Course Description

Designed to re-examine Euclidean plane geometry as a postulational system.  Emphasis is on  formulating definitions and constructing valid proofs.  Topics include mathematical reasoning, postulational method, finite geometries, congruence, similarity, parallelism, and construction with ruler and compass.

Prerequisite:  Mathematics 184

Text: Michael Serra, Discovering Geometry, Key Curriculum Press, Latest edition including a student edition of the Geometer's Sketchpad.

Overview: The major parts of this three credit geometry course are as follows:

Euclidean Geometry and Non-Euclidean Geometry
Concepts of Measurement
Geometric Awareness and Analysis of Shapes
Transformations, Symmetry, Congruence, and Similarity
The use of technology in Geometry.

Each part of the course will be linked to the Virginia Geometry and Measurement Standards of Learning objectives for grades 5-12 and each will stress the van Hiele model for learning geometry. Copies of the SOLs and articles on the van Hiele model will be included as part of your materials. In addition, the following  text materials which will be available in Robinson Technology Rm. 234:

GEO Geometry Focus Issue of Mathematics Teaching in the Middle Grade , NCTM, March-April 1998

GSS Geometry and Spatial Sense: Addenda Series, Grades K-6, Del Grande et al, NCTM,1993.

GMG Geometry in the Middle Grades: Addenda Series, Grades 5-8, Geddies et al, NCTM, 1991.

MMG Measurement in the Middle Grades: Addenda Series, Grades 5-8, Geddes et al, NCTM, 1994.

PPG Patty Paper Geometry, M. Serra, Key Curriculum Press, 1994.

The course will mix presentations with workshops and all sessions will stress hands-on discovery learning. A project to make a substantial classroom unit will be part of this course.

Our principle  goal in this course is to learn informal geometry that relates to our curriculum in grades 5 through 12. The informal geometry taught in these years is a foundation that our students will need to understand the physical world they live in. This material has long been neglected in our curriculum. When it is taught well, children find subject matter to be interesting and motivating. When our students do not develop an intuitive understanding of the way our world fits together, i.e. of geometric attributes, proportions and relations, they do not succeed in later science, technology, and mathematics studies. Today our students are also required to pass state mandated tests on geometry.

We believe that teachers need the geometry content that they will learn in this class, but that they also need to understand how students learn geometry. Our course's second goal is to develop an understanding of how children learn geometry and to explore a variety of effective teaching strategies. The fact that geometry has been neglected in the schools is largely a consequence of our lack of success in teaching geometry. We all avoid repeating failures, and when our students do not learn what we are trying to teach, it is a failure and something to avoid. Consider the following data published in 1988:

What does it mean when more than half of our seventh graders can not or do not count the squares or multiply the lengths of the sides to find the area of a 5 by 6 rectangle? I think that it does not mean that they cannot count to 30, and I do not think it means that they do not know the formula "Area = length x width" or that they cannot solve the multiplication problem 5 x 6. But it does mean that many do not know what area is!

When Sally taught these children in the 5th grade, she complained about them not knowing what area was. She taught them and passed them all and they came into your 6th grade class. You complained and taught them again and passed them into Curt's 7th grade class where the process was repeated. Years later these children graduate and come to college where I teach them calculus and complain that they don't understand area. Then they take physics and my colleagues complain. In fact, they are quite insulting about it and say, What do you people teach in the math department anyway? Are we are going to have to start teaching math in physics?" These kids were not so challenged that they could not learn area in 12 years of schooling. What happened is that we thought we were teaching area, but something else occurred. In this class we will learn a lot that will help us understand this and how we can better succeed.

A smaller third goal will be to explore the use of technology, calculators and computers in learning and teaching geometry. You won't become an expert, but you should learn enough to do some things well and be better able to judge appropriate uses of technology.  A working knowledge of  Geometer's Sketchpad is required for completion of the course.

We will pursue these goals through hands-on discovery workshop activities using lots of manipulative We aim to wed learning geometry with learning how geometry is learned and where it fits into the school curriculum. We want you to leave this course as much more powerful geometers and teachers of geometry.

The class is divided into two basic components, lectures and workshops. These will be held on alternating weeks. The lecture portions will be led by the professor. This component will include lectures, discussions, small group activities, and demonstrations. On the alternating weeks, participants will be divided into smaller groups for workshops. These workshops will include computer and/or calculator training, hands-on activities and problem-solving.

Grading: Grading will be principally be based on

1.    Completion of a reflective notebook. The notebook must include your sketchpad constructions.  Guidelines, suggestions, and a grading rubric will be distributed. 10%

2.    A group project of a school geometry unit will be prepared. Projects will be reviewed by the faculty before completion. During the last class session projects will be presented to the entire class. 15%

3.    Chapter Tests and Final (A chapter test will be given after completion of four chapters). 75% Each test will be of the same weight.  See the class procedures handout for specific information.

4.    Attendance and good humor.

EFFECTIVE USE OF  NOTEBOOKS AND JOURNALS

Notebooks and journals have three primary functions: recording information for later reference, providing structure and guidance for your future use of the materials, and helping you think about what you are learning. The first two functions are, in my mind, principally note book functions while the third is primarily a journal function. I urge you to organize your notebook with these dimensions in mind.

1. In class and while studying, write down the essential features of what you are learning. This will produce a notebook that will be useful during the course, but will be too big and disjointed to be of much value later.

2. At least once or twice a day, go over your recent notes. Think about summarizing or writing short references that will help you find useful information later and put these entries in a second summary section.

3. At the same time, think about items such as your interpretations of what you are learning, the significance of the material and their place in the curriculum. These reflective entries go in a third section, your journal.

In this course, I hope you do these three activities each day. The result can be more useful than any outline or summary that you could make on the spot. You will find that if you do it well, your mathematical and pedagogical understanding will grow significantly.

Your notebook should include at least three sections, one of which will be your journal. The exact format is up to you. Write a few sentences each day.

1. You may want a separate section for the Geometer's Sketchpad materials and commands, for the graphing calculator, or possibly for Logo.

2. Write about your work on tough challenge problems. What is the problem about? What information do you know that will be necessary solve it? What is your plan of attack? Did it work? If it did not work, where did your problems? Summarize what you learned in your journal.

3. When you solve a problem, learn a new concept, have an interesting discussion, or see a new activity, write what you learn. Discuss the difficulties. What did you learn of interest for your teaching?

4. Put your homework in your notebook.

Some suggested topics for your journal entries are as follows:

1. What is the van Hiele model and how does it apply to my classroom?

2. How will I change my teaching to reflect the van Hiele model?

3. What did I learn about geometry today and how/where will I use it next year?

4. What would I change about today's class?

5. How is our group project going? What help or resources do we still need? What is your goal?

It is most important that you use the notebook and journal . Think about what you are learning and write about it. Our job is to become powerful geometers and geometry teachers. One of the ways we can do this is through our notebooks and journals.

Our objective here is for you to write a unit of geometry and measurement lessons making use of what you learn in this course. Ideally, this will start with a unit that you now teach. You will do this as part of a team of teachers who teach similar content and grades.

Requirements:

1. The finished unit must be a coherent set of lessons that fit into your curriculum and grade level and have clearly stated objectives which are addressed by the project activities. A unit on symmetry built around tessellations might be fine, but a unit on tessellations that never gets beyond making attractive patterns is not acceptable.

Include a summary of the unit with remarks on what you have been doing and how this differs from that. The summary needs to include a description of the content and a statement of your objectives. The objectives should be embodied in the unit.

2. Relate the unit to the Virginia SOL and the NCTM Standards. State which of the standards are addressed by the unit. This should be tied to requirement 1 above. Does your unit meet the first four NCTM Standards: mathematics as problem solving, as communication, as reasoning, and connections?

3. The unit needs to be consistent with the van Hiele model. Describe the van Hiele levels of the activities and objectives with sufficient detail that it is clear that these are consistent.

4. The unit must use some manipulatives and hopefully, some technology. Explain the purpose of each. How is the lesson enhanced by these materials and technology?

5. Follow a format similar to that of the sample projects. If possible, submit the unit in a word processor format.

6. It is desirable for your unit to make connections with other subjects .

7. Try to include references to related materials that could be used to extend the unit or adapt it to other audiences.

Satisfactory projects will address the above standards 1 through 4. Grades will be determined by averaging how well these standards are met.

Writing a Project: Start as soon as possible. First study the SOL and the source materials that you received. Look at the Georgia lessons and the Addenda volumes. You need to choose a topic. Then think of what you now teach and where you would like an improved unit. Talk with your team members and compare notes. Talk with your instructors. Touch all these preliminary bases during the first few weeks of class.

Get approval for your project before beginning work! Once the outline of your unit has assumed a clear shape, you need to turn in a rough preliminary draft for comments and final approval. Your instructor will inform you of deadline dates.

Topics: Topics may be adapted to a variety of levels and can be combined with other topics. The following is a list of possible topics. Many are broad and contain sub-topics which could be explored. The list is not at all complete .

1. Measuring lengths: Units. perimeter, formulas, estimation, regular polygons, the perimeter of circles, length of circular arcs, measuring distances on a sphere, similar triangles, applications, the size of the earth, history and fractals.

2. Measuring angles: Pre-angle concepts, right angles, angle comparisons, measuring with wedges and building a protractor using a protractor, length of circular arcs, estimation, constructing angles, shadows, similarity applications, and history.

3. Area: Pre-area concepts, determining areas with square tiles, units, approximating areas, estimation, surface area of polyhedra and regular prisms, paint brush area, formulas, area of a circle, similarity,  and history.

4. Volume: Pre-volume concepts, determining volumes with cubes, units, approximating volumes, volumes of regular prisms, formulas, playing card volumes similarity.

5. The Pythagorean theorem with proofs, history, applications, and similarity.

6. Coordinate geometry.

7. Triangles and quadrilaterals.

8. The analysis behind building and classifying prisms and platonic solids is rich. Visualization and drawing could be part of a unit here.

9. Symmetry, the mathematics of reflections and kaleidoscopes, the classification of transformations, and the analysis of tiling are all related.

10. Sperical geometry can be connected to geography and science to create an interdisciplinary unit.

ACADEMIC INTEGRITY POLICIES:

Students are expected to attend all class sessions. Each session is 3 hours with a break.  This is the equivalent of 1 week of classes. Consequently, missing 2 classes is the same as missing 6 regular classes.  Missing 3 or more sessions may result in an automatic failing grade. Further information regarding academic or academically related misconduct, and disciplinary procedures and sanctions regarding such misconduct, may be obtained by consulting the NSU Student Handbook.

Outline
Week1        Introduction and Van Heile,  Networks, Chapter 0
Week 2      Chapters 1 & 2, Logic
Week 3     Chapters 3 & 4, Sketchpad
Week 4     Measurement of Length, Test Chapters 1-4, Chapter 5
Week 5    Chapter 6 & 7, Sketchpad
Week 6    Chapters 8 & 9, Area,
Week 7    Test Chapters 5 - 8, Chapter 10,  Tesselaations
Week 8    Chapters 11 & 12, Volume
Week 9    Chapters 13 & 14
Week 10   Test Chapters 9 - 12, Chapter 15, Logic
Week 11    Chapter 15
Week 12    Chapter 16
Week 13    Chapter 16, Logo
Week 14    Logo and Sketchpad
Week 15     Final