Halpern (2014) describes critical thinking as “purposeful, reasoned, and goal-directed” (p. 8) and contends that many definitions of critical thinking in literature use the term “reasoning/logic” (p. 8), so being able to apply the rules of logic can be seen as a requirement for critical thinking. Many studies report difficulties with logical reasoning for different age groups (e.g. Daniel & Klaczynski, 2006; Galotti, 1989; O’Brien, Shapiro, & Reali, 1971; Stanovich, West, & Toplak, 2016). Because of those difficulties, it is by no means certain that secondary school students are able to reason logically and thus develop their critical thinking abilities autonomously.
The area of logic can be divided into formal logic and informal logic. Aristotle already differentiated between formal logic with syllogisms described in Analytica Priora and ‘dialectics’ in his combined work Topica exploring arguments and opinions (Aristotle, 2015 version). Almost 2000 years later, Gottlob Frege (1848 – 1925) studied and developed formal systems to analyse thoughts, reasoning, and inferences. Also, he developed the so-called ‘predicate logic’, inspired by Leibniz (1646 – 1716), which is more advanced than the ‘propositional logic’ (Look, 2013; Zalta, 2016). Nowadays, those types of systems are often called ‘symbolic logic’ with strict validity as a key aspect (De Pater & Vergauwen, 2005) in which formal deductive reasoning is applied.
In general, formal systems contain a set of rules and symbols and the reasoning within these systems will provide valid results as long as one follows the defined rules (Schoenfeld, 1991). The corresponding reasoning is often called formal reasoning and “characterized by rules of logic and mathematics, with fixed and unchanging premises” (Teig & Scherer, 2016, p. 1). The same use of formal procedures can be found in definitions of logical reasoning as well. For instance “Logical reasoning involves determining what would follow from stated premises if they were true” (Franks et al., 2013, p. 146), and “When we reason logically, we are following a set of rules that specify how we ‘ought to’ derive conclusions” (Halpern, 2014, p. 176).
However, there is no consensus on the term reasoning and it is not exclusively used for formal deductive reasoning or mathematical situations only. Although reasoning in mathematics differs immensely from everyday reasoning (Yackel & Hanna, 2003), even reasoning in mathematical proof is not only a formal procedure, but involves discussion, discovery, and exploration (Lakatos, 1976) and shows us a need for more informal methods when approaching formal reasoning problems.
In the previous section, we indicated that, dependent on the situation, reasoning demands more than applying rules of logic. For instance, the importance of transforming information as stated by Galotti (1989): “[Reasoning is a] mental activity that consists of transforming given information … in order to reach conclusions” (p. 333) and the role of samenesses as stated by Grossen (1991): “Analogical and logical reasoning are strategies for finding and using samenesses. … logical reasoning applies these derived samenesses in order to understand and control our experience” (p. 343).
The notion of broadening formal methods with more informal methods is not new. Toulmin already discusses the limitations of formal logic for all sorts of arguments in his famous book The Uses of Argument (1958). He distinguishes different logical types to emphasise how logic is used in different fields, such as law, science, and daily-life situations. In his layout of an argument, he schematises the grounds for a claim balanced with reasons that rebut a claim. He also uses qualifiers to indicate the probability of a claim.
Philosophers and educators were also dissatisfied with the dominance of formal logic, that they considered as inappropriate for evaluating real-life arguments, and started in the 1970s an informal logic movement for another approach of analysing arguments stated in ordinary, daily-life language (Van Eemeren et al., 2014). One of the major textbooks still in print today is Logical Self-Defense (Johnson & Blair, 2006), which covers an introduction in “logical thinking, reasoning, or critical thinking … that focuses on the interpretation and assessment of ‘real life’ arguments” (p. xix). In literature, this is often indicated as informal or everyday reasoning, but this term has various meanings, from reasoning originating from formal systems to all reasoning related to everyday life events (Blair & Johnson, 2000; Voss, Perkins, & Segal, 1991). Different from formal reasoning, the reasoning and the conclusions depend on the context and can be questioned on their validity as already shown by Toulmin. Therefore, the topics are open for debate and invite to ponder on justifications and objections. The argument, as the result of the reasoning, often concerns open-ended, ill-structured real world problems without one conclusive, correct response (Cerbin, 1988; Kuhn, 1991). For this, Johnson and Blair (2006) use “acceptable premises that are relevant to the conclusion and supply sufficient evidence to justify accepting it” (p. xiii). The use of acceptable premises can arise from practical reasons to reach a certain goal and often includes presumptions or presuppositions. Walton (1996) uses the term ‘presumptive reasoning’ for this kind of arguments, which he sees as dialogues.
Although presumptive reasoning is not always conclusive or accepted by everyone, it is, in particular if full knowledge is unavailable or unobtainable, according to Walton, the best supplement to describe and discuss everyday life reasoning, for which he uses argumentation schemes. Even though Blair (1999) acknowledges the importance of presumptive reasoning for describing human reasoning and the strength of conclusions derived from the premises, he questions if all arguments are dialogues and discusses the completeness of the schemes.
To sum up, we define informal reasoning as reasoning in ordinary language to construct an argument which requires a critical review of the given premises and transforming of information, as well as finding additional or similar information provided by the problem solver or by external sources.
Towards a Definition for Logical Reasoning in This Study.
Now, we have seen that for well-founded reasoning, formal and informal methods are useful, we need to formulate a definition of logical reasoning for this study, which captures both aspects. A definition of logical reasoning should contain both the context and the way of reasoning, which can consist of formal and informal strategies. In other words, a definition of logical reasoning should not be synonymous with formal deductive reasoning. Important key words taken from the previous sections are ‘derive conclusions’ from Halpern and ‘transforming information’ from Galotti. That can be done with rules derived from formal systems, but that is not a necessity, so informal reasoning will also be part of our definition and thus seen as a valid reasoning process. Therefore, we conclude that logical reasoning involves several steps and define logical reasoning for this study as selecting and interpreting information from a given context, making connections and verifying and drawing conclusions based on provided and interpreted information and the associated rules and processes.
Formal and Everyday Reasoning Tasks.
Until now, we focused on the ways of reasoning and stressed the importance of the context. If we want to study how students reason in a variety of contexts, we have to differentiate between closed tasks with one correct answer and more open tasks. For this, we will use Galotti’s (1989, p. 335) division: ‘formal reasoning tasks’ and ‘everyday reasoning tasks’. Formal reasoning tasks are self-contained, in which all premises are provided. For those tasks, established procedures are often available which lead to one conclusive answer. In everyday reasoning tasks, premises might be implicit or not provided at all. For those tasks, established procedures are often not available and it depends on the situation when an answer is good enough. In daily-life situations, everyday reasoning problems “are [often] not self-contained” and “the content of the problem typically has potential personal relevance” (Galotti, 1989, p. 335). For both types of tasks, but for everyday reasoning tasks in particular, selecting and encoding relevant information is of great importance. We will call that the interpretation of the task.
Formal reasoning tasks may be provided in different forms: with symbols and completely in ordinary language without symbols. As shown in Fig. 1, we differentiate formal reasoning tasks in formally stated and in non-formally stated tasks. Formally stated tasks are stated with a certain set of symbols, for example a task with the premises ‘(1) All A are B. (2) All B are C.’ Non-formally stated tasks are tasks stated in ordinary language, for example a task with the premises ‘(1) All mandarins are oranges. (2) All oranges are fruits.’ For each task, students’ reasoning starts with an interpretation of the given information. That might be either a formal interpretation, in other words, an interpretation within a certain set of symbols (e.g. A ⊆ B ⊆ C ⇒ “All A are C”), or an informal interpretation in ordinary language.
Everyday reasoning tasks are not translatable to formal reasoning tasks and often contain implicit premises as, for instance, in everyday life stories. Like in formal reasoning tasks, students will need to interpret the information in everyday reasoning tasks as well. That can be done completely informally, but a formal representation, such as a schematic overview, might help students to get an overview of the given situation. In this study, we focus both on students’ interpretation and the reasoning strategies that follow from there.
From prior research among university students (e.g. Lehman, Lempert, & Nisbett, 1988; Stenning, 1996), we conjecture that reasoning in all kinds of situations will benefit from the use of formal representations or formalisations. We will use the term formalisation in its broadest sense, including all sorts of symbols, schematisations, visualisations, formal notations and (formal) reasoning schemes. Stenning (1996) gives support for the role of (elementary) formal notations and rules by mentioning that “learning elementary logic can [emphasis added] improve reasoning skills” (p. 227) and can help to understand formal thoughts and arguments. Also, Lehman et al. (1988) found support for the notion that reasoning in general can improve as a result of teaching formal rules within a particular field. Nonetheless, this does not imply that every formalisation is helpful: The chosen representation should support the thinking process for the specific context, rather than that it should capture all aspects (McKendree, Small, Stenning, & Conlon, 2002). In this study, we will investigate which formalisations are used by the participants and if those formalisations are beneficial.
Since little is known about the reasoning processes of 16- and 17-year-old students in logical reasoning tasks, our aim is to explore their reasoning strategies. Because of its exploratory nature, we selected, according to the division provided in Fig. 1, three elementary types of reasoning tasks: two formal reasoning tasks, to be presented with (formally stated) and without (non-formally stated) symbols, and an everyday reasoning task. Our exploratory study was guided by the following research questions: (1) How do students reason towards a conclusion in formal reasoning and everyday reasoning tasks, whether or not by using formalisations? And (2) what kind of reasoning difficulties do they encounter when proceeding to a conclusion?