Math Representation in Language

The following matrix summarizes the representation of mathematics as a language.

The mathematical objects (nouns) being operated upon by a mathematical action which can be represented as a manipulation (verb) in a manner in which we believe is appropriate for elementary students and therefore a necessary part of an elementary curriculum. See additional resources

Mathematical Objects (nouns)   Mathematical Actions (verbs)  

Types of objects

Properties of objects

 Practicle use

Operations on objects

Number/quantity

  • integers
  • rational numbers
  • real numbers
  • measures of:
    •   length
    •   area
    •   volume
    •   time
    •   weight
  • order
  • between-ness
  • part/whole relationships
  • units
  • dimensions
  • counting
  • measuring

Arithmetic operations

  • addition
  • subtraction
  • multiplication
  • division
  • expotential

Shape/space

  • metric - value between two points
  • lines/segments
  • polygons
  • circles
  • conic sections
  • spherical shapes
  • topological properties - geometric properties and spatial relations unaffected by the continuous change of shape or size of figures.
  • distance
  • location
  • symmetry
  • similarity
  • connectedness
  • enclosure
  • mapping and traveling
  • designing and building objects
  • translation
  • rotation
  • reflection
  • covering, packing, & tessellating
  • scaling
  • projection
  • inversion
  • conformal mapping
  • homotopies/deformations

Pattern/function

  • linear
  • quadratic
  • power
  • rational
  • periodic
  • transcendental
  • input/output
  • domain/range
  • continuity
  • boundedness
  • rate of change, curvature, etc.
  • maxima and minima
  • rate of accumulation
  • linear-root, slope/intercept
  • quadratic-roots, axis of symmetry
  • power-roots, asymptotic behavior
  • rational-roots, singularities, asymptotic behavior
  • periodic-frequency, phase
  • transcendental-"growth" constant
  • expressing how something depends on another - one or more other things
  • resolving constraints
  • (solving equations and inequalities)
  • identifying and describing repetitive relationships
  • arithmetic operations (functions on Rn)
  • comparison
  • equations
  • inequalities
  • identities
  • composition
  • translation
  • reflection
  • dilation/contraction

Arrangements

  • combinations
  • permutations
  • graphs
  • networks
  • trees
  • adjacency
  • enumeration
  • vertices and edges of graphs/networks
  • organizing discrete information
  • tends to be closely connected to the idea of repetitive pattern at this level

Chance/data

  • discrete
  • continuous
  • determinism
  • randomness
  • relative frequency
  • distribution
  • moments
  • dealing with uncertainty
  • dealing with lack of precision
  • sampling (by counts and/or measures)
  • composing
  • representing

 

Dr. Robert Sweetland's notes
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