1. SYMBOLIC SYSTEMS Number as case study

2.  Transparency of Symbolic Systems  Acquisition of Language  Transparency of Symbolic Systems  Acquisition of Concepts  Naturalness of the Symbolic Systems to Subserve Computation  Speed

3.  History (happenstance, evolution) of symbolic systems  Youtube:  http://www.youtube.com/watch?v=wo-6xLUVLTQ http://www.youtube.com/watch?v=wo-6xLUVLTQ

4.  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)

5. 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)

6.  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 III Above Twenty (slide from Kevin Miller)

7. A longitudinal view (slide from Kevin Miller)

8.  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?

9. 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

10. Experiment 1  Test children at 4 and then 5 yr-old  Lo-CS-av-IQ  Hi-CS-av-IQ  Hi-CS-hi-IQ

11. Experiment 1: at 4

12. Experiment 1: at 5

13. Experiment 2: Chinese vs. English  Matched IQ  Chinese Hi CS  Chinese Lo CS  English Hi CS  English Lo CS

14. Experiment 2: Results

15. Experiment 2: Results by Y (near/far)

16. 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.

17. Miura et al. (1993)

18.  1 st 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)

19. Miura et al. (1993)

20. Another Example: Time

22. Kelly & Miller (1999)  Participants:  Ages: 2 nd graders, 4 th 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)

28.  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.”