The document discusses research into how numerical concepts and representations develop. It proposes that new concepts are formed through extracting operations from existing concepts, coordinating those operations, and applying and evaluating them through a process of trial and error. An exploratory study found that participants used known operations like counting and arithmetic to develop a system of representations for a fictional base-4 number system. The document concludes that conceptual understanding is grounded through explicitly constructing new operations from older ones.

1. The Development of Counting and
Numerical Representations
Stefan Schneider, Benjamin Angerer, Sven Spöde, Alexander Blum
stefschn@uos.de, bangerer@uos.de, sspoede@uos.de, ablum@uos.de
October 3, 2010

2. Institute of Cognitive Science
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BACKGROUND / RESEARCH QUESTION
General Questions: What are concepts?
How do they come about?
We suppose that:
→ concepts derive from other concepts
→ concept formation uses former (pre-)concepts as “raw material“
→ To answer both questions we have to look at the concepts' grounding,
→ more specifically: we have to answer the epistemological question of how
concepts are grounded in experience.
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3. Institute of Cognitive Science
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BACKGROUND / RESEARCH QUESTION
Where does this raw material come from?
"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)
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4. Institute of Cognitive Science
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BACKGROUND / RESEARCH QUESTION
We are interested in a general mechanism that allows for this construction of
new concepts.
→ therefore the choice of raw material is not crucial, and should be
interchangeable (although it is not arbitrary)
Candidates for this mechanism:
→ Fauconnier's & Turner's Conceptual Blending
→ Piaget's Reflective Abstraction
→…?
So far, we concentrated on Piaget, as he was one of the few who tried to capture
the whole of cognitive development with his model. However, it still lacks detail.
Schneider, Angerer, Spöde, Blum The Development of Counting 4/11

5. Institute of Cognitive Science
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WHY NUMBERS?
- the development starts early, lasts long, results in complex and abstract concept
- numbers are used in a broad variety of contexts, thus providing a lot of material
- numbers are clearly definable and less fuzzy than many other abstract
philosophical concepts
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6. Institute of Cognitive Science
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METHODS
Accumulation of several approaches:
Using research from Developmental psychology / Maths Education:
- Looking at how abilities develop may give insight in how they work
- work from e.g. J. Piaget, K. Mix, I. Schwank
Theoretical psychological and philosophical analysis:
- What has to be possible, how can't it be under any circumstances
- e.g. One cannot "store" infinitely many representations of individual
numbers, but one can generate arbitrarily many numbers
- Systematicity: One can combine numbers and operate with them
(cf. Fodor/Pylyshyn)
Problem-solving tasks / Interviews with students:
- observing people solving problems and coming up with solutions
- esp. the structure of their argumentation in correspondence with their
performance
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7. Institute of Cognitive Science
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EXPLORATORY STUDY
Setup:
- subjects: a-level students
- 1 interviewer, 1 interviewee, video recording
- interviewer poses problems, subject should solve it
- subject was frequently asked to explain his answers ("Why?")
Tasks:
- base 4 number system, symbols: A,B,C,D A
- participants were not told that it is a number system B
- first task: "What comes next?" (and why)
- then simple arithmetic tasks (+,-,*) in the system C
(also with different symbols and without knowing the rule) D
BA
→ in more general terms:
all of the different small tasks were about BB
deriving general rules from given examples
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8. Institute of Cognitive Science
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OBSERVATIONS
Extraction: Many known operations “pop up” (are being extracted) 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 (subject already knows 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. which switches between increasing digits and enlarging the
sequence
- after one operation has taken place, the next one can apply
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9. Institute of Cognitive Science
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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
- e.g. is generalisation possible? Can I repeat that type of operation?
- “Occam's Razor”
→ A dynamic process of testing, observing, and reordering.
→ Very finegrained equilibration of operations (many details to be studied!) (cf.
Dubinsky, Piaget).
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10. Institute of Cognitive Science
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WRAPPING IT UP
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.
but understanding a domain is also
- a smorgasbord of tricks. Subjects develop “shortcuts”, e.g.
knowing that A # C → C and A is like zero, BA # C → BC comes easily
- having implicit understanding. One does not have to know how the number
system works to correctly apply it. But we believe that implicit understanding
comes through explicit construction and only then is becoming automatised.
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11. Institute of Cognitive Science
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CONCLUSIONS
→ Grounding is achieved through developing new operations from older ones
through explicit construction.
→ A multitude of established, transparent operations serves as material for the
formation of new ones.
→ A concept consists in operations of what can be done with it (or: of what
it can do). A number, thus, is not an individual, but a system of operations that can
be applied to anything that is recognized as such (e.g. our ABCD-system).
As a consequence we think that research on concepts and representations has to
focus on processes, which embody regularities.
→ For maths education this means that research on ordinality is more
promising than on cardinality (focussing on individual numbers without their
relation to others, cf. Brainerd).
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