# Math Book using Spiral Theory

## 17 posts in this topic

As the end of my senior year in high school draws near, I've decided to being a sort of "re-education" of myself in math, science, history, and literature, since I feel so unfulfilled with the education I've received for the past twelve years. I made this thread specifically because I noticed what looked like a (probably inadvertent) implementation of the Spiral Theory of Knowledge in a math book I was considering buying. The book is called Mastering Technical Mathematics (click), and here is the table of contents from the book:

Part I: Arithmetic as an outgrowth of learning to count.

2. Subtraction

3. Multiplication

4. Division

5. Fractions

6. Area: the second dimension

7. Time: the fourth dimension

Part II: Introducing algebra, geometry, and trigonometry

8. First notions leading to algebra

9. Developing "school" algebra

11. Finding short cuts

12. Mechanical mathematics

13. Ratio in mathematics

14. Trigonometry and geometry conversions

Part III: Developing algebra, geometry, trigonometry, and calculus

15. Systems of counting

16. Progressions

17. Putting progressions to work

18. Putting differentiation to work

19. Developing calculus theory

20. Combining calculus with other tools

21. Introduction to coordinate systems

Part IV: Developing algebra, geometry, trigonometry, and calculus as analytical methods in mathematics

22. Complex quantities

23. Making series do what you want

24. The world of logarithms

25. Mastering the tricks

26. Development of calculator aids

27. Digital mathematics

Notice how it goes over algebra, geometry, trig, and calc multiple times, progressively getting more advanced each time. Is this not an example of education with the Spiral Theory?

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Notice how it goes over algebra, geometry, trig, and calc multiple times, progressively getting more advanced each time. Is this not an example of education with the Spiral Theory?

I simply cannot see that from the chapter headings alone. I do note that the author has written several self-teaching type books in different areas of math and physics, so if you benefit from this one you might have something to look forward to in the others. I applaud you for taking the intitiative to learn that which your regular education did not provide.

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I simply cannot see that from the chapter headings alone. I do note that the author has written several self-teaching type books in different areas of math and physics, so if you benefit from this one you might have something to look forward to in the others.

Yeah, I figured I was judging it on too superficial a level. I will try it anyway, though, because one of my main grievances with my math education was how each subfield (algebra, geometry, etc...) was taught in an isolated fashion, like different chapters in a book.

I applaud you for taking the intitiative to learn that which your regular education did not provide.

Thanks! This is what my wishlist looks like for this summer so far:

Math & Science:

Mastering Technical Mathematics by Stan Gibilisco

Conceptual Physics by Paul G. Hewitt

Selected topics in the Philosophy of Science (Audio) by Harry Binswanger

History:

[still looking for good textbook]

Introduction to Intellectual History (Audio) by John Ridpath

Literature:

The Romantic Manifesto by Ayn Rand

Philosophy:

How to Study Ayn Rand's Writings (Audio) by Harry Binswanger

[Ayn Rand's non-fiction works]

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... because one of my main grievances with my math education was how each subfield (algebra, geometry, etc...) was taught in an isolated fashion, like different chapters in a book.

But "chapters in a book" is often a proper way to describe many subject fields we study. Clearly there are connections between each, but most often it is best to first learn the foundations upon which a particular subject is built. It is not an accident that learning algebra, geometry, and trigonometry prepares you for your study of calculus.

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I've not used them, but I've heard good things about Saxon Math texts. They have a series of books right up through calculus.

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Notice how it goes over algebra, geometry, trig, and calc multiple times, progressively getting more advanced each time. Is this not an example of education with the Spiral Theory?

What exactly is the "Spiral Theory"? Does it actually originate from Ayn Rand and Objectivism? Where can I read more about it in Objectivist literature?

I am curious to find the answers to these questions, because so far the only epistemological "spiral theory" I have heard about is the one proposed by Jerome Bruner in the 1960's.

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But "chapters in a book" is often a proper way to describe many subject fields we study.  Clearly there are connections between each, but most often it is best to first learn the foundations upon which a particular subject is built. It is not an accident that learning algebra, geometry, and trigonometry prepares you for your study of calculus.

The problem I found was that, for example, after learning about geometry for a year (in 9th grade), we never used it again. As a resuly, I only vaugely remember the things I learned. I understand that they are seperate fields, but it doesn't seem very ineffective to teach them once and never confront them again. I would've rather been given a very broad, integrated understanding of math in general, and then have specific fields like algebra and geometry layered on top, revisited repeatedly.

BTW, as you can see in the contents of MTM above, calculus is first introduced in part 3, while algebra, geometry, and trigonometry were first introduced in part 2. So it still seems to respect the need to teach some subjects before others.

What exactly is the "Spiral Theory"? Does it actually originate from Ayn Rand and Objectivism? Where can I read more about it in Objectivist literature?

I recommend you read The Montessorian's posts on this forum, here:

http://forums.4aynrandfans.com/index.php?showtopic=355

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Typo: I understand that they are seperate fields, but it doesn't seem very effective to teach them once and never confront them again.

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Oakes,

The problem I found was that, for example, after learning about geometry for a year (in 9th grade), we never used it again. As a resuly, I only vaugely remember the things I learned. I understand that they are seperate fields, but it doesn't seem very ineffective to teach them once and never confront them again. I would've rather been given a very broad, integrated understanding of math in general, and then have specific fields like algebra and geometry layered on top, revisited repeatedly.

You might want to read the link to Saxon math books I gave above.

Here is the relevant quote:

Continual Practice Distributed Across the Level

Practice of an increment is distributed continually across each grade level. Continual, distributed practice ensures that concepts are committed to students' long-term memory and that students achieve automaticity of basic math skills. Several research studies show that students who are taught with a mathematics curriculum that uses continual practice and review show greater skill acquisition and math achievement (Good & Grouws, 1979; MacDonald, 1984; Hardesty, 1986; Mayfield & Chase, 2002; Usnick, 1991; Ornstein, 1990; Hardesty, 1986; MacDonald, 1984; Good & Grouws, 1979). Additional studies have concluded that spaced (distributed) practice results in higher performance than massed practice (Dhaliwal, 1987; Proctor, 1980).

It sounds like they continually make you use earlier material.

I hear that you have to be careful, because they wrote a dumbed down series for the public schools recently. The original series has been given high ratings.

For instance, Dr. Arthur Robinson homeschools his kids, his website is here clicky. The two oldest boys scored very high on their SATS (maybe perfect, I don't recall), and one went through Cal Tech, and was more advanced than most of the other students in math, which gave him a big edge. He has his own curriculum, called the Robinson Curriculum for home schooling. He's a chemist by profession, but he's also religious, just to warn you.

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You might want to read the link to Saxon math books I gave above.

<snip>

I hear that you have to be careful, because they wrote a dumbed down series for the public schools recently.    The original series has been given high ratings.

Thank you very much for the information. Do you know where the original series can be found? On their website, all I see is a list of K-12 math books:

http://www.saxonhomeschool.com/math/index.jsp

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The problem I found was that, for example, after learning about geometry for a year (in 9th grade), we never used it again. As a resuly, I only vaugely remember the things I learned. I understand that they are seperate fields, but it doesn't seem very ineffective to teach them once and never confront them again. I would've rather been given a very broad, integrated understanding of math in general, and then have specific fields like algebra and geometry layered on top, revisited repeatedly.

But revisiting material is a separate issue from whether or not the subjects are taught in an independent or integrated manner. Also, I am not even sure what a "very broad, integrated understanding of math in general" actually means at an introductory level. There are college courses that try to do that; "Math for Poets" and the like, but I think those courses are mostly bull.

In my experience, a "very broad, integrated understanding of math" only comes after many years of extensive study of many of the sub-fields within the subject. In fact, in order to integrate it all together, you need to see the connections between the sub-fields -- to see how they relate to each other and grasp the unifying principles -- and for that you first have to master the sub-fields themselves. I would be very suspicious, in general, about claims to integration minus the detailed material to integrate. I think an hour lecture on the subject of math, talking about its history and overall structure, should be enough to get any serious student going in his detailed studies of the subject.

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Is this not an example of education with the Spiral Theory?

This looks more like a straight hierachical presentation than than the spiral technique or method which moves up and down the hierarchy. I would also have to see the book to know.

My understanding of the spiral techinique is that one starts with the less complex and moves to the more complex. When things get difficult or hard to understand, one goes down to a less complex level and makes sure that it is well undstood and then tries to move upward again, thus making sure that the more complex materials, or higher level abstractions are always grounded solidly in the lower level concepts that they are based upon. This keeps the higher level concepts from being "floating abstractions".

I use this technique in teaching music theory. First I teach notation, then triads and scales, and then inversions of triads. If the students are having trouble with triad inversions, it is usually because they did not really learn triads. So I go back down the spiral and review triads until they have it and then move up to inversions again. If they seem to have that, I move on to chord progressions (a higher level abstraction). If they have trouble with chord progressions, back to triads and inversions again...and then up the spiral again to chord progressions and even higher level music abstractions, going up and down the hierarchy, eventually ending up with highly abstract forms like the Sonata Allegro.

This is my understanding of it. I would welcome any comments or corrections. I believe Dr. Piekoff goes over it thoroughly in his Philosophy of Education taped lecture series.

gmartin

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Thank you very much for the information. Do you know where the original series can be found? On their website, all I see is a list of K-12 math books:

http://www.saxonhomeschool.com/math/index.jsp

The public school series is here http://www.saxonpublishers.com/

The link in your quote above is for homeschoolers, and the last five books in the series, at least, have dates prior to 2004, so are probably the right ones.

The K-12 should cover everything from the basics up through calculus. They have a placement test so you can figure out where best to start.

Since I'm not certain if those are the right books, I've emailed Dr Robinson to see if he knows. I can't gurantee that I will get an answer from him. In the mean time, I will look around elsewhere for that answer.

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Oakes,

I just got word back from Dr. Robinson. He said he was not sure about that website, but that he sells the better books himself. They are also at a discount, I note, since the prices are lower than the homschooler site's.

Best of luck.

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Oakes,

I just got word back from Dr. Robinson.  He said he was not sure about that website, but that he sells the better books himself.  They are also at a discount, I note, since the prices are lower than the homschooler site's.

Thanks a lot for your help, Thales. A brief look on that website grew my interest in his work even more. I love his reply (click) to one parents who said "I want my child to learn social skills." Here's just the first few sentences of it: "I rarely meet an adult who cannot articulate and relate to others. Yet a great many adults will not or cannot think."

After reviewing what he has to say about multiculturalism, here's my next question: Is this guy a closet Objectivist or what?

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After reviewing what he has to say about multiculturalism, here's my next question: Is this guy a closet Objectivist or what?

Robinson is on the right side of many issues, but an Objectivist he is not. For one thing, religion. Thales mentioned him to me a number of years ago, and coincidentally a friend had just sent me an article Robinson wrote. I appreciated the attitude he expressed towards science. I also heard him on a local radio show, arguing against global warming if I recall correctly. An interesting guy, but not an Objectivist.

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Oakes,

You're welcome.

Stephen has him assessed properly. He's definitely religious.

He's a very accomplished chemist, who was educated at CalTech, and taught chemistry at a California university (can't recall which). He was also the science director for the Linus Pauling Institute, and now, as you can see, is doing his own research in health and aging.

He educated his own kids, and turned it into a lucrative business!

Dr. Robinson took over the news letter Access To Energy (ATE) from the late Petr Beckmann, who was very accomplished in his own right. ATE is a newsletter that debunks environmental myths and presents various writings on scientific matters. It was first published, if I'm right, in the early seventies, or late sixties by Petr Beckmann. There are several older issues on line which you can read. My Dad started getting ATE in the early 1980s, and I read it occasionally.

Robinson is the guy who put together the petition against hyped global warming fears, and garnered the signatures of thousands of top flight scientists. He has a video somewhere on his website on global warming. Capmag.com referenced it a couple of months ago.

He is a neat guy, with many insightful views as you point out, but, sadly, he's not an Objectivist.