Sažetak (engleski) | Mathematics is a key educational area and the acquisition of mathematical competences has far-reaching effects on an individuals’ academic and professional development. The basic aspects of mathematical competence are conceptual knowledge, which represents the understanding of concepts and the relationships between these concepts (Baroody & Dowker, 2003), and procedural knowledge, which refers to the application of procedures in order to solve a problem or task (Rittle-Johnson, Siegler, & Alibali, 2001). While conceptual knowledge means "knowing that", procedural knowledge means "knowing how" (Byrnes & Wasik, 1991). Both types of knowledge are important for the development of adaptive expertise and success in mathematics. Since the 1980s, these two aspects of mathematical competences have been thoroughly studied (Hiebert & LeFevre, 1986; Rittle-Johnson & Schneider, 2015). Specifically, researchers were focused on the relationship between conceptual and procedural knowledge, as well as the teaching methods that could improve these forms of knowledge. There are four perspectives from which the relationships between conceptual and procedural knowledge have been examined and which aim to explain the order of acquisition of these two forms of knowledge: concept-first (Byrnes, 1992); procedure-first (Siegler & Stern, 1998); inactivation view (i.e., independent development; Haapasalo & Kadijevich, 2000); and iterative view (i.e., causal relationships are bidirectional; Rittle-Johnson et al., 2001). At present, the iterative perspective is most accepted in the literature, although research has shown that conceptual knowledge has a stronger influence on procedural knowledge than vice versa (Rittle-Johnson & Schneider, 2015). In order to encourage the acquisition of conceptual and procedural knowledge, it is useful to adjust teaching methods in accordance with approaches that have been shown to be effective in research and practice. The two teaching methods for which positive effects on conceptual and procedural knowledge have been confirmed in recent research are contrasted comparison (or comparing solution methods/problems) and productive failure. In contrasted comparison, students compare one procedure with a new procedure and, through this comparison, detect the key features of each procedure and differentiate between the two procedures (e.g., Rittle-Johnson & Star, 2007). Compared to the sequential presentation of procedures, the contrasted comparison method has been shown to be more effective on measures of procedural knowledge, procedural flexibility and conceptual knowledge. Productive failure is based on the integration of guided discovery and direct instruction and is implemented in two phases: 1) students (individually or in a group) discover the solutions to problems; and 2) during instruction provided by teachers, students compare their solutions with the correct solutions, which leads to the discovery and correction of negative knowledge and an understanding of the basic concepts included in the lesson (Kapur, 2014, 2016). This method often includes contrasted comparison in the first or second phase and has been proven to be particularly effective in acquiring conceptual knowledge. Since productive failure is a new approach in teaching mathematics and has been implemented most often in topics related to statistics (e.g., Kapur, 2014; Kapur, 2015a), it would be important to explore the contexts in which this approach has positive effects and whether it reduces or eliminates various systematic mathematical errors. One of these errors is the illusion of proportionality or linearity, which represents a propensity to comprehend certain sizes as proportionally related, even in situations where such understanding is not justified (De Bock, Van Dooren, Janssens, & Verschaffel, 2007). Proportionality is omnipresent in everyday life, as well as in various areas of mathematics. It is mostly intuitive and is taught before nonproportionality in mathematics education (Bocke et al., 2007; Ebersbach & Resing, 2008; Ministry of Science, Education and Sports, 2011). Therefore, increasing experience in solving proportional problems leads to automation in the application of proportional models. Automatic processes have been thoroughly examined within dual-process theories, which can add valuable insight into how this error in mathematical reasoning emerges and is maintained. Dual process theories are focused on two modes of information processing, labelled Type 1 and Type 2. Type 1 processing occurs automatically and engages minimal working memory. It is fast, effortless, inflexible and not influenced by verbal instructions (Evans & Stanovich, 2013; Stanovich & Toplak, 2012). Type 2 processing is slow and requires cognitive effort, thus engaging working memory. It is controlled, flexible, and younger in an evolutionary sense, can be taught verbally and is based on conscious thinking. According to the defaultinterventionist model, both Type 1 and Type 2 processing are activated every time people encounter information. Type 1 is activated automatically, while Type 2 follows afterwards and intervenes in four ways (Thompson, 2009). The first intervention indicates little or no further analysis of the initial response. The second type is rationalization, through which Type 2 processing analyses the initial response and subsequently justifies this response, regardless of its accuracy. The third intervention is the most demanding in light of its improvements to the initial response. Finally, the fourth intervention is the failure of Type 2 processing because it generates an alternative response that is less satisfying than the initial response. The theory of metacognitive reasoning assumes that the type of intervention applied depends on metacognition, often labelled Type 3 processing. More precisely, it has been proposed that metacognition is the mediator between Type 1 and Type 2 processing (Thompson, 2009). Among the various dimensions of metacognition, researchers have typically explored metacognitive experiences as Type 3 processing, and the metacognitive feelings of rightness (FOR) and confidence in particular (where FOR is offered for an automatic answer, while judgment/feeling of confidence is estimated for a reflective answer). Recent studies have demonstrated that, when an individual has a strong FOR that the response that first comes to mind is correct, it is less likely that (s)he will reconsider that response later (Shynkaruk & Thompson, 2006; Thompson, Evans, & Campbell, 2013; Thompson & Johnson, 2014; Thompson, Prowse Turner, & Pennycook, 2011). In other words, Type 2 processing will only accept the initial response, without further analysis. Research examining dual-process theories and metacognitive reasoning have usually used problems in which the first, intuitive and heuristic answer is logically or rationally incorrect (i.e., non-congruent problems). In this situation, participants mostly respond in line with the intuitive, rather than the logical or rational, answer (e.g., De Neys & Glumicic, 2008; Thompson et al., 2011). Although most people err in non-congruent problems, studies examining conflict detection have revealed that, while people do detect conflict between their heuristic answer and logical or rational principles, they do not disregard the heuristic answer due to inhibition failure (De Neys, 2012, 2014; De Neys, Moyens, & Vansteenwegen, 2010; De Neys, Vartanian, & Goel, 2008). In other words, people possess a good system for monitoring information processing. Moreover, De Neys (2012) suggests that Type 1 processing generates both a heuristic answer and an intuitive logical answer and, depending on the situation, one answer will prevail over the other. Confirmation of De Neys’s (2012) ideas of intuitive logic is offered by research conducted by Bago and De Neys (2017), which demonstrated a high rate of logically correct answers (around 50%) generated by Type 1 processing. Experimental manipulations with time constraints and a secondary problem revealed that proportional reasoning has heuristic characteristics (Gillard, Van Dooren, Schaeken, & Verschaffel, 2009a, 2009b). As such, it is not surprising that people often ignore mathematical principles and instead tend to respond in accordance with an intuitive, proportional answer. Moreover, research conducted by Putarek and Vlahović-Štetić (2019) demonstrated that students can detect conflict between a heuristic or proportional answer and a mathematically correct (non-proportional) answer. However, the role of metacognitive feelings in the formation and maintenance of the illusion of proportionality should be further explored. Finally, the methods that are effective in improving mathematical competences (e.g., productive failure) have not been examined in the context of the illusion of proportionality. The current study The aim of this study was to examine the postulates of dual-process theories and metareasoning in the context of mathematical reasoning about proportionality and nonproportionality among second- and third-grade high school students (aged 16-17 years). Moreover, the aim was to examine whether productive failure can reduce the illusion of proportionality. A two-response paradigm (e.g., Thompson, 2009; Thompson et al., 2011) was used. Namely, when a proportional or non-proportional problem was presented for the first time, students were requested to provide their first or intuitive answer (i.e., although there was no time limit, students were under time pressure to respond). When the problem was presented a second time, students were told that they could take as much time as they needed to respond and were encouraged to think carefully and to provide the correct answer. Three groups of students were exposed to the two-response paradigm. Students in the control group did not receive any information beyond the basic instructions, while the remaining two groups were exposed to productive failure. Specifically, one group of students exposed to productive failure received instructions and solved examples that were aimed at encouraging procedural knowledge, while the other group received instructions and solved examples that were aimed at encouraging conceptual knowledge. In order to examine whether first responding altered second responding, another three groups of students responded only once, without time pressure. Here, one group did not receive any information beyond the basic instructions while productive failure was used in the other two groups (in the same manner as in the groups exposed to the two-response paradigm). For each problem, three possible answers were offered: the correct answer, a distractor (in non-proportional problems, the distractor represented a proportional answer to a nonproportional problem) and a “neither of these answers” option. The correct answer and the distractor were presented randomly as the first or second offered answer, while “neither of these answers” was always presented as the third offered answer and was used to reduce guessing among students. Finally, all students participated in a pre-test and a post-test that examined students’ procedural and conceptual knowledge about proportionality and non-proportionality and, at post-test, degree of transfer. In short, we examined whether students would be prone to the illusion of proportionality at pre-test, during the main experiment and at post-test. We hypothesized that students would mostly err in non-proportional problems and that their incorrect answers would be consistent with proportional reasoning. However, we assumed that students in groups exposed to productive failure would have more correct answers in non-proportional problems than students in the control groups and that they would not be prone to the illusion of proportionality. Furthermore, we examined students’ FOR and FJC, whether Type 2 processing would depend on FOR and whether students would detect conflict between heuristic and mathematically correct answers at the first and second response times (i.e., when they were exposed to time pressure and when they were given time to provide the correct answer). The hypotheses were as follows: 1) students would have higher FJC than FOR; 2) slower response time would be accompanied with lower FOR and FJC; 3) students would detect conflict between heuristic and mathematically correct answers at the first and second response times (i.e., they would exhibit lower FOR and FJC and a slower response time when selecting an answer consistent with proportional reasoning in non-proportional problems than when they selecting an answer consistent with proportional reasoning in proportional problems); 4) students with lower FOR would be more likely to change their answer and have a lower response time at the second response time than students with higher FOR. Moreover, we examined whether students exposed to productive failure would demonstrate a greater gain in their procedural and conceptual knowledge from pre-test to posttest, as well as greater transfer in post-test, than students in the control groups. We assumed that students exposed to productive failure, and those receiving instruction that encouraged conceptual knowledge in particular, would improve their mathematical competencies more than students in the control groups. Methods Participants Participants were 1083 high school students from four academic-track schools in Zagreb (67.7% were girls). A total of 694 students participated at all three points [pre-test, main experiment, and post-test]. Students were enrolled in the second and third grades (53.2% were second-grade students) at the time of participation. Participants’ mean age was 16.32 years (SD = 0.65, range 15-18). Materials Pre-test and post-test In the pre-test and post-test, students solved the same two problems, both of which measured procedural knowledge. One problem involved proportional relations between objects, while the other problem involved non-proportional quadratic relations. At post-test, there was an additional non-proportional problem involving cubic relations. The purpose of this problem was to examinee transfer. Two tasks were used to measure conceptual knowledge, where students compared proportional and non-proportional problem(s) and explained which mathematical concepts were represented by various parts of these problems. Main experiment At the beginning of the experiment, students exposed to productive failure solved a problem that involved non-proportional quadratic relations, while students in the control groups did not solve any problem nor read any already-solved examples. Afterwards, students in each experimental and control group solved 10 problems (5 proportional and 5 non-proportional quadratic problems; examples are presented below). Upon completion of each problem, students estimated FOR (for fast responding) and FJC (for reflective responding) on a scale from 1 (not at all confident) to 7 (highly confident). After solving all 10 problems, the sum of all correct answers was presented on the computer screen. Next, students responded to a number of sociodemographic questions, which included questions about general academic achievement and achievement in mathematics at the end of the previous school year. Finally, students’ motivation to participate in the study and any anxiety or boredom experienced during the study were estimated using responses to questions on a scale from 1 (“do not agree at all”) to 5 (“completely agree”). Example of a proportional problem: Workers need 3 days in order to build a fence around a square-shaped playground where one side is 60 m long. How much time do the workers need to build a fence around a square-shaped playground where one side is 120 m long? The speed of work is the same in both cases. a) 9 days b) 6 days (correct answer) c) neither of these answers Example of a non-proportional problem: There are 10 apple trees in a square-shaped orchard where one side is 5 m long. How many apple trees can grow in a square-shaped orchard where a side is 10 m long? The distance between the apple trees is the same in both orchards. a) 40 apple trees (correct answer) b) 20 apple trees c) neither of these answers Procedure All data were collected in schools between December, 2017 and May, 2018. The researchers informed students that all data will be collected anonymously and that they are allowed to terminate their participation at any time during the assessment. Following a brief introduction, all students agreed to participate. Students solved problems individually in classrooms. The pre-test and post-test were administered in paper-pencil form, while a computer program was developed for the main experiment. This program contained all instructions and productive failure, 10 problems, sociodemographic questions, as well as questions about motivation and emotions. Students solved the problems in 15 to 30 minutes. The time lag between pre-test and main experiment and between main experiment and post-test was between seven and 10 days. Results and discussion At pre-test, results suggest that students were susceptible to the illusion of proportionality, where 29.2% of students did not correctly solve the non-proportional problem and most incorrect answers were consistent with proportional reasoning. In contrast, nearly all students solved the proportional problem correctly (93% correct answers). In regards to conceptual knowledge, students demonstrated surface information processing and a low level of connection and integration between conceptual knowledge and procedural knowledge. These latter results about conceptual knowledge are consistent with international PISA research (Braš Roth, Markočić Dekanić, & Markuš-Sandrić, 2017) and previous analyses of the Croatian educational system (Garašić, Radanović, & Lukša, 2018; Bregvadze & Jokić, 2013). In the main experiment, students were mostly accurate in solving proportional problems, but inaccurate in solving non-proportional problems. Students exposed to productive failure (experimental groups) were equally likely to choose the correct answer and the answer in line with proportional reasoning, suggesting that students in these groups were less prone to the illusion of proportionality than those in the control group. Moreover, students in the experimental and control group increased their accuracy in non-proportional problems at the second response time, with students in the experimental groups demonstrating greater improvement than students in the control group. For proportional problems, students in control group demonstrated equal success rates as students in experimental groups, both in fast and reflective responding. It seems that productive failure had positive effects on students’ nonproportional reasoning and deceased their tendency to use a proportional model for nonproportional problems, a finding consistent with our hypothesis. Given that students in the experimental groups were able to increase their accuracy under fast responding conditions, our results are also consistent with De Neys’s (2012) model of intuitive logic, which proposes that Type 1 processing can generate not only a heuristic, but also a logical answer. Students demonstrated higher FJC than FOR, a finding that is consistent with previous research (e.g., Ackerman, 2014; Shynkaruk & Thompson, 2006) and can be explained by students’ beliefs about the relationships between response time and accuracy. While students demonstrated higher FOR and FJC when they solved problems faster, the correlation between these metacognitive feelings and response time was low. Therefore, it can be assumed that while answer fluency (represented by response time) is an important determinant of metacognitive feelings, “top-down” processes and other metacognitive feelings (e.g., feeling of difficulty) also have an effect on FOR and FJC (Ackerman, 2014). In both fast and reflective responding to non-proportional problems, students detected conflict between a heuristic answer and mathematical principles. This confirms the assumption that, while students have a “gut feeling” that they are wrong, they nevertheless err in nonproportional problems due to the salience of the proportional model and inhibition failure. The postulates of the theory of metacognitive reasoning (Thompson, 2009) were also confirmed. Specifically, students with lower FOR were more likely to change their answer at the second response time than students with higher FOR. As a result, students who were more accurate at the first response time and had lower FOR were less accurate at the second response time compared to students who were equally accurate, but had higher FOR. In contrast, students who were less accurate during the first response time and had lower FOR were more accurate at the second response time compared to students who were equally accurate, but had higher FOR. Finally, students with lower FOR were slower to solve problems at the second response time than students with higher FOR. These results confirm the importance of metacognitive feelings in information processing and solving mathematical problems. At post-test, students improved their accuracy in solving a non-proportional quadratic problem (50% correct answers). However, only 30.5% of students correctly solved the nonproportional cubic problem, suggesting that they were not successful in transfer. Experimental and control groups did not differ in their accuracy in solving problems at post-test, suggesting that productive failure had only short-term positive effects on deceasing the illusion of proportionality. Conclusion This research has demonstrated that, while productive failure has a positive effect on decreasing the illusion of proportionality, this approach should be modified (e.g., instructions should be provided by teachers and not only on a computer screen) in order to improve its longterm effects. In order to examine the illusion of proportionality, the theory of metacognitive reasoning was implemented and its postulates were confirmed. Namely, the metacognitive feeling of rightness was demonstrated to be an important determinant of Type 2 processing. Furthermore, the level of FOR can explain why students change their answers or stick with their original answers in a repeated response situation. Our results suggest that deep information processing strategies could be improved among students. For instance, more problem-based learning might be useful for supporting conceptual change, the detection of anti-knowledge and for encouraging adaptive expertise. During lessons, teachers should also emphasize the importance of metacognition. In other words, students should not only have knowledge of theoretical facts and procedures, but also know how to learn and how to self-regulate their learning. |