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The standard algorithm for multiplication is bound up in notions of place value, and the taking of partial products, which are then totalled.
Mathematics teaching and learning has progressed in many ways since 1989, when NCTM reform began. conceptual understanding debate as a “false dichotomy.”A lengthy quotation will help make this point: In mathematics education, this debate takes the form of “basic skills or conceptual understanding.” This bogus dichotomy would seem to arise from a common misconception of mathematics held by a segment of the public and the education community: that the demand for precision and fluency in the execution of basic skills in school mathematics runs counter to the acquisition of conceptual understanding.
It has also progressed in many ways since the 1960s, when the first “New Math” curricula were tried (and eventually abandoned). The truth is that in mathematics, skills and understanding are completely intertwined.
One the one hand, we have a real or perceived loss of “basic skills”, usually signified in the debate through times instant recall, and on the other hand, we have the idea that our students are not good enough at problem-solving for the “modern world”, or for “the future”.
It is very worth noting that this debate is several generations old.
Most excitingly, both seemed to support better problem representation.
Representation is an enactment of thinking; students must have ways to think about mathematical concepts.
How do procedures and concepts work together, not against each other?
Accepting that they do not have to work against each other, and indeed, that they can and must work together, would be a start. Developing conceptual understanding and procedural skill in mathematics: An iterative process.
It is my belief that educators, researchers, and those who write articles for newspapers need to drop the belief that one must precede the other.
Future research studies could test the acquisition of what we could broadly call the “procedural condition”, and the “conceptual condition”.