outline  - the object:     numbers & positional number systems                    -> question 1: what is a concept?  - que...
number representation “The way we do arithmetic is intimately related to the way we represent the numbers we deal with.”  ...
number representation         0, 1, 2, 3, ...
number representation         0, 1, 2, 3, ...         “zero”, “one”, “two”, “three”, ...
number representation         0, 1, 2, 3, ...         “zero”, “one”, “two”, “three”, ...         0 = ∅, 1 = {∅}, 2 = {{∅}}...
number representation         0, 1, 2, 3, ...         “zero”, “one”, “two”, “three”, ...         0 = ∅, 1 = {∅}, 2 = {{∅}}...
number representation         0, 1, 2, 3, ...         “zero”, “one”, “two”, “three”, ...         0 = ∅, 1 = {∅}, 2 = {{∅}}...
operating with number representationhave to allow for what weactually do with numbers indaily life, e.g.:- mental arithmet...
operating with number representation
positional number systems     ... + a3 · 103 + a2 ·102 + a1 · 101 + a0 · 100    with an is a digit from a finite set of at...
how does counting actually work? Principles/requisites that make up counting
how does counting actually work? Principles/requisites that make up counting ●   number symbols ●   incrementing digits ● ...
how does counting actually work? Principles/requisites that make up counting ●   number symbols ●   incrementing digits ● ...
answer 1: the concept of number ➔ we do not have every number (infinite instances) represented ➔instead, we have procedure...
research question 2 - what cognitive mechanisms allow people to learn number systems? - what strategies do they use to cop...
theoretic assumptions "If we do not want to believe that ideas are innate or God-given, but the result of subjective think...
theoretic assumptions                                                           Anschauung                                ...
theoretic assumptions - in the light of our discussion of positional number systems, learning numbers requires the underst...
the experiments - 12 qualitative case studies (video and tablet recordings) - quantitative online study (so far 58 subjects)
the qualitative studies - 30-40 min. sessions - interview situation (as little guidance as possible, as much as necessary)...
the qualitative studies           A           B           C           D          BA          BB  1. “What comes next?”  2....
the qualitative studies           A           B           C           D         BA         BB          ...         DD     ...
the qualitative studies
A   BC B   BD C DBABB     CA   {6}                  DA {5}                   CAA {1}     CB                        DB     ...
A   BC                                   B   BD                                   C                                   D   ...
A   BC                                   B   BD                                   C                                   D   ...
A                     BC                                   B                     BD                                   C   ...
the qualitative studies  points of interest:  - the aspects subjects mention  - problems that they mention  - solutions an...
the qualitative studies - observationsExtraction: Many known operations “pop up” and are used whilesubjects try to find a ...
the qualitative studies - observationsApplication & Evaluation of ones ideas. “Running” the coordinated operationand check...
the quantitative study  How can we quantify the investigation of cognitive mechanisms?  What kind of experimental set-up i...
the quantitative study  How can we quantify the investigation of cognitive mechanisms?  What kind of experimental set-up i...
A      B      C      D           BA    L1.1                 AA   L1.2                     E   L1.3               AA   L1.4...
the quantitative study
the quantitative study  Results training-phase I:
the quantitative study  Results training-phase I:
the quantitative study  Results training-phase I:
the quantitative study  Results training-phase I:
the quantitative study
Lauf 1.1                                 A B C D > BA BB BC BDkeine der Aussagen scheint mir sinnvoll                     ...
Lauf 1.3                                    ABCD > EFGHkeine der Aussagen scheint mir sinnvoll                            ...
the quantitative study  How can we quantify the investigation of cognitive mechanisms?  What kind of experimental set-up i...
the quantitative study
the quantitative study  Results consolidation phase:
the quantitative study  How can we quantify the investigation of cognitive mechanisms?  What kind of experimental set-up i...
the quantitative study
the quantitative study  Results testing-phase:
preliminary results from the quantitative study
the quantitative study further ideas for analysis: -   look at the system aspects -   correlation of “correct” system aspe...
system aspects                                               system aspects                                  A ist ein Pla...
A few explanations- „A = 0 B = 1 C = 2 D = 3 Rechnen Base 4“- „base(4) = { A, B, C, D }; erster Stellenübertrag verwendet ...
A few explanations Wenn rechts kein D steht, dann ändere diese Position in den nächsthöheren Buchstaben. Wenn rechts ein D...
A few explanations        zuerst: folge A,b,c,d rechts,        dann an nächster stelle links eins weiterzählen (in der    ...
your speculations about the online experiment        Ich habe die Vermutung, versagt zu haben. Das        Experiment geht ...
discussion
discussion cycle of abstraction (“construction & trial & error & correction”)    - extracting of operations (one „sees“ pa...
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  1. 1. outline - the object: numbers & positional number systems -> question 1: what is a concept? - question 2: what cognitive mechanisms allow us to understand such a system ? - assumptions: - people learn by adapting their understanding, building on previously learnt structures - people continuously “construct” an ad hoc understanding and test its effective power, only over time leading to stable structures - the approach: confront subjects with example “problems” and look how they “make sense” of it qualitative interviews + online study - some results
  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. operating with number representationhave to allow for what weactually do with numbers indaily life, e.g.:- mental arithmetic- strategic decomposition Mental Calculations. Nikolay Bogdanov-Belsky. 1895.
  9. 9. operating with number representation
  10. 10. positional number systems ... + a3 · 103 + a2 ·102 + a1 · 101 + a0 · 100 with an is a digit from a finite set of at least two symbols
  11. 11. how does counting actually work? Principles/requisites that make up counting
  12. 12. how does counting actually work? Principles/requisites that make up counting ● number symbols ● incrementing digits ● position expansion ● carry-over ● the special role of zero
  13. 13. how does counting actually work? Principles/requisites that make up counting ● number symbols ● incrementing digits ● position expansion ● carry-over ● the special role of zero The crucial point with base notation is the repeated application of “incrementing digits” at different positions.
  14. 14. answer 1: the concept of number ➔ we do not have every number (infinite instances) represented ➔instead, we have procedures to generate numbers and operate with them So whenever we face a number symbol, we know what can be done with it.
  15. 15. research question 2 - what cognitive mechanisms allow people to learn number systems? - what strategies do they use to cope with problems? - how do people learn to “orient themselves” in systems like base notation?
  16. 16. theoretic assumptions "If we do not want to believe that ideas are innate or God-given, but the result of subjective thinkers conceptual activity, we have to devise a model of how elementary mathematical ideas could be constructed - and such a model will be plausible only if the raw material it uses is itself not mathematical." (von Glasersfeld, p.64, 2006)
  17. 17. theoretic assumptions Anschauung Zahlkonzept Verfahren - where we began Produktivität Operationale Linking Verknüpfung metaphors Systematitzität Conceptual Symbolizing Blending Erhaltung Kompositionalität Reflektierende distributing one-one Abtraktion correspondence early arithmetic turn-taking (relation/subit.) Vielheitsubitzing tagging Objektidentität Objektpermanenz alignment Einheit SNWS prä- numerisch
  18. 18. theoretic assumptions - in the light of our discussion of positional number systems, learning numbers requires the understanding of a system of regularities, and cannot merely be an upscaling of an innate “number sense” - Piagets notion of “SCHEMA” - mechanisms of “ASSIMILATION” and “ACCOMODATION” - Dubinskys differentiation: - extraction - coordination - encapsulation - generalisation
  19. 19. the experiments - 12 qualitative case studies (video and tablet recordings) - quantitative online study (so far 58 subjects)
  20. 20. 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) The essential idea of the experiment was to let subjects construct a coherent system themselves. This approach reflects the idea that learning and understanding is essentially a construction by the individual, in the conflict between the schemes he already possesses and a problem that cannot be readily assimilated. We therefore let subjects substantially elaborate on their ideas. - “obfuscated” quaternary system, using symbols {A,B,C,D}
  21. 21. the qualitative studies A B C D BA BB 1. “What comes next?” 2. “Why?”
  22. 22. the qualitative studies A B C D BA BB ... DD BAA B°B→C C°B→D D ° C → BB BB ° B → ?
  23. 23. the qualitative studies
  24. 24. A BC B BD C DBABB CA {6} DA {5} CAA {1} CB DB CBB CC use next symbol, DC “A” was omitted, so CCC 2nd place without A CD vary rightmost DD also omit "C" CDD 3rd place without A,B DA place 4th place w/o A,B,C DB DC "accept exception" differing variants for other places DD
  25. 25. A BC B BD C D BA BB CA {6} DA {5} CAA {1} CB DB CBB CC use next symbol, DC “A” was omitted, so CCC 2nd place without A CD vary rightmost DD also omit "C" CDD 3rd place without A,B DA place 4th place w/o A,B,C DB DC "accept exception" differing variants for other places DD E new symbol, new sequenceEA new symbol, new combinationsDE append new symbols
  26. 26. A BC B BD C D BA BB CA {6} DA {5} CAA {1} CB DB CBB CC use next symbol, DC “A” was omitted, so CCC 2nd place without A CD vary rightmost DD also omit "C" CDD 3rd place without A,B DA place 4th place w/o A,B,C DB DC "accept exception" differing variants for other places DD E new symbol, new sequenceEA new symbol, new combinationsDE append new symbols AA {4} complete "missing" combinations
  27. 27. A BC B BD C D BA BB CA {6} DA {5} CAA {1} CB DB CBB CC use next symbol, DC “A” was omitted, so CCC 2nd place without A CD vary rightmost DD also omit "C" CDD 3rd place without A,B DA place 4th place w/o A,B,C DB DC "accept exception" differing variants for other places DD E new symbol, new sequenceEA new symbol, new combinationsDE append new symbolsAAA {2} BAA {1} AA {4} CAA {1} omitting AA after D was omit AAA complete "missing" next place, next "one-off" exception combinations symbol always B in the leftmost place pyramid-like growth
  28. 28. the qualitative studies points of interest: - the aspects subjects mention - problems that they mention - solutions and respective explanations problems our subjects face: - missing AAs - order of variation in multiple digit sequences (BAA → BBA, BAA → BAB, …) - A = 1? (0-omitting habit)
  29. 29. the qualitative studies - observationsExtraction: Many known operations “pop up” and are used whilesubjects try to find a „good“ continuation; e.g.: - lexical order - repetition (in cycles of 4) - enlarging string ( e.g. BA → BAA) - implicit counting (automatic, without explicit understanding) - explicit counting (knowledge about the system) - usage of known tools, e.g. counting with fingersCoordination: Operations are being ordered sequentially andhierarchically, e.g.: - increasing digits. A then B then C then D - turntaking, e.g. switch between increasing digits and enlarging the sequence
  30. 30. the qualitative studies - observationsApplication & Evaluation of ones ideas. “Running” the coordinated operationand checking whether it works or „makes sense“; via some kind of judgementabout e.g.: - interviewers reaction - recognition value - homogeneity / systematicity of the invented system - “strong solutions” / “Occams Razor” - e.g. is generalisation possible? Can I repeat that type of operation?→ A dynamic process of testing, observing, and reordering.
  31. 31. the quantitative study How can we quantify the investigation of cognitive mechanisms? What kind of experimental set-up is needed? ● Investigate problems people had in the case studies (corroborate qualitative analysis) ● 20-30 min. online experiment ● “supervised” control group
  32. 32. the quantitative study How can we quantify the investigation of cognitive mechanisms? What kind of experimental set-up is needed? Training-phase: ● participants have to see the system and try to understand it by rating how much sense certain continuations make, here participants need to detect and extract certain regularities in the system ● Implicit feed-back is given by using one of the possible continuations in the last block as the next “given” Measured data of the participants: ● Time needed ● Rating of the continuations ● Aspects clicked as being part of the continuation Cognitive mechanisms: ● Extraction/Internalization: Detecting relevant aspects of the process of continuation, mental re-enactment
  33. 33. A B C D BA L1.1 AA L1.2 E L1.3 AA L1.4 BB LB: AB LB: F LB: BB LB: BC LA: AC LA: G LA: CC LA: BD AD H DD CA L2.1 DA L2.2 E L2.3 CAA L2.4 CB LB: DB LB: F LB: CBB LB: CC LA: DC LA: G LA: CCC LA: CD DD H CDD DA L3.1 AA L3.2 BAA L3.3 DAA L3.4 DB LB: AB LB: BAB LB: DBB LB: DC LA: AC LA: BAC LA: DCC LA: DD AD BAD DDD BAA L4.1 AA L4.2 BBA L4.3 AAA L4.4 BAB LB: AB LB: BBB LB: AAB LB: BAC LA: AC LA: BBC LA: AAC LA: BAD AD BBD AADSolution types ABCD - System A - problem new symbols new position precox middle variation
  34. 34. the quantitative study
  35. 35. the quantitative study Results training-phase I:
  36. 36. the quantitative study Results training-phase I:
  37. 37. the quantitative study Results training-phase I:
  38. 38. the quantitative study Results training-phase I:
  39. 39. the quantitative study
  40. 40. Lauf 1.1 A B C D > BA BB BC BDkeine der Aussagen scheint mir sinnvoll A fehlt verdoppeln alle Kombinationen verwenden D löst Übergang aus einen Buchstaben auslassen links Zeichen anfügen links versetzt Folge A bis D rechts immer A bis D blockweise zusammenfügen neue Zeichen des Alphabets 0 5 10 15 20 25 30 35 40 45 50 Lauf 1.2 A B C D > AA AB AC ADkeine der Aussagen scheint mir sinnvoll A fehlt verdoppeln alle Kombinationen verwenden D löst Übergang aus einen Buchstaben auslassen links Zeichen anfügen links versetzt Folge A bis D rechts immer A bis D blockweise zusammenfügen neue Zeichen des Alphabets 0 5 10 15 20 25 30 35 40 45
  41. 41. Lauf 1.3 ABCD > EFGHkeine der Aussagen scheint mir sinnvoll A fehlt verdoppeln alle Kombinationen verwenden D löst Übergang aus einen Buchstaben auslassen links Zeichen anfügen links versetzt Folge A bis D rechts immer A bis D blockweise zusammenfügen neue Zeichen des Alphabets 0 5 10 15 20 25 30 35 40 45 Lauf 1.4 A B C D > AA BB CC DDkeine der Aussagen scheint mir sinnvoll A fehlt verdoppeln alle Kombinationen verwenden D löst Übergang aus einen Buchstaben auslassen links Zeichen anfügen links versetzt Folge A bis D rechts immer A bis D blockweise zusammenfügen neue Zeichen des Alphabets 0 5 10 15 20 25 30 35 40 45
  42. 42. the quantitative study How can we quantify the investigation of cognitive mechanisms? What kind of experimental set-up is needed? Consolidation-phase: ● Participants have to continue the given sequence on their own ● Implicit feed-back is given by using one of the possible continuations as the next “given” Measured data of the participants: ● Time needed ● Continuation chosen (possibility of scoring) Cognitive mechanisms: ● Extraction/Internalization: Detecting relevant aspects of the process of continuation, mental re-enactment ● Coordination of detected aspects to create a solution
  43. 43. the quantitative study
  44. 44. the quantitative study Results consolidation phase:
  45. 45. the quantitative study How can we quantify the investigation of cognitive mechanisms? What kind of experimental set-up is needed? testing-phase: ● participants have to give the successor/predecessor for a given item Measured data of the participants: ● Time needed ● Last/next item answered Cognitive mechanisms: ● Coordination of detected aspects to create a solution ● Encapsulation of coordinated detected aspects (?)
  46. 46. the quantitative study
  47. 47. the quantitative study Results testing-phase:
  48. 48. preliminary results from the quantitative study
  49. 49. the quantitative study further ideas for analysis: - look at the system aspects - correlation of “correct” system aspects with performance - cluster-analysis of the data; distance btw. subjects with respect to certain dimensions -> division into groups
  50. 50. system aspects system aspects A ist ein Platzhalter als erste neue Stelle immer B links versetzt A, B, C, D wiederholen rechts A, B, C, D wiederholen jedes zweite Zeichen überspringen einen Buchstaben auslassen blockweise Zusammenfügen von Teilfolgenalle Kombinationen von Buchstaben werden verwendet keine der Aussagen scheint mir sinnvoll 0 10 20 30 40 50 60
  51. 51. 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 machtmich wahnsinnig... ansonsten wie normale Zahlenbasis.“- „polyadisches System, mit den Zeichen Zeichen B, C, D, mit Ausnahme, anrechtester Stelle fängt es immer mit dem Zusatzzeichen A an.“
  52. 52. A few explanations Wenn rechts kein D steht, dann ändere diese Position in den nächsthöheren Buchstaben. Wenn rechts ein D steht, dann 1. Wenn links daneben kein D steht, dann ändere das D in ein A und den Buchstaben daneben in den nächsthöheren Buchstaben. 2. Wenn links daneben ein D steht, dann ändere DD in BAA. (Das geht so nicht, da dann alle Zeichenketten länger als 3 nur Bs links hätten. So ungefähr habe ich zuvor fortgesetzt. Vielleicht sollte man eher sagen, dass 1. Wenn links neben dem D nicht nur Ds stehen, dann zähle wie im 4er-System eins weiter. 2. Falls dort nur Ds stehen, dann tritt an die Stelle die Zeichenkette der gleichen Länge nur aus As mit noch einem B links davon.)
  53. 53. A few explanations zuerst: folge A,b,c,d rechts, dann an nächster stelle links eins weiterzählen (in der Folge a,b,c,d), dann erst wieder rechts durchzählen von a bis d. dann kann links wieder eins weitergezählt werden. (Diese Beschreibung passt für die ersten zwei Stellen). Insgesamt: Es wird immer ganz rechts von A bis D durch gezählt, zählt man dann noch eins weiter muss eine Stelle links weitergezählt werden, dann wieder rechts. Dieses Weiterzählen wird eventuell noch weiter nach links verschoben, wenn an der ersten stelle links schon D steht.
  54. 54. your speculations about the online experiment Ich habe die Vermutung, versagt zu haben. Das Experiment geht bestimmt darauf ein, welche Regeln Menschen nutzen, um Folgen zu konstruieren, obwohl es stets viele Möglichkeiten gibt, eine Folge weiterzudenken. Wollt ihr wissen wie wir mathematisch vorgehen? Ich vermute, dass es darum geht, wie sehr man von seinen Erfahrungen im Dezimalsystem verblendet ist. Funktionsweise Stellenwertsystem. Vor- und Nachteile davon. => durch Experiment viel sichtbarer als nur anhand der Zahlzeichen
  55. 55. discussion
  56. 56. discussion cycle of abstraction (“construction & trial & error & correction”) - extracting of operations (one „sees“ patterns) - coordination of these operations (hierarchical and sequential order) - through ongoing application of the operations - and checking for problems → results in a system of operations, that realises a successor function.

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