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We aim to graphically analyze the depth of conceptual understanding behind the Force Concept Inventory (FCI) responses of students, focusing on three questions (questions 1, 15, and 28). In our study, we created and implemented subquestions to clarify and quantify the students’ reasoning steps in reaching their responses to the original FCI questions. We used alluvial diagrams to visualize the responses of students on the subquestions and original FCI questions, representing the transition rates across the question options as flows. The flows represent common patterns of consistency and inconsistency in the students’ conceptual understandings across the original FCI question and the subquestions. By combining subquestions and alluvial diagrams as we did, we are able to efficiently analyze the depth of understanding for a given situation and to visualize the existence of a specific misconception. For example, we found that even among those who correctly answered FCI question 15 about Newton’s third law, many students incorrectly answered a corresponding subquestion asking about whether Newton’s third law holds at the end of the interaction between the two objects. This misconception turned out to be fairly robust, as these same students had a similar answering pattern for question 28. Alluvial diagrams can be used for a broad range of applications, including analysis of multiple-tier tests, testlets (e.g., FCI questions 8 through 11), tests for representational consistency (e.g., a representational variant of the FCI), and so on.

This paper presents improvements made to a computerized adaptive testing (CAT)-based version of the FCI (FCI-CAT) in regards to test security and test efficiency. First, we will discuss measures to enhance test security by controlling for item overexposure, decreasing the risk that respondents may (i) memorize the content of a pretest for use on the post-test or (ii) share information about the items with their classmates who take the assessment later. Second, we will discuss measures to enhance test efficiency, so that a shorter test length can yield a desired accuracy and precision of the measurement. Specifically, we utilized collateral information in the form of a pretest proficiency estimate of each respondent for selecting items and estimating respondent proficiency level in the post-test. To shorten the total testing time further, we also allowed the test lengths to be different for the pre- and post-test. To analyze how these improvements affect the accuracy and precision (which we measure in terms of root-mean-square error) of Cohen’s d, we conducted a Monte Carlo simulation and a post hoc simulation. Then, we calculated the minimal test length of the FCI-CAT whose accuracy and precision are equivalent to that of the paper-and-pencil version of the FCI. Consequently, we obtained the following three findings: (i) By using collateral information, we can achieve the accuracy and precision of the full-length FCI with fewer items via the FCI-CAT. (ii) For a class size of 40, we can control for test security while still reducing the sum of the pre- and post-test lengths of the FCI-CAT to a total of 33 items (17 items on the pretest and 16 items on the post-test), thereby reducing the testing time to 55%. (iii) If one’s goal is to maximize test efficiency, the pretest length should be slightly larger than the post-test length. On the other hand, if the goal is to maximize test security, the pretest length should be smaller and the post-test length should be larger. If one desires a balance of these two goals, it would be reasonable to choose equal pre- and post-test lengths.

As a method to shorten the test time of the Force Concept Inventory (FCI), we suggest the use of computerized adaptive testing (CAT). CAT is the process of administering a test on a computer, with items (i.e., questions) selected based upon the responses of the examinee to prior items. In so doing, the test length can be significantly shortened. As a step to develop a CAT-based version of the FCI (FCI-CAT), we examined the optimal test length of the FCI-CAT such that accuracy and precision [which we measure in terms of bias, standard error, and root-mean-square error (RMSE)] of Cohen’s d would be comparable to that of the full FCI for a given class size. First, we estimated the item parameters of the FCI items based on the three-parameter logistic model of item response theory, which are used in the algorithm of CAT. For this estimation, we used 2882 responses of Japanese university students. Second, we conducted a Monte Carlo simulation to analyze how the bias, standard error, and RMSE of Cohen’s d depend upon the test length. Third, we conducted a post hoc simulation to examine the consistency of the Monte Carlo results with what would have been obtained using empirical responses. For this comparison, we used 86 pairs of pre- and post- test responses of Japanese university students. As a result, we found that for a class size of 40, we may reduce the test length of the FCI-CAT to 15–19 items, thereby reducing the test time of the FCI to 50%–63%, with an accompanying decrease in accuracy and precision of only 5%–10%. The results of the Monte Carlo study and the post hoc simulation were consistent, which supports the adequacy of our Monte Carlo study and its relevance in terms of administering the FCI-CAT in real classrooms.

We address the validity of the FCI, that is, whether respondents who answer FCI questions correctly have an actual understanding of the concepts of physics tested in the questions. We used sub-questions that test students on concepts believed to be required to answer the actual FCI questions. Our sample size comprises about five hundred respondents; we derive false positive ratios for pre-learners and post-learners, and evaluate the significant difference between them. Our analysis shows a significant difference at the 95 % confidence level for Q.6, Q.7, and Q.16, implying that it is possible for post-learners to answer three questions without understanding the concepts of physics tested in the questions; therefore, Q.6, Q.7 and Q.16 are invalid.

In this study, we analyze the systematic error from false positives of the Force Concept Inventory (FCI). We compare the systematic errors of question 6 (Q.6), Q.7, and Q.16, for which clearly erroneous reasoning has been found, with Q.5, for which clearly erroneous reasoning has not been found. We determine whether or not a correct response to a given FCI question is a false positive using subquestions. In addition to the 30 original questions, subquestions were introduced for Q.5, Q.6, Q.7, and Q.16. This modified version of the FCI was administered to 1145 university students in Japan from 2015 to 2017. In this paper, we discuss our finding that the systematic errors of Q.6, Q.7, and Q.16 are much larger than that of Q.5 for students with mid-level FCI scores. Furthermore, we find that, averaged over the data sample, the sum of the false positives from Q.5, Q.6, Q.7, and Q.16 is about 10% of the FCI score of a midlevel student.

In this study, we evaluate the structural validity of Q.16 and Q.7 in the Force Concept Inventory (FCI). We address whether respondents who answer Q.16 and Q.7 correctly actually have an understanding of the concepts of physics tested in the questions. To examine respondents’ levels of understanding, we use subquestions that test them on concepts believed to be required to answer the actual FCI questions. Our sample size comprises 111 respondents; we derive false-positive ratios for prelearners and postlearners and then statistically test the difference between them. We find a difference at the 0.05 significance level for both Q.16 and Q.7, implying that it is possible for postlearners to answer both questions without an understanding of the concepts of physics tested in the questions; therefore, the structures of Q.16 and Q.7 are invalid. In this study, we only evaluate the validity of these two FCI questions; we do not assess the validity of previous studies that have compared total FCI scores.