

3blue1brown recommended a book recently that I've been reading which makes these points better than I can called Infinite Powers. I highly recommend it if you'd like to gain more inspiration around the importance and wonder that Calculus presents. It's a very easy and quick read.

Edit: rereading your question, maybe you're asking "why is it dis-proportionally represented in the US when compared to the UK?" ; my doesn't really help there, but I'd still stand by the book recommendation!

Trigonometry was developed for astronomy, so the history of trigonometry is more or less coextensive with the history of astronomy. I haven’t read it but this book looks promising, https://amzn.com/0195095391

Or you could try finding a book about the history of celestial navigation, something like https://amzn.com/1575240955

Or for something with a more explicit focus on trigonometry, https://amzn.com/0691175993

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    Strogatz has a new pop math book about Calculus, Infinite Powers* https://amzn.com/dp/1328879984
    
    
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    It’s not quite what you are asking for, but let me recommend Lockhart’s book Measurement*, https://amzn.com/dp/0674284380