On the Resilience of Misconception in Mathematics Learning : Focusing on 'the law of small numbers'

Abstract

In previous studies, it has been reported that misconceptions have some resilience. The resilience is recursivity with misconceptions or fragility with conception, and one of the important natures in mathematical recognition process. Many studies in mathematics education, however, have not regarded this nature as the subject matter of researches. The purpose in this paper is to address the resilience of misconception. Particularly, this paper focuses on misconception of the law of small numbers (MLSN), which makes learners believe that a small sample will be representative a population.
　　For this purpose, a tetrahedral model of static aspect of conception (TMSC), which is based on Steinbring's framework (i.e., the epistemological triangle) and on Mizoguchi's model (i.e., the C(C,N,E) model), is used as interpretative framework. TMSC consists of four components which are 'object/reference' (O), 'sign/symbol' (S), 'notion' (N) and 'conviction' (C) (the C and N in these four letters have different meaning from those in the C(C,N,E) model). N means learner's concept, ambiguous idea, knowledge, or mental model. C means learner's attitude towards mathematics or mathematical knowledge. O means learner's practical experience on object or reference context, which are laden with N and C. S then means learner's practical experience on sign or symbol, which are laden with N and C. One solid line and three broken lines show whether four components connect directly or indirectly with each other. TMSC as a whole means that O, S and N are nothing but connected indirectly by C. With this model, misconceptions are characterized as "non-shift of only C." MLSN is then identified by TMSC: [O] small sample; [S] population; [N] the law of large numbers; [C] determinism.

Fig. 1.　A tetrahedral model of static aspect of conception

　　The resilience of MLSN results from a kind of firmness. One of sources bringing firmness to misconceptions is a traility of C. In case of 'MLSN', which has probability as O, the C has three aspects as the traility: 1) nature of knowledge of probability; 2) attitude towards phenomena; 3) symbolic notation of number. The shift of C in 'MLSN' can be characterized as Fig. 3. That is, transitions of three aspects of C need shifting C itself, so it is difficult. The determinism as C in 'MLSN' means that a transition of one aspect of 1)〜3) does not occur at least.

Fig. 2.　A traility of ‘conviction’: ‘MLSN’

Fig. 3.　An interpretative framework for shift of ‘conviction’: ‘MLSN’