This is mostly just empty vagaries that amount to little more than "our program is great". All the publishers say stuff like this. Come on now... everyone is going to say this. Or, do you think that Houghton-Mifflin will say: "We actually strive to introduce material when the student is ill-equipped to handle it in an effort to prevent them from acquiring any sort of conceptual understanding of the mathematics. We find that the best way to teach math is to sabotage the student at every step under the motto that 'what doesn't kill us only makes us stronger'. It is our Spartan way, actually." Of course not. No one does. Everyone says that they cover everything, introduce material at just the right time and in just the right way, and that their students really learn math conceptually. About the most you can infer from this much of the response is that they attribute what appear to be charges that their program doesn't cover everything to sequencing differences. But, the charges aren't just "my favorite topic X isn't in your algebra Y book." Some folks have gotten on here and said their "Algebra II" is really "Algebra I" by most people's standards. That's different from Saxon. After three years of Saxon (Algebra I, Algebra II, and Advanced Math), you have apparently covered a lot more than three years of TT (Algebra I, Geometry, and Algebra II). That is a disturbing trend any way you look at it, and he is failing to respond to it properly. He is responding with one suggestive general comment about sequencing and a paragraph of a protracted sales pitch of vagaries about how he likes his program. Again, so he says. The guys at Houghton Mifflin or Saxon or where ever beg to differ. Everybody says they explain what a technique means, why its important, what it's used for and how it relates to what the student has already learned. This paragraph is largely a non sequitur slur against public school textbooks. So, does he or doesn't he, though, cover the very same things. He sounds like he has just followed suit based on his opening remarks. And, at any rate, he wants to sell one book nationally, as well. He just doesn't want to go to all the trouble of getting it approved by school boards. I don't necessarily disagree with his lack of concern for some bureaucratic accomplishment like that, but he has largely just followed the same old paradigm. There is nothing special about the content of his program that causes him to really digress from Saxon, say. As he says -- they all cover the same content in the long run, anyway, right? It just takes him longer, apparently. Well, he's right about that, but his program is not even remotely any kind of a panacea for that. It is hard to appreciate this completely separate matter going on in K-14 math ed if you don't have a BS in math. In fact, it may even be hard to appreciate for someone with a BS. What the fields of math are -- what constitutes mainstream math -- is really not a very contentious matter at all. And, what the nature of the fields are is even less contentious. I've said before that the three main fields of math are abstract algebra, analysis and topology. In fact, you might even be able to kind of narrow it down to abstract algebra and real analysis as far a people just being introduced to the subject are concerned. Most people outside of the field don't even know what these subjects are. And, that is precisely because "our students aren't learning the major math concepts that they need to know, ever!" But, it probably isn't at all what Sabouri has in mind. We just don't teach math. What we teach are the mathematical methods of engineering, business and science. And, Sabouri is most certainly doing just that in his program. He is not at all breaking away from that paradigm. He contradicts his opening remarks. So, he doesn't cover everything, then...? Every program has someone that went to Harvard on it and someone that became an eighth grade drop out on it. The simple fact is that there is a strong objective difference between the difficulty of TT and its peer. A lot of people have really tried it and regretted it (apparently). You cannot respond to that kind of an issue with "trust us!" Complete with the 800 number and all! Don't get me wrong. I do not disparage choosing TT. I just dispute that it is a great program for really mathy kids. I would take Saxon over TT any day just based on what the students seem to be able to do at any given stage in the program. And, I would take Singapore's NEM over Saxon any day. I would take almost any mainstream program over TT: Foerster, Dolciani (though, I've only seen the old texts), Saxon, Singapore (which would be my top pick of such programs), Jacobs. You guys need to start coming to terms with the possibility that maybe your kid "gets math" now just because it is a whole lot easier than the math he would be presented with in other programs. Especially if you are not very mathy, yourself, and you look over your kids shoulder and you "get it", too, does that not really prove that this is the case?? I mean you know how much you don't know and if Saxon has a whole lot of that kind of stuff in it but TT doesn't, doesn't that strongly indicate an immediate lack of content?

This is mostly just empty vagaries that amount to little more than "our program is great". All the publishers say stuff like this. Come on now... everyone is going to say this. Or, do you think that Houghton-Mifflin will say: "We actually strive to introduce material when the student is ill-equipped to handle it in an effort to prevent them from acquiring any sort of conceptual understanding of the mathematics. We find that the best way to teach math is to sabotage the student at every step under the motto that 'what doesn't kill us only makes us stronger'. It is our Spartan way, actually." Of course not. No one does. Everyone says that they cover everything, introduce material at just the right time and in just the right way, and that their students really learn math conceptually. About the most you can infer from this much of the response is that they attribute what appear to be charges that their program doesn't cover everything to sequencing differences. But, the charges aren't just "my favorite topic X isn't in your algebra Y book." Some folks have gotten on here and said their "Algebra II" is really "Algebra I" by most people's standards. That's different from Saxon. After three years of Saxon (Algebra I, Algebra II, and Advanced Math), you have apparently covered a lot more than three years of TT (Algebra I, Geometry, and Algebra II). That is a disturbing trend any way you look at it, and he is failing to respond to it properly. He is responding with one suggestive general comment about sequencing and a paragraph of a protracted sales pitch of vagaries about how he likes his program. Again, so he says. The guys at Houghton Mifflin or Saxon or where ever beg to differ. Everybody says they explain what a technique means, why its important, what it's used for and how it relates to what the student has already learned. This paragraph is largely a non sequitur slur against public school textbooks. So, does he or doesn't he, though, cover the very same things. He sounds like he has just followed suit based on his opening remarks. And, at any rate, he wants to sell one book nationally, as well. He just doesn't want to go to all the trouble of getting it approved by school boards. I don't necessarily disagree with his lack of concern for some bureaucratic accomplishment like that, but he has largely just followed the same old paradigm. There is nothing special about the content of his program that causes him to really digress from Saxon, say. As he says -- they all cover the same content in the long run, anyway, right? It just takes him longer, apparently. Well, he's right about that, but his program is not even remotely any kind of a panacea for that. It is hard to appreciate this completely separate matter going on in K-14 math ed if you don't have a BS in math. In fact, it may even be hard to appreciate for someone with a BS. What the fields of math are -- what constitutes mainstream math -- is really not a very contentious matter at all. And, what the nature of the fields are is even less contentious. I've said before that the three main fields of math are abstract algebra, analysis and topology. In fact, you might even be able to kind of narrow it down to abstract algebra and real analysis as far a people just being introduced to the subject are concerned. Most people outside of the field don't even know what these subjects are. And, that is precisely because "our students aren't learning the major math concepts that they need to know, ever!" But, it probably isn't at all what Sabouri has in mind. We just don't teach math. What we teach are the mathematical methods of engineering, business and science. And, Sabouri is most certainly doing just that in his program. He is not at all breaking away from that paradigm. He contradicts his opening remarks. So, he doesn't cover everything, then...? Every program has someone that went to Harvard on it and someone that became an eighth grade drop out on it. The simple fact is that there is a strong objective difference between the difficulty of TT and its peer. A lot of people have really tried it and regretted it (apparently). You cannot respond to that kind of an issue with "trust us!" Complete with the 800 number and all! Don't get me wrong. I do not disparage choosing TT. I just dispute that it is a great program for really mathy kids. I would take Saxon over TT any day just based on what the students seem to be able to do at any given stage in the program. And, I would take Singapore's NEM over Saxon any day. I would take almost any mainstream program over TT: Foerster, Dolciani (though, I've only seen the old texts), Saxon, Singapore (which would be my top pick of such programs), Jacobs. You guys need to start coming to terms with the possibility that maybe your kid "gets math" now just because it is a whole lot easier than the math he would be presented with in other programs. Especially if you are not very mathy, yourself, and you look over your kids shoulder and you "get it", too, does that not really prove that this is the case?? I mean you know how much you don't know and if Saxon has a whole lot of that kind of stuff in it but TT doesn't, doesn't that strongly indicate an immediate lack of content?

Well, what's kind of dumb about that is that the parents could have said anything. Is it just their policy to make all homeschoolers take subject area tests? It's just kind of lame any way you look at it, though, because high schools could really be doing virtually anything, for that matter. You certainly cannot rely on one (even if it is really super low) universal standard coming out of thousands of high schools in the state. You just have to accept them or not based on an IQ test (aka the SAT) and then place them specifically into your program with your own placement tests. If you have a super bright student, it really doesn't matter if they lack education beyond basic SAT-type stuff. You pretty much have to structure your courses that way anyway. For instance, I took freshman Latin, History, Chemistry and Physics in college. None of these assumed any particular prior knowledge. (Technically speaking, I placed into second semester Latin, but based on that, I seriously doubt the first semester assumed anything.) About the only things that do really assume actual knowledge are English and Math, and the former only because they didn't teach me any grammar and just expected me to write essays straight away. (I don't know if that even counts -- I guess if it does, then maybe history has that same "prerequisite". Come to think of it, they all expect you to be able to read....?) At any rate, I'm just sayin'... if you let high schoolers in from more than a few high schools without checking, then I don't see why you wouldn't let just anyone in that makes a good score on the SAT proper. In fact, just imagine the absurdity of "I see here you made an 800 on the math portion of the SAT and placed into our second semester calculus, according to the placement test we gave you. But, we were a little concerned about the fact that you used Teaching Textbooks, so we're asking you to take the SAT math subject test...."

No I don't think you are accounting for most of the issue here, Robin. For instance, read this. (I think this may have been posted already.) One major problem with this article is a false dichotomy. He makes it out like modern logic is just symbolic logic. In fact, "logic with words" is plenty much included in mainstream modern philosophical logic. Traditional logic is not actually supposed to just be modern philosophical logic, but rather more of a competing alternative to it with a specific disdain for symbolic logic. And, the way he talks about it, it really sounds like he really does mean it that way. On the other hand, Venn Diagrams are modern not traditional, so I'm not sure what that means. There is a very good answer to his question of "why we would want to" (replace traditional logic with modern logic). Traditional logic is nothing more than modern philosophical logic sans some pretty major developments over the last couple centuries. You should no more exclude symbolic logic from a treatment of modern philosophical logic than you should exclude latin grammar from your latin program. Modern logic doesn't make additional assumptions. If anything, it is the other way around. Especially modern symbolic logic is just the essence of the syllogism without the additional baggage of things like the Categories, for instance, or the handicap of an imprecise natural langauge. You have to go through exactly the same logic to do it with imprecise natural language. Truthfully, there are a lot of good reasons to include modern symbollic logic and not doing so is up there with trying to use Aristotle's theory of the spheres for astronomy. I'm really not exaggerating that point at all, and, again, modern symbolic logic is not "the modern system" which includes informal logic and "logic with words" and so on. And this: "If logic is not math, then what is it? The answer, of course, is that logic is about finding truth with words, not symbols and with language, not math." is just false. It isn't about finding the truth, as he, himself, said in some other quoted passage. And, even if it was, it wouldn't be about doing it with words and specifically not with symbols. And this: "The difference between the two systems of logic is quite dramatic, but most people can recognize the modern system because of its prolific use of symbols, in addition to common modern fixtures, such as truth tables and Venn diagrams." is only superficially true. Modern symbolic logic is just Aristotelian Logic. So is Traditional Logic. The basic laws of identity, excluded middle, and contradiction are all the same. And, you say that he even has Venn Diagrams in his books, so what's the deal with that? What is really called for here is a standrad course of philosophical logic which includes "logic with words" and symbolic logic. Cothran's specific anti-math ranting and focus does him, I think, a great disservice. That was almost specifically why I bought Nance's book rather than his. At the time, I wasn't looking for mathematical logic, per se, but I certainly wasn't looking to exclude it either like Cothran seems clearly intent on doing if the above article is any indication. (And, incidentally, there is a whole lot more to his bringing up Bertrand Russell than just a name associated with modern symbolic logic. It is clear that Cothran's real beef is more with modern secular analytic philosophy. He has done just what I say he has -- thrown the baby out with the bath water.)

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Oh no... I just switched it to a PDF just recently without telling anyone. LOL. I just told Myrtle -- she'll be updating her blog at some point, I'm sure.

http://www.oplink.net/~adrian/gelf_alg.pdf I've actually been working on beautifying it. I made some sort of computaitonal error on 90(h) and 122(b) could be factored out further for a nicer end result. I still need to fix those and will do so as I get to them. There is a large chunk of the solutions that are in straight text without formatting. Most of it is still formatted like you can format it in HTML. So, that means it is kind of ugly. I have just yesterday gotten, for instance, to problems where it is really helpful to have a big fraction bar. In HTML, you have to do it a little uglier than that using parentheses and a simple "/" to get the job done. In the uglier parts, I didn't even, at the time, know how to make superscripts and subscripts in HTML, so x squared would literally appear like "x^2". (Super ugly, especially with complicated polynomials.) I've gone back and forth on just how I wanted to do it. At one point, I was going to do it all in LaTeX. It actually all has to do with how enamored with linux I am at any given point in time. But, now I am in my non-linux phase and am making a pdf in Word. So, I can actually pretty much get all the formatting I really need in a simple, direct, point-and-click sort of way, including big fractions, whatever symbols I might need like summation, and so on. And, as I say, I am making progress through it -- I worked on it just yesterday, even, to beautify it some more. (I think I'll just stick with pdfs from now on.) I do think Gelfand has the best problems around. They're just good hard math problems and some of them are problems you might actually find in real math text books. However, it does bring up problems that cannot really be handled properly for the level of the student. (For instance, there are a number of problems where you really ought to use Mathematical Induction.) I've also found, in my many New Math texts (especially that Oakley and Alendoerfer book, for instance), a lot more problems like that. (Several of the Mathematical Induction problems, for instance, in the Alendoerfer book can be found right inside Herstien's Abstract Algebra book.) But, mostly, while Gelfand represents a single book with a lot of great problems, the New Math sequence presents a program that can lead (if you get the right books, once again) to normal math as done by mathematicians and math majors. (For instance, given what I just said about Alendoerfer, how crazy would it be to do Herstien after that? And, if it's not that crazy, then I guess you just have to know what is contained in a normal math degree at the bachelors and post-baccalaureate level to really appreciate that.) (Also, note that by "New Math", I am not referring to a recent program or educational paradigm. I am referring to a paradigm shift in math education driven by mathematicians in the 60s and that had run its course and was almost completely dead by 1972.)

Modern symbolic logic is Aristotelian logic. That is to say, it is a subset of it. You really should do both "logic with words" and symbolic logic. And, truthfully, "formal logic" really should mean symbolic logic. Sure, when there was no such thing as symbolic logic, then when you are talking about the formal aspects of Aristotelian logic, you would be talking about his syllogisms and such (as opposed to, say, informal fallacies) all done in a natural language. But, symbolic logic is just the outcome of a bunch of work people have done in that specific area. It isn't like anyone rejected Aristotle's Laws of Excluded Middle or Contradiction. We should no more reject symbolic logic than we should start trying to use Aristotle's theory of the spheres again to do astronomy. That said, certainly philosophical logic is more than just formal logic. If you just do what essentially looks like math all the time, then you will not necessarily walk away being very good at something like rhetoric. So, if you have an eye toward that sort of thing, then you would rightly be interested in "logic with words", too. More generally, as I say in my other post in this thread, when dealing with problems where philosophical vagueness plays a major role, you really need a lot of experience with "logic with words". You simply cannot "formalize" the objects you are dealing with and deal with them symbolically. Forcing the formality often tends to just end up begging the question. (That hasn't stopped people from trying to do such a thing with moral philosophy, though, for instance.) Nevertheless, though, you simply should not just reject out of hand some of the largest advances in logic in all of history (including Aristotle). It is true that a lot of modern empiricism is heavily associated with symbolic logic -- it's a real science-y thing to do and has been wielded in the past seemingly in an effort to do away with normal (informal) philosophy. I would agree that all of that sort of thing is wrong-headed. But, let's not throw the baby out with the bathwater, here. That doesn't mean that the logic, itself, is the bad thing, so much as, in this case, people's interpretation of it. I think it is a big mistake to literally try to turn back time to 1800 and pretend that symbolic logic never happened by relegating it to some other subject as if "symbolic logic is really math and not logic at all, you see...." I mean, if you are going to do that, then do you just limit yourself only to the Organon, then? No one -- not even Ockham or any of the medievalists that came after Aristotle -- can add to it? Modern symbolic logic did not change Aristotelian logic. It's not like the Hegelian dialectic that actually rejects the Law of Contradiction or maybe popular ideas of "quantum logic" used to discuss events that both happen and don't happen all at the same time. Symbolic logic really just largely represents modern developments in Aristotelian logic. It's not analogous at all, for instance, to choosing to do modern languages over Latin or Greek.

First of all, let me just give my $0.02 on the whole formal/informal thing. The formality is generally associated with the degree of precision involved in the subject. Specifically, when there is a great deal of philosophical vagueness in whatever you are talking about, you can no longer focus on just the form of your argument. (Because you are constantly at risk of equivocating.) So, what I think has kind of started happening is that the term "formal" is getting wrapped up in just that distinction. So, that is how we end up with "formal" largely just meaning that it uses a formal language -- because of the role this issue of philosophical vagueness has played recently philosophical matters. At any rate, the standard philosophy explanation of all this is to generally characterize "knowledge" as "justified, true belief". (I always get heckled when I say that -- yes, I am aware of the Gettier Problem.) The point is that knowledge entails at least justification and logic is the study of what constitutes good justification. Epistemology is the study of what knowledge is, and so when looked at this way, it makes a clear distinction between semantics, the study of what truth is/means, and logic as two separate ares of epistemology. Another distinction that ought to be more clearly made, incidentally is between metaphysics, the study of what is real, and ontology, the specific subarea of metaphysics that is concerned with what exists. We are running headlong into both empiricism and foundationalism in some of these quotes here where we are wondering how to come to Truth (with a capital T), and I think that the lack of distinction between "what is real" and "what exists" as at least possibly disparate matters is part of that. At any rate, consider the followiong proposition: "I think." Is it true? Can it ever be false? Do we know it empirically? And, yet, "I" and "think" don't seem to contain the truth of the statement simply in their definitions, like "bachelor" does for a statement like "Bachelors are unmarried males." This sort of thing is often used as an example of a possible synthetic a priori proposition. We are talking ourselves right out of the possibility of such a thing in this thread's general approach to the matter of logic. It is because we (very naturally) imagine some sort of a foundation from which we proceed using valid formal deductive logic to make further conclusions. We imagine this foundation of Truth (since it doesn't come from logic) to then have to be arrived at ("I guess") from our empirical observations of that which exists. We've failed to make a number of very slight subtle distinctions between what exists and what is real because we have trouble imagining how we could possibly talk meaningfully about something that does not exist. And, we have trouble imagining how we could "back in" to truth using reason or that reason might be bigger than logic. We just sort expediently equate all these things. But, consider, as I say, "I think." Is it analytic? More generally, do numbers "exist"? Isn't what we say about them true or false? Is the concept of number just a convention, perhaps? (At any rate, you may have noticed that I am sympathetic to some particular conclusions about all this. I won't pretend I am giving an unbiased account of any of this....)

Myrtle thinks that the first book will be easy after Allen I. The second book naturally follows the first book. It is kind of redundant, actually. Theoretically, you could just do the second book ab initio. In fact, that is what that book is for -- just written to older students. The reason I wouldn't do it with a younger student is just because it is too fast and too hard probably to just jump into from scratch like that. Plus, the redundancy is pedagogically kind of a good thing when it comes to something like logic with a view to doing math. (You can never drill on too much logic.) So, the first book is just a way to get started, and, with a lot of overlap to the first book and ending with a lot of intuitive set theory, the second book is sort of where you want to end up. After that, jumping into Herstein shouldn't phase you one little bit, especially if you have done all of Allendoerfer's mathematical induction problems as well. (The last two starred problems in Allendoerfer, for instance, are to prove that Mathematical Induction is equivalent to the Well Ordering Principle. Also, several of the problems in Allendoerfer show up again in Herstein. So, this Allendoerfer book is awesome!) At any rate, we plan to just do it on the side, concurrently with our math. And, we are going to start the first book yesterday! LOL. We can start the first book at any time, I guess. With the next child we will plan on starting it earlier, perhaps, or maybe we will just save it for concurrent with Moise and Downes. I really doubt it will take as long as three years to get through them both. And, Myrtle just said she wants to see what Allendoerfer recommends, so maybe we'll end up substituting one of those books. (But, I must say, I do like Suppes' second book, specifically because it is written with an eye for doing math and because of the part two and specifically the fact that part two is naive set theory.)

This is no accident. The real math that graduate students and mathematicians do is something very unique indeed. It is the rare case of a formal a priori subject. Philosophy is, by and large, an informal a priori subject. Math and philosophy are, thus, sister disciplines. (Not math and science.) Also, while the debate between rationalists and empiricists continues (...I guess... :rolleyes:), the fact is that most of what is important -- most of the important and hard insights -- are a priori. They may or may not be "ultimately" tied to some sort of empirical "experience". It really doesn't matter. It is the getting from one step to the next that in some hard cases is what makes the difference. That is why math is special. Not "math" -- like teaching textbooks or even singapore or really even Gelfand's Algebra book. But, math like what R L Moore does in his graduate classes as well as the math that is in the various Bourbaki texts. That kind of math, where the proof is more important than the theorem, the kind of math that people put up a million dollar reward for -- like the millenium problems -- that math is something that anyone should feel lucky for the opportunity to beat themselves up with.

Math Sequence: Modern Algebra (Book One) by Allen and Pearson Geometry by Moise and Downes Modern Algebra (Book Two) by Allen and Pearson Principles of Mathematics by Oakley and Allendoerfer Abstract Algebra by I N Herstein Principles of Mathematical Analysis by Walter Rudin Logic/Philosophy Sequence: First Course in Mathematical Logic by Patrick Suppes Introduction to Logic by Patrick Suppes (The second book of logic should/must be done by the time you start Herstein.) **** Now let's look at why this is the answer. Well, if it really does end in Herstein and Rudin, then there is no doubt about it -- it is definitely "math for math majors". Both of those books are along the lines of honors senior college courses, so if you can do that, you win. The shakiest part of it all is the lack of very much matrix algebra, actually, which is largely taken for granted in various parts of both Herstein and Rudin. However, Myrtle, for instance, just got done doing a litany of mathematical induction problems out of Allendoerfer, and I showed her the very same problems right there in Herstein. So, that New Math is no joke. And, now that I am seeing it really work over time on both Myrtle and on our oldest, I am about ready to drop Gelfand and everything else just to do that. What's our pedagogical approach? We memorize and do problems. We memorize axioms. We memorize "scripts" for proving things. After they memorize a fair amount, it gives them a framework within which to figure things out. They rarely can just figure everything out on the fly the first time around. And, usually they need the other pieces that they haven't figured out yet in what they are doing for them to be able to do it at all. So, they just memorize that part and keep moving. By "keep moving", I do not mean move on to the next topic. I just mean keep trying to do what they are doing -- to do problem after problem until it starts to click with them. And, then the other part of this is doing the problems. In the end, there is one and only one thing that matters -- what problems can you do. Period. Don't kid yourself about any single other thing mattering, here, other than that. They have to do the problems themselves without coaching. So, that's what you do and how you do it. Of course, it is possible to trade out some of these books for true alternatives. You might be able to trade out Allendoerfer for Docliani's Modern Introductory Analysis. You might be able to switch out Herstein for Gallian or Rudin for Bartle. If you switch out real analysis for a calculus text, you lose. If you switch out a new math text for non new math or a not sufficiently hard on logic and set theory program, you will not be able to touch books like Herstein or Rudin in a hundred million years (which, again, means that you lose). Also, just to be clear, it is not even remotely disputable what the subjects of modern mathematics are. The three main subjects, as any mathematician should tell you, are abstract algebra (e.g. groups, rings, fields), topology, and analysis (e.g. rigorous calculus). Topology is usually chapter 2 of analysis (as it literally is for Rudin). That's why it is sometimes given short shrift and not done all on its own. But, if you are looking for the book on that, it is probably the one by Munkres. (Again, there are alternatives like Armstrong and others -- for the these three, any math department or mathematician can easily recommend a book. But, make sure they do not know what your true intentions are. Just say you want an introduction to the subject at the senior college level and there are tons of books.) No one does this inside or outside of America. Not the Russians nor the Japanese nor the Singaporeans. There is probably something to the fact that you will basically have to both tutor someone through it and really kind of "make" them do it in a way you just can't do with the general public. So, everyone else, at best, does a really good engineering math program. There are some pockets of Russian Math Circles or something else with a few very special and very motivated students that do something else, but even then it usually isn't the systematic training in math as its own subject. It is normally just some really good "real math"-type problems and the chance to talk with real (in many cases first rate) practicing mathematicians both of which are invaluable in their own right. And finally, there is probably one realistic alternative to this: Euclidean Geometry. It is "real math", but it really is profoundly detached from modern math. Basically, everyone does analytic geometry in reality. However, the axiomatics of it and the antiquarian nature of it as well as its historical status make it something kind of special. So, in particular, these crazy ideas of "I'll just do Euclid with my kids" are actually not that crazy at all. After spending some time on the matter, my real recommendation is to go find a book from the 19th century that literally goes through the Elements and gives exercises. I'm kind of down on even Kiselev nowadays. Solomonovich is just Kiselev on steriods, and to really be able to do all the stuff he touches on, it takes a lot more than is there, I think. In other words, I just don't think the student is walking away having mastered things like proof by mathematical induction or even the idea of the Method of Exhaustion as Eudoxus conceived it or anything like that. You're just going to have a nice conversation with a mathematician that maybe inspires you to figure out what the hell he was talking about, anyway. Birkhoff is, indeed, kind of like cheating or something. So are the SMSG axioms in the Moise and Downs book I have listed in The Answer, above. Actually, if you want to get technical, you need to use Tarski's axioms (not even Hilbert is good enough), but you are really missing the point at that point. So, just get a good 19th century text. (I was looking at this one in Google Books by Potts, the other day, for instance.) That would give you a truly classical education right there, and if you close your eyes and concentrate, you might, just for a moment, feel like you are actually standing in Plato's Academy over 20 centuries ago. At any rate, there you have it. Mathematicans, math educators -- probably no one will tell you this. It is ludicrous to suggest the possibility of texts like Herstein or Rudin as well as some sort of gaff to act like they are even meaningful without years of calculus and differential equations. But, I am not exaggerating and I am not just making this up. I've even tested a lot of the most important aspects of it empirically. I guess I won't "know for sure" until I take a few kids with a variety of "special ed" issues all the way from start to finish through it all. Now, give me your obolus or I'll beat you with my oar just like I do everyone else!

One ingredient essential to the very idea of unschooling is openness to a fairly wide variety of activities or interests as being "educational". I'm just not -- not even in a normal sense. Things like being widely read or knowing a lot of specific details about history are the halmarks of an educated person not what makes them educated. There is a very short list of subjects that are acceptable to me. Anything else certainly isn't garbage or anything like that, but it isn't educaitonal, either. Not really. That list basically boils down to learning how to prove theorems and learning Latin -- essentially Latin and Math where "Math" is interpretted to be math like Euclidean-style Geometry as opposed to calculation oriented math. Now, I do want to try to prepare my kids for the world, too. So, I don't just drop everything and do only these two in the most direct and natural way like there isn't even a world out there with SATs and colleges and so on. Why Latin and Math? The Math I'm describing is actually kind of special and though requiring some explanation as to why it is special, it becomes clear once one appreciates why it is special as to why someone might think it is one of the subjects. The other subject is really philosophy, but informal philosophy is hard to teach formally. So, Latin is something that can be readily formally taught and becomes a proxy for being articulate that hits all these different things of practical value. (Is a second language, has bearing on English and grammar and so on.) Plus, it is the only second language I, myself, studied in high school and college -- maybe that has something to do with it, too. At any rate, it is the "verbal" component. So everything else isn't bad. It's not like you don't become more articulate reading a lot of great literature, for instance. And, it isn't like you don't get anything out of studying history or science, by any means. But, it just isn't the same, frankly. You get something out of learning how to be handy with electronics, too -- not just specific practical skills and knowledge but also general analytical reasoning skills. I got a lot out of reading comic books when I was a teenager. There were some pretty wildly imaginative ideas expressed in them that truly exceeds a lot of things that are considered "literature". In fact, while we are doing useful and interesting things, maybe we can try to cultivate some auto mechanic interests, or how about interest in sweeping the floor? The point is that as useful and/or intellectually challenging as some of these things can be, not just anything really counts, here, as being "educational". If I had a long list of things that I could enumerate and feel good about my child doing as much as they wanted of any one or all of them, then I would unschool. "You know, you might laugh, but you learn a lot of hand-eye coordination, analytical reasoning skills for that matter, ... from playing video games,...." That may be -- that may really be true. But, that's still just not what I am trying to do, here. So, I take more of a direct traditional approahc to teaching essentially the two subjects I want to teach and otherwise just doing the minimum to prepare them for life as an adult.

Math is not a science. But, it is intellectually repugnant to watch someone imagine that this must mean it is an "art" like painting or something. There is an art to mathematics as there is an art to painting -- just as there is an art to doing science. A physicist could come along and spew the same old hackneyed crap: "science is art!" "You must appreciate the beauty of physics!" (In fact, they do it all the time.) He is also right that we don't do any math with people until they are seniors in college at the earliest. And, he is right that most people don't have a clue what math really is. He also does a great job of extracting the idea out of the imperfect empirical depiction of it, for that matter. He is half right that elegance, for instance, in a proof matters. Well, it is absolutely true that it does matter and someone can get published by proving an old result in a new way. But, the way he's talking, it is all about the poetry of the proof or something. Now, you know that's BS. It is far more important to just come up with one no matter how ugly it might be. The theorem is going to be named after the guy that got the job done in the first place, not someone that had a more elegant proof a few decades later. And, anyway, if you think that calculus as it's taught at the university is more like real analysis and we need to actually dumb it down some more and get rid of all that "formalism", then you just need to be put down. That's all there is to it. Especially when that is what real math is -- the real analysis not the calculus. In fact, what he is saying there is such an obvious and outrageous lie that you cannot call it a difference of opinion or simply a mistake or even a misinterpretation. He is unequivocally trying to just gain popularity with a bunch of self-serving bull****. Everybody knows that the real subject is real analysis and that is precisely what is being refered to when we say that you don't start really learning math until you are a senior or a graduate student -- because you don't start doing math that way -- like real analysis -- until then.

No -- his article is the standard student led discovery reform math BS. You don't need to be teaching alternative geometries to students, for that matter. Going "Aha! But, in non-Euclidean space....!" is about the most hackneyed pile of bull**** online. Yes, those problems are great and all, but he clearly has no intention of teaching his students any of that. He might bring those problems up, but he is clearly all in favor of just bull****ting the students like normal about them, playing a game of chess and calling it a day. He is a turkey because he is selling out his own field with a whole lot of bull**** rhetoric. He has earned his place on my list right beside Morris Kline. He's just trying to sell his royal road to geometry. That's why he's a turkey. And everything he says in that article is almost entirely just a bunch of cheap, hackneyed, self-aggrandizing rhetoric. What's wrong with math ed has never been not disclosing "the beauty of math" or failing to foster a "students love of the subject". It has always been that people start spewing that crap right before they take out the rigor and the standards just like what Paul is doing in this article.

I don't agree with anything he says. He basically thinks that we need to take rigor out of math and that Euclidean Geometry is the worst course we teach in K-12 because it is so dogmatic and not "charming" enough. I bet if you read far enough he'll be calling for us all to sit in a circle and sing Kumbyah. Perhaps we should teach what an all inclusive and happy subject math is and not so ethnocentrically focus on western thought and start teaching about the mathematics of the aboriginal tribes of New Guinea. We should try to emphasize the role of women and minorities in math. We should ask students questions like "If math was a color, what color would it be??" No. The reason no one knows what math is is because of turkeys like this that want to turn math class into the chess club. If he has his way, everyone will think that math=go in 10 years because playing games like go is "real math". We want people to know what math is? How about we cut the crap and start teaching some? The reason no one knows what math is is because we don't actually start teaching it until about your senior year in college if you happen to major in the subject. And, it is just this kind of nonsense that has led us to this point.

Here is one of the most fundamental formulas in the business world: P[1-v^(n+1)]/(1-v) = P + Pv + Pv^2 + ... + Pv^n This formula gives the present value of a stream of payments of P dollars n periods into the furture where P can be invested into a savings account earning a rate of interest of i per period payable at the end of the period and where v = 1/(1+i). This is a polynomial of degree n. And, if you just "foil it out", you will see that (1-v)(1+v+v^2+...+v^n) = 1-v^(n+1). At any rate, the point just is that this sort of thing is just the sort of thing you will learn in the right algebra program. Even beyond algebra, 1+v+v^2+...+v^n is a finite geometry series. You use some theorems and the algebra to determine, for instance, that 1+v+v^2+... converges to 1/(1-v) for |v|<1 (or a rate of interest greater than 0). Indeed, the present value of a perpetuity of P payable at the end of the period is simply P/i where i is the interest rate because v+v^2+v^3+... = v(1+v+v^2+...) = v/(1-v) = [1/(1+i)]/[(1+i)/i] = 1/i. In short, just to be a banker, it seems you need to be able to understand calculus. To actually manufacture and disribute a product, you very well may need to know how to optimize a linear programming problem which already uses linear algebra. The Black Scholes model for pricing options starts getting pretty complicated as a case of brownian motion with drift mu and volatility sigma and turning into a PDE. In short, "business math" gets pretty complicated pretty fast. Pricing the the instruments used in garden variety investments is pretty complex. The mutual fund you put your 401(k) in, for instance, may well use options to achieve its investment goals. And to properly manage it to your retirement needs, you will certainly need to understand the present value of a stream of future payments. So, personally, even just for "business math", I would keep him in algebra.

Do it! lol. No -- it isn't training on New Math, per se. It is just some good regular math is all. It isn't quite at the level of Abstract Algebra so much as more New-Math-esque and at the level of "college algebra" (ie before calculus but definitely probably a step up from most high school courses). Could you do it after Algebra I? Well, it looks kind of like you could theoretically do it as Algebra I or probably Algebra I and II combined or something. My guess, though, is that it probably goes too fast for a kid that age. Theoretically, you could do abstract algebra with 5 year olds -- I mean, it's not like there are any prerequisites. But, on a more realistic note, you probably have to wait for a lot more intellectual maturity to develop before you can really do that kind of stuff. This book looks like it is for college algebra. I would say that from what you have posted, it looks like it has some really good math in it -- proving that the multiplicative inverse of the multiplicative identity is the multiplicative identity, for instance (i.e. that 1*1^(-1) stuff).

I don't really make recommendations for [1] and [2], mostly because you don't need one if you fall into that category. Do teaching textbooks or something easy and just forget about it -- stop even worrying about whether your math program is good or bad at all. If you fall into that category, and you really still want advice, then my advice is to get out of that category. It's not that hard -- it isn't as hard as learning Greek or Latin -- that's for sure. At any rate, my recommendations fall into three categories of my own (which may seem like "hard", "harder" and "harder than that" in no particular order): 1) You want a normal program and want your kid to be successful with any math-related endeavors they may persue outside of the program. Then do Singapore up through NEM 4 and put your kid in Community College. It is a "no brainer". It pretty much covers everything (though it might be a little light on trig or this or that -- if it really bothers you, you shouldn't have to supplement very much to completely remediate it). The brain comes in when you teach/TA it. You are going to have to be able to solve those problems and explain them to your kid. But, outside of that, you should have no anxieties like "But, what if they bomb the SAT??" or "What if they can't handle a normal Calculus class??" 2) You don't care about normal or college or the SAT. You just want the straight dope, no bull**** answer that doesn't depend on a whole lot of social crap to be even remotely valid. Then, what you do for arithmetic is practically irrelevant -- do something, though, like up through Singapore 6B -- don't just skip it. You will start out around 9-12 years old with "mathematical logic" ala Patrick Suppes' An Introduction to Mathematical Logic . Do a couple of his other books all the way up to axiomatic set theory. Add in some number theory from the Art of Problem Solving, perhaps (real early on -- perhaps in your arithmetic). Add in some kind of Euclidean Geometry like Kiselev's Geometry . Do all that until your child is ready for something like Joeseph Gallian's Contemporary Abstract Algebra or I N Herstein's Abstract Algebra . Now a lot of mathematicians, and definitely a lot of math teachers, will screech like a pack of harpies at you for doing something like that, but rest assured, they are all mostly full of it. All they are doing is observing the fact that you are bucking the system and want to know just who the hell you think YOU are?? However, it is true that you will have to stop along the way at times if you hit some intense algebraic manipulations since your child will not have much, if any, experience with that while these texts are written for people that really have. At that point, you may have to to just make up some problems to get the practice in on the relevant mechanics for a while. This kind of thing will only happen a few times before your student is pretty handy with a formula. Once you are at that point, you should be able to follow that up with a stronger Algebra text (like Abstract Algebra by Dummit and Foote) and something like Topology by James Munkres. Finally, wrap it all up and call it a day with Principles of Mathematical Analysis by Walter Rudin. You may have to go really super slow at first, but it can be done this way, and it may well take you until the kid is around 20 to pull it all off. No one really seems to know for sure because it appears as though no one has ever had the stones to do this. Also, go ahead and follow all this up with a really tough Analytic Mechanics or something -- some senior level physics text on classical mechanics -- just to make sure the material is really made applicable for your student. 3) Okay, so you do care about college and the SAT, but you don't care about normal. And/or, you can't do the real no bull**** program -- no really, you really cannot do it, as in, not able to as opposed to just feeling a little too neurotic or lazy to because it probably really does take a PhD in math to do well, for one thing. Then, that's a toughie, but I really like these books by Allen and Pearson from the New Math era. Pretty much everything I'm doing here is in my tag line. Though it is often criticized as such, it really isn't Abstract Algebra. Nevertheless, there is some pretty good stuff like proving that the inverse is unique (like you would do normally only in an abstract algebra class). What other high school algebra does two column proofs? Probably 1960s Docliani or something (in other words, some other New Math text from the 60s). I am totally infatuated with Kiselev's Geometry -- a classical Euclidean-style Geometry text. That kind of geometry is always a good one for the axiomatic method. In fact, Euclid basically invented axiomatics in the Elements -- that's why it was such a big deal. And, of course, I like and have always liked Gelfand's books because the problems are really pretty d*** good -- better than Frank Allen and probably better than NEM. In fact, I really can't think of a book with better problems. I found at least one of his problems in a graduate text on complex analysis, for instance. (But, it's totally "doable" in the context it appears in Gelfand.) The problem with Gelfand is that there really are problems in there that if you were to do it truly rigorously, you would need to use Mathematical Induction, say, which he doesn't teach you and is a bit too hard porbably for most American 9th graders. In fact, for that matter, Allen probably isn't really completely rigorous. Ordering, for instance, is better done in Beckenbach's book (also listed in my tag line and something we plan to do after or towards the end of Allen, but that we have nonetheless started informally introducing already). If you can get your kid doing S I Gelfand's Sequences, Combinations, Limits -- all the problems on their own , then there is little doubt in my mind they could just skip the calculus and go straight to something like Lay's Analysis. Just aside from all of this -- forget about high school, having a "rigorous program", or anything like that for a moment. The three main subfields of math are Abstract Algebra, Topology and Analysis. Okay: not calculus. This race to calculus is really all about a race to engineering not to mathematics. You would think that the freshman calculus sequence would have some kind of bearing on math as a field. But, it really profoundly doesn't. In fact, the whole of undergraduate math lives in the service of other fields, most notably physics and engineering. Some students go to graduate school in math never having taken a course in real analysis! That is outrageous! Because when they get there, they will be in for a rude awakening when their whole success or failure is based on a bunch of stuff that bears almost no resemblence to anything like freshman calculus and looks like real analysis all day every day. That was the big weed out qualifying exam at Ohio State. And, it probably still is. (Not that calculus is a bad course or anything -- just that it isn't like "what mathematicians do". It is more like "what physicists and engineers do". It's all the theorems of real analysis without the proofs, so it isn't doing math so much as using math -- usually to shoot a cannon ball or something -- some sort of a physics problem. Hopefully, I haven't slighted calculus too much....)

It looks like some kind of college level text maybe -- like pre-calculus of some sort or just an alternative to freshman calculus. No book that does mathematical induction and systems of linear equations is a pre-algebra text, I don't think. I think if you look at it more closely, you'll find you have to already know algebra for this. Maybe it's some sort of "pre-calc" or something.

No. He is 11. But Myrtle, nonetheless, thinks Suppes' Introductory Logic will be easy for him. That book ends with symbolic quantified logic. She thinks he could do that and a couple more books in three years time and all that ends in axiomatic set theory. My first thought was "Great! I really will be able to do Lay's Analysis when the time comes and he'll have studied yet another axiomatic system." (Lay's book is like the easiest real analysis text possible and it has this big logic and set theory beginning to it.) But, truthfully, if you are talking about being able to do axiomatic set theory and having the logical wherewithall to negate elaborate quantified sentences by the time he is 14, then you could do regular math. What's stopping you? Maybe a little prerequisite knowledge. But, yeah -- somehow, I just think it will all get bogged down real fast somewhere -- that axiomatic set theory won't work or something. He'd get stumped for months and months and just never be able to do proof by mathematical induction -- something like that. (But, like I say, if the logic really is there, then I don't think he'd get stuck on stuff like that.) And besides that, like you say -- we still have a whole lot of Latin to do. We were doing Henle but ended up just letting it slide last year. We'll probably have to just start from scratch again. I was thinking this logic and set theory is good enough for any sort of "philosophy" I would have ever wanted to do (as opposed to that Lipman series I am always promoting). Actually, it is just the sort of thing that philosophy majors have to prove themselves on, anyway, I think. At any rate, I think there is just a limit of emotional energy someone has. You just can't do the most intense thing all day, every day even if he could theoretically do each individual piece on its own. But, I seriously was telling Myrtle about topics and such and how between algebra, analysis and a course of introductory (calculus-based) physics, you'll get everything out of math and 50% of science anyone else gets. (Plus, of course, you would literally be practically skipping straight to graduate level work in math.) Myrtle's response was "Okay, try it on me, then. If it doesn't work on me then it isn't going to work on the kids." So, now I am literally thinking about getting this Dummit and Foote book or something that comprehensive but that is easier maybe or some such thing. Myrtle can actually do a little set theory and even a little mathematical induction. I think we both would deserve some kind of medal if she actually starts working at the level of Dummit and Foote as someone that never even took calculus in college. (At least someone could say I was right for a change about some of the stuff I spout about math ed.) Maybe they'll just let her sign up for a graduate program around here. Dummit and Foote's about at my limit, I think. I would have to concurrently do my own thing in Lang or something just to competently TA her through it. (I don't even like abstract algebra, for crying out loud!)

Well it's basically because she thinks she can teach the kids some serious logic. So, yeah, I'm thinkin' "If she can do that, then I can...?" Consider this. What if you could teach your kid a solid sequence of abstract algebra followed by a solid sequence of real analysis? Maybe it takes like four years to do the whole thing and it includes polynomial rings and vector fields and all of a typical real analysis. And do a year of physics on the last year. What will they not be able to do? Manipulate x? Solve a physics problem? Hell yeah I'd do that if I thought I could! They'd walk into college being qualified to teach calculus. ;o) (But, let's just say I'm not ready to bet the farm on that one yet....)

...and this will sound paradoxical coming from a math nazi like me, but I don't care so much about science. Don't get me wrong, I like physics as much as the next guy -- black holes and stuff are totally cool. On a personal level, I like science. However, outside of a basic one year of biology and one year of chemistry, I don't think it is crucial to do it -- not even for future physicists or engineers. It's not that science isn't crucial for those disciplines so much as that they will do it all again in college. I took a ton of physics, a year of chemistry, and AP biology when I went through the program. Chemistry was literally a blow-by-blow rehash of high school chemistry only just about exactly twice as fast. (But, all of college is twice as fast, so hopefully the pace doesn't shock anyone.) I guess I don't know what biology would have been like in actual college, but once again, the AP biology I took was literally just like the regular biology only it consumed two class periods instead of one and we covered twice as much stuff. So, that's almost the same sort of outcome. I didn't take physics in high school at all, and, although the freshman sequence I took in college was super lame, I did do just fine in it. (I think I made a B or something.) It's kind of like asking "But, what if I want to be philosopher?? Should I take high school philosophy?" Well, in some sense, sure. But, more realistically, no body does, and they can't really assume too much when you get to college about what you really know coming out of high school. I think everyone should know some basics about how the world works, but I just am not going to emphasize science, myself. (We "box check" it with Singapore Science.) And, I think that with a super strong math background, they will be able to walk in to any discipline and do fine on their freshman course sequence. (And conversely, I kind of think it is a lot harder to get along in a calculus-based physics class if you can't do calculus but you have tons of conceptual physics under your belt rather than the reverse of no physics but strong on calculus.)

...because I am male. :rolleyes:

Incidentally, out of all those books and all the books on my website, I think that Beckenbach is special. For one thing, the subject of inequalities is actually pretty key to real analysis (i.e. calculus done correctly). So, there may be something to the topic in that regard. But, what really makes it so special is the fact that it is accessible to a typical parent in a way that other books treating axiomatics really aren't. But, it is simultaneous a real example of axiomatics. It is completely correctly axiomatic about the notion of ordering. Not pretty close. Not an attempt to push the student in that direction in the hopes that one day maybe they can finally do it right. This is it -- the correct axiomatic method for developing that. And, it isn't "presented in a way that is accessible" -- it just is an accessible topic to present. It really has provided a clear example for Myrtle and me to talk about programs with -- how Jacob's just shows the student a number line as opposed to what it takes to really do it correctly. It is also an example of how not every axiom that the student is going to have to use is spelled out -- just the relevant ones for the book and the rest are just taken for granted. And the student doesn't just do axioms all day every day. They do the axioms and then they do the results. It's all done axiomatically but not just as a big exercise in proving the obvious. It really is a very good example of what it's really like to do math. You solve problems rigorously. I feel like this provides the most concise example possible, probably, of "what I'm talking about" all the time on these boards and elsewhere. At any rate, I just thought I'd mention that about that book, in particular. (I really didn't appreciate it, myself, at first until Myrtle started working through it, and I started really examining those first few chapters and then the following chapters. That's really a great book by a great mathematician.)