Intuitive Geometry: Where Mathematics, Visualization, and Writing Meet

1. Intuitive Geometry: Where Mathematics, Visualization, and Writing Meet Teresa D. Magnus Rivier College tmagnus@rivier.edu Viewpoints Grand Reunion, June 14, 2008

2. 2004 • Rivier College adopts WAC/WID program • Viewpoints! • Wanted to adapt our Intuitive Geometry course so that it would meet the requirements of the WAC/WID program and be mathematically meaningful to the students.

3. Personal Philosophy • Discovery/Inquiry-based learning gives students a richer and more memorable experience. • Observing patterns is an accessible and engaging mathematical activity for most students. • Students need to move beyond the observations to an understanding of the big ideas. • Students should have the opportunity to experience mathematical reasoning.

4. Why writing? • Hone writing skills through the study of mathematical ideas. • Develop a more thorough understanding of geometry through writing. • Enable students to write with precision and clarity and to use visual diagrams and tables in papers.

5. Additional goals • Teach geometrical concepts that will be useful to students majoring in elementary education, art, and other humanities courses. • Have students experience mathematics as a creative art. • Give non-math majors a chance to recognize that they can think mathematically.

6. Planning Process • Looked for ways to bring Viewpoints ideas into the course. • Began to develop and write my own labs and materials. • In the end, chose not to recreate the wheel, but used a textbook with some labs of my own.

7. Symmetry, Shape, and Space: An Introduction to Mathematics through Geometry -- L. Christine Kinsey and Teresa E. MooreKey Curriculum Press • Discovery/Inquiry based. • Without writing component, students seemed to miss the big ideas. • Used a writing assignment to help students focus on the big questions while doing the lab and encourage them to discuss the reasons behind their conjectures.

8. Writing in the Course Most assignments took the form of written summaries/explanations of the discoveries made in class. Needed to transition students into mathematical writing.

9. Measurement Pythagorean Theorem Regular Polygons and Tiling Billiard Patterns Compass Constructions Star Polygons Semiregular Tilings Polyhedra Symmetries The Course

10. Measurement and Basics • Ping pong ball problem: How many ping pong balls will fit into this classroom? • How much carpet? Paint? • Reviewed visual/mathematical arguments of how the formula for the area of a rectangle gives rise to area formulas for other shapes. • Also discussed volumes of prisms and cylinders, pyramids and cones. • Computational homework with emphasis on adding and subtracting regions.

11. Pythagorean Theorem • Students arranged four right triangles and a square to come up with the formula.

12. Transition into Mathematical Writing • Day one in-class writing assignment on a mathematical turning point in his/her life. • “How to” paper (introduction to precision writing). • In-class “blind copying” activity. • Provided students with sample summary papers (good and bad) introducing and proving area formulas for parallelograms, triangles, and trapezoids.

13. Regular Polygons and Tiling • Guided explorations to discover the formula for the vertex angle measure of a regular polygon and the only regular polygons that give a monohedral tilings. • Writing assignment: Write a paper that identifies the three regular tilings and explains why these are the only three. Use the vertex angle measure formula and explain why it works.

14. Billiard Patterns • Used NCTM web applet to explore the path of a billiard ball shot from the corner of an mxn-table with pockets only at the corners. Kept records. • Looked for patterns and made conjectures. Dividing dimensions by greatest common factor played a major role. • Wrote paper summarizing their discoveries. http://illuminations.nctm.org/Lessons/imath/Pool/PoolTable/pool.html

15. Compass Constructions • Covered traditionally for the most part. • Students were challenged to discover how to construct particular lengths, angles, and polygons. • Research paper on either the history or application of classical constructions.

16. Star Polygons • Explore number and lengths of cycles formed by connecting every kth point of n equally spaced points on a circle. • Homework involved stating conjectures. Some students wrote this up as a paper.

17. Semiregular Tilings • In class exploration of when two or more regular polygons can be used in a semiregular tiling. • Students wrote a paper explaining the rules they used to narrow the possibilitiess and showing the resulting patterns.

18. Polyhedra • Traditional look at prisms, pyramids, and Euler’s formula. • Construct regular and semiregular polyhedra from tagboard and Polydron shapes. • Investigate the role of vertex angle measures and odd/even numbers of sides in determining whether certain semi-regular polyhedra can be built. • Paper summarizing which semi-regular polyhedra can be constructed and why.

20. Symmetry and Final Paper • Explore the four types of symmetry and identify whether certain patterns are preserved by them. • Develop and apply a group table for the symmetries of a square. • Create a GSP rosette pattern and kaleidoscope. • Final paper: Revise one of the previous summary papers or write a paper on the patterns connected to star polygons.

21. Challenges • Irregular attendance of some students made it difficult to get labs done. • Many students were weak at both writing and mathematics. • Only a few students have room in their schedule to take this course.

22. Successes and Benefits • Students developed an awareness for and appreciation for the use of precision in writing. • Students saw mathematics as a creative discipline rather than manipulation of formulas. • Students worked as a group to discover the mathematics. They even took over the blackboard so that they could share and observe the patterns. • Students learned how to effectively test conjectures. • We had fun!

23. Student quote • “Writing mathematical papers made me a more detailed and precise writer. When it comes to writing these papers you need to explain what everything is and what everything stands for …if you leave out the slightest step, it can throw the whole paper off.”

24. Student quote “When writing papers in other classes, you are basically writing information from other materials or your opinions. However, in math, you need to supply pictures to support your thoughts and help explain what you are trying to say.”

25. Student quotes “Unlike written papers in other courses, writing in math must be precise and without opinions…[It] must also be in some kind of order…certain concepts must first be explained…before other concepts. And unlike some writing courses, writing in mathematics courses does have a right and a wrong answer…

26. Student quote continued …And most obviously, writing in a mathematical course includes numbers, algebraic and geometric concepts. In order to write a math paper, a person must be able to explain math concepts using not only numbers but words as well.”

27. Student quotes “ By writing mathematical papers, I expect it will be easier to teach math. I foresee this because by writing papers, the processes of the math problem become more apparent to me and help me to understand the problem better…Math is very methodical and having experience writing math problems in the same step-by-step manner will help me to better convey my knowledge to my students.”

28. Later Summer 2004 Dalsnibba, Norway

29. Viewpoints and Parenting Paul touches the top of the Alnes Lighthouse near Ålesund, Norway

30. Paul’s incomplete sketch

31. Paul enjoys Amanda Serenevy’s origami workshop

32. Thank you.

33. Contacts and Web Sources Teresa Magnus tmagnus@rivier.edu NCTM Illuminations Pool Table Activity: http://illuminations.nctm.org/Lessons/imath/Pool/PoolTable/pool.html Textbook: http://www.keycollege.com/catalog/titles/symmetry_shape_space.html

34. Assessment and computation of grades: • Homework assignments (writing and computational) 10 points each (80-130 points total) • Researched mid-semester paper 30 points • Tests 40 points each (80 points total) • Participation in class activities and peer writing reviews 20 points total • Final revision of paper 40 points • Total 250-300 points