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Cogmath count Presentation Transcript

  • 1. COUNTSTUDY PROJECT
  • 2. number representation “The way we do arithmetic is intimately related to the way we represent the numbers we deal with.” Donald Knuth, TAOCP Vol. II, p.195
  • 3. number representation 0, 1, 2, 3, ...
  • 4. number representation 0, 1, 2, 3, ... “zero”, “one”, “two”, “three”, ...
  • 5. number representation 0, 1, 2, 3, ... “zero”, “one”, “two”, “three”, ... 0 = ∅, 1 = {∅}, 2 = {{∅}}, ...
  • 6. number representation 0, 1, 2, 3, ... “zero”, “one”, “two”, “three”, ... 0 = ∅, 1 = {∅}, 2 = {{∅}}, ... 0 = ∅, 1 = {∅}, 2={∅,{∅}}, ...
  • 7. number representation 0, 1, 2, 3, ... “zero”, “one”, “two”, “three”, ... 0 = ∅, 1 = {∅}, 2 = {{∅}}, ... 0 = ∅, 1 = {∅}, 2={∅,{∅}}, ...
  • 8. cognitive accounts of number representation
  • 9. cognitive accounts of number representationhave to allow for what we actually dowith numbers in daily life, e.g.:- mental arithmetic- strategic decomposition Mental Calculations. Nikolay Bogdanov-Belsky. 1895.
  • 10. cognitive accounts of number representation
  • 11. cognitive accounts of number representation - we do not have every number (infinite instances) represented - instead, we have procedures to generate them and operate with them So whenever we face a number symbol, we know what can be done with it
  • 12. positional number systems
  • 13. A trivial counter
  • 14. A base-notation counter
  • 15. A base-notation counter
  • 16. A base-notation counter
  • 17. A base-notation counter State 1: increases digits, goes left if 9 State 2: finalizes, goes right till “–“
  • 18. A base-notation counter State 1: increases digits, goes left ● number symbols, if 9 ● position expansion, ● carry-over, State 2: finalizes, goes right till ● the special role of zero “–“ The crucial point with base notation is the repeated application of “increasing digits” at different positions.
  • 19. A quaternary base-notation counter
  • 20. question - how do people learn to “orient themselves” in systems like base notation? - how do they “really” blend concepts in doing so? - what strategies do they use to cope with problems?
  • 21. the experiments - 12 qualitative case studies (video and tablet recordings) - quantitative online study (so far 58 subjects)
  • 22. the qualitative studies - 30-40 min. sessions - interview situation (as little guidance as possible, as much as necessary) - let the subjects construct their own solutions (if possible) - “obfuscated” quaternary system, using symbols {A,B,C,D}
  • 23. the qualitative studies A B C D BA BB 1. “What comes next?” 2. “Why?”
  • 24. the qualitative studies A B C D BA BB (...) BAA B°B→C C°B→D D ° C → BB BB ° B → ?
  • 25. the qualitative studies
  • 26. the qualitative studies “big problems”: - missing AAs - order of variation in multiple digit sequences (BAA → BBA, BAA → BAB, …) - A = 1? (0-omitting habit)
  • 27. the quantitative study - investigate problems people had in the case studies (corroborate qualitative analysis) - 20-30 min. online experiment - “supervised” control group
  • 28. the quantitative study
  • 29. the quantitative study
  • 30. the quantitative study
  • 31. the quantitative study
  • 32. preliminary results from the quantitative study - in general, performance was good (in successors, antecedents, continuations) - rating was inconclusive - The “A-problem” was abundant
  • 33. A few explanations- „A = 0 B = 1 C = 2 D = 3 Rechnen Base 4“- „base(4) = { A, B, C, D }; erster Stellenübertrag verwendet B statt A, das macht michwahnsinnig... ansonsten wie normale Zahlenbasis.“- „polyadisches System, mit den Zeichen Zeichen B, C, D, mit Ausnahme, an rechtesterStelle fängt es immer mit dem Zusatzzeichen A an.“
  • 34. A few explanations- „Die Reihenfolge beruht auf dem Alphabet nur bis zum Buchstaben D nach D wiederholtsich die Reihenfolge wieder wenn D erscheint verändert sich der vorherige Buchstabe zumdarauffolgenden“- „bei A anfangen und bis D durchzählen, danach macht man ein B vor das A und zählt damitdurch bis D dann ein C und so weiter nach dem D kommt ein BA vor die Ursprungsfolge.”- „Die Folge ist in Viererbloecken organisiert. Ganz rechts sind immer die Buchstaben A bisD. Links werden immer blockweise die Buchstaben B bis D angefuegt. Dazwischen dieBuchstaben A bis D.”