Making Comparisons
    The Spectrum of Concreteness

    1. People (students stand up in groups, and are counted)
    2. Objects (Cuisenaire rods; counters; pennies, etc)
    3. Cardboard/cutouts of number lines
    4. Number lines as symbols on page.
    5. Numbers as symbols on page

    Disclaimer #1: your spectrum may look different.

    Disclaimer #2: Other levels; ambiguities, etc (E.g., numeral shaped manipulatives with varying weights, etc).

    Advantages and Limitations..?

    1. Concrete enough for child's present cognitive state?

    2. Motivation and relevancy? (People, money, etc)
      Possible bridges: stories; from pictures to * * * to "3".

    3. Generalizability? (e.g, from Natural to Real numbers)
      Un-marked rods and commutativity of addition.

    4. Extendibility? (e.g. from Addition to Multiplication)
      Commutativity: from array of objects, to "area".

    5. Connections to other parts of mathematics?




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