Learning:
The Journey of a Lifetime
A Cloud Chamber of the Mind

February 2006 Mathematics Notebook

An Example of a "Learning Process" Journal

The following comments are simply a collage of notes that I found while going through some loose papers that I brought with me from Lethbridge.

• What is mathematics? Clearly a multi-faceted answer is required. Here are two such components: (1) a search for pattern, and (2) a verb. The second highlights the importance of the process. Mathematics is a way of thinking about certain ideas. I wonder what answers a grade 3 class would give? A grade 10 class?

• What are some of the essential ideas of mathematics?

  • Number is one. And then a moment's reflection indicates that there are many kinds of number. Have we identified them all, or are there other families of numbers lurking in the mists of the future?
  • The idea of proof is also important.
  • Infinity is on my list,
  • as is limit,
  • and variable (and function),
  • probability,
  • and fractal.
    
    
• Mathematics is also an attitude, and a language. Both of these dimensions have a developmental aspect. Language tends to improve with age, at least during the school years. And attitude?
  
  
• How many dead mathematicians can you think of? Besides their name, what else can you add? How many living mathematicians can you name? How many female mathematicians?
  
  
• What constitutes a good question in mathematics? What questions would a grade 5 class generate? A grade 12 class? This opens the door to children's mathematics. This is not a question of what children can do as much as it is a question of their understanding.
  
  
• "The brain is a three-pound mass you can hold in your hand that can conceive of a universe a hundred-billion light-years across." (source ?)
  
  
• "Time and space (and number) are modes of thought. They are not something we find ourselves embedded in." (Einstein) (source?)
  
  
• Think of ways kids have changed in the last 20 years. Parents. Teachers. (source?)
  
  
• Name 5 books that you would recommend for a mathematics teacher. Would any of these be mathematics books?
  
  
• The emphasis should be on what YOU want to learn more about, not what someone else decides is good for you. One should initiate their own program of study (much like I have created this web site for my personal purposes.)
  
  
• Three types of knowledge:
  • Platonist: mathematical ideas exist and wait to be discovered.
  • Formalist: axioms and logic lead to new theorems
  • Intuitionist (constructivist): our brain build ideas from what it has already learned.
    
    
• Notational conventions:
  • "at or ta" - a letter has one meaning, whereas 12 or 21 - a symbol's meaning depends on its position.
  • APL
  • Logo
  • StarLogo
  • Maple
  • Mathematica
• Graphical conventions:
  • Euclid - compass and straight edge constructions
  • d'Oresme - use of graphs to represent data
  • Tukey - box & whisker plots
  • Descartes - coordinate geometry (combine algebra and geometry)
  • graphing calculators
  • spreadsheets
  • Maple & Mathematica (algorithms and displays)
  • Logo & StarLogo
  • STELLA
    
    
• Intuition is largely based on experience, yet our experience is usually simple.
  
  
• Getting answers versus open-ended exploration.
  
  
• Future:
  • fast powerful display
  • overlaying displays
  • dynamic displays (Monte Carlo vesus V-8 pistons)
  • networks (displays of the Web)
  • new symbolisms
  • new kinds of diagrams (e.g. Scientific American, Hawking's "Universe in a Nutshell")
    
    
• A good idea is one that is generative.
  
  
• If you think you understand something, create a model of it.
  
  
• Reality is complex, brains are complex, minds are simple (7 + or - 2)
  
• The future belongs to people interested in mathematics education: teachers, students, parents, mathematics educators, mathematicians, psychologists. We need a forum where they may all communicate with one another, and hopefully they will generate new topics, new goals.
  

1:20 PM