Mauricio Herrera, Rubén Preiss and Viviana Schiappacasse
Facultad de Ingeniería, Universidad Diego Portales
Instituto de Ciencias Básicas, Santiago, Chile

The resolution of mathematical problems is a complex and demanding mental activity and often involves several associated tasks. People who are occupied in such activities are much more likely to respond to some of these tasks by blurting out whatever comes to mind. According to Kahneman and Tversky (2002) the rationality of thought is bounded by certain heuristic shortcuts which are applied in resolution of complex tasks and that in certain cases can lead to systematic errors. We use the so termed Dual Processes Theory (DPT) (Kahneman, 2002; Stanovich, 1994) to analyze, with some examples, the interrelation between intuitive and analytic thought in mathematical problem – solving situations and how this often conflicting interrelation can be affected by the introduction of computational tools like CAS calculators. According to DPT, our cognition and behaviour operate in parallel in two quite different modes, called System 1 (S1) and System 2 (S2), roughly corresponding to our common sense notions of intuitive and analytical thinking. Besides typical errors by carelessness (mistakes in algebraic manipulations, sign's change, etc.) or by simple ignorance of the subject matter, there is other kind of student’s errors, which could be related to the way in which the S1 and S2 systems work. Many researches (see for example Confrey, 1990; Tirosh et.al 1999) have supported the fact that students react in a similar way to a wide variety of conceptually non related problems which share some external common features. This fact could suggest that many responses that literatures describe as alternative conceptions (misconceptions) could be explained as evolving from the way in which mind process information. According to DPT we could assure that due to the lack of experience and knowledge the students often give solutions to a mathematical problem with fast and intuitive answers typical of the S1 system, without the controls and regulations that are characteristic of the S2 system. In simplifying their reasoning in an effort to reduce the burden of information processing, students may make a number of errors, which in turn lead to poor performance during the assessments. Often student’s intuition about certain mathematical concepts and ideas are not in line with accepted scientific frameworks in which are based the operation principles of computational tools used in the classroom. In order to assure an “intelligent dialogue” with these tools is necessary on one hand train students to be aware of the way S1 and S2 operate and to include this awareness in their problem - solving toolbox; on the other hand to insist in the correct use of definitions of mathematics’ objects and concepts. In an attempt to clarify the ways in which students think and reason when resolving mathematical problems we use some “cognitive mapping techniques” to explore the structure and content of student mental representations in relative direct fashion. Differences in the relative amounts of type S1 and type S2 processing employed during the resolution of the mathematical problem will affect the complexity of the student mental representation. Different outcomes in mathematical task will hold different mental representations by the student.

Confrey J., (1990). A review of research on student conceptions in mathematics, science and programming, C. Cazden (Ed.), Review of Research in Educations 16, Washington, DC; American Education Research Association, pp. 3 – 56.

Kahneman, D., (2002). Nobel Prize Lecture, December 2002, 8. Obtained on October, 2007, http://www.nobel.se/economics/laureates/2002/kahnemannlecture.pdf.

Stanovich, K.E.& West, R.F. (1994). Individual differences in reasoning: Implications for the rationality debate. Behavioural and Brain Sciences, 23, 645-726.

Tirosh D.& Stavy R., (1999). Intuitive rules: A way to explain and predict students’ reasoning. Educational Studies in Mathematics. 38: 51-66.