Paradoxes and student conceptual difficulties in probability theory

Vjerojatnost je gradivo koje se u Republici Hrvatskoj obraduje u osnovnoj i u srednjoj školi. Učenici se pritom susreću s konceptualnim poteškoćama koje mogu biti specifične za vjerojatnost. Konceptualne poteškoće se razlikuju od nesporazuma ili nedostatka znanja po tome što predstavljaju dubinsko nerazumijevanje koncepta u vjerojatnosti te su teške za ukloniti. Zato konceptualne poteškoće možemo nazvati miskoncepcijama. Miskoncepcije koje se najčešće javljaju u vjerojatnosti su pristranost jednakoj vjerojatnosti (equiprobability bias), pristranost reprezentativnosti (representativeness bias) i orijentacija na ishode (outcome orientation). Pristranost jednakoj vjerojatnosti je sklonost učenika da automatski smatra da su u nekom pokusu svi dogadaji jednako vjerojatni, iako nema razloga za to. Pristranost reprezentativnosti je sklonost učenika da očekuje da rezultati pokusa moraju odgovarati nekom uobičajenom uzorku ili poznatom uzorku. Orijentacija na ishode je sklonost učenika da se fokusira na ishod pokusa, a ne na vjerojatnost ishoda pokusa, pa time odbija prihvatiti da je pokus slučajan i smatra da na rezultat pokusa može utjecati vanjska sila. Istraživanja pokazuju da se navedene miskoncepcije pojavljuju u svim dijelovima obrazovanja, dakle u osnovnoj i srednjoj školi te u višem obrazovanju. Moguće miskoncepcije nastaju zbog učeničkog oslonca na intuiciju. Budući da se u vjerojatnosti često javljaju problemski zadatci s kojima se učenik možda susreo u životu, učenik može na temelju iskustva ili osjećaja procijeniti vjerojatnost događaja, a pritom zanemariti matematičku logiku iza rješavanja zadatka. Osim toga, učenik može jedno pitanje zamijeniti drugim, na koje mu je lakše odgovoriti, pa time dobiti krivo rješenje. Miskoncepcije i oslonac na intuiciju se mogu ukloniti susretanjem sa zadatcima koji navode učenike na razmišljanje i propitkivanje svog razmišljanja, jer učenik često nije ni svjestan koliko ne razumije gradivo. Jedan od načina da se učenika dovede do razmišljanja je proučavanje paradoksa. U ovom radu su opisani paradoks rođendana, paradoks Montya Halla, Bertrandov paradoks i Simpsonov paradoks. Neki paradoksi se mogu zvati i problemima ili dilemama, jer nije problem u zadatku koliko u učenikovim miskoncepcijama. Drugi paradoksi su primjeri kontradiktornih izjava iz istih premisa, pa time nije problem u učeniku nego u samoj definiciji problema. Obje vrste paradoksa potiču učenika na razmišljanje i fokusiranje na razumijevanje problema. Da bi učenici razumjeli rješenja zadataka, potrebno je dobro razumijevanje vjerojatnosti. U drugom poglavlju se obrađuje dio teorije vjerojatnosti koja je potrebna za zadatke u srednjoj i osnovnoj školi. U samom stvaranju teorije vjerojatnosti su matematičari imali poteškoća, jer klasične definicije vjerojatnosti (a priori i a posteriori) nisu bile precizno definirane i imale su ograničenja. Nakon uvodenja Kolmogorovljeve aksiomatike, funkcija vjerojatnosti i vjerojatnosni prostor su se mogli precizno definirati te njima rješavati zadatci.In the Republic of Croatia, probability is covered in primary and secondary schools. In doing so, students encounter conceptual diffculties which are specific to probability. Conceptual diffculties differ from lack of knowledge in that they represent a profound misunderstanding of the concept in probability and are diffcult to overcome. That is why we can call them misconceptions. The misconceptions that occur most often in probability are equiprobability bias, representativeness bias and outcome orientation. Equiprobability bias is a student’s tendency to automatically assume that in an experiment all events are equally probable, although there is no reason to do so. Representativeness bias is the tendency of students to expect that the results of an experiment must match a common pattern or a known pattern. Outcome orientation is the student’s tendency to focus on the outcome of the experiment rather than the probability of the outcome of the experiment, thus refusing to accept that the experiment is random and believes that the outcome of the experiment may be influenced by an external factor. Research shows that these misconceptions occur in all parts of education; primary and secondary school and in higher education. Possible misconceptions arise due to the student’s reliance on intuition. Because a student may have encountered in life problems that occur in probability, the student can assess the probability of an event based on experience or feelings, rather then the mathematical logic behind solving the problem. In addition, the student can replace one question with another easier to answer, and thus get the wrong solution. Misconceptions and reliance on intuition can be overcome by encountering tasks that lead students to think and question their thinking, because the student is often not even aware of how much he or she does not understand the material. One way to get students to think is to study paradoxes. This paper describes the birthday paradox, the Monty Hall paradox, the Bertrand paradox and the Simpson paradox. Some paradoxes can also be called problems or dilemmas, because the problem is not as much in the task as in the student’s misconceptions. Other paradoxes are an example of contradictory statements coming from the same premises, so it is not a problem in the student but in the very definition of the task. Both types of paradoxes encourage students to think and focus on the definition of the problem. In order for students to understand the solutions of the problems, a good understanding of probability is required. The second chapter in this paper deals with part of the probability theory in high school and elementary school. Mathematicians had diffculty in creating probability theory itself, because the classical definitions of probability (a priori and a posteriori) were not precisely defined and had limitations. After the introduction of Kolmogorov’s axioms, the probability function and the probability space could be precisely defined and thus used for solving problem