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I'm looking to books introducing different topics of math with fun for my son, to give him a taste of different areas.

My son roughly understands A-level topics until single variable calculus (of course, need exercise to get familiar to all the tricks).

Dover Books on Mathematics is a well-known series, on Amazon there are now 306 items listed. However, they are a bit too long (normally about 200 pages), with many topics too deep, and , honestly, the text might be a bit too serious for a kid of 9.

Martin Gardner's books are quite fun, with many challenging puzzles, maybe more attractive, however, they are normally not focusing on a particular topic.

Evan Chen started a Napkin project introducing more than 80 topics, but are a bit "formal" and more on the pure math theories instead of motivation, application, and funny puzzles.

Any suggestion please?

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    $\begingroup$ Many of the books by Raymond Smullyan should work. "What is the Name of This Book?" or "A Mixed Bag" or "The Lady or the Tiger?", and so on. $\endgroup$ Dec 28, 2022 at 22:27
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    $\begingroup$ Dover Books are almost all reprints. (Very rarely they have an original) This means original publisher has lost interest in it, so the rights are being passed to Dover to make books to be sold cheap. Other than that there's no theme or series editor. Audience, quality, style, age all vary considerably. (A bunch of Martin Gardner is now available from Dover) $\endgroup$
    – user71659
    Dec 29, 2022 at 6:05
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    $\begingroup$ [1] MAA New Mathematics Library series and possibly some selected volumes in these other MAA book series. [2] English translations of the Russian books from the series Topics in Mathematics and Popular Lectures in Mathematics and Little Mathematics Library. $\endgroup$ Dec 29, 2022 at 14:23
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    $\begingroup$ See also the many expository articles published in Pi Mu Epsilon Journal and Mathematical Spectrum and Quantum. $\endgroup$ Dec 29, 2022 at 14:29
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    $\begingroup$ See also the books listed in this answer. I was planning to extend this sometime, but apparently that time has not yet arrived. $\endgroup$ Dec 29, 2022 at 21:05

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Have a look at the series of books originally published starting in the 1960's under the series title of "Anneli Lax New Mathematical Library" by the Mathematical Association of America (MAA). These are now available from the American Mathematical Society (AMS) and Cambridge University Press (CUP).

The series is described as:

Featuring fresh approaches and broad coverage of topics especially suitable for high school and the first two years of college, the volumes in this series are an excellent source of enrichment material for teachers and students. Good mathematical reading with lively exposition.

Some of my favorites:

Grossman, Israel, and Wilhelm Magnus. Groups and their Graphs. Vol. 14. New York: Random House, 1964.

Niven, Ivan. Mathematics of Choice: Or, How to Count Without Counting. Vol. 15. MAA, 1965.

Niven, Ivan. Numbers: Rational and Irrational. Vol. 1. New York: Random House, 1961.

Ore, Oystein, and Robin J. Wilson. Graphs and their Uses. Vol. 34. Cambridge University Press, 1990.

I was reading these books by the time I was in high school. In general, they should be accessible to interested students who are studying US high school level mathematics.

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    $\begingroup$ Thank you! Just browsing through it I found something interesting -- Uses of Infinity by Leo Zippin, seems a pretty nice supplementary reading. I'll check the other ones. $\endgroup$
    – athos
    Dec 29, 2022 at 20:33
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    $\begingroup$ Definitely second the "Anneli Lax" volumes - and they're quite varied. I had quite a bit of fun/learned quite a lot from many of them. I especially liked the 3 volumes by I.M. Yaglom Geometric Transformations I,II,III - and combine that with dynamic geometry software (esp. Cinderella, which supports these transformations directly) and you're really ready to explore. $\endgroup$
    – davidbak
    Dec 31, 2022 at 5:11
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I would go with Art of Problem Solving and/or Beast Academy if he has some primary gaps.

https://artofproblemsolving.com/

There should be enough fun factor, that he's engaged. And lots of interesting problem solving. But enough of a structure and curriculum so he moves through and masters basic subjects.

Your 9 year old (I checked your previous questions) is bright and has been exposed to a lot of stuff ahead of time. But is not really experienced the way a conventional student ready to go into calculus is (at age 18ish).

So, keep it fun and keep letting him have exposure, but don't convince yourself he's really at the level of a strong 17 year old. For comparison, I spent Christmas with my relative's boys. They're bright 9 and 11 year-olds and have had a lot of exposure pushed at them by their dad ("we know trig", "we know calculus"), but it's really exposure, not mastery. In terms of real school, the 6th grade 11 yo is doing 9th grade (US system) algebra 1. Still very good...but the real practiced level is not as high as his exposure.

Note also, Dover is not really a coordinated series, but reprints of old classics. Which vary a lot in quality, scope, approach, solutions enclosed, or even print quality. It's a great resource of old classics. But you need to check the individual titles (look at Amazon reviews, skim them, ask about them) to see if they match a specific need.

A few other suggestions:

Calculus Made Easy and Calculus for the Practical Man (and the other "for the practical man") are relatively fun (and easy prose). Were used by Feynman to learn young (but he eventually mastered normal texts also.

For an easy prose, non doorstop review of high school math (algebra, geometry, trig, precaculus, intro to calculus), I like Frank Ayres First YEar College Math Schaum's Outline. Has the solutions. I have the first edition, which I like. You can get that or the second edition on the net, used. Probably pretty similar but I always worry about regression...but use your judgment.

I don't know of a good baby physics (project based or book based) curriculum. But if you could find one, that would be good. the intuition of angles and position/speed/acceleration are helpful to informing and motivating math study.

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  • $\begingroup$ Thank you very much for the suggestions. Many good points. $\endgroup$
    – athos
    Dec 29, 2022 at 13:05
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    $\begingroup$ Take a look at this Dover: amazon.com/Moscow-Puzzles-Mathematical-Recreations-Recreational/… Fun puzzles. Not based on high end math, but tricky aspects of common math. And engaging art, for a kid. $\endgroup$ Dec 29, 2022 at 19:12
  • $\begingroup$ Love the decisiveness! $\endgroup$ Dec 29, 2022 at 19:27
  • $\begingroup$ One more thought is something on tangrams. An article, a book, a physical set. Whatever. Kids love the manipulative aspect (logical dominoes) and it sort of mimics origami. $\endgroup$ Dec 29, 2022 at 19:28
  • $\begingroup$ yeah, hands-on with the shapes, I bought some puzzles as this but time was more diverted to Calculus. i know, i'm a bit too practical, boring.. $\endgroup$
    – athos
    Dec 29, 2022 at 20:41
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So let me answer your request for non-Dover books by recommending you some Dover books! First of all I agree with your overall assessment of them, and the descriptions on the backs are often misleading ("only requires familiarity with high school calculus" my a**). Also you've got some very nice non-Dover suggestions already.

Anyway, there are indeed a few Dover books that are beginner(ish)-friendly. Anything by W.W Sawyer fits the bill, along with Excursions in Geometry and Excursions in Number Theory by Ogilvy. I also like Morris Kline's Calculus: An Intuitive and Physical Approach. It may be longer than you'd like but for anyone wanting to learn or brush up on calculus who also likes physics it's great.

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  • $\begingroup$ Thank you very much! picking the pearls from the sea. Indeed Morris Kline's Calculus is quite attractive. I wrote a note to teach my kid calculus, dropping $\epsilon-\delta$ language, mixing math and mechanics (basically Newton's laws of motion), trying to show the motivation and application. However, many loose ends are left, so I call it Book 0 of calculus, a trailer more than a proper introduction. Prof Kline's book sounds very much a nice Book 1. Oh I have Excursions in Geometry, also a nice one. I'll check other Dover books too. Thanks again. $\endgroup$
    – athos
    Dec 30, 2022 at 10:22
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Dover books tend to be from early generations of math when most topics were taught at a higher level and as such the text books are often more sophisticated. That trend paired with the general concept that the reader should fill in the gaps makes Dover texts on the average a somewhat dangerous path for the independent self-learning enthusiast. Yet, they are cheap. Certainly I looked a lot at Dover texts when I was an undergraduate. In retrospect, I would have done better to get my hands on some undergraduate Springer texts. In any event, the texts available now are generally much more accessible than what you find in Dover (with a few exceptions, there are Dover reprints of things from the 90's and later and my comment here is not really aimed at those)

Ok, so what to do ? I would consider getting an old edition of a few different topics which fit your child's interest. For example:

  • an old edition of Susan Colley's Vector Calculus
  • an old edition of a Friendly Introduction to Number Theory by Silverman

Also fun, "A Manga Guide to Linear Algebra", or one of Brian Greene's books on String Theory. Perhaps "Road to Reality" by Penrose (that will give them a lot to chew on, it's over 1000 pages).

Anyway, as a rule, pick a book which is currently popular in courses, but get a copy from a few editions back. For not too much money you can generate a library for your child to poke around as their interest leads them.

For me, getting a copy of "Hyperspace" by Michio Kaku was very eye opening at one point in my education. There are many such books now.

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  • $\begingroup$ Indeed, the interest is the most difficult to nourish... $\endgroup$
    – athos
    Dec 29, 2022 at 13:07
  • $\begingroup$ Is the reason for suggesting old editions just cost? Or are the older editions better? $\endgroup$
    – user18187
    Jan 16, 2023 at 6:01
  • $\begingroup$ @user24096 The older editions of popular textbooks are often better. New editions are pushed to be dumber by the powers that be... I'm sure there are exceptions, but that seems to be the pattern. $\endgroup$ Jan 17, 2023 at 3:44
  • $\begingroup$ However, the first edition of any book is dangerous, maybe edition 2 or 3 is best. In particular, I think Susan Colley's book is better after the first edition, it has a lot more material if memory serves me correctly. $\endgroup$ Jan 17, 2023 at 4:15
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Group Theory in the Bedroom by Brian Hayes, a great writer. His articles are educational, informative, and often entertaining. I linked to this collection in this earlier MESE question.

   

Contents here.

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  • $\begingroup$ thank you for the recommendation, let me read it. $\endgroup$
    – athos
    Dec 31, 2022 at 10:52
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The Man Who Counted, by Malba Tahan

Written in Brazil, set in the Middle East, these stories follow the adventures of Beremiz, an accomplished mathematical problem-solver. He uses math to settle disputes, solve riddles and mysteries, and entertain his hosts. The series of 34 adventures, each with a math puzzle, is reminiscent of the Arabian Nights.

So it's not developing a particular topic.

How to Count Like a Martian, by Glory St.John

A really good way to understand place value is to work with other number bases. How to Count Like a Martian is a detective story in which the history of other number systems plays a starring role. “Out of the depths of the dark and starry night come the first of the faint and mysterious sounds … At your radio telescope, you are expertly tuning the dials.” You have just received a message from Mars. “You know that this is not a message in words. Martians and Earthlings would have too much trouble trying to find the same words to succeed that way. But there is another kind of language that both Martians and Earthlings understand.”

Numbers… And so you research the number systems that have been used on Earth, hoping that will help you decipher this message. The book proceeds to explain eight different counting systems, including the abacus, and computers.

In the process, the concepts of place value (she just calls it place), base, and zero are explored. By the end of the book, you can see that the beeps and bee-beeps of the message you received are just the counting numbers, Martian style. How to Count Like a Martian was written in 1975, when there were still dials and tape recorders. those two items may be the only evidence of its age.

The Number Devil, by Hans Magnus Enzensberger

The Number Devil visits Robert in his dreams, and gets him thinking about the strangest things! Rutabaga numbers and prima donnas (roots and primes) are just the beginning. This one includes Pascal's (aka Jia Xian's) triangle.

Mathematics: A Human Endeavor, by Harold Jacobs

This is a delightful textbook. It explores logic, symmetry, functions, ...

Who Is Fourier? by Transnational College of LEX

Fourier series, approachable with very little background.

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    $\begingroup$ Thank you very much for the suggestions. The Number Devil is really good. I'll try other books. $\endgroup$
    – athos
    Dec 29, 2022 at 13:05
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    $\begingroup$ Who Is Fourier? is pretty good! $\endgroup$
    – davidbak
    Dec 31, 2022 at 5:04
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Bert Mendelson's Introduction to Topology

I began working through Mendelson's Introduction to Topology in high school, but could have begun even sooner. It begins with a perfectly accessible introduction to set theory and proofs, and then moves on to the basics of point-set topology without requiring any knowledge of abstract algebra or real analysis. I know it sounds crazy to recommend a topology text to a 9 year old, but I wish I had been exposed to the abstract thinking of set theory sooner, and I think Mendelson's text is an excellent choice for that. Once you've got a grasp of the fundamentals of sets, Mendelson's approach to topology is really just goofing off with set theory relationships.

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    $\begingroup$ Topology could be fun! Prof Mendelson's book seems an accessible invitation, from set to metric spaces to topology spaces, then discussed connectedness and compactness. My only concern is that it seems a bit "pure" on math, not that "tangible", but Let me read it first. Thanks for the recommendation. $\endgroup$
    – athos
    Dec 29, 2022 at 20:57
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    $\begingroup$ @athos I think exposure to abstract thinking could be a valuable thing. One of the most common challenges I've encountered with high school and college students is an almost complete inability to work with abstract concepts. $\endgroup$ Dec 29, 2022 at 21:02
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    $\begingroup$ Consider also the three volumes of A Mathematical Gift - I, II, III by Ueno, Shiga, and Morita (of the AMS Mathematical World series). Subtitled: "The interplay between topology, functions, geometry, and algebra". $\endgroup$
    – davidbak
    Dec 31, 2022 at 5:12
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    $\begingroup$ @athos: For topology books that are appropriate for (good) high school students, see this MSE answer. $\endgroup$ Dec 31, 2022 at 10:00

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