"Mathematics is more than just
a collection of right answers to questions posed by someone else.
Knowing mathematics is doing mathematics. We need to create
situations where students can be active, creative, and responsive to
the physical world. I believe that to learn mathematics students must
construct it for themselves. They can only do that by exploring,
justifying, representing, discussing, using, describing,
investigating, predicting. In short, by being active in the
The principal of my school is a former English teacher, and he asked if anyone in the Mathematics department was interested in teaching a writing-intensive mathematics class. I eagerly volunteered, not having any idea how to do that but determined to learn how. I started on a path that challenged every assumption that I had previously made about teaching, and everything that I had ever seen in school. My first step was to read Joan Countryman's book. I then set out to re-think every aspect of my teaching.
In developing my writing-intensive Geometry class, the first thing I realized is that in order to ask students to write, I need to give them something to write about. There is not much for them to say or do when presented with lists of theorems and properties. But, if we set the textbook aside, give them opportunities to explore mathematical concepts, and ask them guided questions, they can begin to discover the principles of geometry for themselves, and learn to write mathematics.
Using hands-on materials, sticks, string, compass and straightedge, as well as interactive geometry software, my students explored the properties of geometric figures and made conjectures. They wrote their conjectures, explained why they came to their conclusions, and proved what they had discovered. They verified their findings in explanatory sentences, paragraph proofs, and two-column proofs.
A question that interests me, and that I asked my students, is "Why do we prove things?" The answer is NOT "because the teacher told us to"! In order to get them thinking about this, I offered an "extra credit" challenge: to find examples of things that were once believed to be true but were later found to be false. The students did some research, and came up with some interesting examples: "People used to believe that the world was flat and that if you sailed a boat too far, you would fall off"; and "Some people still believe ithe Loch Ness Monster".The students, in their reflections, agreed that there may be hundreds, if not thousands of myths, incorrect beliefs, and misconceptions in this world. The following website will show you some of the current ones: http://hoaxbusters.ciac.org/HBUrbanMyths.shtml
We discussed these myths, including the new myths and "urban legends" of today, and the students began to understand why we prove things in geometry. In the end, the students realized that we prove things not only to verify that they are true, but in doing so we find out WHY they are true. So, although the students did find writing proofs difficult sometimes, they did see the value in doing them.
In this class, the students did a number of different types of writing, besides writing traditional geometry proofs. Some of the types of writing in this class were: expository writing, self-examination, philosophical writing, and reflective writing. Each student kept a "portfolio" of their work, which contained a variety of types of writing assigments.
In order to explain this "portfolio" class to my students on the first day of school, I prepared and gave them the following written introduction:
Students learn in a variety of ways: some are good at taking tests, others can explain verbally with skill, some can draw diagrams that make complex problems clear, and others communicate effectively in writing. Some students can work well with others; some students prefer to work individually. Many find computers fascinating, others would rather build a model.
The goals of this writing-intensive geometry class are to provide a wide variety of mathematical experiences to teach the geometry curriculum and to assess the whole student, not just their test-taking ability. Each student's grade is determined by his or her work on writing assignments, explanatory papers and proofs done in and out of class, projects, presentations, as well as tests. There are many opportunities for bonus projects related to the course work, often pursuing geometric topics to greater depth. Activities include individual and group work. Each student will keep a portfolio of his or her best work, with reflections on what you learned from each piece.
When I first set out on this path, that of teaching a mathematics class in a very different way than it is usually taught, I did not know exactly what the results might be. I had nothing to guide me except a sense of what I valued in teaching, and what I believed was important in learning mathematics. It seemed to me that "what" was not as important as "why", and that understanding the concepts was more important than just performing the calculations. I believed that the reasons, the relationships, and the connections were more important than the right numberical answer and that that learning why is more important than just learning how. I valued the process more than the answer. I wanted my students to become mathematicians, not good memorizers; to question not to passively accept, and to become an active participant in their own mathematical life.
I set out with an open mind, and dared to take chances. My students showed me time and time again that they could live up to my expectations; they accepted the challenge and went beyond. And they taught me as much as I taught them.
I learned that mathematics is a living thing, that it is still changing, and that students can be part of this change. I learned that it is not all written in the book, and that it is not best just to teach it ithe way it's always been taught. I learned that kids have lots to say about learning, and that they want to be part of the search, the struggle, and the discovery. I learned that questioning can be more valuable than answering, and that the process can be more valuable than the product.
There are frustrations and disappointments on this path, and sometimes it seems easier to go back to teaching straight out of the book, tellong the students exactly what they should know, theorem 47 page 182. But when I read a ten page student project on the relationships between proving triangles congruent and proving quadrilaterals congruent, a one page reflection on the value of doing the "Clock Problem", or explore a student's web site on discovering the center of a triangle, then I know that it is worth the time and effort.
As the school year progressed, the students chose a variety of assignments to place in their portfolio. My students' comments when asked to evaluate their portfolios validates my belief that they have been enriched and challenged, and that they too believe that it was worth the effort. They were proud of their work, and their comments reflected this sense of accomplishment, and their feelings about this course:
"The contents of this portfolio show the many variations of activities we've done during the first quarter. It shows my demonstrations of mathematical, cooperative and creative strengths and weaknesses. I am proud of the pieces; I think they show diversity, and I worked hard on all of them. Although some assignments were more challenging than others, each required different types of thinking, and I feel they show my best work," Kim K.
"The following pieces are my achievements and my obstacles. But I appreciate all of them and the challenges they gave me." Mary T.
"Putting this portfolio together was an enjoyable experience as it allowed me to reflect back on previous works. As I looked back on my successes and failures, I learned and relearned some things which I had either skipped or forgotten, and through it all, came a sense of satisfaction on a job well done." David A.
"Of all the work we've done this year, I think the projects were the best. I really like the hands on stuff because it helps me visualize what I learn in class. Also, I can explore different concepts to discover theorems and properties." Nicole S.
"I chose this for my portfolio because it came to me easily and reminded me of how math can be fun. I enjoyed doing this because I could feel my mind working and solving problems." Paige M.
"I have improved the most this year from tests and the hands-on projects. The projects have helped my to see concepts of geometry that I would not have understsood by just reading the book. Experimenting with my group has helped me to improve my comprehension of Geometry." Michael D.
"This quarter I've learned many things in geometry. I have also been introduced to the idea of exploring math for myself, by hands-on techniques. It is great to work with my fellow students, and the projects we did together helped us understand things better. This has benefited me because it has opened my eyes to a whole new world of math which can be helpful and fun. I've enjoyed learning all these things in math." Elyse W.
"I enjoyed the projects this quarter and thought they were really fun because we got to explore things and discover properties that we could use. I felt this made remembering the properties easier and really helped me understand them better." David A,.
"I benefited most this quarter from the hands-on work, where we can actually see what is going into the work and apply the knowledge elsewhere. I have liked the quadrilaterals project and the midpoint project especially because of this. They have given me reasons for why theorems work and why we do certain things in proofs. Also, actually doing something with the knowledge has helped me to remember the information later on." Elizabeth M.
"This math portfolio is to review on the work that I've done in geometry. It is to reflect on how hard I've worked and how well I've done and by doing reflections, or evaluations on some of the work I've done, I also see some of my mistakes and can try not to make those same mistakes in the future. The work I've done has been complete and I've usually tried my hardest to do the best that I can do. It pleases me to see the accomplishments that I have done and in this portfolio I can see the very best things that I've done." Taufique K.
In conclusion, and on a more personal note, I would like to answer a question that was asked of me, in a math workshop. The question was "How do you find time, with all the material that must be covered, to do all of these extra things like projects, writing in mathematics, and explorations?" My answer was "I guess I just MAKE the time, I squeeze it in wherever I can, in 10 minutes or in a half an hour if I can find a half an hour. I make the time because the alternative is not possible for me. I have often thought of how much easier it would be if I just stuck to the textbook and did what it says, and if I did only what I did last school year. And I've even tried to just stick to the required curriculum, and the problems in the textbook . . . but I can't, because I get bored! When I find a new application of math, when I happen upon an interesting activity in an article in a teacher magazine that relates to what I'm trying to teach, I get excited about it and I just have to try it. I find geometry so interesting that I just keep getting sucked back into it. And the reason that I teach at all is that I want my students to feel this way too.
I know we all have to "Cover Stuff"! And I do; we keep our noses to the grindstone and we cover it ALL. I expect a lot of my students, and of myself. But I teach because I believe that years from now they will remember the castle they constructed using Sketchpad, if not the formulas involved in the construction, and the pride they felt in writing a particularly difficult proof. And maybe at least one student will pursue a career in math because they learned to love it themselves, or become an architect because we explored some of the applications of math in architecture, or become a graphic designer because they discovered their own creativity could be put to practical use. And I believe that the skills learned in pursuing the logic of a geometric construction and a mathematical proof will help a future lawyer to write a clear and logical legal brief, or an architect to design a better set of house plans through the ability to write precise instructions for others to follow. I believe that math is fascinating, and I hope that my enthusiasm will kindle their own.
I find the time because the connections are as important to me as the theorems, and I find the time because I can't imagine teaching any other way than with enthusiasm, excitement, and passion. And it is my greatest wish that somehow this will kindle enthusiasm, excitement, and passion in my students.
I hope that these web pages will offer options for both teachers and students to see another way of teaching and learning mathematics, and will inspire others to consider new methods and mindsets.
Go To Homepage Go To Introduction
1) Constructions 2) Clock Problem 3) Test Corrections 4) ASN Explain 5) Thoughts About Slope 6) What is Proof?
7) Similar Triangles 8) Homework Corrections 9) Quads Midpoints 10) Quads Congruence 11) Polygons
12) Polygons Into Circles 13) Area and Perimeter 14) Writing About Grading 15) Locus 16) Extra Credit Projects
17) Homework Reflections 18) Students' Overall Reflections 19) Parents' Evaluate Method 20) In Conclusion