Jump to content

Menu

Charon

Registered
  • Posts

    111
  • Joined

  • Last visited

Everything posted by Charon

  1. It is the view that some structures that would otherwise seem to occur naturally (i.e. were not created by man), specifically with regard to biology, are so complex and/or sophisticated that they must have been designed by an intelligent designer (which we refer to as God or some have imagined that it was multiple such designers). (This is a common and ancient argument for the existence of god.) What do you think it is?
  2. Intelligent design represents little more than a certain kind of argument for the existence of god. If you taught it, you could probably teach an enormous volume of philosophy as it turns out. So, it is hard to answer just what you would teach beyond just questioning certain arguments for atheism, materialism, empiricism, and/or postivism. It could be all sorts of things from ancient times to this very day. Now, I should come out and confess that I do not actually advocate intelligent design, myself. However, its intellectual roots are indisputable. It may not be science, but science is an unreliable Jonny-Come-Lately to intellectual life. It really is. I'm not talking about Newtonian Mechanics being overturned by Einstein's Theory of Relativity. I'm talking about a great deal of pseudo-science becoming widely accepted into mainstream "science", about how science is strongly susceptible to political and social influences, about stuff that scientists say and that their books say that are fallacious and internally inconsistent. But, above all, I am talking about how the advocates of science, materialism and empiricism seem to go around absurdly dismissing long well-established intellectual and academic traditions and entire fields of inquiry that have more than adequately withstood scrutiny and the test of time in a way that many of their ideas cannot and have not. I would say that there does seem to be a larger agenda of the Intelligent Design movement. It is not to teach young earth creationism. It is much more about stamping out positivism. And, personally, I don't really contest that. In my opinion, positivism is a bunch of bad, poorly thought-out ideas based on philosophical skepticism which is ironically the antithesis of science and knowledge usually dessed up to look like science. In fact, positivism is one of the few things in the world that makes me wonder if there really isn't a diabolical demon or devil insidiously working behind the scenes to orchestrate mankind's downfall. (No offense to any positivists out there.)
  3. It is not re-packaged creationism. Creationism is really the christian fundamentalist view that the world is only however old it supposedly is in the bible based on a literal interpretation of the bible. That is "young earth" creationism and I am aware of other versions of it, but that is the source of creationism. It is an invention of certain groups of protestant Christians and it is derived from the bible and it is recent, relatively speaking. Intelligent Design is more like re-packaged 3000 years of theology and metaphysics. Some people that adovocated intelligent design: Plato Aristotle Cicero (author of the Sundial Analogy now going by "Watchmaker") Augustine Aquinas Newton's watchmaker was blind but existed nevertheless Descartes Really, for that matter, almost everyone prior to 1800, say, unless they were noted atheists would probably agree with some sort of intelligent design theory of the universe. In fact, we should probably all read the Discovery Institute FAQ and get what Intelligent Design means from those that advocate it. Incidentally, you know why the early church made that mistake? It was because they were dogmatically using Plato's Theory of the Spheres. And the reason they were doing that is because they were largely noeplatonists (e.g. Augustine was a neoplatonist). All the way back in 200 BC Aristarchus of Samos had a heliocentric theory. It just didn't get selected for in the market place of ideas at the time. That may be because the market place wasn't as good back then at selecting good ideas. (In other words, it may be that people just went with Plato because he was the best all the way around as opposed to today where we can pick and choose more and go with Plato on Epistemology or something, maybe, but Godel on logic and Steven Hawking on cosmology and so on.) In other words, it wasn't some sort of religious quackery that caused it. It was widely regarded as true contrary to other, better theories at the time and for some time, and they just built it into church doctrine. If you have to pick one person to follow, then I'll certainly tolerate something like geocentrism over not even knowing what constitutes knowledge, for instance. I would certainly take the mathematics and philosophy of the Platonists over the largely pervasive pseudo science and huckstery of the day. And, the choices were a lot more like that back then because there were no well established peer reviewed academic journals and what not. There wasn't a well established way of determining authorities or experts on things like there is today. And, everything tended to be highly philosophical.
  4. Incidentally, this would not be the first time someone has said that math should include a little more math history. I had no idea who Cajori was until Myrtle turned me on to him. That is a miscarriage of justice! ;oP Cajori is a good read when it comes to math education and math in general -- it is both relevant and important to this very day. I wish I had time to read more of that guy. Math isn't supposed to be some cult like it is these days. It is supposed to be transparent. And, I think the lack of history is one key ingredient that enables the mystics to obfuscate what math is and has always consistently been for longer than even the classics have existed. Most highly trained mathematicians are woefully ignorant of even the last 50 years of math education or even mathematics! Even people out of Princeton or MIT often just know the current state of mathematics. Professors at Princeton or MIT just know how it all happened during their career as a mathematician. In philosophy, by contrast, the History of Ideas is just as prominent as the Problems of Philosophy. We all, especially mathematicians, need to stop taking this for granted. There is more to math than just the next big theorem. But a million dollars for anything from P vs NP to real progress on Navier Stokes Equations is a hard thing to resist. Unfortunately, no one is there to tie Odysseus to the mast....
  5. What A D said, only in a less snarky manner. Day trading is gambling. Investing in a well-managed, diverse portfolio of stocks is just good business. Cash, as opposed to a promise to pay cash at some future date, has value. So, when you invest in highly volatile stocks, what you are really doing is just loaning your money out at a really high rate of interest. (At least that is one way to think of it, though it certainly is not just exactly that. This is more of a high level macro-economic perspective on it.) You're portfolio has good years and bad years, but, on average, it should bring in a good yield (e.g. the S&P 500 has yielded something like 10% over its lifetime) if you leave it in a mell managed mutual fund or something like that for a long time. If you start raiding the 401(k) every few years, though, all bets are off. Incidentally, as for inflation, we know that inflation exists because we keep printing and/or loaning money to banks faster than money is destroyed. That it is "only" 3.4% shouldn't be a super shock. That's normal. Maybe it is higher, but I certainly haven't noticed it, personally, yet, if it is. (I don't really keep tabs on such things.) I do know that interest rates are super low, and interest rates are normally thought of as a real rate of interest plus inflation. So, I don't know how high people can really imagine inflation to be. At any rate, if you just do the most "conservative" thing, and invest in cash, say, you will end up losing money to the very inflation this thread was started on. Your dollars will only buy 96% as much food, gas, or shelter a year from now as it does now. So, you need to keep it in some instrument that will at least keep up with inflation. But, just keeping up with inflation is like buying a brand new car and just parking it in the garage. It depreciates all the same while it is parked there (albeit a little slower if it is just parked). You should use it as a car and drive it around. Otherwise, you have to spend a bunch of money on taxis or bus fare or whatever to get around, anyway, so it is just a waste to park it like that. The same is true for large sums of money. You should invest it appropriately in something that puts your money to use and not just maintains it. So, "investing your money conservatively" means putting it in a mutal fund or something along those lines that yields like 8-10% when you are in your 20s and 30s and then slowly moving it over into lower yield fixed income securities in your 40s and fifties whenever the market is on an up swing. You may well still have some of your money aggressively invested even in retirement.
  6. Companies pay pension actuaries big bucks to calculate the employer contribution for a defined benefit plan. For one thing there is this entire body of code called ERISA that governs it and a very well defined way to go about doing it determined by the IRS for tax purposes. (That is, the minimum amount a company must contribute and the maximum amounts a company is allowed to contribute are determined this way.) There are similar (though probably more flexible) rules for calculating liabilities associated with a defined benefit plan for accounting purposes. The reason it is so complicated is because in a simplistic way of looking at it, there is no contribution a company has to make for a particular employee. The way it works is the company is just on the hook for providing a certain benefit at a certain time to employees. There is no funding requirement, per se -- that comes from the IRS's requirements to recognize it as a "qualified" pension plan. The US's social security plan is pay-as-you-go, for instance, which is why it is in so much trouble whereas most legal pension plans (are required to) pre-fund their liabilities. So, nevertheless, there is a typical manner in which companies assign a liability to a particular emloyee for these kinds of valuation purposes. You would probably be interested in knowing something like your projected accrued liability or something like that except you would want it based on some special assumptions about withdrawal and other decrements. (For instance, I can tell you right now, his liability is probably zero from an actuarial perspective because he isn't even vested in it until 5 or more years of service. And, there is similar "vesting" for employer contributions to a defined contribution plan, too, by the way.) To pull this off, it would take some fairly sophisticated financial modelling, involving mortality tables, interest rate assumptions, inflation assumptions, a model of your expected future salary and so on. So the punchline is: don't do it this way. Go out to Vanguard or a similar place and look up some nice looking annuities (and how much they cost, in particular, perhaps trying to match benefits to the DB plan as best you can). Project how much you would end up with under the defined contribution plan and translate that into a monthly stream of income based on a typical annuity you might buy (from Vanguard or some other place). Then, compare the monthly streams of benefits rather than the accrued liability/asset like you are thinking about doing. (For instance, say you are going to accrue a million dollars in this DC plan by the time you get ready to retire based on your best guess about what your salary will be and so on. You can figure it in today's dollars -- so you probably do not need to think too much about inflation or that sort of thing. Say an annuity that pays you $1 a month for the rest of your life costs $500. Then, the monthly stream would be $1,000,000/500 = $2,000. Is that what the defined benefit plan would get you under similar salary assumptions?)
  7. I kind of doubt it. It's not like I have gone through NEM with a fine toothed comb or anything, but I think what makes Allen challenging is more in the way of the axiomatic method you use all along the way. (But, I wouldn't characterize the problems as easy either.) So, there is some value-added in that regard to doing Singapore. In fact, I suspect that our kids will not do as well on the SAT being trained on Allen as opposed to NEM, for that matter.
  8. I'll also second Jane's post. However, I use Frank Allen instead of Dolciani and will end with a book by Allendoerfer and Oakley rather than the Dolciani, Beckenbach, Wooton, etc book. But, mine are even more scarce than hers, so even less recommendable to someone else since you probably cannot get your hands on the texts themselves, let alone such things as solutions manuals or teachers manuals. I would also say that in my (albeit vastly minority) opinion, physics majors need it more than math majors. A math major gets to just live in a fairly contrived world of whatever the math department is doing. And, they will eventually get to the rigor. A physics major (and the physics department) is constrained by the needs of the physical problems they encounter. The world doesn't care if you don't have the math to handle this now or not -- it just is what it is. And furthermore, once they struggle through something enough to ostensibly "get it" (e.g. convincingly BS their way through it, say), they have very little incentive to go back and "do it right". So, there is no reason at all they will get the rigor eventually. And so, a mathematically rigorous understanding calculus, for instance, is something that they could really benefit from that most of them probably never get. Nevertheless, I do realize that no body does it that way, so who am I to buck the system....
  9. When would you need to know the why? How about multiply (x+3)(x+2)? Multiplying it out you get: 6+3x+2x+(x^2). Replacing x with 10, what did you have? 6 (3x2) 30 (3x10) 20 (10x2) + 100 (10x10) ------ 156 "The long way" is getting you ready for polynomials. The short way is just some algorithm.
  10. Well what does it say, anyway? Who does it hit, first of all? And what does it say abou them? What does it say about Kant, for instance? (Just curious -- I've seen it mentioned a lot.)
  11. I bought Nance almost entirely because of Cothran's bad attitude about modern developments in logic. No, modern symbolic logic is not some sort of atheist conspiracy to make logic difficult nor is it just good for getting rid of philosophy or religion and replacing them with science. It is just progress. With that said though, I do not think that Cothran is the teaching textbooks of logic. (I am becoming famous for my disdain for that math program.) I really do appreciate where Cothran is coming from in his attitude, but I think he is just kind of wrong, unfortunately. (I am something of an anti-positivist, myself, and I'd like to think that I would know one if I saw one. If I do -- Cothran is definitely one. But, he is still wrong about math and logic.) At any rate, you could do much much worse, I think than to use Cothran's program. I bought Nance but did also flip through Cothran. Personally, I didn't and don't do either. I am doing something even more mathematical (and probably harder and probably at the expense of other subjects) than either of these.
  12. One day, I am going to write a book called simply "MATH". It is going to be one long series of of five problem groups with little more than a paragraph every now and then. It will be even more sparse than Schaums but designed in five problem sets to have standard problems, additional problems and the next problem set. Every now and then there will be a string of problems that starts from some point and ends at the derivation of the Quadratic formula, say. It will go from elementary arithmetic all the way through a rigorous treatment of calculus. It will be one book and it will have probably a little more than twenty thousand problems. I will dedicate the book to my friend Song Xi from when I was in school who went into actuarial science and once said of the (infamous) 150 exam (with a reasonably thickish chinese accent): "6 exams. That's all the problems. About six exams and you have done them all -- probably some of them twice.... OK, see you later...."
  13. Now that I read that, it comes off a little arrogant. I should point out that I have an MS in Math from Ohio State. I actually am somewhat of a "competent professional" in terms of evaluating that kind of mathematical content. The fact is that every program usually has a PhD Mathematician or other credentialed expert endorsing it. That's why you need to just take and make a direct comparison of the problems between programs. Some PhD endorses (indeed helps to write) Saxon. And, the Sabouri brothers have whatever endorsements they have. Now, let's look at the problems specifically. Bottom line: TT's problems are far easier than everyone else's.
  14. No looking at the problems is not an anecdote, especially when it is someone like me looking at them. I think the main thing lacking in my own assessments of TT is completeness. I suppose I literally have to buy all of the texts and go through them with a fine toothed comb. If I was actually being paid (enough) to write an evaluation of them, then maybe I would do that. But, then again, you would have to wonder about the motivation of the people that paid me. ;) I am basing what I am saying on specific evidence originally fairly systematically acquired through their website designed by them specifically for people to evaluate their product as well as on specific things that others have said. I am not sure what more (other than more thoroughness) you would want. Statistics are always problematic and generally a little abused by whoever presents them. Although, some basic ones wouldn't be bad here. I think, in the end, though, there really is no substitute for simply directly evaluating the problems students can do.
  15. No. I don't think I said that at all. At one point in my post I said that you must provide proof of your program along the lines of specific problems your students can solve if they go through your program. But, I have already fairly adequately investigated that on a number of occasions. Also, my findings are corroborated by others who have used the program before. Selected anecdotes like yours are not evidence. Even the worst public school programs have such an anecdote. It becomes statistically likely with an increasing number of people using the program. And, it's vague. So, you're saying your kid is in the 7th grade, using the 7th grade TT and they made in the 60th percentile on the ACT? On the whole, 60th percentile is not all that great, per se, but for a 7th grader it is pretty awesome, I would think. (What do 7th graders normally make on the ACT?) If your kid was 16 I would say "So what -- that's closer to average than to good." But, since they are 11 or 12, that is a good sign that they are bright. However, it doesn't necessarily indicate much about the program they are in, and even if it did, it is not clear what it would indicate. "That it's good," is almost completely meaningless unelss you are to infer that students out of the program can solve some hard problems covering typical topics. And, especially since it is one outcome selected not at all at random, it really doesn't indicate anything. Given what I do actually know about the programs, it sounds like your kid would have scored even higher had they been in 7th grade Singapore which I believe is NEM 1 but it is probably 6B if you go according to his age. In either case, I am quite confident that the problems of 6B blow the problems of TT7 out of the water in terms of difficulty.
  16. Myrtle has spent days thinking about a problem. She was reminding me that she spent 6 hours proving that the geometric mean is less than the algebraic mean. I, myself, certainly have spent days thinking about a single problem. Our kids have spent hours on a hard problem. Myrtle imagines that she really probably could not handle a normal college paced course in the kinds of things she is doing, but she doesn't back down one bit from the rigor or the difficulty. She just takes as long as it takes to do it. And, now, she just got finished doing a series of mathematical induction problems in about two or so days time and a number of those problems appear in a senior college abstract algebra text (by Herstien). She never made it beyond Trig in college. You can do it if you want to. And, you're kids can, too.
  17. Basically, to answer your question: no, not really. There certainly may be something to that effect. It may be that the Singapore students all do twice as good as the TT students on average but that Singapore is not really quite twice as good of a program as TT, say. But, I've been over the online problems and placement tests, myself. In fact, that was directly where I went the first time I ever looked at it (years or a year (?) ago). The problems just aren't there. I've looked at a lot of material by now. And watched a number of videos. If I were really formally reviewing it, then I would spend hundreds of dollars on texts and go meticulously through them in every last detail. But, I've really seen enough and I have compared it to both Saxon and Singapore. Like I say, it just isn't as hard -- that's all. A number of other people who have actually bought and used the program have come back corroborating my assessment. So, we kind of really know that TT is not as "rigorous" as Singapore. It's not just anecdotes or isolated test scores. It is a direct comparison of the problems. And, the anecdotes aren't just "I used TT and it didn't work out for us." They are "I used TT and the problems were all really easy compared to Saxon" -- stuff like that. That's not just an anecdote. That's real information that doesn't change from family to family.
  18. Your kids will encounter a whole lot of stuff along the way that they just aren't ready for. It will happen just because they don't take the sequence they were supposed to. It will happen because they did take the sequence they were supposed to but it just doesn't prepare you like it should. It will happen because courses in the sequence they took ended up being weak for some reason -- perhaps because they "lucked out" and ended up with a really easy professor that semester. Their success is going to be determined a lot more by how they get through those rough spots. When they come they are going to be a lot rougher than simply not factoring enough difficult cubics or something like that. It is going to be something more like them being expected to handle a certain kind of second order linear ordinary differential equation in their circuits class long before they are even told what a "differential equation" is just because they have to cram a litany of engineering into 4 years. That's a helluva prerequisite to miss! And, it was baked into the program at the university I attended! (My brother got his degree in EE.) So, I don't really know how your children will do in college because so much of it is going to be determined by how they handle that kind of situation, and it will play out at a much higher level than high school. But, to give you an anecdote on how reality often doesn't conform to our expectations, my brother was a high school drop out. He dropped out in like the 9th or tenth grade and is 9 years older than me. When I came up through the ranks, I went to an elite private school -- perhaps the best in town. I then transferred to a pretty top notch public school and entered their "level 4" track -- the highest level. I took all kinds of level 4 english and history and math. I took AP biology and made a 5 on the exam which was all essays. You know what that high school drop out did -- he crushed the SAT and made a 1430. I only made a 1340 and largely the difference was in the verbal! (I remember our scores distinctly because the middle numbers were transposed and our math scores were either exactly the same or ten points off from each other.) He placed in second semester calculus, even. He ended up take first semester freshman calculus and just sucked it up and filled in any gaps he may have had on his own. He has his EE degree now.
  19. Well, it isn't just Saxon it is all of the mainstream programs seem to cover a lot more. Some folks have come on here and said that TT Algebra II is really more like Algebra I by most curricula standards. And, it isn't just the fact that they are doing something else like mathematical logic or something. They just are moving a lot slower and the problems are a lot easier and the material is just a lot more spoon fed and students coddled all the way around. That fact is disturbing if you are concerned about the content of TT. And, it warrants a much better response than some dismissive "well, they're cramming all kinds of superfluous crap in texts these days." I don't necessarily dispute that, actually, but it is non sequitur to the real concern. The concern is that crap or no crap, this is what everyone is doing and I don't want my dc to be left behind. (Actually, I do something a lot different from all this, so maybe I don't have quite the same concerns as other parents, but that is the general angst people have over this issue, at any rate.) All of that requires some sort of a "respose" in one's mind. If I were using TT, my response would just be that "Fine -- you're right. I'm simply not covering as much math. I don't feel like I really need to." If that was my response, though, I wouldn't really turn around and simultaneously imagine that my math program was just as good as the mainstream. I would think that I am using a somewhat mediocre math program because I don't really value math, per se, that much -- similar to what is probably, in fact, the case for English literature and history around here. I'm not goign to come around here saying what a great English Lit program when I have next to nothing on my list of books I want to cover. Now, if I was planning on my kids reading a bunch of Of Mice and Men, Brave New World or whatever it is -- Beowulf or Shakespear, perhaps -- that might be different. Also, specifically with regard to advincing a child through it -- like you have a 12 yo that is in TT Algebra II, say -- I, personally wouldn't do that. I think it is at best redundant -- you'll have to rehash the very same material only harder later on. It could also end up undermining them later on if you end up not covering topics with sufficiently difficult problems and they are later expected to have a high level of facility and sophistication with that topic. With that said, though, I will concede that it is probably different to take what might normally be considered an inferior program and accelerate a student through it. You are making up for lack of content with speed of coverage. In other words, a 9 year old doing TT Algebra I is probably a lot more like a 12 yo doing NEM I than not. That 9 yo is probably on track, for instance, to do NEM I when he is 12 (from what I can tell). In a similar fashion, I have often tried to figure out how to do Analysis with an Introduction to Proof by Steven R. Lay. There is no doubt about it that this text is vastly inferior to Principles of Mathematical Analysis by Walter Rudin. However, not all students can handle "Baby Rudin", as it's commonly called. And doing Lay's book with a 16 yo, say, would be a striking achievment, indeed. So, while I, personally, do tend to dislike the same idea applied to TT, it is probably true that a child that has been accelerated through it is not receiving some sort of an easy unchallenging math program.
  20. Don't you think Saxon, Singapore, Dolciani, Foersters, and Jacobs all have similar stories associated with them? The point is that Singapore seems to have more stories associated with it. And if you compare placement tests, for instance, Singapores are much harder. And, if you start comparing problems between, say, TT Algebra II and NEM 3, what do you find? Reality is not one thing for one family and another thing for another family. TT is either a good program or a not so good program for all families. Choosing to use it or not is a matter of style and priorities. No amount of "it works for my family" is going to turn TT into something it's almost surely not: the top program that all the "mathy" kids do.
  21. You do TT and be done with it. Or, you decide, like Myrtle did, that you want to do math now and really learn you some math and teach it to your kids. Math is just a subject. Just one subject. Now, I do, personally, think it is a very important one. It took Myrtle, who never made it beyond Trig, to really renew my interest in the subject for its own sake. And so, we do it beyond most any homeschoolers that I know of. And, yet we don't really do a lot of Engineering Math, so I am not even altogether certain just how much this will all necessarily translate even to SAT scores. However, was Albert Jay Nock really good at math? Or, for that matter were any of the Roman emperors? (I bet even Marcus Aurelius wasn't that good at math.) It's not like you can't be a man of letters and not be able to barely do arithmetic. There's plenty of room in the world for people that have trouble so much as balancing their check book. (Actually, in an ironic twist of fate, a lot of mathematicians figure into that category.) I still think you are missing out on something special and unique if you don't do math, but you are missing out on that whether you do TT or Singapore, actually. (You really have to do something different to get what I'm thinking of.) I'm just saying that you should know the limitations of what you are doing. TT probably does "cover everything" if you go all the way through to the end. You've probably gotten at least as far as most programs' Algebra II which is probably as far as you need to get for someone who doesn't like math and intends to go on to college and major in the classics or in English or something.
  22. You are confusing the sensitivity of a lot of the people that use the program with actual harshness of the comments made about it. This is not subjective. It is a simple matter of fact as to whether or not a student of TT can do the same kind of problems as a student of Saxon or Singapore. It seems almost certain that they can't. That's it. Maybe I'm wrong -- then prove it with problems. Show me the problems that students can do after TT. Show me something besides the vague assurances of its author or charges of antidiplomacy on my part. And as far as what we need in math programs, what is wrong with math ed, at its root, has nothing to do with any of these things. It has nothing to do with pedagogical methods. It has nothing to do with conceptual understandings or any of that. Or even brainedness or some such. The problem is that we continually want to teach material that students aren't really ready for. We want a short cut to advanced material. And we want to do it with everyone, most of whom don't really care -- they don't care and their parents don't really care that much. This problem is as old as dirt. There is no royal road to geometry. If we did this the right way, it would all be about 10 times as hard as it is and most people wouldn't get through one onehundredth of the material they currently do because not everybody needs to do a lot of math. But, everyone should do the math that they do end up doing correctly. But, that's not what we do and that's not what TT does either. So, fine. Just box check it and be done with it. TT is great for that. Don't come up with an infomertial email from the publisher and act like TT is a super good program. Not every program is "the best". On an ancient Simpsons episode where Homer and Barney undergo a grueling competition to see which one NASA will select as the token blue-collar slob on its next manned mission to space, a NASA official tells them "Gentlemen, you've both worked very hard and in a way, you're both winners. But in another more accurate way, Barney's the winner." I know people say it a lot -- "Everybody's a winner". But, I am afraid it just isn't true. In most of the important ways but one, TT loses to its competition. It does seem to provide the most eshaustive and thorough video explanations of any program and it is the easiest and so most doable. But, it seems a near certainty that it just doesn't have as good of problems and the students coming out of it are less equipped to work hard problems than they are coming out of most of the other programs. That is not a matter of opinion nor is it a matter of student or parental preference. It is a simple matter of fact. It is much better to know this and accept it and plan around it than to talk yourself into thinking that TT is going to produce students that can really solve problems the way programs like Singapore or supplements like Gelfand do.
  23. I think most of that "and PS texts cram a bunch of BS in there just to get approved" was more of a hedge to make you feel less concerned about the possibility that his program doesn't cover enough. It seems to contradict via tone and/or insinuation is very strong and unequivocal opening remarks that his program is just like all the rest in terms of scope and content. (That alone, frankly, is enough to get me to not buy it right there, but I will admit that I am somewhat eccentric in my math choices....)
×
×
  • Create New...