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).
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.
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.
Math is often considered an esoteric set of information that is disjointed from the reality people face, aside perhaps from money. Sadly, in school, especially in older grade levels, math is indeed presented this way.
A situation as simple as riding an elevator provides opportunities to show and engage a student with math applied in authentic and common situations. For example, the elevator buttons address counting and cardinality (4 indicates a total of 4 floors – ignoring the R), comparison (if we are on floor 2 and need to go down, which floor do we go to?) and measurement (height above ground floor measured in floors). Such situations also provides opportunities for generalization into other settings – the important settings of every day life!
