Measuring meta-ignorance through the lens of confidence: examining students' redox misconceptions about oxidation numbers, charge, and electron transfer

Alexandra R. Brandriet and Stacey Lowery Bretz *
Department of Chemistry and Biochemistry, Miami University, Oxford, OH 45056, USA. E-mail: bretzsl@miamioh.edu

Received 18th June 2014 , Accepted 27th July 2014

First published on 28th July 2014


This manuscript describes the relationship between students' redox understandings and confidence as measured by the Redox Concept Inventory (ROXCI) which assesses symbolic and particulate redox concepts. The ROXCI was administered to two samples of 1st- and 2nd-semester general chemistry students after the students were taught and tested on redox concepts in their classrooms. Cluster analysis was used to identify groups of students with similar response patterns, based upon both total scores and average confidence on the ROXCI. Three clusters of students were identified in both samples: students with (1) moderate total scores and high confidence, (2) low total scores and low confidence, and (3) low total scores but high confidence. Clusters were further analyzed at an individual item level using average confidence, individual item difficulties, and the Confidence Discrimination Quotient (CDQ). Findings align with the Dunning–Kruger effect, i.e. in which students demonstrated a false sense of confidence regarding their own poor performance, and therefore, exemplify meta-ignorance. Descriptions of the clusters, example misconceptions held by the students regarding oxidation numbers and electron transfer, and the implications of this research are discussed.


Introduction

Students' understandings

Students construct knowledge based upon information gathered from their environments by their senses. Constructivist theories describe knowledge as viable, adaptive, and founded within the learner's prior knowledge (von Glasersfeld, 1984, 1995; Bodner, 1986), and that students construct meaningful ideas when an integration of students' thoughts, feelings, and actions exists (Novak, 2010). However, when students construct meaningful knowledge, this does not necessarily mean that the new knowledge aligns with scientifically accepted ideas. When knowledge is incorrect, it can impede upon a deep understanding of scientifically correct ideas (Chi and Roscoe, 2002; National Research Council, 2012). Some of these incorrect ideas are seen as robust within students' mental frameworks because they are highly plausible and intelligible to students (Nakhleh, 1992; Strike and Posner, 1992). Because of this, they can be very difficult to remediate through formal instruction. These incorrect ideas are often described as “misconceptions” (Strike and Posner, 1992; Chi et al., 1994; Chi and Roscoe, 2002). Students hold misconceptions in a wide variety of chemistry content domains (Peterson and Treagust, 1989; Sanger and Greenbowe, 2000; Taber, 2002; Chandrasegaran et al., 2007; Smith and Nakhleh, 2011; Linenberger and Bretz, 2012, 2014; Naah and Sanger, 2012; Luxford and Bretz, 2013; Cruz-Ramirez de Arellano and Towns, 2014). As documented in the literature, one potential source of misconceptions is a student's inability to transition between the wide variety of chemical representations found in textbooks, lectures, laboratories, and scientific journals. Johnstone (1982, 1991, 1993, 2006, 2010) has categorized chemists' representations into the symbolic (e.g. chemical equation), the macroscopic (e.g. demonstration), and the particulate (e.g. atoms, molecules, or ions) domains. While experts may be able to effortlessly transition between domains, novices often struggle (Nakhleh, 1992; Davidowitz et al., 2010; Kelly et al., 2010; Kern et al., 2010; Nyachwaya et al., 2014).

Of particular interest to this study are students' understandings about oxidation–reduction reactions. Redox content is especially difficult for students (Johnstone et al., 1971; Butts and Smith, 1987), and several studies have identified misconceptions or difficulties that students have with redox concepts (Allsop and George, 1982; Garnett and Treagust, 1992; De Jong et al., 1995; Schmidt and Volke, 2003; Stains and Talanquer, 2008; Österlund and Ekborg, 2009; Österlund et al., 2010; Jaber and BouJaoude, 2012; Rosenthal and Sanger, 2012; Øyehaug and Holt, 2013; Brandriet and Bretz, 2014). In a fundamental study, Garnett and Treagust (1992) identified that students could not distinguish the concepts of oxidation numbers and charges such as assigning oxidation numbers to entire polyatomic molecules or ions, and using changes in the charges of polyatomic species to identify redox reactions. For example, in a reaction of CO2−3 + 2H+ → H2O + CO2, students identified that the charge changed from −2 for carbonate ion to 0 for carbon dioxide, and therefore, decided that the carbonate ion was oxidized. Additionally, in a study by Schmidt and Volke (2003), German students were surveyed and interviewed in order to determine if the students could identify acids, bases, oxidizing agents, and reducing agents in several reactions, as well as what mechanisms the students used to identify the reaction species. As with the findings of Garnett and Treagust (1992), these students also used changes in charges on polyatomic ions and molecules to identify oxidized and reduced species.

While a majority of the previous literature on students' understandings of redox reactions has focused on students' understandings in the symbolic domain (e.g., using chemical equations as prompts, applying oxidation numbers, etc.), fewer studies have identified students' misconceptions in the particulate domain. In a study by Rosenthal and Sanger (2012), students' misconceptions were elicited using two computer animations of varying complexity representing a reaction of solid copper and aqueous silver nitrate. Several student misconceptions were elicited during this study, including that cations and anions are attached or bonded together, an unspecified number of electrons transfer from copper to silver, and nitrate ions are more attracted to copper ions than to silver ions and thus drive the reaction.

In order to quantify the presence of misconceptions about redox chemistry, Brandriet and Bretz (2014) developed the Redox Concept Inventory (ROXCI) to measure students' understandings about redox concepts. The ROXCI is an 18-item multiple-choice assessment that measures students' ideas about oxidation numbers, surface features of chemical reactions, electron transfer, the role of the spectator ion, the dynamic reaction process, and electrostatics & bonding. These 6 major themes were identified during interviews with students, and individual item responses were drawn from students' quotes during the interviews. The ROXCI uses both symbolic and particulate prompts in order to elicit students' misconceptions within the symbolic domain, particulate domain, and the connections between the two. Each of the 18 items also includes an associated confidence tier that asks students to report their confidence about their responses by marking an ‘X’ on a 0% (Just Guessing) to 100% (Absolutely Certain) scale. Through the use of the confidence tier, the ROXCI not only detects students' redox misconceptions, but also identifies the robustness of the misconceptions. Example ROXCI questions and the associated confidence tier can be found in Fig. 1 and 2. Additional information about the development of the ROXCI, including the validity and reliability of the data generated is discussed elsewhere (Brandriet and Bretz, 2014).


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Fig. 1 Question 13 and associated confidence tier on the Redox Concept Inventory.

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Fig. 2 Question 7 and associated confidence tier on the Redox Concept Inventory.

Meta-ignorance

Several recent chemistry education research studies have recently reported that not only do students have misconceptions, but they are also largely unaware of their own misconceptions (Caleon and Subramaniam, 2010a, 2010b; McClary and Bretz, 2012; Sreenivasulu and Subramaniam, 2014). These results demonstrate the Dunning–Kruger effect, a well-known psychological phenomenon in which poor performers are largely unaware of their own deficiency (Dunning, 2011) and may exhibit greater confidence than their more able peers. Essentially, poor performing individuals lack the metacognitive resources which are necessary to recognize their inabilities, and therefore, these individuals embody meta-ignorance, i.e., ignorance about one's own deficits in knowledge (Dunning, 2011). Dunning (2011) describes three kinds of deficits: (1) instances of “unknown unknowns”, (2) instances where knowledge deficits are concealed by “misbeliefs” that individuals believe is true knowledge, and (3) instances where individuals “construct responses that are relevant and reasonable” to the individual, but in reality are incorrect. The latter two instances are most powerful for explaining students' misconceptions which are often intelligible, plausible, and highly integrated in students' mental frameworks. Therefore, it comes as no surprise that students may feel quite confident about their misconceptions. However, identifying deficits in one's own knowledge is an intrinsically difficult task, and students are unlikely to identify these deficits without intervention (Carter and Dunning, 2008; Dunning, 2011). Based on the emerging literature on the Dunning–Kruger effect in chemistry education research (Bell and Volckmann, 2011; Karatjas, 2013; Pazicni and Bauer, 2014), the problem of meta-ignorance seems to be a pervasive issue in the classroom, and therefore, suggests that students may be “poorly calibrated” (Caleon and Subramaniam, 2010b).

One way that instructors can help students identify deficiencies in their own understandings is to use formative assessments, such as concept inventories, in the classroom. A variety of chemistry concept inventories have been created (Treagust, 1988; Voska and Heikkinen, 2000; Mulford and Robinson, 2002; Chandrasegaran et al., 2007; Caleon and Subramaniam, 2010a; Villafañe et al., 2011; Bretz and Linenberger, 2012; Luxford and Bretz, 2014; Wren and Barbera, 2014). To a large extent, concept inventories use multiple-choice questions to elicit students' understandings. However, multiple-choice questions have been criticized because they allow for the possibility that students may respond by guessing rather than report their candid understandings (Burton, 2001; Roediger and Marsh, 2005; Caleon and Subramaniam, 2010a, 2010b). Researchers have sought to reduce this limitation by incorporating confidence tiers in order to gain an understanding of how confidently students rate specific responses (Clement et al., 1989; Hasan et al., 1999; Caleon and Subramaniam, 2010a, 2010b).

Recently, confidence testing has begun to appear in the chemistry education research literature (Caleon and Subramaniam, 2010a, 2010b; McClary and Bretz, 2012; Brandriet and Bretz, 2014). In a series of studies by Caleon and Subramaniam (2010a, 2010b), the authors developed the Wave Diagnostic Instrument in order to identify alternate conceptions that students held about waves. In these studies, genuine alternate conceptions were identified when students chose an incorrect distracter at a frequency of greater than 10% above chance and when the average confidence rating was greater than 3.5 on a 1–6 scale (greater than 50%). In these studies, a Likert scale was used for the confidence tier moving from “Just Guessing” at one end through “Very Unconfident,” “Unconfident,” “Confident,” “Very Confident,” to “Absolutely Confident” at the other end of the scale. In these studies, Caleon and Subramaniam (2010a, 2010b) introduced the Confidence Discrimination Quotient (CDQ) as a statistic for the standardized mean difference between the average confidence of the correct and incorrect students for each item. In this study, Caleon and Subramaniam (2010b) identified several items in which the CDQ values were negative, meaning that students who chose incorrect responses had greater confidence than their correct peers.

When creating a measurement, several decisions must be made regarding the scales used. Controversy exists within research regarding the appropriateness of conducting interval and ratio level analyses on ordinal data (e.g. Likert scales). The degree of difference between the categories in an ordinal scale may not be uniform (e.g., the difference between ‘Strongly Agree’ and ‘Agree’ may not be equal to the difference between ‘Agree’ and ‘Neutral’) (Stevens, 1946; Knapp, 1990), which could potentially be problematic when conducting statistical analyses, such as means and standard deviations. In a separate study, McClary and Bretz (2012) developed a concept inventory called ACID I to identify organic chemistry students' alternate conceptions and associated confidence about acid strength. In this study, the authors attempted to reduce these limitations by using an interval confidence tier that asked students to mark their confidence with an ‘X’ along an continuous scale from 0% (Just Guessing) to 100% (Absolutely Certain). McClary and Bretz (2012), identified several genuine student alternate conceptions and negative CDQ values, suggesting that the ACID I measures several robust student misconceptions. Based on this argument, the ROXCI also uses a continuous confidence scale similar to that used on the ACID I (Brandriet and Bretz, 2014).

Study significance

Dunning (2011) argues that there are many instances in life where individuals exhibit meta-ignorance and that some of these instances have no repercussions on their lives. However, in instances where knowledge deficiencies have implications for individuals' day-to-day activities, it becomes imperative to determine what deficiencies exist. However, it is impossible to expect individuals to identify these deficiencies on their own (Dunning, 2011). This outlook is not only true in the field of psychology, but also for chemistry students. The goal of this study is to characterize students' meta-ignorance regarding their understandings of redox concepts, and to provide chemistry instructors with examples of misconceptions that students hold post-instruction and post-assessment on redox content. Misconceptions that students hold are detrimental to future learning processes, and redox concepts are foundational to many chemical and biological processes that are recommended for an American Chemical Society approved Bachelor's degree curriculum (ACS, 2008). Students who are meta-ignorant are burdened by not only holding misconceptions, but also by being grossly unaware that their mental models are faulty, as they will continue to rely upon such misconceptions during their learning processes (Dunning, 2011).

Research questions

This research uses cluster analysis to explore the extent to which students who have been taught and tested about redox reactions are meta-ignorant regarding their understandings of redox concepts. A robust description of each cluster in terms of the students' average confidence, item difficulties, and Confidence Discrimination Quotients (CDQ) (Caleon and Subramaniam, 2010b) are described below. Finally, examples of students' misconceptions with high confidence are discussed in order to help instructors understand the difficulties that students have in the chemistry classroom. The research questions that guided this study include:

1. What is the nature of the relationship between students' understandings and confidence as measured by the ROXCI?

(a) Are students meta-ignorant about their understandings of redox concepts?

2. What misconceptions do students hold regarding electron transfer and oxidation numbers?

Methods

Student samples & data collection

Institutional Review Board (IRB) approval for human subjects research was obtained from the colleges and universities sampled for this study prior to collecting students' data. Data was collected from two distinct student samples. Sample 1 consisted of 1083 students in the first-semester of first year university chemistry called general chemistry I (GC1) in the United States. The students in Sample 1 were from 10 colleges/universities from 9 states within the United States. Instructors were recruited primarily through a chemistry education research listserv and by advertising the research study at conferences. In order to standardize the data collection methods, instructors were given detailed instructions for administering the ROXCI to their students and were given a script to read to students before the students took the assessment. The instructors were asked to implement the ROXCI to their students after they had been taught and tested on first-semester redox concepts (e.g., electron transfer and oxidation number concepts, often couched within the major classes of chemical reactions). All students answered a paper-and-pencil version of ROXCI, however, only those students who consented to allow their data to be used in the research study and did not have any missing data were included in the analysis. Students in Sample 1 marked their confidence by placing an ‘X’ on the scale from 0% to 100% (see Fig. 1 and 2). Sample 1 was 58% female and 42% male (3 students did not report their sex), and the students reported a variety of different academic majors (Fig. 3).
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Fig. 3 GC1 students' academic majors.

The ROXCI was also administered in an online format to Sample 2 who were students in the second-semester of first year university chemistry (GC2) at a large, Midwestern research university. No students from this university were part of Sample 1. Students answered the ROXCI using a department computer lab near the end of second-semester general chemistry, immediately prior to any instruction on electrochemistry or any experiments regarding electrochemistry and electrolysis. The continuous confidence tier scale was modified for use in online data collection software; students in Sample 2 did not mark an ‘X’ on the continuous scale shown in Fig. 1 and 2. Rather, they indicated their confidence using check-boxes from 0% to 100%, incremented every 5%. The option of 50% was not included, however, in order to address the local instructor's concern that students might choose a ‘neutral’ confidence response for each question. A total of 554 students consented to participate and 510 students remained for analysis after removing missing data. Demographic information was not collected for these students.

Data preparation

After the GC1 students completed the ROXCI, the paper-and-pencil assessments were mailed back to the researchers for scoring. A template ruler was created to simplify the scoring process for the confidence tier and estimated to nearest 1%. The students' multiple-choice and confidence responses were entered into a Microsoft Excel spreadsheet for data analysis. The accuracy of both the confidence measurements and the entered data were assessed by re-evaluating a random 10% of data from each college/university. For the GC2 students, the data was exported from survey software within a course management system, scanned for missing data, and prepared so that it was in a form practical for data analysis. Data analyses were conducted through the use of SAS 9.3, SPSS 21, and Microsoft Excel 2013.

Cluster analysis

Cluster analysis is a generic term to describe a variety of statistical techniques that can be used to identify and describe homogenous groups of entities, namely, clusters within a dataset (Mirkin, 2005). A chemistry analogy to cluster analysis would be classifying reactions as redox, combustion, precipitation, or acid–base. Reactions can be categorized or “clustered” according to empirical similarities, e.g., when electrons transfer or protons transfer. Clustering methods are particularly useful for exploring large datasets and are a convenient way of organizing data into smaller groupings (Everitt et al., 2011). Cluster analyses suggest structure within a dataset based upon relationships between variables.

Cluster analysis techniques are often considered exploratory and within a family of statistical techniques used for data mining (Everitt et al., 2011). Accordingly, cluster analysis does not provide the researcher with pre-defined clusters. While, the cluster solution identifies students who respond in a similar manner, the researcher must interpret these response patterns. That is, the researcher must craft an explanation of why the students are so clustered and determine the significance of the cluster's existence within, in this case, a chemistry context. In this study, students were clustered based upon patterns in both their total scores and their average confidence.

Results & discussion

Descriptive statistics

The descriptive statistics for the students' total scores and average confidence, as well as the Cronbach-α (Cronbach, 1951) and Ferguson-δ (Ferguson, 1949) for the ROXCI data can be found in Table 1. Scores on the ROXCI can range from 0 to 18; the mean score was 5.7 for GC1 and 6.2 for GC2. Therefore, ROXCI was quite difficult for both samples as can be seen by the total scores in Fig. 4 and 5. By contrast, students' average confidence was skewed toward the higher end of the 0–100% confidence range (also shown in Fig. 4 and 5), with an average confidence of 55.9% for GC1 and 57.5% for GC2. Clearly these histograms depict a disconnect between students' total scores and average confidence. However, histograms are insufficient for characterizing a possible relationship between total scores and average confidence about redox concepts, so students' total scores and average confidence were examined using scatterplots (Fig. 6a and 7a). Because the data was skewed, a Spearman Rank-Order correlation (also known as Spearman's rho) coefficient was used to evaluate whether or not a relationship existed. A medium positive correlation for both the GC1 (ρ = 0.433) and GC2 (ρ = 0.325) students suggests that there is a moderate relationship between students' total scores and average confidence. However, inspection of Fig. 6a and 7a reveal large variability within the students' average confidence, especially for the students with low total scores. Such variability warranted additional investigation in order to characterize the students' responses.
Table 1 Descriptive statistics for GC1 and GC2 ROXCI samples
  GC1 GC2
Total score (0–18) Average confidence (0–100%) Total score (0–18) Average confidence (0–100%)
N 1083 1083 510 510
Mean 5.7 55.9% 6.2 57.5%
Std. dev. 3.1 19.7% 2.9 21.6%
Minimum 0.0 0.2% 1.0 0.0%
Median 5.0 56.9% 6.0 60.6%
Maximum 16.0 99.8% 14.0 100.0%
Cronbach-α 0.68 0.94 0.62 0.96
Ferguson-δ 0.95 0.99 0.95 0.99



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Fig. 4 GC1 students' total scores and average confidence.

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Fig. 5 GC2 students' total scores and average confidence.

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Fig. 6 GC1 students' average confidence plotted against total scores for: (a) the aggregate of the sample, (b) the cluster solutions.

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Fig. 7 GC2 students' average confidence plotted against total scores for: (a) the aggregate of the sample, (b) the cluster solutions.

Cluster analyses

TwoStep cluster analysis. In order to further investigate the variability in students' responses in Fig. 6a and 7a, a TwoStep cluster analysis was conducted to classify the students' based upon both their total scores and their average confidence. The TwoStep method uses a two-step sequential clustering approach (Chiu et al., 2001; SPSS Inc., 2001; IBM SPSS Statistics, 2011, 2012a). In order to reduce a large dataset into smaller pieces, the first step considers the cases (i.e. students in the dataset) one by one, and the algorithm decides (based on a measure of distance as described below) whether the students should be merged with a previous group of students or if a new group of students should be started. These groupings are called preclusters (Chiu et al., 2001; SPSS Inc., 2001; IBM SPSS Statistics, 2011, 2012a). In the second step, an agglomerative hierarchical cluster analysis is conducted using the preclusters created in the first step. In this step, the preclusters are recursively merged until only one cluster remains. Since all of the cases in the precluster are now considered a single entity, it reduces the size of the distance matrix in the hierarchical step because the distance matrix is now dependent on the number of preclusters which is fewer than the number of students. This makes the TwoStep method ideal for clustering large datasets (SPSS Inc., 2001; IBM SPSS Statistics, 2011). Finally, the Bayesian Information Criterion (BIC) fit index (Schwarz, 1978; IBM SPSS Statistics, 2012a) is used to determine the optimal number of clusters produced by the hierarchical clustering. This is a highly desirable feature of the TwoStep clustering method (SPSS Inc., 2001; IBM SPSS Statistics, 2011).

The TwoStep method uses a log-likelihood distance measurement that works well for both categorical and continuous variables. The distance measure assumes that the variables in the model are independent, that each continuous variable is normally distributed, and that each categorical variable has a multinomial distribution (IBM SPSS Statistics, 2011; Norušis, 2012). However, this is rarely the case in practice, and the literature describes that the clustering algorithm is fairly robust to violations in these assumptions (IBM SPSS Statistics, 2011; Norušis, 2012). Further, the order in which the cases are arranged in the dataset can influence the final solution (IBM SPSS Statistics, 2011). Therefore, the stability of the cluster solution was assessed and is described below.

Cluster solution. A three cluster solution emerged for both the GC1 and GC2 student samples Fig. 6b and 7b. The three clusters are summarized in Table 2. The first cluster of students received the highest average scores (10.0/18.0 for the GC1 students and 9.6/18.0 for the GC2 students) with highest average confidence (73.0% for the GC1 students and 72.2% for the GC2 students); however, even these highest scoring groups of students still had average scores only near the mid-point of the point scale. Therefore, this cluster of students has been descriptively described as the “moderate scores/high confidence” group. The second cluster of students received the lowest average total scores (4.2/18.0 for the GC1 students and 4.9/18.0 for the GC2 students) and reported low average confidence (32.7% for GC1, 30.0% for GC2). These results suggest that this group of students seemed aware of the fact that they were likely not choosing the correct ROXCI responses. This group of students has been qualitatively labeled as the “low scores/low confidence” cluster. The third cluster of students also scored poorly on the ROXCI (4.4 for GC1, 4.5 for GC2), but reported a high average confidence (62.7% for GC1, 66.0% for GC2). This group of students has been qualitatively labeled as the “low scores/high confidence” cluster.
Table 2 Descriptive summary of the cluster solution
Cluster Cluster description GC1 GC2
N Average total score (SD) Average confidence (SD) N Average total score (SD) Average confidence (SD)
1 Moderate scores, high confidence 264 10.0 (2.0) 73.0% (13.0%) 158 9.6 (1.7) 72.2% (13.2%)
2 Lowscores, low confidence 334 4.2 (2.0) 32.7% (11.1%) 148 4.9 (1.9) 30.0% (12.0%)
3 Low scores, high confidence 485 4.4 (1.7) 62.7% (10.7%) 204 4.5 (1.6) 66.0% (11.7%)


Analytic interpretation of the cluster solution offers a meaningful characterization of the student sample that neither the histograms nor the correlations could provide. In both samples, more than 1/3 of the students responded with high confidence about their responses, despite scoring very poorly on the ROXCI assessment (i.e., Cluster 3). This is highly suggestive of the aforementioned Dunning–Kruger effect in which students hold illusory confidence and rate their abilities higher than is accurate (Kruger and Dunning, 1999; Dunning, 2011). Similar to Cluster 3, the students in Cluster 1 also responded with high confidence about their responses to the ROXCI, and while Cluster 1 on average scored higher than Cluster 3, their average scores were still only at that of the mid-point of the total possible scale. This again suggests that these students held illusory high confidence in comparison to their ROXCI responses, but to a lesser degree than the students in Cluster 3. While the students in Cluster 2 scored poorly on ROXCI, unlike Cluster 1 or 3, these students responded with low confidence which suggests they were aware that that they were performing poorly.

Validity, reliability, and stability of the cluster solutions. While several pieces of evidence can be gathered to evaluate the validity, reliability, and stability of a cluster solution, there essentially is no one correct answer regarding how many clusters might constitute “the best” solution. Establishing the validity of a cluster solution requires finding an interpretable and meaningful solution with a reasonable number of fairly homogeneous clusters (Norušis, 2012). The cluster solution presented above possesses face validity as a parsimonious solution (only 3 clusters) with a substantial number of students in each cluster, and the clusters can be interpreted such that they have the potential for implications in chemistry education (Mooi and Sarstedt, 2011). Additionally, the Average Silhouette Coefficient (SC) was used as a measure of the degree of cohesion and separation for the clusters in the solution (Kaufman and Rousseeuw, 1990). Both cluster solutions (i.e., for GC1 and for GC2) exceeded the threshold value for an SC of 0.50, which suggests a reasonable balance between cohesion of the clusters and separation amongst the clusters (Kaufman and Rousseeuw, 1990; IBM SPSS Statistics, 2012b). Finally, since the results of this study support the presence of the Dunning–Kruger effect (Kruger and Dunning, 1999), this provides additional evidence for the validity of the cluster solution. The reliability of a cluster solution is often assessed by examining the degree to which the solution is stable over time (Mooi and Sarstedt, 2011). While this study did not replicate data collection with the sample of GC1 students, the study was replicated with a sample of GC2 students. Similar results were obtained across student samples, and therefore, this study uses this as evidence for reliability of the cluster solution.

As noted above, a limitation of the TwoStep cluster method is that the results are dependent upon the order in which the objects (i.e. students) exist in the dataset. In order to examine the stability of the solution, IBM SPSS Statistics (2011) recommends repeating the cluster analysis with the objects (i.e. students) sorted in different random orders. Therefore, the cluster analysis was replicated 9 additional times. The order of the students in the dataset was randomly redistributed each time based on a random number generator. A 3 cluster solution was generated 7 out of 10 times for the GC1 student sample and 8 out of 10 times for the GC2 student sample. Furthermore, the replicate solutions produced scores and confidence results very similar to that summarized in Table 2. (For additional information, see the Appendix.)

Item analyses

Item difficulty and confidence. In addition to examining the results based on total scores, individual item analyses were also conducted in order to determine the extent to which the Dunning–Kruger phenomenon occurred per individual item. In order to do this, item difficulties were compared to the average of the students' reported confidence (Fig. 8 and 9). Difficulty is calculated as the fraction of the students who correctly answered each individual item. This value is known as the difficulty index (p) which is an item statistic commonly used when constructing assessments using Classical Test Theory (Kline, 1986; Ding and Beichner, 2009; Rust and Golombok, 2009; Adams and Wieman, 2011). The difficulty index ranges between 0 and 1, and easier items have larger difficulty indices while more difficult items have smaller difficulty indices. If the students can distinguish between the questions they could and could not answer, then it would be expected that very difficult items (p < 0.30) would have low average confidence, very easy items (p > 0.80) would have high average confidence, and moderately difficult items (0.30 < p < 0.80) would be near the mid-point of the confidence range. However, there were several instances where students had high average confidence for items with a low difficulty index. (The item numbers for the results shown in Fig. 8 and 9 can be found in the Appendix.) Each of the three clusters is discussed below.
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Fig. 8 GC1 students' average confidence vs. item difficulty.

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Fig. 9 GC2 students' average confidence vs. item difficulty.

Cluster 3. As can be seen in Fig. 8 and 9, the students in Cluster 3 had the greatest mismatch between item difficulties and average confidence. For both the GC1 and GC2 samples, 13 of the 18 items had difficulty indices below 0.30 (very difficult items) with average confidence greater than 50%. Therefore, students were highly confident despite that the items were very difficult. A total of 11 of these 13 items were common across both the GC1 and GC2 student samples. While the cutoffs for difficulty and confidence are somewhat arbitrary designations established in the research literature, such cutoffs do help to facilitate an interpretation of the data. This analysis suggests that not only is there a large group of students (Cluster 3 contains nearly 50% of the sample in both samples) who are essentially unaware that they are choosing responses which are misconceptions, but also that this effect is fairly stable across GC1 and GC2 students. Also in Fig. 8 and 9, only a very small incline is observed in the relative slope of the items which can be interpreted to mean that the students in Cluster 3 were generally unable to distinguish whether an item was relatively difficult or easy. (Note that even the easiest items for both samples still had less than 80% of the students responding correctly.) Consequently, students' were unable to recognize whether their understandings were valid knowledge or misconceptions, and the ROXCI was able to detect the meta-ignorance exhibited by these students.
Cluster 1. On average, the students in Cluster 1 answered more questions correctly than the students in either Cluster 2 or 3. Despite being the strongest students, instances of mismatch between confidence and item difficulty were still observed in which 3 items had difficulty indices below 0.30, but average confidence greater than 50%. Therefore, even the strongest students in the samples still held robust misconceptions about redox concepts, and the ROXCI can be used to detect these misconceptions. Note that 2 of these 3 items were common to both the GC1 and GC2 student samples.
Cluster 2. The items tended to be very difficult for the students in Cluster 2, but for this group of students, their associated confidence was also quite low. This group did not seem to exhibit meta-ignorance about their responses as Cluster 1 and 3 did, because their low confidence matched with their poor performance. While these students were not meta-ignorant, the items on the ROXCI do encompass basic redox concepts (Brandriet and Bretz, 2014), and the students in Cluster 1 did not have a strong conceptual grasp of these concepts. Because this effect was also observed in the GC2 sample, it is likely that such students carry this deficiency forth from GC1 to GC2 while trying to learn electrochemistry concepts in second-semester general chemistry (GC2).
Confidence Discrimination Quotient. The Dunning–Kruger effect postulates that many individuals hold illusory confidence about their own poor abilities, and these individuals may even have confidence that is equal or greater than their more able peers (Kruger and Dunning, 1999). In order to compare the confidence of the poor and high performing students, the Confidence Discrimination Quotient (CDQ) originally described by Caleon and Subramaniam (2010b) was calculated for each item (Fig. 10 and 11). The CDQ is calculated as the difference between the average confidence of the correct and incorrect students, divided by the standard deviation of all the students' confidence (Caleon and Subramaniam, 2010b). Positive CDQ values indicate that the average confidence of the correct students was higher than that of the incorrect students. Near zero and negative CDQ values indicates that the average confidence of the incorrect students is similar to or higher than that of the correct students, and hence, the CDQ is a direct indicator of the Dunning–Kruger effect. (The average confidence of the correct and incorrect students can be found in the Appendix.)
image file: c4rp00129j-f10.tif
Fig. 10 GC1 students' CDQ values ordered by results for the aggregated sample. (No CDQ value exists for Item 1 in Cluster 1 because all students responded correctly.)

image file: c4rp00129j-f11.tif
Fig. 11 GC2 students' CDQ values ordered by results for the aggregated sample.

One difficulty with interpreting CDQ values, however, is that no benchmarks exist for interpreting the magnitude of the CDQ. Just because a CDQ value is negative is not sufficient evidence to claim that a meaningful difference exists between the average confidence of correct and incorrect students. Because the CDQ is equivalent to a standardized mean difference effect size statistic (Ferguson, 2009), this study used Cohen's (1988) benchmarks for standardized mean differences to signify the degree of the differences. Therefore, the magnitude of the difference was identified as small (0.20), medium (0.50), or large (0.80). Positive values of 0.20 or greater indicate the correct students were more confident than the incorrect students, negative values of −0.20 or less indicate the incorrect students were more confident than the correct students, and values between −0.20 and 0.20 indicate that the difference between the correct and incorrect students was trivial (near zero). Therefore, benchmark values of 0.20, 0.00, and −0.20 are included in Fig. 10 and 11 to aid with interpretation of the CDQ values for each cluster of students.


Cluster 3. For the GC1 students in Fig. 10, a total of 12 items had CDQ values that indicate that the correct and incorrect students' average confidence were similar (CDQ value between −0.20 and 0.20), while 1 item had a small negative difference to suggest that the incorrect students were more confident than their correct peers. Similar results were also found with the GC2 students (Fig. 11) with 10 items having trivial differences in confidence, and 3 items indicating the incorrect students were more confident than the correct students. Because the students in Cluster 3 were also fairly confident in their responses (Fig. 8 and 9), these students were confident independent of whether they chose correct responses or redox misconceptions.
Cluster 1. Similar to Cluster 3, the students in Cluster 1 also exhibited instances of the Dunning–Kruger effect; however, there were fewer instances in Cluster 1 than in Cluster 3. For the GC1 students there was 1 item where the difference between correct and incorrect students was trivial and 2 instances where the incorrect students were more confident than the correct students. Interestingly, this effect was not just invariant across the GC1 to GC2 students, but rather the effect seemed to increase. There were 6 instances where the CDQ was trivial and 1 instance where the CDQ was negative but small. Since the students in Cluster 1 had the highest scores on the ROXCI and they were highly confident about their responses, Fig. 10 and 11 suggest that even the strong students can hold robust misconceptions about redox concepts. However, for a majority of the items the CDQ values were larger for Cluster 1 than for Cluster 3, which suggests that students in Cluster 1 were better aware when their responses were incorrect.
Cluster 2. The students in Cluster 2 responded with low average confidence across the ROXCI items. Caution should be taken when interpreting the CDQ results because a higher probability of guessing is inferred from the students' reported low confidence. Higher probability of guessing suggests that students may not be accurately placed into correct and incorrect student groups, as is necessary for the CDQ. For the GC1 students, there were 9 instances of a trivial difference between correct and incorrect students and 4 instances where incorrect students were more confident than the correct students. For the GC2 students, there were 7 instances of a trivial difference between correct and incorrect students and 2 instances where incorrect students were more confident than the correct students.

The CDQ results provide further support for the argument that students' confidence about their understandings of redox is often independent of whether or not their understandings are correct, and therefore, students hold robust redox misconceptions. Since the students who chose the ROXCI distractors were often just as confident or more so than their correct peers, these results further confirm the response process validity of the ROXCI distractors (Brandriet and Bretz, 2014).

Students' misconceptions

The previous analyses clearly show that not only do students have incorrect understandings about redox concepts, but also that they are often unaware of their misconceptions. However, investigating students' meta-ignorance is more than just exploring whether students were correct or incorrect. Concept inventory questions like those on the ROXCI can provide some of the richest data by carefully examining students' responses to individual items. Students' misconceptions regarding oxidation numbers, charge, and electron transfer are discussed below in the context of confidence data. As a reminder, the data below reflect students' ideas after classroom instruction and testing about redox concepts at their local institutions.
Oxidation numbers and charge. The concept of an oxidation number is key to understanding the electron movement that occurs in redox reactions. However, it is also a concept that is highly challenging for students. Question 13 asks students to differentiate between oxidation numbers and charges by identifying whether sulfur and sulfate have an oxidation number, a charge, or both (Fig. 1). Of particular interest are the students who chose response D that sulfur and sulfate both have oxidation numbers and charges, as these students could not differentiate between the concept of an oxidation number and a charge. Also of interest is response D to question 7 (Fig. 2). The interviews conducted in order to develop items for ROXCI (Brandriet and Bretz, 2014) suggest that the students who chose this response likely did so because they identified changes in charges of the polyatomic species in Fig. 2. They believed that the charge changed from −1 on NO3 to 0 on NO2 (i.e., oxidation) while the charge changed from +1 on H+ to 0 on H2O (i.e., reduction).

Table 3 summarizes student response data for these two misconceptions. For question 13, 31.4% of GC1 and 28.2% of GC2 students chose response D, and they did so with 53.7% and 56.3% average confidence, respectively. For question 7, 34.2% of GC1 and 28.0% of GC2 students chose response D, with 54.0% and 53.0% average confidence, respectively. These results indicate that students not only incorrectly applied oxidation numbers, but they also failed to differentiate them from charges. Once again, when partitioning these results by the cluster solution a pattern emerged in which Cluster 1 and 3 held misconceptions with high confidence (with a greater percentage of Cluster 3 than Cluster 1 choosing the misconceptions described), and Cluster 2 choosing distractors but with correspondingly low confidence.

Table 3 Students' misconceptions and confidence about oxidation number and charge
  In CuSO4(aq), sulfur and sulfate have both oxidation numbers and charges [question 13, response D] In the reaction shown in Fig. 2, NO3 is oxidized and H+ is reduced [question 7, response D]
GC1 GC2 GC1 GC2
a N GC1 = 264, NGC2 = 158. b N GC1 = 334, NGC2 = 148. c N GC1 = 485, NGC2 = 204.
Cluster 1a Number of students (% of cluster) 53 (20.1%) 34 (21.5%) 36 (13.6%) 22 (13.9%)
Mean confidence (%) 70.4% 70.0% 73.9% 69.3%
Cluster 2b Number of students (% of cluster) 116 (34.7%) 41 (27.7%) 147 (44.0%) 53 (35.8%)
Mean confidence (%) 33.4% 27.9% 34.1% 27.4%
Cluster 3c Number of students (% of cluster) 171 (35.3%) 69 (33.8%) 187 (38.6%) 68 (33.3%)
Mean confidence (%) 62.2% 66.3% 65.7% 67.7%


Electron transfer. Because students held misconceptions about oxidation numbers and charges, it was no surprise to find that students also struggled with the concept of electron transfer. Question 9 on the ROXCI, asks students to describe which species are involved in the electron transfer in the reaction Fe(s) + CdSO4(aq) → FeSO4(aq) + Cd(s). The correct response was the one most frequently chosen (electrons transfer from iron to cadmium) by 33.2% of GC1 and 31.2% of GC2 students with average confidence of 66.8% and 66.5%, respectively. In question 10, the students were asked to describe how the electrons transfer. Overwhelmingly, students chose the response that electrons transfer as the bond between cadmium and sulfate breaks, and as the iron bonds with sulfate. This misconception was chosen by 69.7% of GC1 and 68.6% of GC2 students with 54.7% and 60.0% average confidence, respectively. These results suggest that the students were better able to identify where the electrons transferred, but struggled to describe the particulate process underlying the symbolic equation. Despite that the correct response was the most frequently chosen for question 9, students still had limited success in identifying the substances involved in electrons transfer (note that only 30% of the students were able to identify the substances involved in electron transfer). The remaining 70% of students preferred a response that electrons transferred from cadmium to iron, or from cadmium to sulfate and from sulfate to iron, or from iron to sulfate and from sulfate to cadmium. As seen in the previous section, when partitioning the results by the cluster solution, Clusters 1 and 3 had high confidence with a higher percentage of Cluster 3 choosing distractors than Cluster 1, and Cluster 2 performed poorly with low confidence (Table 4).
Table 4 Students' understandings about electron transfer in a reaction of Fe(s) and CdSO4(aq)
  Electrons transfer from iron to cadmiumd [question 9, response B] Electrons transfer as the bond between cadmium and sulfate breaks, and as iron bonds with sulfate [question 10, response A]
GC1 GC2 GC1 GC2
a N GC1 = 264, NGC2 = 158. b N GC1 = 334, NGC2 = 148. c N GC1 = 485, NGC2 = 204. d Correct response.
Cluster 1a Number of students (% of cluster) 186 (70.5%) 88 (55.7%) 176 (66.7%) 115 (72.8%)
Mean confidence (%) 80.1% 82.0% 64.4% 67.6%
Cluster 2b Number of students (% of cluster) 60 (18.0%) 37 (25.0%) 219 (65.6%) 90 (60.8%)
Mean confidence (%) 28.5% 34.5% 35.3% 32.2%
Cluster 3c Number of students (% of cluster) 114 (23.5%) 34 (16.7%) 360 (74.2%) 145 (71.1%)
Mean confidence (%) 65.3% 61.5% 61.8% 71.3%


Conclusions

The results of this study demonstrate that not only do students hold misconceptions about redox concepts, but also that they are often unaware that they hold such misconceptions. However, asking students to identify deficiencies in their own knowledge by themselves is an intrinsically difficult task that will likely be unproductive (Carter and Dunning, 2008; Dunning, 2011). Without proper intervention, students may continue with this thinking and reasoning during the learning process. This is especially problematic with redox concepts because they are fundamental to so many chemical and biological processes. The ROXCI assesses both students' understandings and confidence about symbolic and particulate redox concepts. Through the use of cluster analysis, groups of students were identified based on their responses to the ROXCI, resulting in an in-depth understanding of the student sample that would not have otherwise been accessible with an analysis of only the aggregate sample. A 3 cluster solution was identified in which Cluster 1 had moderate total scores and high average confidence, Cluster 2 had low total scores and low average confidence, while Cluster 3 had low total scores, but high average confidence. While Cluster 3 exhibited the most instances of meta-ignorance, both Clusters 1 and 3 displayed instances of meta-ignorance based on the results of the analyses of students' average confidence, item difficulty, and CDQ values.

One similarity that existed across the 3 clusters was that the students held deficiencies in their understandings of redox concepts, despite having been taught and tested by their instructor on redox concepts prior to taking the ROXCI. This study specifically describes students' misconceptions involving oxidation numbers and electron transfer concepts including students' inabilities to distinguish between oxidation numbers and charges, changes in charges on polyatomic ions and molecules can be used to identify oxidized and reduced species, and electron transfer occurs as the bond between cation and spectator ion breaks/forms. These misconceptions are not unique to one specific pedagogy or curriculum. The data in this study were obtained from approximately 1600 first year university chemistry students from a variety of colleges/universities as well as different classroom environments. Redox misconceptions are clearly ubiquitous in the first year university chemistry classrooms and were prevalent in both the GC1 and GC2 samples. This suggests that students may carry these misconceptions from GC1 to GC2 and use them as prior knowledge upon which they may build electrochemistry concepts.

While the term misconception was specifically used to describe students' incorrect ideas for the purposes of this study, it is difficult to truly identify whether a students' incorrect understanding is a misconception, naïve idea, alternate conception, etc. This is especially true since there are many competing views of what constitutes a misconceptions (Strike and Posner, 1992; diSessa, 1993; Chi et al., 1994; Chi and Roscoe, 2002). However, the addition of the confidence tier provides an additional piece of evidence that students robustly hold their incorrect ideas. Future research should continue to evaluate the relationship between students' misconceptions and their confidence, such as identifying specific confidence cutoffs for determining what constitutes a robustly held misconception. Future studies should also continue to investigate students' misconceptions in additional samples of GC1 and GC2 students, as well as students beyond first year university chemistry.

Implications for teaching

Instructors need to help students gauge the effectiveness of their understandings so that students do not erroneously use redox misconceptions as a foundation for building future knowledge structures. The ROXCI can be used to assess students' understandings and confidence of symbolic and particulate redox concepts. (Colleagues who are interested in using the ROXCI in their classroom should contact the corresponding author to obtain a copy.) In order to help shed light on students' meta-ignorance, instructors could use the questions on the ROXCI in tandem with clickers to help students not only evaluate the accuracy of their own understanding, but also evaluate their confidence. Techniques such as think-pair-share may help students uncover potential flaws in reasoning through discussion with their classmates. Lastly, instructors should attempt to align their pedagogical goals (e.g., conceptual understanding) with their local assessments. If the instructors do not assess deep conceptual understandings, the students will be less motivated learn the concepts.

A deep understanding of redox requires more than the rote memorization and application of oxidation numbers. Instructors should focus on emphasizing the concepts involved in redox reactions. This includes what substances collide, how the electrons transfer, what makes an oxidation number different from that of charge, and how oxidation numbers and electron transfer relate to one another. In order to facilitate students' conceptual understanding, a variety of representations should be used by instructors. Representations that depict redox reactions often contain implicit features that may be obvious to the experts, but students struggle to decode such information (Stains and Talanquer, 2008). In this study, some students believed that the electrons transfer through bond breaking/forming between the cation and spectator ion because they saw the spectator ion direct adjacent to the aqueous cation in the chemical equation. This suggested to some students that a physical bond exists between these two entities, despite the phase indication of (aq). Instructors should help students encode the information shown in representations and challenge them to make connections across the symbolic, macroscopic, and particulate domains.

While this study did not collect empirical evidence for what specific strategies will and will not work to remediate the misconception presented in this study, the authors speculate that students who are highly confident in their misconceptions may require different strategies for conceptual change than students who are aware that their knowledge is flawed. It is possible that students who are aware their knowledge is flawed, may be receptive to conceptual change merely by having the correct conception described to them. However, students with robust misconceptions may require significant instructional intervention such as the learning cycle, discrepant events, etc. Instructors may find that inducing cognitive dissonance in students' mental models may help the students see the limitations of their own understandings. For example, the students who believed that the electrons transfer through bond breaking/forming between the cation and spectator ion could examine the conductivity of aqueous solutions versus just ionic solids. This may help students begin to realize that cations and anions are not physically bonded in aqueous solutions, and therefore, electrons could not possibly transfer in this manner.

Implications for research

The use of the confidence tier for concept inventory design and development is a novel and newly emerging research practice within chemistry education (Caleon and Subramaniam, 2010a, 2010b; McClary and Bretz, 2012; Brandriet and Bretz, 2014). The confidence tier helps identify whether students' multiple-choice responses are a result of genuine understandings or guessing (Caleon and Subramaniam, 2010a). This may help the researcher provide additional evidence for the response process validity of concept inventory data. However, with the development of any assessment tool, many decisions have to be made regarding how to collect students' data. The ROXCI uses an interval confidence tier originally published by McClary and Bretz (2012), in order to reduce the potential limitations created by ordinal scales (Stevens, 1946; Knapp, 1990). Additionally, the authors believe that the 0% to 100% scale may be more meaningful to students since colloquially it is not uncommon for individuals to describe their confidence in terms of a percentage (e.g. I am 70% confident that…). However, there are limitations to the interval confidence measurement. The authors do not presume to claim that students' confidence ratings are different when measured to ±1% confidence. Certainly, the ones' place is the uncertain digit for the measurements made in this study. However, the authors believe that the 0–100% scale is an improvement upon previously used Likert scales when measuring confidence. Future studies could further investigate the methods used to measure students' confidence.

This study makes an important contribution to both the concept inventory and the chemistry education research literature by not only identifying students' redox misconceptions and their associated confidence, but also by using cluster analysis to distinguish patterns in students' responses in order to identify meta-ignorance. Because cluster analysis provides a tool to partition the students into groups based upon patterns in their responses to ROXCI items – and their confidence – a different and more detailed analysis can be achieved about students' understandings than can be generated by simply examining the sample in aggregate. When developing pedagogy to target students' misconceptions, creating a “one-size-fits-all” pedagogical strategy will not likely be optimal for a classroom with varying levels of meta-ignorance (e.g., Clusters 1, 2, and 3). However, tailoring pedagogy to each individual student is also not plausible, especially in a large lecture classroom. Therefore, the results of the cluster analysis suggest a more manageable estimation of the meaningful differences amongst the students that likely exist within a general chemistry classroom. Cluster analysis is a tool that could be used to examine a variety of research questions in chemistry education, and future studies should consider using it to help further recognize the natural groupings in student data. Future studies should continue to expand upon the best practices for using cluster analysis and for determining the validity, reliability, and stability of the cluster solutions.

Appendix

Stability of the cluster solution

The observations (i.e. students) in both the GC1 and GC2 datasets were reordered based on a random number generator in order to evaluate the stability of the cluster solution. The data was reordered 9 times beyond that of the initial solution for a total of 10 cluster solutions. For the GC1 students, a 3 cluster solution was identified in 7 out of 10 of the replicate solutions, and for the GC2 students, a 3 cluster solution was identified in 8 out of 10 of the replication solutions. Because the 3 cluster solution was identified as the most common solution, this was the solution accepted for the manuscript. Furthermore, the replicate solutions produced scores and confidence results very similar to that of the primary solution described in the body of the manuscript (Tables 5 and 6).
Table 5 Replication cluster solutions for GC1 stability analyses
Solution N Average total score Average confidence (%)
1 2 3 1 2 3 1 2 3
a 3 cluster solution.
Primarya 264 334 485 10.0 4.2 4.4 73.0 32.7 62.7
2a 249 411 423 10.1 3.7 5.1 73.9 36.8 64.0
3a 272 334 477 9.9 4.2 4.3 73.5 32.7 62.3
4a 227 423 433 10.4 4.0 4.9 73.4 36.4 65.9
5a 250 292 541 10.0 3.7 4.8 74.9 31.4 60.4
6a 262 404 417 10.1 4.1 4.5 72.9 35.6 65.0
7a 267 339 477 10.0 4.2 4.4 73.0 32.9 62.8
8 413 670 8.5 4.0 72.8 45.5
9 402 681 8.4 4.1 73.9 45.3
10 541 542 7.4 4.0 71.3 40.6


Table 6 Replication cluster solutions for GC2 stability analyses
Solution N Average total score Average confidence (%)
1 2 3 4 1 2 3 4 1 2 3 4
a 3 cluster solution.
Primarya 158 148 204 9.6 4.9 4.5 72.2 30.0 66.0
2a 159 148 203 9.6 4.9 4.5 72.2 30.0 66.0
3a 159 148 203 9.6 4.9 4.5 72.2 30.0 66.0
4a 185 143 182 9.2 5.0 4.1 71.5 29.6 65.2
5a 158 148 204 9.6 4.9 4.5 72.2 30.0 66.0
6a 158 148 204 9.6 4.9 4.5 72.2 30.0 66.0
7a 159 148 203 9.6 4.9 4.5 72.2 30.0 66.0
8a 158 148 204 9.6 4.9 4.5 72.2 30.0 66.0
9 214 296 8.9 4.3 70.5 48.1
10 106 212 111 81 10.2 4.7 4.0 8.0 77.9 67.3 28.4 45.0


Item difficulty and confidence

Fig. 12–17 provide the individual item numbers for the GC1 and GC2 students' average confidence vs. item difficulty shown in Fig. 8 and 9.
image file: c4rp00129j-f12.tif
Fig. 12 GC1 students' average confidence vs. item difficulty for Cluster 1.

image file: c4rp00129j-f13.tif
Fig. 13 GC1 students' average confidence vs. item difficulty for Cluster 2.

image file: c4rp00129j-f14.tif
Fig. 14 GC1 students' average confidence vs. item difficulty for Cluster 3.

image file: c4rp00129j-f15.tif
Fig. 15 GC2 students' average confidence vs. item difficulty for Cluster 1.

image file: c4rp00129j-f16.tif
Fig. 16 GC2 students' average confidence vs. item difficulty for Cluster 2.

image file: c4rp00129j-f17.tif
Fig. 17 GC2 students' average confidence vs. item difficulty for Cluster 3.

Confidence Discrimination Quotient

The Confidence Discrimination Quotient (CDQ) is calculated by taking the difference between the average confidence of the correct students (CFC) and incorrect students (CFW), and dividing it by the standard deviation (SD) of the students' confidence (Caleon and Subramaniam, 2010b). The CFC, CFW, SD, and CDQ values for the GC1 and GC2 students can be found in Tables 7 and 8.
Table 7 Values for calculating the CDQ for GC1 sample
Items Cluster 1a Cluster 2 Cluster 3 All students
CFC CFW SD CDQ CFC CFW SD CDQ CFC CFW SD CDQ CFC CFW SD CDQ
a All of the students responded correctly to question 1, and therefore, a CDQ value cannot be calculated.
1 92.0 12.6 54.5 47.4 27.1 0.260 79.4 70.9 18.4 0.460 78.0 58.8 25.6 0.748
2 92.1 71.9 12.8 1.574 57.0 40.0 26.8 0.637 82.0 67.0 20.8 0.719 81.2 55.0 27.3 0.959
3 86.3 80.2 16.6 0.369 49.8 47.0 24.0 0.120 71.9 75.6 19.0 −0.196 72.0 66.5 24.8 0.220
4 83.1 72.5 20.2 0.524 40.2 40.1 23.2 0.003 69.7 69.5 20.6 0.009 72.4 59.3 26.6 0.493
5 76.5 64.9 23.8 0.487 25.6 33.4 21.8 −0.354 60.1 59.6 22.9 0.021 64.8 51.4 27.3 0.491
6 77.7 66.2 25.5 0.455 23.2 28.3 21.5 −0.239 52.4 56.9 26.3 −0.173 59.8 48.4 30.2 0.377
7 78.9 68.3 22.8 0.465 31.4 32.1 22.9 −0.033 63.4 63.7 23.2 −0.012 67.9 52.8 28.6 0.530
8 66.2 72.5 21.0 −0.300 27.8 30.4 23.1 −0.115 59.4 61.7 23.1 −0.099 53.8 53.9 27.9 −0.005
9 80.1 66.4 23.6 0.582 28.5 29.0 22.8 −0.023 65.3 57.4 23.1 0.343 66.8 47.6 29.3 0.656
10 54.2 61.9 26.5 −0.291 17.6 30.3 22.0 −0.576 48.4 59.2 23.1 −0.468 44.8 50.8 27.4 −0.218
11 63.9 58.8 24.9 0.203 25.7 30.9 23.1 −0.225 58.6 61.1 22.5 −0.114 50.2 51.1 27.4 −0.034
12 71.4 63.6 23.5 0.332 38.9 33.9 23.7 0.212 65.8 62.5 21.9 0.151 63.5 52.3 26.8 0.417
13 72.0 70.3 21.6 0.082 32.6 29.6 24.3 0.126 62.6 60.1 25.0 0.103 58.3 52.3 29.0 0.209
14 82.5 74.3 20.2 0.407 38.7 29.5 26.6 0.345 74.3 64.8 25.7 0.369 70.4 54.2 31.1 0.519
15 91.2 71.3 15.8 1.260 43.7 29.4 29.9 0.480 79.4 61.3 25.9 0.699 76.9 47.3 32.9 0.902
16 62.6 55.9 26.6 0.251 21.0 22.7 20.6 −0.080 52.3 54.1 24.2 −0.076 50.3 44.5 28.2 0.206
17 59.3 48.1 26.9 0.415 20.7 18.9 19.0 0.091 48.3 48.3 24.0 0.002 44.0 38.9 27.3 0.187
18 66.3 58.4 27.7 0.285 18.9 20.6 21.5 −0.081 51.4 49.8 25.8 0.063 44.1 43.2 29.8 0.031


Table 8 Values for calculating the CDQ for GC2 sample
Items Cluster 1 Cluster 2 Cluster 3 All Students
CFC CFW SD CDQ CFC CFW SD CDQ CFC CFW SD CDQ CFC CFW SD CDQ
1 90.1 79.4 14.4 0.740 47.8 34.8 24.1 0.540 76.2 76.3 17.1 −0.007 74.3 60.0 26.2 0.545
2 90.5 68.1 15.8 1.422 47.4 40.8 26.4 0.250 78.3 76.2 17.3 0.123 77.8 60.2 27.0 0.654
3 84.8 78.9 16.9 0.350 39.7 41.4 22.8 −0.076 74.5 73.6 18.0 0.053 69.8 64.4 26.0 0.208
4 85.0 71.2 17.4 0.791 37.4 35.7 23.2 0.076 72.7 68.5 19.2 0.217 71.4 57.9 26.9 0.504
5 71.5 56.1 28.9 0.532 26.3 21.6 20.0 0.232 65.4 57.0 27.3 0.306 67.2 44.0 31.5 0.735
6 79.3 50.5 30.6 0.941 24.7 20.4 20.8 0.209 61.9 54.2 28.4 0.269 68.0 41.7 32.9 0.800
7 75.2 69.3 21.2 0.278 26.7 28.4 20.2 −0.081 64.2 65.3 21.9 −0.052 63.6 53.6 28.2 0.355
8 63.5 71.4 21.3 −0.373 23.8 32.1 20.9 −0.400 67.0 67.8 21.5 −0.037 49.8 59.1 27.5 −0.339
9 82.0 67.6 21.6 0.668 34.5 26.1 20.6 0.407 61.5 62.9 22.8 −0.063 66.5 52.2 28.9 0.496
10 69.2 66.5 19.9 0.133 52.5 28.3 20.0 1.213 51.7 69.2 18.9 −0.926 61.4 56.4 26.5 0.188
11 61.6 59.6 24.5 0.082 27.1 29.6 20.0 −0.122 60.3 64.9 20.9 −0.218 48.7 54.2 26.7 −0.207
12 66.3 68.8 21.6 −0.115 26.1 29.4 21.2 −0.157 64.3 64.8 20.7 −0.025 58.1 53.5 27.1 0.172
13 71.0 71.1 21.6 −0.003 30.0 28.3 20.8 0.083 67.0 66.3 21.3 0.030 61.7 55.0 27.9 0.239
14 76.7 72.9 22.4 0.170 38.0 28.2 23.8 0.412 74.9 71.7 21.5 0.146 67.6 57.4 29.7 0.342
15 87.8 74.7 17.3 0.758 44.1 32.8 26.3 0.427 79.2 68.9 20.6 0.498 75.2 55.3 29.0 0.688
16 56.0 59.0 25.4 −0.118 17.7 23.4 20.9 −0.270 51.3 60.1 24.0 −0.367 47.4 48.8 28.7 −0.049
17 61.1 54.7 24.2 0.266 20.3 20.1 17.3 0.012 52.4 55.1 22.8 −0.120 48.6 44.2 27.1 0.162
18 67.6 62.7 24.2 0.200 26.1 19.1 17.5 0.400 66.5 55.9 24.1 0.438 55.1 47.0 28.9 0.283


Acknowledgements

This material is based upon work supported by the National Science Foundation under the Graduate Research Fellowship Program (NSF-GRFP) and the NSF DRK-12 Grant No. 0733642. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the authors and do not necessarily reflect the views of the National Science Foundation.

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