I have been searching for a Geometry text to use with my students. There is a current debate over what approach to geometry is best. If I may request, please read all this post before you respond since I will try to address many issues at the end of my post. My criteria for an ideal text are: 1*) Academically rigorous (text targeted towards advanced placement students) 2*) Must stress formal proof throughout text 3) Exploratory exercises and experiments 4) Has creative thought problems 5*) Text is easy to read 6) Historical and other notes/sidebars 7) Examples of geometry in Art/fractals/etc. 8) Has computer programming examples The *'s indicate what are crucial criteria for me. The others would be nice but not necessary since I supplement these areas with my own materials. I didn't expect to find a text that fit everything :) My educational goal is to prepare the students for majors in math/hard sciences/engineering/any mathematically demanding field. I know that many of my students will not go on in these fields! But what I have found is that when I set the educational bar at this level my students rise to the challenge and they have amazing gains in their ability to use their minds mathematically/creatively. It is truly an awesome phenomenon to watch 6th and 7th graders debate if one infinity is larger than another or be able to derive the quadratic equation on their own (more on this later). And in the end, students who thought that math was boooring fall in love with the subject.
I finally narrowed down my search to 2 texts Geometry Moise, Downs Hardcover (January 1991) Addison-Wesley Pub Co; ISBN: 0201253356
So I can say that in all the essential things I am looking for, Geometry for Enjoyment fits the bill nicely and I am pleased. I will post when we are completed with the course to tell you how it went!
Now to the wonderful geometry debate that has been raging. One side says that a book dedicated to proof is dry, turns students off and isn't the correct approach. For this camp, they prefer an inductive/exploratory approach. Another side says that since formal rigorous proof is the foundation of math, it needs to be stressed and that the inductive approach lacks rigor. But we shouldn't forget that every teacher is different. An approach that works well for one teacher may fail utterly for another. The key points are: Are the students learning the math? Are the students learning to think logically/mathematically? Are the students learning to think mathematically out of the box? Are the students learning to think creatively? Are the students excited about math? Do the students feel mastery of the subject? So from my point of view, if a teacher is doing the above things they have succeeded. If it turns out that one teacher uses a different approach from another and achieves basically the same result then why argue with the approach since it works. I know that in many teachers' hands, a proof laden book will turn the students off and they will not learn the math. But for my teaching style this type of book is the perfect foil for what we do in class. Let me explain. When I teach I teach Socratically. This means that in class the students derive EVERYTHING themselves with me as their guide. The class doesn't move forward unless the students come up with the answers/ideas to my questions. So class time is 100% creative/exploratory. Then students go home and do the homework. Just like when you take piano you need to do scales and practice, I have found that my students need to do mathematical scales to burn in the math. Repetition of concepts is essential, without this the math doesn't stick as well. I also have a ton of creative puzzlers for homework which adds a crucial dimension. But what I like to say is the proof is in the pudding. I can talk all day about philosophy of teaching but what are my results? Let me give you examples from my Algebra class as proof that for me my approach works. I started with 6th and 7th grade students. The students are being home-schooled and the parents pay for the class (we are not affiliated with a school). Most of them had no previous experience with algebra and most of the class would say they didn't like math when we started. I choose a textbook that is only used at the college level (in fact one of the parents had used the book in college). This book is very difficult and covers all of Algebra I and a fair amount of Algebra II. This book is totally drill and kill and as dry as a bone in the summer desert. They were assigned between 150-200 problems a week for homework. A recipe for disaster? Quite the contrary! I watched EVERYONE of my students bloom mathematically/intellectually. How many students at this grade level could solve this problem: 1+1/(1+1/(1+1/(1+...)...)) the answer to this infinite fraction is: (square root of 5 + 1)/2 Or discuss Cantors proof of why the set of fractions is the same order of infinity as the integers and THEN derive the mathematical formula that relates the positive integers to the fractions (can you do this?) When I teach it is not good enough to just do the problem you have to understand why what you are doing works mathematically. That is, you have to understand math from a fundamental perspective. For example we all know that: 2 raised to the 0 power = 1 (I will write this 2^0) But why is this so? What is the reason for this? Just saying that you were told this is not enough! In fact, many math teachers don't know why we choose 2^0 to be defined as 0. My students can give you 2 different logical arguments as to why this should be so. The quadratic equation is another example. Memorizing a formula won't cut it can you derive it? My students can and not by memorizing the steps in the derivation but logically putting the proof together. Over 60% of my student came in with low math self esteem and everyone of them now knows that they can do it! When I gave the final exam, the students could pick a regular 1 hour exam or they could opt to take a 3 hour college level exam. We discussed that there was no advantage in taking one exam over the other. I explained that I wanted them to have the opportunity to experience what a college test is like if they so choosed. They debated which exam they would take. When the day came everyone of them took the 3 hour exam voluntarily. In fact, when they finished the section that comprised the 1 hour exam, they exclaimed that this was too easy! Every student scored 90%+ on the exam. (FYI the exam was taken from the textbook and was what the author considered to be a standard exam for the book) One of the most gratifying things has been to watch the student's excitement about math grow. In the beginning most of the students didn't do extra homework or brainteasers. By the end of the class 100% of them were doing extra work outside of class on a regular basis. Some students now complain when we have vacation days and miss a class. They all want to continue during the summer. Students have announced that they want to become mathematicians. Did I mention that we meet once a week for a 3 hour math class? That's right! And most of the time, my students remain sharp for the 3 hours. When I teach physics at college, my students don't have that kind of attention span! By word of mouth I have students and parents banging on my doors to get in. So after 36 weeks of doing algebra, every student has excelled and is looking forward to geometry. As a foot note, at this point the students are 1-2 years ahead of grade level in math. My plan is to have them finish pre-calculus by 10th grade and then go to the local university (UC Berkeley) to take calculus and differential equations for the 11th-12th grade. Wish me luck!
A little about me and the math program. If you talk to many University professors they will tell you that their foreign students are much better prepared in math even compared to the ones from private schools. Routinely, Japanese students take differential equations in high school, something that is unheard of here. European students in general have a much higher standard. So I want to close the gap. I am currently developing a curriculum starting with algebra that will take average students and accelerate them mathematically to the above levels. It is important that the math not become a grind and that I don't overwork the students (in Japan many of the students complain of overwork and I would agree with them). I believe that what I am developing could NOT be used effectively in a standard school environment for these reasons: 1) I graduated summa cum laude from Cornell and have a master in physics from UC Berkeley. I am constantly using my mathematical knowledge in class to bring in advanced concepts and analyze student's responses to find the mathematical gold in a "wrong" response or to amplify a concept to a higher level. Without this degree of education my class would lack an essential depth that is a major reason for my students developing their mental facility. Obviously most math teachers don't have this level of knowledge. 2) My curriculum takes a tremendous amount of time, energy and commitment on my part. Teaching math is a calling for me and again you can't expect the average teacher or any person to have this type of commitment. 3) I have 100% backing from the parents and they are involved in cheering the students on. They come to every class and help me when rough spots occur with their child. Again this kind of support is not common. 4) By 6th grade, many students are watching loads of TV and have short attention spans. In general, the students in the home-school community I deal with have excellent attention spans. This is crucial to the class being a success.
Does this mean that the curriculum would not be useful to a wider audience? No, I think that there are many things I am developing that would be helpful but I would think when applied to a standard situation you would not get the results I get. What I am hoping to do is to develop a solid curriculum/game plan for like minded teachers. Maybe this will start a small revolution!