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09 Oct 2005 - 20:26
From the Comment thread about Lone Ranger's approach to teaching an 8-year old why it's OK to write the number 5 as 5/1: I mentioned that Saxon Math uses four-fact families to teach the operations of arithmetic, while both constructivist curricula and Singapore math seem to use 'number bonds.' Here's an example of a number bond flash card:
You can download these cards from DonnaYoung.org, a homeschooling resource that looks pretty good, and has a page of mostly terrific paper math manipulatives, including lots of circular fractions, terrific large-print math facts drill sheets, graph paper, play money, scale paper for household furniture arrangements, and some cool-looking empty worksheets with number lines on top. It also has triangular addition and subtraction flash cards (pdf file).
from the directions:
To use the cards, hide one of the corner numbers with your thumb or finger and let the child tell you what the hidden number is.
2 + 1 = 3
3 - 2 = 1
3 - 1 = 2
Same deal with multiplication and division. Here's a typical four-fact family problem from Lesson 2 in Saxon 7/6:
Christopher can do that in his sleep. So can I. I probably have done it in my sleep. I've been doing so much grade school math I sometimes dream about it.
a post I wrote on this subject awhile back:
About a month after Christopher and I began working with Saxon Math 6/5, he told me,Multiplication and division are the big brothers, and addition and subtraction are the little brothers.Then he said,And multiplication and division are cousins.This is a 9-year who, just 6 weeks earlier, had been flunking math.
You have to do a lot of four-fact fact families to come up with a thing like that.
Not nearly as beautiful as Doug's number lines, but a good idea.
Curricular Game Playing
Curricular Game Playing, part 2
number bonds vs. 4-fact families
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Great website! That's a keeper. I agree about the fact families. I never would have thought so, but after watching my son do them ad nauseam during almost every set, it's easier to use the inverse of something as a reminder teacher tool when he gets stuck (which is alot.) It just quietly drums it in. It seems so obvious to me, but as an adult I think we just sometimes forget what they are really thinking. -- SusanS - 09 Oct 2005
It's one of those things you have to see and/or do to believe. I couldn't forget that multiplication and division are inverse operations now if I wanted to. -- CatherineJohnson - 09 Oct 2005
CatherineJohnson - 09 Oct 2005
Plus, I was thinking today about the whole procedural knowledge business.......procedural knowledge is usually defined in terms of motor skills. Baseball, riding a bike, etc. Well, you know how Carolyn's always talking about the 'craft of math,' and getting math 'into' Ben's hands....that was one of her very first posts: Swoop and Swoop. I'm thinking she's using the classic form of procedural memory there, the motor form. My feeling about math is that it's so hard to remember, you ought to use everything you can. So why not enlist procedural memory in the sense of motor memory?? Doing a gazillion pencil-and-paper four-fact fact families puts inverse operations in your hand. Staring at triangular number bonds is just weird. Plus when you write in the operation signs, the things look like something from Battlestar Galactica. -- CatherineJohnson - 09 Oct 2005
Here's Carolyn's post on the craft of math. -- CatherineJohnson - 09 Oct 2005
And........I'd put money on it that the normal transition from: 2 + 3 = 5 to 2 + a = 5 is not that smooth for most kids. For Christopher it was nothing. And he has to work to learn math; he's definitely not a kid who can learn it 'by smell.' I'm repeating myself! But it's pretty amazing. (I guess what I mean is: it's pretty amazing when you see something go easily. Not too much does.) -- CatherineJohnson - 09 Oct 2005