Entries from AboutBooks

The kids pick the number of digits (we usually start with 5). They put 5 dashes on their paper. I turn over 5 cards in a deck one by one. They have to decide where to put the numbers. Then each kid reads their number to me while I put it on the white board. The kids with the highest number wins.

For some reason, they love this game. On the next round, we go up one digit. Today, we went all the way up to 100 million.

It's a great game.

• They gain familiarity with large numbers. They get a lot of practice with reading large numbers out loud and hearing large numbers read out loud while it is being written on the board.

• They have to use strategy. In some games, we have a lot of high numbers at first which every kid puts in the same place. Then, they winner is the determined by the numbers in the ones and tens place. Conversely, sometimes we have a lot of low numbers in the beginning. Then the winner is determined by the highest digits. Much more interesting is when we have medium and low cards. Then, they have to do a lot more thinking about where the cards go.

• There are very concrete results from this game that allow us to explore numbers even further. In one game, 5 out of 8 kids had the same highest number. So we talk about why and when does this happen? In one game, we had one winner that was a lot higher than anyone else. When does this happen?

We have a gang of kids that run semi-wild in our neighborhood in the summer. They are very mixed in age (ranging from 7 through 11). I have thought about corralling the whole lot of them and bringing them in to teach them all some math together; it would do them all some good to work on it over the summer, and Ben would enjoy his math sessions more if he shared them. I'm a little stumped, though, about how to teach a wide range of ages and interest levels simultaneously.

I'd love to collect some more math games that are as simple and elegant as this one is, especially games that might appeal to a broad range of ages, and (like this one) start a math session off on the right foot.

This reminds me that I never got around to reading The Cuckoo's Egg: Tracking a Spy Through the Maze of Computer Espionage, so I'm ordering that, too!

I think Clifford is right about computers in classrooms.

The research I've seen makes me think that Computers are Calculators writ large, with many of the same negative effects on learning.

Even if I hadn't seen the research, the fact that we have Mystery NGOs actively promoting the use of computers in classrooms--and being cited as authorities by Steve Leinwand--would make me leery.

I'll get around to posting the studies I've found on this question sooner rather than later, I hope.

update

I already did post the Israeli study of computer use in the classroom.

I've been wanting to get this book for a while, but funny -- it's never at our (beloved) local used bookstore. That is usually the sign of a good book -- people are hanging on to their copies. Of course, it's too bad for me. I may have to break down and buy it new. Everything I'm seeing about this book is telling me it's a great read.

I am a fan of the whole concept of Core Knowledge, being (I suppose) an educational traditionalist. Ben went to a Core Knowledge elementary school, and he learned quite a bit in spite of himself.

Thomas/Vorblik writes:

The philosophy behind these movements is often described in the literature as "constructivism''. Thus, Jack Price, President of the NCTM, says in [6]: "the standards are based on research and on a constructivist theory of learning . . . Critics may not agree with the theory, but they cannot say that the standards are not research based.'' But, as The Schools We Need shows, they can and do.

Is it possible that the ideas recommended by the NCTM are the very ideas that already pervade the schools they are supposed to reform?
Such a hypothesis is reinforced by the teaching methods that the NCTM and other reform groups advocate for achieving higher-order thinking skills. These "new'' methods include attention to individual needs and learning styles, discovery learning, and thematic learning. But these teaching techniques are essentially the project-oriented, child-centered methods that have long dominated educational thought and have prevailed for decades in our schools.
--Hirsch p. 132

If Hirsch is right about the entrenchment of constructivism, then we are the radical reformers; and it feels kind of nice for a change.

This supports my feeling that constructivism, in some form, is (like the poor) always with us, at least since Rousseau. And I suspect before Rousseau as well ... perhaps Ed knows what predated Rousseau's ideas, or perhaps Hirsch has even written about it; the reviews say that he treats the historical genesis of constructivism at length.

Fie on Rousseau, anyway. One reason I tend toward educational traditionalism is that, deep down, I believe mankind has not evolved or changed that much since we were all tribesmen. I really do feel that civilization is precariously thin, and not to be taken for granted. I guess Lord of the Flies made a big impression on me (either that, or I'm just paranoid).

Hirsch recommends a math curriculum which he says is "reasonably close to what research is telling us about how students learn". Surprise: it's Saxon math.

I've often complained that Ben's Prentice-Hall Mathematics Course 1 textbook, apart from being a basically decent text, is too full of pointless, distracting colors and pictures and charts. I wrote in the comments in this post that I suspected someone was actually counting the numbers of pictures of Asian and black and Hispanic and white and disabled kids to ensure they were all roughly equal (although, at the risk of being tasteless, I note that there are no facially deformed or cerebral palsied or even blind kids' pictures among the disabled kids; just perfectly typical-looking smiley kids on crutches and sitting in wheelchairs).

JdFisher, who hosts MathAndText, put up this post today that confirms my sense that this head-count is something textbook publishers pay a lot of attention to.

Why the distracting pictures of things that have no connection to the text, not even anything as tenuous as a link to an irrelevant aspect of the word problems in the text?

Why do we have to have a head count of photos of kids -- why not skip the photos completely?

J.D. writes:

The desire on the part of publishers to include images and "real-world connections" is so strong that publishers will face all kinds of headaches to put them in their books. Teachers and administrators want to motivate their students to approach mathematics, and publishers, to compete in today's textbook market, need to try to help teachers and administrators do this. These are the reasons for what some would call glitz.

It sounds as though the source of this problem is a misunderstanding, on the part of teachers and administrators, of what makes a successful textbook for the kids (the publishers, of course, know that from their perspective, what makes a successful text is whatever causes consumers to buy it).

Do they really suppose that kids come into math class with their little eyes glowing because their texts have 5 colors of text, and totally unrelated pictures of Amish people having communal barn-raising?

I think it's more likely that they are a huge source of distraction -- and kids from all walks of life are a lot more distractable when they're finding something difficult already.

Better a very clean presentation, I think, than a busy one; and that goes for all sorts of textbooks, not just math.

This textbook is our effort to construct a mathematics course for elementary teachers that incorporates Liping Ma's insights about effective teacher knowledge. It is a mathematics book designed to set prospective teachers on the road to developing what Ma termed a "Profound Understanding of Fundamental Mathematics.''

So, for someone like Catherine, who was fascinated by Liping Ma's book and trying to attain a deep understanding of fundamental mathematics for the first time as an adult, P&B fills a real vacuum. It did for me as well, because I was shamed by Liping Ma's book. It led me to realize that, in spite of all my many years of studying math, elementary math teaching offered conceptual and pedagogical challenges that I couldn't measure up to.

And I've taught more than my share of elementary math. I taught huge chunks of elementary math at LSU, where Scott Baldridge now teaches. It was very disconcerting to read Liping Ma and realize that elementary math teaching was a calling, a craft, that I hadn't been sufficiently aware of or respectful of. I didn't make the errors that Liping Ma's American teachers made, but when I ran out of teaching methods and understanding and patience for students who were struggling, her Chinese teachers were just getting warmed up. They could go the distance with the deepest questions and misunderstandings that their students could have.

Take, for example, something Catherine complains of frequently; that she hasn't a visual image to go with dividing a fraction by a fraction. She hasn't a visual intuition of why multiplying the numerator by the reciprocal of the denominator is the right thing to do, and it's been driving her nuts.

I don't have a visual image to go with dividing a fraction by a fraction, either; but I don't have any problem with that, or at least I never did before. I just applied the rule, and moved on.

And once, if I'd had Catherine in my math class, I would have told her to do the same.

But if you've got someone like Catherine -- the ideal student, who knows what she's doesn't know and demands to be taught it -- or if you have a child that you care about and want to help, then you have to be intellectually honest, set aside your ego (if you're me), and build up your own deepest levels of understanding in preparation. This is what P&B can help you to do.

More on this topic -- and the answer to Catherine's conceptualization problem -- tomorrow.

Parker and Baldridge introduce the topic beautifully, emphasizing for their teacher-students the concepts that are most likely to trip up students. Kids tend to be stunned by the terms 'numerator' and 'denominator' at first, so P&B suggests that teachers eschew it at first, substituting 'top' and 'bottom'. Kids tend to think of a fraction as somehow representing two independent numbers, and so fail to see that a fraction is a single number. This results in the common misconception/error:

a/b + c/d = (a+b)/(c+d).

P&B encourage the teacher-student to think of a denominator, instead, as representing the fractional unit into which the whole has been divided. In fact, it can be useful to think of different denominators as being like different units entirely. It doesn't make sense to add quantities represented in feet and meters without doing a conversion to a common unit first; similarly, fractions must be converted to a common unit (i.e., denominator) before they can be added. The Singapore-style line drawings that illustrate this conversion in P&B are well-done and get the point across clearly.

One tidbit that I got out of P&B was an answer to Catherine's question: why is multiplying the numerator by the reciprocal when dividing fractions the right thing to do?

The easiest fraction-division case to understand, they point out, is the case where both the numerator and denominator have the denominator in common, as in the problem:

6/2 3/2.

In this case, one should envision a bar representing 1/2; if we have 6 of these bars, and divide them into groups of 3 such bars, clearly we will have two groups. But whether a bar represents 1/2 or a whole is irrelevant, as long as the bars represent the same quantity; and so this is really the same problem as

6 3.

Therefore, fraction division is pretty easy to understand if you have common denominators. But, of course, any two fractions can be converted to have common denominators. So now look at the general problem:

a/b c/d.

To figure out what we get, we convert both fractions to have a common denominator, b x d:

(ad)/(bd) (cb)/(db).

Now we have a common denominator bd, and so this problem is equivalent to the problem of dividing their numerators::

(ad) (cb) = ad/bc.

But note that this is exactly the usual multiplication-by-the-reciprocal formula:

a/b x d/c.

And that is why multiplying by the reciprocal is the right thing to do (incidentally, in a comment on this thread, J.D. Fisher mentioned that the key to understanding practically everything about fractions is to think about fractions that have a common denominator. Once again, he turns out to have been right).

This is one of the very best books I've read on math education. Wonderful. Well worth the price.

Here she is on middle school math:

The current middle school curriculum as described in the TIMSS data lacks intellectual rigor. In fact, the topics covered in the United States' seventh- and eighth-grade classrooms are much like those covered in third and fourth grades--lots of arithmetic (Schmidt et al., 1999, p. 49). In Japan and Korea, arithmetic is taught for mastery in those early grades and students then move on to a more algebra- and geometry-centered curriculum. One of the most disappointing aspects of the TIMSS report as it described the United States was what a small amount of new learning actually occured during the eighth grade. Since both seventh- and eighth-graders took the same test, researchers had the unique opportunity of creating a quasi-longitudinal study. Sadly there was no significant difference between the scores of U.S. students at the end of seventh and eighth grades.

'no significant difference between the scores of U.S. students at the end of seventh and eighth grades'

school starts soon
a gift for your principal

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GiftForPrincipals 15 Aug 2005 - 12:27 CatherineJohnson


I think it's worth posting the one reader review of Elaine McEwan's The Principal's Guide to Raising Math Achievement:

Having read this book as a parent and a school board member, I am giving it to both the principals in my district. This book explains both many of the things that are done badly in many schools in the country and shows the path for how to do them well. I found the comparisons with the Japanese and Chinese methods of teaching particularly helpful. This book was pleasant to read as well as enlightening in how to promote the effective teaching of mathematics.

That is not a bad idea. I was on the verge of buying a copy for our principal (whose wife is a high school math teacher) all year long. I didn't do it, ultimately, because the book is awfully pricey ($28 for a paperback).

It's still a good idea.


Principal's Guide to Raising Math Achievement
school starts soon

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HaroldStevensonRIP 15 Aug 2005 - 20:07 CatherineJohnson


I've just had one of those strange synchronicity moments.

Last night, after talking to Caroline about E.D. Hirsch's The Schools We Need and Why We Don't Have Them (which Caroline was raving about), I went to my bookshelves & pulled out Hirsch's book, determined to read it at last.

But then I pulled out Harold Stevenson & James Stigler's The Learning Gap: Why Our Schools Are Failing and What We Can Learn from Japanese and Chinese Education, which I've been reading off and on today. This morning, too, I returned to Nisbett's Geography of Thought....and was in the midst of writing my follow-up post on Asians and math when I discovered that Harold Stevenson has died, 3 weeks ago, at the age of 80.

from his obituary in the GLOBE:

The book punctured stereotypes of Asian elementary schools as high-pressured learning factories and illuminated what many specialists came to agree were grave deficiencies in the US education system, including weak academic standards, overburdened teachers, and misguided cultural beliefs about parental roles and the importance of individual student effort....

Although educators had known as early as the 1960s that Japanese and other Asian students ranked higher than Americans on international assessments of academic achievement, the explanations were ''too often cloaked in speculation," said Jack Schwille, assistant dean for international studies in education at Michigan State University. ''Stevenson collected data on classroom teaching and learning [that] could help explain the differences," Schwille said, ''and he got educators and laypersons to pay attention to them."

Dr. Stevenson's work was often cited during the national debate over education standards in the late 1980s and early 1990s, particularly in discussions of US students' poor mastery of math. He argued that US educators would do well to emulate the systems in Japan and Taiwan, where learning goals are carefully plotted and clearly defined, and creative hands-on exercises are considered crucial.

At the core of the Asian schools' success in math, Dr. Stevenson believed, were thoroughly trained teachers who were given ample support during the school day to craft lessons and share ideas with colleagues.

''Stevenson's work made clear the kind of education that was really going on in Asia . . . and helped pave the way for some improvements we see now, especially in California," said David Klein, a mathematics professor at California State University, Northridge, who has been active in the movement to strengthen math teaching in the United States....

The researchers eventually focused their inquiries on math achievement because the gap between American and Asian students in that subject was so wide. By fifth grade, Dr. Stevenson and Stigler found, the lowest-scoring Japanese classroom still outperformed the highest-scoring US classroom.....

In contrast to the Japanese, American teachers were loathe to submit their students to public scrutiny out of fear it would damage the youngsters' self-esteem, Dr. Stevenson and his colleagues found. Moreover, US teachers often segregated students into low- and high-ability groups, a practice that Stevenson said reflected a deeply held belief that not all students could succeed.

Another important difference he found was that Japanese and Chinese teachers received considerably more time during the school day to prepare lessons, discuss goals with other teachers and work with individual students. On average, they spent only three to five hours a day in front of a classroom.

In the United States, however, ''we keep teachers busy in front of the classroom all day long," Dr. Stevenson told The Dallas Morning News in 1993. ''We deprive teachers of opportunities for . . . extending their knowledge, both in the subject area they're teaching and also in methods, so that it's very difficult for American teachers to do a good job."

Ed and I knew James Stigler a little at UCLA, and we saw his videos of Japanese math classes there. He was a terrific guy.


how Asians and Westerners think differently
how Asians and Westerners think differently, part 2
How Asians & westerners think differently, part 3
Harold Stevens, RIP
describe this picture
creativity gap, part 2

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GeorgeOrwellPartTwo 15 Aug 2005 - 22:39 CatherineJohnson


an online searchable database of George Orwell's 1984!

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MoreMetacognition 23 Aug 2005 - 00:25 CatherineJohnson

THE SAD TRUTH is that there are only three kinds of financial prognosticators: those who don't know, those who don't know they don't know, and those who know they don't know but who get paid big bucks to pretend they know.

from The Random Walk Guide to Investing by Burton G. Malkiel

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MontyHallPart3 28 Aug 2005 - 02:39 CatherineJohnson


A ktm reader (I'm sorry--I've forgotten who it was) mentioned that Mark Haddon has a nice illustration of the Monty Hall problem in his novel The Curious Incident of the Dog in the Night-Time.

He does, and it's terrific:

update

Wow. Carolyn's search engines are fantastic.

I searched Comments & discovered that it was Greta Frohbieter who left the tip about Curious Incident.

Thanks, Greta!

update update

One of the things I like about this chart is that you can see that you are 'still in the same event' from start to finish.

The reason people think the odds change from 1 in 3 to 1 in 2 is that they see the second choice (stick or change) as a secont event, with a second set of odds.

This visual representation makes you feel that the event is ongoing. You haven't changed odds because you haven't changed events.

It's the Unbearable Seamlessness of Being.


low birth weight paradox (& Monty Hall)
Monty Hall, part 2
Monty Hall, part 3
false positives
false positives, part 2
Doug Sundseth on Monty Hall
John Kay: We are likely to get probability wrong (subscription only)
Monty Hall diagram from Curious Incident
probability question from Saxon 8/7

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BookRecommendationAboutReading 09 Sep 2005 - 17:55 CarolynJohnston


Ben has an independent reading project for school. He is in a class called "Reading Clinic", which is supposed to be for weak readers, and in fact he is a weak reader of sorts. However, he can read at speed, and he'll actually get most of the story; it's the subtleties he misses -- the inferences, the innuendoes, and sometimes the main idea of the story. You'd be amazed how much of readng requires you to make inferences.

The kids went to the library the other day and picked out books to read. Ben has been enjoying Goosebumps, so one of his buddies helped him find an R. L. Stine book. Unfortunately, the Stine book wasn't the usual Goosebumps sort of book -- i.e., it wasn't about vampires or ghosts or zombies -- it was about some guy who was stalking babysitters. I don't really want to explain about stalking babysitters to Ben.

So I want to find something else for him to read, something in the horror genre that's maybe a little bit of a departure from the Goosebumps thing -- something by a different author, perhaps. So tonight I went digging around the house looking for one of my favorite parent books on kids' reading -- Parents Who Love Reading, Kids Who Don't, by Mary Leonhardt.

Ms. Leonhardt is a teacher who developed a simple approach for getting kids to become avid readers. It's actually more of a philosophy than a teaching approach. Her attitude is that you let them read whatever appeals to them, whether you personally think it's trashy or not. So comics are in, celebrity tabloids are in, Danielle Steele is in. Reading all the books by a single author that they love is fine. You try to hook them into reading, and then you count on their branching out on their own.

I've been using this philosophy for years with Ben, who is an especially tough case because of the autism spectrum disorder; he has a much greater tolerance for repetition than most of us do, and won't necessarily branch out on his own. He has to be gentled along. All through elementary school, though he hated reading, he was at least reading comics; particularly Calvin and Hobbes (which I love) and Garfield (which I don't -- and you'd be amazed how many Garfield comic collections there are in print). Last year he branched out a bit and started enjoying the Foxtrot comic; I tried him on Bloom County, but he didn't like it. This year he's loving his first chapter books, the Goosebumps series.

There is more to love about Ms. Leonhardt's book than her attitude toward kids' reading. I like her advice to parents about dealing with big problems their kids are having at school. Some teachers, she says, do enjoy emotionally battering children; if your kid gets one of these, move heaven and earth to get the kid out of their classroom. She has extensive advice on how to help your child if he is doing poorly in school (based partly on some insight a friend gave her into how to deal with panic attacks --often, simply knowing there's a way out will calm a person enough to allow them to carry on!).

It's got 5 stars from 4 reader reviews at Amazon.

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PaulosBooks 12 Sep 2005 - 00:54 CarolynJohnston


I went to the library today, among other things to look up a book I've been curious about reading -- Innumeracy, by John Allen Paulos. They didn't have it, and instead I ended up picking up another similar book by the same author, A Mathematician Reads the Newspaper.

I was wondering if anyone had read either of these books? There are 26 reader reviews of "A Mathematician Reads the Newspaper"; some love it, and some do not. I get the impression that a lot of it covers the same ground as books like "How to Lie With Statistics", and "Lies, Damned Lies and Statistics", not to mention "Innumeracy" by the same author.

By the way, if you enjoyed Richard Feynman's "Surely You're Joking" book, try Adventures of a Mathematician, by Stanislaw Ulam. Ulam's book was published well before Feynman's, and he was also in Los Alamos for the Manhattan Project; the first description I ever read of Feynman picking the locks at Los Alamos was from Ulam. I read it when I was young and impressionable, and it left me with the attitude that being a mathematician is Really Cool (thereby, no doubt, sealing my subsequent fate).

Coda

There is also a website called innumeracy.com (it's unrelated to John Allen Paulos). It bills itself as a collection of links to articles and sites pertaining to numeracy and critical thinking. I haven't checked it out (it's rather disorganized) but there were a couple of very interesting-sounding articles linked on it.


books by Paulos
book rec: What the Numbers Say
false positives & Bayesian statistics

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BookRecommendation 11 Sep 2005 - 18:27 CatherineJohnson


V doesn't think too much of John Allen Paulos's Innumeracy, which got me to thinking: I don't believe I've ever read, all the way through, an entire book devoted to debunking the misconceptions of the American Reading Public. (Or even the American Writing Public, for that matter; I can't get throuh entire books on the many grievous errors committed by the press.)

And now that I think about it a bit more, there's a reason for that. It's a waste of time.

I don't need a mathematician to tell me most people don't understand math. I'm aware most people don't understand math; I don't understand math myself.

I need a mathematician to help me join the tiny group of people who do understand math.

So it looks like I'll be reading every last page of a fantastic book published just last year: What the Numbers Say : A Field Guide to Mastering Our Numerical World by Derrick Niederman, David Boyum. Check out the reviews on Amazon. 6 5-star reviews (including one by Arnold Kling) and 1 4-star review.

It's wonderful.


I've been planning to put up some posts about the FIELD GUIDE. Here's a link to the one I wrote a few weeks ago on false positives.








books by Paulos
book rec: What the Numbers Say
false positives & Bayesian statistics

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CalculusBookRecommendationNeeded 15 Sep 2005 - 17:27 CatherineJohnson


A lot of good stuff in the comments I want to get pulled up front, but since I have to go into the city today, there's no time at the moment.

I'll just get this posted, from Anne:

Speaking of good books, does anyone have a recommendation for a good calculus book? I still have mine from college, but I have a hard time understanding the proofs. It seems like there are steps missing.

In the book Countdown about the math Olympiad, the author mentions that one of the mathematicians who constructs the problems is Russian. This mathematician says that his calculus book was only 100 pages long, but it was excellent and had no extraneous information. I wonder if anyone has translated a Russian calculus book.

I've been wondering the same thing, and ktm needs a recommendation to post as well.

So if you've got suggestions, please let us know.

I have two, potentially.

Calculus Made Easy by Sylvanus P. Thompson (and updated/revised by Martin Gardner) This is a classic (always a good sign), and people rave about it. I don't know whether it has proofs, or whether the idea is to give people conceptual understanding without formal proofs.

Also, believe it or not, the University of Chicago School Mathematics Project, the same folks who are responsible for EVERYDAY MATH, had a longrunning project translating foreign math textbooks into English. I'm not sure I can track down what's happened to the list; it seems to have moved to the American Mathematical Society, but I can't find it there at the moment.

I know I did once track it down...so I assume it's still findable.

If someone else comes across it before I do, could you post the link?

Thanks.

translation from the Russian

Calculus of Variations by I.M. Gelfand & S.V. Fromin

Is this the one?

update

Bernie & others say the Gelfand book is an advanced text. (I didn't have time to read the blurb yesterday.)

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CalculusRecommendations 22 Dec 2005 - 16:49 CatherineJohnson


OK, I've collected a handful of recommendations.

Michael Spivak

First, check out the Comments thread on calculus books.

Here's one interesting comment:

Michael Spivak's books are good, as is Tom Apostol's Calculus. Personally, I prefer Spivak. They are both Americans by the way. G.H. Hardy's A Course of Pure Mathematics, and Richard Courant's Differential and Integral Calculus are both classics which are very good, but probably not for everyone. Those are all longer than 100 pages. If you are looking for brevity then you can try out Dan Bernstein's(another American) "Calculus for mathematicians" which is only 12 pages. Find it here: More Mathematics .

None of these books are typical of what you will find in the modern science/engineering calculus courses. If you want something along those lines, then I'd recommend Salas, Hille, and Etgen's Calculus: One and Several Variables.

Fomin and Gelfand's book considers calculus of variations as opposed to calculus of real variables(i.e. "standard" calculus). It's a good book, but probably not what you are looking for.

People love Spivak.

oops. Just clicked on 'See all 60 customer reviews.' Some people love him, some hate him.

Here's Apostol.

Purcell, Varberg & Rigdon

I've asked both David Klein & Barry Garelick for recommendations.

Here is Klein:

I'm not up on calculus texts. I use a standard book (one of many) along with others at CSU Northridge called, CALCULUS WITH ANALYTIC GEOMETRY, 8th ed., by Purcell, Varberg, and Rigdon. It has its faults, but isn't bad. The theory part is good, but it needs more medium level difficulty problems and more graphing examples (without calculator assistance). [One Amazon reviewer loathes it; the other likes.]

Worth avoiding in my opinion is the so-called "Harvard Calculus" books:

Calculus Reform—For the $Millions by David Klein and Jerry Rosen (you'll have to register to open this pdf file, but registration is free)
WHAT IS WRONG WITH HARVARD CALCULUS? by Jerry Rosen and David Klein

Subsequent editions have remedied the worst of the deficiencies, but I would still avoid it.

(I should add that I think Carolyn somewhat liked reform calculus. She's in transit at the moment, but when she chimes in, I'll either edit out this comment, or add hers as needed.....)

Ivan Niven

Barry's first suggestion, which comes from Dick Askey of the University of Wisconsin, is Calculus: An Introductory Approach by Ivan Niven.

I'm sorry to say I've bought the one and only used copy available at Amazon, but there are 2 copies available at Alibris.

Niven wrote his book in 1961, before graphing calculators.

Lipman Bers

Another recommendation from Barry:

Calculus by Lipman Bers, which I ordered the minute I read this Amazon review:

I had come across this book in the university library.

Before that I had been getting excellent marks in Calculus by mechanically going thru the rules in my mind. This book changed all that and gave me a proper perspective on the discipline.

The explanations are clear and this book is eminently suitable for self study.

Recommend this book whole-heartedly at least for the first and second years of calculus.

This was about twenty-five years ago ! But it's just as relevant now.

Barry says Bers was a Russian mathematician emigre to the U.S. who was familiar with Russian textbooks.

Thomas's Calculus

Another possibility might be an early edition of Calculus by George B. Thomas, now in its 11th edition.

Barry says Thomas's Calculus was a college staple for years, and is not easy.

I'm having trouble finding out when Thomas died, so I have no idea which editions of Thomas' Calculus were revised after he was gone.

.....oh, here's a clue, in an Amazon reader review:

I've used both Stewart's Calculus and Thomas'. Interestingly, Thomas has been writing calculus books for a LONG time and i've picked up several editions in the used book stores, because from the first time i bought a Thomas calc book back in Jr. High for my own self interest, i was a fan of his style.

His style is that of the old-school American text book authors who wrote in a clear, concise manner of English, using tangible and visual examples. Those old writers still thought of much of the material as novel, and were appealing to a more agrarian society of students.. especially the young and booming field of engineers. This is lacking in today's texts. The only drawback is that some old texts are much too impersonal and use the passive voice for everything, which can make them very difficult to read at times.

Thomas' recent editions (at least - i can not recall for the 60's era editions) are not only formally clear, but easy to understand and read. Here are the ways in which Thomas' book beats Stewart's book....

[snip]

Thomas' book is in fact probably the best calculus textbook around. I've looked at many many of them, and fraknly, none of them are this complete and well developed... The funny thing is, Thomas' book was one of the best decades ago. It has only gotten more exhaustive and more mature!

This reminds me of Carolyn's post about the early books in a field, Don't teach in a monotone

Thomas has 5-star & 1-star reviews. Very mixed.

James Stewart

Lastly, Barry reports that James Stewart's texts, which teach graphing calculators, are being used a great deal. Barry says Stewart's books are 'fairly good.'

The two big ones seem to be:

Calculus : Concepts and Contexts (with CD-ROM, Make the Grade, and InfoTrac)

and

Calculus

Mixed reviews, expensive as the dickens.

off-topic: Arnold Kling

I just found all of Arnold Kling's Amazon reviews....

'the calculus page'

No idea if this is worthwhile: calculus.org: THE CALCULUS PAGE

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BernieOnCalculus 14 Sep 2005 - 22:37 CatherineJohnson


First off, I've become very wary of Amazon's reader reviews ever since I realized that they remove negative comments in order to boost the ratings of the books. That's not kosher. [Catherine speaking: I posted 2 5-star reviews on Amazon that have disappeared, so I'm not sure Amazon has a systematic policy against negative reviews....]

Ok, what's the big deal about Calculus? Why are there thousands of Calculus books and none of them any good?

The reason is that the subject is simultaneously too big and too deep. And there's really no good way to split it up into manageable digestible pieces.

If you want to understand a computer, say, you can split it into pieces (power, case, motherboard, plug-in cards) which are you can then study and understand separately. But with Calculus, learning the subject is more like approaching a huge ship in the fog. At first you don't have any idea what is there. Then a few points become clear, but they are disconnected and make no sense. Then a few structures show themselves, and gradually, very gradually, the whole thing starts to come together. It takes much more energy and much more determination to carry through with such a program than with simpler subjects. So most people don't carry through with it, and it becomes a filter, a flunk-out class.

Linear algebra is a much more useful subject which is amenable to being broken into manageable chunks, and perhaps for this reason it doesn't carry the same mystique as Calculus.

Let's lay out what Calculus is in order to make this clear. It consists of two new operations called "differentiation" and "integration"--roughly analogous to subtracting and adding--both of which are based on a totally new view of the world, called "limits". Limits are a pretty deep concept, much deeper than is generally supposed or understood by most people taking Calculus. In fact, I would venture to say that most people taking Calculus never really grasp limits and, as a result, end up more confused and resentful about mathematics than when they started. Moreover, limits cannot be tackled until one has already achieved a certain mastery of both algebra and geometry, for they entail a melding of these two subjects. Both subjects must have been learned down to the "have it at my fingertips" level before limits will start to make sense.

To be perfectly honest, the problem is even worse than that, because I think it's fair to say that in some sense the human race doesn't really understand Calculus yet. This is because, although there is complete agreement on what basic Calculus is and how to use it, there is still sharp disagreement on what the logical underpinnings of it should be. It's really kind of like Quantum Mechanics in this regard, and that makes it quite unlike all the other kinds of mathematics young students have ever seen, which is all cut and dried.

So, to take the larger view once more, Calculus has three aspects which the student must master more or less simultaneously: 1) the mechanics of integration and differentiation and limits, 2) a philosophical understanding of limits, 3) the thing we discussed yesterday--an understanding of the underlying meaning of the formalism of Calculus in terms of real-world problems. Because there is so much interconnected stuff to learn, the connection between formalism and real-world meaning is even more tenuous, and must be held in even greater abeyance, than is the case with standard school mathematics. The student must suspend disbelief for a much longer period than ever before. Which means that there are inevitably many more Calculus students who get left by the wayside than occurs in elementary mathematics.

It is generally accepted among mathematicians that the hardest part of learning Calculus is 2), the philosophical part, and therefore the teaching of Calculus is usually broken into two subjects, taught to two different groups. "Mechanical Calculus" (high-school Calculus) is ta