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20 Jul 2006 - 17:25
Preface to Saxon Algebra 2 This is the second edition of the second book in an integrated three-book series designed to prepare students for calculus. In this book we continue the study of topics from algebra and geometry and begin our study of trigonometry. Mathematics is an abstract study of the behavior and interrelationships of numbers. In Algebra 1, we found that algebra is not difficult—it is just different. Concepts that were confusing when first encountered became familiar concepts after they had been practiced for a period of weeks or months—until finally they were understood. Then further study of the same concepts caused additional understanding as totally unexpected ramifications appeared. And, as we mastered these new abstractions, our understanding of seemingly unrelated concepts became clearer. Thus mathematics does not consist of unconnected topics that can be filed in separate compartments, studied once, mastered, and then neglected. Mathematics is like a big ball made of pieces of string that have been tied together. Many pieces touch directly, but the other pieces are all an integral part of the ball, and all must be rolled along together if understanding is to be achieved. A total assimilation of the fundamentals of mathematics is the key that will unlock the doors of higher mathematics and the doors to chemistry, physics, engineering, and other mathematically based disciplines. In addition, it will also unlock the doors to the understanding of psychology, sociology, and other nonmathematical disciplines in which research depends heavily on mathematical statistics. Thus, we see that mathematical ability is necessary in almost any field of endeavor. Thus, in this book we go back to the beginning –to signed numbers—and then quickly review all of the topics of Algebra 1 and practice these topics as we weave in more advanced concepts. We will also practice the skills that are necessary to apply the concepts. The applicability of some of these skills, such as completing the square, deriving the quadratic formula, simplification of radicals, and complex numbers, might not be apparent at this time, but the benefits of having mastered these skills will become evident as our education continues. We will continue our study of geometry in this book. Lessons on geometry appear at regular intervals, and one or two geometry problems appear in every homework problem set. We begin our study of trigonometry in Lesson 43 when we introduce the fundamental trigonometric ratios—the sine, cosine, and tangent. We will practice the use of these ratios in every problem set for the rest of the book. The long-term practice of the fundamental concepts of algebra, geometry, and trigonometry will make these concepts familiar concepts and will enable an in-depth understanding of their use in the next book in the series, a pre-calculus book entitled Advanced Mathematics. Problems have been selected in various skill areas, and these problems will be practiced again and again in the problem sets. It is wise to strive for speed and accuracy when working these review problems. If you feel that you have mastered a type of problem, don’t skip it when it appears again. If you have really mastered the concept, the problem should not be troublesome; you should be able to do the problem quickly and accurately. If you have not mastered the concept, you need the practice that working the problem will provide. You must work every problem in every problem set to get the full benefit of the structure of this book. Master musicians practice fundamental musical skills every day. All experts practice fundamentals as often as possible. To attain and maintain proficiency in mathematics, it is necessary to practice fundamental mathematical skills constantly as new concepts are being investigated. And, as in the last book, you are encouraged to be diligent and to work at developing defense mechanisms whose use will protect you against every humans’ seemingly uncanny ability to invent ways to make mistakes. One last word. There is no requirement that you like mathematics. I am not especially fond of mathematics—and I wrote the book—but I do love the ability to pass through doors that knowledge of mathematics has unlocked for me. I did not know what was behind the doors when I began. Some things I found there were not appealing while others were fascinating. For example, I enjoyed being an Air Force test pilot. A degree in engineering was a requirement to be admitted to test pilot school. My knowledge of mathematics enabled me to obtain this degree. At the time I began my study of mathematics, I had no idea that I would want to be a test pilot or would ever need to use mathematics in any way. I thank Tom Brodsky for his help in selecting geometry problems for the problem sets. I thank Joan Coleman and David Pond for supervising the preparation of the manuscript. I thank Margaret Heisserer, Scott Kirby, John Chitwood, Julie Webster, Smith Richardson, Tony Carl, Gary Skidmore, Tim Maltz, Jonathan Maltz, and Kevin McKeown for creating the artwork, typesetting, and proofreading. I again thank Frank Wang for his valuable help in getting the first edition of this book finalized and publisher Bob Wroth for his help in getting the first edition published. John Saxon
Beautiful. The third editions of the Saxon books seem to have done away with John Saxon's prefaces; at least, that's the case with the 3rd edition of Algebra 1/2. Thanks to our ktm Book Fairy, I have a copy of the 2nd edition of Algebra 1/2, so I'll post that preface, too. The books themselves don't seem to have been changed in other bad direction. If you're interested in buying the 2nd edition, though, Rainbow Resource seems still to have them. So does Seton Books. I'm sure other homeschooling stores do as well.
Wilfried Schmid on procedures and understanding
''I'm a professional mathematician, and I myself very often use mathematical methods that I understand only imprecisely,'' he said. ''It is while I use them that I begin to understand. After a while, the use and the understanding are mutually supporting.''
Carolyn on procedures and understanding Carolyn has said more than once that she believes in teaching procedures first. Conceptual understanding follows. (I can't find any of her posts on this, so if I've misremembered I'll delete this.) I was always a little skeptical of this, although my working assumption is that where Carolyn and I disagree, Carolyn is right. I've now spent enough time working my way through Saxon to see what Saxon, Schmid, and Carolyn are talking about. When you practice a procedure you don't understand over and over and over again, at some point it "naturalizes." It seems right and inevitable. And it makes sense. John Saxon stresses this idea in book after book. Math isn't hard; it's different. It's unfamiliar. When you've done so much math that it no longer seems strange, it starts to seem easy — or at least not harder than other subjects. Of course, the irony is that this naturalizing process leaves me unable to explain procedures to someone for whom math is still strange. It does, however, make me understand why "math brains" tend to say things like, "It just is" when I ask for an explanation! I'll add that Saxon (and probably Carolyn & Schmid, too) rarely teaches a concept stripped of all meaning or explanation — though he does do so far more often in Algebra 1 than in the earlier books. A student using Algebra 1 must take a lot on faith. If nothing else, meaning helps memory; it's easier to remember a procedure you understand. (I have references for this observation, but don't want to spend the time to dig them up just now.) I'd be willing to bet that meaning increases student motivation, too. I recall Steve H saying that students always want an explanation if they can get one. (Steve - am I remembering that correctly?) Every one of Saxon's explanations in 6-5 through 7-6 has been pure pleasure to read, and has made me want to learn more math. In contrast, my motivation sometimes flags as I work with Saxon's highly abstract Algebra 1, my motivation sometimes flags. In short, I think it's probably always good to try to teach some conceptual understanding along with procedure. I also think, after living through Ms. K's Phase 4 math class, that it's essential to include mini word problems — although Saxon does not do so in Algebra 1. But John Saxon can get away with it, because he's a genius math textbook writer. If you're not a genius math text writer, or a genius math teacher, you can't. Nevertheless, these caveats aside, math is first and foremost something people do. Barry says that constructivist math ends up teaching math appreciation, not math, and I agree. Teach procedures supported by meaning where possible, and, where not possible, teach the procedure and practice it to mastery. Understanding will follow as "totally unexpected ramifications appear."
John Saxon & John von Neumann on math
preface to Saxon Algebra 2
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johnsaxonalgebra2preface johnsaxonpreface saxonpreface algebra2 -- CatherineJohnson - 15 Jul 2006
"It is while I use them that I begin to understand. After a while, the use and the understanding are mutually supporting." Linkage. "Carolyn has said more than once that she believes in teaching procedures first. Conceptual understanding follows." I'm with Carolyn here too, but I will elaborate. This is not the same as rote learning. Unfortunately, Ed School types don't seem to understand that. I have mentioned before that it took me until Algebra II (and perhaps into Trig.) before I felt that I could do (and really understand) any algebraic manipulation. This doesn't mean that I had no understanding of what I was doing until then. Understanding first is not only a vague concept, it is a vanity of modern low content and skills education. "I'll add that Saxon (and probably Carolyn & Schmid, too) rarely teaches a concept stripped of all meaning or explanation — " There is only one thing to call teaching without any understanding - bad teaching. There are different levels of understanding and a good teacher/curriculum will take into account the need for skills and practice along that path. "I recall Steve H saying that students always want an explanation if they can get one. (Steve - am I remembering that correctly?)" Duh ... That doesn't sound familiar. I would say that you always have to give an explanation. For things like multiplying negative numbers together, the simple explanation first should suffice. Explanations first, however, sometimes don't sink in. Practice a little (or a lot) and then talk about it. There are different levels of understanding. There is no such thing as understanding first. Experience is not just about doing things faster. Practice counts. Linkage. -- SteveH - 16 Jul 2006
This is not the same as rote learning. It's SO hard for me to make that distinction when talking to other people... Ms. K's class was pure rote learning, not procedural. I know the difference, because I can "see" it. But I can't explain it. I'm like people who say they know pornography when they see it. -- CatherineJohnson - 19 Jul 2006
I have mentioned before that it took me until Algebra II (and perhaps into Trig.) before I felt that I could do (and really understand) any algebraic manipulation. That's gotten to be one of the fun parts of learning math for me - finding out that I don't understand something I thought I did, that there's more to it. At first, that experience was quite difficult, and I would feel a bit of panic. Now I like it. -- CatherineJohnson - 19 Jul 2006
Carolyn told me early on that, with math, you have to develop trust that understanding will come in time. It's hard to take such an attitude on faith, but now that I've experienced "deepending understanding" for myself, I don't have any problem assuming that at some point I'm going to understand what I'm learning & doing better than I do now. I also have the feeling that I don't have to understand everything....that you could probably keep on discovering new aspects of what you already know forever, just as you do with any other topic. It's funny that I never made that connection..... -- CatherineJohnson - 19 Jul 2006
johnny von neumann is supposed to've something somewhere
to the effect "you don't understand things
in mathematics ... you just get used to them".
von neumann quotes -- VlorbikDotCom - 20 Jul 2006
"you don't understand things in mathematics ... you just get used to them". Boy, there's a lot of truth to this. And in the world of mathematical statistics (which I've been teaching myself since January), I can tell you that I'm just not sufficiently used to things yet. But I'm still plugging away at it. -- CarolynJohnston - 20 Jul 2006
V oh, that's incredible! That's EXACTLY what I've been experiencing. Some concept or procedure will come to seem obvious to me, and then when I think about it I realize I don't have the first clue why it is the way it is. (Not always the case, but often enough.) Absolutely, you get used to it. You habituate! -- CatherineJohnson - 20 Jul 2006