I posted about my son skipping out of geometry earlier on this blog. (Thanks to all who commented!) Both my 8th & 9th grade sons
These were Wednesday's homework assignments. The first two pages are from the book: Geometry by McDougal Littell (2004 version). They were the 8th grader's homework assignments.
The second two pages are the first 2 of 3 created by the 9th grader's geometry teacher. The school uses the newest version of Prentice-Hall Geometry & Algebra 2 books. The pages were a take-home quiz and he could work with partners. Just the first 26 questions.
I see more challenging problems in the 6B Singapore Math workbook.
The older son starts Algebra 2 on Monday. He's already read through the first few chapters that he's missed and over the weekend we'll be working problems similar to what he did last year in Paul Foerster's Algebra.
(FYI- I didn't check my son's work, he did the assignment in 10 minutes in the car. He asked me to black his name out. - Anyone want to talk about years of failed handwriting instruction?)
"I see more challenging problems in the 6B Singapore Math workbook."
ReplyDeleteYikes!
I'm teaching my son with the Glencoe Geometry book. It's more like the first two pages, but I will have to check. We're off to a slow start - still dealing with things like p->q and truth tables.
After spending three days at the Realistic Mathematics Education Conference, it's hard to want to admit that there are still plenty of schools presenting math to students this way. The problems are dumbed-down and lack context, the classic symptoms of busy work. I'm not sure what the teacher expects to learn from the 26 problems in the second assignment that some teachers wouldn't already learn in 10 or fewer problems.
ReplyDelete"The second two pages are the first 2 of 3 created by the 9th grader's geometry teacher."
ReplyDeleteSigh.
My 8½ year old is doing problems similar to these (a *tiny* bit simpler, but not much).
9th grade geometry, you say?
:-( :-(
-Mark Roulo
The first two pages seem very reasonable. (A bit of hand holding in constructing proofs is probably worthwhile and possibly necessary for beginning geometry students.)
ReplyDeleteAs others have noted, though, the second two pages are awfully basic and repetitive, though.
But isn't Geometry just a matter of vocabulary and recognizing the attributes of the various shapes?
ReplyDeleteMath in the US - a survey course of concepts to be familiar with. Perhaps so they'll know what to google.
It makes you sad when you compare this with Abe Lincoln carrying around Euclid to teach himself logic.
"We're off to a slow start - still dealing with things like p->q and truth tables."
ReplyDeleteThose things are useful ... for theorems and building basic proofs, but after a while I think linear algebra is more useful. At that age, I feel truth tables are very useful only if you have a 9th grader designing functions or experimenting with circuits or something. (Or maybe the nervous system of C. elegans in a biology lab...)
Great comparison, Cassy.
ReplyDeleteThe proofs on the 3rd page of the 8th grader's assignment get even more intense...less "hand-holding". I looked long and hard for that textbook to use in our Geometry curriculum. It is not perfect, but I think it is an excellent resource for both teacher and student alike, while still being rigorous enough to challenge even the brightest math student. Lots of proofs, good skill practice problems, good challenging conceptual problems, not a lot of fluff. There is also extra stuff in there about interesting math topics like non-Euclidian geometry, topology, etc, that we have fun with during CSAP time. We are only in our 2nd year with the book in our curriculum, but I'm pretty happy with it so far.
Good luck to your 9th grader...he is always welcome back at his former school:-)
-T Love
Want see a horrendous Geometry text?
ReplyDeleteCheck out the website below:
http://aleph0.clarku.edu/~djoyce/java/elements/geotfacw.html
Happily I found this review because the administration of my school wanted to adopt this text. When I showed this review to my department head he forwarded it to the central office CAO. Another text was adopted as a result, but only after some heated discussion.
You would think the CAO would at least consult the the high school math teachers who have at least have some degree of expertise in the subject area.
I'm running into a problem with the Glencoe Geometry text I'm using. We're getting into formal proofs, but the book is not clear about what definitions can be used. They begin with "paragraph proofs" and start pulling definitions out of thin air, or at least from somewhere else in the book. Is there a formal list of definitions somewhere that I can print out? I only found a "partial" list online. How about undefined terms and postulates?
ReplyDeleteUgh. I haven't done proofs in ages and I want to do this right.
This comment has been removed by the author.
ReplyDeleteDefinitions are arbitrary; axioms (otherwise known as postulates) have mathematical content. The axioms are things that were assumed true without proof, upon which all of the theorems are derived. The definitions are just shorthand for the things you're referring to. So the way to think about a "definition" as the explanation in a 2 column proof is that it just restates something without increasing the expressiveness of what you've got.
ReplyDeleteDoes that help?
So while maybe there's a list of Glencoe definitions, they are truly arbitrary, and you shouldn't be using definitions from some other text, because they may not match up. (Though the obvious ones will, like an angle bisector. An angle bisector is just a definition of that ray that bisects an angle; you could keep calling it "that ray that bisects an angle", but it's easier to say "ray ACE is the angle bisector".)
There's no rule in Euclidean geometry as to which truths are the postulates and which are the theorems. Some things must be assumed, but whether you start with Euclid's 5 or a different set of 10, you still can eventually prove all of the same geometry.
I take it the back of the geometry book does not list the postulates (or axioms) ?
Hm hm hm.
ReplyDeletethis paper
says:
the ‘new math’ era developed
geometry texts which used Birkhoff’s approach of building in the real numbers....Faced with the inability of students to do (algebra) proofs, the publishers
reaction was to remove (geometry) proofs from the curriculum. [This] has resulted in texts which flatten out the geometry and destroy any notion that geometry is proved from a few basic principles. For example Glencoe [BCea05] has 24 postulates; these include SAS, SSS, ASA, and HL (if
two right triangles have a hypoteneuse and a leg congruent then the triangles are congruent.) Three of the postulates (the ruler, protractor, and segment addition postulates) tie the geometry to the real numbers so the axioms for the reals are a suppressed additional set of hypotheses. (The derivation of the other 4 congruence theorems from SAS (with no reliance on the mobility postulate) is routine and I think part of my high school education.) The difficulty of teaching from such a text is exacerbated by the difficulty of trying to instill intuitions for geometric notions such as congruence while being hamstrung by locutions which require one to say ‘the measure of angle A equals the measure of angle B’ to express equality of angles.
The use of 24 postulates is ridiculous. Euclid managed 5. My high school text had 10. We derived all of the triangle congruence ones from SAS I believe.
Testing 6*7=43
ReplyDeleteOK. That posted fine.
ReplyDeleteThat's odd. I just tried it again, but it didn't work. The post isn't that long.
ReplyDeleteHm, hm, hm, is right.
ReplyDeleteI found the list in the back; 24 postulates, divided up by chapter.
Thank you for the article. Although I specialize in geometry and topology (NURBS), I haven't done formal proofs since I was in 9th grade. My immediate problem is to deal with his textbook and the need to do the tests at the end of the chapters. In general, however, I'm struggling with how to teach my son so that it makes sense without having to use a parallel course.
The textbook seems to be trying to have it both ways; include formal proofs, but make it easier. I don't see the easier part because they seem to pull things out of the air without proper explanation. It's easy for him to understand a proof, but he has little knowledge of how to do one himself. He doesn't know where to begin. I need to figure out how to fix that.
This is the last part...
ReplyDelete"Some things must be assumed, but whether you start with Euclid's 5 or a different set of 10, you still can eventually prove all of the same geometry."
That's nice to know, but I guess I will be starting with 24 (Glencoe). However, my son already thinks that whenever you need something, you just define it or make it up. It seems like it would be better to start with 5. All I know is that I have to do a lot more preparation.
Thanks again, Allison.
I think I know what it was. I took out something that had quotes and the angle bracket to use for angle "a". I think it got hung up thinking is was some sort of html instruction. I just did it again and it gave me an error message. It didn't do that before.
ReplyDeleteSteve and Allison,
ReplyDeleteI don't mean to change topics much, but if you have a second, can you guys (or anyone--Barry?) give me reasons why proofs in geometry are important, reasons that a 15 yr. old math type kid will get?
My son is struggling in honors geometry right now because proofs are "stupid" and it isn't "real math." He has a tough teacher who doesn't spend a lot of time explaining anything, so he's left with his non math mother to try to give him a reason to try harder.
I just need a couple of decent answers because I don't really know what to say to him when he complains. He is more of an algebra type, but he has never had any problems with any geometry that's been taught to him before.
Any tips would be greatly appreciated....
SusanS
"...can you guys ... give me reasons why proofs in geometry are important...."
ReplyDeleteWell, since this is one of my pet crusades, I should probably reply. 8-)
A geometric proof requires that you identify your prior assumptions (called axioms in geometry), and then, using no other information, reason your way to a conclusion using transparent steps. What that means is that anyone who examines your argument can determine whether your priors are reasonable without proof and whether your individual logical steps make sense.
The formalism at the heart of these proofs, is, in fact, "real math" in a way that none of the rest of math before college is. Real math is fundamentally formal proofs.
Beyond math, though, formal proofs both require and help to teach a style of thought that is not taught anywhere else in primary or secondary school. Understanding this sort of rigorous proof is also key to making and understanding arguments in other areas of life.
Steve,
ReplyDeleteHowever, my son already thinks that whenever you need something, you just define it or make it up. It seems like it would be better to start with 5.
I think it's possible for you to help him by proving some of the assumed postulates in Glencoe. Maybe this is too time consuming for you, it may be too sophisticated a thought for him, but you could explain that the book didn't prove these postulates because they thought it was easier. And then show him it's possible to prove these postulates (and in fact, for euclidean geometry, you can prove anything from 5, and one of those (parallel lines don't intersect) defines planar geometry, and if you remove it, you get cool noneuclidean stuff).
But mostly it IS a big leap to understand why some obvious things need proofs, while other obvious ones are definitions. So the best thing is just to find the smallest proofs you can.
Beating home the difference between a definition and a postulate, though, is HUGE. He couldn't have done the proof without the postulate, but he didn't need the definition if he was willing to be very pedantic. That takes practice too.
When doing proofs, thinking backwards is the most helpful way to begin. Instead of forcing the thought of "I've got p, what qs does it imply?", thinking about "what ps are sufficient to get to q?"
To prove all of the angles in a triangle sum to 180, you think about what other things you know sum to 180: two angles defined by a line and a ray on that line (or the intersection of two lines). And then you try to see how given you've got a triangle, what lines and rays you could have defined whose sums you know are 180; and then you look and see you'd need to prove congruence of certain angles with others in the triangle, etc.
Susan S,
ReplyDeleteI will follow up Doug's comment with a couple examples later tonight. But maybe your son thinks proofs aren't real math because they gave him 24 postulates too.
Opening up Euclid's work, or Apollonius', or Newton's, can really show how much "real math" is in a proof. So can number theory. So geometry may not seem "real" but proofs should. Let me think of books that might help.
Steve,
ReplyDeleteThis might be a time for you to read Polya's How to Solve It. He has dialogues with students in there, and he shows what to say to help them get started thinking about what they know and how to use what they know.
Also, part of the reason I've found kids have difficulty with proofs is that the first proofs in Geometry that students encounter would have to be proven by "reducto ad absurdum" or proof by contradiction. The old saying, "it takes a while to make progress" tends to confuse and sometime frustrate the students, and definitely slows down the pace.
ReplyDeleteFor example:
POSTULATE: Thru two distinct points there is exactly one line.
From this postulate you can prove:
THEOREM: If two distinct lines intersect they intersect in one point.
This theorem is proven by supposing two lines intersect in two points and arriving at a contradiction to the given postulate. Reducto ad absurdum.
It is an exercise in logic and if written, is in paragraph form, not in traditional two-column format.
Some geometry textbooks will avoid this will and go so far as to classify this theorem as a postulate!
Thus textbooks will have a different number of postulates.
I'm also dealing with what Susan is talking about, but I don't think justifications will make him happy. What my son wants is a carefully laid out path of explanation and practice. There doesn't seem to be one. He can follow the logic in one proof, but he sees little connection to how he should approach another proof.
ReplyDelete"When doing proofs, thinking backwards is the most helpful way to begin. Instead of forcing the thought of 'I've got p, what qs does it imply?', thinking about 'what ps are sufficient to get to q?'"
Great advice! However, this seems like a chicken and egg skill.
"it takes a while to make progress"
I'd like to find ways to minimize this sort of thing. I think I will have to force the issue now or else it will be a bigger problem later on. I suppose that the authors of the textbook feel that they are making life easier with more postulates. I don't know about that.
Let me just say again that my son won't dispute that this is all important, but he just wants a clear and proper explanation. His frustration comes not from thinking it's stupid, but from the confusing explanations. How much of this goes with the territory and how much is just bad explanations? I have to get back up to speed and not just rely on the textbook. I wish I had all of my old textbooks.
ReplyDeleteThanks Doug,
ReplyDeleteTying things into logic always helps with him somewhat. He still has your logical fallacy bingo on his board. It made grade school bearable.
I'll have to check what textbook he is using.
I'm going to cut and paste your responses for him. Maybe something will click. Oh, why can't you people live on my block?
Susans
Steven said...
ReplyDelete"I'd like to find ways to minimize this sort of thing."
Euclid said, "There is no royal road to Geometry."
You're right. Authors try to simplify things by making more postulates.
Okay, my son actually has no geometry homework for the first time ever, so he left his book at school.
ReplyDeleteI asked him if he remembers any of the postulates in the back of the book and he said, "Yeah, there's around 50."
That can't possibly be right, can it? He's exaggerating, right?
SusanS
SusanS: "That can't possibly be right, can it? He's exaggerating, right?"
ReplyDeleteThere's no upper limit on axioms, since axioms are the rules you assume without proof. That said, all of plane geometry can be derived from the five Euclidean axioms:
1. A straight line can be drawn between any two points
2. A finite line can be extended infinitely in both directions
3. A circle can be drawn with any center and any radius
4. All right angles are equal to each other
5. Given a line and a point not on the line, only one line can be drawn through the point parallel to the line
Normally, you start with this limited set of axioms and then start proving theorems. Once you have a formal proof of any theorem, you can use that theorem as a unitary step in any future proof. Since theorems have fairly standard names (which might vary by teacher), it's possible that he's thinking of some standard list of common theorems as a list of axioms.
Or maybe the book is really bad. 8-/
SteveH: "How much of this goes with the territory and how much is just bad explanations?"
ReplyDeleteMuch of it goes with the territory. Each theorem proof is a logic puzzle that must be solved with the available tools. When the teaching is well done, you should need only a limited number of insights (preferably one) for each new theorem, at least near the beginning of the sequence, but it still comes down to flashes of insight when you're working the proof. Until that flash happens, you will likely feel like you are stumbling around in the dark, but when it does happen, the feeling is pretty great.
The reason I think this is so important is that this same process is core to most creative jobs. There's really no cookbook way to write a new computer program, for instance; you figure out what you have to start with (information, clock cycles, whatever) and what you need to end with, then you fill in the middle. You might start at the end and work toward the beginning, start at the beginning and work toward the end, start in the middle or both ends and work both ways, or do something even more complex.
The other bit about this process that I think is useful to understand is that there are often multiple valid ways to arrive at the same result. One is likely to be more efficient than the others, but many may be equally correct. Again, this is like many non-trivial problems in the real world.
SusanS: "He still has your logical fallacy bingo on his board. It made grade school bearable."
ReplyDeleteOK, that's funny, gratifying, and depressing all at once.
Bonus!
8-)
Thanks Doug.
ReplyDeleteOne of the problems is that I'm not back up to speed on proofs. The second is that the book starts pulling things out of thin air. (or seems to)
In section 2-5 on Postulates and Paragraph Proofs (they start with these), they talk about postulates or axioms like:
"2.1 Through any two points, there is exactly on line."
and
"2.2 Through any three points not on the same line, there is exactly one plane."
The very first proof they do asks:
Given that M is the midpoint of PQ, write a paragraph proof to show that PM (is congruent to) MQ.
They do this by evoking the definitions of midpoint and congruence. My son wants to know what list of definitions he can use. I told him that since the problem uses the terms "midpoint" and "congruent", he needs to dig up those definitions. He still wants to see a list of definitions he might need to use.
My son can easily follow proofs, but he is lost when he has to do one himself. He can deal with issues of trying to figure out an answer, but he doesn't know what tools or skills he needs to have to do that. He wants to see a list of what he can or can't use to solve a problem. This brings up the issue of context. You can't just pull something out of the air and say prove it (for students, anyway). They have to know where they are in the book.
I feel like I need to provide him a starting list of definitions and then start adding postulates and theorems (from the book, of course) in sequential order. By chapter 4, we are up to 16 postulates and 22 theorems. Perhaps I should have him put them in a notebook so that for each new proof, he can see what tools he has. Perhaps I can have him sort them into different categories (e.g. lines versus angles).
This is the sort of thing that I'm trying to figure out and explain to my son; that the proof might not be obivous, but that shouldn't hold true for the process.
OK. What's the big deal about "congruent" versus equal? Is this a way to formally relate shape (geometry) with a number?
ReplyDeleteWhen I was in 9th grade, I probably thought "whatever", but now I'm trying to figure out what's really important and what level of understanding my son needs to have. It's one thing to learn logic, but another to "get" what's going on. Perhaps, I don't get it yet after all of these years, but then again, I'm an applied guy.
I'm working through an old Dolciani Geometry text, and that's exactly what I've done. I typed up the properties of real numbers, the postulates (22), and theorems (67, with proofs). I refer to them a lot. I wish the book had a list in the back, but it was probably a good exercise to make the list, anyway.
ReplyDeleteHere is the deal with "congruent" versus "equal". As always, the notion of equal means "are the same within a category" So 1 and 2/2 are equal as numbers (but not as strings of symbols).
ReplyDeleteWhat is an angle? Two rays that intersect at their endpoint form an angle. Suppose angle A and angle B are equal. What does that mean? Well, angle A has two rays; angle B has two rays; but if the angles are equal (the same angle) then each of angle A's rays is in fact one of angle B's rays. So the total number of rays is 2, not 4. The angles are the same angle.
You can discuss congruence by way of measuring angles. Two angle are congruent if they have the same measure (in degrees or whatever). You can also discuss congruence in terms of translations, rotations, and reflections. Suppose you have two distinct angles (they are not equal -- we have more than 2 rays!). If I can think of picking one up and setting it on top of the other so they all match up, they are congruent.
It is easier to start by thinking of something like squares. Draw two squares, one over here and one over there. Make both squares have the same side length. They are congruent. Are they equal? No, they are not the same square, since one is over here and one is over there. They are two distinct squares (but congruent). Likewise with angles.
I wonder whether the reported proliferation of postulates might not contribute to the difficulty. Beyond some basic set like the Euclidean set, all the rest of geometry is derivable, and thus should be theorems, not postulates. When you don't do those proofs, the process of proving later theorems might be less understandable.
ReplyDeleteFWIW, having a list of known postulates and theorems might be useful. You can examine each one and see whether you can apply it to the instant problem. If you can't apply it, it's not useful, so at least you can eliminate (or at least delay using) some large part of the available tool set.
"whether the proliferation of postulates might not contribute to the difficulty"
ReplyDeleteSomething is difficult about the textbook, but I can't quite pin it down. After learning math and technical material for many, many years, I have a sense (not always correct) about whether there should be an easier explanation. I'll have to let you know after I'm done with the course.
"having a list of known postulates and theorems might be useful"
I think that I need to have my son do this himself. I will have him fill an empty notebook only with the things that have been defined or proven. My feeling is that the sequence and process of filling the notebook will be critical.
It seems to me that he can't just look at a list in the back of the book; that it matters where he is in the sequence. This brings up another question. For a chapter test or a mid-term test, when a student has to do a proof, is it clear what can be used and what can't be used? In other words, do students know where they are in the sequence?
"They are congruent. Are they equal? No, they are not the same square... "
ReplyDeleteBut 5 = 5? I can't have two separate 5's? What is the significance of congruence? What about other types of transformations, like shearing or scaling? Are those different types of congruence?
Maybe it will be easier for my son when the textbook gets to a level where he can use theorems that don't cause him to say "whatever", but that's not my goal. That's what I did in school.
It seems that the lowest level theorems are the most critical (and difficult) for understanding, but those are the ones that they get first or are assumed.
What's different about geometry? We don't do this for algebra, although his textbook has a section on algebraic proofs. (That seems to only confuse the issue.) You develop a lot of logic when you get good at algebra. There is no one way to move terms around to solve an equation.
So, is geometry being used as a vehicle for learning formal logic, or is the goal much deeper. What is that goal and how does it apply to other areas of math?
My concern is that the textbook I'm using is neither here nor there. They go through the motions but no context (other than logic) is applied.
While it has been a long time since I took Geometry (the Ford administration 8-0 ), IIRC, you could use any postulate or any theorem you had previously proven to justify a single proof step. (Something of a subroutine call, as it were.)
ReplyDeleteIn fact thinking of geometry proofs as modular programs might be a useful analogy. Postulates are analogous to the basic syntax of the language* and proven theorems are subroutines.
* Particularly a language like Forth, if you have any experience with that.
"I can't have two separate 5's?"
ReplyDeleteNo, SteveH. There is only one five. We are all using the same five. Over and over again. There are many marks on paper, however, and many of these have represented the number five.
Congruence has only to do with rotation, translation, and reflection. Intuitively, can you pick this one up and set it down on that only and they completely line up? Are they copies of each other (but possibly distinct as sets of points)?
Add dilation and you get similarity; add shears and you get some other notion of equivalence which you might be interested -- it would classify ellipses and circles the same but different from triangles, different from quadrilaterals, and so on. Does this equivalence have a name? Not that I know of. Equality, congruence, similarity, are all equivalence relations, and in that order they allow larger and larger equivalence classes.
Is geometry different from algebra? It's not just the emphasis on proofs. For one thing, you could push all this emphasis on proofs down to beginning algebra. Then you would be teaching what in college is called "abstract algebra" to people who don't even know "high school algebra" yet. You could do that with a tiny fraction of the population, and for the rest of us it would have been a fruitless exercise.
In algebra you can manipulate equations. In geometry you have diagrams. Here is the tricky part. The diagram is not the real thing. Ultimately, nothing is "to scale". The diagram helps organize the student's reasoning, but the reasoning is supposed to result in a series of statements backed up with previous results. Algebra is not like that. The equation or expression you are working with is not a diagram of some platonic ideal, it's the real thing. This might be where your son's difficulty lies.
"Postulates are analogous to the basic syntax of the language..."
ReplyDeleteI think it is more accurate to think of postulates as the built-in words (or functions) you get. Theorems are like the words (or functions) you build out of the built-in words (or functions) and other functions you have already written. The syntax is more like the rules that you can use when doing your proofs.
-Mark Roulo
"...it would have been a fruitless exercise."
ReplyDeleteI guess some are claiming this for geometry too.
But are you saying that there is nothing special about learning proofs with geometry except that it is the easiest vehicle?
My worry is that the textbook we are using is neither here nor there. It has to be more than an exercise in logic. If they are not formal enough with the process, then why bother at all? It has to be more than proof appreciation.
Well, I don't believe in platonic ideals and geometry and algebra are still both okay for me.
ReplyDeleteThere is nothing wrong with saying that two congruent triangles are "the same" triangle.
They are the same. They may have different names, but they are the same triangle.
Same for congruent angles, shapes, rays, whatever is congruent.
The morning star is the evening star.
Congruence is about picking up the triangle and moving the whole thing in such a way as to fit cleanly on top of the new one.
Think tangrams. This is what they were invented for.
The "picking up the whole thing" is the set of operations that admits rotation and translation.
Shearing or scaling is different. They do not form the same triangles. Those operations deform.
Similar triangles are related by scaling. You can define this scaling if you can define a point on the plane (somewhere on the plane), and you scale EVERYTHING on the plane relative to that point. Then you will see similar triangles.
This is a place where Geometer's Sketchpad can help you (though probably not your son until you've done the exercises yourself and tightly controlled them...)
Shearing is another kind of transformation. It's not worth thinking about just yet.
--So, is geometry being used as a vehicle for learning formal logic, or is the goal much deeper. What is that goal and how does it apply to other areas of math?
ReplyDeleteThere are two goals of a good geometry class:
to teach truths about the world, truths which were subtle and important and could not be known at the time through exhaustive search or modelling yet were knowable thousands of years ago;
to teach real, formal mathematics.
You should think of the existence of geometry in high school as an anachronism that came from a time when "algebra II" in high school was a SUBSTANTIAL course, so substantial that it required sophisticated knowledge of math and proof.
Why? Because the geometers --Euclid, Appollonius, etc. knew EVERY SINGLE THING we know about algebra through geometry. EVERY SINGLE THING.
Factoring? Derived geometrically. quadratic rule? They derived it geometrically. Solutions to cubics? Done geometrically. Everything you know about vectors, parallelograms, parallelpipeds, ellipses, orbits? geometrically?
Everything you can prove using integration --areas under curves, volume of a cone or sphere, etc.? Done geometrically.
EVERYTHING. Any set of the Great Books pre-2006 has volumes of Appollonius. Go skim it and be astonished.
--My concern is that the textbook I'm using is neither here nor there. They go through the motions but no context (other than logic) is applied.
There really was no reason for us to believe that high school was any better than K-8, was there? And yet, we are constantly disappointed.
Find a better book. Maybe you can find some way that isn't you teaching two courses in parallel...
--What's different about geometry? We don't do this for algebra, although his textbook has a section on algebraic proofs. (That seems to only confuse the issue.) You develop a lot of logic when you get good at algebra. There is no one way to move terms around to solve an equation.
ReplyDeleteThere is no one way to prove a theorem in geometry, either.
We don't do two column proofs in algebra because a) once you're familiar with algebraic manipulation, you don't need to justify your proof step with a "reason", and b) hs algebra is watered down awfulness, mostly.
If you were really good at geometry, having taken pre-geometry for two years and learned all about congruence and similiarity, having learned a hundred constructions using straight edge and compass, you would know all of the definitions and rules already. You wouldn't need to formally write down a "reason" for that step in your proof, because it'd be obvious where your midpoint was defined, what an angle bisector is, that you could use your prior theorem "congruent parts of congruent triangles are congruent" off the top of your head, the way we use the distributive property.
But of course, our kids don't even understand that we can DO the manipulations we do algebraically because of a REASON, a reason that's true for all time and space. So maybe what you're saying really points to the idea that algebra should require two column proofs at the beginning, too.
--If they are not formal enough with the process, then why bother at all? It has to be more than proof appreciation.
ReplyDeleteHas to be more than that to WHOM?
You're not suggesting that the current authors of Glencoe's book, or the math ed folks out there, have a clear vision of what they are teaching geometry for, are you?
SteveH -- is there anything special about geometry for teaching proofs? Well, you could use another subject to teach proofs. For example, you could start with field axioms and then use this to prove things about arithmetic and algebra. Or you could use simple propositional logic as a tool to teach about proofs. Ideally a curriculum would introduce the idea of proving things in measured doses so that by the time a student is learning geometry they don't have this confusion about whether proof is a characteristic of geometry but not of mathematics as a whole.
ReplyDeleteHaving said all that, there is still something different about geometry. There is a diagram, but the diagram only guides reasoning, sort of as a memory aid. In proving things about points and lines, it is helpful to recall that points and lines are not always what we think they are but the theorems derived from axioms will apply to them also. It might be helpful to introduce basic examples from non-euclidean geometry early in the curriculum so that it is clear that inspecting a diagram is not enough to ascertain truth -- you need to keep in mind what axioms you have.
Allison -
Is there are anything wrong with conflating the terms "congruent" and "equal" (or "the same")? Precision in language is a good thing in general but not always crucial, so sure, people will say "these squares are the same", like they will say of two Honda Civics, "our cars are the same". They are the same! But there are two distinct cars. Because they are not equal in the strict sense.
Objects in geometry are sets of points. Two objects are equal if they are the same sets of points, otherwise they are not equal. This precise meaning is sometimes important. Often not. But there is nothing to be gained by asserting that all equivalence is equality.
"This is a place where Geometer's Sketchpad can help you ..."
ReplyDeleteI write and sell a CAD program that specializes in NURB curves and surfaces. My son uses it and has no troubles with 2D or 3D shapes and transformations. It allows him to dynamically apply transformations to the geometry on the screen.
"to teach real, formal mathematics."
This is what I'm concerned about. The problem is that I don't think I learned it properly when I was in school. The other problem is that I really have to make the textbook work. It's what the high school uses and he has to take their tests. I think it's close enough, but I'm trying to fill in the key ideas.
"So maybe what you're saying really points to the idea that algebra should require two column proofs at the beginning, too."
Yes, but maybe not as formal as in a proper geometry course. It would make the transition to geometry easier for them.
"...have a clear vision of what they are teaching geometry for, are you?"
He, he, he. No. The immediate problem is me. I need to figure out what key ideas and skills I need to teach. I like your comments on how historically, everything has been derived from geometry. I think I will pursue that angle with him. do you have some link or source that provides an historical perspective?
Other KTMers probably have better ideas. I've read a lot of popular math books, but I didn't actually make this realization until I (while unemployed) decided to do Newton's Principia myself. That is, prove all of the proofs in order.
ReplyDeleteI was immediately overwhelmed. The things Newton already knew geometrically from Euclid, Appollonius (he did the work in conic sections), etc. were so sophisticated, so I went back to read those. the more I read, the more I realized they knew.
In all of my years in school, in ugrad (with a math degree) and grad school (in theory CS), in all of my reading of popularized math books, I had never had any idea that the Greeks had done so much math. I was shocked how little I knew.
I think reading a cleaned-up Euclid yourself would be beneficial. There must be some good popularizations out there of the geometers. I will look through what I own to see...
Because the geometers --Euclid, Appollonius, etc. knew EVERY SINGLE THING we know about algebra through geometry. EVERY SINGLE THING.
ReplyDeleteI'm stunned. Really stunned. Wow.
So, being someone who's generally pretty bright and well-educated, but hasn't gone that far in math (Calc II in college), and has a son who will likely go further, is there anything I can/should do to get more up-to-speed to guide his development? I feel we've done okay by making sure he has his facts down and getting him involved in problem-solving early, but I'd like to know that when he gets out of the more trivial procedural math I can still be of some help in guiding him.
- Sam