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. 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
8. operating with 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. 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. how does counting actually work?
Principles/requisites that make up counting
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. 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. 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. 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. 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. 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.)
Vielheit
subitzing tagging
Objektidentität Objektpermanenz
alignment
Einheit SNWS
prä-
numerisch
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”
- Piaget's notion of “SCHEMA”
- mechanisms of “ASSIMILATION” and “ACCOMODATION”
- Dubinsky's differentiation:
- extraction
- coordination
- encapsulation
- generalisation
19. the experiments
- 12 qualitative case studies (video and tablet recordings)
- quantitative online study (so far 58 subjects)
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}
24. 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
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 sequence
EA new symbol, new combinations
DE append new symbols
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 sequence
EA new symbol, new combinations
DE append new symbols
AA {4}
complete "missing"
combinations
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 sequence
EA new symbol, new combinations
DE append new symbols
AAA {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. the qualitative studies
points of interest:
- the aspects subjects mention
- problems that they mention
- solutions and respective explanations
problems our subjects face:
- missing AA's
- order of variation in multiple digit sequences
(BAA → BBA, BAA → BAB, …)
- A = 1? (0-omitting habit)
29. the qualitative studies - observations
Extraction: Many known operations “pop up” and are used while
subjects 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 fingers
Coordination: Operations are being ordered sequentially and
hierarchically, e.g.:
- increasing digits. A then B then C then D
- turntaking, e.g. switch between increasing digits and enlarging
the sequence
30. the qualitative studies - observations
Application & Evaluation of ones ideas. “Running” the coordinated operation
and checking whether it works or „makes sense“; via some kind of judgement
about e.g.:
- interviewers reaction
- recognition value
- homogeneity / systematicity of the invented system
- “strong solutions” / “Occam's Razor”
- e.g. is generalisation possible? Can I repeat that type of operation?
→ A dynamic process of testing, observing, and reordering.
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. 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. 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 AAD
Solution types
ABCD - System A - problem new symbols new position precox middle variation
40. Lauf 1.1
A B C D > BA BB BC BD
keine 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 AD
keine 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. Lauf 1.3
ABCD > EFGH
keine 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 DD
keine 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. 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
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 (?)
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. 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 Teilfolgen
alle Kombinationen von Buchstaben werden verwendet
keine der Aussagen scheint mir sinnvoll
0 10 20 30 40 50 60
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 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.“
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 B's
links hätten. So ungefähr habe ich zuvor fortgesetzt.
Vielleicht sollte man eher sagen, dass
1. Wenn links neben dem D nicht nur D's stehen, dann zähle wie im
4er-System eins weiter.
2. Falls dort nur D's stehen, dann tritt an die Stelle die Zeichenkette
der gleichen Länge nur aus A's mit noch einem B links davon.)
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. 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
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.
