symbolic systems number as case study. transparency of symbolic systems acquisition of language ...
Post on 19-Dec-2015
225 views
TRANSCRIPT
SYMBOLIC SYSTEMS
Number as case study
Transparency of Symbolic Systems Acquisition of Language
Transparency of Symbolic Systems Acquisition of Concepts
Naturalness of the Symbolic Systems to Subserve Computation Speed
History (happenstance, evolution) of symbolic systems
Youtube: http://www.youtube.com/watch?v=wo-6xLU
VLTQ
Numeral 11 12 13 14 15 16 17 18 19 20 Chinese (written)
十一 十二 十三 十四 十五 十六 十七 十八 十九 二十
Chinese (written)
shi yi shi er shi san shi si shi wu shi liu shi qi shi ba shi jiu er shi
English eleven twelve thirteen fourteen fifteen sixteen seventeen eighteen nineteen twenty
Chinese has a clear base-ten structure– similar to Arabic numerals: 11 = “10…1”
English lacks clear evidence of base-ten structure– Names for 11 and 12 not marked as compounds with
10.– Larger teens names follow German system of
unit+digits name, unlike larger two-digit number names
compare “fourteen” and “twenty-four”
Number names in Chinese & English - Part II
From Ten to Twenty
(slide from Kevin Miller)
Language and Learning to Count
Children need to learn a system of number names as they learn to count
Not a trivial task
(slide from Kevin Miller)
Language Rule Example Chinese (written)
三十七
Chinese (written)
Decade unit (two,three,four,five,six,seven,eight,nine) + ten (shi) + unit
san shi qi
English Decade names (twen,thir,for,fif,six,seven,eight,nine) + ty + unit thirty-seven
Both languages share a similar structure– similar to Arabic numerals: 37 = “3x10 + 7”
For Chinese, this extends previous system
For English, it represents a new way of naming numbers
Number names in Chinese & English - Part IIIAbove Twenty
(slide from Kevin Miller)
A longitudinal view
1 2 3 4 5 6 7 8 9 1011120
10
20
30
40
50
60
70
80
90
100
110
Med
ian
Ab
stra
ct C
oun
tin
g
Month
China
US
1 2 3 4 5 6 7 8 9 1011120
10
20
30
40
50
60
70
80
90
100
110
Month
China
US
1 2 3 4 5 6 7 8 9 10 11120
10
20
30
40
50
60
70
80
90
100
110
Month
China
US
2-year-olds 3-year-olds 4-year-olds
(slide from Kevin Miller)
Q ‘bout data so far: Does the ability to recite up to a higher number by Chinese children say anything about numeracy and or mathematical ability?
Ho & Fuson (1998)
IQ Counting Sequence
Prompt with “1, 2, 3” if necessary If stop, “what comes after x?” If still no response, “x-2, x-1, x…?”
Hidden Object Addition X + Y; 4+y;10+y; 2+1 (warm-up) First I put x blocks into the box, then I put y more
blocks in it. How many blocks altogether in the box now?
Feedback by counting
Experiment 1
Test children at 4 and then 5 yr-old Lo-CS-av-IQ Hi-CS-av-IQ Hi-CS-hi-IQ
Experiment 1: at 4
Experiment 1: at 5
Experiment 2: Chinese vs. English Matched IQ Chinese Hi CS Chinese Lo CS English Hi CS English Lo CS
Experiment 2: Results
Experiment 2: Results by Y (near/far)
Miura et al.
Part 1: Base 10 block understanding Out of 100 units and 20 10-unit blocks, make
11, 13, 28, 30, 42 3 coding schemes
1-to-1 collection (e.g. for 42 = 42 unit blocks) Canonical base 10 (e.g. 10-unit blocks & 2 unit blocks) Non-canonical base 10 (e.g. 3 10-unit blocks & 12 unit blocks)
Part 2: Five Place-value questions in random order See number (32).Show with blocks the 10s place, 1s place. Shown blocks (40 10-units, 4 unit), say number; Shown number (44).
Point to place, ask which of two 4 ten blocks or 4 unit blocks. Shown 13 blocks, asked to group them into 4 blocks each with 1
remaining. What number does this make? (Misleading perceptual Q) Shown 26, and same procedure as 13 blocks above Shown 3 10-unit blocks and 12 unit block, write number. Then ask
about relation to 4 and 2.
Miura et al. (1993)
Miura et al. (1993)
1st grader Monolinguals Base 10 block understanding
1-to-1 collection (e.g. for 42 = 42 unit blocks) Canonical base 10 (e.g. 10-unit blocks & 2 unit
blocks) Non-canonical base 10 (e.g. 3 10-unit blocks &
12 unit blocks)
Miura et al. (1993)
Another Example: Time
Kelly & Miller (1999)
Participants: Ages: 2nd graders, 4th graders, Adults Language Grp: English vs. Mandarin
Six Conditions Weekday naming Month naming Weekday forward (+4) Weekday backward (-4) Month forward (+7) Month backward (-7)
Kelly et al: “Symbol systems such as calendars are
learned in order to serve as tools for solving basic problems…. How such a system is organized has consequences for the ability of its users to perform the tasks for which it was acquired in the first place.”