April 8, 2009  


  
Mental Processes
when
Learning Mathematics


On this web site we will give a brief overview:

Which are the mental concepts 

behind mathematical activities? And how do they develop? There are numerous investigations and we only can summarize some aspects. 

Children develop their mathematical abilities step by step. Elementary skills grow together and create new concepts and abilities. Bauersfeld (1983) introduced the term Subjektive Erfahrungsbereiche (SEB), in English Subjective Domain of Experiences, to describe developmental phases of concept formation. We all develop our own individual SEBs. And when we get confronted with a problem one of several related SEBs automatically get activated. The once chosen SEB then remains dominant as long as possible. Problem solving and learning in this sense means to manage the activation and the competitions between SEBs and to coordinate, adapt, select, and reconstruct the related inter­actions.

Through the individual subjective experiences including social interactions and communication processes with peers, teachers, parents, or others the children develop their own individual mental concept images, in German Vorstellungen. Vorstellungen cannot be seen, but they stimulate how learners or problem solvers react or answer or solve mathematical problems. These reactions or answers or solutions we call Darstellungen. Darstellungen we can see and interpret, they are external representations of mental processes or concepts. Also questions from teachers, written problems, graphs, etc. are Darstellungen. We carefully distinguish between Vorstellungen and Darstellungen. For more details see Meissner (2002)


There are two types of Vorstellungen

According to Vygotzky (1962) school learning leads to a more general and abstract thinking, dominated more by language and less from representations of reality. He therefore distinguishes natural concepts which are developed from representations of reality in more inductive forms, and scientific concepts which are learnt from general symbolic definitions later applied to concrete situations.

Ginsburg distinguishes between informal mathematics and formal mathematics. Informal mathematics involves techniques for solving quantitative problems developed by children before entering school or outside the school setting. Formal mathematics includes written symbols, algorithms, and explicitly stated principles generally taught in school.

Also Strauss (1982) describes that duality: Before schooling young children have a global non-differentiated concept of a certain domain which is appropriate to solve operations or tasks adequately within that domain. The concept is biological in origin and refers to a common sense knowledge. But then schooling starts and another cultural concept develops. 

Often there is a gap between formal and informal knowledge and this disparateness of knowledge is more common than rare.  

    

Dual Process Theories

That gap between intuitive and analytical thinking is of fundamental concern and it is not only related to mathematics education. Mainly experts from cognitive and social psychology are discussing such dual process theories. They analyze the thinking processes from professional decision makers (management, economics, justice, etc.) and argue that decision makers come to good decisions by efficiently combining intuitive and deliberate processes. 

Some dual-process theories are concerned with parallel competing processes involving explicit and implicit knowledge systems, others are concerned with the influence of preconscious processes that contextualize and shape deliberative reasoning and decision-making. But all these theories have in common the distinction between cognitive processes that are fast, automatic, and unconscious and those that are slow, deliberative, and conscious (Evans 2008).

Ejersbo, Inglis and Leron have brought to attention that dualism to the community of mathematics education researchers and teachers in their Working Session on Intuitive vs. Analytical Thinking at the international PME conference 2006 in Praha (see also Leron/Hazzan 2006). Using a dual process theory view we must be aware that working on a mathematical problem may happen in parallel in two different modes where a spontaneous or intuitive thinking (S1 mode) may interfere with the analytical or reflective thinking (S2 mode). Consequently we will need for the development of powerful mathematical concept images the development of two types of Vorstellungen, and criteria how they should  interact.

Team Work Results

In 1973/74 Hartwig Meissner was a mathematics teacher in a first grade in a primary school for one whole school year. The experiences collected here changed his view about mathematics learning dramatically. Thus he and his team and all preservice students in his projects became especially sensitive for mental processes when learning mathematics. During the last 30 years they developed many projects (Meissner 1995, 2003a). They went into schools to observe learning processes or to introduce new topics or to evaluate new teaching strategies.

They studied very early the use of calculators and computers in mathematics education (Lange/Meissner 1980, 1983). One of the first observations was to realize that the use of these machines stimulates a lot of guess and test activities. They also studied problem solving processes. They detected that also without using calculators or computers a lot of problem solving activities are dominated by an often unconscious guess and test behavior (Meissner 1985b). Meissner combined these two observations and created a new teaching strategy, the One-Way-Principle (in German Prinzip der Einbahnstrasse). This principle was used in many investigations:

Lange (1979, 1984) studied the effect of playing calculator games in arithmetic education to train mental arithmetic and number sense. Mueller-Philipp (1994) used the computer and the One-Way-Principle to develop a “function sense”. There also is a detailed study of how powerful strategies develop unconsciously when playing the calculator game Hit the Target (Meissner 1985a, 1987). We also had several investigations how to teach percentages in using the One-Way-Principle (Meissner 1982). And there is a summary analyzing the development of number sense (Meissner/Diephaus 2009).

Experimental investigations in schools give tremendous opportunities to observe learning processes. This is especially true when children discuss with peers or interact with machines or manipulatives. Here we get a diversity of individual Darstellungen of the children’s Vorstellungen which we then can interpret. This is  true also in geometry teaching when the students get objects, drawings, nets, etc. to manipulate with. Thus we discovered also in learning geometry interesting mental processes, a summary is given in Pinkernell (2003).
 

References  

The following list includes also references which were not mentioned above.

Bauersfeld, H. (1983): Subjektive Erfahrungsbereiche als Grundlage einer Interaktionstheorie des Mathematiklernens und –lehrens. In: Lernen und Lehren von Mathematik, Bd. 6, pp. 1-56, Aulis Verlag, Koeln Germany

Evans, J. St. B. T. (2008): Dual-Processing Accounts of Reasoning, Judgment, and Social Cognition. Annual Review of Psychology 59, p. 6.1-6.24.

Ginsburg, H. (1977). Children’s Arithmetic. Van Nostrand, New York USA

Gray, E. M., Tall, D. O. (1991): Duality, Ambiguity and Flexibility in Successful Mathematical Thinking. In: Proceedings of PME-XV, vol. II, pp. 72-79, Assisi Italy

Kienel, E. (1977): Datenverarbeitung und die algorithmische Methode im Mathematikunterricht (Dissertation). Muenster Germany

Lange, B. (1979): Schneller Kopfrechnen mit dem Taschenrechner. In: Sachunterricht und Mathematik in der Primarstufe, 7. Jg., H. 11, p. 430-441. Aulis Verlag Deubner & Co KG, Koeln Germany

Lange, B. (1984): Zahlbegriff und Zahlgefuehl (Dissertation). Lit-Verlag, Muenster Germany

Lange, B.; Meissner, H. (1980): Taschenrechnerspiele. In: Praxis der Mathematik, Vol. 22, p. 174-176 (Zielwerfen), p. 245-248 (Die grosse Null), p. 308-311 (Die grosse Eins) and p. 373-375 (Faktorfinden). Aulis Verlag Deubner & Co KG, Koeln Germany

Lange, B.; Meissner, H. (1983): Zum Lernprozess im Bereich Arithmetik. In: Zentralblatt fuer Didaktik der Mathematik, Vol. 15, H. 2, p. 92-101. Klett Verlag, Stuttgart Germany

Lawler, R. W. (1981): The Progressive Construction of Mind. In: Cognitive Science 5, pp. 1-30. Ablex Publishing Corporation Norwood, New Jersey USA

Leron, U.; Hazzan, O. (2006): The Rationality Debate: Application of Cognitive Psychology to Mathematics Education. Educational Studies 62/2, p. 105-126

Meissner, H. (1978). Projekt TIM 5/12 - Taschenrechner im Mathematikunterricht fuer 5- bis 12-Jaehrige. Zentralblatt fuer Didaktik der Mathematik, Vol. 10, no. 4, p. 221-229, Klett, Stuttgart Germany

Meissner, H. (1979). Problem Solving with the One Way Principle. In: Proceedings of PME 3, p. 157-159, Warwick England

Meissner, H. (1982): Eine Analyse zur Prozentrechnung. In: "Journal fuer Mathematik-Didaktik", Jg. 3, Heft 2/1982, S. 121-144, Verlag Ferdinand Schoeningh, Paderborn Germany

Meissner, H. (1985a): Selfdeveloping Strategies with a Calculator Game. In: Proceedings of PME 9, p. 53-58, Noordwijkerhout, The Netherlands

Meissner, H. (1985b): Versuchen und Probieren - Beobachtungen zum mathematischen Lernprozess. In: "Empirische Untersuchungen zum Lehren und Lernen von Mathematik", Schriftenreihe Didaktik der Mathematik, S. 175-182, Hoelder-Pichler-Tempsky, Wien Austria

Meissner, H. (1986a): Cognitive Conflicts in Mathematics Learning. In: "European Journal of Psychology of Education", Vol. 1, No. 2/1986, p. 7-15, I.S.P.A. Lisboa, Portugal

Meissner, H. (1986b): Kognitive Konfliktsituationen beim Mathematiklernen. In: "Grundfragen der Entwicklung mathematischer Faehigkeiten", IDM-Schriftenreihe Untersuchungen zum Mathematikunterricht, Bd. 13, S. 46-58, Aulis Verlag Deubner & Co KG, Koeln Germany

Meissner, H. (1987): Schuelerstrategien bei einem Taschenrechnerspiel. In: "Journal fuer Mathematik-Didaktik" Jg. 8, Heft 1-2/1987, S. 105-128, Verlag Ferdinand Schoeningh, Paderborn Germany

Meissner, H. (1992): Was heisst Verstehen? In: "Beitraege zur Didaktik der Mathematik, Heft 1/1992: Verstehen und Beweisen", S. 17-31, Schriftenreihe des Fachbereichs Mathematik und Informatik der Universitaet GH Essen, Germany

Meissner, H. (1994): Graphical and Symbolical Representations of Functions by the Use of Computers. In: „Proceedings of the 46th CIEAEM Meeting“, Tome 2, p. 97-104, Toulouse France

Meissner, H. (1995): Teacher Students as Researchers. In: „European Research Conference on the Psychology of Mathematics Education“, p. 50-53, Osnabrueck Germany

Meissner, H. (2002):  Einstellung, Vorstellung, and Darstellung. In: Proceedings of PME 26,  Vol. 1, pp. 156-161, Norwich UK

Meissner, H. (2003a): Beispiele fuer Projekte zum empirischen Forschen mit Studierenden. In: "Beitraege zum Mathematikunterricht 2003", S. 433-436, Franzbecker Verlag, Hildesheim Germany

Meissner, H. (2003b): Constructing Mathematical Concepts with Calculators or Computers. In: Proceedings of CERME 3. Bellaria Italy

Meissner, H. (2006a). Taschenrechner in der Grundschule. mathematica didactica, 29. Jg., Heft 1, pp. 5-25. Franzbecker Verlag, Hildesheim Germany

Meissner, H. (2006b): Creative Use of Calculators. Proceedings of the Fourth International Conference "Creativity in Mathematics Education and the Education of Gifted Students", p. 31-34. Ceske Budejovice, Czech Republic

Meissner, H. (2008): Mental Processes: Intuitive versus Analytical. Contribution to: Hoyles, C.; Lagrange, J.-B. (Eds.): ICMI Study 17 “Digital Technologies and Mathematics Teaching and Learning: Rethinking the Terrain”. Springer, New York USA

Meissner, H.; Diephaus, A. (2009a): The Development of Number Sense. Will appear in the Proceedings of SEMT 2009, Praha Czech Republic, see also web page http://wwwmath.uni-muenster.de/didaktik/u/meissne/WWW/mei150.doc

Meissner, H. (2009b): Paper & Pencil Skills in the 21st Century, a Dichotomy? Will appear in the Proceedings of the 10th International Conference “Models in Developing Mathematics Education”. Dresden 2009, Germany, see also web page http://wwwmath.uni-muenster.de/didaktik/u/meissne/WWW/mei153.doc 

Meissner, H.; Mueller-Philipp, S. (1993):  Teaching Functions. In: Proceedings of PME 17, Vol. II, p. 89-96, Tsukuba Japan

Mueller-Philipp, S. (1994). Der Funktionsbegriff im Mathematikunterricht (Dissertation). Waxmann, Muenster Germany

Pinkernell, G. (2003): Raeumliches Vorstellungsvermoegen im Geometrieunterricht - eine didaktische Analyse mit Fallstudien (Dissertation). Franzbecker Verlag, Hildesheim Germany

Schrage, K.-P. (1988): Bruchrechnung International – Eine Bestandsaufnahme in vier Laendern. (Dissertation). Muenster Germany

Strauss, S. (Ed., 1982): U-shaped Behavioral Growth. Academic Press, New York, USA

Vygotsky, L. S. (1962): Thought and Language. MIT Press, Cambridge MA, USA

 

See also List 3 of References and the following web pages:

(*) Discussion Forum on the Future of Paper & Pencil Skills:
http://wwwmath1.uni-muenster.de/didaktik/u/meissne/WWW/Forum-P&P.htm  

(*) Investigations with Calculators in Primary Schools:
http://wwwmath1.uni-muenster.de/didaktik/u/meissne/WWW/TRint.htm

(*) Summary report on TIM Calculator Projects in Germany:
http://wwwmath1.uni-muenster.de/didaktik/u/meissne/WWW/TR.htm