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Symbolic SystemS

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

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Symbolic SystemS

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  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

  4. Number names in Chinese & English - Part IIFrom Ten to 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” (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. Number names in Chinese & English - Part IIIAbove Twenty • 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 (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. 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)

  19. Miura et al. (1993)

  20. Another Example: Time

  21. 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)

  22. 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.”

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