Developing critical thinking through probability models

In light of the importance of developing critical thinking, and given the scarcity of research on critical thinking in mathematics studies in the broader context of higher-order thinking skills, we have carried out a research that examined how teaching strategies oriented towards developing higher-order thinking skills influenced the students' critical thinking abilities. The guiding rationale of the work was that such teaching can foster the students' skills of and dispositions towards critical thinking. In this research, a primary attempt has been made to examine the relations between education for critical thinking and mathematics studies through examining teaching and learning critical thinking according to the infusion approach, which combines critical thinking and mathematical content ("Probability in Daily Life" learning unit).The main contribution of this work and the innovations it is expected to introduce lie in elucidating the connection between critical thinking and the study of mathematics and creating insights into the mechanisms of critical thinking development, and its place and importance in the study of mathematics, in spite of the uncertainty whether critical thinking skills acquired in studying one field will necessarily be applied by students in other fields, referred to as "the transfer problem." In this way it will be possible to strengthen the status of mathematics studies in imparting higher-order thinking skills in various frameworks, in parallel with and beyond the formal program of studies. The purpose of this research is to examine how and to what extent it is possible to develop critical thinking by means of the learning unit "Probability in Daily Life" using the infusion approach. The research questions that guided it are: (1) To what extent does the study of "Probability in Daily Life" in the infusion approach contribute to the development of critical thinking dispositions? (2) To what extent does the study of "Probability in Daily Life" in the infusion approach contribute to the development of critical thinking abilities? (3) What are the processes of construction of critical thinking skills (e.g., identifying variables, postponing judgment, referring to sources, searching for alternatives) during the study of the "Probability in Daily Life" learning unit in the infusion approach? The present research involved nine groups of gifted and high-achieving mathematics students in eleventh grade from all the social groups and strata of Israeli society. The students studied the learning unit "Probability in Daily Life" modified by the researchers to include critical thinking teaching in the infusion approach. The students were then tested in two critical thinking tests, CCTDI and the Cornell Critical Thinking Test, the results of which were statistically analyzed, and also selectively interviewed, with subsequent qualitative analysis of the interviews and lesson transcripts. Thus the research combines quantitative and qualitative methods. The research findings can be summed up in the following categories: (i) In all three iterations of the experimental teaching, a moderate improvement was detected in the critical thinking dispositions of all experimental groups. (ii) Throughout these iterations, a moderate improvement was also detected in the students' critical thinking abilities. (iii) Teaching critical thinking contributed to the construction and use of these skills in the framework of mathematics. Thus, when teachers consistently emphasize critical thinking skills, the students are more likely to succeed in the subject of mathematics. (iv) This research did not detect a clear-cut distinction between the critical thinking abilities and dispositions of excellent and average mathematics students. That is, no direct correlation has been found between the development of mathematical knowledge and the development of critical thinking. On the basis of these findings, the following recommendations for further research can be made: (1) A more comprehensive examination of the processes of critical thinking: to what extent could the students describe, orally and in writing, the processes of thinking, activate them and apply the thinking skills they studied on the procedural and meta-cognitive level? Did they make an informed use of terms and strategies of higher-order thinking, including critical thinking? In other words, it should be examined what use the research participants make of the "language of thinking," or, in the words of Costa and Marzano, "do they speak thinking?" (Costa & Marzano, in Harpaz, 1997). Developing such a language involves, on the part of the teacher, such skills as using precise vocabulary, presenting critical questions, presenting data rather than answers, aspiring for exactness, giving directions, and developing meta-cognition. (2) Examination of the attitudes and perceptions of education students in colleges for teacher training, practicing teachers and researchers of mathematical education with regard to teaching that develops critical thinking in mathematics; evaluation of these students' and professionals' critical thinking functions in teaching, learning, and research. (3) Teaching "Probability in Daily Life" and conducting the same research among all the strata of the students' population and not only among those who study mathematics at the higher level. It definitely seems that in the last decade, there has been a rapidly growing awareness of the importance of promoting the development of thinking skills in the Israeli educational system, and the system has been making considerable progress towards integrating the curriculum learning materials that contribute to the development of higher-order thinking skills1. In 1994, the Ministry of Education recognized thinking skills as a distinct subject of studies. This recognition lead to the establishment of a Subject Committee for Thinking Skills, which is in charge of consolidating appropriate didactic materials, as is the case with the rest of the academic subjects in the school system. The complex and ceaselessly changing contemporary reality, which requires independent decision-making on a daily basis, makes it extremely important to impart tostudents the ability to think critically. Critical thinking is needed in every field of activity, as it allows the individual to deal with reality in a reasonable, mature and independent way (Lipmann, 1991). The need for developing critical thinking in different disciplines is anchored in the ideals of education for democracy, as our freedom to think about and criticize the reality and society in which we live is a form of expression of our autonomy as individuals. Today this idea is even more vital, because of the growing need to be capable of engaging in inquiry and evaluation based on rational considerations regarding the various messages we are exposed to in different areas of life (Feuerstein, 2002, Perkins, 1992, Swartz, 1992). In the field of education, mathematics has traditionally been considered a branch of knowledge particularly suited to the teaching and learning of higher-order thinking skills, such as critical thinking. Mathematics curricula all over the world, including Israel, identify the acquisition of these skills as one of their goals. The idea that mathematics is a discipline suited to teaching critical thinking also appears in the research literature2. However, in spite of this assumption, very few empirical studies to date have engaged with the question of whether the study of mathematics indeed develops or even requires this mode of thinking. The answer to this question is far from being clear. The present research tackles precisely this basic question, "Is it possible to develop critical thinking in the framework of mathematics studies?".