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This posting came up in response to a discussion on over reliance on

tools such as hand held calculators, and the possiblilty of the tool

obscuring the actual knowledge it allegedly supports/embeds, i.e.,

calculators preclude math sense (numeracy, for instance).

Given our current and recent discussion on artifacts and zpds, I thought

the list might find this interesting.

Lindax

Begin forwarded message:

From: "Claude Almansi (BW)" < claude.almansi@BLUEWIN.CH

<mailto:claude.almansi@BLUEWIN.CH> >

Date: March 22, 2006 2:44:42 AM PST

To: ITFORUM@LISTSERV.UGA.EDU <mailto:ITFORUM@LISTSERV.UGA.EDU>

Subject: Re: student dependence on calculators and other technology

Reply-To: Instructional Technology Forum < ITFORUM@LISTSERV.UGA.EDU

<mailto:ITFORUM@LISTSERV.UGA.EDU> >

Johansson, Melinda wrote:

(...)

I agree that functioning mathematically in the world is essential, and

certainly I hope my own kids can perform basic 4-function math. But

doing arithmetic and understanding the size of things are not the same

thing. Many of us that have good arithmetic skills have a poor grasp of

concepts like geologic time or the distance to the sun

(...)

May I branch off again from Melinda's words, out of a few mom's

memories?

Memory 1, re 4-function math: My daughter was 5, still in kindergarden,

a long time ago. We were shopping. Out of the blue, she asked "Mom, are

5 times 7 34?" Other shoppers laughed, but I was intreagued:

multiplication is not taught at kindergarden.

"Almost but not quite: how did you get that result?", I said.

She spread her fingers, folding 3, and started tapping the rest on her

thighs, saying "1, 2, 3, 4, 5, 6, 7," then she tapped her foot and said

"1"; then tapping her fingers again "8, 9, 10, 11, 12, 13, 14" foot tap:

"2", etc. This time she got "35"

"Who showed you this?" I asked.

"No one. N. (an older friend of hers, already in primary school) does

multiplications so I wanted to do them too."

She was using her body as a calculator, as humans have done until fairly

recently. When we went home, I retrieved an abacus she had banged on as

a toddler. "Sure, it's easier this way", she said after trying it, "but

I can't go around carrying this thing, I'd look silly." However, the

interesting thing to me was her motivation: she was not trying to figure

out something concrete, like getting an idea of how long she should wait

for something that would happen "in 5 weeks time". She was envious of a

skill her friend had and she didn't, and which seemed fun.

She was also, in a way, counting in base 7 without being aware of bases:

Memory 2, still about arithmetic, a few weeks later, in the train.

"What are our seat numbers?", she asked.

"26 and 27"

"Why is 26 written 2 6 and 27 2 7?"

"Because it would be impossible to remember words and signs for all the

numbers if they were all different, so people decided that when they got

ten things, they'd make a heap and put it to the left, then they went

on counting and when they reached 10 again, they made a second heap and

put it to the left, and so on."

"But why 10?"

I spread my fingers and added "but people didn't always use heaps of

ten. We still say "quatre-vingt" (80, literally 4 times 20 - this

conversation was in French), maybe because the Gauls and Romans wore

sandals and used their toes as well."

Of course, it is difficult to have this kind of kid-triggered

discussions with 20+ students in a school context. But you can still

relate arithmetic to the personal, experience of each student. When my

daughter started primary school the following year, they made number

lines, starting with time lines of their, their parents' and

grandparents' life spans so far in different colors. More affectively

appealing than the cabbages I was asked to jugggle with at the same age

(I hated cabbage).

Envy, fun, vanity (her "I'd look silly" about using the abacus in

public), affects, personal tastes - these subjective things have

nothing to do with math and arithmetics seen by adults, but they do

influence the way kids approach math and arithmetics. Maybe if we took

them more into account at school, then calculators and other technology

could be integrated as what they are: tools that are useful, but not the

only solution to depend on.

Best

Claude

Claude Almansi

www.adisi.ch

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