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Levels of Learning

Learning is not a singular threshold to be met. There are different levels of learning – a continuum (see photo below taken from the book Teaching Mathematics Meaningfully).

continuum of learning

A student demonstrating proficiency (fluency) is far different from a student simply showing some level of understanding (acquisition). I remember learning to drive a car with a stick shift. During acquisition (initial understanding) I was looking down at the pedals and the stick shift as I thought through the steps. It is not surprising that many students who only show acquisition of a math topic soon forget it. Despite this, the acquisition stage is often were math in schools resides.

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This extends beyond math fact fluency to all math topics and the students should take the next step and demonstrate maintenance. To do this, I recommend that a curricular based assessment be given a couple of weeks after a student initially showed what is considered mastery – the student successfully performing problems aligned with a given math objective.

Below is a excerpt from the book with an explanation of the topics. I use this text in the math for special ed courses I teach at different universities.

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