(although which kind of raw material is necessary for the development of a specific concept is of course not arbitrary, and finding these is one of the tasks of developmental psychology)
extraction conscious /unconscious coordination hierarchical and sequence checking with interviewer, sequence, intuition component operations come and go
further qualitative studies in a hermeneutic cycle sharpening ideas based on the studies, yielding new questions for further experiments We believe that qualitative investigation does a great job in structuring questions, insights and so on. But are there also possibilities to quantify the claims we derive from qualitative studies?
Kogwis count final
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The Development of Counting and Numerical RepresentationsStefan Schneider, Benjamin Angerer, Sven Spöde, Alexander Blum stefschn@uos.de, bangerer@uos.de, sspoede@uos.de, ablum@uos.de October 3, 2010
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Institute of Cognitive Science Study Project COUNTBACKGROUND / RESEARCH QUESTIONGeneral 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. Schneider, Angerer, Spöde, Blum The Development of Counting 2/11
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Institute of Cognitive Science Study Project COUNTBACKGROUND / RESEARCH QUESTIONWhere does this raw material come from?"If we do not want to believe that ideas are innate or God-given, but the result ofsubjective thinkers conceptual activity, we have to devise a model of howelementary mathematical ideas could be constructed - and such a model will beplausible only if the raw material it uses is itself not mathematical." (von Glasersfeld, p.64, 2006) Schneider, Angerer, Spöde, Blum The Development of Counting 3/11
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Institute of Cognitive Science Study Project COUNTBACKGROUND / RESEARCH QUESTIONWe are interested in a general mechanism that allows for this construction ofnew concepts.→ therefore the choice of raw material is not crucial, and should beinterchangeable (although it is not arbitrary)Candidates for this mechanism: → Fauconniers & Turners Conceptual Blending → Piagets Reflective Abstraction →…?So far, we concentrated on Piaget, as he was one of the few who tried to capturethe whole of cognitive development with his model. However, it still lacks detail. Schneider, Angerer, Spöde, Blum The Development of Counting 4/11
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Institute of Cognitive Science Study Project COUNTWHY 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 abstractphilosophical concepts Schneider, Angerer, Spöde, Blum The Development of Counting 5/11
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Institute of Cognitive Science Study Project COUNTMETHODSAccumulation 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 cant 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 Schneider, Angerer, Spöde, Blum The Development of Counting 6/11
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Institute of Cognitive Science Study Project COUNTEXPLORATORY STUDYSetup:- 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 Schneider, Angerer, Spöde, Blum The Development of Counting 7/11
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Institute of Cognitive Science Study Project COUNTOBSERVATIONSExtraction: Many known operations “pop up” (are being extracted) and are usedwhile 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 thesequence - after one operation has taken place, the next one can apply Schneider, Angerer, Spöde, Blum The Development of Counting 8/11
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Institute of Cognitive Science Study Project COUNTOBSERVATIONSApplication & Evaluation of ones ideas. “Running” the coordinated operation andchecking 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? - “Occams Razor”→ A dynamic process of testing, observing, and reordering.→ Very finegrained equilibration of operations (many details to be studied!) (cf.Dubinsky, Piaget). Schneider, Angerer, Spöde, Blum The Development of Counting 9/11
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Institute of Cognitive Science Study Project COUNTWRAPPING IT UPcycle 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 numbersystem works to correctly apply it. But we believe that implicit understandingcomes through explicit construction and only then is becoming automatised. Schneider, Angerer, Spöde, Blum The Development of Counting 10/11
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Institute of Cognitive Science Study Project COUNTCONCLUSIONS→ Grounding is achieved through developing new operations from older onesthrough explicit construction.→ A multitude of established, transparent operations serves as material for theformation of new ones. → A concept consists in operations of what can be done with it (or: of whatit can do). A number, thus, is not an individual, but a system of operations that canbe 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 tofocus on processes, which embody regularities. → For maths education this means that research on ordinality is morepromising than on cardinality (focussing on individual numbers without theirrelation to others, cf. Brainerd). Schneider, Angerer, Spöde, Blum The Development of Counting 11/11
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