Be sure to check out Text Savvy's recent posts about Framing. He also has a link to an excellent article by Keith Devlin titled, "How do we learn math?"
I really like the way Devlin put it -- that understanding emerges as a consequence of use.
Understanding cannot be constructed separately from use, or be constructed from only one specific use.
One of my recent insights is that TERC aims for conceptual understanding, but neglects to include anything children can practice "to perfection." If children used to be taught procedures-without-understanding, TERC now teaches concepts-without-adequate-procedures-to-practice.
What's going into long-term memory?
Traditional math ties concepts to procedures so that children may at least practice the procedures. Traditional math gives children something to hang their conceptual hat on.
"Traditional math gives children something to hang their conceptual hat on."
Well put--I honestly don't know if I would have survived math if I hadn't been taught procedural fluency. As it was, I somehow managed to get through Calculus in college on my way to an Accounting degree.
My daughter who is a graduating high school senior went K-4 in Indiana. (We're now in central Illinois.) There was "some" fuzzy math in K-4, but it was predominantly based on the traditional algorithms. It was that procedural fluency that gave her the foundation to succeed in her subsequent courses.
I had this conversation with our district's curriculum director (who has since retired) here in Illinois several years ago, and it just fell on deaf ears. We parted ways, agreeing to disagree. My point was that some kids must have procedural fluency in order for understanding to follow; her position was that conceptual understanding ruled the day. I have yet to recover from that conversation. . . .
But Devlin also says that "doing math" doesn't require sentience.
He's doesn't use terms quite the way the rest of us do, for example, he's got lobsters and birds "doing math."
"What then if some non human living creature solves the same problem? The Indigo Bunting, for example? Is there any justification for denying that it too is doing mathematics? "
http://www.maa.org/devlin/devlin_03_05.html
"If you want to find the world's greatest mathematician, you don't need to travel to Harvard or Stanford or Princeton. Just visit the ocean or look up at the birds in the sky"
That sort of rhetoric makes me nervous and I would need to see more articles to see if he's really got a point or if he is just sort of kooky. You know, I just want to figure out if I should teach cross multiplication as an algorithm in the fifth grade or if I should wait and teach it two years later when the kid can derive it. This talk about lobsters and bird brains isn't useful for making such decisions ;-) And I haven't seen Devlin discuss these sorts of things other than in super general terms. I guess it comes off more as philosophical musings rather than any real interest in math ed.
Here is the link for his site:
ReplyDeletehttp://www.textsavvyblog.net/
I really like the way Devlin put it -- that understanding emerges as a consequence of use.
ReplyDeleteUnderstanding cannot be constructed separately from use, or be constructed from only one specific use.
One of my recent insights is that TERC aims for conceptual understanding, but neglects to include anything children can practice "to perfection." If children used to be taught procedures-without-understanding, TERC now teaches concepts-without-adequate-procedures-to-practice.
What's going into long-term memory?
Traditional math ties concepts to procedures so that children may at least practice the procedures. Traditional math gives children something to hang their conceptual hat on.
"Traditional math gives children something to hang their conceptual hat on."
ReplyDeleteWell put--I honestly don't know if I would have survived math if I hadn't been taught procedural fluency. As it was, I somehow managed to get through Calculus in college on my way to an Accounting degree.
My daughter who is a graduating high school senior went K-4 in Indiana. (We're now in central Illinois.) There was "some" fuzzy math in K-4, but it was predominantly based on the traditional algorithms. It was that procedural fluency that gave her the foundation to succeed in her subsequent courses.
I had this conversation with our district's curriculum director (who has since retired) here in Illinois several years ago, and it just fell on deaf ears. We parted ways, agreeing to disagree. My point was that some kids must have procedural fluency in order for understanding to follow; her position was that conceptual understanding ruled the day. I have yet to recover from that conversation. . . .
But Devlin also says that "doing math" doesn't require sentience.
ReplyDeleteHe's doesn't use terms quite the way the rest of us do, for example, he's got lobsters and birds "doing math."
"What then if some non human living creature solves the same problem? The Indigo Bunting, for example? Is there any justification for denying that it too is doing mathematics?
"
http://www.maa.org/devlin/devlin_03_05.html
"If you want to find the world's greatest mathematician, you don't need to travel to Harvard or Stanford or Princeton. Just visit the ocean or look up at the birds in the sky"
That sort of rhetoric makes me nervous and I would need to see more articles to see if he's really got a point or if he is just sort of kooky. You know, I just want to figure out if I should teach cross multiplication as an algorithm in the fifth grade or if I should wait and teach it two years later when the kid can derive it. This talk about lobsters and bird brains isn't useful for making such decisions ;-) And I haven't seen Devlin discuss these sorts of things other than in super general terms. I guess it comes off more as philosophical musings rather than any real interest in math ed.
Myrtle (Pot calling the kooky kettle black)
I have to say, Myrtle, you picked the perfect article to highlight Devlin's "kookiness."
ReplyDelete"One of my recent insights is that TERC aims for conceptual understanding, but neglects to include anything children can practice "to perfection."
ReplyDeleteThis is it exactly. The only practice my third grader has received is from home and by attending KUMON. His school uses TERC.
A different sheet of homework every night on a different topic and a few worksheets in the classroom.
When I complained to my son's teacher about the lack of practice, she gave me a list of websites I could go to.
That reminds me -- Devlin has an interview on NPR that Ed said was great.
ReplyDeleteI'll have to find it.