The Importance of Fuzzy Mathematics

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The Importance of Fuzzy Mathematics

• Kay Owens
• Charles Sturt University
• Dubbo
• kowens_at_csu.edu.au

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Key Ideas

• A place to start conceptualising
• Beginning imagery and using imagery
• Noticing
• Problem solving a new tack
• Providing aha moments that can turn negative
  feelings into positives for problem solving
• Overcoming the rigidity of procedures and
  procedural learning in mathematics

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A place to start conceptualising

• Battistas summary
• physical features to relational features
• a word can help
• moving away from procedures
• role in modelling

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Beginning imagery and using imagery
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Noticing

• Attention drawn to relationship by words,
  questions
• Attention drawn by material
• Gazing
• Re-seeing
• Using other ideas
• Letting the mind think alternatively rather than
  following a set procedure that was not developed
  from ones own experience (ie the teachers
  procedure)
• Noticing properties

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Problem solving a new tack

• Real life problems are fuzzy
• Having the disposition to think of alternative
  approaches when faced with a problem
• Not having a mind-set to following set procedures
  in problem solving
• Knowing processes and strategies that might be
  used
• Planning the chunks for problem solving (teacher
  helps initially)
• Having aha moments
• Thinking creatively when faced with a challenge
• Investigating and going beyond the particular
  problem

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Reflecting on Thinking During Problems

• The group will be split into small groups of 3 to
  work on problems different groups will start on
  different numbers and finish with sharing
• Problems are mostly secondary level and varied
  but can be used for a variety of stages

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1
GRAPH 1. RAINFALL, ADELAIDE, 2005 and 2006
What do you think about when studying the first
graph and what questions do you ask yourself?
What is the annual average rainfall in 2005 and
2006? Why has the consumption rate dropped?
Source Bureau of Meteorology, Archive of SA and
Adelaide Monthly Climate Summaries
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1
Do you think fuzzy to get a handle on the graphs
in the first instance? Why does the fuzzy
thinking assist with redirecting your attention
and questioning? How does it fit with the areas
presented on the importance of fuzzy mathematics?
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2

• Allen and his friends are sitting at a large
  round table playing a card game. In this game,
  there are 25 cards in the deck. The cards are
  passed around the table, and each player takes 1
  until there are no cards left. Allen takes the
  first card and also ends up with the last card.
  He may have more than the first and last cards.
  How many people could be playing cards? (Charles
  et al, 1985)

How did you get started? How did you plan a
strategy to find solutions? Did you suspect
their may be more than one solution? Did you
think to investigate further? Or ask what if?
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3

• ABCD is a square. Let E be the midpoint of side
  DC. Using E as the centre and EB as the radius,
  construct a quarter-circle arc that intersects
  extended side DC at point F. Construct a line
  perpendicular to DF at F. Extend AB to intersect
  that perpendicular line at G. ADFG is a perfect
  golden rectangle.
• If three identical golden rectangles intersect
  each other symmetrically, each perpendicular to
  the other two, the corners of the rectangles will
  mark the 12 corners of a regular icosahedron
  (20-sided figure, each side being an equilateral
  triangle). The corners will also coincide with
  the centres of the sides of a regular
  dodecahedron (a 12-sided figure, each side being
  a regular pentagon). (Garland, 1987)

Thinking and drawing this description requires
visualising and that requires some fuzzy
mathematical thinking to get started. How are you
thinking? Try drawing.
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4

• A golden triangle is an isosceles triangle with
  sides 3, 5 or 5, 8 or 8, 13. Sketch.

Why do these work? How can you find more? Are
you thinking fuzzy? What type of thinking?
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Use the similar triangles from the common 7-piece
tangram, continue the division and arrange to
form an attractive spiral design.
Did you expect the result you saw? Was it
pleasing? Is this precise or fuzzy?
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6

• First develop a strategy and then check with
  the calculator.
• Using the digits 8,7,6 (each once only) set out
  like ?? x ?what is the largest possible
  product?
• What is the largest possible product using the
  digits 8, 5, 4 in a similar way?
• Using any four different digits, what is the
  highest possible product set out like (i) ?? x
  ?? (ii) ??? x ?
• Using any five different digits (each once only),
  what is the largest possible product?
• Using any five different non-zero digits (each
  once only) what is the smallest possible
  product? (Clements Ellerton, 1991).

What strategies are you using? Is their a final
solution? How do you think fuzzy mathematics
helps you make these decisions.
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8

• Take any number, square it. Now take one up and
  one down, multiply them. What do you notice?
  (from BBC Horizons, 1985)
• Make diagrams of the square numbers and link to
  sequences of adding numbers. Which numbers?
• Now take cuisinaire rods, make two staircases,
  turn one upside down and join to form a mat. How
  many units all together? How many in the
  staircase? What does 1 2 3 . 9 10
  equal?

How important is experience in making, seeing,
using patterns? Is there a set procedure for
working with patterns?
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9

• On squared paper, draw a rectangle 5 squares long
  and 3 squares wide. Draw a diagonal. Through how
  many squares does it pass? Investigate other
  sizes of rectangle and analyse your results.
  (Bastow, Hughes, Kissane, Mortlock, 1984)

How did you develop your investigation? What kind
of strategies helped you to extend your thinking
about this problem? Did you find that your
initial thinking was fuzzy enough?
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10

• There are two white rabbits and two black rabbits
  in a hat. If two rabbits are drawn out of the hat
  one at a time, what is the probability that they
  are both white?
• You are told that the probability of drawing two
  white rabbits from another hat full of rabbits is
  ½ How many white and how many black rabbits could
  there be in the hat for this to be true?
• Can you find more than one solution to this
  problem?
• Investigate the number of rabbits (black and
  white) if the probability of drawing two white
  rabbits is 1/3, ¼, 1/5, 1/n (Victorian
  Curriculum and Assessment Board, 1989).

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11

• The Gyrocopter
• Take a piece of A4 paper and fold in half. Fold
  in half again to form strip. Tear carefully down
  the fold
• Tear carefully down the centre of the strip to
  half way.
• Leave just over a centimetre space under the
  centre tear and tear in one third from either
  side
• Fold in thirds on one side flap and again of the
  other side
• Turn up the bottom to form a flap about 0.5 cm
• Make sure the blades at the top are learning in
  opposite directions, hold it as high as you can
  and allow to fall.
• How can you change the directions to make the
  gyrocopter work better?

How did this example require a different form of
fuzzy thinking?
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References

• Anderson, J. (2004. Problem solving, patterns and
  probability Opportunities for working
  mathematically. Reflections 30(1), 16-20.
• Bastow, B. Hughes, J., Kissane, B., Mortlock,
  R. (1984). 40 mathematical investigations.
  Perth, WA Mathematical Association of Western
  Australia.
• Becker, J P, Jacob, B. (2000). The politics of
  California school mathematics The anti-reform of
  1997-99. Phi Delta Kappan, 81(7), 529-32, 534-37.
• Charles, R., Mason, R., Nofsinger, J., White,
  C. (1985). Problem-solving experiences in Grade 8
  mathematics. Menlo Park, Ca Addison-Wesley .
• Clements, M., Ellerton, N., (1991). Polya,
  Krutetskii and the restaurant problem
  Reflections on problem solving in school
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  Press.
• Garland, T. (1987) Fascinating Fibonaccis.
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  Seymour.
• Goldin, G. A. (2000). Affective pathways and
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  Mathematical Thinking and Learning, 2(3), 209-19.
  
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  the PirieKieren Theory. The Journal of
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• Pirie, S., Kieren, T. (1992). Watching Sandy's
  understanding grow. Journal of Mathematical
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• Southwell, B. (2004). Investigating problem
  solving. Paper presented to Problem Solving
  working group, ICME-10, Copenhagen.
• Southwell, B. (2004). Sweet problem solving.
  Reflections 30(1), 13-16.
• Victorian Curriculum and Assessment Board (1989).
  VCE Mathematics 1989 CAT 2 Challenging problems
  Reasoning and Data problems.
• Zimmerman, W., Cunningham, S. (Eds.). (1991).
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  mathematics. Mathematical Association of America.