The issue of balance between procedural fluency and conceptual understanding in mathematics has served as a dividing line in education. Some believe that understanding of a procedure or algorithm must precede the procedure/algorithm itself—and if it doesn’t precede it, it should come about quickly. Failure to do this results in students who some call “math zombies”. Others believe that procedural fluency and conceptual understanding is an iterative process where one feeds the other.
The learning process encompasses a spectrum that begins at the novice level and extends to expert levels. Students benefit by seeing worked examples which shows how to think about the problem before actually working it. Students then imitate the procedure, which ultimately becomes imitation of thinking. As anyone knows who has learned a skill through initial imitation of specific techniques, such as drawing, bowling, swimming, dancing and the like, what seems like it will be easy often is more challenging than it appears. So too with math.
Most of the time, the expert level does not resemble the means of its nurture. Just as footballers and athletes do numerous drills that look nothing like playing a game of football or running a marathon, so the building blocks of final academic or creative performance are small, painstaking and deliberate. As one goes up the scale from novice to expert,imitation of thinking includes many levels of understanding. Even at the most basic levels, the procedural understanding of novices is the foundation that allows them to reason mathematically, to solve problems and to build upon in developing conceptual understanding.
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5 thoughts on “Misunderstandings about Understanding”
Reblogged this on Nonpartisan Education Group.
Absolutely correct. That said, how receptive was your audience? Were there a bunch of math ed experts who never learned (and never will learn) any mathematics beyond the algebra level much less a good proof-based Euclidean geometry course or anything but the most monkey-see/monkey-do calculus?
The audience was very receptive. researchED is unlike the conferences that groups like NCTM hosts. The premise of researchED is finding evidence and real research that identifies effective practices–and ineffective ones. The people attending the conf in Vancouver were relieved to hear what most people know but is never said; things like the “core competencies” are nonsense–you can’t teach “skills” like creativity and collaboration. They are domain specific. My talk had a very positive response. If a researchED conference happens in the LA area (and I’m pushing for that) I’ll not only try to get you to attend, but actually get you to speak!
There was an excellent Q&A session after Barry’s talk which seemed to indicate many teachers, and parents (such as myself) were grateful to hear their own thoughts on math education validated. There was a lovely young trainee teacher, who grew up in China. She gave a very meaningful, and vivid illustration about how math was never difficult for her in school, but her mother insisted that the basic facts were already instilled BEFORE she started school, so she could then focus on the more abstract aspect of mathematics in the classroom. She also said there was one aspect of a problem that she struggled with. So over the course of a month, she spent 1000s of hours of practicing this one aspect of the problem, in order to really understand what it was about. And didn’t think twice about dedicating that much time to the problem, because that is what mathematics, like many abstract subjects, require.
Reblogged this on The Echo Chamber.