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How discouraging. No wonder kids hate proofs--they get tested on them without ever having a chance to learn them (OK--this isn't true of all schools/teachers). I love proofs and love teaching them, but there's nothing I know that can be effectively done by Thursday.
There's just one word for this comment: droll.
I love droll!
Speaking of what can and cannot be done by Thursday, if I hurry up and order now I could, by Thursday, have 5 or 6 or 7 books about proofs.
Any suggestions?
Also, does anyone know anything about Math Dictionary With Solutions by Chris Kornegay? It looks and sounds fantastic (way expensive, but fantastic - exactly what a person at my stage of the game desperately wants and needs).
27 comments:
I ordered several books that are used in college "bridge" courses. I wasn't able to work through them.
The dealio with proofs is that it isn't a "topic" in math, it is the methodology by which math is done. I can not remember the person to whom to attribute this quote but it went like this, "If mathematics were correctly taught, bridge courses would not be needed. Who knew?
Ultimately I found the most useful approach was to go back and relearn high school math using rigorous (in math rigorous doesn't mean what it does to everyone else, it means "proofy")textbooks.
And in the end my blog seems to be only about this one thing.
In summary, it looks like the best way to learn how to prove things is to know what axioms you are working with (bridge courses don't always clearly spell these out and any ole book which has proofs doesn't list them out either)and to have a good foundation in logic. The logic amounts to the rules for moves you can make.
Historically, the easiest axiomatic system for kids was in Geometry. See Mark Solomonovich's website for the low down on junk geometry.
Dolciani's Modern Analysis from the 1960s has a lot of proofs in high school level algebra and this textbook can still be found used.
The books for bridge courses may get you bogged down in set theory and group theory and you'll be faced with learning two things at once: content and methodology.
The other problem with beginners is not knowing when something has been proved or what counts as a proof (Say you are doing proofs in Mathematical Circles) and that is the $64 million dollar question in math. While a solutions guide may have one outline of a proof, it is in fact possible to come up with others. It's also possible to have something very similar to the proof in the solution but missing the detail that makes it work. You really need to have a person who can grade your work.
If you don't want traditional geometry a charming book is Beckenbach's Introduction to Inequalities, it is written at the high school level and it is a rigorous treatment of the properties of inequalities. I worked a few chapters in this book and got many pleasant "Ah ha! So that is why this works the way it does" moments.
Another high school level book is Patrick Suppes Introduction to Mathematical Logic and by the end of it you are proving some of the theorems used in ninth grade algebra.
Dolciani's Modern Analysis from the 1960s has a lot of proofs in high school level algebra and this textbook can still be found used.
I had no idea!
I've seen that book many, many times...but somehow I'd thought it was ---- calculus?
I should probably be mortified by the fact that I've written this comment, but there you are.
The other problem with beginners is not knowing when something has been proved or what counts as a proof
This is a huge problem for me, diving into the middle of things as I am. I was taught to do proofs in high school so the concept of a mathematical proof isn't foreign to me, which is good. BUT, because I have NO grasp of the axioms or of the theorems, I can't tell what I can use in a proof, whether something's been proved, etc.
Just as you say.
THANK YOU!
If you're still around, and don't mind a re-phrased form of the question: given the situation I'm in, where I'm going to have to get C. through a TEST ON PROOFS ON THURSDAY, are there "bridge"-type books that will help in this project.
Since I don't have a teacher who can grade my work (well, actually, I bet people around here could or would do that for me....) I need simply to read and study and then try to reproduce a lot of completed proofs.
I need worked examples.
I agree, absolutely, that this is not a good way to go about things.
But I'm not in a good situation.
OK, now I'm going to BOAST.
I found out, just by accident, that C's new school next year USES DOLCIANI MATH TEXTS.
I discovered this when I downloaded a copy of the school calendar and found, on one page, a large color photograph of a page in a math textbook that looked suspiciously like Dolciani's Algebra and Trigonometry Strcture and Method Book 2.
Sure enough.
The Dolciani text was photographed open on a desk, beside a copy of Virgil's Aeneid.
The Jesuits are going to have to change their motto.
from "Men for others" to "Last man standing"
I'm confused, and I have forgotten what grade C. is in. Is he taking 9th grade geometry already? Or is this a geometry unit in a muddly 7th grade spiral?
I don't remember encountering proofs until 9th grade geometry, and after that I never touched them again until honors calculus in college (which was not required for the engineering track that I was in).
He's in New York's Math A, which is an integrated 3-year sequence of algebra 1, geometry, and algebra 2.
Because this class is accelerated, he's taking all of Math A, normally a year and a half sequence, in 1 school year.
The algebra part of the course has been fine because the kids spiraled through so much algebra in 6th & 7th grades & I spent so much time giving C. extra practice (& trying to fill gaps) over the summers. He's really pretty good on literal equations; he completely gets how you derive the distance formula from the Pythagorean theorem; he probably won't have a lot of trouble with the quadratic formula (which I assume they'll get to this year & which I will insist that he practice deriving from the quadratic equation); he picked up the idea of and procedure for completing the square immediately when I taught it to him a couple of weeks ago...
I'm not the best judge, obviously, but I'd say he's in decent shape on algebra 1 topics.
But I've told him nothing about proofs (and don't know enough about proofs myself, although I did do simple proofs in high school) and neither has anyone else.
The topic was introduced for the first time on Monday & C. says they're having a "test on proofs" on Thursday.
I used that Dolciani book, Algebra: Structure and Method, as a high school student in the mid 80s.
It's nothing to write home about. Professor Wu rated it a D+ or a C-, I think.
But it is an actual book, with actual problems, and half of the answers in the back. It's the least bad of the options out there for general use, and orders of magnitude better than most.
Well, one often quoted statement is that the business of mathematics is proving theorems. So, you probably can't just "pick that up real quick" if it is, in some sense, precisely what math students spend years trying to learn the art of to become mathematicians. There isn't really a formula for it, and really knowing what it takes to prove something is almost what it means to be a mathematician and what mathematicians do as much as actually prove it at all -- establish the standards for what it takes to prove it (i.e. really establish that result). It's really kind of a philosophical question as to what it really takes to prove something and it takes a lot of that mysterious quality known as "mathematical maturity" (which is really just a kind of philosophical maturity/sophistication) to answer it.
At any rate, I doubt modern Dolciani or probably any modern in print text is that good for teaching secondary students to prove theorems. Something like Mathematical Circles or Gelfand's Algebra might have some good problems in that vein. But, you really need to go back to the 60s and get a New Math text for something that directly teaches students to prove theorems using modern techniques in mathematics. Essentially, every math major must go through that New-Math-esque training which amounts to learning a little naive set theory, mathematical induction, probably some general combinatorial reasoning, and epsilon-delta-style arguments.
Even 60s Dolciani is not all that great for that, either. We use Frank Allen (which you really cannot find which is probably why Myrtle suggested Dolciani). The book that is really good by Dolciani from that era is really that Modern Introductory Analysis that Myrtle referenced which was co-authored by Edwin Beckenbach. Edwin Beckenbach -- that guy has some really good stuff. The Introduction to Inequalities that Myrtle mentioned is one of the singularly best books that I can think of for presenting axiomatics to interested parents (and bright high school students). I posted something proofy a while back that was essentially lifted out of there (or at least the axioms I used were). It's so good because you really only deal directly with two axioms (though you implicitly use the full compliment of the normal field axioms which he takes for granted). So, it gives you a very manageable introduction to the axiomatic method. And, then you build up some typical tools you might use out of those axioms, and then you use those tools in the normal way to produce some significant results.
That's how math get's done, right there! But, it's all very manageable. And, if you compare it to the normal axioms of ordering, you even get to see why set theory makes everything better. Furthermore, no other book that I have seen, outside of Frank Allen and maybe this 1960s Dolciani Algebra II text we have, has a good treatment of order. When you get to inequalities, they typically show you a number line and try to heuristically discuss it that way (which completely begs the question). The only draw back to Beckenbach is that it really doesn't treat mathematical induction very well at all (and yet, he does use it at some points). But, not many books do -- Myrtle has had a lot of success with Alendoerfer and Oakley (another New Math alternative to that Dolciani text she recommended).
Pardon me while I write way too much about proofs:
It is hard to learn or teach proofs long distance because they are, like all forms of communication, socially constructed (see, I’ve learned something from being married to an English major). This means that to know what a proof is, when you know you have a proof, how much detail is required for a proof to be valid, and so forth, what you really need is to spend time creating proofs, and then discussing them with someone who understands the standards for proofs. The detail Euclid provided is different from the detail Hilbert provided, which is different from what a mathematician now writing a paper provides, which is only slightly different from what a college math major would be required to provide, which is somewhat different from what a high school student would provide. It’s possible to learn proofs from a book, and by studying examples, but its a lot more efficient if you can ask someone: do I need this detail? Am I missing something?
Proofs are always: using reasoning to get from things you know (axioms and previously proven theorems) to things you don’t know yet (theorems, or, more correctly, conjectures that have not yet been proven). [Exception: sometimes, with particularly intractable problems, you prove that if one conjecture is true, then another one has to be, even though you don’t know for sure if the first conjecture is true or not (there was a theorem that went into proving Fermat’s last theorem that was of this type)]. The only way you can know what axioms and theorems you are allowed to use in a geometry course is to consult your book/notes and see which axioms and theorems are given prior to where your problem/theorem/conjecture appears. No, theorems don’t have consistent numbers, and no, they aren’t always given in the same order in every book, so you have to use _your_ book to know what your allowed assumptions are. With rare exceptions, you can always assume that arithmetic and basic algebra are known and do not have to be proven.
Proofs always require that reasoning you use to get from the known axioms and theorems to what you are proving be sound and logically correct. A lot of the reasoning you use must be explained and justified. A safe thing to assume is that every time you make a statement, you have to give a justification for it (for example, if you say two triangles are congruent, then you should tell which triangle congruence theorem you used to conclude that). If you are doing two-column proofs this is mostly codified in the format: statement 1 on the left, justification 1 on the right, etc.. If you are doing paragraph proofs, you have to include the same information, but you have to write it in a sentence (eg. ______ is true because _______, by theorem_____.) This, is where the socially constructed-ness comes in. How much detail do you need to use in your justification: is it OK to just say SAS, or do you have to specify which sides and angles? Which format is OK for a proof: is 2-column OK? How about a paragraph? How you need to know about flow chart proofs? This is the stuff that varies depending on who you are and who your audience is. Sadly, my impression is that a lot of teachers are not particularly aware of the nuances of this--and it’s hard to be precise about just what you want.
Sadly, I don’t have a copy of Glencoe sitting on my shelf, or I could fritter away more time writing you a couple of relevant examples, but I do have “Discovering Geometry”, so instead you get an example from there.
This theorem follows the triangle congruence theorems SAS, ASA and AAS
Note, this problem also follows several similar proofs that were partially done, and students filled in missing steps.
Conjecture to prove: The bisector of the vertex angle in an isosceles triangle is also the median to the base.
Given (with accompanying diagram): triangle ABC with AC congruent to BC, and CD is the bisector of angle C (where D is a point on AB)
To Show: CD is a median.
Paragraph proof:
Angle ACD is congruent to angle BCD because CD is the angle bisector of angle ACB. It is given that AC is congruent to BC, and by the reflexive property CD is congruent to itself. Thus, by SAS, triangle BCD is congruent to triangle ACD. So, AD is congruent to BD because they are corresponding parts of congruent triangles, and thus CD is a median by the definition of median.
In real life, of course, this would have a lot more notation and fewer words (even in paragraph proofs, I would use the symbols for angle, triangle probably for congruence).
Notice that there are only a few real steps (this is a HS proof, after all), but that every step has a justification.
Good luck. I hope you find a good source of examples.
Regarding the social constructedness of proofs, in some sense the acceptable standards are always a matter of some sort of social convention. So, in other words, what level of detail is required to pass MATH 547 in college, say, vs your high school geometry vs getting published in the Transactions of the AMS are all different socially established norms.
However, there is an objective intellectual standard it is all based on. It is not a scientific one and not even a mathematical one, probably, but rather a philosophical one. It really hasn't changed much since the Platonists established it over 2000 years ago. Our social norms have gone way down, for instance, in Newton's day, when they were demonstrably insufficient, and have come right back up to what they are in this day and age (which I, personally, think is probably mostly attributable to Gauss and his mathematical descendents of which there are a multitude, including both Hilbert and Weierstrass, for instance). I don't think there is much of a difference between our notions of rigor in this day and age and those of Eudoxus, Euclid or Archimedes, for instance. We just know more now than we did back then. ("Rigor" often gets confused with "knowledge of foundations" which are two quite distinct things, actually.)
At any rate, I just wanted to say that while, on a practical level, one is generally faced with using social norms to guide them because it isn't a formulaic thing you can read about in a book, anymore than you could learn martial arts, say, from a book, what those standards are and the reason they are that way are not social conventions.
In order to get proofs students need a "mindset" change, especially since at primary school, you're not used to innovating new solutions or deriving new formulas.
Part of the problem as a primary school kid is that you think that adults have already discovered everything about mathematics and that there isn't anything new to discover -- it's just all being fed to you. Thus, if you were to be a mathematician, you would be basically working on old problems. That was the vibe I got as a primary school kid.
The book that changed my mindset in secondary two and made me conceive of mathematics no longer as a necessary evil but a field that sustained active research was a book called "Fermat's Last Theorem" by Simon Singh. Before that, I hadn't been aware there were unsolved problems in mathematics. [It was silly to think otherwise, but due to the culture that the adults pushed, I hadn't realised...) Also, the historical intrigue in which the knowledge of mathematics unfolded and evolved as bit by bit the required prerequisites of the theorem were proven suddenly made me a lot more interested in proofs.
So if your math class has to be like an English class, I definitely recommend that book for other students.
--, I hadn't been aware there were unsolved problems in mathematics. [It was silly to think otherwise, but due to the culture that the adults pushed, I hadn't realised...)
I had a similar misunderstanding of mathematics even as an undergraduate physics major at MIT. It wasn't until my senior year that I saw there were not only problems left to prove in math, but problems that had no known closed form solutions.
I was so used to being fed physics problems where you couldn't solve them in any closed form, and being fed math problems that were all number theory or topology statements of truth, that I had no idea there were entire fields of (applied, discrete, as well as more theoretical ones etc.) math where things were still not known, and active proving was still happening. In fact, I think I was at that time a math major and still did not understand, even though I knew kids going to math grad school, but it just seemed so unclear what that really meant was left "to do" in my head.
I think between now and a couple hours from now, you might be able to decipher whatever it is C's been told will be on the test, and that's about it.
Are they even doing two column proofs?
Coming to this thread late (desperately trying to revise my book) - Frank Allen is not findable.
Or wasn't the last time I tried.
VERY frustrating.
Are they even doing two column proofs?
It turns out I had no idea that what they're doing is a proof, so C. and I kept having miscommunications where I kept trying to get him to at least memorize the one two-column proof the textbook had and he kept saying "that's not going to be on the test" and then I would say, "What's on the test?" and he would say, "Proofs" and I would say "Who's on 3rd?" etc.
What the teacher wanted them to do was to demonstrate that four points on a coordinate plane made a parallelogram.
I asked a friend whether that was a proof & he said it was.
I had no idea.
I did do some proofs in high school geometry. I have no idea how many, or how much I learned -- none at all.
But I probably have some notion of what a proof is.
Proofs always require that reasoning you use to get from the known axioms and theorems to what you are proving be sound and logically correct.
He has absolutely no idea what this means.
I wish I'd had a video camera.
my god
It was the legally blind leading the blind.
I was trying to figure out what they were doing in the class, also what C. knew about what they were doing in the class, and I was saying things like, "Do you know what an axiom is?"
"What's an axiom?"
"You've never heard of axioms?"
"No."
Thank God I've read Karen Pryor.
It could have gotten ugly.
Having read Karen Pryor, however, I did NOT start shrieking, "What do you mean you've never heard of axioms and you're having a test on proofs???? Have you ever heard of theorems???? How are you going to prove stuff if you don't have axioms and theorems????? WHAT'S ON THE G*D TEST???
Someone was telling me about a Simpson's episode where Marge decides to start giving piano lessons. She doesn't know how to play piano herself (I think that's the story) and she reasons she can just stay a couple of lessons ahead of the kids and it'll be OK.
That's the situation around here.
I have one obvious problem with geometry proofs, which is that...I don't think I know the various theorems well enough -- and I don't "see" the possibilities when I look at a figure....it doesn't jump into my mind that I can construct a diagonal and voila I'll have two triangles and then I have 360 degrees because triangles have 360 degrees etc --- it's thinking too many steps into the future.
I should probably have learned to play chess if I was going to be doing this.
A safe thing to assume is that every time you make a statement, you have to give a justification for it (for example, if you say two triangles are congruent, then you should tell which triangle congruence theorem you used to conclude that).
That's the part of proofs that I LOVE!
A good friend of mine back in NAAR days (Natl Alliance for Autism Research), who was a scientist at Princeton, told me that every time he wrote a sentence in a paper he would ask himself, "Do I actually know this" and "How do I know it?"
That's pretty much how I write -- it's pretty close to every sentence.
I think this probably leads to a fair amount of dread when I'm starting a new project and am facing the prospect of asking and answering that question for 10,000 sentences or so.
I think Dolciani is still findable -- I'm pretty sure I tracked it down the other day before I got too many windows open and I had to Force Quit.
Various editions of Dolciani are findable, though sometimes it's hard to get accurate publication dates, but amazon.com lists several for under a few bucks. I just bought one myself, a few weeks ago to see what was in it. That was when I realized on looking at the cover, it was the same ed. as my high school book.
The major problem I have is realising which theorems the others build on and therefore can't use to prove my assigned theorem, when I decide how to attack an assigned proof. For example, you can't exactly prove that the limit to zero of sin x/x is 1 by using L'Hopital's rule, because the rule that d/dx sin x = cos x was proven using the fact that lim->0 [sin x / x] = 1 (using the definition of a derivative). So you would be in fact using circular logic.
I mean, after accumulating a good bunch of problem-solving techniques in your head, unless you take the history of math it's hard to remember which technique came first, especially if you've forgotten how some formulas were derived even though you've become familiar with them by working with them every day and that you know that if the formula weren't true, numerous other formulas built on that one wouldn't be true.
I think I could actually prove some of the congruence theorems using linear algebra for instance. But many linear algebra theorems rely on existing trig and geometry theorems.
"You've never heard of axioms?"
Ah, yes. This is the socially constructed part. What are the facts you are allowed to use? What are they called? This is most unpredictable without looking at the textbook. High school geometry books today often don't have axioms. Some of them don't call theorems theorems (I've seen some that are reasonably careful about calling such facts either theorems or conjectures, but that doesn't mean yours will be equally well explained). So how do you know what you can assume is true to construct your proof? If you are fortunate, then it's in the book, and its fairly easy to find. If you are less fortunate you have to guess a bit, and rely on the less than stellar memory* of your offspring.
*My offspring's memory can rarely be relied upon to hold any such information. Your mileage may vary.
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