For fun, I decided to do a quick comparison of the Major Arcana of the Tarot with the chapters of the first book I came across with approximately 22 chapters. Since I am an eclectic soul, the first such book that came to hand was Calculus Made Easy by Silvanus P. Thompson. This book was originally written in 1910, revised and enlarged in 1914, and the third edition was edited posthumously in 1946. It’s still in print because it is a concise summary of the geometric approach to calculus. I was astounded at how well the Major Arcana related to the various chapters, so I present a quick summary for your amusement and possibly edification:

  1. Card: The Magician. Book chapter: To deliver you from the preliminary terrors.
    Thompson’s whole point in this book is that once a student’s intuition is properly engaged, calculus becomes much more easy and natural. Thus he starts out by explaining that dx means “a little bit of x” and the integral symbol means “add up all the little bits.” This is in fact a neat way of introducing students to the basics of calculus, worthy of a Magician, and an example of the magical powers that calculus promises to the student.
  2. The High Priestess/Different degrees of smallness.
    Thompson enlarges (sorry) on the idea of dx by using real-world examples to convince readers of some algebraic properties, especially that dx of dx is negligibly small. Since the High Priestess (or Papesse) is usually about internal knowledge or hidden information, as dx is about functions, the comparison is apt.
  3. The Empress/On relative growings.
    The Empress, a card about fertility, presides over Thompson’s chapter on comparing one rate of growth to another.
  4. The Emperor/Simplest cases.
    Basic differentiation is the first example of students gaining the power or authority of calculus, when they begin to become Emperors over relationships between functions and rates of change.
  5. The Hierophant/What to do with constants.
    The chapter about basic rules of differentiation shows up at the same point as the card that alludes to hierarchical power structures and established authority – the constants in our lives.
  6. The Lovers/Sums, differences, products, and quotients.
    When two (or more) functions are involved in a differentiation, how we handle them in calculus becomes more complicated; thankfully it’s still less complex than the kind of human relationships the Lovers card usually refers to.
  7. The Chariot/Successive differentiation.
    Learning about second (and more) derivatives means that students start making lists of successive derivations and charging right through them – although sometimes one of the horses pulling the Chariot goes off in a different direction and they get confused.
  8. Strength/When time varies.
    With physical examples, Thompson explains the idea of a rate and introduces equations describing acceleration, force, and work, while the Tarot shows a woman closing a lion’s mouth – but gently.
  9. The Hermit/Introducing a useful dodge.
    In this chapter, the chain rule lets students ignore one part of an equation while working on another, then return to the ignored part to get the whole solution – rather as the Hermit takes time away to focus on some things first.
  10. Wheel of Fortune/The geometrical meaning of differentiation.
    Just as the wheel shows people at different points in its rotation, this is the chapter when we finally start looking at curves and how the curves change between different points. This is also a level when students generally start to internalize (or not) the intuitive aspects of calculus; if they do, they stand a good chance of completing the course successfully, but they’re only halfway through.
  11. Justice/Maxima and minima.
    The lady with the sword and scales takes a hard look at what’s going on as students learn to find the turning points – where a function is at its highest and its lowest.
  12. The Hanged Man/The curvature of curves.
    In this chapter, students have to take a different perspective on differentiation and its geometrical meaning, rather as the Hanged Man is all about a changed perspective.
  13. Death/Other useful dodges – partial fractions and inverses.
    This correlation is appropriate on two levels: partial fractions are one of the more technically difficult algebraic techniques taught in basic calculus, and can be the “death” of many students’ patience, but the process of using partial fractions or inverses is also all about transforming from one form into another.
  14. Temperance/On true compound interest and the law of organic growth.
    Temperance is usually depicted pouring liquid from one vessel into another. In my favorite deck, he’s juggling. These two very practical applications of calculus are all about pouring and juggling – money and populations and how they are constantly in flux even when they seem to stay still.
  15. The Devil/How to deal with sines and cosines.
    Enough said.
  16. The Tower/Partial differentiation.
    Here I actually dislike Thompson’s technique, because he starts in to a topic usually reserved for Calculus III in today’s teaching style, and it runs the risk of shattering students’ still-precarious understanding and self-confidence without ever getting to the second half of basic calculus, integration, so I find the card appropriate.
  17. The Star/Integration.
    Some readers see the Star as a kind of healing experience after the tumultuous change of the Tower; she is usually pictured as pouring out two vessels of water. The idea of reuniting what has been broken is not a bad metaphor for integration (adding up all the “little bits” that were broken up in differentiation). It’s also comparable to the way tiny individual drops of water can make an entire sea when added together.
  18. The Moon/Integrating as the reverse of differentiating.
    In some ways of teaching calculus, this is “the big secret” – that integration and differentiation are not just seeming opposites but literally opposite processes that reverse each other. Most students start to pick up on this a little earlier, and have subconscious suspicions of the relationship even before it’s presented as a theorem. Since the Moon is about hidden information and the subconscious, it’s appropriate.
  19. The Sun/On finding areas by integrating.
    Finally, students put it all together and start doing problems that can have real-world equivalents. The ability to find areas under curves is another of the major accomplishments for students of calculus, so it certainly can feel like a triumph, or coming back out into the sun after a long night.
  20. Judgment/Dodges, pitfalls, and triumphs.
    Thompson pulls together a useful assortment of ways to transform an insuperable problem into a solvable one, “resurrecting” it or giving the student ways to emerge triumphant.
  21. The World/Finding solutions.
    Here Thompson pulls together the techniques of the entire subject so far to launch the student on the next major topic of mathematical studies: solving differential equations. As in the Tarot, this is both an ending, the completion of the journey that began at #1, and the beginning of a new exploration. I feel justified in combining chapters 22 and 23 under this card as well, because they too are side-notes or brief introductions of other directions the study of calculus can be expanded.

0/22: The Fool. Prologue/Epilogue and apologue. I cannot put it better than Thompson himself did in his Prologue:

Considering how many fools can calculate, it is surprising that it should be thought either a difficult or a tedious task for any other fool to learn how to master the same tricks.

Some calculus-tricks are quite easy. Some are enormously difficult. The fools who write the text-books of advanced mathematics – and they are mostly clever fools – seldom take the trouble to show you how easy the easy calculations are. On the contrary, they seem to desire to impress you with their tremendous cleverness by going about it in the most difficult way.

Being myself a remarkably stupid fellow, I have had to unteach myself the difficulties, and now beg to present to my fellow fools the parts that are not hard. Master these thoroughly, and the rest will follow. What one fool can do, another can.

In his Epilogue, he explains why mathematicians may condemn the book and admits the charge that he has simplified the subject beyond what professional but impractical mathematicians regard as necessary. He defends his deliberate choices as being the best way to  introduce “fools” like himself to the subject – a journey that any of us can find ourselves taking part in.

9 thoughts on “The Foolish Student’s Journey through Calculus

    1. Thanks! I’ve got the book out because I intend to give it to one of the students I’m tutoring right now. She’s only in geometry, but she’ll get to calculus eventually, and she and I both have the kind of visual intuition that the book addresses better than many regular texts.

      I think it matters that the author wrote with the particular intent he did – a “standard” math text wouldn’t have worked as well, I think.

    1. Hugs for the “allergy” – although I doubt it. Reading Slacktivist isn’t easy to begin with! And thanks; one of the reasons I like the intersection is that it is perfectly in keeping with the author’s intent – and mine! – of making even complicated ideas more approachable and accessible.

  1. This is weirdly appropriate. I just picked up a “Refresher Calculus” book last week so I’m familiar with all the processes you were describing. So as a fan of both Calc & Tarot, well done.

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