Category Archives: Textbooks

More enVisionMATH: Adding “near doubles”

The last post about enVisionMATH and how I, as a math person and dad, go about trying to make sense of what my 6-year old brings home from first grade seems to have struck a chord among parents. The comments have been outstanding and there seems to be a real need for this kind of conversation. So I have a few more such posts coming up soon, starting with this one.

The 6-year old brought this home on Monday. Click to enlarge:

It’s about adding “near doubles”, like 3 + 4 or 2 + 3. In case you can’t read the top part or can’t enlarge the photo, here are the steps — yes, there are steps, and that’s kind of the point of this post — for adding near doubles:

  1. “You can use a double to add a near double.” It gives: 4 + 5 and shows four blue balls and five green balls.
  2. “First double the 4”. It shows 4 + 4 = 8, and the four blue balls, and four of the green balls with the extra green ball sort of falling to the ground.
  3. Then it says: “4 + 5 is 4 + 4 and 1 more.” At this point you really have to look at the worksheet itself, because it’s hard to put into words what is going on:

And from there, in the fourth frame, one of the girls in the earlier frame concludes that 4 + 5 must be 9 because 8 and 1 more is 9.

The Guided Practice section has the kids doing four near-double sums. Clearly, the way the worksheet wants kids to learn how to do this is not simply to add 2 + 3, but (1) to recognize that 3 is 2 plus 1 more, (2) add 2 + 2, and (3) then add 1 to the result of 2 + 2:

There’s a thing at the bottom asking kids to explain the process and then a bunch of near-double sums to practice — presumably kids are supposed to use the method described above, but there’s nothing forcing them to do so — and some “algebra” questions with blanks in the place of variables.

I’m not sure exactly how my brain goes about adding near doubles — whether it just somehow does the addition in ways that are almost automatic thanks to 35 years of repetition, or whether there are little tricks it employs — but I am absolutely certain that  I don’t do it the way enVisionMATH is telling kids how to do it. I tried it. When I read the worksheet, I thought about near-doubles that aren’t so easy, like 121 and 122. Quick! Add those together. Did you think, “122 is 121 plus one more; 121 + 121 = 242; 242 and 1 more is 243” ? I didn’t — not by a long shot. I just added the numbers together. No methods, no tricks; just old-school addition. There may be some tricks that my brain invokes to “just add the numbers” — for example, I tend to visualize the two terms of the sum stacked atop each other in the classic vertical arrangement for adding, and then visually add the digits — but I am most definitely not going through the four-step process on this worksheet.

In fact, the four-step process complicates matters so much that it’s inexplicable why they are even bringing it up. Most kids at this stage can add 2 + 3 or 5 + 6 in one step. But by introducing this method, there are four operations: comparison (find the larger of the two near-doubles), subtraction (take 1 from the larger number), addition (add the two duplicates), and another addition (add 1 to the result). Technically there is a fifth operation kids have to perform, namely recognize that the two numbers they are adding are near-doubles in the first place.

One might argue that doubling a number (in the third step) is easier than adding it to itself — kids just recognize that doubling 5 gives 10, for instance — and subtracting 1 is a very easy special case of subtraction in general that nearly everybody at this age can do without thinking, similarly for adding 1 at the end. That may be so, but it can’t be so much easier that adding in steps 1, 2 and 4 results in a net reduction in complexity or a net gain in conceptual understanding.

But what about kids who can’t add two one-digit numbers together in one step? There are some of those out there, including probably a few in my daughter’s class. This method doesn’t help those kids. Again, we may argue that adding 4 + 5 is considerably harder than the combined process of comparison, subtracting 1 from 5, doubling 4, then adding 1. But I don’t think so. A four-step process is no less cognitively demanding than a single-step process, even if the four steps are easy. And besides, life does not throw near-doubles at you to add. How is a kid going to learn to add 2 + 5, or 2/5 + 7/8, or 123.38 and 99.99 this way?

If there is some research that suggests that people really do add near doubles this way, I would love to see it. Otherwise it’s hard for me to believe that any more than a tiny fraction of the human population actually does it this way. Is there going to be some mind-blowingly cool way to do complicated arithmetic in one’s head farther down the road that uses this idea, like multiplying numbers that are near-squares or something? Perhaps I should be more patient. But for the time being, I told the 6-year old just to add the numbers together like she already knows how to do, the old-fashioned way.

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Filed under Early education, Education, enVisionMATH, Math, Teaching, Textbooks

Calculus and conceptual frameworks

I was having a conversation recently with a colleague who might be teaching a section of our intro programming course this fall. In sharing my experiences about teaching programming from the MATLAB course, I mentioned that the thing that is really hard about teaching programming is that students often lack a conceptual framework for what they’re learning. That is, they lack a mental structure into which they can place the topics and concepts they’re learning and then see those ideas in their proper place and relationship to each other. Expert learners — like some students who are taking an intro programming course but have been coding since they were 6 years old — have this framework, and the course is a breeze. Others, possibly a large majority of students in a class, have never done any kind of programming, and they will be incapable of really learning programming until they build a conceptual framework to  handle it. And it’s the prof’s job to help them build it.

Afterwards, I thought, this is why teaching intro programming is harder than teaching calculus. Because students who make it all the way into a college calculus surely have a well-developed conceptual framework for mathematics and can understand where the topics and methods in calculus should fit. Right? Hello?

It then hit me just how wrong I was. Students coming into calculus, even if they’ve had the course before in high school, are not guaranteed to have anything like an appropriate conceptual framework for calculus. Some students may have no conceptual framework at all for calculus — they’ll be like intro programming students who have never coded — and so when they see calculus concepts, they’ll revert back to their conceptual frameworks built in prior math courses, which might be robust and might not be. But even then, students may have multiple, mutually contradictory frameworks for mathematics generally owing to different pedagogies, curricula, or experiences with math in the past.

Take, for example, the typical first contact that calculus students get with actual calculus in the Stewart textbook: The tangent problem. The very first example of section 2.1 is a prototype of this problem, and it reads: Find an equation of the tangent line to the parabola y = x^2 at the point P(1,1). What follows is the usual initial solution: (1) pick a point Q near (1,1), (2) calculate the slope of the secant line, (3) move Q closer to P and recalculate, and then (4) repeat until the differences between successive approximations dips below some tolerance level.

What is a student going to do with this example? The ideal case — what we think of as a proper conceptual handling of the ideas in the example — would be that the student focuses on the nature of the problem (I am trying to find the slope of a tangent line to a graph at a point), the data involved in the problem (I am given the formula for the function and the point where the tangent line goes), and most importantly the motivation for the problem and why we need something new (I’ve never had to calculate the slope of a line given only one point on it). As the student reads the problem, framed properly in this way, s/he learns: I can find the slope of a tangent line using successive approximations of secant lines, if the difference in approximations dips below a certain tolerance level. The student is then ready for example 2 of this section, which is an application to finding the rate at which a charge on a capacitor is discharged. Importantly, there is no formula for the function in example 2, just a graph.

But the problem is that most students adopt a conceptual framework that worked for them in their earlier courses, which can be summarized as: Math is about getting right answers to the odd-numbered exercises in the book. Students using this framework will approach the tangent problem by first homing in on the first available mathematical notation in the example to get cues for what equation to set up. That notation in this case is:

m_{PQ} = \frac{x^2 - 1}{x-1}

Then, in the line below, a specific value of x (1.5) is plugged in. Great! they might think, I’ve got a formula and I just plug a number into it, and I get the right answer: 2.5. But then, reading down a bit further, there are insinuations that the right answer is not 2.5. Stewart says, “…the closer x is to 1…it appears from the tables, the closer m_{PQ} is to 2. This suggests that the slope of the tangent line t should be m = 2.” The student with this framework must then be pretty dismayed. What’s this about “it appears” the answer is 2? Is it 2, or isn’t it? What happened to my 2.5? What’s going on? And then they get to example 2, which has no formula in it at all, and at that point any sane person with this framework would give up.

It’s also worth noting that the Stewart book — and many other standard calculus books — do not introduce this tangent line idea until after a lengthy precalculus review chapter, and that chapter typically looks just like what students saw in their Precalculus courses. These treatments do not attempt to be a ramp-up into calculus, and presages of the concepts of calculus are not present. If prior courses didn’t train students on good conceptual frameworks, then this review material actually makes matters worse when it comes time to really learn calculus. They will know how to plug numbers and expressions into a function, but when the disruptively different math of calculus appears, there’s nowhere to put it, except in the plug-and-chug bin that all prior math has gone into.

So it’s extremely important that students going into calculus get a proper conceptual framework for what to do with the material once they see it. Whose responsibility is that? Everybody’s, starting with…

  • the instructor. The instructor of a calculus class has to be very deliberate and forthright in bending all elements of the course towards the construction of a framework that will support the massive amount of material that will come in a calculus class. This includes telling students that they need a conceptual framework that works, and informing them that perhaps their previous frameworks were not designed to manage the load that’s coming. The instructor also must be relentless in helping students put new material in its proper place and relationship to prior material.
  • But here the textbooks can help, too, by suggesting the framework to be used; it’s certainly better than not specifying the framework at all but just serving up topic after topic as non sequiturs.
  • Finally, students have to work at constructing a framework as well; and they should be held accountable not only for their mastery of micro-level calculus topics like the Chain Rule but also their ability to put two or more concepts in relation to each other and to use prior knowledge on novel tasks.

What are your experiences with helping students (in calculus or otherwise) build useable conceptual frameworks for what they are learning? Any tools (like mindmapping software), assessment methods, or other teaching techniques you’d care to share?

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Filed under Calculus, Critical thinking, Education, Educational technology, Math, Problem Solving, Teaching, Technology, Textbooks

The Kindle evolves again

Image from

Update: Here’s an overview video of the new Kindle.

Amazon today unveiled the third generation of its Kindle ebook readers. The new devices, which will ship beginning August 27, will be smaller (21% reduction in size, while keeping the same size screen) and lighter (8.7 ounces) than the current generation of Kindles, with double the storage capacity, improved contrast and fonts, and built-in WiFi. Most importantly is the price point: $189, with a $139 WiFi-only model also being offered.

When Amazon first sold the Kindle, I roundly criticized it (here, here, and here; and then here for the second generation Kindles) as a good idea but lacking several deal-breaking features that should have been obvious, and would have been inexpensive, to include. I also thought the price point — which at the time was in the $359 range! — was way too high. I don’t think Jeff Bezos has been reading this blog, but I must applaud Amazon for addressing most of the issues I’ve brought up.

It took them long enough, but clearly the rapidly-expanding competition in the ebook reader market — not least of which is the iPad — has forced Amazon to make a better mousetrap. We now have native PDF support; WiFi in addition to WhisperNet; a better user interface and sturdier physical design; integration of social networking tools; and a reasonable price tag. The only thing they haven’t done that I first wished they had is made the screen touch-sensitive and in color, but after using the Kindle app on my iPhone and other ebook readers, I’m inclined to think that this isn’t such a big deal after all.

Additionally, Amazon has employed a pretty smart marketing strategy, which is to focus on the content rather than the hardware. If I own a Kindle, buy a bunch of books with it, and then decide I don’t like the Kindle any more or if the Kindle breaks, I’m not screwed — just use the iPhone or Mac Kindle app. For that matter, I don’t have to own a Kindle device at all to read Kindle books. That gives readers more freedom (which is good) and it’s also probably what allows them to drop the price on their hardware so much — more people are buying Kindle books without the Kindle reader, so the demand for the device is lower.

The one thing that seems curious in this announcement is that I would have expected Amazon to go full-throttle into the academic textbook market. Colleges and universities are beginning to adopt the iPad as the hardware platform of choice, and the lower price of the Kindle, availability of prominent textbooks (like Stewart’s Calculus) as Kindle editions, and the generally lower price of Kindle books over their print editions would seem to be big selling points. But there was no big announcement aimed at students and educational institutions to accompany the Kindle announcement itself. And the August 27 ship date is just a little too late for students entering the Fall semester. I wonder if Amazon believes they have a shot in that market; I happen to think they do, but they’ll have to get a move on if they want to compete with the iPad.

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Is the iPad really what students need?

Dave Caolo believes that students are one of the four groups of people who will make the iPad huge, because:

Students are on a fixed budget, and e-books are typically cheaper than their paper-based counterparts. Also, consider all of the money publishers lose when students buy used books from the campus bookstores. Additionally, Apple can distribute textbooks through iTunes U — an established and proven system that students, faculty and staff already know how to use.

Suddenly the iPad is a device that follows a student from his/her freshman year of high school all the way through graduate school. Why buy a laptop when every student has a device that can be a textbook, reference tool, Internet appliance and whatever else the imaginations of developers can dream up?

I do believe that the iPad’s success will be closely tied to its success in the EDU sector, but Caolo’s analysis misses some important points about students and their educational computing needs.

  1. The argument about used books explains precisely why students, and conscientious faculty, will resist textbooks on an iPad. Already textbook companies charge full (and overly high) price for products that are speciously “revised” every couple of years, even though the revisions are virtually identical to the prior versions. If using the iPad as a sort of universal textbook locks students in to using only the most recent version at the highest possible price, then how is this a step forward? Students would be better off purchasing used versions of textbooks.  (One way to ameliorate this problem is for textbook companies to take my advice and give away previous versions of their textbooks whenever a new revision comes out.)
  2. Students need more from their computers than just email clients, ebook readers, and web access. They need to be able to run spreadsheets and word processors simultaneously. They need to be able to run sophisticated scientific computing software. They need to be able to install and run legacy software that their universities may have purchased — or even developed in-house — decades ago. (For example, in our math courses alone at my college, we use Minitab, Winplot, and even Derive. The chances of these being ported to the iPad are basically zero.) They need to be able to do video chats with Skype. These are just a few of the things that the iPad cannot do right now.
  3. The above argument assumes that textbooks are the center of a student’s education. I would argue that the best thing about an iPad in education is that it provides a great platform for getting away from textbooks as the center and focusing on existing, web-based information sources instead. Why invent a whole new class of technology only to have it perpetuate a rapidly-outmoded means of instruction?

I think the iPad is a neat-looking device, and it does have the capacity to change the entire landscape of computing from a user interface point of view. The next time I’m up for an upgrade to my work machine (in 2014, sadly) I fully expect to be getting an Apple device that has all the guts and power of my new Macbook Pro but with a sleek form factor and intuitive touch interface like the iPad (apparently) has. This kind of device is probably what students need. The first-generation iPad, not so much, not right now at least. Although I am sure students will buy it.


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Calculus reform’s next wave

There’s a discussion going on right now in the Project NExT email list about calculus textbooks, the merits/demerits of the Stewart Calculus textbook, and where — if anywhere — the “next wave” of calculus reform is going to come from. I wrote the following post to the group, and I thought it would serve double-duty fairly well as a blog post. So… here it is:


I’d like to add my $0.02 worth to this discussion just because (1) I’m a longtime Stewart Calculus user, having used the first edition (!) when I was an undergrad and having taught out of it for my entire career, and (2) I’m also a fairly consistent critic of Stewart’s calculus and of textbooks in general.

I try to see textbooks from the viewpoints of my students. From that vantage point, I unfortunately find very little to say in favor of Stewart’s franchise of  books, including the current edition, all of the previous five editions, the CCC version (which is almost exactly the same as the non-reform version of the book but with less clarity in its language), or the “Essential” calculus edition. Stewart has a relentlessly formalistic approach to calculus that, while admirable in its rigor, renders it all but impenetrable to students who are not used to such an approach, which is certainly nearly every student I teach and I would imagine a large portion of the entire population of beginning calculus students.

If you don’t believe me, go check out his introductory section on the definite integral (Section 5.2 in the sixth edition). Stewart hopelessly confuses the essentially very simple idea of the definite integral by hitting students with an avalanche of sigma-notation right out of the gate. Or, try the section on exponential functions (1.5), in which Stewart for some reason feels like it’s necessary to explain how it is we can define an exponential function at rational and irrational inputs. This is all well and good, but does the rank-and-file beginning calculus student need to know this stuff, right now?

As a result, I find myself having to tell students NOT to read certain portions of the book, and then remixing and rewriting large parts of the rest of it. But that leads to the ONE thing I can say in the positive sense about Stewart, which I can’t say about many “reform” books: Stewart is what you make it. The book does not force me to teach in a certain way, and if I want to totally ignore certain parts of it and write my own stuff, then this generally doesn’t cause problems down the road. For example, at my college we don’t cover trigonometry in the first semester. In most other books we’ve examined, trig and calculus are inextricable, and so the books are unusable for us. With Stewart, though, given a judicious choice of exercises to omit, you can actually pull off a no-trig Calculus I course with very little extra work on the prof’s part.

I can also say, regarding Stewart CCC, that the ancillary materials are excellent. The big binder of group exercises that comes with the instructor edition is much better than the book itself.

I don’t think that I have yet seen a calculus book that is really fundamentally different from the entire corpus of calculus textbooks, with the possible exception of Hughes & Hallett. They all cover the same topics in the same order, more or less, and in the same ways. If you’re looking for the next wave of calculus reform, therefore, you’ll have to find it outside the confines of a textbook, or at least the textbooks that are currently on the market. Textbooks almost by definition are antithetical to reform. Perhaps real reform will come with the rejection of textbooks as authoritative oracles on the subject in the first place. That could mean designing courses with no centralized information source, or using “inverted classroom” models utilizing online resources like the videos at Khan Academy ( or iTunesU, or some combination of these.

Actually, more likely the next wave of reform will be in the form of reconsidering the place of calculus altogether, as the CUPM project did several years ago. Is it perhaps time to think about replacing calculus with a linear combination (pardon the pun) of statistics, discrete math, and linear algebra as the freshman introduction to college mathematics, or at least letting students choose between calculus and this stat/discrete/linear track? Is calculus really the best possible course for freshmen to take? I think that’s a discussion worth having, or reopening.

Robert Talbert
Franklin College
Peach Dot 1997-1998

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Filed under Calculus, Education, Linear algebra, Math, Teaching, Textbook-free, Textbooks

Resources for the MATLAB class

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We’ve had one full meeting of Computer Tools for Problem Solving (the MATLAB course I’ve blogged about). According to the survey I’m having students fill out on our Moodle site, it went pretty well, even if it was a little like drinking from a fire hose. This first meeting was a lengthy guided tour of all the core features of MATLAB, assuming no prior knowledge of computer algebra systems or programming. Subsequent meetings will be a lot more hands-on, with students working in groups on lab activities centered around a particular topic or problem. This next week it’s graphing, for instance, and students will be creating all kinds of different plots of data and functions.

Students prepare for these activities through out-of-class reading and viewing assignments and through homework assignments that are intended both to pull together the material they learned in week n, and to get them to teach themselves the basics for what they will need in week n+1. Here’s the homework I gave last week, for instance. Self-teaching is at the heart of the pedagogical design of the course. This past week I stressed the centrality of the doc and lookfor commands as the primary way they will learn what commands are available in MATLAB and the syntax for each, and I laid down the rule that I will never answer a question of the form, “How do you do this in MATLAB?” I want students to be self-feeders. (I will be fielding questions from the reading/viewing each week prior to the activity starting, so the students are not totally on their own.)

Which brings us to the question of resources. As I mentioned in a previous post, none of the wide array of entry-level MATLAB textbooks really hit the right notes for my student audience.  So I ended up going with no textbook at all. I realized that in learning MATLAB myself, I’d come across a wealth of resources on the web that were completely free, and if properly integrated into the course, they could serve just as well for students as they did for me.

I find myself gravitating towards three main sources:

  • The page of tutorials for MATLAB at This contains two hours’ worth of professionally-produced, appropriately-pitched interactive video tutorials starting from the beginning and hitting all the important topics one would need to get started with MATLAB. There are very few places in these videos that are not excellent, and there are even built-in multiple choice quizzes to assess your learning of the material as you watch. For me, these videos are the main “textbook” for the course.
  • Cleve Moler‘s free e-book, Experiments with MATLAB. When I first drew up this course, Moler’s book was going to be the textbook. It is said to be designed for high school students taking calculus or a science course. Unlike the bone-dry technical approach that most introductory MATLAB books seem to take, Moler teaches MATLAB via the problems he sets out. It’s a great idea. Unfortunately, the book quite often badly overestimates the math and computing background — and comfort level — of the typical college student, and it just doesn’t work as a primary source for my students. Nonetheless, it’s free, and it is well-written when the level of writing is audience-appropriate, and the problems are fun. We’ll be using 3-4 of the early chapters later in the course for activities.
  • The MIT course “Introduction to Computer Science and Programming” available through MIT OpenCourseWare. This is actually a course on basic computer science using Python and has no MATLAB in it, but it’s so well-done and the problems so well-conceived that I am finding much that I can retrofit for the MATLAB course. When we get to MATLAB programming in week 5, I will be basing much of my approach to teaching the material on the outstanding teaching of the MIT profs, Eric Grimson and John Guttag. And many of the problems I’ll be giving are just appropriated from their problem sets with Python being replaced by MATLAB. (All OCW courses are under a Creative Commons license that allows this kind of remixing.)

The really nice thing about these three sources is that they are all free for students and me to use. Since students aren’t paying $60+ for a textbook, they can consider spending $100 to purchase the student version of MATLAB and not be tied to the campus network for their classwork.

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A simple idea for publishers to help students (and themselves)

OXFORD, ENGLAND - OCTOBER 08:  A student reads...
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I’m doing some research, if you can call it that, right now that involves looking at past editions of popular and/or influential calculus books to track the evolution of how certain concepts are developed and presented. I’ll have a lot to say on this if I ever get anywhere with it. But in the course of reading, I have been struck with how little some books change over the course of several editions. For example, the classic Stewart text has retained the exact wording and presentation in its section on concavity in every edition since the first, which was released in the mid-80’s. There’s nothing wrong with sticking with a particular way of doing things, if it works; but you have to ask yourself, does it really work? And if so, why are we now on the sixth edition of the book? I know that books need refreshing from time to time, but five times in 15 years?

Anyhow, it occurred to me that there’s something really simple that textbook companies could do that would both help out students who have a hard time affording textbooks (which is a lot of students) and give themselves an incentive not to update book editions for merely superficial reasons. That simple thing is: When a textbook undergoes a change in edition, post the old edition to the web as a free download. That could be a plain PDF, or it could be a  Kindle or iBooks version. Whatever the format, make it free, and make it easy to get.

This would be a win-win-win for publishers, authors, and students:

  • By charging the regular full price for the “premium” (= most up-to-date) edition of the book, the publisher wouldn’t experience any big changes in its revenue stream, provided (and this is a big “if”) the premium edition provides significant additional value over the old edition. In other words, as long as the new edition is really new, it would cost the publisher nothing to give the old version away.
  • But if the premium edition is just a superficial update of the old one, it will cost the publisher big money. So publishers would have significant incentive not to update editions for no good reason, thereby costing consumers (students) money they didn’t really need to spend (and may not have had in the first place).
  • All the add-ons like CD-ROMs, websites, and other items that often get bundled with textbooks would only be bundled with the premium edition. That would provide additional incentive for those who can afford to pay for the premium edition to do so. (It would also provide a litmus test for exactly how much value those add-ons really add to the book.)
  • It’s a lot easier to download a PDF of a deprecated version of a book, free and legally, then to try your luck with the various torrent sites or what-have-you to get the newest edition. Therefore, pirated versions of the textbook would be less desirable, benefitting both publishers and authors.
  • Schools with limited budgets (including homeschooling families) could simply agree not to use the premium version and go with the free, deprecated version instead. This would always be the case if the cost of the new edition outweighs the benefits of adopting it — which again, puts pressure on the publishers not to update editions unless there are really good reasons to do so and the differences between editions are really significant.
  • The above point also holds in a big, big way for schools in developing countries or in poverty-stricken areas in this country.
  • Individual students could also choose to use the old edition, and presumably accept responsibility for the differences in edition, even if their schools use the premium edition. Those who teach college know that many students do this now already, except the old editions aren’t free (unless someone gives the book to them).
  • All this provides publishers and authors to take the moral high road while still preserving their means of making money and doing good business.

Some individual authors have already done this: the legendary Gil Strang and his calculus book, Thomas Judson and his abstract algebra book (which I used last semester and really liked), Fred Goodman and his algebra book. These books were all formerly published by major houses at considerable cost, but were either dropped or deprecated, and the authors made them free.

How about some of the major book publishers stepping up and doing the same?

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