In mathematics, probability is the field that explains the randomness (Moore, 1990). Understanding probability subjects sufficiently and recognizing the importance of probabilistic thinking helps us to better comprehend what is happening around us and allows us to examine whether the information reached is correct. Over the five decades, research on probability has grown tremendously (Shaugnessy, 2003). The development of probabilistic thinking and the misconceptions about probability were the focuses of research literature. To improve instruction of probability in school practice, Jones and his colleagues (1997) developed the Framework for Probabilistic Thinking that represents a coherent frame about how students think in probabilistic phenomena. This framework was enhanced for conditional probability and independence by Tarr and Jones (1997).

Conditional probability is one of the fundamental stochastic ideas, which has helped to develop probability theory and probabilistic thinking throughout the history, in Heitele’ (1975)’s list. However, compared to the studies on sample space, probability comparisons, and probability of an event, which are the three key constructs of probability thinking, the studies on conditional probability are quite a few in the literature (Jones, et al., 1997). Common findings of these several studies showed that this concept was marked as hard to understand by students (e.g. Fishbein & Gazit, 1984; Diaz and Batanero, 2009). In theoretical probability, probabilities of events can be calculated with the help of counting techniques such as permutations and combinations without trials. On the other hand, conditional probabilities could be more complicated to understand, since it depends on the conjunction event and restricts the overall sample space (Watson & Kelly, 2007). As called representative heuristics, students try to make quick judgement without considering to the role of all parts in shaping event and organize categorical data according to the prototype in their mind in conditional probability (Kahneman & Tversky, 1972).  However, recent studies suggest that use of natural frequencies (Martignon & Wassner, 2002) and visualization (Sedlmeier, 1999) can make it simpler to understand the process and to perform computations.

While some areas of mathematics such as geometry or functions acknowledge the use of representations as an inherent part, other areas like probability do not include it in the domain of specific knowledge (Zahner & Corter, 2010). Previous studies suggest that use of representations such as Venn diagrams, contingency tables, tree diagrams, and etc. can facilitate problem solving in probability (e.g. Corter & Zahner, 2007; Rossman & Short 1995). For example, analyzing categorical data that represented in two way tables could be essential for conditional probability to understand the underlying logic and to interpret necessary conditions (Rossman & Short 1995).

In the literature, there are limited studies that investigated the role of two-way tables in probabilistic problem solving. These limited studies suggest that two-way tables could help students to understand conditional probability and to discover Bayesian reasoning (Rossman & Short 1995; Watson, 1995). However, the question of how two-way tables can be useful in students’ problem solving process is not clear. This study aims to investigate the role of two-way tables in developing prospective teachers’ probabilistic thinking in the case of conditional probability and to explore prospective teachers’ views about the role of two-way tables in their problem solving processes.

In this case study, four participants were selected from sophomore students (2nd year) enrolled in the teacher education program at one of the largest public universities in Ankara, Turkey. All participants took the related courses that included probability in their curricula such as MATH 112 Discrete Mathematics and STAT 201-202 Introduction to Probability and Statistics I-II. Interviews were conducted to collect data since it allows the researcher to collect highly personalized data, furtherly probe the participant, gather rich and in-depth information about the topic in question, and give simultaneous feedback during the process (Creswell, 2007). Each interview approximately took 120 minutes. The first part of the interview, which involved a think-aloud process, was conducted to understand the role of two-way tables in participants’ probabilistic problem solving processes in the context of conditional. In the interview protocol, two items of the Conditional Probability Reasoning (CPR) test, which developed by Diaz (2007), were used. In the CPR test, all items referred certain knowledge, ideas, and biases for conditional probability and aimed to explore students’ conditional probability reasoning processes. For this study, item 1 related to computing conditional probability, joint probability, and inverse conditional probability from two-way tables and item 5 related to computing conditional probability from joint and compound probability were selected. In addition to these items, two problems were constructed with same targeted questions in the similar context as item 1 and item 5 by the researchers. The main difference between the questions was whether a two-way table represented in the main question or not. The participants were asked to express their solutions verbally. The researchers asked necessary questions to understand their thinking processes during the process. In the second part of the interview, students are asked to explore their own solution processes for all questions. Semi-structured clinical interviews were conducted to explore participants’ perspectives about the role of two-way tables in their problem solving processes. In the analysis of the students’ responses, all interviews were transcribed. After the open coding was performed, the general codes were specified. Two researchers showed 90% similarity in data coding.

The analysis of the think-aloud processes showed that in general, two-way tables, whether it is given in the problem or constructed by students, helped students in the solution processes. The participants used Bayesian reasoning in their solutions when a two-way table is represented in the question or constructed by students. However, in the case of not using two-way tables, they made subjective judgement or used representative heuristics. Thus, as similar with the previous studies (Rossman & Short 1995; Watson, 1995) participants could identify the data as a whole and also its parts and employ their intuitive sense for conditional probability with the analysis of two-way tables. In the consideration of these findings and regarding our research question, the results of the semi-structured interviews showed that two-way tables are helpful in probabilistic problem solving. Three of the students identified the role of two-way tables as useful whereas only one student (ParticipantB) stated that tables do not have an influence on his solution process. Still it was observed that ParticipantB employed his intuitive sense for conditional probability and applied precise numerical reasoning in problem 2 that included a two-way table, while using subjective reasoning in the solution of item 5. The findings regarded the prospective teachers’ views about two-way tables suggested that: representing the information as a whole, showing the distinctions and the intersections of categorical variables, and helping to observe the composition of sample space make two-way tables a facilitator of solution processes in conditional probability problems. On the other hand, two participants suggested that they can be hindrance in solution processes. They criticized that when two-way tables are given in the problem, numerical probabilities can be assigned by making transition from relative cell numbers into ratios or percentage judgements, without having to consider the meaning of the problem.

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