Weellll...:o I did some form of Latin nearly every year until this year and that was because the algebra was sucking up a huge amount of time and I needed to remediate some basic reading comprehension issues as well. I don't know what label to put on our home school but our ultimate goal is reading fluency in Latin for all of the kids with Attic Greek thrown in as well. (And somebody is over my shoulder telling me that this sounds defensive and that at any rate it doesn't count why I do, but I do with the kids :D ) However, Henle, Wheelock, Athenaze, Hansen and Quinn, Mastronarde are the grammars on my shelf. I have several dozen Latin and Greek texts in translation such as Hesiod, Pindar, Euripedes, Sophocles, Aristophanes, Aeschylus, Xenophon, all of Aristotle, all of Plato, Thucydides, Herodotus, Ovid, Pliny, Plutarch, Virgil, Lucretius...most of these I have read in their entirety, or at least large chunks, in order to "stay ahead of the kids." Remember, my oldest is in sixth grade in algebra and the other younger ones are in the first and second grade! Whether we actually achieve fluency or not, that is my goal for the kid by the time they turn 18. I also have many more general treatments of ancient history and education by Victor Davis Hanson, Bruce Thorton, Edith Hamilton, Will Durant, Tracy Lee Simmons, etc which have inspired me. The kind of math that we are after is "useless" pure math rather than math methods in the sciences as it is currently taught. More specifically, we've adopted Plato's definition of knowledge as "justified true belief" and made our ultimate goal in math education the ability to justify mathematical assertions. I'm not sure that it can be considered "classical" since the fields of math in which we are pursuing this approach didn't even exist in the days of the Romans or Greeks, but this is just our interpretation of how the ancients might approach math if they were alive today. You know, "What would Plato do?" Historically speaking, the more I read the more I am convinced and the more it seems obvious to me, that Euclid's Elements were foundational to any mathematical discussion, it's the style of argumentation used at least until the 18th century. So if you have any desire to read Archimedes or Newton's Principia, the Elements seems a good place to start. I think we've strayed from WTM in the science department because I'm using a boxed curriculum for that, but it remains to be seen if I can't incorporate more WTM-style grammar stage science with my two younger ones next year along with their boxed curriculum. And we really are not doing history as faithfully as perhaps we ought to be either.

I have had some awful intellectual changes since I was in college, so much so that I suspect I must have some underlying problem. Some days I find it difficult to speak and I have to stop to find the correct word to use. I mean words like "door," not big words! I will tell you something funny. When my grandmother became elderly she had suffered from some small strokes which affected her memory. When my father would tell her a particular joke she would howl with laughter at the punchline. The next day he'd speak to her he'd tell her the same joke and she'd laugh just as hard because she wouldn't remember the joke from that last time he told it. This made his job of entertaining her very easy. While my memory is not quite that bad, I do keep running across things in math, such as the geometric series in one form or another, and I won't recognize it as such or how I handled it the last time. Although after a review I still marvel at how wonderful it is. The good thing about a bad memory is that the beautiful music, stories, and math just don't get "old." Working on my own is so much more pleasurable than being forced to do it on someone else's schedule. While there were topics that I enjoyed studying in college, it mostly seemed like work. Even if I enjoyed the topic, I didn't appreciate having to study it when I wasn't in the mood or being rushed through it without being able to explore questions that naturally came up. There are some advantages to self-study.

I'd let her learn at her own speed and worry about what happens when she gets stuck--if and when that happens. My son was read for algebra more or less in the fifth grade and he hasn't had particular difficulty with dealing with the abstraction of it now that we are into it in the sixth grade although he works much slower than an 8th or 9th grader would. The biggest problem that I have with him working at an advanced levels is that he doesn't grasp the adult applications of the math. For example, if he had to calculate compound interest involving mortgages his hang up would be with, "what's a mortgage and why do people take loans out on houses to begin with?" rather than the compound interest itself. I put more anxiety into "he's going to hit a wall" than I should have back when he was in the first and second grades. There are arithemetic supplements that can bridge the gap between arithmetic and algebra if it comes down to that. My older son is aware of the fact that his younger brother is making better grades in second grade math than he did when he was in the second grade (we've have kept all his tests) and he dismisses that as, "That's because he likes it more." And that probably is the truth of the matter. I have wondered why math is different from art or music. What happens to the ego if a younger sibling draws better, plays better, or can do more graceful cartwheels in gymnastics? It still gives me great pleasure to plunk my way through Moonlight Sonata although many piano students even at young ages can play this much more skillfully than I. That is a value that I wish to transmit to my children when it comes to pouting because brother does it better. If you like the activity then do it for that reason and don't worry about others. If you don't like the activity then it's no surprise that those who do like it spend more time on it, concentrate more, remember more, and gain more skill. It probably is also helpful if the older child has activities which only he does in which the younger sibling does not participate.

I have no idea how to search specifically for children. Hmmm. I came across general histories such as "Stories of the Maple Land" And McIlWraith's Canada. I do understand the "pejorative issue," there are very few 19th c history books for children that I've read that I haven't seen one thing or another that I wish they would have phrased a bit differently.

"First Course on Mathematical Logic" in the mail yesterday. I was expecting it to be all symbols and really hard, but it's not all symbols. It was written back in the 1960s and the preface says that it was field tested in various classrooms. The author claims it to be a course for high school students or first year college students but after having flipped (see, flipping is a very methodical and authoritive way in which to ascertain the contents and nature of a book ;-) I think my son will be ready for this next year. Just to recap, I haven't used this book yet, I just got in the mail yesterday but if someone out there just can't wait to add another logic book to their collection for consideration for a future course, here's another title. And Laura, I can't speak to how to coordinate Geometry and logic, but Suppes' book seems to be a very gentle lead-up to mathematical proofs. The proofs that he has at the very end of the book my son can already do. Patrick Suppes Home Page

I don't belong here because I do belong here but since everyone else here doesn't belong here then I guess I don't after all. Hey, remember the Monty Python Scene: "You are all individuals!" "I'm not!"

I had to take a look at Bowditch and it's far more technical that anything I'm ready for right now! Do you sail?

I've been reading mine online, but it's quite comfortable doing so on a laptap. My daughter reads online too. The science reader that Jessica pointed out to me I downloaded as a PDF file. There is an option in the right column that allows this and I'm thinking about perhaps printing out select pages to use during school.

I liked the Chambers Science Readers and bookmarked those. I also added your "Seven Sisters" book to my library. I think my daughter would like both of these.

This year I used both with my first and second grader. I found that there were not enough of lessons in MCP to cover the entire year and in the beginning they worked multiple lessons. They have now finished their grade level of MCP and I put the first grader in ETC 5 and the second grader in ETC 5 1/2. The advantage of MCP is that it reviews previously learned phonics while the advantage of ETC is that it seems to offer more in depth practice. I will continue having them work through different levels of ETC until the fall and then I'll put them into their corresponding grade level of MCP. At the K level I had them do ETC since I felt that levels 1 & 2 of ETC were much simpler for them and that MCP was too busy and had too many sight words.

For those not in the know, Harvard is taking all of the books in their library that have expired copyrights and putting them online. You go to the google page, then click on "More" to select books. Use the "full view" search feature and you'll start seeing results. I've got some technical stuff that might not be interesting to anyone but me, but I also have, 1. Lewis Carol's Curiousa Matematica 2. A Beacon Introductory Second Reader: Animal Folk Tales By James Hiram Fassett which happens to be right at the reading level of my daughter and when she reads it she thinks she's doing something special online. 3. Florian Cajoris arithmetic for primary students up to 3/4th grade.and his grammar school book which goes up to pre-algebra. Cajori is in wiki and wrote a enough math texts to go from first grade to college level work. I'm not actually doing these with my kids; I just keep looking at them and thinking along the lines of possible supplement for extra problems and word problems. I've seen several people have been recommending google books from time to time and I'm nosey. What's in your library?

I suppose it depends on what your goals in math are. This is a repost of something that I said in another thread, In general, I will say that my big enlightenment was that it is one thing to train a student to use math as a tool for his future vocation and for daily use--say engineering and accounting-- but it's an entirely different thing to study math as a subject. Many parents imagine the study of music, art, or literature is done simply for the sake of those fields and not for any specific vocational or utilitarian needs. For example, when a kid studies biology, while there are utilitarian needs addressed in the curriculum, the curriculum includes a sampling of the major areas of biology that a professional biologist is involved in. The same is true for psychology, your psychology 101 class gives you a sampling of all the areas of psychology. History and philosophy give you overviews of those fields but the traditional math sequence of high school does not give the student a sampling of what math is as an academic field: topology, analysis, algebra (not elementary algebra), number theory, etc. nor the methodology used in those fields. The interest in such a pursuit is really subjective, no one is going to fail the SAT because they didn't prove that the multiplicative inverse is unique, or because he didn't read Ovid in Latin, for example. In the context of a classical education we are interested in proof because it is the "justification" in Plato's defintion of knowledge as being justified true belief. It is true that cross multiplication works. But believing that it works because it always has so far, or because the book says so does not constitute valid justification and therefore, philosophically speaking, according to Plato's definition, the student doesn't have knowledge of this theorem even though he can skillfully apply it. In one popular algebra text that I have sitting on my shelf the transitive property of inequalities is treated in the following manner, "...if x<y, and y<z, then it follows that x<z. It is easy to see with a number line why this is true. The inequalities x, y and y , z mean that x is to the left of y and y is to the left of z." That is a lie. The graph does not show why it is true, it merely illustrates that it is true in a particular context. This can be detected by the use of circular reasoning in their graphical "proof/definition". How does the student know where to place the number on the numberline without evaluating beforehand which is greater? This perhaps doesn't seem like such a problem when working with integers, the student knows where to put five and three on a number line because he already knows which is greater, but how about determining the order of a/d and b/c if a>b and c>d? Where they appear in an illustration isn't what determines their order. One counterexample for the child that thinks that greater than "means that" the number is on the right is to flip his graph with his numbers on it 180 degrees and he will see that now the greater number is on the left so that definition doesn't hold. How did we get fooled then? It is true that 5 is greater than 3. It is true that on a particular kind of number line 5 is to the right of 3. It is not true that one of those statements follows from the other, (if p then q) yet in many textbooks the student is invited to jump to such a conclusion. It's not so much that a rigorous program trains the mind, so much as that a non-rigorous program perhaps facilitates and encourages fallacious reasoning. Here is a good, if not long, article written by a Princeton mathematician for a general non-specialist audience on the history of proof in math. He gets off to the typical start with the Greeks but his treatment of the 19th and 20th century is more interesting. Here is a tirade by a mathematician about "proof by picture" and other picky pedantic gripes about how Calculus is taught.

Some time ago I was following the blog of a graduate student in math who was from Singapore. He had a post up about the deficits of Singapore math as he percieved them and here are some of the more relevant points and quotes from others who commented on his blog: It is very difficult finding open discussion about the flaws of Singapore math. It seems that it's illegal to speak out against the government or its programs there in any way, and in fact, this particular blogger was later sued by the Spore government for libel (but not for this particular post but of another that was critical of an entirely different topic) H Wu also has publically said that there are flaws in Singapore math that deal with content but he did not elaborate on what that might be. It is difficult for me to judge the merit of the above criticisms because when a mathematician or math major refers to "how the principles work" they usually mean something on an entirely different level than the rest of us do. Furthermore, I wonder if the same things can't be said about any American math program. (I've seen the same sorts of things said about Saxon, for example.) So there is what I found related to content for what it's worth. While we eeventually did find an old out of print proof-based algebra curriculum the difficulty of the mechanical calculations that are in the exercises isn't anything like what NEM offers.

I think that reflects the general trend of the philosophy of education in the public schools, in that context one hears more about getting the children to "like" a subject rather than actual achievement. At the risk of overgeneralization the ed school message is that the only way that children can learn is if they are "engaged." In the extreme this is catering to instant gratification. If the kid doesn't find it interesting or relevant (oh, and that's a whole ball of wax for those us interested in a dead language, is it not?) they steer clear of it. And I don't suppose I entirely fault someone for avoiding programs which require a lot of ungratifying work since many students really have been exposed to relentless, mindless, and purposeless busy work and why should it be any different this time? I think there must be some small subset of parents that really did shlog through a year or two of work, perhaps as undergraduates or perhaps in graduate school, that had no immediate reward but did finally enjoy the fruits of their labor at the end and it's just easier for us to trust this kind of an approach. One of the more helpful suggestions in Well-Trained Mind is to keep a portfolio or record of work accomplished in a binder and then the child can reflect back and see how far he has come. This has been useful for me when it comes to encourage my son to keep on and not get discouraged. If the kid goes on to college, no one will care about how much they "like" Spanish and much fun they think it is to talk to the carpet installer-- their test paper is just another one of hundreds and they'll be counted off for incorrect mood on their verbs along with the kids that don't particular like Spanish but do it because it's required. I never liked my major, I just realized that I was better at it than almost anyone else and got a degree in it. The one class that I really hated and found difficult turned out to have a surpise ending when I was finished with it and worth it when I was done. I smile everytime I think of GH Hardy saying that as a boy he never liked math but he was just competitive and did better than the other boys. Because of my experience I'm not terrified of giving my kids work that they don't particularly like.

When did you start Latin? What texts did you use? In retrospect, when would have been a good time to have started Greek?

"FWIW, no, I haven't rated my own post. -nt- " I did.:D I tried just giving you a positive rep privately for simply bringing up a neat discussion topic but a message came up and said I had to give points to someone else so I guess I must have already repped you earlier and so I just repped the whole thread instead. From Tracy Lee Simmons pgs 14-15, And although it is not apparent from that quote and Simmons makes this clear in the rest of the book, that while the study of Latin and Greek is necessary, it is not sufficient to making an education a classical education. For example, the local public high schools where I live all offer Latin to high school students but it's not a "classical education." For example, the following lists the classical course of study in a Boston high school about mid 19th century, Anyone here doing Caesar the sophomore year and Virgil and Xenophon the junior? I'd love to read through some threads on how that's going. We dropped Latin this year but will definitely pick it back up again next. I downloaded Xenophon from textkit.com with the best of intentions of making that my goal in high school. For right now I've been getting my classical education fix from reading the blogs of classicists and high school Latin teachers. Carry on.

I am sure there are anecdotes of individuals making it through any program that's ever been marketed since there are always kids that succeed, but statistically speaking the average high school student does not even get a college degree, much less a degree in math and science. Here's what I mean. In the state in which I live half of the students do not graduate high school. Out of the students that do graduate about half go on to attend college, and about half of those graduate. Out of the students that graduate the majority of them are education and liberal arts majors. I have seen statistics online that say that if a student places directly into Calculus in their freshman year that they have a much, much greater chance of finishing their math/science degree than a student who placed back into algebra, and the chances are indeed very slim for a student who can't even place into algebra. While textbooks may be labeled, "college" algebra, there really is no such thing. Algebra is not a college level course, it is a high school level course. If it were "college" level you'd see AP exams in which students could earn college credits toward algebra. Similarly, calculus really is a college level course even though it's offered in high schools. Most math and science degree programs that I have seen do not count algebra as a credit and consider it remedial work. The moral is that it really doesn't matter what your textbook author says about the quality of the high school math program that they themselves created, what matters is how the university evaluates your students' preparation. Can you contact the math department to find out what their expectations are? Can you find out what placement test they use? Can you find samples to practice? By the end of high school can your student do the problems they give their students in the textbook they use for their course? Does the department have released copies of finals from their algebra course? What exactly are the math requirements in the major that your student wishes to pursue?

I'm speculating, but this program was invented for dyslexics, was it not? (Do I remember this correctly) I have used it with my dyslexic son specifically because he was weak in his ability to visually recognize when two words matched and when they didn't. When doing an exercise with him I made sure to write the word rather than spell it out orally. So, that required him to very carefully and methodically examine letter by letter what the problem might be. (He is eleven and still reverses b/d, by the way). I think that it helped to develop a habit that he was able to apply in other situations. For example, say when he's writing an answer in science, he's more likely to carefully recheck the word that he's writing against wherever it is that he's copying it from than he was before. I never used a white board, I just wrote it out on a sheet of paper like he was doing. I would be very interested in what the author of this program has to say about this.

Let me throw this idea out there as another possible interpretation. :) Perhaps the trivium was not about math and historically the mathematical sciences were part of the quadrivium, not the trivium. Although if you want to apply the trivium as "stages" rather than as content areas then it might go something like this: Grammar stage - Basic mastery of the algorithms of arithmetic. Logic stage- Learning why those algorithms work and learning mathematical justification. For example, rather than only applying "invert and multiply" to solve a problem about pies or plumbing, the student in the logic stage studies why this algorithm works the way that it does and can derive it from first principles. Rhetoric stage would build on the student's knowledge of first principles and the student would make original mathematical arguments in support of the truth of a propositions about mathematical objects. The culmination of this in a classical education was in fact synthetic geometry and not anything like Calculus. At the time when classical education and Latin grammar schools were popular the foundations of algebra and calculus had not been discovered (they still relied on Euclidean axioms of equality for justification) and entire fields of math, such as topology, didn't even exist. When mathematicians say that students learn math "heuristically" they mean that the student is not learning the proper mathematical justification for what they are doing. When they use the word "rigor" or "rigorous" to describe a math book they mean that the student is learning the proper mathematical justification that motivates an algorithm. When the rest of us use the word "rigorous" we mean that the program has lots of hard problems.

"If I've told you once, I've told you k + 1 times."

Jane, it sounds like you might be talking about taking a "philosophical" approach to math. From what I have seen this is in fact how mathematicians conduct class at the higher levels and whenever I've had questions about something it's how people interact with me as well. Are you familiar with "The Moore Method" by the way? It was the teaching style of RL Moore.

Hi Rhonda, An absolutely fabulous resource for anyone planning on majoring in physics is www.physicsforums.com It's exactly the same board system as this one and has extremely active posters from a broad background--applied mathematicians, physics majors, as well as high school students--all discussing what has worked and what hasn't.

The best resource on how to approach math as a subject is a content expert--a research mathematician. Before I found any "support" groups online, before I came across Well-Trained Mind, I was emailing unversity mathematicians and asking questions. In general, I will say that my big enlightenment was that it is one thing to train a student to use math as a tool for his future vocation and for daily use--say engineering and accounting-- but it's an entirely different thing to study math as a subject. Many parents imagine the study of music, art, or literature is done simply for the sake of those fields and not for any specific vocational or utilitarian needs. For example, when a kid studies biology, while there are utilitarian needs addressed in the curriculum, the curriculum includes a sampling of the major areas of biology that a professional biologist is involved in. The same is true for psychology, your psychology 101 class gives you a sampling of all the areas of psychology. History and philosophy give you overviews of those fields but the traditional math sequence of high school does not give the student a sampling of what the subject area of math is: topology, analysis, algebra, number theory, etc. nor the methodology used in those fields. It has been of great interest to me to see that PhD mathematicians do want their kids to have a math education that has nothing to do with future vocational training and everything to do with studying the subject for its own sake--no surprise there really. So, I have noted that people like Mark Solomonovich wrote a book specifically for teaching his daughter, that Alexander Givental did a marvelous translation of Kielev's Geometry, that Robert Talbert is considering an old 1960s text that he thinks gives a broad overview of math to liberal arts majors rather than keeping them plugging away on the path to engineering calculus, I watched videos of university mathematicians involved in the direct education of high school students and paid attention to what content they were emphasizing. Having a husband with an advanced degree in math and a father in law who is a retired mathematician has made getting opinions of curricula a bit easier for me. Their first and foremost concern is that a book be correct and lead the student further down the path towards the study of math for its own sake. They really do not care at all if the presentation is in and of itself entertaining and are occassionaly privately derisive about claims that, "the students love math now since we changed to XYZ program" responding with, "the kids don't like the math, they like the video, the personality, the cartoon characters," etc and it is their most earnestly held belief that the way to get a kid like math is to give him interesting problems to work on. In fact, they will not infrequently propose a problem for the kids (and even to me!) to think about and it's all done orally while driving down the road with no video game, no worksheet, no time limit, just the kid turning over an idea in his head.

In the 1800s the teacher would begin by having the students memorize word by word all the definitions and axioms at the beginning of each book. They would then proceed on to the proofs and the kids would have to prove the question at hand. In other words, they would try to, on their own, come up with all the steps. If they couldn't do it they would ultimately memorize the steps and move on to the next proof. Your first problem would be, "to describe an equilateral triangle upon a given finite straight line." You will have to come up with a way of doing this as well as come up with a proof, or a chain of syllogisms, using the things that you "know" (given axioms and definitions) and show how your conclusion follows from those first principles. If anyone is interested in how math was taught in the US during the 1800s, there is a very good book online that you can read by Florian Cajori in which he gives anecdotes of students in college courses and summarizes what was happening mathwise and how it was taught. Anyone looking fondly back on some Golden Age of Math education will not find it here! (I did just a blog entry on this). If you are interested in what texts were used in high schools, especially classical education tracks, check out The Rise of the High School in Massachusetts. Some of the texts which they list out are found in Google Books. The advantages of teaching geometry are that it allows the student to work within an axiomatic system and acquire knowledge through justification of their reasoning. The key piece to this training of the mind lies in the parsimonious set of axioms that are used which are slowly built up into proving quite sophisticated propositions. Today there are many modern geometry books, some do not teach proof at all and some claim to teach proofs, but give you an "infinitude of axioms" to work with which some might argue can create difficulties for the student.

What if the lattice points on a graph are not orthogonal? Would we refer to the area of a figure as being "parallelogrammed"?