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- 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 representation have to allow for what we actually do with 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
- 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 → ?
- 26. the qualitative studies “big problems”: - missing AA's - 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
- 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 mich wahnsinnig... ansonsten wie normale Zahlenbasis.“ - „polyadisches System, mit den Zeichen Zeichen B, C, D, mit Ausnahme, an rechtester Stelle 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 wiederholt sich die Reihenfolge wieder wenn D erscheint verändert sich der vorherige Buchstabe zum darauffolgenden“ - „bei A anfangen und bis D durchzählen, danach macht man ein B vor das A und zählt damit durch 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 bis D. Links werden immer blockweise die Buchstaben B bis D angefuegt. Dazwischen die Buchstaben A bis D.”