- Thinking visually and intuitively with hand-drawn diagrams and by folding paper
- Thinking in a Euclidean framework with straightedge and compass and geometry software
- Thinking in a hyperbolic geometry framework with geometry software
Overarching Goals
As a result of your efforts in this course, you should strengthen your skills in:
Central Theme, Connections, Media
The central theme to this course concerns geometry as a way of knowing and a way of representing. In Math 120 you'll explore such questions as: How do we come to "know" something in mathematics? How can we be sure a geometric theorem is true? Why and how did it come to be that mathematics is used for human exploration of the universe? Can there be more than one geometry? If so, which one is right? We'll also explore connections between geometry and art and between geometry and the sciences. All of the above will be explored in concrete ways, using paper, computer software, walks through campus, and kits like Zome Tools.
Topics
Categorizing Geometric "Objects"
Geometry and Paper Folding
The Geometry of Euclid and Constructions by Ruler and Compass
Constructions Using Geometry Software
Inversions, Reflections, and Reflection on Inversions
Finite Geometries
A Geometry with Many Parallels
Your Journey Reader
Journey into Geometries by Marta Sved - ISBN: 0-88385-500-3 - published by the MAA (Mathematical Association of America)
Here's another way to look at this explorations class: You'll explore geometric thinking in three major ways:
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Each of the three parts will be organized around construction problems. In the beginning weeks of the course constructions will be done using paper folding and other non-computer tools, while in the remainder of the term most of the work will be aided by using the Geometer's Sketchpad software. For the hyperbolic geometry portion, you will study the disk model of the hyperbolic plane, investigating a microworld in which constructions like the perpendicular (hyperbolic) through a point to a (hyperbolic) line are accomplished by scripts written by the instructor. The curriculum will present opportunities for students to make investigations, form conjectures, and prove a few things along the way.
First Principles
It is my responsibility to make a reasoned estimate of your progress in the Math 120 based on the work you present. Making such an estimate is not easy since each of us has understandings in many spheres that are ever-changing, and how well we demonstrate our understanding is highly dependent on the ways in which we are asked for such a demonstration. For these reasons, I will ask you to present various kinds of work over the course of the semester in the hope that I can get as full a picture of your understanding as I can. The formats through which I will evaluate your work will be taken from: individual portfolios containing questions, lab responses, homework, problem solutions (at least attempts at solution), and your reflections on all of these, and a team or individual project. In this way you will have a variety of ways to demonstrate your understanding.
Honor
It is your responsibility to present your work with honesty and integrity in accordance with the Mount Holyoke Honor System. Such responsibility takes on particular importance in the area of the collaboration you do with one another in Math 120. I am encouraging you to collaborate with one another, in the spirit of the MHC Honor Code, which includes the statements, "Students are expected to comply with the rules and regulations of the College. Beyond minimal compliance, students under an honor code grow to understand, through course work, collaborative learning, and living together, that honor is achieved with diligence, commitment, and courage. The Mount Holyoke College honor system relies on the integrity of each student. The strength of the system derives from the honesty and self-discipline of every student and from the faith of the faculty and the administration in the student body." When you present work that is based on collaboration with others or consultation with others, it is your responsibility to note that collaboration or consultation in writing on the presented work. Please talk to me and refer to the Mount Holyoke College Honor Code whenever you have questions regarding issues of honesty.
I am sure that, with hard work, each of you will make great progress in your understanding of geometry. It is my responsibility to help you to do so.
What's Different about this Course?
Probably lots of differences between this course and other math courses you've taken: No tests, no teacher telling you how to perform procedures, an emphasis on process over product, ... But most importantly, I believe that asking good questions is much more important to intellectual growth than answering questions. So, if you'll give free rein to your curiosity, you'll probably do just fine in Math 120!
Next Steps
We will work out details of determining a grade, including the formats by which you can demonstrate your learning, as we move through the course. If you have questions or suggestions right now send me a message!
1/25/08: Portfolio Information
As a starting point, your portfolio is a place to put all the work, processes and products of your thinking about geometry (classwork, including labs, and homework) this term. So, the portfolio should include, but not be limited to, your homework and lab work, your reflections on these first two items, and the questions you have asked, even if you never voiced those questions in class. Specific types of things like folded paper, sketches, and scripts should be included. The portfolio starts as just a folder that is a dumping ground for all those items mentioned above.
A portfolio should then be organized, removing class notes and other distracting materials, in such a way that your own original work is prominent, readable, and reflective of your thoughts and your understanding of geometry as it emerges in this course. Sectioning and headings are helpful as organizing tools. Reflection on what you have done is essential. As a default option for organization, you could have five sections, one each for questions, reflections, lab work, problem solution attempts, and "other homework."
Portfolios don't need to be limited to work on paper; you could augment work on paper with electronic files and web pages if you want.
More on Assessment and Portfolio at GRADING DETAILS and Notes for Portfolio Check 1
If you would like to read more about your instructor's ideas about this course, click here: Background Information