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Mathematical structure#

See also: teaching-mathematics

Nascent idea about the importance of understanding mathematical structures to being able to teach it effectively. But begs the question which mathematical structure? There's probably not one?

Watson & Shipman - learner generated examples#

Watson & Shipman (2008) cite Davis and Hersh (1980) arguing that mathematical structure is manifested "through relationships among variance/invariance and similarity/difference". By understanding/identifying those relationship we begin to

  • define classes of mathematical objects - which can be something physical, a symbolic representation, an abstract idea (classification or relationship)
  • understand the effects of operations
  • express and manipulate relationships
  • etc

In teaching/learning#

US-based common core standards identified avenues for mathematical thinking (see this)

  • Make sense of problems and persevere in solving them

    The core of the mathematical practice, engagement in which can include... - Reason abstractly and quantitatively - quantities and relationships - Look for and make use of structure - organisation or behaviour of number and space - Look for and express regularity and repeated reasoning - repetition in processes or calculations

CSER MOOC#

CSER maths-in-schools MOOC offers the following "common structures"

  • number systems
    • decimal system (base 10) using the digits 0 to 9
    • natural numbers
    • negative numbers
    • binary system (base 2) using only the digits 0 and 1 and used for programming
    • Roman numerals
  • language conventions
  • algebraic rule
    • pronumerals and variables
    • expressions and equations
  • measures
    • length
    • unit
    • scale

  • symbolic representation

    • less than \(<\) and and more than \(>\)
    • \(\div \times = \)

  • spatial operations

    • associative and commutative properties

References#

Watson, A., & Shipman, S. (2008). Using learner generated examples to introduce new concepts. Educational Studies in Mathematics, 69(2), 97--109. https://doi.org/10.1007/s10649-008-9142-4