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ereks mom

Foerster or Jacobs or Dolciani for Algebra 1??

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Since you all talked me out of using MUS for Algebra 1 ... ;)

 

Now I'm deciding between Foerster, Jacobs, and Dolciani. Here are my requirements for a math program:

 

1) Must be mastery approach. We don't do well with the incremental/spiral method.

2) MUST have a solutions manual for Geometry, Algebra 2, & higher levels. For Algebra 1, a solutions manual would be great, but we'll be okay as long as there's at least an answer key.

3) Must have lots of step-by-step examples in the text, along with explanations about why each step is done (which property, etc.)

4) Must be visually appealing, with at least some color in diagrams, charts, etc. Photos are a plus, but not a necessity.

 

Below are my options, as I see them now, for the high school sequence (which EK will actually begin next year in 8th grade).

 

Option 1

 

Algebra 1 -- Dolciani's Algebra Structure & Method Book 1

Geometry -- Jurgensen's Geometry

Algebra 2 -- Dolciani's Algebra and Trigonometry Structure & Method Book 2

Precalculus -- ??

 

 

Option 2

 

Algebra 1 -- Foerster's Algebra 1: Expressions, Equations, and Applications

Geometry -- Jacobs' Geometry

Algebra 2 -- Foerster's Algebra and Trigonometry

Precalculus -- ??

 

 

Option 3

 

Algebra 1 -- Jacobs' Elementary Algebra

Geometry -- Jacobs' Geometry

Algebra 2 -- Foerster's Algebra and Trigonometry

Precalculus -- ??

 

 

I have not actually looked at (as in, held in my hands) any of these except Jacobs' Elementary Algebra. So please tell me which, if any, of these options do NOT meet my criteria?

Which do you like best? Why?

Please tell me what you like and what you don't like about the individual books/courses.

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I picked option 3 for my eldest dd.

 

With my second I took a look at Foerster's Algebra I because I've been very impressed with his Algebra and Trig but decided that it was just too mathematical too early. We're again going with option 3.

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Since you all talked me out of using MUS for Algebra 1 ... ;)

 

Now I'm deciding between Foerster, Jacobs, and Dolciani. Here are my requirements for a math program

Which do you like best? Why?

Please tell me what you like and what you don't like about the individual books/courses.

 

Well, you really can't go wrong with any of these three. What it comes down to is personal style and which text you and/or your student clicks with. I didn't like Jacob's, but there is really no reason for it. Probably for precisely the same reason it appeals to other people. It's very wordy and liberal artsy in feel. Most people have a gut reaction to Jacobs, either positive or negative. I like both Forrester's and Dolciani, but we used Dolciani in my Catholic high school, so it was nostalgic for me. Not very scientific, I know, but there you go.

 

My suggestion would be to find the cheapest copies you can of each book then look for yourself. Jacobs will probably be pricey, but if you choose not to use it, you can always resell it on Half.com or Amazon. You may be lucky enough to find a review copy of one or two in your library or maybe your local school district. It's an important decision and IMO cheaper to blow a little money on some used texts rather than a lot of time and money on possibly choosing the wrong curriculum.

 

Barb

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OK, I am going to try this. I know other people use this same option, so they can support or refute me as needed!

 

We use Option 1. We are midway through Geometry and so far it has been fantastic for dd.

 

It meets requirement 1. If needed, there are little self-tests and mixed review exercises that are separate from the section's written exercises spattered throughout the chapters. At the end of every chapter there is a cumulative and an algebra review (in the geometry book). At the end of the book are extra practice problems or extra tests or additional, fun-type, mega-hard problems depending on the book.

 

It meets requirement 2. At least as far as I am aware, there are solutions manuals. I have never actually looked at one, so I cannot comment on them other than that they exist. I have ordered directly from the publisher before and had no trouble, but we all know that means nothing. I have the TE for Algebra 2/trig, but it has little more than answers.

 

Yeah... About number 3... I dont really see this requirement as being fulfilled. There are definitely examples and there are definitely examples that are worked out step by step, but they are not really accompanied by explanations about each step. This book really, really, really needs a good foundation. I cannot go so far as to say it needs a great pre-algebra program because dd did not use any pre-algebra book at all, but there were some areas that needed much input from me at some points. Also, as the problems get more difficult (B & C level), there is an assumption that you have some clue as to what you are doing, so there may not be specific examples of the more difficult problems. Did that make sense at all?

 

As for number 4 there are color pictures, diagrams and mini-biographies. Keep in mind, though, that it is a somewhat serious book, and it maintains a to-the-point nature. DD loves the little facts and biographies, though.

 

I am more than willing to take pictures or scan some sections so that you can see what they are like. I know that I really like to have something in my hands to go through at my leisure and examine completely, so any way I can help, just let me know!

 

Also, in my totally worthless opinion, the Algebra 2/trig book IS precalc. I am planning on a separate semester of trig at this point because I think dd may need it (and it's fun!), but the topics covered are the same topics covered in the college algebra books I have laying around, and in more detail than the precalc book that our CC uses. I believe, though, at this point, that we will have to have something between that and calculus. I know there is something, but I cannot remember what it is called, and I have to run out the door! No time to edit!!:eek:

 

 

HTH!

 

 

PS, I have seen these books CHEAP!

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Also, in my totally worthless opinion, the Algebra 2/trig book IS precalc.

 

Yeah, you're right. I came on here a few years ago completely bumfuzzeled about the difference between Alg. II, Trig, and PreCalc. I've come to the conclusion that PreCalculus is the same course as some Algebra II review, trig, and a couple of early Calculus topics thrown in at the end. PreCalc is sort of like Pre-Algebra...a great way to segue for kids who need the extra practice, but wholly unnecessary for kids who are ready to move on. So a transcript could look like this:

 

1 credit of Algebra II and 1/2 credit of Trig -or-

1 honors credit of Algebra II/Trig between Geometry and Calc -or-

1 credit of Algebra II and one credit of Pre-Calculus -or-

1 credit of Algebra II, one half credit or trig, and one half credit of Pre-Calc.

 

I'm sure there are other permutations I've not thought of.

 

Barb

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But I can tell you that option does not meet the color requirement. Unless, the newer edition of Jacob's geometry has color. The 2nd edition which we used does not.

 

Did you try getting them through interlibrary loan? I looked at Foerster's that way.

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PreCalc is sort of like Pre-Algebra...a great way to segue for kids who need the extra practice, but wholly unnecessary for kids who are ready to move on.

Barb

 

I certainly think that is the case in this situation, but I know that other algebra I/II books do not cover nearly the same amount of stuff, so it would be all new information! This Algebra I book covers material through much of the algebra II books I have seen here and the majority of algebra II books I have seen used at these public schools dont even try and think about trig. All I know is that after finishing the first book, dd took the ASSET test to get into the cc here and it placed her into college algebra/ trig (which I totally disagreed with just for the record). It doesn't just cover a topic, it thoroughly and completely covers a topic.

 

In fact, dd was having a conversation about proofs (and how mean I am for demanding so many of them) the other week with a boy we know who is on the honors track at his HS and he told us that his teacher never required proofs or even went over any logical thinking in geometry. She glossed over reasoning skills and stuck to the basic geometric rules that would be on a standardized test. Of course it is Arizona, where we were, what, 48th in education last year?:rolleyes:

 

I never even thought about what it would look like on a transcript, though. I just figured if I had to use them, I would put Algebra I, Geometry and Algebra II since I have to have detailed course descriptions, and hope that admissions people were familiar with the texts!

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If you begin with 8th grade next year, you will be exactly on the same track as us, which in 11th grade is Foerster's Precalculus course.

 

I think the books in option three meet your requirements. For Al I, the answer key or solutions book is less detailed, but, hey it's Al I, and it really is not very hard to figure it out when you run into a problem. ( I don't mean that nastily, by the way, I was absolutely terrified that I wouldn't be able to do it and we did just grand). For Jacob's geometry, we went with the third edition and the solutions are pretty solidly explained. Since much of it is proofs, the solutions have to be step by step or they can't show you the proof...

 

We used Foerster's Al II/trig last year, but just the AL II portion (with a side trip into thinkwelll because we were curious). The text examples in Foerster are outstanding IMO....Now I am not a "real mathy person, even though many people here think I am. I have become a mathy person because I have learned that I actually like math because of working through this stuff with ds. Really, you can easily spend a year on the AL II portion of Foerster.

 

Now this year it is Foerster's Precalculus with Trigonometry. Again it is an excellent course. I really like the online dynamic presentations that you can use through Key press's site. We have had a little bit of a struggle in learning to use our graphing calculators, but that struggle has been worth it. Expect that you will have to do some internet searching to learn how to use some of the features...but you really should do it.

 

As for the color issue....well....Not a lot of color, but great illustrations and photos in all of the above.

 

I'd especially agree with Moira's comment about Foerster's book for Al I being too mathy too soon. I literally compared Foerster's AL I to Jacobs chapter and verse and went with Jacobs for this very reason. A little humor goes a long way in easing into higher math IMO. Foerster is a little deeper, but Jacobs is friendlier. You have plenty of time to go deep with Foerster...you don't have to have it in 8th grade. It is minimally different.

 

Okay...I blathered way too much....sorry....

 

HTH

:o

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I certainly think that is the case in this situation, but I know that other algebra I/II books do not cover nearly the same amount of stuff, so it would be all new information! This Algebra I book covers material through much of the algebra II books I have seen here and the majority of algebra II books I have seen used at these public schools dont even try and think about trig.

 

That's true...I've conveniently forgotten how bad many school texts are. Now you're giving me the vapors. :D

 

Barb

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Like some of you, I thought "Why precalculus?" My dd was good in math and cruised through algebra 2.

 

A friend of mine (MIT alum, taught math at a small Christian school) made me reexamine this. She had me actually sit down with the Brown precalculus book and the Dolciani algebra 2 book and compare them -- not the table of contents (which does look very similar) but the actual material covered.

 

The precalculus textbook re-covered a lot of the material in algebra 2/trig, BUT at a much more advanced level. It is NOT the same material all over again.

 

My kids found pre-calculus to be an easy course, but precalculus really solidified their understanding of advanced math. My kids entered their calculus course completely ready for the material -- and calculus is weird enough that most kids can use all the background they can get to prepare them!

 

Would they have survived calculus 1 without precalculus? Probably -- but many people do founder on calculus. I would rather take my time getting there and have my kids succeed when they do get there. And there is something really satisfying when they come home from the William & Mary calculus 1 class with a 100% on the pretest and the comment from the prof, "You must have had an excellent math teacher!" :D

 

I would exercise great caution in skipping precalculus. Undoubtedly some students can and should skip precalculus, but most can benefit from the increased preparation that precalculus/trig provides.

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Like some of you, I thought "Why precalculus?" My dd was good in math and cruised through algebra 2.

 

The precalculus textbook re-covered a lot of the material in algebra 2/trig, BUT at a much more advanced level. It is NOT the same material all over again.

 

I would exercise great caution in skipping precalculus. Undoubtedly some students can and should skip precalculus, but most can benefit from the increased preparation that precalculus/trig provides.

 

I was seriously considering skipping the second half of the Foerster's Pre-calc/trig text and moving right on into Calculus. But then I looked at the actual material in the text more closely...there is considerably more depth and meat in there than what is covered in Al II or Thinkwell's College Algebra. We decided that we will do the whole book. I don't want ds to miss these greater foundations.

 

I sometimes have to remind myself that this is not a race....there is no prize for getting to Calculus first. It is much more important that he get there well prepared.

 

I would be very careful about deciding to skip pre-calculus. I really think it will be a big help for my ds.

 

HTH

Sharon

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I usually preview my posts before hitting submit....I should not have broken my pattern....

 

Sorry Gwen and Barb....:eek:

 

LOL, that's okay. I dunno, a couple of years ago we were advised to head on to Calculus right after completing Dolciani. We had no problems heading straight into Calc I. Maybe it's the book, or it's the particular kid. My oldest two are math brains...I'll keep your comments in mind as my 11yo begins to head through the math sequence.

 

Barb

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No experience at all here, but from all my reading and researching, I would pick option 2.

 

I would actually do the following as espoused by Math Without Borders:

 

Algebra I: Expressions, Equations, and Applications, by Paul A. Foerster

Geometry-A Guided Inquiry, by G. D. Chakerian, Calvin D. Crabill, and Sherman K. Stein

Algebra and Trigonometry: Functions and Applications, by Paul A. Foerster

Precalculus with Trigonometry: Concepts and Applications, by Paul A. Foerster

 

but you didn't have that option :) and it seems that Jacobs' Geometry text is a really good one.

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but this is a very personal issue. I recommend taking a look at Jacobs by borrowing it from the library through interlibrary loan if necessary. For us, looking at it was the clincher. I hated the look of it and the wordiness. (I never looked at Dolciani's but borrowed Lial's.) If the visual aspect is important to you, you really should look at them all.

 

I still have Jacob's on my shelf because I have another dd coming up in several years who I think might like Jacobs.:tongue_smilie:

 

Best of luck!

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For my oldest, I did

Jacobs Elementary Algebra

Jacobs Geometry, 2nd edition

Kinetic Books Algebra II (beta version, not completed yet)

 

For my middle, I did

Jacobs Elementary Algebra (for the first 8 chapters only) and Kinetic Books Algebra I (almost done with chapter 10)

 

I plan to have her start Jacobs Geometry (3rd edition) when she finishes Kinetic Books Algebra I. When she finished geometry, she'll do Kinetic Books Algebra II.

 

KB takes just about everything in Jacobs a step or two further. You don't have to go that step or two further (like solving systems of 3 equations for 3 unknowns instead of stopping at systems of 2 equations for 2 unknowns).

 

It has color, sound, animated models, immediate feedback, and stepped solutions (available for many of the immeidate feedback problems). There is only an answer key for the end of unit problems (and the answers are only for the odds), but by the time my kids have gotten to the end of unit problems, they've had so much practice that they rarely miss anything anyway.

 

They have a 3-day trial so you can check it out.

 

http://www.kineticbooks.com

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Now I'm deciding between Foerster, Jacobs, and Dolciani.

 

 

 

I am interested that Dolciani is even mentioned in the same breath as Foerster and Jacobs. Dolciani is at the heart of the '60s "New Math" establishment. It is solid, mathematically, but it suffers from the excesses of the period. (Read "Why Johnny Can't Add." I'm not saying users of Dolciani can't add, but this excellent book by mathematician Morris Kline gives an insightful analysis of the movement.)

 

I graduated from high school in 1966 and used Dolciani's pre-calculus text myself, as a student. I also taught out of Dolciani's geometry text. It's definitely one way to go, but it's a prime example of "textbook by committee." I find it reliable, but sterile.

 

Both Foerster and Jacobs, on the other hand, bring a sense of personal authorship to their work. Studying (or teaching) these texts feels like you are interacting with a (mathematically competent) human being with a personal style, a sense of purpose, and the ability to convey some of the joy of discovery and invention that are part of the mathematical enterprise. As a teacher, I was absolutely amazed when I happened upon my first Foerster textbook, quite a few years ago. You won't find a better collection of application problems at this level anywhere...period! I diverge from the author on a few points, but I've also learned from him on others.

 

My answer to the question posed is answered by my choice of texts. Each of my video lesson projects is an investment of 1-2 years of work. I chose the Foerster texts because I saw them as the best there was. If a student is free to choose, why not choose the best?

 

I've seen Foerster described as "honors" level texts. I'm sure that is because of the huge number of application problems. But that's what it's all about, isn't it? Isn't the goal of Algebra to be able to use it fluently as a tool? Foerster's explanations are exceedingly clear, which is a plus for students of any ability level. He also shows good sense about the kinds of difficulties real students will encounter, because he put it through its paces with real students.

 

I just finished recording the Algebra II lessons for Foerster a couple weeks ago and I'm suffering withdrawal symptoms! He (as he comes through his text) is a great colleague to work with!

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I am interested that Dolciani is even mentioned in the same breath as Foerster and Jacobs. Dolciani is at the heart of the '60s "New Math" establishment. It is solid, mathematically, but it suffers from the excesses of the period. (Read "Why Johnny Can't Add." I'm not saying users of Dolciani can't add, but this excellent book by mathematician Morris Kline gives an insightful analysis of the movement.)

 

IBoth Foerster and Jacobs, on the other hand, bring a sense of personal authorship to their work. I just finished recording the Algebra II lessons for Foerster a couple weeks ago and I'm suffering withdrawal symptoms! He (as he comes through his text) is a great colleague to work with!

 

fwiw, my dd has tried no less than 5 Algebra texts (she prefers to learn from a book, and does so.) We also listened to Chalk Dust & VideoText samples to see about learning that way. Jacobs was way too slow and boring for her, even though I'd read great reviews of it. We have Foerster's sitting on our shelf collecting dust. But she enjoys the 1965 Dolciani Structures and Methods book, and prefers the writing of that one over the later Dolciani texts. Not that it's exciting writing. She also loves Gelfand's Algebra, which she is very, very slowly working through, for the theory & application. As for Dolciani, there are some older posts on this forum you may wish to read by Jane in NC, Charon, et al that discuss this, particularly Charon & Jane who have degrees in mathematics (hope I didn't miss any other mathematicians here). Most of the problems with new math stemmed from poorly done imitations from the original. What I think is that there is no one tried and true Algebra text that is going to work equally well for all Algebra students.

 

I did new math and did very well with math (the schools in our area switched c. 1970). I think there are good things and bad things to say about various methods of teaching Algebra. Some of us who have used or are using the older Dolciani's will stand by it because we've seen it work well.

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Some of us who have used or are using the older Dolciani's will stand by it because we've seen it work well.

 

Thank you, Karin, for stating this. One grows weary of repeating oneself. Suffice it to say that I respectfully disagree with Mr. Chandler's conclusion.

 

Jane

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Thank you, Karin, for stating this. One grows weary of repeating oneself. Suffice it to say that I respectfully disagree with Mr. Chandler's conclusion.

 

Jane

 

I'm not trying to quibble but I'm not sure what part of my conclusion you're disagreeing with. I want to elaborate on the positive side of my experience with the Dolciani texts: They are very competent, thorough, and systematic in their approach to the subject matter. The problems are carefully graded in difficulty. If a student responds well to the text, I have no doubt that he/she will get a thorough grounding in the material. Like I said, I studied out of the original version of Dolciani's pre-calculus text in high school in 1965-66 and actually liked it.

 

What motivated my post was not that Dolciani is bad, but it seemed incongruous to put it up beside Foerster and Jacobs. They are poles apart stylistically. Dolciani texts are quite literally written by committees. Foerster and Jacobs are both single-author works where their personalities come through in the writing. I haven't worked with the Jacobs books, but I have loved working with the Foerster books. I'm sure part of the affinity is that Foerster and I both come from applied math backgrounds: engineering in his case and physics and applied math in my case. (I have a BS in Physics from Harvey Mudd College, and a MS in Applied Mathematics from California Polytechnic University.) I've been teaching for ~30 years and have seen it all! (...so to speak.)

 

As for your repeating yourself, I am a relatively new visitor on the WTM forums and I haven't seen your previous posts. I'll search out a few. It sounds like this discussion has gone around the block a few times.

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Our backgrounds are similar. I have both a BS and an MS in Mathematics, used the Dolciani Modern Introductory Analysis text in high school (this what I believe you are calling the precalculus text) as well as the earlier books in the series, have taught mathematics for years, etc.

 

In terms of content, I see the Dolciani books of the '60's and '70's as more rigorous texts. For example, most precalc students do not learn induction and other techniques of proof. A student working his way through the Dolciani Analysis book walks away with a foundational understanding of what mathematics truly is.

 

Many of the more "modern" texts prepare students to be engineers by their algorithmic approach to the subject. Jacobs and Foerster are fine books, but I do not feel that they are superior to Dolciani. As you admit, Foerster is strong in applications, stronger than Dolciani in that regard. My goal for my son is learn mathematics. I am giving him the tools to understand the theorems that he will encounter and the techniques used in proving them. This goes beyond algorithms and applications.

 

And yes this discussion has been happening for years on this board and its older version. You can comb the archives and read older entries on the blog (Drat These Greeks) kept by one of our participants (Myrtle--oh where are you?) for further discussion.

 

Best,

Jane

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I am interested that Dolciani is even mentioned in the same breath as Foerster and Jacobs. Dolciani is at the heart of the '60s "New Math" establishment. It is solid, mathematically, but it suffers from the excesses of the period. (Read "Why Johnny Can't Add." I'm not saying users of Dolciani can't add, but this excellent book by mathematician Morris Kline gives an insightful analysis of the movement.)

 

I graduated from high school in 1966 and used Dolciani's pre-calculus text myself, as a student. I also taught out of Dolciani's geometry text. It's definitely one way to go, but it's a prime example of "textbook by committee." I find it reliable, but sterile.

 

Both Foerster and Jacobs, on the other hand, bring a sense of personal authorship to their work. Studying (or teaching) these texts feels like you are interacting with a (mathematically competent) human being with a personal style, a sense of purpose, and the ability to convey some of the joy of discovery and invention that are part of the mathematical enterprise. As a teacher, I was absolutely amazed when I happened upon my first Foerster textbook, quite a few years ago. You won't find a better collection of application problems at this level anywhere...period! I diverge from the author on a few points, but I've also learned from him on others.

 

My answer to the question posed is answered by my choice of texts. Each of my video lesson projects is an investment of 1-2 years of work. I chose the Foerster texts because I saw them as the best there was. If a student is free to choose, why not choose the best?

 

I've seen Foerster described as "honors" level texts. I'm sure that is because of the huge number of application problems. But that's what it's all about, isn't it? Isn't the goal of Algebra to be able to use it fluently as a tool? Foerster's explanations are exceedingly clear, which is a plus for students of any ability level. He also shows good sense about the kinds of difficulties real students will encounter, because he put it through its paces with real students.

 

I just finished recording the Algebra II lessons for Foerster a couple weeks ago and I'm suffering withdrawal symptoms! He (as he comes through his text) is a great colleague to work with!

 

My DH and I spent last spring looking at many math textbooks (Foerster, Lial, Jacobs, Larson, new Dolciani, old Dolciani, Chalkdust, Videotext, Teaching Textbooks, etc.). My DH is a mechanical engineer, and in my former life, I was a CPA.

 

The biggest problem we had with the newer editions across the board was the chatty language used. I found it highly distracting, but obviously, that's personal preference, not a sign of superiority.

 

We chose to use the older edition Dolciani series because of the reasons you've already cited: it's systematic, logical approach and precise use of language. Foerster and Jacobs tied for 2nd place, but ultimately, we found them to be too overwhelming in the application questions, and not enough explanation to begin with. As an accountant, I had trouble following their explanations (moreso for Foerster), and then applying that to the problems. My DH didn't have that problem, but he did take more higher level math courses in college. He could make the connections that the author made that I couldn't see or understand.

 

The bottom line for me was that the newer editions taught the student how to work the problems, rather than why they are working a problem a certain way. I learned math by example in school. I had no idea why I was doing it, and I definitely couldn't apply it to another problem from a different angle. In my opinion, the Dolciani texts teach the 'whys' so that the student can work any problem given.

 

I did keep the Jacobs Algebra book because I felt that the visual presentation of polynomials may be more beneficial to one of my non-math sons, but for our math studies, we will use all four of the 1960's Dolciani math books. I'm very confident that this series will prepare my sons for college.

 

Of course, someone else can look at the same math books and come up with a different conclusion for their homeschool. My point here is that Foerster, Jacobs, and Dolciani are on the same level, but the choice is a matter of style and preference, not quality.

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The bottom line for me was that the newer editions taught the student how to work the problems, rather than why they are working a problem a certain way.

.

 

This is critical. When I asked my brother, who teaches post secondary physics (with a Ph.D. plus he did a postdoc), and has taught a few calculus courses extra when needed, what he thought was missing in high school education today, he said, without hesitation, logic and thinking skills. He teaches in Canada, where Calculus isn't introduced until university/college. His example of the lack of thinking skills had to do with how math is taught. He said he'd have students come in who could factor beautifully in Algebra, but when it came to Calculus, they couldn't see when they needed to factor. ie, they knew how to work problems, but had no idea WHY they were doing what they were doing.

 

I keep the Foerster's on the shelf for reference, in case my dd, or either of my younger ones, ever don't understand the explanations in Dolciani or Gelfand's.

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The bottom line for me was that the newer editions taught the student how to work the problems, rather than why they are working a problem a certain way.

.

 

This is critical. When I asked my brother, who teaches post secondary physics (with a Ph.D. plus he did a postdoc), and has taught a few calculus courses extra when needed, what he thought was missing in high school education today, he said, without hesitation, logic and thinking skills. He teaches in Canada, where Calculus isn't introduced until university/college. His example of the lack of thinking skills had to do with how math is taught. He said he'd have students come in who could factor beautifully in Algebra, but when it came to Calculus, they couldn't see when they needed to factor. ie, they knew how to work problems, but had no idea WHY they were doing what they were doing. My goal isn't just to teach application, but for my dc to understand the theory, be able to do proofs, and to know WHY they're doing what they're doing--that way they can not only apply it IRL, but learn if/then thinking.

 

I keep the Foerster's on the shelf for reference, in case my dd, or either of my younger ones, ever don't understand the explanations in Dolciani or Gelfand's.

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