Please Help with Advanced Math Curriculum!

[greta_elisif]

I need help planning an advanced math curriculum. I made a post (http://www.welltrainedmind.com/forums/showthread.php?t=40143) a little over a month ago, but nobody has replied, so I thought, “Maybe the title isn’t clear enough so I should try again.”

As this is to be a college-preparatory curriculum of the highest level, I would like advice on making sure every item required (explanation in previous post) is covered, and in the right order. My daughter is exceptionally gifted, so I believe she will progress through coursework faster (maybe much faster) than is normal. Thus, the video-lecture instructional method, which seems popular on this message board, would probably not be effective (and I’m trying to keep kids away from TV—also if there were a video on how to teach that I had to watch, I wouldn’t have the patience to sit through it; I’m busy and a fast reader, and learn more easily by reading). We would prefer something more old-fashioned, but need to take into account new developments in math theory at higher levels of study.

Does anybody have anything to say about textbooks by George Albert Wentworth? (they are on Google Books (http://books.google.com/books?as_q=&num=10&as_brr=1&btnG=Google+Search&as_epq=&as_oq=&as_eq=&as_brr=1&lr=&as_vt=&as_auth=George+Albert+Wentworth&as_pub=&as_sub=&as_drrb=c&as_miny=&as_maxy=&as_isbn=)) I think they look very good and straightforward. They range from introductory arithmetic to plane trigonometry.

[Brenda in MA]

have you posted your question on the Accelerated Learners Board? Perhaps someone there will have suggestions for a student who is very quick with math.

HTH,
Brenda

[Jane in NC]

Personally I do not think that the Wentworth texts will help you achieve your goal of a rigorous and superior mathematics education. Granted, I have not read every page of every book in the list but a cursory glance suggests that modern mathematical notation and terminology is lacking. Further, the author presents lists, not an axiomatic approach.

To achieve your goal, you might want to consider some of the Russian texts (Gelfand is used by some on these boards). I am a fan of the '60's and '70's Dolciani texts--search through posts to read discussions. What are you using for your now elementary aged student?

The Cambridge list presents a number of excellent supplements for the mathematically curious. But they are not mathematical texts in terms of rigor--many of these books attempt to express mathematically ideas to the layman. The Mechanics book that you mention would not be for a typical US university freshman nor would it be an exciting book for all mathematics majors--Newtonian physics is not everyone's cup of tea. That is one book on the list that has a prerequisite: vector calculus. This is an achievable high school goal for the mathematically minded, but consider that it is not a necessity. Calculus is a stepping stone in one branch of mathematics, analysis. Your mathematically gifted student might be more interested in group theory at a young age--I would not force her to read mechanics in these circumstances.

With more information on your current text, supplments and fun reading list, perhaps other posters and I can be more specific with suggestions.

Jane

[KAR120C]

I don't know anything about Cambridge and their requirements, but we've been very happy with Singapore as a rigorous math curriculum for elementary and secondary levels for a quick student. It's a good combination of standard algorithms and thorough understanding.

I'm just guessing here, but I imagine that if you went through the Primary books and the NEM books, that at the end of it you could pick up wherever that put you in the progression to the Cambridge requirements and do fine.

[MaryM]

Seconding Singapore, enthusiastically! You can always add in extra problems from other books to provide all around experience with problem presentations. Then there is EPGY which is an excellent math program for the gifted. in the higher levels. It gives a strong math student the opportunity to move into university level courses at any age. We found that the presentation of the material was well paced for our ds although the administration will make you pull your hair out if you aren't a patient person. I'm not. Many of the other talent search programs have alternative math offerings as well like John Hopkins, Duke, Denver etc.. I might suggest that you hook up with one of those....

Mary

[Ailaena]

I don't know where everyone else is because they explain things so much better (:glare:) but...

Algebra (http://www.mcdougallittell.com/ml/math.htm?lvl=4&ID=1005500000030772): Structure and Method
Geometry (http://www.mcdougallittell.com/ml/math.htm?lvl=4&ID=1005500000030773)
Algebra 2 & Trig (http://www.mcdougallittell.com/ml/math.htm?lvl=4&ID=1005500000030774)
Introductoy Analysis (http://www.amazon.com/Modern-Introductory-Analysis-Mary-Dolciani/dp/0395350484/ref=sr_1_13?ie=UTF8&s=books&qid=1218297229&sr=1-13)

I know that even though I use the modern versions, the older versions (that would be the 60s and 70s Dolciani texts) are still amazingly awesome (some say much, much better). These books are used for honors courses at many private schools.


And the ever wonderful Art of Problem Solving (http://www.artofproblemsolving.com) books are really good, but have a different sort of focus. Great for kids who love the math contests (not so great for parents who aren't ex-math geeks).



eta: I didnt know your daughter was so young, were you asking for recommendations for lower-school math?

[greta_elisif]

Thanks for all the suggestions! The Art of Problem Solving and Gelfand books look very good. We’ll look into them more closer to the necessary time.

Regarding the Cambridge reading list, you’re right: not every last book is necessary, and some books are for the layman or everyday reader. For instance, my daughter and I could start reading together a lot of that type right now, but some, like analysis, are beyond me! I still like the idea of having a guide like that, to give us some direction, and some idea of what kinds of things advanced future-math-majors study.

I can’t remember if I posted on the Accelerated Learners board, so I’d better do that soon, just in case.

I have done a lot more research and studying since first posting. Sorry I didn’t clarify the problem better. I’m trying to achieve this (impossible? :D) goal with only resources using a classical methodology, i.e. that used before about 1900–1910. So the books wouldn’t have to be from before that time, but the method of instruction would be of the same type.

So, nothing: progressive, outcome based, standards based, child centered, constructivist, incremental, or integrated. No assessments, spiraling, Direct Instruction, new math, new new math, books full of colors and cartoons and photographs and racial/disability quotas, fraction strips, blocks used for counting (except at the very beginning), or calculators to be used before completion of higher arithmetic. I believe all those things are distracting and damage or weaken children’s minds. Sorry I’m picky! :)

The elementary-school student was using 4th-grade Calvert Math, then Prentice Hall Mathematics: Course 1. He’s gifted, but was forced to do (for online school) things that would have been too easy the previous year, yet his basic skills were not yet solid. These and similar books, and their approach (matching those items in the previous paragraph about exactly) are what solidified my low opinion of all progressive methods. One of the kids had a Dolciani textbook assigned one year and it wasn’t outrageous in its progressiveness like the others at least, but still pretty progressive.

I’ve found plenty of suitable and free arithmetic, mensuration, and geometric-construction e-textbooks from the 19th century online (probably going with Ray’s mainly because a complete set with answer keys is easy to find), and lots of free, modern, college- and graduate-school-level math e-textbooks of this type online, but little for levels in between, even to buy. :001_huh: I think the following books are probably better than Wentworth’s though:

Algebra I: I just love Ball’s Elementary Algebra (http://books.google.com/books?id=XxYWAAAAYAAJ). Very challenging.

Algebra II: I’ve been studying it with Cohen’s Precalculus with Unit-Circle Trigonometry (http://www.amazon.com/Advantage-Precalculus-Trigonometry-David-Cohen/dp/0534352758#) (unfortunately pre-calc is integrated, so maybe it should be used supplementarily, but the book is clearly written and the problems start off as moderately challenging and based directly on the lesson, then move up a step in complexity and in concept level, and then up another) and with Hawkes, Luby, and Touton’s Second Course in Algebra (http://books.google.com/books?id=VNhEAAAAIAAJ). Some explanations in the latter aren’t the clearest though. Two other books that look promising, which were for remedial Algebra II college classes in 1947 and 1948 (so also appropriate for mature, college-material students?), respectively, are Underwood, Nelson, and Selby’s Intermediate Algebra (http://www.archive.org/details/intermediatealge033585mbp) and Hart’s Intermediate Algebra for Colleges (http://www.archive.org/details/intermediatealge030183mbp).

Geometry: So far, I have this volume (http://www.amazon.com/Thirteen-Books-Euclids-Elements/dp/0486600882) of Heath’s Euclid, and am considering supplementing it with problems from Hall and Stevens’s Euclid (http://books.google.com/books?id=AUgqAAAAYAAJ). I am also considering using Kiselev’s Geometry (http://www.sumizdat.org/).

The biggest problem with using these books though is a lack of answer keys. My math abilities are good, but not so great that I’m sure I’m right every time in high-school math. I should at least buy copies of the best ones we don’t yet have, to use for fun. (Yeah, we like math that much!) I’m also planning on getting a slide rule and manual eventually. (We must be total nerds! :D)

Other Math Subjects: Still not sure, and looking for suggestions.

[fractalgal]

Thanks for all the suggestions! The Art of Problem Solving and Gelfand books look very good. We’ll look into them more closer to the necessary time.

Regarding the Cambridge reading list, you’re right: not every last book is necessary, and some books are for the layman or everyday reader. For instance, my daughter and I could start reading together a lot of that type right now, but some, like analysis, are beyond me! I still like the idea of having a guide like that, to give us some direction, and some idea of what kinds of things advanced future-math-majors study.

I can’t remember if I posted on the Accelerated Learners board, so I’d better do that soon, just in case.

I have done a lot more research and studying since first posting. Sorry I didn’t clarify the problem better. I’m trying to achieve this (impossible? :D) goal with only resources using a classical methodology, i.e. that used before about 1900–1910. So the books wouldn’t have to be from before that time, but the method of instruction would be of the same type.

So, nothing: progressive, outcome based, standards based, child centered, constructivist, incremental, or integrated. No assessments, spiraling, Direct Instruction, new math, new new math, books full of colors and cartoons and photographs and racial/disability quotas, fraction strips, blocks used for counting (except at the very beginning), or calculators to be used before completion of higher arithmetic. I believe all those things are distracting and damage or weaken children’s minds. Sorry I’m picky! :)

The elementary-school student was using 4th-grade Calvert Math, then Prentice Hall Mathematics: Course 1. He’s gifted, but was forced to do (for online school) things that would have been too easy the previous year, yet his basic skills were not yet solid. These and similar books, and their approach (matching those items in the previous paragraph about exactly) are what solidified my low opinion of all progressive methods. One of the kids had a Dolciani textbook assigned one year and it wasn’t outrageous in its progressiveness like the others at least, but still pretty progressive.

I’ve found plenty of suitable and free arithmetic, mensuration, and geometric-construction e-textbooks from the 19th century online (probably going with Ray’s mainly because a complete set with answer keys is easy to find), and lots of free, modern, college- and graduate-school-level math e-textbooks of this type online, but little for levels in between, even to buy. :001_huh: I think the following books are probably better than Wentworth’s though:

Algebra I: I just love Ball’s Elementary Algebra (http://books.google.com/books?id=XxYWAAAAYAAJ). Very challenging.

Algebra II: I’ve been studying it with Cohen’s Precalculus with Unit-Circle Trigonometry (http://www.amazon.com/Advantage-Precalculus-Trigonometry-David-Cohen/dp/0534352758#) (unfortunately pre-calc is integrated, so maybe it should be used supplementarily, but the book is clearly written and the problems start off as moderately challenging and based directly on the lesson, then move up a step in complexity and in concept level, and then up another) and with Hawkes, Luby, and Touton’s Second Course in Algebra (http://books.google.com/books?id=VNhEAAAAIAAJ). Some explanations in the latter aren’t the clearest though. Two other books that look promising, which were for remedial Algebra II college classes in 1947 and 1948 (so also appropriate for mature, college-material students?), respectively, are Underwood, Nelson, and Selby’s Intermediate Algebra (http://www.archive.org/details/intermediatealge033585mbp) and Hart’s Intermediate Algebra for Colleges (http://www.archive.org/details/intermediatealge030183mbp).

Geometry: So far, I have this volume (http://www.amazon.com/Thirteen-Books-Euclids-Elements/dp/0486600882) of Heath’s Euclid, and am considering supplementing it with problems from Hall and Stevens’s Euclid (http://books.google.com/books?id=AUgqAAAAYAAJ). I am also considering using Kiselev’s Geometry (http://www.sumizdat.org/).

The biggest problem with using these books though is a lack of answer keys. My math abilities are good, but not so great that I’m sure I’m right every time in high-school math. I should at least buy copies of the best ones we don’t yet have, to use for fun. (Yeah, we like math that much!) I’m also planning on getting a slide rule and manual eventually. (We must be total nerds! :D)

Other Math Subjects: Still not sure, and looking for suggestions.

In addition to what has already been mentioned, I suggest Suppes' and Hill's First Course In Mathematical Logic. I've been working through it making a solutions manuel for my daughter to use.

It takes the student deeper into math and introduces rigorous mathematical logic. Students translate English sentences into logic and mathematical symbols.

It came out in the 1960s - it's good stuff. ;)

[kiana]

Cohen's precalc is an excellent book. I learned precalc from a first edition and at my current university we use the latest edition. You should at least be able to buy the student solutions manual, which would help with the odd problems; many even problems are immediately followed or preceded by an odd problem which is quite similar.

[Nan in Mass]

Since you are reading about math education, you might be interested in reading this: http://www.singaporemath.com/Mathematics_s/1.htm
The book is expensive and if you aren't doing Singapore, you might not want to buy it, but reading the description and looking at the sample lessons gives you a good overview of why people choose to use Singapore. It might even give you some ideas to use with your own child if she gets stuck, especially now when she is still probably mostly doing her math in her head.
-Nan