Myrtle, Charon, Jane in NC, and other math people that are into "Math as a subject"

[Kimber]

What do you look for in curriculum other than proofs? Is your goal to make your students really struggle through the process of learning? Is it that they discover some of these connections and concepts on their own? Or do you try to clarify and shovel in as much knowledge and logic as their little brains can take, sort of smoothing the way for them?

 

I've got all of these math books coming so that my dh and I can give our kids the option to study math as a subject in and of itself, but never having done it myself, I am sort of confused on the approach. I usually like to do things in the most efficient way possible simply because I have a 2 year old terror that thinks that people can fly.

 

 

And another question, Am I right in thinking that the way the US currently studies math is akin to studying writing without ever studying grammar?

 

I'm a little slow at putting new ideas together sometimes, so thanks for the indulgence and the help.

 

 

Kimberly

[Jane in NC]

Hi Kimber,

 

Frankly I did not put much thought into this issue until my son was at the point in the American progression at which he would take Algebra. I collected a stack of books from library sales and charity shops, then spent a few hours looking at them. Ugh. I did not like a single one.

 

So I ordered a copy of the book that I had used for Algebra, an antique Dolciani. My husband and I then compared this book with others in the stack. It was more "mathy". Yeah, there were some proofs. Yeah, there were more interesting problems. But what ultimately sold us was the use of proper mathematical notation, the statement of axioms, and the interesting material in the "Extra for Experts" sections.

 

My son has been using a mid 70's Dolciani for Algebra II/Trig this school year, but I just jumped ship to a separate '70's era volume called Modern Trigonometry. It too is a Dolciani book, but it is co-authored by Beckenbach, who co-authored the book Modern Introductory Analysis with Dolciani. The latter, a book that I cherish, led me to become a math major.

 

So perhaps it is more nostalgia than mathematics which leads to my book choices? One wonders. I can say that I have taught from a variety of precalculus texts, none of which I have liked. Having my son use the aforementioned Analysis text is a no brainer for me: I view it as a precalculus text which will prepare him for either engineering Calculus or an honors Calculus for Math majors.

 

Back to questions asked in your post: To me mathematics is more than a series of algorithms. It is a way of thinking. I have my son read his math book before tackling his problems. (This is a skill that most kids in college lack. They use their math texts only for the problem sets but rarely read them!) If my son is stumped on something, I provide clues of varying levels. Sometimes I might offer the first step of a proof or suggest a path that he should follow. With word problems, I may walk him through the set of variables and picture demonstrating the problem by asking a series of leading questions. I do want him to scratch his head, learn to read the section, look back at previous material, and use the index. This process does indeed involve discovery--sometimes shoveling of concepts, I suppose. But I do not believe that students should bang their heads on walls for too long. Head scratching and a bang or two is fine. More than that leads to frustration!

 

Jane

[Charon]

Math Sequence:

 

Modern Algebra (Book One) by Allen and Pearson

Geometry by Moise and Downes

Modern Algebra (Book Two) by Allen and Pearson

Principles of Mathematics by Oakley and Allendoerfer

Abstract Algebra by I N Herstein

Principles of Mathematical Analysis by Walter Rudin

 

Logic/Philosophy Sequence:

 

First Course in Mathematical Logic by Patrick Suppes

Introduction to Logic by Patrick Suppes

 

(The second book of logic should/must be done by the time you start Herstein.)

 

****

 

Now let's look at why this is the answer. Well, if it really does end in Herstein and Rudin, then there is no doubt about it -- it is definitely "math for math majors". Both of those books are along the lines of honors senior college courses, so if you can do that, you win. The shakiest part of it all is the lack of very much matrix algebra, actually, which is largely taken for granted in various parts of both Herstein and Rudin. However, Myrtle, for instance, just got done doing a litany of mathematical induction problems out of Allendoerfer, and I showed her the very same problems right there in Herstein. So, that New Math is no joke. And, now that I am seeing it really work over time on both Myrtle and on our oldest, I am about ready to drop Gelfand and everything else just to do that.

 

What's our pedagogical approach? We memorize and do problems. We memorize axioms. We memorize "scripts" for proving things. After they memorize a fair amount, it gives them a framework within which to figure things out. They rarely can just figure everything out on the fly the first time around. And, usually they need the other pieces that they haven't figured out yet in what they are doing for them to be able to do it at all. So, they just memorize that part and keep moving. By "keep moving", I do not mean move on to the next topic. I just mean keep trying to do what they are doing -- to do problem after problem until it starts to click with them. And, then the other part of this is doing the problems. In the end, there is one and only one thing that matters -- what problems can you do. Period. Don't kid yourself about any single other thing mattering, here, other than that. They have to do the problems themselves without coaching.

 

So, that's what you do and how you do it. Of course, it is possible to trade out some of these books for true alternatives. You might be able to trade out Allendoerfer for Docliani's Modern Introductory Analysis. You might be able to switch out Herstein for Gallian or Rudin for Bartle. If you switch out real analysis for a calculus text, you lose. If you switch out a new math text for non new math or a not sufficiently hard on logic and set theory program, you will not be able to touch books like Herstein or Rudin in a hundred million years (which, again, means that you lose). Also, just to be clear, it is not even remotely disputable what the subjects of modern mathematics are. The three main subjects, as any mathematician should tell you, are abstract algebra (e.g. groups, rings, fields), topology, and analysis (e.g. rigorous calculus). Topology is usually chapter 2 of analysis (as it literally is for Rudin). That's why it is sometimes given short shrift and not done all on its own. But, if you are looking for the book on that, it is probably the one by Munkres. (Again, there are alternatives like Armstrong and others -- for the these three, any math department or mathematician can easily recommend a book. But, make sure they do not know what your true intentions are. Just say you want an introduction to the subject at the senior college level and there are tons of books.)

 

No one does this inside or outside of America. Not the Russians nor the Japanese nor the Singaporeans. There is probably something to the fact that you will basically have to both tutor someone through it and really kind of "make" them do it in a way you just can't do with the general public. So, everyone else, at best, does a really good engineering math program. There are some pockets of Russian Math Circles or something else with a few very special and very motivated students that do something else, but even then it usually isn't the systematic training in math as its own subject. It is normally just some really good "real math"-type problems and the chance to talk with real (in many cases first rate) practicing mathematicians both of which are invaluable in their own right.

 

And finally, there is probably one realistic alternative to this: Euclidean Geometry. It is "real math", but it really is profoundly detached from modern math. Basically, everyone does analytic geometry in reality. However, the axiomatics of it and the antiquarian nature of it as well as its historical status make it something kind of special. So, in particular, these crazy ideas of "I'll just do Euclid with my kids" are actually not that crazy at all. After spending some time on the matter, my real recommendation is to go find a book from the 19th century that literally goes through the Elements and gives exercises. I'm kind of down on even Kiselev nowadays. Solomonovich is just Kiselev on steriods, and to really be able to do all the stuff he touches on, it takes a lot more than is there, I think. In other words, I just don't think the student is walking away having mastered things like proof by mathematical induction or even the idea of the Method of Exhaustion as Eudoxus conceived it or anything like that. You're just going to have a nice conversation with a mathematician that maybe inspires you to figure out what the hell he was talking about, anyway.

 

Birkhoff is, indeed, kind of like cheating or something. So are the SMSG axioms in the Moise and Downs book I have listed in The Answer, above. Actually, if you want to get technical, you need to use Tarski's axioms (not even Hilbert is good enough), but you are really missing the point at that point. So, just get a good 19th century text. (I was looking at this one in Google Books by Potts, the other day, for instance.) That would give you a truly classical education right there, and if you close your eyes and concentrate, you might, just for a moment, feel like you are actually standing in Plato's Academy over 20 centuries ago.

 

At any rate, there you have it. Mathematicans, math educators -- probably no one will tell you this. It is ludicrous to suggest the possibility of texts like Herstein or Rudin as well as some sort of gaff to act like they are even meaningful without years of calculus and differential equations. But, I am not exaggerating and I am not just making this up. I've even tested a lot of the most important aspects of it empirically. I guess I won't "know for sure" until I take a few kids with a variety of "special ed" issues all the way from start to finish through it all.

 

Now, give me your obolus or I'll beat you with my oar just like I do everyone else!

[Charon]

To me mathematics is more than a series of algorithms. It is a way of thinking.

 

 

 

This is no accident. The real math that graduate students and mathematicians do is something very unique indeed. It is the rare case of a formal a priori subject. Philosophy is, by and large, an informal a priori subject. Math and philosophy are, thus, sister disciplines. (Not math and science.) Also, while the debate between rationalists and empiricists continues (...I guess... :rolleyes:), the fact is that most of what is important -- most of the important and hard insights -- are a priori. They may or may not be "ultimately" tied to some sort of empirical "experience". It really doesn't matter. It is the getting from one step to the next that in some hard cases is what makes the difference.

 

That is why math is special. Not "math" -- like teaching textbooks or even singapore or really even Gelfand's Algebra book. But, math like what R L Moore does in his graduate classes as well as the math that is in the various Bourbaki texts. That kind of math, where the proof is more important than the theorem, the kind of math that people put up a million dollar reward for -- like the millenium problems -- that math is something that anyone should feel lucky for the opportunity to beat themselves up with.

[Kimber]

No payments yet, although I owe y'all big time. :) I've spent all of my coins on researching math texts.

 

Now I have to go and look up every author you've mentioned. I'm starting from ground zero with this stuff. I'm going to try Dolciani through to Modern Introductory Analysis. I think, unless we have a genius in the family that I'm unaware of and maybe even if we do, that I'll dual enroll them from then on.

 

I was blessed enough to find teacher's editions and solutions to the books up through MIA.

 

Last question, at least before I put the littles to bed :) , when do you exactly squeeze in the Logic books? Are they targeted for a certain age or grade level? Also, I did a quick search on line and the introductory text by Suppes has no solution manual. I've never had a logic course, so I think I need the answers. Are these books superior to a modern day math based logic text?

 

I plan to finish 6th grade math and pre-algebra over the next year with my dd. She'll be almost 11 by then. Math isn't really her strong suit, at least according to her.

 

 

Thanks,

 

Kimberly

[Kimber]

Jane, why did you switch text for Trig? I searched and saw that a few of these are still around.

 

Also, thank you for the response. I'm trying to teach my dd to actually start reading her text. Truthfully, we had the conversation for the first time this morning. But I know that I wasn't doing for the reasons that you mentioned. I'm just trying to get her follow directions.

 

We'll be facing Algebra in a year and a half or so. So I'm trying to do better job for high school than I did with my kids in elementary. I totally wasn't prepared. Thank God I know better.

 

Thank you for the info,

 

Kimberly

[Jane in NC]

Jane, why did you switch text for Trig? I searched and saw that a few of these are still around.

 

Also, thank you for the response. I'm trying to teach my dd to actually start reading her text.

 

Kimberly

 

On your second comment: In college math classes that I taught, it was such a pleasure to have the rare student at office hours not whine but ask for clarification on something in the text. Granted, learning to read a math book is made more challenging when the text is busy with lots of irrelevant sidebars. Extra fluff is contrary to mathematics as a discipline yet "fluffy" math books seem more the norm than the exception.

 

To answer your question: I had picked up this old "Modern Trigonometry" text at a library sale and had assumed that it was the same material that was in the Algebra II/Trig text since they were both Dolciani. (My new mission in life is to acquire cheap Dolciani texts and then proselytize.) When my son was on the verge of starting trig, I took a second look and realized that it was something completely different. Trig functions are usually defined via triangular relationships on the unit circle. This book began with the trig functions as periodic, circular functions. It is a slightly different approach.

 

Perhaps it is because Beckenbach is in the picture--I really don't know why--but there is a richness to the material that is lacking in the other book. For example when vectors are introduced in an Algebra II or Precalculus book, we see them as tool for use in physics. Any book will list properties of vectors, but this book demonstrates why the set of vectors is a commutative group under addition and a vector space when you throw in multiplication by a scalar. "So what?" you might ask. When you know that something is a particular structure, then you know how that structure should behave. Most people associate mathematics with getting the right number for the answer, but, as Charon will tell you, mathematicians seek to demonstrate existence and uniqueness of solutions, as well as develop a knowledge base of generalized structures and an understanding of the spaces in which these structures operate.

 

My goal for my son is to provide a foundation that allows him to see what mathematics really is, while also demonstrating how mathematics can be used. But neither my husband nor I wish to do the latter in the generally sloppy way that applications are taught in most math classes. I have no problem with students learning about physical applications in mathematics, but physical applications should not be the reason that we study mathematics. Judging from most modern mathematical curricula which lead to and terminate with Calculus, the study of physical change, one would think that this is the only reason that mathematics was created.

 

That was a long winded answer! Should I have just said that I like the second book better? A true statement.

 

Jane

[Charon]

No payments yet, although I owe y'all big time. :) I've spent all of my coins on researching math texts.

 

Now I have to go and look up every author you've mentioned. I'm starting from ground zero with this stuff. I'm going to try Dolciani through to Modern Introductory Analysis. I think, unless we have a genius in the family that I'm unaware of and maybe even if we do, that I'll dual enroll them from then on.

 

I was blessed enough to find teacher's editions and solutions to the books up through MIA.

 

Last question, at least before I put the littles to bed :) , when do you exactly squeeze in the Logic books? Are they targeted for a certain age or grade level? Also, I did a quick search on line and the introductory text by Suppes has no solution manual. I've never had a logic course, so I think I need the answers. Are these books superior to a modern day math based logic text?

 

I plan to finish 6th grade math and pre-algebra over the next year with my dd. She'll be almost 11 by then. Math isn't really her strong suit, at least according to her.

 

 

Thanks,

 

Kimberly

 

 

Myrtle thinks that the first book will be easy after Allen I. The second book naturally follows the first book. It is kind of redundant, actually. Theoretically, you could just do the second book ab initio. In fact, that is what that book is for -- just written to older students. The reason I wouldn't do it with a younger student is just because it is too fast and too hard probably to just jump into from scratch like that. Plus, the redundancy is pedagogically kind of a good thing when it comes to something like logic with a view to doing math. (You can never drill on too much logic.) So, the first book is just a way to get started, and, with a lot of overlap to the first book and ending with a lot of intuitive set theory, the second book is sort of where you want to end up. After that, jumping into Herstein shouldn't phase you one little bit, especially if you have done all of Allendoerfer's mathematical induction problems as well. (The last two starred problems in Allendoerfer, for instance, are to prove that Mathematical Induction is equivalent to the Well Ordering Principle. Also, several of the problems in Allendoerfer show up again in Herstein. So, this Allendoerfer book is awesome!)

 

At any rate, we plan to just do it on the side, concurrently with our math. And, we are going to start the first book yesterday! LOL. We can start the first book at any time, I guess. With the next child we will plan on starting it earlier, perhaps, or maybe we will just save it for concurrent with Moise and Downes. I really doubt it will take as long as three years to get through them both. And, Myrtle just said she wants to see what Allendoerfer recommends, so maybe we'll end up substituting one of those books. (But, I must say, I do like Suppes' second book, specifically because it is written with an eye for doing math and because of the part two and specifically the fact that part two is naive set theory.)

[Kimber]

Kimber,

 

Thanks Jane!

[Kimber]

Thanks Myrtle and Charon!