Intuitive Geometry: Where Mathematics, Visualization, and Writing Meet

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Intuitive Geometry Where Mathematics,
Visualization, and Writing Meet

• Teresa D. Magnus
• Rivier College
• tmagnus_at_rivier.edu
• Viewpoints Grand Reunion, June 14, 2008

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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.

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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.

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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.

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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.

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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.

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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.

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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.

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The Course

• Measurement
• Pythagorean Theorem
• Regular Polygons and Tiling
• Billiard Patterns
• Compass Constructions
• Star Polygons
• Semiregular Tilings
• Polyhedra
• Symmetries

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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.

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Pythagorean Theorem

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

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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.

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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.

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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/P
oolTable/pool.html
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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.

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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.

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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.

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Polyhedra

• Traditional look at prisms, pyramids, and Eulers
  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.

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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.

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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.

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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!

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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.

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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.

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Student quotes

• Unlike written papers in other courses, writing
  in math must be precise and without opinionsIt
  must also be in some kind of ordercertain
  concepts must first be explainedbefore other
  concepts. And unlike some writing courses,
  writing in mathematics courses does have a right
  and a wrong answer

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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.

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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 betterMath 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.

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Later Summer 2004
Dalsnibba, Norway
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Viewpoints and Parenting

• Paul touches the top of the Alnes Lighthouse
• near Ålesund, Norway

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Pauls incomplete sketch
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Paul enjoys Amanda Serenevys origami workshop
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Thank you.
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Contacts and Web Sources

• Teresa Magnus tmagnus_at_rivier.edu
• NCTM Illuminations Pool Table Activity
• http//illuminations.nctm.org/Lessons/imath/Pool/P
  oolTable/pool.html
• Textbook
• http//www.keycollege.com/catalog/titles/symmetry_
  shape_space.html

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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