Talk me down please... "geometry for dummies"

Parent Cafe
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<p>Well, let me give a shout out to modern geometry curriculum. … as a math lover and economist I was also befuddled by my daughter’s geometry class last year…what happened to proofs? I was totally stumped at first at what the book and teacher were trying to accomplish. Well…I have to say that it is actually a great way to teach geometry AND logic through this more open ended way of demonstrating geometric relationships. The terminology like “conjecture” is a bit silly…but it actually allows kids to learn to convince themselves by convincing others that certain relationships hold and that there are ways to prove something without saying “it just is!”. This valuable lesson in logic is not best achieved by teaching kids through the very structured logic of a old fashioned proof. As the product of traditional geometry/logic/proof curriculum , in graduate school I had a hard time doing “proofs” for highly complex relationships in math or econ. I felt compelled to lay out all the little bits of logic and would lose sight of my initial intuition of why I knew that the truth was a truth. I had a professor tell me to write out my “proof” in an essay or describe my logic to a non-mathematician and only then try to formalize the logic. That is what the new geometry is asking of kids…try and understand what you are trying to prove, why you intuitively feel it is true (or not) and then formalize this thinking. </p>

<p>Unfortunately, the OP’s kid seems to have a lazy and uninspired teacher so he would be better with the old fashioned, closed proof approach to geometry. Not everything new and confusing is bad, sometimes it is just implemented badly.</p>

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I have a Masters in mathematics and never had a geometry course in high school or college. We had to take a Jr. college level course in proofs. It didn’t help me a whole lot, as I never got too adept at the proving part. Better with application.</p>

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<p>well its been a long time since I took trig or Calc, so I must be misremembering - i thought you did at least some proofs in them, and I vaguely seem to recall DD doing proofs in them. I could be wrong.</p>

<p>I will say that I while inductive reasoning is important in economics, I do think that knowing the basics of deductive proof was helpful, if only as a way of thinking.</p>

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<p>The curricula has changed in many cases, things like learning how to prove a theorem, inductive and deductive reasoning, have gone by the wayside. Today geometry is basically memorize that alternate interior angles on a transverse are equal, and use that concept on some diagram on a standardized test, along with knowing the sum of the angles in a triangle is 180 degrees (without knowing or proving why) and so forth.</p>

<p>Ironically, someone mentioned the infamous tests used to compare the results of students in different countries, when those I believe are the problem. In many countries, for years their teaching has been based around standardized tests, that are based in rote memorization of formulas to solve problems on standard tests. Even back in my day, when geometry was taught along classical lines, where you learned the underlying reasons, US students didn’t do well on those tests, where kids from places like Singapore scored highly…the problem is learning math and science like that isn’t really teaching the ability to use math, it is teaching to do well on tests.</p>

<p>I remember in one of Richard Feynman’s books he was talking about being a visiting professor in Brazil in physics, and being shocked at the way the kids were taught. For light entering a fluid (like water), there is a principal that says the angle of incidence (angle of the light to the water) is equal to the angle of refraction (the angle the light is bent when it enters the fluid, such as water, and slows down). Kids could state that, but when asked to explain what it meant, they couldn’t…</p>

<p>And sadly, the US has decided that the way to teach is to these tests, and a lot has been lost because of it. Working out theorems, showing how to bisect an angle using a straight edge and a piece of string, we about thinking things out; today what we have is what many places in the world did, memorize stuff to get good grades on tests. Problem is, without the other part kids are missing major pieces of learning, about puzzling things out or going beyond what is on a test, and it is why Singapore, for example, that generally scores high on math and science tests, isn’t exactly known as a hotbed of innovation and new and original ways of doing things. </p>

<p>As far as textbooks go,my big hope is that they go digital. Besides the groaning backpacks, it also hopefully means that maybe, just maybe, schools can find textbooks that haven’t been watered down by idiot places that have decided creationism is science, history is what you claim it is and math is 1+1=2 but get textbooks written ‘their way’ because they unfortunately do bulk buying.</p>

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<p>Unfortunately, too often today people who understand math are not teaching it. When those that do not understand it teach it all they can do is repeat the text book they are handed. If reading the book did not help the kid understand it, they are toast.</p>

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<p>Wow, that’s really bad. Even though it doesn’t prove anything, it really helped me at least to have to do proofs in my 9th grade honor geometry class. We learned why things were that way, and that kind of thinking helped me on the SAT/ACT, or at least I think so. </p>

<p>Inductive reasoning means nothing in math.</p>

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There were quite a few proofs along the way in calculus, linear algebra, differential equations, and so on, but I just had to muddle through without that background stuff that most people got in their geometry courses.</p>

<p>Here, geometry is a grade 10 subject, and DD and DS both had it as freshmen in 10H sections. They had a lot of proofs, etc., so maybe it depends on what grade they are getting it in or whether it is an advanced/honors section or not?</p>

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<p>I just want to throw in here that not ALL schools are messing up the geometry curriculum. I dont think it is necessarily this day and age, so much as the particular schools. My class had my writing proofs every night. I thought it was an excellent class.</p>

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<p>OT: Sorry Pinot for stealing part of your thread title for my thread. I must have seen your title earlier and it must have stuck in the back of my head when I went to post mine. Didn’t mean to ‘plagiarize.’ :o</p>

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<p>I haven’t had to help a kid in Geometry in years, but wow, I wish this website had been available when mine were in middle and high school:
[Khan</a> Academy](<a href=“http://www.khanacademy.org/]Khan”>http://www.khanacademy.org/)
step by step through the stuff we all forgot after 8th grade.</p>

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<p>I had to learn proofs in 8th grade geometry. I had a good teacher, but I must admit he wasn’t incredibly helpful in explaining how to go about doing anything that wasn’t really obvious.</p>

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<p>It’s hard to see inductive reasoning ever counting as proof. Are you thinking of induction proofs? (Where you prove that something is true for n=1 then you prove that if it’s true for n=k it follows that it’s true for n=k+1, allowing you to conclude that it’s true for all natural numbers)</p>

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<p>You may be interested in the standards put forth by the National Council of Teachers of Mathematics:
[Standards</a> for School Mathematics: Geometry](<a href=“http://standards.nctm.org/document/chapter3/geom.htm]Standards”>http://standards.nctm.org/document/chapter3/geom.htm)</p>

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<p>Maybe that’s it. In which case it doesn’t sound like somthing from geometry. I have a fairly vague recollection of induction or inductive logic, reasoning or proofs. What I recall iis a method where you hypothesize something as true, and then look for whether making that assumption violates some other known law. Sort of like hypotheses testing.</p>

<p>But I am probably jumbling a lot of things together. It’s been a long time since I had to prove anything other than my name, address, and insurance to a traffic cop.</p>

<p>I certainly remember a lot of step-by-step proofs from when I took geometry 40 years ago.</p>

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<p>I took geometry in eighth grade (about 10 years ago) and it was a hard class. Most of what we did were proofs, if I’m remembering right. It doesn’t sound like your son is really learning it thoroughly. Does he like math? Would he be open to doing self-studying with you? (And would you be?) My father is a math person and certainly understood the concepts, but I would never have done that. I do have friends who always had parents assign them extra homework beyond what they were getting in class. If he isn’t even understanding what he is supposed to be learning, it seems like a waste. Also, does he always do his homework? I’ve found that math is one subject where if you don’t really thoroughly do your homework, and work through each problem and make sure you understand it, it’s really easy to get behind.</p>

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<p>ThisCouldBeHaven & bovertine - the induction my son’s class is using can be defined as “inference of a generalized conclusion from particular instances.” It is not the inductive reasoning TCBH described. They basically note that the drawings in the book make it seem that all angles of an equilateral triangle seem equal, and then accept that all angles of any equilateral triangle will always be equal. Sigh.</p>

<p>levirm - good idea to check the standards. this is what our state standards say: “Geometry
Using their knowledge of shapes and the coordinate system, students learn to construct proofs to prove mathematical theorems. Students determine the measure of sides and angles using proofs to justify their methods.”</p>

<p>I think I’m not in the mood to fight against the teacher, principal, and school system about whether or not this textbook and this class meet that standard. I think I will just enjoy my time with my son doing geometry on the side at home, and try not to focus too much on the frustration that comes with having to do the school’s job for them, and with having to painstakingly review what he’s doing in class and cross reference two geometry books at home to stay in sync and find the right section of the good books to back up the bad book. </p>

<p>The good news is, I find the geometry itself fun, so hopefully we will have fun doing it at home!</p>

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<p>Yeah, - “assume, guess and generalize” - that’s not a good technique for a math class.</p>

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<p>To the contrary, guess and generalize are wonderful techniques for a math class, but only if this leads to the insight to formally prove statements.</p>

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<p>It’s interesting that those students are told to induce like that, because in my math classes I have always been told to NEVER assume the graph is to scale unless explicitly noted.</p>

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Are you just trying to be argumentative about a silly post?</p>

<p>If I intended that to include a formal proof along with the “assume, guess and generalization” I would have said so. And since we are talking about the situation in the OP, which doesn’t include such a proof, I would have thought that was apparent.</p>

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<p>Jacob’s Geometry, 2nd edition, 1987 by Harold Jacobs. I have not looked at the third edition but the 2nd edition will provide you that old-school geometry. Proofs, constructions, etc.</p>

<p>Son’s CS undergrad program has proofs all over the place. Discrete Structures I, II, Foundations, Mathematical Statistics, Multivariable Calculus, Differential Equations, Linear Algebra, Algorithms, Database, etc.</p>

<p>An interesting and perhaps alternate experience for middle-school and high-school kids would be to go through the first chapter of Spivak.</p>