Northern Virginia Community College

MTH 150 Topics in Geometry Fall 2006

T 4:30 – 7:10 PM AA 345

INSTRUCTOR: Ms. Trudy Streilein

OFFICE: 352 Bisdorf Phone: 703-845–6522 (leave a message)

E-MAIL: tstreilein@nvcc.edu

WEB PAGE: www.nvcc.edu/home/tstreilein

OFFICE HOURS: MW 7:30-9 AM, 11AM-12 PM, 1:30–2:30 PM, T 2 - 4:30 PM, R 12-2 by appointment

TEXT: Geometer's Sketchpad, version 4 ( http://www.keypress.com/ or bookstore)

Exploring Geometry with Geometer’s Sketchpad, Bennett, Key Curriculum Press, 2002

ISBN# 1-55953-581-4

This is an informal course in Geometry that is primarily for math teachers at any level through high school. For those teaching below the high school level, this course will adequately prepare you to teach geometric concepts. For those whose teaching will include high school mathematics, this course should be followed by the more formal and rigorous MTH 250, College Geometry.

This is not a course comparable to a traditional high school or other geometry course. If at any time you wish to review the content of a typical high school geometry course, you might want to purchase a basic geometry book for reference. Recommended texts include:

Essentials of Geometry for College Students. Lial, Brown. 2^nd ed. 2004. Addison Wesley,

ISBN 0-201-74882-7

Geometry to Go, A Mathematics Handbook, Houghton Mifflin Company, ISBN 0-669-48129-7

Schaum’s Easy Outlines: Geometry, McGraw Hill, ISBN 0-07-136973-2

Areas covered in this course include (1) geometry and measurement, (2) geometric awareness and spatial reasoning, and (3) transformations: symmetry, congruence, and similarity. We will have a mix of lectures and hands on in-class activities. You will also be introduced to the SOLs that to pertain to geometry; we will discuss the connection between the topics in this course and the related SOLs.

Many of the course activities will be appropriate for the middle school classroom. Loren Pitt, a mathematics faculty member at UVA, has provided the bulk of the course materials. He has been developing and teaching a comparable course at UVA for several years. While most of the activities will be low-tech, we will also be introducing Geometer's Sketchpad, a dynamic geometry software package. There will be infrequent use of graphing calculators, such as the TI-83.

Withdrawal Policy:

You may withdraw from the course up to September 5 and still receive refunds, up to October 27, without grade penalty. It is your responsibility to initiate this action. I will NOT sign withdrawal slips after October 27, don’t even ask. I will not approve any audits; you have to actively participate in order to learn from this course.

Special Needs and Accommodations:

Please address with the instructor any special problems or needs at the beginning of the semester. If you are seeking accommodations based on a disability, you must provide a disability data sheet, which can be obtained from the counselor for special needs, who is located in Room 148 of the Bisdorf Building, telephone number 703- 845-6301.

Technology:

Please turn OFF the ring on all beepers, phones, etc. when in class. The interruption of the rings and beeps is very annoying. Any phone calls will be for me!

Fire Exit Route:

We will review the appropriate exit route for our room on the first day of class. If you are physically disabled please see me the first day of class for specific instructions for you to follow.

Grading Policy:

The course contains some lecture material and lots of in class, workshop, and homework activities. The workshop type components will include small-group activities and demonstrations. In the workshops participants will be divided into smaller groups. These workshops will include computer and/or calculator training, hands-on activities, Sketchpad, and problem solving.

Grading: Grading will principally be based in equal parts on:

1. Completion of a reflective notebook

2. Preparation and presentation (during the last class sessions) of a project that consists of a classroom geometry lesson. (Details below)

3. Attendance, active participation, and good humor.

4. Completed homework assignments: always due the following week, mailed or emailed if you are absent; late papers are not accepted. If you are absent, you must download that weeks packet from my website and complete and turn in everything, what we did in class as well as the new homework.

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 principally notebook functions while the third is a journal function. I urge you to organize your notebook with these dimensions in mind.

Your notebook should include everything reasonable with at least three sections, one of which will be your journal. Journals may be collected at mid-semester.

1. The first section will be class notes and homework. In class and while studying you should write down the essential features of each class. 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 unless finely referenced.

2. The second section will be summaries. At least once a week go over your recent notes and summarize them by writing a short synopsis of that work so that you will be able to find useful information later.

3. Finally, the third section will have you reflect on that material. Think about items such as your interpretations of what you are learning, the significance of the material and its place in the curriculum. Was the homework particularly difficult one week? How did you break it down to complete it? Has the Van Hiele model finally made sense? Have you been abusing it with your students? Once you start your final project, be sure to start a thread of reflection on it as well. What is working, frustrations, problems, etc? Do you have an idea that would cover the class material in a better fashion?

The journal may be the toughest part, to relax and actually reflect on the impact of the material. It should be given thought, not just thrown together at the last minute.

In this course you do these three activities at least weekly. The result can be more useful than any outline or summary that you could make on the spot. And, if you do it well, your mathematical and pedagogical understanding will grow significantly. Note: The second two should always be typed.

• You may want a separate section for the Geometer's Sketchpad materials and commands-its up to you-or just include them weekly when given in your first section.

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

The Math Archives web site: http://archives.math.utk.edu/ Other interesting web sites are linked from my home page. Check them out.

ABSOLUTELY NO FOOD OR DRINK IN ANY COMPUTER LABS

Tentative Schedule:

22 Aug Introductions

Measurement - Perimeter

Making a royal ruler

29 Aug Angles and measuring angles

Introduction to Geometer’s Sketchpad

5 Sept Area

Area activities

12 Sept Dot paper area & Pythagorean theorem

Square Twits

19 Sept Cut & reassemble area activities

Building squares around a right triangle

Pythagorean activities and problems

26 Sept Surface Area Activities with cubes

Polyhedron

3 Oct Surface area for Simple Closed surfaces

Discovering

10 Oct No classes

17 Oct Polyhedron and volume of pyramids and cones

24 Oct Volume-cutting solids into other shapes

Geometry awareness – Scavenger hunt

31 Oct Visualizations with cross sections

Triangle and Quadrilateral Activities

7 Nov Intro to geometric analysis – concepts and vocabulary

Parallel lines and Euclidean geometry

Geometry on a sphere

14 Nov Elementary Constructions

Patty paper/mira/compass & straight edge

21 Nov Transformation, symmetry, and similarity

28 Nov Symmetry problems

Tessellations and Transformation

5 Dec Project presentations

12 Dec Project presentations

About This Course

Our focus in this course is the informal geometry that relates to our curriculum in grades 5 through 8. The informal geometry taught in these years is a foundation that our students need to understand the physical world they live in. Although this material has long been neglected in our curriculum, the material is essential for our students understanding of their physical world. When the material is taught well, students find this subject matter to be interesting and motivating, but when students do not develop an intuitive understanding of the way our world fits together, i.e. an understanding of geometric attributes, proportions and relations, these students do not succeed in later science, technology, and mathematics studies. In addition our students are now required to pass SOL tests in geometry.

To teach geometry successfully our classroom teachers need the geometry content contained in this course as well as a deeper understanding of how students learn geometry. Our course's primary goals are to teach this informal geometry and measurement content and to foster an understanding of how children learn geometry. In addition, the course models effective teaching strategies. We believe that geometry has long been neglected in large part as a consequence of our lack of success in teaching geometry. Supporting evidence for this belief may be found in the following data first published in 1988:

One can not help but ask the question, 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? It certainly does not mean that they cannot count to 30 nor that they do not know the formula "Area = length x width" or that they cannot solve the arithmetic problem 56. But it does mean that many do not know what area is! They simple did not have enough meaningful experiences with area and perimeter that the concepts became meaningful to them.

When our hero, Sally the fifth grade teacher, taught these children she undoubtedly complained that they did not know what area was. She taught them and passed them all and they came into your 6th grade class and you complained and taught them again and passed them into Burt's 7th grade class where the process was repeated. Years latter these children graduate and come here where I taught them calculus and complained again that they don't understand area. Perhaps they then took physics and my colleagues complained again. In fact, they were probably quite insulting about it and said, "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 cannot learn about area in 12 years of schooling. What is happening here is that we all thought that we were teaching area, but something quite different occurred. We all continued to repeat the same lesson that failed from grade 5 through grade 12. In this class we hope that you will learn a lot about the concept of area that will help you teach it more effectively.

A final goal of the course 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.

These goals will be pursued using effective hands-on activities that use many physical objects and manipulatives. The aim will be to wed learning geometry with learning about how geometry is learned and where it fits into the school curriculum. I hope that all will leave this course being much more powerful geometers and teachers of geometry.

The Projects:

The objective here is for you to write a lesson for geometry and/or measurement making use of what you learn in this course. If possible, this will start with a lesson that you now teach, but will now take a significant change in presentation.

Requirements:

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

Include a summary of the lesson with remarks on what you have been doing and how this lesson 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 lesson.

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

3. The lesson 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 lesson must use some manipulatives and hopefully some technology. Explain the purpose of each. How do these materials and technology enhance the lesson?

5. To the extent that it is possible, submit the lesson in a word processor format.

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

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

Satisfactory projects must address at least standards 1 through 4. Grades will be determined by averaging how well these standards are met.

Writing your Project:

Start as soon as possible. First study the SOL and the source materials that you received. You need to choose a topic. Then think of what you now teach and where you would like an improved lesson. Talk with your instructor. 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 lesson has assumed a clear shape you need to turn in a rough preliminary draft for comments and final approval. I will inform you of deadline dates.

Possible 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 that could be explored. The list is not at all complete.

1. Measuring lengths: Lessons, 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 even 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, history.

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

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

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

6. Coordinate geometry.

7. Triangles and quadrilaterals could produce many different lessons.

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

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

10. Spherical geometry can be connected to geography and science to create an interdisciplinary lesson.

LEARNING GEOMETRY AND THE VAN HIELE MODEL

Some of us have experienced situations where our best teaching efforts failed and although we prepared and presented an excellent mathematics lesson our students did not learn. At the end of the class period in which we had given our lesson on measuring angles, we asked a bright and interested student to measure a particular angle of 45°. This student picked up his protractor, measured, smiled, and reported his answer of 135°. We know how he arrived at this answer. He misread the protractor. It is a simple mistake. We calmly explain again the correct way to read the protractor. We then dismiss class and start preparing tomorrow's lesson on SOL number 6.21, but we don't feel good. In our hearts we know Johnny just did not get it. Johnny did not know what we were talking about.

There are reasons for why this happens and sometimes the reasons are better understood than others. In learning geometry there is a model that helps explain the learning process more completely than in any other area of mathematics.

This model is not complete nor is it absolutely correct. It will not solve all of your classroom problems, but it can give you a way to think about teaching and learning geometry that will help your students understand and enjoy geometry.

In the 1950's Dina van Hiele-Geldof and Pierre van Hiele were a married couple of Dutch mathematics teachers that were also graduate students of a well-known Dutch mathematician and mathematics educator Hans Freudenthal. Their dissertations grew directly out of classroom experiences like that described above and they reached the conclusion that learning geometry involves a developmental sequence and that this sequence is essential. If our students try to skip levels they are doomed to fail. It is imaginable that Janet and Johnny can run to the top of the ladder taking two steps at a time and not fall, but we know that if ladder climbing were taught by insisting that students run up ladders that are missing half their steps, a majority of your students would fall, fail, and be injured. A majority of our students have been falling on the geometry ladder and the van Hiele model indicates it is because the ladder is missing some steps.

The van Hiele model consists of five levels of understanding and claims that for most people levels cannot be skipped. Going from level 1 to level 3 does not work, or at least will not work unless the students are so very clever - so gifted and talented - that their pure powers of mental agility allow them to overcome gaps in their understanding. To assist in confusion the five levels are labeled as:

0 or 1. Visualization

1 or 2. Analysis

2 or 3. Informal Deduction

3 or 4. Formal Deduction

4 or 5. Rigor

Both numbering systems will occur in your readings, but we will follow Dina and Pierr and use the 0 - 4 system. The model asserts that learners move sequentially and discretely upwards starting with visualization. The transitions only occur if they are accompanied with appropriate educational experiences. It will take a while to bring this into focus so let's begin with examples:

Level 0: Visualization: Learners interpret and react to space and geometrical objects without actual analysis. Properties, components and attributes of figures, if explicitly recognized, will be incidental to the shape. Triangles, squares, and rectangles may be identified and their names learned because "these all look alike." At this level a person will be able to put the square peg in the square hole, but will not recognize that rectangles have all right angles. They may not recognize that a square that has been turned 45° is still a square.

Level 1: Analysis: This is the stage when formal geometric concepts begin to emerge. The properties, components, and attributes of shapes and figures that were not explicitly recognized before now become explicit and important. A person at this level is able to recognize rectangles by their properties - right angles and parallel sides. He can identify equal angles in a variety of figures, but he may not know that squares are also rectangles. Interrelationships between figures will generally not be seen. Definitions and deductions are not generally understood.

Level 2: Informal Deduction: Here students begin to understand interrelationships between figures and recognize common properties. The students would know that triangles and rectangles were both polygons and they would know that a square is also a rectangle because it has all of the properties of a rectangle. Students can understand and begin to formulate informal deductive arguments. Definitions can be understood, but the role of deduction is not understood. Students will mix empirical reasoning with deductive reasoning. They cannot generally extend arguments to new situations and they do not understand the role that axioms play in the formal system of high school geometry.

Level 3: Deduction: This is the traditional level of high school geometry. It is not necessarily what actually happens in the classroom, but it is what we traditionally have hoped would happen. Students at this level will understand the process and role of deduction, the place of axioms, and the need for definitions. Students in Levels 2 and 3 will see that the diagonals of a square are perpendicular bisectors of each other; only the Level 3 students will recognize the need for a formal proof. They can construct proofs and understand that many deductive arguments or proofs may lead to the same conclusion.

Level 4: Rigor or Abstraction: This is the level of much advanced college level mathematics where many of the activities are dominated with formal definitions, constructs, and different axiom systems. Level 4 has received little formal study.

There are three important points to make here.

If these ideas are new to you, then you would surely be surprised at the low level of geometric understanding of your students, their parents, and even your coworkers. A large percentage of the

population never rises above the first two levels. This is related to experiences, not maturity.

It is difficult to communicate across levels. If you talk at level 2 with a student thinking at level 0 very little good will come of it. More than likely, this is what has been happening when our students tell us the angle has 135°. Students who enter a high school geometry class functioning at levels 2 and 3 are very successful. Students functioning at levels 0 and 1 have very low success rates.

For the classroom teacher the first four levels are of primary concern. How do they relate to your

students, the SOL that they are responsible for, and your student's success in more advanced mathematics?

We do not need to know if the model is exactly right, or what precisely separates levels I and 2. The essential issue is that our students are functioning at levels 0 and 1 and by the time they get to high school they need, if they are going to succeed, to function at levels 2 and 3. We can look at this and see that there is a world of difference between these levels, and that a student working at level one does not have a fair chance for success in a high school geometry course operating at level three.

There is a lot that can be said about the process of moving a child from level 0 to 3. All that we will claim here is that this model can give you a useful framework for thinking about teaching geometry. If used with common sense and hard work it can help your students succeed in their geometry studies.