I finally got my class schedule today (yes, class starts in 6 days), and apparently I'm teaching 3 Geometry classes. Which is fine, but all summer I was told I was teaching only Algebra and Algebra was all I prepped for...so I'm trying to pull a lot of things together quickly! I've never taught Geometry before, and am mostly wondering if anyone has any great ideas on how to keep all those postulates/theorems organized, understandable, and easy to recall at the needed times. As a student, I personally struggled with formal proofs, so I really want to find a way to make them more understandable for my students. So, Geometry people: any advice? Do you let them use postulates/theorems notes on quizzes and tests, or do you make them memorize them? Any help in anything related to teaching geometry will be VERY VERY appreciated!!

Stick to the books and plan by the week to stay on track. If you're doing McDougall-Littell, you have absolutely nothing to worry about. Proofs don't show up until congruent triangles. This subject has been watered down so much in the last few years, it's barely recognizable anymore.

I just sent you a private reply, through your file on this website, because I realized my reply was too long (smile!)

This is the first year in several that I'm not teaching geometry. Instead, I'm teaching 6th grade math. But when I did teach geometry... I never made the kids do them from scratch. Instead I would give them proofs with the statements and reasons scrambled, or with partial blanks (sometimes with a word bank).

I'm Brazilian. We're a week in. The first things to do are to introduce the undefined terms (point, line, plane), and explain the differences such as rays, segments, etc.. My dept. wants us to go right into the distance formula, which I think is pretty wierd, but whatever. The classic first week has them drawing the diagrams you write out in words, i.e. line AB is collinear with C, but not with BD. Draw ray BD so that E is between B and D, etc. Also go into the deeper definitions - lines are collections of infinite points, planes are an infinity of rays, maybe show how you can define n-space through these definitions if you really want to scare them. Also review the intersections: two lines create points, two planes create lines, etc.

My professor on Plane Geometry really taught us well. She gave us ideas on how to teach geometry to our students. What she did was she used simple words and terms to make us understand the lessons.Even slow learners can quickly keep up with the topic. It will be more helpful if there are fill in the blanks in the proving statements since this is easier. You have to explain the flow of the proving process, item by item. Postulates and theorems notes can be used during major examinations but you have to make sure that the students can explain each postulate/theorem just be reading at each statement.

Honestly, I think you're looking for a shortcut where none exists. The kids need to KNOW the theorems and postulates. There's no way around it. I don't make them know the number, or even whether it's a theorem, postulate or definition. And they're allowed to paraphrase, as long as they get all the info in there. (in fact, after I give the textbook explanation, I always include a paraphrased one of my own.) Additionally, they're allowed to say "definition of perpendicular" instead of actually citing the definition. But beyond that, they really do have to study.

I never said that there should be a shortcut - I was just looking for advice on the best way to have them remember them and know when to use them. *Luckily* for me, proofs were introduced in the 2nd chapter, so I've already been teaching them. I think some of the kids are getting the hang of the things now and most importantly, I understand them now - which is something I never quite did when I was in school! So I feel much more confident of the things and I think it shows when I'm trying to teach the kids. I've started off with a fill-in-the-blanks type of thing, so that they understand how proofs work. Slowly, I add a few more blanks, and I now teach the proof with every new theorem introduced. I do make the kids write notecards for each theorem/postulate and I think I'll let them use each new chapters notecards on that test, but then after that, they'll have to learn them...

Don't be surprised when some smart kids simply struggle with proofs. It's a totally different kind of "smart" than most kids are used to. In fact, some kids who have always struggled in math are going to find success for the first time in geometry, and wonder what all the fuss is about. For those kids who do struggle: take a look at your syllabus. In my school, the proofs REALLY slow down after about Christmas, once we get past the quadrilateral chapter. And once you hit the similar triangles chapter, the proofs are easier to do from the bottom up-- lots and lots of partial credit even for those kids who struggle with proofs. Teaching geometry means you need to be part teacher, part cheerleader. It's a combination of cheering on those who get it right away, and persuading the others that it will click for them too. Please ask any questions that pop up. As I said, I LOVE geometery

That was ME! I always had a tough time with math despite having some really great teachers and working with a tutor one-on-one all through middle school. But geometry just made SENSE to my right brain! It was awesome. I learned later that reading and geometry generally use the same parts of the brain, so it's not unusual for "language" people to do well in that subject.

We have had so much trouble with proofs that the dept. chair has instructed us only to do them until chapter 6 (half way through term 2).

In my department, they're trying somethng new this year--pushing proofs off until the spring. They want the kids to have better background knowledge on the geometric figures, as well as some success in geometry, before introducing the more difficult material. (Plus, from a practical standpoint, it means that teachers can include some of the easier stuff as review questions on tests, and keep the grades at a decent point.) I'm not teaching it this year, but I'll let you know how it goes.

Hmm...thats intresting. Keep in mind though the proofs will be harder becuase they are with the harder material in the later chapters.

As I understand it, they won't see proofs at ALL until after Christmas. They'll still see many of the theorems, particularly those dealing with triangles, but they won't apply them to proofs yet. Then, when they do start proofs, they'll go back to the more elementary ones and build from there. It was an idea that had merit; whether or not it's workable is still to be seen.

It's so funny that you say that. I don't personally remember much about geometry because it was so long ago BUT at the time I took the class I remember thinking this is the best Math class I have ever taken and it was easy for me. I'm used to struggling in Algebra type classes.

Strange, isn't it?? It's a totally different kind of thinking. People who do well in geometry tend to be the same kind who like jigsaw puzzles-- it's that "pulling info together from different sources" kind of thinking. But it really kills those kids who are used to following a procedure from point A to point B to point C. Some of them really struggle in Geometry, no matter how much they study.

Does anyone use the McDougal Littell book from the early 90s/late 80s? It either is a short stubby blue book or a regular sized text with a white/grey cover....

No. My school uses the same textbook I used in high school (and that was during the Nixon presidency ) the black Amsco Geometry text by Dressler and Keenan.

I have a legitimate question fueled by ignorance. Do you even have to teach proofs? I took geometry in high school (and finished with a 110 average) and the only time my teacher ever mentioned proofs was in the beginning of the year when she said that problems like x-y were proofs and we wouldn't be doing them. So long as you teach the rules and the students get it, why should they have to 'prove' them? I promise I'm not trying to be argumentative! I honestly don't know and would like to hear from someone who knows more about this than I do

It depends on the course you're teaching, probably as well as the state guidelines. I'm in a private Catholic High school. We want the kids to know proofs-- to understand the reasoning that takes you from point A to point B. It's not so much about knowing how to prove a figure is a parallelogram. It's about knowing how to take separate pieces of information and combine them in order to reach a logical conclusion.

background... A lot of the algebra-in-geometry debate stems from the progression of the subject through the years. The basic level, which is still seen as gospel by the pros, is euclid. 5 axioms, collapsable compass constructions, and no units. Over the years, people incorporated things like the ruler and protractor postulates for applications. Then when testing became an issue, they put algebra in there to keep the skills alive, especially after the wierd majority decision (which I still don't understand) to plunk geometry square in the middle of Algebra 1 and Algebra 2...even though testing proves that students do better with the algebra years being contiguous. When I learned geometry, the algebra was in there as an afterthought, usually 10/50 points on each test. These days I see it front and center in most department tests, and proofs are fill-in-the-blank at best. Thanks to my dept head's notes, I can get away with proofs while teaching, but the students are still not expected to complete them from start to finish. Another nice thing is that logic gets a chapter on the editions of these books, and receives a fair treatment after some supplements. This is actually the one thing I want students to take out of the course, actually. The reasoning is really what you're trying to get at in these courses; the material is just there as an aide to deliver that instruction. Never "put down" the basic ideas of proofs being conditionals, biconditionals, conjunctions, etc., and that the arguments are basic logic. Use two column proof, but also expose them to paragraph and flow-chart proof as well (most importantly for beginners). Word banks are good for a lot of these reasons. Now that I'm beginning to learn what I can get away with at this place, I'm doing a lot more of this.

More about the necessity of proof comments: A lot of the beauty of mathematics as a subject is how few assumptions can be put together to show some not-so-obvious facts. That's why Euclid is still a favorite with his 5 axioms (and you really only need 4, since 5 only works on the plane; not spheres, hyperbolic surfaces, projective planes). Same goes for algebra...although its grade level excludes the proof, taking a rigorous algebra course, structured according to the different number systems, assumes practically nothing and leads right into tools so powerful, you can see why the stuff is the language of the universe. If it weren't for standards and dull problem sets, I would teach it this way. Unfortunately, everyone wants to get "the answer" without knowing what's behind it, and misses the point of the entire discipline. We always tell students what the pros do - the writer's job is clear, the psychologist's is clear, etc., but few understand that mathematicians don't crunch numbers; they actually write these theorems! About 30% of the known mathematics is actually USED. As science progresses, it always seems to use more, but there the progression requiring nothing but reason is pretty impressive. I'm thinking about group theory from abstract algebra used in quantum mechanics...learned as a prerequisite to most math majors! Most math teachers are bad, because we are forced to be. Even at the community colleges, it's frowned on if we don't just "give them the formula," yet when people fail to memorize the formula, and bomb the tests because they can't DERIVE it or pull it together from the underlying concept, they assume that this straw man that they're calling mathematics is boring. It's disheartening, because most other subjects actually TEACH THE SUBJECT. Mathematics is always under the gun to perform, so we shortcut the subject to death.

sensijoao--- it is so near to hear someone who actually still values proofs!! My HS is one of the very few left in the area that still teaches rigorous proofs without any of that fill in the blank stuff.....and I think our students really learn a lot of reasoning and logic as a result. We also were the lone school in NJ to teach Alg 1 > Alg 2 > Geometry until this year when my dept head switched due to pressures from the hspa. I totally agree that Geometry should come after Alg 2 ... and I have no idea why all these other schools teach Geometry in the middle....does anyone have some insight on that???? Thanks!

I think this is the reason: most Algebra II classes also do a lot of trig. But trig is based on the idea of similar triangles--geometry. Plus, once you really get into it,trig can be kind of difficult. (Not as much as before calculators when we had to interpolate, but still tricky.) So I think many schools prefer that the older kids take Algebra II & Trig.

I "taught" trig once in Geometry at my last school and another time in Algebra 2 during my student teaching. This year it's lumped in with algebra 2, but in both cases, I find that it's not much more than some dopey "SOHCAHTOA" followed by lots of complaining. No identities, no waves, no equations. I taught some of it anyway and will continue teaching it, against my peers and administration.

In my school, Junior math is Algebra II & Trig. The Algebra II is lots of word problems, some functions, logs, and I forget what else, the trig includes graphs, inverse and reciprocal functions, veryfying identities, Laws of Sines and Cosines, solving triangles, and probably a bunch of other stuff. I've never included most of that stuff in Geometry, just right triangle trig. I know they'll get it next year, and this is their one shot at proofs