I think the Singaporeans acknowledge that it is "engineering math". It is plenty good at that. Everything is "engineering math" unless you either get high enough up (like graduate math texts or the right kind of senior undergraduate text) or you really find something exotic like the New Math from the 60s or something. So, unless you are really trying to do something like what I am doing (teach rigor in a way that leads up to rigorous calculus, for instance), I really don't know of much of a better program than NEM for teaching you engineering math. (Although, I have to admit that I haven't really thoroughly purused it, since, after all, I'm not using it. I have looked at it quite a bit, though.)

It doesn't seem to have any multivariable calculus or differential equations or anything like that. That must be why everybody wants to do something else like Apostol or Courant.

In fact, let me just use this post to point something out. When I was in high school, I never got national merit scholarships. I never took AP calculus. I was on the AP track, but I just made it up through precalculus. And, I placed into honors calculus at a second teir university. In fact, that sort of thing was quite normal at the time. Okay, the AP Calculus is becoming notorious among mathematicians for how bad it is. Doing advanced material poorly is not going to help you. You need to do the elementary prerequisites really, really well. This is how doing something like accelerating through TT can actually end up hurting you. You end up at a high level but playing a very weak game. When you get out of your sheltered environment of TT, you can end up getting hammered by "the real world". Hopefully, you can just take your lumps and drop down low enough to redo the material you have to do more carefully and thoroughly, but you might just be stuck having to perform at a level you can't possibly hope to perform at. Best case you just wasted a lot of time doing a bunch of stuff you have to go back and repeat. Worst case you have to drop out.

No -- I was able to get to it on google. I think I wasn't able to get to it in Amazon. It's not Baby Rudin -- definition-theorem-proof... now prove the Baire Category Theorem for an "exercise". But, I really do think he is trying to get his students to do epsilon-delta proofs, and at any rate, they are there in the exercises. It looks like maybe even every other exercise or something. It's pretty solid. I don't think you would walk away from spivak unable to do an epsilon-delta proof, for instance. I think you really could do that with Apostol, thinking to yourself "Well I did the vast majority of the exercises, so I'm pretty solid". We'll have to see what the rest of the book looks like. So far, I definitely like it better than Apostol, hands down. I'm not altogether sure Courant isn't as good in terms of what you can do after getting through the exercises.

His section on limits is awesome. I really was expecting another Apostol like outcome. But, at least for that section, he really does go into great detail about the idea of a limit and how it is formalized and why. And, he backs it up in the exercises with a bunch of good problems. Problem 3 on page 106 is essentially 6 epsilon-delta proofs. By contrast, despite all the wonderful development of the material within Apostol, when you get to his section on limits you get a very sexy looking discussion of epsilon neighborhoods. You get exotic theorems for such a section, like that the indefinite integral is continuous which is something that only with Apostol's integration first approach could you do. (Otherwise, you would just have to do it later when you cover integrals.) There is nothing at all lacking in the presentation, and if anything, it beats Spivak hands down on neat-o-ness. But, you get to the problems and it is just a bunch of "use the limit theorems to calculate the following limits". I don't want to be too harsh -- the last eight or so problems are pretty good. They are kind of hard, though. You shouldn't have to skip, for instance, to working through Apostol's outline of an alternate proof of the fact that sine and cosine are continuous straight out of doing just a bunch of computational problems. Nevertheless, this section is not altogether that bad if you make sure and do all the problems. (It would be really easy, though, for a student to do the first 25, get stuck on the last eight and decide he had more less done enough for that section.) But, while Spivak lacks a certain amount of sizzle, I have to say he really is taking a no nonsense direct approach to really getting the student to prove something with epsilons and deltas. And, you have to start doing it on problem 3 not on problem 26. I have to say I am pleasantly surprised. I'll have to get the book and check it out more thoroughly. Thanks, Jane!

I would say, Katia, that you absolutely have to start planning on placing into calculus if what Jane is saying is true. (Who knew that biology was so math intensive!) If you don't place into calculus -- if your daughter isn't playing at that level -- then it will be a miracle if she actually manages to graduate in a field like that (that needs diff eq's, for instance). No doubt. You cannot do TT. Maybe if you could cruise through as easy as possible and perhaps accelerate the program so that you actually make it to calculus, and if calculus was just some requirement that you could just get through and never have to worry about again, then maybe you could kind of slop it up a little with the math. If you really are going to have to be doing differential equations and stuff like that all the time, though, there is no way your daughter can be "bad at math" and do this. Maybe your daughter should rethink just what she wants to do. If this is what is entailed in Marine Biology, then is she really sure she wants to be one? Doing diff eq's isn't training dolphins. In fact, even without this diff eq issue, training dolphins is something else entirely from marine biology, anyway, I think. It is probably all rolled up into some big mental package -- "I'll be a marine biologist and train dolphins... yeah...." Well, the truth is probably that those are in reality two different activities and now she should figure out what really appealed to her about it -- the dolphins or the idea of studying marine organisms. If she really is emotionally "all in" on the studying of organisms, then she should figure out why the math is so important to that and maybe the math will start to matter to her more and she'll become a lot more mathy. If she just wants to be an animal handler and train dolphins, then she probably doesn't need to be majoring in marine biology.

Yes, I do know of Spivak's text. Those seem to be the big three: Apostol, Courant and Spivak. My dad recommended Apostol to me and I eventually got it and was initially excited about it. But, my excitement faded when I realized what was going on with the problems. Just for example, Thomas and Finney has the rigorous definition of a limit in it. It's not like these texts aren't written by mathematicians. The point is that Thomas and Finney clearly doesn't require the student to actually use epsilons and deltas abstractly to prove a result. In fact, both Thomas and Finney and Courant do this silly business of "if epsilon is 1/1000 then find an delta such that...." The epsilon-delta argument is not supposed to be used for some kind of approximation techniques like that. Or, at any rate, you are not doing the epsilon-deltas if you are doing that kind of thing -- that completely misses the point. I can't believe these turkeys (yes, I just called Courant a turkey) actually try to pass that off like that. (Although, it may be that Courant is just trying to ease into real epsilon-delta proofs -- I need to look at the book more closely.) So, Apostol just having a rigorous presentation but not the problems to back it up is a pretty **** serious ommission. (I' sure his book is better than Thomas and Finney but probably not because of all the rigor the student is not expected to know.) Spivak is the only one of the three I don't have a copy of. I've seen Spivak listed as the top choice of the three, and I've seen it listed as the last choice of the three. Is there anyway you could look in your copy and tell me what you think about how the limit is handled? Or, how Rolle's Theorem and/or the Mean Value Theorem is done, for instance? It is okay if he does a proof by picture first or looks at a sequence of points for a while to heuristically talk about limits or continuity or whatever. What I want to know is if he ever does give a slew of problems that go like "Use the formal definition of a derivative to prove that the derivative of x^2 is 2x," or "Prove Theorem 1.3" (and when you look at that theorem it is real theorem)... that kind of thing. If it is just one starred problem at the end of some section. That's not enough. If it is use theorem <blap>, theorem <blap> and theorem <blap> to show that <blah blah blah> -- where you apply several theorems and then it just becomes a simple (or even if it is not so simple) calculation, then that isn't a "proof". (That's just another hairy calculation.) Just before I buy it (and I'll probably end up buying it anyway), I just want to know what to expect there. I care a lot less about whether or not Spivak can do an epsilon-delta proof. (For crying out loud!) I want to know if the student is expected to.

Well, actually more like a third, but let's not go there. Math Ed I still need to clean up the ole website a lot. But, the basic schedule is out there up through "Algebra III". I have not made a "Pre-calculus" schedule in which we plan to use Cletus's book (listed on my website there). (Yes, I absolutely must refer to it as "Cletus's book".) We'll probably put off S I Gelfand's book until after Cletus and put BeckenBach's <i>Introduction to Inequalities</i> in there before Cletus. Myrtle and I argue over geometry on a regular basis, but I think we are converging on Kiselev, actually, which is a lucky outcome since I already had made the schedule around it. She had always wanted to do 1964 Moise and Downs (a New Math text that more or less fits into our "spine", so to speak). I feel like I ought to want to do Birkhoff and Beatley which is what is listed in my tag, but originally I chose Kiselev over Birkhoff and even made the schedule for it that is on my website probably more for "aesthetic" reasons. Actually, before any of that, I was planning on doing this physicist/mathematician's book (Solomonovich, who is also listed on my website). I just think that his book is more like a first year college text, and I don't want to spend quite that much time on synthetic geometry. (I do think one can definitely do it -- it's just going to be difficult and probably take a lot more than a year to get through it with a younger kid. And, our kid is going to be 12 -- maybe 13 at the oldest -- when we get there.) That said, though, neither of us has really systematically gone through the book. My schedule could be totally unrealistic and I could be wrong about it being easier than Solomonovich. (Solomonovich is largely based on Kiselev.) At any rate, it will all get resolved soon -- our oldest is more than half way through Allen 1 right now. If you get through all that, then I have been searching for a calculus text. So far, the best I have found is this text by Courant. I ended up not liking Apostol (which I used to have on my website) because of a dearth of a certain kind of problems. I do like his general approach -- it's very much a shadowing of even measure theoretic real analysis. And, he is completely rigorous in his presentation of the material from what I can tell. But, his problems just don't exercise the student on that. So, I just don't believe the student is really going to walk away with it at the end of the day. Courant does calculus the more normal way by introducing limits first and it does look like he might have some epsilon's and deltas that he expects students to wield. I need to look through the book more thoroughly. I think the Hardy book listed on my website is even more like that -- really expecting the student to be able to do the rigor. But, the real truth is if you can do something like Lay's Analysis or probably my all time first choice would be to somehow be able to do all of Georgi Shilov's books, then that is what you really need to do. (But, at that point you are committing some sort of cardinal sin of actually doing the rigor which somehow ruins the "beauty" of calculus and/or is an all around Satan-worshipping, evil thing to do, apparently. So, that's probably why I totally want to do it.) If you did all of Shilov's books, you would have a truly mathematically rigorous and fairly complete basis for doing physics, for instance. (Basically, a typical physics major should take calculus, linear algebra, ODEs, PDEs, vector "analysis" and complex "analysis". Shilov hits most of that. And, what he doesn't technically cover, would tend to be child's play to someone that was actually able to do all the problems in his book, I think.) At any rate, there you go. That's the current state of my thinking on math programs in K-14 education. (It is largely centered around avoiding having to have a vague, shaky, heuristic calculus foundation for further study in math or physics.) I strongly doubt I'll be able to do it all. Frankly, I will be happy to go up through Cletus and S I Gelfand. I will even be happy to make it through the Algebra III schedule, for that matter. "Shilov?!? Yeah -- keep dreamin' buddy!" That's what I keep thinking. On the other hand, I didn't expect to be doing Frank Allen and Gelfand with an 11 yo math-hating, ADHD, dyslexic child, either. So, you never know.

I would be inclined to agree. (Of course, by "rigorous", here, we don't mean mathematically rigorous but rather just as a program of study.) If you want a no brainer top notch math program, couldn't you just do Singapore all the way through NEM 4 and put them in quackulus in the community college? (The program ends when their 16.) I would think that would tend to maximize their SAT scores, give them a first rate treatment of engineering math, mesh well with Singapore's top notch science program if you did that, too.... I, personally, don't actually do that, as it turns out. (In fact, I really do something completely different.) But, most people don't like my math program. If you are looking for something "normal", just doing Singapore pretty much without supplementing it and racing straight up through calculus at a community college would probably be my first choice. (I wouldn't even teach it myself, but rather try to get it done through a college/university and I would try to avoid AP.)

I study math because it is brutal and ugly. I find joy in having hapless engineering students driven before me in chains, brought to their knees with my indomitable epislons and deltas! WHOO! (just kidding) Actually, I just wanted to point out an interesting side issue that I have come to realize while looking through Apostol's calculus. "The fundamental concept on which the whole of analysis ultimately rests is that of the limit of a sequence." -- Courant (from his famous calculus text) One interesting aspect of Apostol's approach is that he puts integration first as opposed to most modern texts that put differentiation first and older texts like Courant that do them almost at the same time. (Although, Courant does technically do integration first.) What is so ingenious about doing something like what Apostol is doing is that to develop the integral as he does it, all he needs is the Archimedean Property to do it with. He doesn't need a full blown limit and indeed hasn't even covered limits by that point. In fact, all the limit really is is a couple applications of a general tactic based on the Archimedean property. So, I will say that, in fact, if you are really clever about it, you can arrange a lot of it to be done that way -- with the Archimedean Property rather than the limit. Or, in other words, it isn't the limit that is the fundamental concept but the Archimedean property or more specifically, a particular application of it that is more or less equivalent to saying that the lim(1/n)=0 (which is, as a matter of fact, the very first limit discussed in Courant). The reason I bring all this up (saving my conclusion for the very end to build maximum suspense) is because who really first articulated that along with the method of exhaustion? No -- it wasn't actually Archimedes which would probably make the point anyway if it were. It was Plato's student Eudoxus. So, even our precious modern invention of Calculus really does (no really) go back to none other than Plato's Academy (or at least the key insight of it -- what makes the difference between a mere finite sum and an actual integral, for instance).

LOL! My office was the "bridge office", too! Didn't you say your husband went to Ohio State? I'm starting to realize now that I think I would have gotten along with Mityagin, infamous at OSU. My understanding was that he only takes the stick out of his arse when he needs to beat one of his students. He actually flunked his own student on their Comprehensive exams (which, Mityagin, himself, recommended the guy for). But, however legendary his brutality may sound, I really think he just has standards -- too high of standards for OSU. But, you sure could learn some functional analysis from the guy. (His lineage goes: Kolmogorov -> Gelfand -> Shilov -> Mityagin!) Higher standards is something OSU needs. Had I known then what I know now, I would have stayed and tried to become his student. The cure for that kind of BS is harder problems. Use your calculator all you want -- it won't help you. In fact, that's why I could almost endorse the widespread use of calculators or even computers -- it's just going to cause me to make my section harder, giving you series that converge on the computer but not in reality, problems that only involve symbols and not any numbers, and so on. We could actually start doing real math eventually.... My motto seems to have been: "Grill them hard for justification when they get the answer wrong and grill them harder when they get the answer right."

...I TA it. I don't strictly teach anything. I just show up with a low tolerance for bull****, an expectation that people will listen to me and take me seriously, and possibly a grimace. I imagine R L Moore to be that way. In fact, the one overriding theme I see in these legendary thinkers is a low tolerance for bull****, at least when it comes to their subject matter. But, then again, I am not strictly in charge of the daily lessons, either (which may be a good thing). Occasionally, we do have those moments that sounds like it came straight out of one of Plato's dialogues which makes it all worth doing.... Frankly, we all know that it really all just comes down to what kind of problems you can do. I'm just here to make sure they do them.....

Incidentally, and I don't want to rile anyone up any further, but why do you think it will be different with TT? That's the point of saying that it isn't "rigorous" -- that students will have precisely the experience your daughter did. They wil go through the program, ostensibly have covered things like the quadratic formula or solving a system of two equations with two unkowns or whatever, but when they get to college, they'll start flunking placement tests, not knowing how to do the algebra they need to do in their calculus class and stuff like that. That's what people mean -- they specifically mean that the problems in TT are not hard enough, for instance. So, while it may also be true that their Algebra II has a lousy table of contents -- that is, they only cover half the topics of a standard Algebra II -- I am saying that, based on what I have seen, they have a pretty mediocre problem set. Frankly, I don't care about the table of contents -- that can, indeed, be made up for by just accelerating through the program. In other words, so it's covered in Algebra III, then -- big deal. The problem is how well it is covered when it is covered. Does the student have to solve some hard problems on their own or don't they? I think that if your daughter ended up this way with a PS curriculum, then the odds are extremely likely this exact same outcome would have happened with TT and homeschooling. And, I am basing this on my own purusal of their material online, the reports of others on WTM both positive and negative as well as everything else I have seen about this program. And, I am also speaking, here, purely from the perspective of going on to college algebra or calculus in college. (In other words, I am not saying any of this from a perspective of "but they don't learn to prove theorems...," or something like that. I am just talking about the practical reality and outcome of working within the normal system.)

I doubt the Elements were really meant as a textbook. They are more like a compilation of the mathematical knowledge of Euclid's day. All of the Elements together cover a lot more than just geometry. The high school course you would be interested in is the geometry of a plane. And, I think the main interest there should be to start from a few axioms (true/false statements that you can assume without havign to justify) and eventually work up to proving the Pythagorean Theorem. That's probably the best way to characterize it right there. (Of course, if you get through all of that then there is still plenty of material to take out of the elements to fill up a year or even all of high school.) Mainly, I wanted to just point out, though, that Euclid wasn't some high school teacher from Ancient Greece. His Elements includes material that is taught to graduate math students right now today, open problems in mathematics, for that matter, and has been a source of inspiration and the very model of doing math for all sorts of great mathematicians in history. It's not just the geometry of the plane but also solid geometry, number theory, geometric sequences and series, the method of exhaustion (which is proto-calculus),.... I have come across people in my many travels that seem to think that Euclid is crap or something. Nothing could be further from the truth. And, incidentally, just to put it a little more in perspective, Euclid is probably largely an outgrowth of Platonist geometry (in case you might wonder how these things are all connected). In fact, if, say, you are totally into calculus (as some of the people I have seen disparage Euclid seem to be), then that Method of Exhaustion or, more particularly, what has now come to be called the Archimedean Property of the real numbers, was first articulated by one of Plato's students, Eudoxus. That Archimedean Property is probably one of the first and most fundamental things a student learns when they do calculus rigorously (i.e. correctly and usually in a much more advanced class called "Real Analysis"). That's probably a lot to digest if you didn't know of the Elements to begin with. I guess I'm just saying that this particular "rabbit hole" is infinitely deep, arguably deeper than any other one in all of academia or education. It doesn't just amount to some old high school class that is almost not even taught anymore.

Yeah, I looked through their website some time ago and a few times since. Basically, I think the program sucks, too. But, in all fairness, it sucks just like every other program out there. If you home school your kid with it, I bet they wouldn't fair any worse than your average public schooled child -- that's about the level of it from what I can tell. It's probably no worse than Every Day Math. If you don't want to think too hard about program choices or are just not confident in your own ability to devise a program of study on your own, then just do Singapore up through NEM 4 and then put them in community college for calculus. (Personally, I also think just about every sort of calculus sucks, too, but that's another matter.) That said, though, I would probably use it if I didn't care much about math. For instance, we probably do that sort of thing with art or something (which we don't care much about). We just "box check" it. It's all a question of priorities. Maybe you just want to do more Latin and Greek or maybe you just want your kids to have an easier time of it.....

Much better to find out your algebra program sucks on the front end than on the back end. Personally, the TOC looks pretty mundane and conventional. I doubt it's super special, but there is no reason to think it is teaching textbooks or something, either, just based on the TOC.

Take the placement test. Don't be shocked if it is 2A or lower.

I'm fixing that, by the way, as well as some other errata. I actually learned some HTML so I can properly put exponents in the upper right corner now and stuff like that. My biggest problem is that at one point I decided to turn over a new leaf and stop doing Linux and started doing it all in MS Word. Boy did that put a litany of junk in my HTML! I go on these long hiatuses where I don't do anything to my website and then go on a big work-on-the-website binge. I am entering one of those binges, now. I may be able to get through most of those solutions reformatting everything this time around. Eventually when I get that all fixed up, I will probably do all the problems in his Trig book. I think I might even do all the problems in Baby Rudin and a few other books, too. (But, that's definitely beyond the scope of high school at that point and well into the future.)

When I was a TA at Ohio State, I was perfectly willing to let people use calculators for quizzes and such. To the chagrin of a large portion of my students, on one assignment, they all got a point deducted for writing down "1.999" instead of "2" for the answer. All of those guys apparently thought that the square root of two was equal to 1.414. Personally, I never used a calculator until I started taking Actuarial Exams. And, what I really use for that kind of stuff is Microsoft Excel (or Gnumeric, actually). There was a big debate about the role of calculators when I was still in academia. Frankly, I never cared. You can use all the calculators you want in my class. It won't help you.

Then, we switch to something else for algebra-geometry-trig-etc. How good of a fit it is or if our kids "like math" or any of that has nothing to do with it. Singapore is basically just the best program for what we use it for. And, our students have to just conform to the requirements of the material rather than trying to make the material conform to their "needs" as a student. Pedagogy simply is not nearly as big of an issue as people make it out to be. The issue is and has always been content. We're willing to take as long as it takes to really do it and even sacrifice other subjects for it if we have to. Unfortunately, a lot of kids and/or their families simply don't want it that bad. To draw an analogy, it's like Hamlet vs Laertes in their duel. Just because you go study fencing under some fancy French master, don't think that Hamlet's not going to have his way with you when you get back home. Reality has a way of working out like that. And, anymore, you can't rely on "experts" because they all drastically contradict each other, and usually just a simple direct interest in or even understanding of the actual subject matter is the furthest thing from their minds. All the algebraic manipulations in the world won't ever equal even a single solid geometry proof. I don't care if my kids ever do calculus if they never do an epsilon-delta proof. And so on. Unfortunately, Singapore does not offer that sort of thing in NEM, so that's why we switch. But, it is pretty much right on for things like arithmetic and, for that, we use it exclusively.

All of mathematics is one big exercise in formal logic. The example I gave in the other thread of such a statement was actually the definition of a limit. All of calculus is based on that. All of the mathematics of physics and engineering, if done correctly, is done by manipulating such statements. In fact, that hippy, New Math set theory stuff (closely related to mathematical logic) is the first thing you cover in abstract algebra, for instance, which has big and very direct applications to physics (and so engineering), computer science, math.... Brownian motion is a physical phenomenon the mathematics of which is used to price options in financial economics. Rigorous calculus is easy compared to something like probability theory which at least requires rigorous calculus if not the more general version of that using measure theory. Of course, all that probability theory forms the basis for doing the kind of statistical and financial analysis that actuaries do for insurance companies. All of this has to be done by some one the right way by formally manipulating complicated quantified logical statements. The rest of us only think we can get away with not having to do that -- with just knowing the theorems and how to apply them. And, mostly because we simply all agree to do that, it works out socially for us. You can, indeed, not study one lick of measure-theoretic probability or the convoluted abstract algebraic methods of solving combinatorial problems imminently important to your filed or what have you and just run off and start programming or analyzing claims data or pricing options or building bridges or spouting off a bunch of your own bull**** speculation about "how the world works". You can get away with that. Indeed, you probably have to do it that way since if you spend all your time trying to actually know what you're talking about, there won't be enough time left for you to master analyzing the shadows on the wall. And then, you will just make an unmitigated ass out of yourself in front of everyone else who doesn't really know what the hell you're talking about, anyway.

Personally, I don't really take militant Randians that seriously. All of western intellectual life -- all of what makes it special, at any rate -- is essentially the result of rationalists refuting empiricists and defending philosophy against science. Plato started it and Kant continued it. If you are trying to characterize these guys as some sort of skeptics that doubt the possibility of real knowledge, then the amount of spin doctoring that you are capable of rivals even that of the fictitious depictions of H G Wells' 1984. It's up there with people who try to characterize John Lock as some sort of closet socialist. The reason you shouldn't be pushing science so hard has to do with the hollow nature of empiricism. Take something like physics -- the ultimate in empirical pursuits. In the end, it all comes down to a handful of empirical facts that we spend the rest of the time just trying to consistently interpret with a priori mathematical or philosophical reasoning. The real issue isn't coming up with a bunch of facts but making sense out of those facts, which is all a priori. (Don't get me wrong, I'm not saying that physics shouldn't be done any differently than it is which is "empirically".) I'm not any sort of pure rationalist, but, at the same time, I do recognize that all of knowledge is "abstract". Oh, I know what you mean if you say something like "Don't try to teach your kids stuff that's too abstract." But, if you're lucky, you will maybe be able to actually teach your kids some knowledge. And, the few pieces of genuine knowledge you do pass down to your kids will be just that -- abstract. They probably won't be able to really understand it right away on the first day -- most people never do. But, that is the goal you are working towards, and what you really mean when you say "Don't try to teach your kids stuff that's too abstract," is "Don't try to teach your kids stuff that's too sophisticated or too advanced." At any rate, the point just is that a true education is rationalist and not overly preoccupied with The Real World . Knowledge about the external world is not somehow special. And, in most cases, it isn't even that important. The only reason we think it is is because we confuse internal a priori knowledge with external empirical knowledge. And, the kind of positivism/empiricism you see co-mingled with militant Objectivism is the the high point of all ironies given that they essentially got it all from a bunch of Fabian socialists like Bertrand Russell trying to figure out how to do away with philosophy so they could promote their own philosophically untenable social agendas. The irony is that Objectivists, before even being rabid adherents of their bogus take on philosophy, are ardent capitalists trying to call bull**** on those very social agendas. So, by adopting all this "real world" crap, Rand just got totally pwnd by her philosophical adversaries. Put down Rand and read some real philosophers.

For one thing, take a look at the Singapore placement test into NEM3. It covers roughly the same topics as, for instance, the TT placement test into their precalc. But if you look at the problems, TT is just a lot easier. Most other programs -- in fact, virtually all other programs -- stand in such a relation to Singapore which is why people often find that their kids place into a lower level -- sometimes a couple grade levels lower -- when switching to Singapore. And, the reverse doesn't happen nearly as often (maybe not even at all -- at least I don't really hear about it). So, there are two things about this. On one level, you could be real flippant about it and just say "So this is the hardest program -- that's all I need to know." But, truthfully if it was just about that, then you could easily just accelerate your student through an easier program and create a similar outcome. While the Singaporeans really know how to work super tough arithmetic, say, your kid has already been doing algebra for the last two years, perhaps. So, really it kind of comes down to what you think your kid needs here, anyway. Do they need to stop and do the "starred problems" or do they need to just accelerate through to more sophisticated material? Personally, I think, in most cases, they need to stop and do the starred problems. If we were talking about something like whether they need to stop and prove Godel's Incompleteness Theorem before they finish their primer on mathematical logic and move on to more normal subjects, then I would say, move on. But, we are so far from being in a situation like that in K-12 education that I wouldn't be saying something like that in a million years. So, that's where I stand, personally. So, the second point is that even if you do TT -- or compare the appropriate Singapore placement test to BJU or whatever program -- I think you will find that, at best, your kid is just going to be accelerated compared to what you would have had if you did Singapore, but at the cost of their skill with the material they supposedly know being much lower. Frankly, even Singapore has left me with a few noticeable outcomes later on down the road that I would normally want to avoid. (Part of the reason for that is because it isn't just about doing ever harder problems but also about doing the right kind of problems and ultimately just picking the right content. But I'll leave that for another thread.)

...Classical Euclidean Geometry is such a big deal and undoubtedly the motivation for Plato's motto: "Let no one enter without geometry." You can teach a course on logic, but that doesn't directly condition the student to think logically. It just identifies what logical thinking is, in the first place (a related matter to be sure but not exactly to the point). A reasonably interesting subject that makes heavy use of logic is a better choice, and for a long time, Euclidean Geometry was kind of the only one. It may still be, to a large extent, for school aged children. I have a couple of logic books by Nance. I have an introductory college freshman philosophy text. I have several advanced math texts. All of them either already assume a certain level of ability with logic, or they don't exactly directly build that aptitude up. What I really want is something like the Sequential Spelling of logic -- a series of about 3000 worksheets that start at a super low level that a 2nd or 3rd grader can handle and then gradually climb to the level of negating a series of statements like the definition of a limit I gave above. The student just does a couple worksheets a day consisting of truth tables and negations. Or, I also have this idea of an "assertion table" -- instead of filling out a truth table with T's and F's you fill it out with English sentences asserting the correct statement for the given cell of the truth table. That would be great, but I don't know of such a program. I do have the best of intentions of doing Nance, though, and moving up from there. But really, as I say, it is built into our proof-oriented math program. I think that programs like Nance's are helpful, but, truly, my best personal recommendation is don't skip geometry. Incidentally, the guy Ali mentioned, Patrick Suppes, is a very important person from the New Math era. I believe he also ran the Stanford EPGY for some time. And, he is also a serious professional philosopher and big contributor to the the philosophy of science. He's probably not a bad author to look into. (Maybe it is work on distributive justice that is blocking me from wanting to use him.)

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