66

This is a really interesting question, because similar issues---the question of how demanding to be about formatting of answers---come up a lot, at all levels, and the answers aren't always straightforward. In this case I think your colleagues are straightforwardly wrong, for the following reasons: They're wrong about what students who write "x=1" think. ...


27

I would like to encourage consideration of a free textbook. The conventional textbooks are outrageously expensive. (Actually, if your department insists on one of the choices you mentioned, I'd want to go with the cheapest.) Students are suffering from massive amounts of debt these days, with no guarantees of good jobs with which to pay off the debt. ...


24

Ask your pedantic colleagues to reconcile their formatting expectations with the context "solve $F=m\cdot a$ for $a$". Would they prefer to see just $\frac{F}{m}$, or maybe this expression jazzed up with set brackets somehow? This formatting of the answer looks jarring to me. I think most educators and scientists expect to see the answer presented as $a=\...


23

What is a good reason to change calculus texts? Reason #1: the text you're currently using costs money. Reason #2: your current text uses an online homework system such as MyMathLab that costs money, rather than a free one like WeBWorK. Some good free texts: Robbin and Angenent - http://www.math.wisc.edu/~angenent/Free-Lecture-Notes/ Boelkins - http://...


23

Spivak proves $\sqrt{2}$'s irrationality fully, but banishes $a^{1/b}$'s to the exercises. Isn't proving (only) the latter more efficient? Yes, it's certainly more efficient. So what? Efficiency is not a primary goal of a textbook. Generality often makes concepts harder to understand. That's true for professional mathematicians, and it's even more true ...


21

Having answers to all the problems in a book is often inconvenient for teachers. For very elementary topics in which not much work needs to be shown by students (such as what is the domain of $f(x) = \frac{x}{x^2 – 4}),$ allowing students to have the answers makes it more difficult to grade homework. For example, suppose a student has the correct answer and ...


19

This might be taking things too far, but Kiesler's book (available free online) does everything using infinitesimals, which make differentials literally immediate. The rigorous underpinning for infinitesimals is nonstandard analysis, but this book doesn't dwell on that. It just teaches how to use them correctly. I'm guessing this isn't exactly what you were ...


19

What are the most important considerations when writing your own lecture notes for a course? I would say that the single most important consideration is to make life easy for yourself. It is easy to fill your time with writing lecture notes but that isn't always the best use of that time. Nevertheless, it can be useful for the students so here are my tips....


16

This semester I adopted Linear Algebra by Jim Hefferon for the junior-level linear algebra course. I didn't have any particular difficulty adopting it, the bookstore was able to arrange printing through some print shop and it was fairly easy given the free availability of the text. I'm not sure how many students actually purchased a copy. Once nice thing ...


15

I have recently been a homeschool parent, as well as a school maths teacher so I hope I have some perspective that can help. I will suggest a textbook that does what you want, hidden somewhere in this post. But I would like you to understand a few things first. It is very likely that your son has never done any mathematics. He has done years of arithmetic. ...


14

I think you'll find some of what you want on Berkley mathematician H.H. Wu's homepage. More precisely, see: Pre-Algebra (pdf) and Introduction to School Algebra (pdf). Note: I mentioned the same homepage (and the two pdf textbooks) in an earlier MESE post here; I would have just re-posted this as a comment, but I believe it is the actual answer to your ...


14

Not an "opinionated" book per se, but Tom Apostol's Calculus follows up the chronological order of the concepts of the calculus. Hence, it starts with an example of Archimedes exhaustion method and after defines the integrals before limits as: Let $f$ be a function defined and bounded on $[a, b]$. Let $s$ and $t $ denote arbitrary step functions ...


14

Permit me to recycle a portion of my answer to an MSE question: Tristan Needham, Visual Complex Analysis, Oxford Univ. Press. 1997.            This book "brings a beautiful and powerful subject to life by consistently using geometry (not calculation) as the means of explanation." [quoting the jacket] ...


14

Preface. Birkhoff & Mac Lane's Algebra is a brilliant book. I should probably spend some time with it again, actually. Also, I apologize for such a long response. I think too much about algebra pedagogy and textbooks. The short version is I think the book can be used for either undergraduates or graduates with some success, but I think it is less than ...


13

While I haven't used an open-source text for a semester-long university class, I have used an open-source calculus text for a summer program with high school students, and I'm currently modifying a number-theory text for a different summer program. Summary, with extended discussion below: Upsides: I can modify the text to suit my needs. I can modify the ...


13

At my university we were using the 6th edition of Stewart and the publisher informed us that they would no longer be willing to sell this book (even at exorbitant prices). If we wanted to continue with Stewart, we would have to "upgrade" to the 7th edition, wiping out most of the benefits of continuity: students would no longer be able to sell their used ...


13

That is the book that had the most deleterious effect on me in my life! Since the courses in (advanced) calculus I had as an undergraduate were very old-fashioned (emphasis on esoteric criteria for convergence like Raabe's test, nonsensical definition of differentials, ..), a young teaching assistant told me to read Dieudonné's book. I tried to, but the ...


13

An proof is meant to convince a reader of the truth of some statement. When a mathematician is communicating an argument to another mathematician, you only include the level of rigor that you need so that the other mathematician is convinced that they could (in principle) give a fully rigorous argument. Even that isn't quite accurate because no one (short ...


11

A motivation for writing course notes, when textbooks/monographs already exist, is to omit some things, first. This might seem strange, until one observes that many standard texts which go through several editions often succumb to the natural pressure, amplified by publishers, to become ever-more encyclopedic. The latter has its uses as reference, but ...


11

Based on extensive (if anecdotal) experience, undergrads really cannot cope with more than a single reference, which must be traversed in order, possibly omitting some sections. Having any other source is somehow beyond imagination. Subject=course=textbook. Grad students can do somewhat better, but are not happy about having to operate at a higher level. ...


11

There are certainly many more. The following list is not meant to be exhaustive but meant just to give you a selection of the many more books that you have missed: Halmos: Linear Algebra Problem Book, A Hilbert Space Problem Book. George Polya and Gabor Szegö: Problems and Theorems in Analysis - I, II. Ram Murty: Problems in Analytic Number Theory. Jody ...


11

I've found it more natural, and more helpful for the students, to arrange the out-of-class work around the learning objectives you have for your instructional units rather than the number of pages in a book. There's not always a natural linear relationship between page numbers and concepts -- sometimes a concept that takes a couple of pages takes twice as ...


11

There are two formulations for definite integrals: $$\int_{\phi(a)}^{\phi(b)} f(x)\, dx=\int_a^b f(\phi(t))\phi'(t)\, dt$$ and the one you state: $$\int_{\phi([a,b]}f(x)\,dx=\int_{[a,b]} f(\phi(t))|\phi'(t)|\, dt$$ In the second, you do need $\phi$ to be monotone. In the first formulation, you do not need this assumption. Of course when you apply the ...


11

I'll answer the question with respect to the Green's/Stokes' theorem special case. The summary is that, from a pedagogical point of view, little is gained by passing directly to the general case, and much is potentially lost (students for example). If by Stokes' theorem is meant the version relating surface integrals and line integrals, this theorem is ...


11

At the moment, I can answer bullet point two: Are there any high school textbooks that explicitly acknowledge that the methods included in the text are not adequate to solve all 3rd and 4th degree polynomial equations, and that in higher degrees that are no general methods at all? Yes, you can find this on p. 267 of CME Project's (2009) Algebra 2 text. ...


10

It seems to me that the question rests on a false dichotomy: either (1) the book has cartoons or artwork, or (2) the book consists of non-exciting material which the author expects the reader to memorize by rote. However, in mathematical writing I think that (1) and (2) are almost orthogonal. (Diagrams are certainly useful in some fields of math, such as ...


10

My instinctive reaction is that a "category error" is being made here (in the philosophical sense, not the mathematical sense of category). Namely, category theory is an abstraction of (standard, undergraduate level) abstract algebra, which is itself an abstraction of the sort of very concrete mathematical manipulations most students have seen up to that ...


10

In case you have not yet seen it, I thought I would draw your attention to (what is currently) the most recent issue of the American Mathematical Monthly, and, in particular, the article: Leinster, T. Rethinking Set Theory. The American Mathematical Monthly, 121(5), pp. 403-415. An arXiv version can be found here. The abstract says: Mathematicians ...


10

The compartmentalization drives me crazy. I'll have students who can use the distributive law perfectly when we're on the "distributive law" chapter in the book. But if I later give them a problem to solve 37 x 40 + 37 x 48 + 37 x 12 in another context, they will do the 3 multiplications out by hand and never notice that that could have been simplified to ...


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