I apologize in advance for this one sided diatribe, because Saxon is a pet peeve. From a mathematician's viewpoint, Saxon seems to be a book for students who dislike math (and written by such a person), and who may not believe there is anything to like in it. The ones I have seen (algebra) are pure drill without concepts, and make the subject look utilitarian and ugly. It is hard to imagine anyone wanting to be a mathematician after using Saxon for their whole career. A modified quote from Julie Smith's "Ted the Thinker" is appropriate here: "Saxon is to math as typing is to literature". The plus side is that people like my younger son, who is creative but forgets quickly, benefited from the drill in Saxon for this aspect of his math learning. However he found Saxon very boring. For him a mixture of Saxon when necessary, not forever, then some fun problems from Harold Jacobs algebra and geometry worked better. Still he eventually dropped out of math in college, in spite of math SAT scores higher than anyone else in our family including me. As I have said here before, our private school tried Saxon for a few years, probably because it was said to increase standardized test scores, then dropped it because (quote) "the students didn't understand anything". If math is just something to get a basic handle on without really thinking about it, like driving a car, Saxon may be a good fit, but to learn the subject well, or enjoy it for its content, depth and beauty, Saxon may actually be harmful in my opinion, and is certainly inadequate. If you have had different experiences, such as a purely Saxon taught child who became a math lover and went on to really enjoy and master math on a higher level, I would love to know that. You have to go with whatever works. But anything making a student hate the subject should be modified, no? This is just one pair of cases, but our older son, who escaped Saxon in school, and read Harold Jacobs at home with me, won some state wide math competitions, then majored in math at Stanford and works in silicon valley.

Question: if an angle can be precisely measured just by its radian measure, why do we use two numbers, sine and cosine? Answer: it's easier to compute sine and cosine, because they are lengths of straight line segments, whereas radian measure is the length of a circular arc. In fact that is the whole reason for the existence of the subject of trig, to relate "linear" measure, to circular measure.

Maybe I will try to write a short trig book. Basically I hated trig in high school and never learned it until decades later. It seemed like a bunch of meaningless complicated formulas. Then I eventually found out how simple it is. Basically it is a way to measure angles. If you put the eraser of your pencil on the table in front of you and raise the pointed end up off the table, the pencil makes a certain angle with the table. How to measure that angle? One way is to measure how high off the table is the point. Of course that depends both on the angle and the length of the pencil. So use a pencil of length one. Then the height of the point off the table is called the "sine" of the angle. (If you don't have a "length one" pencil, then the height divided by the length, is the sine.) The sine is a vertical measure. The other way to measure the angle is a horizontal measure. Shine a light straight down on the pencil and measure how long the shadow is on the table. If the pencil has length one, that shadow length is the "cosine" of the angle. (If the pencil has any other length than one, the cosine is the length of the shadow on the table divided by the pencil length, as before.) There is one basic fact about sine and cosine that rules all others, the "pythagorean theorem". I.e. because the vertical distance from the point to the table forms a right angle with the shadow on the table, and the shaft of the pencil is opposite that right angle, we get the relation sine^2 + cosine^2 = 1^2 = 1, true for every angle. See if you agree that when the pencil is vertical, sine = 1, and cosine = 0. Thus sine(90degrees) = 1 and cosine(90degrees) = 0. Also sine(0degrees) = 0, while cosine(0degrees) = 1, (when the pencil is lying flat on the table.) That's about all there is to trig, except for a couple of tricky formulas that tell you how sine and cosine change when you double the angle, and that are hard to remember. (But there is a cute trick for remembering those too.) There are also some other angle measures that are not really new, but they are taught in school, like "tangent" = sine/cosine. There is also cotangent = cosine/sine, secant = 1/cosine, and cosecant = 1/sine, but you don't need them as much. If you are familiar with the unit circle in the (x,y) plane, then an angle is represented just by a point on the circle. I.e. the angle is the one made by the positive x axis and the radius from the center of the circle to the point on the circumference. The angle vertex is at the center. Then the x and y coordinates of that point on the circle are exactly the cosine and sine of the angle. This is the right way to do trig, using the unit circle. The size of the angle is measured, in "radians" rather than degrees, by the length of the circular arc cut off by the angle. Thus in radian measure a full 360 degree angle is 2pi radians, since 2pi = the length of a circle of radius one. Thus 90 degrees = pi/2 radians. Hence in radians, which are always used in calculus, sine(pi/2) = 1, and cosine(pi/2) = 0, while cosine(0) = 1, and sine(0) = 0. Homework: The other useful facts are that sine(pi/4) = sqrt(2)/2, and sine(pi/6) = 1/2, and sine(pi/3) = sqrt(3)/2. I did these in my head so see if they are correct, using pythagoras on a "30-60-90" triangle, and a "45-45-90" triangle. And see if you can use the pythagorean formula above to deduce that tan^2 + 1 = sec^2. Suggestion: if this is too opaque now, learn from one of those other longer sources and then come back and read this again. This is really about all there is to it, honest.

I love math but am stumped by physics. For this reason I have a fondness for unintimidating treatments of both subjects. I think Harold Jacobs' books on algebra and geometry are by far the least intimidating sources I know of for those topics, but still mathematically substantial. I highly recommend them to anyone wanting to learn algebra and geometry in a way that is both clear and fun. The chapter introducing volumes of spheres has a photo of a huge ball of string collected by an 81 year old man, and asks how to guess how heavy the ball is. Another picture about volume has two similar bones, one three times as long as the other, and asks how much stronger the longer one is. The section on symmetry has three very different looking pictures all of Edgar Allan Poe. Two of them are made symmetrical by using only the left or right half of his face and reflecting. The oddest looking picture is actually his real, extremely asymmetrical, face. Oh yes, in the section in pythagoras, he asks what happens if instead of squares on the sides of a right triangle we have similar drawings of pythagoras ?! I.e. do the sizes of the two drawings based on the two legs add up to the size of the drawing based on the hypotenuse? As a 60 year old professional mathematician I did not know that, even though I now know it already occurs in euclid's original Elements. In the algebra book he starts out with expressions containing empty boxes instead of letters, clearly conveying the idea that in algebra, letters are to be replaced or "filled in", by numbers. Different letters are represented by different shape boxes. I have used this very successfully in several levels of classes for first time learners of algebra. In physics it would be "Thinking physics" by Lewis Carroll Epstein. I don't know if that's his real name or he chose a nom de plume to make me not be afraid, but that's a good informal book on learning to think about physics. He asks questions like whether or not a bee flying around in a truck will be pushed against the back when the truck starts moving fast.

"I never read comic strips as a kid, so I didn't fully understand CH until the moms here told me the intellectual value of them. We now own a few. __________________" get more.

My older son liked Harold Jacobs too and so did I. it's really fun and insightful at the same time, the perfect combination for me. Our son's private school later used Saxon until they eventually found out that the kids "didn't understand anything afterwards" (quote from the head math teacher). There may be nothing you can do but when I see schools using saxon I have the same feeling as when I see college kids smoking or people jaywalking into a busy street: I feel the need to say something but I suspect it won't do any good.

Unfortunately I do not know well many of the math books and series being discussed here. My impression sometimes however from the comments is that there is sometimes little appreciation for the difference in levels of sophistication of different books. I mean that a geometry course for instance would presumably start off just with familiarizing the student with shapes, as many diverse and beautiful shapes as possible. Constructions would play a key role here, including three dimensional figures, e.g. building and coloring regular polyhedra, which always appeal to children. At some point however, questions should begin to be introduced about the material, and a discussion of how to answer them creatively. I.e. what things we can observe lead us to deduce other true phenomena that are less obvious? I consider this stage the beginnings of mathematics. (How long is that circle? Can we construct a square with the same area as the inside of that circle? What is the relation between the volume of a cylinder and a cone on the same base?) I would suggest that a parent can best enjoy being the guide who determines what level the student can appreciate, if the parent pursues and learns as much as possible of the subject from a higher point of view than the student is on. Then they can experiment with introducing deeper perspectives as the student comprehends them. I.e. no doubt certain well regarded series used widely can simply be followed to their end, but certain courageous individuals persuaded me last summer to teach Euclid itself to brilliant 8-10 year olds, many of whom who proved more ready to appreciate it than many college students in the past. A parent can only prepare to probe the abilities of their child by experimenting in learning material at a higher level themselves I suspect. The "experts" thread here started by proposing such instructional seminars for parents but I am not aware that any have surfaced. would it be a good idea to have a community reading course of some advanced material? or is this too time intensive for already busy parents (and experts)? Perhaps less ambitious would be an experts review of various books that reveal the level of sophistication of those books. E.g. it became clear on a related thread that some parents were misled as to whether AoPS geometry contained "proofs", when what was meant was whether the proofs were written in a certain familiar style. It might of interest and use to some to discuss which geometry books do contain "proofs", in the sense of logical discussions. Even an explanation of what the word "proof" refers to in different contexts could be useful. Since I am new here I am unaware how familiar everyone may be with Terry Tao, a successful example of accelerated education in math. here is an article about him written when he was about 13. He is now a Fields medalist and math professor at UCLA. http://www.davidsongifted.org/db/Articles_id_10116.aspx

Here are some reviews you may have seen of your proposed book. All four are 5 stars. http://www.amazon.com/String-Straight-edge-Shadow-Story-Geometry/dp/1892857073/ref=pd_sim_sbs_b_1

I agree with others that AoPS has real proofs in it, but as you say in a different style from the "2 column" proofs in my high school geometry text. They are in the style a mathematician is used to. I might add I never learned anything from the ones in my high school book as they were sort of set up to memorize rather than understand. I have not seen LoF but I was impressed by the quality of writing in AoPS geometry. Having said that, I want to support you LisaK in your awareness of your son's learning style and your conscious attempt to fit the material to him. Although not everyone agrees on this, I see no harm in relearning material from a more difficult or different point of view after seeing it before. I have myself relearned many subjects numerous times, and even last summer I learned some geometry from AoPS, more than 50 years after first taking the course. Some people have believed (sometimes including me as a frustrated college teacher) that high school students are harmed by receiving shallow treatments of calculus in high school, and then being unmotivated to relearn it more deeply in college. I.e. they have already seen the "fun" parts, and see no reason to master the more difficult parts afterwards. I agree that this poses a motivational challenge for the teacher but it does not have to be bad. Maybe it helps if one teaches the second course by focusing consciously on the aspects that were missing or "left hanging" in the first introduction. I think it is possible to make it interesting again the second time. You can say something like "previously we took this for granted, but how do we really know this is true?" The key I think is what you are already aware of, the child should find it fun and interesting. If that is adhered to, the logical progression is perhaps less crucial, since the motivated child will likely be willing to go back and pick up needed elements in order to learn what he wants to learn now. Just one observer's two cents - you are the ones currently on the "front lines".

lets translate the words into symbols and see what we get. "The sum of A and B is 4215 greater than C." this says that A+B = 4215+C. "C is 1833 less than A." This says that C = A - 1833. Substituting for C gives A+B = 4215+ (A- 1833). "What is B?" simplifying gives: B = 4215 - 1833 as stated above. math is mostly about substituting equals for equals. but i much prefer the previous answer which makes it visible.

the one i have most experience with is saxon. that one i think causes a problem with understanding, and focuses too much on computation drill.

i recommend having this conversation with the tutor. hopefully he/she has some expertise and experience.

Is this a child who enjoys math? Does he know/care about why those area/volume formulas are true? if he knows his facts he might enjoy a different direction, one that justifies the formulas. I am not familiar with xmath, maybe that is already covered there. has he ever heard it said that a circle is a triangle with vertex at the center and base equal to its circumference? (for purposes of area) or that a sphere is a cone with vertex at the center and base equal to the surface area? (for purposes of volume) this is archimedes point of view, and explains "why" the formulas hold. it also helps to remember them. here are some links for this: http://www.basic-mathematics.com/proof-of-the-area-of-a-circle.html http://www.k6-geometric-shapes.com/volume-of-a-sphere.html

my dad let me sit in his lap and read to me, then bought me classics comics and biblical comics. i bought my reluctant reader but sports enthusiastic son a subscription to sports illustrated. he read that. 20 years later he reads a lot!

well that certainly makes sense, but this is a sneaky one. since in fact we have a simplification rule for exponents, namely (a^b)^c = a^(bc), the unadorned symbol a^b^c is always assumed to be the other one! so this one is not left to right, but a^b^c is always assumed to mean a^(b^c). who'd a thunk it?

how about this? 2^3^2? is it 2^(3^2)? or (2^3)^2? (I think i know this one from calculus class.)

I just noticed something - my calculator even used this order! i.e. someone programmed it this way: i.e. it said that 8+32 / 4 = 16, not 10. let me try 32 / 8 . 4, and see if i get 16, not 1. yes!! thank you katie interesting.... I thought it would be left to right.... I am a professional mathematician - for the last what...34 years, and I didn't really know this rule cold.

Of course you are right, there are two ways to do this and it seems plausible to do it from left to right! But...that's not how it is done. They taught us in 8th grade algebra that the agreed upon order in which to do these operations is: first multiplication and division, and then addition and subtraction. It is just an agreed upon convention that saves parentheses, but confuses those who have not seen it. Not only not dumb, but a very intelligent question. This is one disadvantage to isolated schooling, we may not always learn what others have agreed to do! But that's why your friends like Momof3littles are here. By the way, the unequivocally clear way to do it is as you did, with parentheses. i recommend always using them and avoiding this confusion. When I teach I never leave out parentheses and advise students to use them as well.

According to my research, not only is pertussis ("the 100 day cough") a disease that is miserable for adults, but if they have it they can pass it to infants, who are more likely to actually die from it. (It is apparently true that the disease is worse for children under 6 months, but this refers to the fact that it may result in death primarily in this group.) Hence it is recommended that everyone receive a Tdap booster as an adult. I got mine in my 60's, but there is now something of a pertussis epidemic, party because there are places even in the US where people decline this vaccination.

abebooks.com is the premier used book site in the world it seems.

In the original math book of western society, the Elements of Euclid, the word divides is rendered as "measures". The point is that the length of a segment divides the length of another if the first segment can be used to measure the other segment exactly. More generally, the number of times a shorter segment can be laid off inside a longer segment, is the integer part of the quotient of the two segments. From this perspective, division is perhaps best taught by measurement.

In euclid one can see the beginnings of geometric algebra in book 2, where essentially the product of two line segments is the rectangle they form. By sticking two segments together end to end, and A segment and a B segment, and forming the square from two copies of the combined A+B segment, one sees it decomposes into an A square, a B square and two AB rectangles, just the rule we teach in algebra as (A+B)^2 = A^2 + B^2 + 2AB. I really like this way of making algebra geometrically visible. It is also why I think I now like Euclid best of all beginning math books. Euclid contains what we call plane geometry, solid geometry, theory of proportion and pure number theory. They do occur in separate chapters, but they draw on and illuminate each other. E.g. to him a number is a length, and rational numbers are ones that can be expressed as ratios of lengths that can be "measured" with the same unit length. So his word for "divides" is actually "measures". This helped me a lot to understand some aspects of division that had confused me. For instance suppose you have two commensurable measuring sticks and you ask two questions: 1) what is the smallest length that can be measured using both of them, i.e. adding and subtracting them from each other. 2) what is the longest length that can be used to measure both of them? i.e. what is the largest unit length that divides evenly into both of them? The marvelous result is that these two questions have the same answer! I.e. the smallest amount of water that you can measure with two buckets, one holding 12 quarts and one holding 9 quarts, is their greatest common divisor, i.e. 3 quarts, the largest number that divides both 9 and 12. This is called "Euclid's algorithm". An illustration of your point is the fact that Euclid apparently never uses the word geometry anywhere in his book, e.g. his book is called simply "the Elements".

please forgive me if this is way off course, but i love this combination of subjects. like Parker Martin i always liked geometry more than algebra, but found out eventually they could be combined. In "algebraic geometry" ( a continuation of analytic geometry), the geometry allows visual intuition to enrich the algebra, and allows the algebra to give precision to the geometry. there is even a sort of dictionary between the two subjects. The concept of "irreducible" means "does not break into two pieces" in geometry, and means "does not factor" in algebra. But these are equivalent in a sense! [OK here's where it gets technical - my apologies.] I.e. the fact that the X axis is just one line in the plane corresponds to the fact that its equation Y=0 cannot be factored. By contrast the equation XY = 0, factors into X=0 and Y=0, corresponding to the fact that the solutions of the equation XY=0 is two lines, the X axis and the Y axis. Similarly a circle has only one piece, which corresponds to the fact that its equation X^2 + Y^2 -1 = 0 cannot be factored. In this example I myself cannot readily see that the equation does not factor but I see immediately that the circle consists of only one piece. So my geometric intuition helps out my algebraic limitations. The fact that a cubic equation has degree 3 corresponds to the maximum number of points in which it can meet a line, so an algebraic notion like "degree" also has a geometric meaning in terms of geometric intersections. With your indulgence I add one more example. To intersect the curve with equation Y - (X^2)(X-1) = 0, with the X axis, we set Y = 0, and get X^2(X-1) = 0. Since this has a double factor of X, hence a double root at X=0, it means geometrically that the X axis meets the curve "doubly", i.e. tangentially at X=0. Here the degree is three, but two of the three intersections have come together. When this happens the multiple intersection is always tangential. Thus we know these two figures must be tangent even before drawing them. This method allowed Descartes to compute tangent lines before Newton and Leibniz invented calculus. This point of view thus allows one to take his strength in geometry and use it to help him gain access to algebra, or vice versa. So I greatly approve of what you are doing! At a certain level many "distinct" math subjects are intimately related.

tonight there was a movie of stevenson's master of ballantrae with errol flynn on AMC, and I enjoyed comparing the movie version of the story with that in both the original novel and the classic comic. they deviated a bit, but many lines were the same. maybe the ending or the hero's character are made a bit happier in hollywood.

As a kid I always liked (and still do) books and stories by Jules Verne, Alexander Dumas, Robert L. Stevenson, Mark Twain, Washington Irving, Victor Hugo, and anonymous fantasies and classic tales, such as Prince and Pauper, Rip van Winkle, Robin Hood, Iliad, Odyssey, Black Tulip, Cyrano de Bergerac, Ali Baba and the 40 thieves, Aladdin and the magic lamp, the magic Horse (carefully chosen samples from Arabian nights). http://chestofbooks.com/fairy-tale/Arabian-Nights/The-Story-Of-The-Magic-Horse.html In the 40's and early 50's many of these were available in delightfully illustrated "classic comics" versions that just magnetized me, long before I later read the originals. These comic versions made me love even books such as Moby Dick which is tedious for many adult readers in the original (but I still love it), and Count of monte Cristo, which is a bit lengthy and maybe adult for a 6 year old. The wonderful "line drawn" Classic comics were gradually replaced by more high brow "painted" versions, but I am still persuaded the early less sophisticated ones helped me develop my strong visual imagination, which was my main asset as a professional geometer. Of course today's kids, and even ours in the 70's and 80's, have Tintin and Asterix and Calvin and Hobbes, with their amazing illustrations. The classics comics were in the nineteenth century tradition of the finest editions of books such as Les Miserables, which were illustrated, and those wonderful original illustrations sometimes even formed the models for the drawings in Classic Comics. I also enjoyed attempting to read the NY Times at about 6 years, but that was to be like my dad. I had a hard time understanding the words. Your child may also enjoy this exercise in growing up. Our first young child also loved the Hobbit and Lord of the Rings, probably a little older than 6, maybe 7 or 8. I think I was 7 when I read Shane and greatly enjoyed it, of course it is ultimately about a gunfight, which may detract. At not much older than that I recall it became a lot of fun to search the library shelves myself for books. True though, the children's section although harmless, wasn't very challenging. I tended to bring home 7 books at a time such as Curious George, and read them all in a few minutes. I see your problem. But I think the 19th century and earlier classics hold a lot of possibilities.