Name: Numbers and geometry
ISBN: 9780387982892

impendiaJul 2020

I'd consider something by John Stillwell. For example, Numbers and Geometry, which investigates the connections between number theory and plane geometry -- two subjects which your child has probably seen, but not seen related.

Stillwell is a magnificent writer -- he loves to go on digressions, and to talk about the history of the subject. My impression is that his books are a bit rambling for traditional use as textbooks, but perfect for self-motivated reading for exactly the same reason. He makes the subject fun.

(Disclaimer: I haven't read this book in any sort of depth, but I have read another of Stillwell's books cover to cover.)

Concerning your other recommendations: The Princeton Companion to Mathematics is magnificent, but in practice it's something he'd be more likely proudly own and display on his bookshelf than to read; it's quite dense. Spivak's Calculus, from what I've heard, is magnificent. Probably best in the context of a freshman honors class, but I can imagine that someone disciplined could love it for self-study. Don't know Moor and Mertens.

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earthicusJan 2019

If you managed to survive high school algebra, you might be able to 'redo' your basic math education with a geometric bent from Stillwell's book 'Numbers and Geometry', which teaches the classical link between the two subjects that has been eliminated from elementary education to some extent. However, the kind of intuition he presents is still semi-abstract - it's not a super duper picture heavy book - but I think it's about as close to what you're asking for as you are likely to find.

https://www.amazon.com/Numbers-Geometry-Undergraduate-Texts-...

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tokenadultJul 2012

I am wondering what books would be good to recommend to parents whose children are deeply in love with mathematics, especially if the parents themselves have only an average background in mathematics. That was my situation a decade ago, and I found some helpful books I'll mention here, but I think a lot of Hacker News participants can suggest some books that are even better. I'll be speaking to a group of parents whose children are primary-school age but already well familiar with secondary school algebra and geometry, taking an advanced mathematics summer course.

Books that I found helpful for understanding mathematics when my oldest son, a mathematics-lover, was growing up include

Concepts of Modern Mathematics by Ian Stewart (Dover reprint)

Numbers and Geometry by John Stillwell (Springer)

Vision in Elementary Mathematics by W. W. Sawyer (Dover reprint)

The Art of Problem Solving by Sandor Lehoczky and Richard Rusczyk (Art of Problem Solving)

A frequently recommended book for this need is Developing Math Talent by Susan Assouline and Ann Lupkowski-Shoplik, but I haven't made much use of that book.

What am I missing? What do you recommend?

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tokenadultJun 2012

Thanks for the many interesting comments. I'll reply jointly here to a few of the comments, and invite further discussion on some points made in the submitted article that I'm curious about.

From a top-level comment:

http://news.ycombinator.com/item?id=4084559

As someone who was taught the 'traditional' way of mathematics, can someone give a few pointers of de-programming myself from the traditional way that I was taught? (Although maybe it won't be so hard since I feel like I've forgotten quite a bit)

From a second-level comment, which has already received some helpful replies:

http://news.ycombinator.com/item?id=4084426

After Vector Calc, I wanted to go back to the fundamentals, to understand instead of remembering.

There is a FAQ page on the Epsilon Camp site

http://www.epsiloncamp.org/FAQ.php

that includes some Frequently Asked Questions articles about learning mathematics for deeper understanding. The FAQ article "Problems versus Exercises"

http://www.epsiloncamp.org/faq/faq_1.php

relates to what kind of work to set for yourself to build deeper understanding, and the FAQ article "Learning Mathematics"

http://www.epsiloncamp.org/faq/faq_3.php

points to writings by various mathematicians, including the book Numbers and Geometry by John Stillwell, about how to appreciate mathematics as a deep, connected subject.

The submitted article mentioned "Numerous studies over the past thirty years have shown that when people of any age and any ability level are faced with mathematical challenges that arise naturally in a real-world context that has meaning for them, and where the outcome directly matters to them, they rapidly achieve a high level of competence. How high? Typically 98 percent, that's how high. I describe some of those studies in my book The Math Gene (Basic Books, 2000)." The most striking example of this that I remember from a news report was a Wall Street Journal series in the 1990s that followed two young men in an inner city ghetto, one who was a good high school student and the other who was a street criminal. The street criminal usually skipped high school, but happened to show up the day students could take one of the major standardized tests (probably the PSAT, if I remember correctly). The street criminal, who sold illegal drugs among other activities, scored just as well on the test as the more regularly attending student who had learned most of his mathematics from school lessons. That's a rather stark illustration of what's missing in school lessons for children who don't have an outside-of-school environment for learning mathematics.

http://www.ams.org/notices/200502/fea-kenschaft.pdf

The article also says that many students say, "You have to be willing to accept that sometimes things don't look like - they don't see that you should do them. Like they have a point. But you have to accept them." I wonder how that relates to the quotation attributed to John von Neumann,

http://en.wikiquote.org/wiki/John_von_Neumann

"Young man, in mathematics you don't understand things. You just get used to them."

And from a third-level comment:

http://news.ycombinator.com/item?id=4084865

I experience math (and programming) quite differently than learning a language or painting: Once I grasp a concept, I can use it. Before that, it's mostly useless to me.

I ask, because when I studied mathematics in school, I had a drive to understand the general principles first before I launched into working on my homework, while some of my classmates were successful--at least in the context of school--by working on the homework and DEVELOPING some level of understanding as they tried to figure out answers for the homework. (I was in a "tracked" mathematics class, taking algebra in eighth grade in an era when most Americans took algebra in tenth grade, if at all, and most of my classmates had parents who were engineers or medical doctors and could ask their parents for help at home if the school lessons were confusing, as they often were.) I also have a very strongly visual approach to grappling with mathematical problems. So when I first learned algebra, which was presented to me as a bunch of "Do this to the equation, and then do this" with little rationale, I found that very dissatisfying. Later in the school year, we learned about coordinate graphing of systems of equations in the Cartesian plane, and I remember thinking, "Why didn't you tell me this in the first place?" For historical reasons, and perhaps for reasons of what most learners consider most easy, usually purely procedural algebra for solving systems of two equations in two unknowns has been taught in school before graphing systems of equations in the coordinate plane. But for some learners, it would be easier and more accessible to reverse that order. What do you think about the issue of students working first according to instructions, to DEVELOP understanding a la the von Neumann quotation, versus getting the "big picture," perhaps explicitly visually, before working on problems.

I'll comment also that the approach taken to learning mathematics in school in most of the newly industrialized countries of east Asia and southeast Asia is plainly superior to the United States approach for at least two reasons:

1) the school textbooks in those countries explicitly encourage students to THINK about why a procedure will or will not work, and about how many different ways there might be to solve a problem, and

2) the school textbooks show multiple representations of most mathematical concepts, building from "concrete to pictorial to abstract" as in the Singapore Primary Mathematics series

http://www.singaporemath.com/Primary_Mathematics_US_Ed_s/39....

and the follow-up New Elementary Mathematics series

http://www.singaporemath.com/New_Elementary_Math_s/47.htm

which interleave arithmetic, number theory, geometry, and algebra in increasing depth and interconnection throughout all grade levels.

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tokenadultAug 2011

Frankly, I'm a little tired of the position calculus holds

You are not alone. Here is a quotation from John Stillwell that I put in a FAQ for parents about mathematics education:

"What should every aspiring mathematician know? The answer for most of the 20th century has been: calculus. . . . Mathematics today is . . . much more than calculus; and the calculus now taught is, sadly, much less than it used to be. Little by little, calculus has been deprived of the algebra, geometry, and logic it needs to sustain it, until many institutions have had to put it on high-tech life-support systems. A subject struggling to survive is hardly a good introduction to the vigor of real mathematics.

". . . . In the current situation, we need to revive not only calculus, but also algebra, geometry, and the whole idea that mathematics is a rigorous, cumulative discipline in which each mathematician stands on the shoulders of giants.

"The best way to teach real mathematics, I believe, is to start deeper down, with the elementary ideas of number and space. Everyone concedes that these are fundamental, but they have been scandalously neglected, perhaps in the naive belief that anyone learning calculus has outgrown them. In fact, arithmetic, algebra, and geometry can never be outgrown, and the most rewarding path to higher mathematics sustains their development alongside the 'advanced' branches such as calculus. Also, by maintaining ties between these disciplines, it is possible to present a more unified view of mathematics, yet at the same time to include more spice and variety."

That's from the preface to his book Numbers and Geometry (New York: Springer-Verlag, 1998), which is a delight to read and full of thought-provoking problems.

http://www.amazon.com/gp/product/0387982892/

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tokenadultMay 2011

There has been a profound change in mathematics education in the years indicated, and the author of the submitted article is on to something. One of my favorite authors on mathematics, Professor John Stillwell, writes, in the preface to his book Numbers and Geometry (New York: Springer-Verlag, 1998):

"What should every aspiring mathematician know? The answer for most of the 20th century has been: calculus. . . . Mathematics today is . . . much more than calculus; and the calculus now taught is, sadly, much less than it used to be. Little by little, calculus has been deprived of the algebra, geometry, and logic it needs to sustain it, until many institutions have had to put it on high-tech life-support systems. A subject struggling to survive is hardly a good introduction to the vigor of real mathematics.

". . . . In the current situation, we need to revive not only calculus, but also algebra, geometry, and the whole idea that mathematics is a rigorous, cumulative discipline in which each mathematician stands on the shoulders of giants.

"The best way to teach real mathematics, I believe, is to start deeper down, with the elementary ideas of number and space. Everyone concedes that these are fundamental, but they have been scandalously neglected, perhaps in the naive belief that anyone learning calculus has outgrown them. In fact, arithmetic, algebra, and geometry can never be outgrown, and the most rewarding path to higher mathematics sustains their development alongside the 'advanced' branches such as calculus. Also, by maintaining ties between these disciplines, it is possible to present a more unified view of mathematics, yet at the same time to include more spice and variety."

Stillwell demonstrates what he means about the interconnectedness and depth of "elementary" topics in the rest of his book, which is a delight to read and full of thought-provoking problems.

http://www.amazon.com/gp/product/0387982892/

I have a collection of analytic geometry and calculus books, accumulated as used books from various readers, that includes the books used by my late father in his higher education as a chemistry major during the Truman administration, followed by books from other previous owners reflecting "new math," "back to basics," and "reform" approaches to mathematics education. Plainly today's secondary and tertiary students of mathematics need to take advantage of current technology so that they can devote more time to THINKING about the mathematics they learn and less time to what even any mathematician would call "tedious calculation." But too few students have ever been guided to through the kind of insight-producing problems in which the tedious steps themselves and the false starts while struggling with the problem produce deep understanding. Stillwell gives examples of such problems in his books, and the minority of students who participate in math contexts or who voluntarily work the "challenge" problems not assigned in their textbooks may gain such insight, but most school textbook problems of all eras are mere exercises, and too few students do enough of those thoughtfully to have hope of learning mathematical concepts.

See "Basic skills versus conceptual understanding: A bogus dichotomy in mathematics education," American Educator, Fall 1999, Vol. 23, No. 3, pp. 14-19, 50-52 for additional commentary on mathematics education,

http://www.aft.org/newspubs/periodicals/ae/fall1999/index.cf...

and see an earlier HN comment

http://news.ycombinator.com/item?id=2515796

for a FAQ on the distinction between problems and exercises in mathematics education.

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