A Syncretistic Experiment in Mathematics & Visual Arts Education September 2000 - January 2001

1. LEARNINGBETWEEN THE LINES A Syncretistic Experiment in Mathematics & Visual Arts Education September 2000 - January 2001 Daniel H. Jarvis

2. It is the supreme art of the teacher to awaken joy in creative expression and knowledge. Albert Einstein

3. OVERVIEW RESEARCH TITLE RESEARCH QUESTIONS RESEARCH INSTRUMENTS INTEGRATED ASSIGNMENTS RUBRIC ASSESSMENT OPEN HOUSE PRESENTATION OF FINDINGS NEXT STEPS

4. Syncretism is the reconciliation or fusion of differing systems of belief, as in philosophy or religion, especially when success is partial or the result is heterogeneous. • This term has been applied to the title of this research experiment due to: (i) the traditionally acute polarity that is popularly understood to exist between the disciplines of mathematics and visual arts; and (ii) the realistically projected imperfection of this philosophical and practical fusion. RESEARCH TITLE

5. Linedefies simple definition. A dictionary may list over 30 working definitions from a wide variety of professions, traditions, and vernacular expressions. • The title of the proposed study seeks to communicate two key elements of the research: (i) that, as in the common phrase reading between the lines, a higher level of thinking is required to ascertain truth or meaning, so too an integrated curriculum seeks underlying principals and elemental connections; and (ii) that the Ontario Grade 9 curriculum for both mathematics and visual arts is intrinsically tied to the concept andconstruction of the line in two- and three-dimensional space. RESEARCH TITLE

6. ACTION RESEARCH The steps involved in, or the key components of, Action Research have been listed in a variety of different, yet similar ways. Whitehead, an innovator and respected authority on Action Research, listed the following outline: Identify a problem in my practice; Imagine a solution to the problem; Implement the solution; Evaluate the solution; Modify my ideas and my practice in the light of the evaluation.

7. RESEARCH QUESTION • What affect would the implementation of three integrated mathematics and visual arts assignments have on student perceptions and learning in the mathematics classroom?

8. SUPPLEMENTARY QUESTIONS • 1. Were the segments of the integrated curriculum pertaining to the Communication of Mathematics perceived by students as having reinforced the new math learning? • 2. Were the segments of the integrated curriculum pertaining to the Creative Application of Mathematics perceived by students as having reinforced the new math learning? • 3. Were the integrated assignments perceived by students as having had any effect on student motivation in the course?

9. SUPPLEMENTARY QUESTIONS • 4. What are some unique qualities of the integrated learning process that become evident through the various research instruments, and through the writing of the integrated assignments? • 5. What patterns emerge from the assessment of student work, regarding the various Achievement Chart categories?

10. RESEARCH INSTRUMENTS Data for this research was collected using the following seven instruments: Field Notes Three Integrated Assignments Video-taped Presentations Student Questionnaires Final Student Survey/Questionnaire Interviews Open House/Response Form for Parents/Guardians

11. GRADE 11 PILOT STUDY RATIO, PROPORTION & THE GOLDEN SECTION Spring 2000

12. READ GRIFFITHS’ ARTICLE ENTITLED, MATHEMATICS AT THE TURN OF THE MILLENNIUM,JANUARY 2000 • ANSWER QUESTIONS & CHOOSE ONE IDEA THAT INTERESTS YOU THE MOST • COMBINE THE GOLDEN SECTION RECTANGLE WITH THE KEY IDEA FROM THE ARTICLE; ANY MEDIA • SUBMIT A WRITTEN DESCRIPTION OF YOUR PROJECT • PRESENT TO CLASS

13. GRADE 11 ADVANCED CLASS & EXHIBITION OF WORK THE GOLDEN RECTANGLE & THE GOLDEN SPIRAL

14. SW 1

15. “The @ sign is used to represent math in computers; the reason the sign looks sketchy is to portray the uncertainty of where it will take us. The green arrow indicates that although we do not know where math will take computers, we know that it will be taken forward. The human eye depicts the relation between beauty and math, as well as the way math opens your eyes to many different perspectives. The square that has warm colours shaded in the background and an abstract mass of ‘purple squiggles’ represents the long and sometimes senseless path some mathematicians take to solve a math problem. The rectangle with the 8 inside it was made as a print from a cut potato. The beauty of the number 8 displays that math can be beautiful, and that it can be found in nature. One rectangle is flaked with gold powder, representing the golden section. The ‘papier mache’ galaxy around the whole rectangle was used to please the eye as well as to depict the universe.” SW 1

16. SW 2

17. SW 2 “I began with the Pythagorean Theorem, for it was Pythagoras who discovered the musical scale. His principles of geometry formed the basis of mathematics for the next two thousand years. Thus the prominence of Pythagoras in the first square of the golden section rectangle. The music notes and the bow divide two right triangles of the Pythagorean Theorem. The next portion came from two notes on the the latest mathematical discoveries. The first portion looked into the constant nature of linear waves, being such that an ‘E’ (musical note) would still sound like an ‘E’ a block away. Thus the focal point of the spiral in the golden rectangle is centered on the ‘E’ string of the violin. The violin itself was used as a reference to the proposed String Theory as a Unified Field Hypothesis. The idea is that the essential building blocks of matter are small loops or ‘strings’ that vibrate like the strings of a violin. These ‘strings’ are depicted as flowing from the focal point of where the E-string of the violin is centered over the F-opening, and curve around following the spiral. The final portion is a block of ‘Phi’ symbols, the Greek letter that represents the numerical ratio of the golden section.”

18. SW 3

19. SW 3 “Before reading the article, I had no idea that mathematicians challenged each other with major math problems. It made me really realize how dedicated some people are to math. For example, Thomas Hales spent ten years solving Kepler’s Sphere Packing Conjecture. With my art project, I tried to show the three famous solved problems, and the next three that people are probably working on at this moment. The three solved problems lie on top of my sort of messy pile of stuff (made to look like it was an area that was torn up and dug into), while the other three are still buried. I have three golden rectangles in my project. One is vertical,and the other two share the same rectangle, but one is upside down. The three unsolved problems are at the infinite points of the rectangles. To further attract attention to those points, I used red construction paper. I liked the idea of the torn paper because in my mind I picture the mathematicians slowly peeling back layers of ‘stuff’ until they finally can clearly see their solution.”

20. SW 4 “My project was done ‘last minute.’ I did think it over a lot though. I’ve always liked pictures from the Chandra or Hubble [telescopes] and I’ve always thought of math as the universe. It’s very complicated, but if you look at it long enough, you see the patterns and obviously some people take longer to find the pattern, or don’t even more than glance at it.”

21. SW 5

22. SW 5 “I used the piece of bristol board that you supplied to mount my project. The golden rectangle was the board itself. I trimmed it to the right size and coloured the lines with marker. There is a picture of a SPA, representing the philosophers Socrates, Plato, and Aristotle. I found a picture of Albert Einstein and posted him up also. There is a picture of the Greek Parthenon which has many golden rectangles in its architecture. There are four coloured circles near the top that represent the ‘Four Colour Problem.’ I also found a picture of Descartes who has a math contest named after him. On the computer I found some interesting charts like the ancient math characters. I also found a table describing functions. I also made a small graph showing how to graph and solve systems. I really enjoyed this project as it provided a break from dull things and also showed me an interesting twist to math.”

23. SW 6

24. SW 6 “When I first decided to start my project, I knew that I wanted to do something related to the Sphere-Packing Conjecture combined with the golden rectangle. Then I decided that I wanted to do it in three dimensions, to be different. My main problem was deciding how to do that. Then my dad and I went through it to see if we could see anything that would allow us to start the project. When I realized that they both started as perfect squares, my mind started dancing and I knew exactly what I was going to do.”

25. SW 7 “My project was done in pencil crayon. It is very small because I had only a small ruler and compass to do measurements. It’s based around Kepler’s Sphere-Packing conjecture. The orange tree is used because oranges are spheres and are stacked in grocery stores for display. Cannonballs are also spheres that get piled by sailors on ships . . . So I drew a pirate. The orange is in the center of the spiral.”

26. SW 8

27. SW 8 “My art project was the large, heavy one that kept falling down. I thought this was somewhat ironic because math, as always, is a heavy load on my shoulders. When I was reading the article, what struck me was how all these common-sense, simple rules lead towards each other. Addition to subtraction, then to multiplication, then division, and so and so forth, and they all relate to something bigger than themselves. This is why I chose to do a colour spectrum, or my version of one at least. I tried to make each colour blend into the next one. I put white in the middle because when you start math, it’s all so simple and obvious. The black on the outside represents the fact that math leads outward to infinity. But then it looked too plain, so I added my quotation and metaphor. Then I got really creative and made my Phi/Pi (the little yin/yang symbol).”

28. SW 9

29. SW 9 “As I read through the handout, I came to the intriguing ‘N versus NP’ problem. Could N = NP? And consequently, could reality equal the imaginary? Was the world that I see, in fact the same world that everyone else sees? I sat there at my desk and thought about this for quite some time until I concluded that ‘yes, to some extent, my world was your world.’ So, due to my complete lack of a sense of reality, I chose to paint my visualization of reality equaling the imagined. The line which cuts the rectangle to form a new rectangle, and subsequently a perfect square, acts as my ‘edge of reality,’ so to speak. The man crossing this line is in the process of becoming ‘unreal.’”

30. SW 10

31. SW 10 “The golden section that I made relates to great minds. Small thoughts and ideas which seem irrelevant at first sometimes turn out to be the most important. This is the reason that I chose quotations. Great minds came up with these quotations about math. Some are positive and some are negative, but it is better to see both sides than just one. The smallest square of the golden section lies on the hands of Albert Einstein. As you can see, his hands are clasped together, which I think shows unity. That is what math is about. Although math is its own subject, it unifies other things like numbers, words, communication, and art.”

32. SW 11 “From the article, I found the mathematical problems and how people worked so long and hard to figure them out, to be the most interesting part. The whirlpool in my picture represents the problems and the boat represents the people who have to work to overcome them. I used the golden section to find the swirl pattern of the whirlpool and the boat is located at the vanishing point of the golden rectangles.”

33. SW 12 “My project was a small golden rectangle with a different theme in each square section. These included: quantum computers, the Four Colour problem, the Sphere-Packing Conjecture, music, and the Pythagorean Theorem. Also, the mouse pointer on the computer screen is in the focal point of the golden rectangle.”

34. SW 13

35. SW 13 “This assignment shows the author’s [article] concern for math in the future. It also shows a progression, from addition to a whole new sphere of thought. It shows how math is being discovered in all different countries. Another look will show how math can be seen in science, nature, and music. Also, the spiral focal point is covered by a ‘thumbs-up’ … there is hope [for me].”