# A trigonometry text that even Charon will like!

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Silly me. I thought that the trig portion of the Dolciani Algebra II/Trig text that my son is currently using would be the same as the Dolciani Modern Trigonometry text that I picked up at a library book sale last spring. (My new hobby is collecting old Dolciani texts on the cheap.) Imagine when I pulled the latter off the shelf and saw the name Bechenbach on the cover. Understand that the Dolciani/Bechenbach collaboration on Modern Introductory Analysis created a stellar high school math text that I used in the mid '70's. So my curiosity was piqued.

This 1966 text begins where every old Dolciani text does: with sets and axioms. The trig begins in Chapter 2. I have taught trig from what seems to be a countless number of precalc texts. The subject usually begins with a discussion of radians, then defines the trig functions essentially as relationships of sides of the triangle in the four quadrants of the plane. Not this book. Chapter 2, entitled Circular Functions, begins with periodic functions. In the A portion of the first problem set students graph and determine the periodicity of things like f(x) = 2(x - [x]), where the bracket denotes the greatest integer function. The B sections contains proofs on periodic functions. And so it goes.

There is no special chapter on proving trig identities. The proofs are throughout the book. Applications on things like uniform circular motion and simple harmonic motion are not lacking.

But it gets even better. Chapter 6 is on vectors. Within the first section the idea of a commutative group is introduced, as well as a vector space. The engineering mathematics of vector applications to forces is all there but to arrive at it one must first work through theoretical material on inner products.

Of course, finding this old text may be a challenge but there are copies floating around the Internet. This is obviously not going to be everyone's idea of a great trig text. But it looks like a wonderful book for someone who wants to treat trig as something other than a bunch of algorithms that you memorize, regurgitate and promptly forget after the test (which I am afraid is how many trig books treat the subject!) My son will start using this book next week, once he wraps up his logarithm material in Dolciani's Algebra II/Trig. Note: odd answers are in the back, but not any graphs. A solutions manual may be impossible to find!

If anyone is interested in how it goes, I'll be happy to keep you posted either on this board or via a private message.

Jane

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Silly me. I thought that the trig portion of the Dolciani Algebra II/Trig text that my son is currently using would be the same as the Dolciani Modern Trigonometry text that I picked up at a library book sale last spring. (My new hobby is collecting old Dolciani texts on the cheap.) Imagine when I pulled the latter off the shelf and saw the name Bechenbach on the cover. Understand that the Dolciani/Bechenbach collaboration on Modern Introductory Analysis created a stellar high school math text that I used in the mid '70's. So my curiosity was piqued.

This 1966 text begins where every old Dolciani text does: with sets and axioms. The trig begins in Chapter 2. I have taught trig from what seems to be a countless number of precalc texts. The subject usually begins with a discussion of radians, then defines the trig functions essentially as relationships of sides of the triangle in the four quadrants of the plane. Not this book. Chapter 2, entitled Circular Functions, begins with periodic functions. In the A portion of the first problem set students graph and determine the periodicity of things like f(x) = 2(x - [x]), where the bracket denotes the greatest integer function. The B sections contains proofs on periodic functions. And so it goes.

There is no special chapter on proving trig identities. The proofs are throughout the book. Applications on things like uniform circular motion and simple harmonic motion are not lacking.

But it gets even better. Chapter 6 is on vectors. Within the first section the idea of a commutative group is introduced, as well as a vector space. The engineering mathematics of vector applications to forces is all there but to arrive at it one must first work through theoretical material on inner products.

Of course, finding this old text may be a challenge but there are copies floating around the Internet. This is obviously not going to be everyone's idea of a great trig text. But it looks like a wonderful book for someone who wants to treat trig as something other than a bunch of algorithms that you memorize, regurgitate and promptly forget after the test (which I am afraid is how many trig books treat the subject!) My son will start using this book next week, once he wraps up his logarithm material in Dolciani's Algebra II/Trig. Note: odd answers are in the back, but not any graphs. A solutions manual may be impossible to find!

If anyone is interested in how it goes, I'll be happy to keep you posted either on this board or via a private message.

Jane

He's threatening to give up on the traditional sequence altogether after arithmetic and approach math from the foundation up beginning with mathematical logic.

Right now we have intro to a few trig functions in our algebra I book, again in algebra II, and finally a third time in Cletus and Oakley.

I keep telling him that I ought to rework some trig book or another in order to review and he tells me, "Don't bother, if you need to do a trig problem just derive it from first principles, that's what I would do if I couldn't remember!"

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He's threatening to give up on the traditional sequence altogether ...

While I suspected as much, I still thought that he might appreciate a more rigorous course of study for those attempting the engineering route. In fact, I plan to be on the lookout for Beckenbach's Modern Mathematics for the Engineer. While I personally am not that interested in applied mathematics, I think that my son may benefit from it. And my husband would enjoy it as well.

Cheers,

Jane

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While I suspected as much, I still thought that he might appreciate a more rigorous course of study for those attempting the engineering route. In fact, I plan to be on the lookout for Beckenbach's Modern Mathematics for the Engineer. While I personally am not that interested in applied mathematics, I think that my son may benefit from it. And my husband would enjoy it as well.

Cheers,

Jane

Well it's basically because she thinks she can teach the kids some serious logic. So, yeah, I'm thinkin' "If she can do that, then I can...?" Consider this. What if you could teach your kid a solid sequence of abstract algebra followed by a solid sequence of real analysis? Maybe it takes like four years to do the whole thing and it includes polynomial rings and vector fields and all of a typical real analysis. And do a year of physics on the last year. What will they not be able to do? Manipulate x? Solve a physics problem? Hell yeah I'd do that if I thought I could! They'd walk into college being qualified to teach calculus. ;o)

(But, let's just say I'm not ready to bet the farm on that one yet....)

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Well it's basically because she thinks she can teach the kids some serious logic. So, yeah, I'm thinkin' "If she can do that, then I can...?" Consider this. What if you could teach your kid a solid sequence of abstract algebra followed by a solid sequence of real analysis? Maybe it takes like four years to do the whole thing and it includes polynomial rings and vector fields and all of a typical real analysis. And do a year of physics on the last year. What will they not be able to do? Manipulate x? Solve a physics problem? Hell yeah I'd do that if I thought I could! They'd walk into college being qualified to teach calculus. ;o)

(But, let's just say I'm not ready to bet the farm on that one yet....)

Is your oldest an adolescent yet? These teenaged brains seems to go through literal phases periodically--I can't imagine presenting day after day of abstract algebra when one just wants to whack the kid with a stick and say, "Wake up!"

Our plan has been for my son to do the Dolciani/Beckenbach Introductory Analysis book next year and call it precalculus on his transcript. I feel that he will be sufficiently prepared to begin a traditional calculus course in the fall but fail to see the purpose of jumping on that bandwagon just because we can. I would rather prepare him for an Honors type of Calculus course which is precisely what the Intro Analysis text will do.

More on the adolescent brain thing: our lad is gliding through math but has hit a wall in Latin. He can read Latin passages for the gist (something that he has always been good at), but just don't ask him to parse a sentence. Grammatical stuff that he has seen many times over has gone out the window. Note the word roots are there (hence the gist) but gone is all subtlety. This has happened before. It is as though a bank of switches has turned off in some part of his brain. Granted, he is nearing the end of his Latin grammatical studies and has had to process a great deal. It just seems peculiar that while he ebbs in one subject, he flows in another.

And who knows what today will bring.

Jane

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Is your oldest an adolescent yet? These teenaged brains seems to go through literal phases periodically--I can't imagine presenting day after day of abstract algebra when one just wants to whack the kid with a stick and say, "Wake up!"

Our plan has been for my son to do the Dolciani/Beckenbach Introductory Analysis book next year and call it precalculus on his transcript. I feel that he will be sufficiently prepared to begin a traditional calculus course in the fall but fail to see the purpose of jumping on that bandwagon just because we can. I would rather prepare him for an Honors type of Calculus course which is precisely what the Intro Analysis text will do.

More on the adolescent brain thing: our lad is gliding through math but has hit a wall in Latin. He can read Latin passages for the gist (something that he has always been good at), but just don't ask him to parse a sentence. Grammatical stuff that he has seen many times over has gone out the window. Note the word roots are there (hence the gist) but gone is all subtlety. This has happened before. It is as though a bank of switches has turned off in some part of his brain. Granted, he is nearing the end of his Latin grammatical studies and has had to process a great deal. It just seems peculiar that while he ebbs in one subject, he flows in another.

And who knows what today will bring.

Jane

No. He is 11. But Myrtle, nonetheless, thinks Suppes' Introductory Logic will be easy for him. That book ends with symbolic quantified logic. She thinks he could do that and a couple more books in three years time and all that ends in axiomatic set theory. My first thought was "Great! I really will be able to do Lay's Analysis when the time comes and he'll have studied yet another axiomatic system." (Lay's book is like the easiest real analysis text possible and it has this big logic and set theory beginning to it.)

But, truthfully, if you are talking about being able to do axiomatic set theory and having the logical wherewithall to negate elaborate quantified sentences by the time he is 14, then you could do regular math. What's stopping you? Maybe a little prerequisite knowledge. But, yeah -- somehow, I just think it will all get bogged down real fast somewhere -- that axiomatic set theory won't work or something. He'd get stumped for months and months and just never be able to do proof by mathematical induction -- something like that. (But, like I say, if the logic really is there, then I don't think he'd get stuck on stuff like that.)

And besides that, like you say -- we still have a whole lot of Latin to do. We were doing Henle but ended up just letting it slide last year. We'll probably have to just start from scratch again. I was thinking this logic and set theory is good enough for any sort of "philosophy" I would have ever wanted to do (as opposed to that Lipman series I am always promoting). Actually, it is just the sort of thing that philosophy majors have to prove themselves on, anyway, I think. At any rate, I think there is just a limit of emotional energy someone has. You just can't do the most intense thing all day, every day even if he could theoretically do each individual piece on its own.

But, I seriously was telling Myrtle about topics and such and how between algebra, analysis and a course of introductory (calculus-based) physics, you'll get everything out of math and 50% of science anyone else gets. (Plus, of course, you would literally be practically skipping straight to graduate level work in math.) Myrtle's response was "Okay, try it on me, then. If it doesn't work on me then it isn't going to work on the kids." So, now I am literally thinking about getting this Dummit and Foote book or something that comprehensive but that is easier maybe or some such thing. Myrtle can actually do a little set theory and even a little mathematical induction.

I think we both would deserve some kind of medal if she actually starts working at the level of Dummit and Foote as someone that never even took calculus in college. (At least someone could say I was right for a change about some of the stuff I spout about math ed.) Maybe they'll just let her sign up for a graduate program around here. Dummit and Foote's about at my limit, I think. I would have to concurrently do my own thing in Lang or something just to competently TA her through it. (I don't even like abstract algebra, for crying out loud!)

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• 3 years later...

OK, I bumped this up, after discovering it, to report that I actually found the trig book Jane is talking about here. I had no idea it existed, until I discovered it this past Saturday, at a university's book sale room. \$2. I couldn't pass up a 1966 Dolciani book that I didn't know existed. :D And Jane gave me the lowdown on it today, and then I found this thread of hers from three years ago. Yay!

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Colleen, this thread reminds me how much I miss Charon and Myrtle. (Virtual waves if either of you glance at these boards on occasion!)

Secondly, it is interesting to read where we had been and see where we went. My dearest son somehow survived high school mathematics. He earned AP credit for Calculus but has yet to see any of the beauty or joy I have witnessed. Nor have I been able to persuade him to take physics in college.

Oh well...

He carries on in Latin. He reads history. He studies philosophy. But math? Hope springs eternal. That is all.

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