Jane in NC Posted March 25, 2013 Share Posted March 25, 2013 A couple of us were derailing another thread with a discussion on the merits of high school geometry, whether it should come before or after Algebra II, or whether it is a course that can be skipped. Let's move that discussion to a new thread. I have argued that it is best to do a geometry course after Algebra I for a number of reasons, one of which is that some students who struggle with the symbolism of Algebra have an easier time with the verbal and visual aspects of a proof based geometry course. This gives their brains a year to mature before moving on to Algebra II/Trig. Further I believe that my learning the logical flow of the proof one becomes a clearer thinker and a better writer. If your students do not study geometry, do you have them take a logic course? Chat away, my friends. Quote Link to comment Share on other sites More sharing options...

regentrude Posted March 25, 2013 Share Posted March 25, 2013 I have no strong opinion on whether geometry should come after algebra 1 or algebra 2, as long as it is done in time before standardized testing. My kids completed AoPS Intro to Algebra in its entirety before moving to geometry, so the equivalent of a traditional algebra 2 course, but they also had Intermediate Algebra before precalulus. I don't buy into the "forgetting algebra" during a year of geometry; if that is the case, algebra has not been taught to mastery. I do, however, strongly believe in the benefit of a good geometry program that includes proofs. I am always puzzled to see the debates about maturity for geo, and the mere fact that this is saved for 10th grade in a typical school sequence. In my home country, geometry of the triangle, classical formal construction with compass and straight edge, and formal proofs are studied in 6th grade. I find proof based geometry important for several reasons: it is the first time students get to see something that resembles what mathematicians are actually doing; it is an opportunity to introduce formal proofs in a subject that is more visual and less abstract; it trains the mind in logical thinking. (Btw, I have never taken formal logic, but plenty of logic can be learned through math and programming.) As I wrote in the other thread, a good course in higher math should be proof based and not just utilitarian drill of algorithms for cranking out integrals. I would imagine a student who did not have the opportunity to be introduced to the concept of formal proofs in geometry to have difficulties with epsilon-delta proofs in calculus. Quote Link to comment Share on other sites More sharing options...

FaithManor Posted March 25, 2013 Share Posted March 25, 2013 I think it can be done either way as long as it's out of the way by the end of the junior year for those taking the SAT and ACT. DD did her algebras back to back and then geometry and it worked out fine. I am doing this with our 14 year old who just completed algebra 2. However, our other high schooler, not as strong in algebra, needed a break. He's, for the most part, enjoying geometry and since I'm using the Jacob's 2nd edition, there is algebra review in every chapter which he is required to do. Youngest son wants to do algebra 2 and geometry concurrently. We'll see. Eldest boy will be going into computer programming. He'll need calc 1 in college, but no additional math. However, as we've pointed out over and over again, programming is all logic and algebra. So he is embracing both the proofs and his algebra review knowing that these are training his mind for computer science. He is seeing the fruit of this has he learns Java programming. My students are tortured - the kids have one year of formal logic, plus geometry, and Classical Rhetoric with Aristotle their senior year. I leave no stone unturned in the quest for the critically thinking mind! :D Faith Quote Link to comment Share on other sites More sharing options...

Dana Posted March 25, 2013 Share Posted March 25, 2013 I intend to have my son do some formal mathematical logic using the materials that are used in the eImacs logic courses. Geometry is a subject I haven't done any teaching in or studying of in about 15 years. It's what I feel weakest in teaching my son, although we have talked about the parallel postulate and alternate geometries. If we do a separate geometry course, it likely will be AoPS, but I haven't looked at their geometry book yet. I think seeing the idea of a formal proof early is important...it gives reasons other than "because I said so". We were watching a Teaching Company video last week with Neil deGrasse Tyson where he was talking about The Big Bang and the assumptions that are made to arrive at this theory. This is what led back to the geometry discussion :) I think an understanding of what a mathematical proof is is really important. I think far too many high school teachers don't get it. Sigh. Quote Link to comment Share on other sites More sharing options...

regentrude Posted March 25, 2013 Share Posted March 25, 2013 . If we do a separate geometry course, it likely will be AoPS, but I haven't looked at their geometry book yet. The geometry book is fabulous! They prove everything, and they use a more natural narrative proof format, not the artificial two column format that seems to be popular in schools in this country. Quote Link to comment Share on other sites More sharing options...

Dana Posted March 25, 2013 Share Posted March 25, 2013 The geometry book is fabulous! They prove everything, and they use a more natural narrative proof format, not the artificial two column format that seems to be popular in schools in this country. That makes me feel good. Thanks! To derail the thread more, how much algebra is used in their geometry? (roughly) I do see the benefit to a 2 column proof if you're trying to teach what needs justification. I do intend to do some formal logic first, which should make geom much easier. Quote Link to comment Share on other sites More sharing options...

regentrude Posted March 25, 2013 Share Posted March 25, 2013 That makes me feel good. Thanks! To derail the thread more, how much algebra is used in their geometry? (roughly) We just had a thread discussing this (among other things): http://forums.welltrainedmind.com/topic/464384-anyone-do-pre-geometry-before-geometry/ You need mastery of algebra 1, through quadratics. You absolutely need graphing lines and circles and completing the square for the analytical geometry in ch. 17 I consider a previous study of functions helpful for an understanding of trigonometric functions. For my kids, I found it beneficial to complete the entire Intro to Algebra book before doing geometry; other posters' students did well with only algebra 1. Quote Link to comment Share on other sites More sharing options...

quark Posted March 25, 2013 Share Posted March 25, 2013 The geometry book is fabulous! They prove everything, and they use a more natural narrative proof format, not the artificial two column format that seems to be popular in schools in this country. I bolded that last bit because I heard from a father of a junior high student in our math circle that fewer and fewer high schools in his area are requiring proofs--let alone two column proofs--and I believe his son is in one of the higher quality school districts too. Our area in general is teeming with math and science and tech professionals and we have some of the better schools here vs. the rest of the state so it came as a shock to me. I know homeschoolers with advanced, mathy kids who are skipping proofs as well...their reasoning being that they don't need it for the SAT. What are they missing? What am I missing? I am from a different education system and didn't learn proofs the way they are taught in the US. We learned a form of narrative proof in our 10th grade equivalent year but I didn't pursue proof-based math after choosing a humanities track in grades 11-12 so I didn't go into much more detail in proofs. But after watching my son work with his tutor and thrive from learning to write proofs (for the last year and 2 months) I see the benefit and am glad he has time to develop his love of proofs further. I can understand why humanities-loving kids may not want to spend time on it but I think it's a waste for advanced, math-loving homeschooled kids, who have the time to develop proof writing skills because they are accelerated, to skip it. I don't know enough about proof-writing to bring this topic up with my friends who have chosen to ignore proofs. I don't plan to hard sell but I also want to give them another point of view. What are some of the benefits I should mention? I see my son doing very well and being a much better thinker as a result of developing proof writing skills. But it sounds braggy to say something like that kwim? Quote Link to comment Share on other sites More sharing options...

Jane in NC Posted March 25, 2013 Author Share Posted March 25, 2013 I know homeschoolers with advanced, mathy kids who are skipping proofs as well...their reasoning being that they don't need it for the SAT. What are they missing? What am I missing? Sigh... The bottom line is that so many people in this culture are mathematically ignorant. They believe that Calculus is higher math or the end of math or something. They don't understand that Calculus is a tool, a powerful tool, but not the be-all and end-all of mathematics. Educators are often to blame. They have created a rush to Calculus, ignoring other aspects of mathematics. I was fortunate to take a proof oriented Calculus series. Most students in the class now see algorithms that are reduced to calculator programs and then forgotten. It is heartbreaking... Quote Link to comment Share on other sites More sharing options...

Jane in NC Posted March 25, 2013 Author Share Posted March 25, 2013 Earlier in the 20th century, before Calculus was offered at every high school, college bound sciency students often took a class in three dimensional geometry. We have my husband's father's books (he was a science professor). Topics like spherical trigonometry were covered. For people who navigated or designed things, a knowledge of three dimensional geometry was necessary. Now people use CAD. Does anyone know celestial navigation? Maybe I am just being a curmudgeon who is failing to embrace the new fangled. Quote Link to comment Share on other sites More sharing options...

Dana Posted March 25, 2013 Share Posted March 25, 2013 Maybe I am just being a curmudgeon who is failing to embrace the new fangled. Our cc has the TI84 as a requirement in int algebra. I used to teach it and tell students they didn't have to have the calculator. I'm trying to teach them the algebraic manipulation, but also WHY it works. We don't just give the quadratic formula and use it, we derive it! So many students tune out and just want the plug and chug. I wouldn't want to go back to trig and log tables as a rule, but I want more UNDERSTANDING, not "my calculator says". Sigh. Get off my lawn. Quote Link to comment Share on other sites More sharing options...

Sebastian (a lady) Posted March 25, 2013 Share Posted March 25, 2013 I took a semester course in celestial navigation. I don't think it is offered anymore, let alone required at USNA. Once upon a time, this was a robust subject at USNA. I own a copy of the Spherical Geometry with Military and Naval Applications book that was written by a couple of the USNA math profs. It is a wonderful book, that makes it clear that there is a connection between geometry and physics. To be honest, Celestial Navigation was a very tough course, that focused mostly on using tables and proceedure "strips" to calculate the sight reductions. The instructor (that I had) was a surface warfare officer, who wasn't that great of an officer, may not have been an excellent mariner or navigator and gave no sign of being a hearty geometer. I honestly learned more about celestial navigation reading Carry On, Mr. Bowditch than in that course. But I do look longingly at my old book and think that someday I'll have the time to master the contents. Quote Link to comment Share on other sites More sharing options...

quark Posted March 25, 2013 Share Posted March 25, 2013 Hee, hee I do admit to liking some of the new, shiny gadgets (yum!). But I agree with Dana about how they are getting in the way of understanding. Sometimes I feel like I am trying so hard to keep up with the times and also sticking to my guns about traditional ideas and methods that obviously work and also require hard work. It can be hard with one kid...I am constantly asking my son not to memorize...I quiz him about deriving formulas first when he uses them and so far, touch wood, he can. I can see the temptation for schools with 20 kids in a class. But it can't be win-win right? Whichever way I look at it, I think kids are losing out from not drawing graphs by hand. Quote Link to comment Share on other sites More sharing options...

Belacqua Posted March 25, 2013 Share Posted March 25, 2013 Sigh... The bottom line is that so many people in this culture are mathematically ignorant. They believe that Calculus is higher math or the end of math or something. They don't understand that Calculus is a tool, a powerful tool, but not the be-all and end-all of mathematics. I had a woman at a party argue with me that homeschoolers must not take calculus before Grade 12. If they do, they won't have any math in Grade 12 and it will look bad to colleges. She steadfastly refused to accept that there was any math to learn beyond high school calculus. I wanted to ask her just what math she thought was taught in colleges, but I think she'd have brained me with the cheese ball. Quote Link to comment Share on other sites More sharing options...

quark Posted March 25, 2013 Share Posted March 25, 2013 I had a woman at a party argue with me that homeschoolers must not take calculus before Grade 12. If they do, they won't have any math in Grade 12 and it will look bad to colleges. She steadfastly refused to accept that there was any math to learn beyond high school calculus. I wanted to ask her just what math she thought was taught in colleges, but I think she'd have brained me with the cheese ball. LOL. Oh dear, I don't seem to learn my lesson reading these forums. Must not drink coffee while reading posts. Quote Link to comment Share on other sites More sharing options...

Dana Posted March 25, 2013 Share Posted March 25, 2013 Hee, hee I do admit to liking some of the new, shiny gadgets (yum!). But I agree with Dana about how they are getting in the way of understanding. Sometimes I feel like I am trying so hard to keep up with the times and also sticking to my guns about traditional ideas and methods that obviously work and also require hard work. It can be hard with one kid...I am constantly asking my son not to memorize...I quiz him about deriving formulas first when he uses them and so far, touch wood, he can. I can see the temptation for schools with 20 kids in a class. But it can't be win-win right? Whichever way I look at it, I think kids are losing out from not drawing graphs by hand. There's definitely a fine line. Some experiments can be done with the graphing calculators and collecting and analyzing real world data that's really cool, like actually collecting the data for a bouncing ball and analyzing that data rather than just getting it from a table in the text. I never have time to do much of the extra stuff at the cc because I'm just trying to get the students to do the outside homework and be able to do the work they must have for the next course. I do what I can (like last class showing the derivation of point slope form of a line, so they'd have seen where the formula comes from). But I'm aware that at most 10% paid attention and got something from that explanation. I do tell my son to explain his work regularly. The correct answer doesn't mean much without a clear understanding of why it is. I think geometry is a particularly tough course for homeschooling parents to teach...kind of like how they have the sub boards for people to share their kids writing and get comments, something similar for higher level math/science could be useful.... And I'm off to give my test on graphing lines.... Quote Link to comment Share on other sites More sharing options...

Arcadia Posted March 25, 2013 Share Posted March 25, 2013 I think an understanding of what a mathematical proof is is really important. I think far too many high school teachers don't get it. Sigh. I think it is a problem with teacher education and how teachers are assigned. You need a competent teacher to teach/explain/discuss mathematical proofs well. The bottom line is that so many people in this culture are mathematically ignorant. They believe that Calculus is higher math or the end of math or something. They don't understand that Calculus is a tool, a powerful tool, but not the be-all and end-all of mathematics. Educators are often to blame. They have created a rush to Calculus, ignoring other aspects of mathematics. I was fortunate to take a proof oriented Calculus series. Most students in the class now see algorithms that are reduced to calculator programs and then forgotten. There is a hard sell here of Calculus BC being the end point for high school math for advanced kids. After that these kids probably go to engineering or pre-med. We have a copy of Larson's Algebra 1 and 2 and pages are dedicated to instructions on how to use the calculator. Can't high school kids figure that out for themselves without being spoon fed detailed instructions. In Singapore calculator use is never part of the math textbook. As students we had to figure out how to use a scientific calculator ourselves. Programmable calculators and graphing calculators are not allowed all the way to university for exams. Our cc has the TI84 as a requirement in int algebra. I used to teach it and tell students they didn't have to have the calculator. I'm trying to teach them the algebraic manipulation, but also WHY it works. We don't just give the quadratic formula and use it, we derive it! So many students tune out and just want the plug and chug. Our teachers used to give no calculators allowed tests. The teachers would bring reams of blank paper to the exam hall for all the math workings. I still know how to read the statistics tables which we were issued for exams. I guess you might get a mutiny if you give a "no calculator allowed" test. Quote Link to comment Share on other sites More sharing options...

lewelma Posted March 25, 2013 Share Posted March 25, 2013 My eyes have been opened to proofs thanks to Kathy in Richmond's recommendation of The Art and Craft of Problem Solving. We have been working on proofs for the past 4 weeks in preparation for a 1-month long proof-based math exam here. What I finally understand is that proofs are to math as experimentation is to science. In math you investigate the problem, come up with a conjecture, work through a proof, and finally either accept or reject your conjecture. In science, you investigate the problem, come up with a hypothesis, collect data, and finally accept or reject your hypothesis. Hummmm, very similar. If you only study what has already been discovered in science and never touch the scientific method, you are not actually studying science in full. I have come to believe that the same is true for math. IMHO, this approach leads to misconceptions about the fields of science and mathematics. Ruth in NZ Quote Link to comment Share on other sites More sharing options...

Luckymama Posted March 26, 2013 Share Posted March 26, 2013 Just have to say---right now dd is in the very first meeting of her AoPS Geometry class. I walked in from a meeting 15 minutes ago, and she told me they've been doing proofs the whole class :D Quote Link to comment Share on other sites More sharing options...

Dana Posted March 26, 2013 Share Posted March 26, 2013 I think it is a problem with teacher education and how teachers are assigned. You need a competent teacher to teach/explain/discuss mathematical proofs well. .... Our teachers used to give no calculators allowed tests. The teachers would bring reams of blank paper to the exam hall for all the math workings. I still know how to read the statistics tables which we were issued for exams. I guess you might get a mutiny if you give a "no calculator allowed" test. That is SO true! I do give no calculator tests for some tests (basically ones that are testing computation alone... fraction arithmetic, integer arithmetic). Students do NOT do well on these. The reliance on calculators has gotten really extreme. Quote Link to comment Share on other sites More sharing options...

Mukmuk Posted March 26, 2013 Share Posted March 26, 2013 I'm also ignorant of proofs, having come from a similar background as Quark. But it tickles me when ds, after a bout of AOPSs, says, "Big proofs make R(himself) happy", a la Life of Fred. And this is just Intro to Algebra. Quote Link to comment Share on other sites More sharing options...

Arcadia Posted March 26, 2013 Share Posted March 26, 2013 "We prove things in geometry because it is an exercise of logic and discipline which are both necessary in the real world no matter what profession you go into. Geometry is challenging and can be frustrating at times; it requires a strong discipline of mind to stick with frustrating problems, work through them logically, not give up, and write out many-stepped proofs which can be tedious at times. This discipline is needed in any field of study in school or profession. Geometrical proofs are also logical and require the student to think along logical lines and follow a train of thought to derive the proof in a clear, concise manner that others may follow and understand. Logic and reason are definitely important skills in life. They allow us to see when propaganda is being used, when incorrect interpretations of experiments are used, when there is a fallacy in someone's argument, etc. It is necessary in all facets of life, not just math. Logic and reason are tools that are very useful in understanding the world around you and having educated views. Constructing proofs helps build your skill in using logic and reason to forge a chain of thought that can bridge the gap between two sides of a river and help you get across."" quoted from "Chapter 6 - What is Proof" in mathforum It says it so much clearer than I can put into words my stand on learning proofs. Had so much fun with QED (quod erat demonstrandum) in math homework because I had to think. Quote Link to comment Share on other sites More sharing options...

quark Posted March 26, 2013 Share Posted March 26, 2013 Arcadia's mention of QED reminded me...I bought this book by Burkard Polster (just found this fun site of his with a link to his compilation of must-watch math movies) on the pretext of interesting my son to improve his math writing skills but it is so cute and visually attractive that I might just use it for myself. Quote Link to comment Share on other sites More sharing options...

MyThreeSons Posted March 26, 2013 Share Posted March 26, 2013 The geometry book is fabulous! They prove everything, and they use a more natural narrative proof format, not the artificial two column format that seems to be popular in schools in this country. As a Geometry teacher, I like the two column proof format when it comes to grading. The text I use at our co-op (Discovering Geometry) also teaches a flowchart-style proof system, which many students find most helpful. Quote Link to comment Share on other sites More sharing options...

Sebastian (a lady) Posted March 26, 2013 Share Posted March 26, 2013 I took a semester course in celestial navigation. I don't think it is offered anymore, let alone required at USNA. Once upon a time, this was a robust subject at USNA. I own a copy of the Spherical Geometry with Military and Naval Applications book that was written by a couple of the USNA math profs. It is a wonderful book, that makes it clear that there is a connection between geometry and physics. To be honest, Celestial Navigation was a very tough course, that focused mostly on using tables and proceedure "strips" to calculate the sight reductions. The instructor (that I had) was a surface warfare officer, who wasn't that great of an officer, may not have been an excellent mariner or navigator and gave no sign of being a hearty geometer. I honestly learned more about celestial navigation reading Carry On, Mr. Bowditch than in that course. But I do look longingly at my old book and think that someday I'll have the time to master the contents. I should clarify that the book I mentioned above (which is actually titled Spherical Trigonometry with Naval and Military Applications) dates from the 1940s; it wasn't my old textbook. (For the curious, the non-spherical book by the same author(s) is Geometry with Military and Naval Applications.) Navigation and cartography is an important topic, still; even though we tend to think that Google Earth and GPS have rendered it all moot. Quote Link to comment Share on other sites More sharing options...

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