On the other hand, there are those who hold that the initial steps of learning (i.e., imitation) as "mimicking" and "not thinking". Imitation is necessary but not sufficient, it's true. But those in the progressive camp of math education focus on the "not sufficient" part of it, and hold imitation of any type in disdain, with the belief that it eclipses "true understanding" whatever their definition of "true understanding" is.
As anyone knows who has learned to play an instrument, learned a golf step, bowling, a dance step, imitation is harder than it looks. With scaffolding, the imitation of a procedure becomes linked to reasoning with it. It serves as a foundation upon which more sophisticated approaches are attached.
I hear the complaint often that "traditional math" does not teach understanding. It is not the case that we teach procedures without context; i.e., "do this if you want to find the discounted amount" or multiplication facts without teaching that it represents repeated addition, etc. We do teach for understanding; we just don't obsess over it, for fear that we are creating "Clever Hans" students. Kids naturally want to know "how to do it", and focus on the procedure more than the understanding piece of it. It doesn't mean that they are learning only the procedure -- some of it sticks, and as the foundational knowledge is added to, they understand more of what it is the procedure represents.
"With scaffolding, the imitation of a procedure becomes linked to reasoning with it. It serves as a foundation upon which more sophisticated approaches are attached."
Exactly. It reminds me of this false dichotomy of some math "reformers" between automaticity with math facts (and the memorization required to achieve it) and the development of number sense.
As educational psychology professor Brian Poncy and math professor Anna Stokke discussed, memorizing number facts (freeing up working memory) enables children to use mental computation strategies with much larger numbers, thereby enabling them to develop even better number sense.
This seems like a spectrum. Not only for feedback from the teacher/parent that gives the cue to the right answer, but also when giving the right answer without understanding.
"With scaffolding, the imitation of a procedure becomes linked to reasoning with it. It serves as a foundation upon which more sophisticated approaches are attached."
Exactly. It reminds me of this false dichotomy of some math "reformers" between automaticity with math facts (and the memorization required to achieve it) and the development of number sense.
As educational psychology professor Brian Poncy and math professor Anna Stokke discussed, memorizing number facts (freeing up working memory) enables children to use mental computation strategies with much larger numbers, thereby enabling them to develop even better number sense.
On the other hand, there are those who hold that the initial steps of learning (i.e., imitation) as "mimicking" and "not thinking". Imitation is necessary but not sufficient, it's true. But those in the progressive camp of math education focus on the "not sufficient" part of it, and hold imitation of any type in disdain, with the belief that it eclipses "true understanding" whatever their definition of "true understanding" is.
As anyone knows who has learned to play an instrument, learned a golf step, bowling, a dance step, imitation is harder than it looks. With scaffolding, the imitation of a procedure becomes linked to reasoning with it. It serves as a foundation upon which more sophisticated approaches are attached.
I hear the complaint often that "traditional math" does not teach understanding. It is not the case that we teach procedures without context; i.e., "do this if you want to find the discounted amount" or multiplication facts without teaching that it represents repeated addition, etc. We do teach for understanding; we just don't obsess over it, for fear that we are creating "Clever Hans" students. Kids naturally want to know "how to do it", and focus on the procedure more than the understanding piece of it. It doesn't mean that they are learning only the procedure -- some of it sticks, and as the foundational knowledge is added to, they understand more of what it is the procedure represents.
"With scaffolding, the imitation of a procedure becomes linked to reasoning with it. It serves as a foundation upon which more sophisticated approaches are attached."
Exactly. It reminds me of this false dichotomy of some math "reformers" between automaticity with math facts (and the memorization required to achieve it) and the development of number sense.
As educational psychology professor Brian Poncy and math professor Anna Stokke discussed, memorizing number facts (freeing up working memory) enables children to use mental computation strategies with much larger numbers, thereby enabling them to develop even better number sense.
This seems like a spectrum. Not only for feedback from the teacher/parent that gives the cue to the right answer, but also when giving the right answer without understanding.
"With scaffolding, the imitation of a procedure becomes linked to reasoning with it. It serves as a foundation upon which more sophisticated approaches are attached."
Exactly. It reminds me of this false dichotomy of some math "reformers" between automaticity with math facts (and the memorization required to achieve it) and the development of number sense.
As educational psychology professor Brian Poncy and math professor Anna Stokke discussed, memorizing number facts (freeing up working memory) enables children to use mental computation strategies with much larger numbers, thereby enabling them to develop even better number sense.