Wednesday, September 7, 2011

Four Stages of Mathematical Learning

The question becomes, “What leads a student to choose a given style when presented with a new concept?” Variations in learning style are often due to how successful a student has been in translating a new idea into a well-understood concept. Indeed, it appears that each of us acquires a new concept by progressing through 4 distinct stages of understanding:

  • Allegorization: A new concept is described figuratively in a familiar context in terms of known concepts.
  • Integration: Comparison, measurement, and exploration are used to distinguish the new concept from known concepts.
  • Analysis: The new concept becomes part of the existing knowledge base. Explanations and connections are used to “flesh out” the new concept.
  • Synthesis: The new concept acquires its own unique identity and thus becomes a tool for strategy development and further allegorization.

It then follows that the learning style of a student is a measure of how far she has progressed through the 4 stages described above:

  • Allegorizers: Cannot distinguish the new concept from known concepts.
  • Integrators: Realize that the concept is new, but do not see how the new concept relates to familiar, well-known concepts.
  • Analyzers: See the relationship of the new concept to known concepts, but lack the information that reveals the concept’s unique character.
  • Synthesizers: Have mastered the new concept and can use it to solve problems, develop strategies (i.e., new theory), and create allegories.

It also follows that a student’s learning style can vary, although in practice a student’s style tends to remain constant over a range of similar concepts.

Reference: Knisley J, A Four-Stage Model of Mathematical Learning, Department of Mathematics, East Tennessee State University, [Online] Available form :

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