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  # Cogmath count
  
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  ### Cogmath count
  
  1. 1. COUNTSTUDY PROJECT
  2. 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. 3. number representation 0, 1, 2, 3, ...
  4. 4. number representation 0, 1, 2, 3, ... “zero”, “one”, “two”, “three”, ...
  5. 5. number representation 0, 1, 2, 3, ... “zero”, “one”, “two”, “three”, ... 0 = ∅, 1 = {∅}, 2 = {{∅}}, ...
  6. 6. number representation 0, 1, 2, 3, ... “zero”, “one”, “two”, “three”, ... 0 = ∅, 1 = {∅}, 2 = {{∅}}, ... 0 = ∅, 1 = {∅}, 2={∅,{∅}}, ...
  7. 7. number representation 0, 1, 2, 3, ... “zero”, “one”, “two”, “three”, ... 0 = ∅, 1 = {∅}, 2 = {{∅}}, ... 0 = ∅, 1 = {∅}, 2={∅,{∅}}, ...
  8. 8. cognitive accounts of number representation
  9. 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. 10. cognitive accounts of number representation
  11. 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. 12. positional number systems
  13. 13. A trivial counter
  14. 14. A base-notation counter
  15. 15. A base-notation counter
  16. 16. A base-notation counter
  17. 17. A base-notation counter State 1: increases digits, goes left if 9 State 2: finalizes, goes right till “–“
  18. 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. 19. A quaternary base-notation counter
  20. 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. 21. the experiments - 12 qualitative case studies (video and tablet recordings) - quantitative online study (so far 58 subjects)
  22. 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. 23. the qualitative studies A B C D BA BB 1. “What comes next?” 2. “Why?”
  24. 24. the qualitative studies A B C D BA BB (...) BAA B°B→C C°B→D D ° C → BB BB ° B → ?
  25. 25. the qualitative studies
  26. 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. 27. the quantitative study - investigate problems people had in the case studies (corroborate qualitative analysis) - 20-30 min. online experiment - “supervised” control group
  28. 28. the quantitative study
  29. 29. the quantitative study
  30. 30. the quantitative study
  31. 31. the quantitative study
  32. 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. 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. 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.”