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Enactive mathematics pedagogy#

See also: teaching-mathematics, ALIVE, low-floor-high-ceiling-wide-walls

Linked to Jerome Bruner's elaboration of enactive (action-based), iconic (image-based), and symbolic (language-based) modes of representation (of tasks). Not clearly delineated, only loosely sequential since they can be translated into each other. However, moving from enactive > iconic > symbolic can be better for learning.

Effective use of manipulatives#

Henderson et al (2020) offer the following

  • Ensure there is a clear rationale

    The main rationale they provide is "used to provide insights into increasingly sophisticated maths"

    More useful advice would be welcome

    Why not identify the situations/reasons why manipulatives are required, when they provide advantage?

  • Enable students to understand the links between manipulatives and the mathematical ideas they represent

  • Try to avoid students becoming reliant on manipulatives

    i.e. manipulatives are used to help students understand the mathematical ideas, but not to solve problems.

  • Manipulatives should act as 'scaffold', which can be removed

    Is this true with all manipulatives/scaffolding

    This seems to clash with Boaler's advice about using hands/fingers to count. i.e. if it is useful, why not use it?

    How does this apply to calculators and other "technologies" which people use to solve matheamtical problems? How would Conrad Wolfram respond to this?

  • Manipulatives can be used to support students of all ages

They also suggest that evidence for number lines is strong, but evidence is weaker for other representations. (Five relevant meta-analyses support concrete manipulatives, two systematic reviews support visual representations)

References#

Henderson, P., Hodgen, J., Foster, C., Kuchemann, D., Deeble, M., Toon, D., Vaughan, T., & Schoeffel, S. (2020). Improve mathematics in upper primary and lower secondary. Evidence for Learning.