Recent developments of the clinical interview and of the teaching experiment are described. Applications and extensions are discussed in the context of developing models of cognitive mechanisms in a constructivist framework. Purposes for which the clinical method has been adopted are described, and problems associated with it are discussed in relation to these purposes. A teaching experiment designed to reveal constructive mechanisms that children use in establishing knowledge of numeration and addition and subtraction is exemplified. Connections between these two methodologies are examined.
References
1.
ApostelL.Towards the formal study of models in the non-formal sciences. In FreudenthalH. (Ed.), The Concept and Role of the Model in Mathematics and Social Sciences.Dordrecht: D. Reidel, 1961.
2.
BerkeleyG.A Treatise concerning the Principles of Human Knowledge, Part I.Dublin: Aaron Rhames for Jeremy Pepyat, 1710.
3.
BrainerdC. J.Piaget's Theory of Intelligence.Englewood Cliffs, NJ: Prentice-Hall, 1978.
4.
BridgmanP. W.The Logic of Modern Physics.New York: 1927.
5.
CampbellN. R.Physics: The Elements.Cambridge: Cambridge University Press, 1920.
6.
DavydovV. V.The psychological characteristics of the ‘prenumerical’ period of mathematics instruction. In KilpatrickJ., and WirszupI. (Eds), Soviet Studies in the Psychology of Learning and Teaching of Mathematics. (Volume 7). Stanford, Calif.: Stanford University, School Mathematics Study Group and University of Chicago, Survey of East European Mathematical Literature, 1975.
DosseyJ. A.The relative effectiveness of four strategies for teaching algebraic and geometric disjunctive concepts and for teaching inclusive and exclusive disjunctive concepts.Journal for Research in Mathematics Education, 1976, 7(2), 92–105.
9.
DuhemP.The Aim and Structure of Physical Theory. (2nd ed.). (P. P. Weiner, Trans.). New York: Atheneum Press, 1974. (Originally published, 1914.)
10.
El'koninD. B.The problem of instruction and development in the works of L. S. Vygotsky.Soviet Psychology.1967, 5(3), 34–41.
11.
FlavellJ. H.The Developmental Psychology of Jean Piaget.Princeton, NJ: Nostrand, 1963.
Gal'perinP. Ya.On the notion of internationalization.Soviet Psychology, 1967, 5(3), 28–33.
14.
GelmanR., and GallistelC. R.The Child's Understanding of Number.Cambridge, Mass.: Harvard University Press, 1978.
15.
GinsbergH., and OpperS.Piaget's Theory of Intellectual Development: An Introduction.Englewood Cliffs, NJ: Prentice-Hall, 1969.
16.
HesseM.Models and Analogies in Science.Notre Dame: University of Notre Dame Press, 1966.
17.
HuntingR. P.Qualitative compensation thought in children's solutions to fraction comparison problems. In IronsC. (Ed.), Research in Mathematics Education in Australia, 1981: Proceedings of the Fifth Annual Conference of the Mathematics Education Research Group in Australasia (Volume 2). Adelaide: 1981.
18.
HuntingR. P.The role of discrete quantity partition knowledge in the child's construction of fractional number (Doctoral dissertation, University of Georgia, 1980).Dissertation Abstracts International, 1981, 41(10), 4308A–4381A. (University Microfilms No. 8107919)
19.
InhelderB., and PiagetJ.The Growth of Logical Thinking from Childhood to Adolescence.New York: Basic Books, 1958.
20.
KantI.Critique of Pure-Reason. (J. Meiklejohn, Trans.). London: Dent, 1934.
21.
KantowskiM. G.Processes involved in mathematical problem solving.Journal for Research in Mathematics Education, 1977, 8(3), 163–80.
22.
KantowskiM. G.The teaching experiment and Soviet studies of problem solving. In HatfieldL. L. (Ed.) and D. A. Bradbard (Tech. ed.), Mathematical Problem Solving: Papers from a Research Workshop.Columbus, Ohio: Ohio State University, ERIC/SMEAC, 1978.
23.
KrutetskiiV. A.The Psychology of Mathematical Abilities in School Children (Joan Teller, Trans.). Chicago: University of Chicago Press, 1976.
24.
KuhnT. S.The Structure of Scientific Revolutions. (2nd ed.). Chicago: University of Chicago Press, 1970.
25.
LangerJ.Disequilibrium as a source of development. In MussenP., LangerJ., and CovingtonM. (Eds), Trends and Issues in Developmental Psychology.New York: Holt, Rinehart & Winston, 1969.
26.
LovellK.Proportionality and probability. In RosskopfM. F., SteffeL. P., and TabackS. (Eds), Piagetian Cognitive-Development Research and Mathematical Education.Reston, Va.: National Council of Teachers of Mathematics, 1971.
27.
MagoonA. J.Constructivist approaches in educational research.Review of Educational Research, 1977, 47, 651–93.
28.
McLellanJ. A., and DeweyJ.The Psychology of Number.New York: D. Appleton, 1895.
29.
MenchinskayaN. A.The psychology of mastering concepts: Fundamental problems and methods of research. In KilpatrickJ., and WirszupI. (Eds), Soviet Studies in the Psychology of Learning and Teaching Mathematics. (Volume 1). Stanford, Calif.: Stanford University, School Mathematics Study Group and University of Chicago, Survey of East European Mathematical Literature, 1969.
30.
MontangeroJ.Recent research on the child's conception of space and geometry in Geneva: Research work on spatial concepts at the International Center for Genetic Epistemology. In MartinJ. L. (Ed.) and D. A. Bradbard (Tech. Ed.), Space and Geometry: Papers from a Research Workshop.Columbus, Ohio: Ohio State University, ERIC/SMEAC, 1976.
31.
MuangnapoeC.An investigation of the learning of the initial concept and oral/written symbols for fractional numbers in grades three and four. (Doctoral dissertation, University of Michigan, 1975).Dissertation Abstracts International, 1975, 36, 1353A–1354A. (University Microfilms No. 75–20, 415).
32.
NoeltingG.The development of proportional reasoning and the ratio concept (the Orange Juice Experiment), Part I: The determination of stages.Educational Studies in Mathematics, 1980, 11, 217–53.
33.
NovillusC. F.An analysis of the fraction concept into a hierarchy of selected subconcepts and the testing of the hierarchical dependence.Journal for Research in Mathematics Education, 1976, 7, 131–44.
34.
OpperS.Piaget's clinical method.Journal of Children's Mathematical Behavior.1977, 1(4), 90–107.
35.
PiagetJ.Autobiographie In Jean Piaget et les sciences sociales. (Vol. 10). Geneva: Librairie Droz, 1966.
36.
PiagetJ.Autobiographie. In Jean Piaget et les sciences sociales. (Vol. 10). Geneva: Librairie Droz, 1966.
PiagetJ.The Construction of Reality in the Child.New York: Basic Books, 1974.
41.
PiagetJ.Psychology and Epistemology.New York: Penguin, 1977.
42.
PiagetJ., and InhelderB.The Psychology of the Child.New York: Basic Books, 1969.
43.
PoincareH.Science and Method. (F. Maitland, Trans.). London: 1974.
44.
RichardsJ.Epistemology and mathematical proof. In SmockC. F., and von GlasersfeldE. (Eds), Epistemology and Education. (Mathemagenics Activities Program Report No. 14). Athens, Ga: Georgia Follow-through Program, 1974.
45.
RichardsJ.Modeling and theorizing in mathematics education. In FusonK. C., and GeeslinW. E. (Eds), Explorations in the Modeling of the Learning of Mathematics.Columbus, Ohio: Ohio State University, ERIC/SMEAC, 1979.
46.
RichardsJ., and von GlasersfeldE.Jean Piaget, psychology of epistemology: A discussion of Rotman's Jean Piaget: Psychologist of the real.Journal for Research in Mathematics Education, 1973, 11(1), 29–36.
47.
RombergT. A., and WilsonJ. W.The effect of an advance organizer, cognitive set, and post organizer on the learning and retention of written materials.Journal for Research in Mathematics Education.1973, 4(2), 68–75.
48.
SheplerJ. L.Parts of a systems approach to the development of a unit in probability and statistics for the elementary school.Journal for Research in Mathematics Education.1970, 1(4), 197–206.
49.
SmockC. D.Constructivism and principles of education. In SmockC. D., and von GlasersfeldE. (Eds), Epistemology and Education. (Mathemagenics Activities Program Report No. 14). Athens, Ga: Georgia Follow-through Program, 1974.
50.
SmockC. D.A constructivist model for instruction. In OsborneA. R. (Ed.), Models for Learning Mathematics: Papers from a Research Workshop.Columbus, Ohio: Ohio State University, ERIC/SMEAC, 1976.
51.
SteffeL. P.On a model for teaching young children mathematics. In OsborneA. R. (Ed.) and D. A. Bradbard (Tech. Ed.), Models for Learning Mathematics: Papers from a Research Workshop.Columbus, Ohio: Ohio State University, ERIC/SMEAC, 1976.
52.
SteffeL. P., RichardsJ., and von GlasersfeldE.Experimental models for the child's acquisition of counting and of addition and subtraction. In FusonK. C., and GeeslinW. E. (Eds), Explorations in the Modeling of the Learning of Mathematics.Columbus, Ohio: Ohio State University, ERIC/SMEAC, 1979.
53.
SteffeL. P., and SmockC. D.On a model for learning and teaching mathematics. In SteffeL. P. (Ed.), Research on Mathematical Thinking of Young Children: Six Empirical Studies.Reston, Va: National Council of Teachers of Mathematics, 1975.
54.
ThomasH. L.The concept of function. In RosskopfM. F. (Ed.), Children's Mathematical Concepts: Six Piagetian Studies in Mathematics Education.New York: Teachers College Press, 1975.
55.
UprichardA. E., and PhillipsE. R.An intraconcept analysis of rational number addition: A validation study.Journal for Research in Mathematics Education.1977, 8, 7–16.
56.
von GlasersfeldE.Piaget and the radical constructivist epistemology. In SmockC. D., and von GlasersfeldE. (Eds), Epistemology and Education. (Mathemagenics Activities Program Report No. 14). Athens, Ga: Georgia Follow-through Program, 1974.
57.
von GlasersfeldE.Radical constructivism and Piaget's concept of knowledge. In MurrayF. B. (Ed.), Impact of Piagetian Theory.Baltimore: University Park Press, 1978.
58.
von GlasersfeldE.The conception and perception of number. Paper presented at the Conference and Workshop on Modelling Mathematical Cognitive Development, University of Georgia, Athens, May 1979.
59.
VygotskyL. S.Mind in Society: The Development of Higher Psychological Processes.Cambridge, Mass.: Harvard University Press, 1978.
60.
WilsonJ. W.An epistemologically based system for classifying research and the role of clinical intervention research in that system. Paper presented at the meeting of the AERA, New Orleans, February 1973.
61.
ZykovaV. I.Operating with concepts when solving geometry problems. In KilpatrickJ., and WirszupI. (Eds), Soviet Studies in the Psychology of Learning and Teaching Mathematics (Volume 1). Stanford, Calif.: Stanford University, School Mathematics Study Group and University of Chicago, Survey of Recent East European Mathematical Literature, 1969.
