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