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Re: High School Geometry
Posted:
Apr 27, 1993 2:26 PM


In article <danh230493145755@emu.math.ilstu.edu> danh@math.ilstu.edu (Dan Hirschhorn) writes: > >There is nothing so enjoyable as a former teacher of high school geometry, >a current teacher of college geometry and a current author of UCSMP >Geometry (ScottForesman 1993, 1991) to hear high school geometry as the >worst taught course and one for which there has never been a good textbook >written. > >Somehow the previous critics of high school geometry teachers must have >picked something up when they were high school students from their teachers >in order to choose mathematics as their profession with a lifelong >interest in geometry.
Or perhaps not. I, personally, hated math throughout elementary, junior high, and high school. The fact that I turned out to be a mathematics major and then a mathematics educator is due not to my high school geometry teacher (or to any of my high school teachers). Nor is it due to any text I used in high school. It is due to 1. Professor James Sethian at UC Berkeley (my freshman calculus prof.) and 2. The difficulty of OChem (I started out a chem major.)
An interest in geometry developed long after a general interest in mathematics and only after realizing that geometry had little or nothing to do with whatever is was I wasted time on during my high school "geometry" class. I believe this story is similar to those of other people on the Connected Geometry project. And most mathematics majors I knew in college did not start out that way. We all had to be convinced that mathematics was not what we thought from our high school experiences.
> >Let me briefly go into some of the issues surrounding high school geometry. > As was noted in Lee Story's article, the elementary curriculum is biased >towards arithmetic. While there are chapters in geometry, they are usually >at the end of the text and often skipped by teachers. Those teachers who >do address geometry usually do so wonderfully, but they are then undermined >by teachers in later grades who do not build on this geometric knowledge. >Since there is no set expectation in geometry (unlike arithmetic where >certain algorithms are to be mastered), the teacher with an interest in >geometry always has to start from square one.
Agreed, but this seems to be changing. (I hope.) I see many more teachers in elementary and middle school dealing with geometric ideas than I remeber from my school experiences. Also, the spread of Logo (sounds like a disease, huh?) is encouraging more teachers of more levels of mathematics to incorporate geometry.
> >The number one predictor of student performance in a high school geometry >course is the amount of geometry that student knows upon entering the >course. In fact, if a student does not know enough basics about the >shapes, it is invariable that that student will be unable to write a simple >proof by the end of the course. One cannot learn about the shapes, learn >about their properties, and then apply those properties in a proof all in >one year. > >The enrollment data seems to show 2 facts. As with any course from 1st >year algebra on, about half the students who enroll in high school geometry >will take no more mathematics courses. Secondly, of those students who are >in a fullyear geometry course, only 1/3 of them will be able to write a >simple triangle congruence proof by the end of the year. A deemphasis on >proof is seen as addressing these issues. Why turn off kids, and why force >them to do something only 1/3 of them can do? Especially when other >geometric topics such as the use of geometry drawing tools and >3dimensional visualization can be engaging and valuable. >
Certainly the use of geometry drawing tools and 3D visualization are engaging and valuable topics. But to the exclusion of proof? Proof and reasoned argument are at the heart of mathematics. What's more, proof of one theorem can give you ideas of new areas to explore with your "engaging" tools. Disocveries from these tools can lead to conjectures, more proofs, more ideas...It all works together.
And only 1/3 of students "can do" proofs? That's an interesting statistic, and I'd like to see where it came from. Is it from a nationwide survey that included classes where teachers skipped all the proofs (as mentioned by another poster)? Was it from one school? I have a hard time believing that kids who can make it to a high school geometry course can't learn how to write a reasoned argument. Of course, many of the may have trouble with the infamous "twocolumn" proofs. I'd bet that the percentage would shoot way up if kids were taught how to write real proofs  work forward, work backwards, work from the middle. Assume intermediary results and prove them if they do, in fact, help you. Scribble and draw and brainstorm with other students. Then write it up formally when you've really figured it out.
>On the other hand, geometry is the only course in school mathematics that >really addresses reasoning, argument, and proof. If the students don't get >this in geometry, when will they?
According to the new teaching standards, communicating mathematically (including reasoning and arguing and at higher levels proof) is a major goal in *all* areas of the mathematics curriculum and for *all* levels. Why should such a deep and important topic be relegated to one year of the high school sequence?
>[deletions] >There are enough problems in teaching mathematics in any grade K16. At >the college level there is too much formalism and chalk and talk so that >the prospective teachers of mathematics don't learn the mathematics they >need and they don't see innovative and effective teaching modelled. We can >all bash one another or we can realize that we all love mathematics (and >geometry specifically) and we would be glad to try things to give our >students better appreciation of geometry.
I agree, but I refuse to admit that all teachers of mathematics, even all teachers of high school mathematics or all teachers of high school geometry, love mathematics. I've seen too many counterexamples. And they were not all mathematics majors as undergrads either. It may be true that all of us reading this love matheamtics and geometry specifically, but it is also true that there is a problem. If all teachers of mathematics loved math, they would be inspiring teachers, and the majority of our students would not loathe math. They *would* be continuing in their studies past high school geometry. We are not "bashing each other" unless we take generalizations about the state of mathematics education in this country too personally. We may all be doing good jobs, but most of us could probably do better (myself included). There needs to be higher standards for mathematics teachers at all levels, not just the high school level. There needs to be inservice training to teach some real and interesting mathematics to current teachers. And you're right, there needs to be improvements in undergraduate mathematics education so that future teachers see how inspiring mathematics and mathematics teachers can be. That, I hope, is coming ...
michelle
  Michelle Manes Reasearch Assistant, Connected Geometry Project Education Development Center



