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Cognitive Diagnostic Analysis of Students' Mathematical Competency Based on the DINA Model

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Title: Cognitive Diagnostic Analysis of Students' Mathematical Competency Based on the DINA Model
Language: English
Authors: Xu, Tianshu, Wu, Xiaopeng (ORCID 0000-0003-0147-8228), Sun, Siyu, Kong, Qiping
Source: Psychology in the Schools. Sep 2023 60(9):3135-3150.
Availability: Wiley. Available from: John Wiley & Sons, Inc. 111 River Street, Hoboken, NJ 07030. Tel: 800-835-6770; e-mail: cs-journals@wiley.com; Web site: https://www.wiley.com/en-us
Peer Reviewed: Y
Page Count: 16
Publication Date: 2023
Document Type: Journal Articles
Reports - Research
Education Level: Elementary Education
Grade 4
Intermediate Grades
Descriptors: Cognitive Processes, Mathematics Skills, Competence, Grade 4, Models, Student Evaluation, Learning Processes, Knowledge Level, Foreign Countries, Evaluation Methods, Cognitive Measurement
Geographic Terms: China (Shanghai)
DOI: 10.1002/pits.22916
ISSN: 0033-3085
1520-6807
Abstract: Considering the importance of mathematics in modern society, it is crucial to understand the cognitive processes involved in the acquisition of complex mathematical competency. As a new generation of evaluation theory, cognitive diagnosis has its unique advantages in personalized evaluation. Based on the mathematical cognitive framework of Trends in International Mathematics and Science Study (TIMSS)-2011 and the Chinese mathematics curriculum, this research has formed a mathematical competency model composed of seven cognitive attributes. Sixty-seven released mathematical items for the fourth grade in TIMSS-2011 were used as assessment tools in this study. The deterministic inputs, noisy, "and" gate model was selected as the cognitive diagnosis model in this research, and the parameters of the model were evaluated according to the response data, based on which the effectiveness of the assessment tool was further verified, forming the evaluation framework of students' mathematical competency. This framework was used to analyze the data of 573 students' mathematical competency from Shanghai, China, specifically from three aspects: attribute mastery probability, learning path, and knowledge structure. Results show that students' performance in mastering attributes of mathematical cognition is excellent on the whole; some students' learning paths are leapfrog; there are certain differences in students' knowledge structure despite that they have the same total score. This research is performed as a systematic case study of the evaluation of students' mathematical competency and also provides a new perspective in assessing other knowledge and skills.
Abstractor: As Provided
Entry Date: 2023
Accession Number: EJ1386792
Database: ERIC
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  Value: <anid>AN0169726498;pis01sep.23;2023Aug04.05:06;v2.2.500</anid> <title id="AN0169726498-1">Cognitive diagnostic analysis of students' mathematical competency based on the DINA model </title> <p>Considering the importance of mathematics in modern society, it is crucial to understand the cognitive processes involved in the acquisition of complex mathematical competency. As a new generation of evaluation theory, cognitive diagnosis has its unique advantages in personalized evaluation. Based on the mathematical cognitive framework of Trends in International Mathematics and Science Study (TIMSS)‐2011 and the Chinese mathematics curriculum, this research has formed a mathematical competency model composed of seven cognitive attributes. Sixty‐seven released mathematical items for the fourth grade in TIMSS‐2011 were used as assessment tools in this study. The deterministic inputs, noisy, "and" gate model was selected as the cognitive diagnosis model in this research, and the parameters of the model were evaluated according to the response data, based on which the effectiveness of the assessment tool was further verified, forming the evaluation framework of students' mathematical competency. This framework was used to analyze the data of 573 students' mathematical competency from Shanghai, China, specifically from three aspects: attribute mastery probability, learning path, and knowledge structure. Results show that students' performance in mastering attributes of mathematical cognition is excellent on the whole; some students' learning paths are leapfrog; there are certain differences in students' knowledge structure despite that they have the same total score. This research is performed as a systematic case study of the evaluation of students' mathematical competency and also provides a new perspective in assessing other knowledge and skills.</p> <p>Practitioner points: Students' performance in mastering attributes of mathematical cognition is excellent on the whole.Some students' learning paths are leapfrog.There are certain differences in students' knowledge structure despite that they have the same total score.</p> <p>Keywords: Chinese students; cognitive diagnostic analysis; literacy evaluation; mathematical competency</p> <hd id="AN0169726498-2">INTRODUCTION</hd> <p></p> <hd id="AN0169726498-3">TIMSS and mathematical competency</hd> <p>Mathematical competency is one of the most fundamental abilities that students need to master to manage successfully in school and in future careers, particularly in the areas of science, technology, engineering, and mathematics. Students with poor mathematical competency are faced with the danger of being shut off from many lucrative career paths in the 21st century (Leighton & Gierl, [<reflink idref="bib31" id="ref1">31</reflink>]). Consequently, considering the importance of mathematical competency, it is crucial to understand students' mathematical competency and development. Among multitudinous studies on mathematical competency, Trends in International Mathematics and Science Study (TIMSS) is the most typical one. TIMSS, implemented by the International Association for the Evaluation of Educational Achievement (IEA), is one of the most comprehensive international large‐scale studies that investigate students' mathematical competency. IEA has been implementing TIMSS every 4 years since 1995 to monitor trends in students' mathematical and scientific achievements (Mullis et al., [<reflink idref="bib37" id="ref2">37</reflink>]). In addition to comparing the mathematical and scientific achievements of students in various countries and regions, TIMSS also collects a large amount of background information and analyzes the factors affecting students' mathematical and scientific achievements from the perspectives of organization, curriculum, and teaching practice in the education system, so as to provide information for education policy formulation and corresponding educational reform. The purpose of TIMSS is not only to compare the mathematical and scientific achievements of students in various countries or regions but also to conduct longitudinal analysis on students' mathematical and scientific achievement, learning attitudes, curriculum settings, textbook management, and teaching resources through tests and questionnaires, so as to improve the systematic understanding of a series of important factors in the process of education (Mullis et al., [<reflink idref="bib37" id="ref3">37</reflink>]). Since 1995, TIMSS has accumulated data for many years to conduct further longitudinal research on the development of different countries and regions, which expanded its evaluation function and formed research on the development trend of mathematical and scientific achievements. TIMSS's results of students' academic achievements, educational practices, and educational policies in mathematics and science have become the main information resources for evaluating the quality of mathematics and science education in the world (Feniger, [<reflink idref="bib10" id="ref4">10</reflink>]; Hambleton et al., [<reflink idref="bib13" id="ref5">13</reflink>]; Kaleli‐Yılmaz & Hanci, [<reflink idref="bib21" id="ref6">21</reflink>]).</p> <p>Specifically, TIMSS aims to provide regular and timely information for educators and policymakers about the mathematics achievement of fourth‐ and eighth‐grade students in participating countries to improve mathematics teaching and learning. TIMSS believes that taking mathematics as an important part of school education is not for the sake of mathematics itself, but because understanding mathematics and being able to use mathematics play a great role in people's daily life and contribute to their future success in work (Mullis et al., [<reflink idref="bib38" id="ref7">38</reflink>]). The mathematical competency assessment in TIMSS is composed of the two dimensions of content and cognition, focusing on students' basic mathematical knowledge, concepts, and mathematical thinking ability that is closely connected with the school mathematics curriculum (Mullis et al., [<reflink idref="bib38" id="ref8">38</reflink>]). To correctly answer the TIMSS items, students not only need to be familiar with the math content being evaluated but also need to use a series of cognitive skills. In the case of TIMSS, extensive information on student performance in mathematics is reported, including not only the trends over the assessments since 1995 but also data on performance in the mathematics content domains (algebra, geometry, etc.) and on competence in problem‐solving in mathematical contexts (Mullis et al., [<reflink idref="bib37" id="ref9">37</reflink>]). To date, many research has been dedicated to making comparisons of students' mathematical achievement across participating countries based on TIMSS results (Dogan & Tatsuoka, [<reflink idref="bib9" id="ref10">9</reflink>]; Lee et al., [<reflink idref="bib27" id="ref11">27</reflink>]; Um et al., [<reflink idref="bib51" id="ref12">51</reflink>]). Furthermore, the results of TIMSS have been applied to studies in many countries to improve education (Kim et al., [<reflink idref="bib23" id="ref13">23</reflink>]; Leung, [<reflink idref="bib32" id="ref14">32</reflink>]; Macnab, [<reflink idref="bib34" id="ref15">34</reflink>]).</p> <hd id="AN0169726498-4">Cognitive diagnostic assessment (CDA)</hd> <p>Although vital information about students' overall performance and critical curricular, instructional, and resource‐related factors that can impact the teaching and learning process has been furnished by TIMSS reports (Mullis et al., [<reflink idref="bib37" id="ref16">37</reflink>]), providing educational policymakers, administrators, teachers, and researchers with reference for educational reform and improvement, from which more detailed diagnostic information of students' mathematical competency cannot be obtained effectively. In the past few decades, a new assessment method called CDA has received increasing attention in educational and psychological measurement (Leighton & Gierl, [<reflink idref="bib28" id="ref17">28</reflink>]; Rupp et al., [<reflink idref="bib43" id="ref18">43</reflink>]; Tatsuoka, [<reflink idref="bib46" id="ref19">46</reflink>]). As a new generation of assessment theory, CDA originates from cognitive psychology and psychometrics, aiming to determine students' mastery of attributes, extracting fine‐grained diagnostic information (de la Torre & Minchen, [<reflink idref="bib50" id="ref20">50</reflink>]), and providing a personalized evaluation for students. CDA combines the cognitive process of learning with psychometric models to infer the mastery of attributes (Alves, [<reflink idref="bib2" id="ref21">2</reflink>]). The cognitive model is the core of CDA, which equips assessment with diagnostic characteristics (Lai et al., [<reflink idref="bib25" id="ref22">25</reflink>]). It provides a clear framework for linking cognitive‐based inferences with specific test score interpretations (Gierl et al., [<reflink idref="bib12" id="ref23">12</reflink>]; Leighton & Gierl, [<reflink idref="bib29" id="ref24">29</reflink>]). The cognitive model refers to the hierarchically ordered attributes which are the cognitive processes or skills involved in students' problem‐solving (Boora et al., [<reflink idref="bib5" id="ref25">5</reflink>]).</p> <p>Traditional assessment theories such as classical test theory (CTT) and item response theory usually locate and evaluate candidates on a continuum of abilities through total scores. Such evaluation results are often summative and are not adequate for formative diagnostic information on the strengths and weaknesses of learners to provide feedback. Unlike those traditional educational assessment methods, CDA provides specific information about each attribute that students need to master, rather than a single score result. It is a form of examination of the cognitive process necessary for the successful completion of tasks. The assessment results are formative, which makes cognitive diagnosis assessment a more important role in promoting students' learning. CDA is designed to measure students' specific knowledge structures and processing skills that cannot be directly observed, namely knowledge state or cognitive structure (Leighton & Gierl, [<reflink idref="bib30" id="ref26">30</reflink>]), through which we can tell what cognitive attributes students have mastered or have not yet mastered. It can provide fine‐grained and individualized diagnostic information about students' learning, including individual student mastery of each attribute, so as to furnish meaningful feedback about students' strengths, weaknesses, and knowledge gaps in a given domain or skill and guide their learning process (Alves, [<reflink idref="bib2" id="ref27">2</reflink>]; Wu et al., [<reflink idref="bib53" id="ref28">53</reflink>]; Ye, [<reflink idref="bib59" id="ref29">59</reflink>]), based on which appropriate instructional strategies can be designed to tailored pupils' needs (Chin et al., [<reflink idref="bib6" id="ref30">6</reflink>]). There is great potential in providing remedial teaching strategies for students based on the results of cognitive diagnosis to promote students' learning, which is supported by a large number of studies (Ketterlin‐Geller & Yovanoff, [<reflink idref="bib22" id="ref31">22</reflink>]; Russell et al., [<reflink idref="bib44" id="ref32">44</reflink>]). For that reason, the CDA approach has been applied to many research to be more informative about students' cognitive processes and improve student learning (Gierl et al., [<reflink idref="bib11" id="ref33">11</reflink>]; Li et al., [<reflink idref="bib33" id="ref34">33</reflink>]; Paulsen & Valdivia, [<reflink idref="bib42" id="ref35">42</reflink>]; Shih et al., [<reflink idref="bib45" id="ref36">45</reflink>]). For many nondiagnostic large‐scale assessments, CDA has been increasingly applied to the analysis of assessment data in some studies to obtain fine‐grained feedback about students' ability in a given domain. For example, the CDA approach has been applied to TIMSS data in several studies (Birenbaum et al., [<reflink idref="bib4" id="ref37">4</reflink>], [<reflink idref="bib3" id="ref38">3</reflink>]; Dogan & Tatsuoka, [<reflink idref="bib9" id="ref39">9</reflink>]; Im & Park, [<reflink idref="bib17" id="ref40">17</reflink>]; Kabiri et al., [<reflink idref="bib20" id="ref41">20</reflink>]).</p> <p>On the other hand, there is no research on the mathematical competency of students in mainland China in the context of TIMSS since they haven't participated in it. And as we all know, students in Shanghai China have made outstanding achievements in the Programme for International Student Assessment (PISA) test many times (Mervis, [<reflink idref="bib36" id="ref42">36</reflink>]; OECD, [<reflink idref="bib40" id="ref43">40</reflink>]). In addition, a recent similar study used PISA results to compare the mathematical abilities of students in China with those in other countries through cognitive diagnostic analysis and found that the PISA Mathematical Key Competencies of Chinese students were the best (Wu, Zhang, et al., [<reflink idref="bib57" id="ref44">57</reflink>]). As such, the present study aims to apply CDA to analyze Shanghai students' mathematical competency in the context of TIMSS. Specifically, the present study aims to answer these research questions (i) "How to construct the cognitive attributes of students' mathematical competency based on TIMSS‐2011 framework?" (ii) "How about the performance of Shanghai students' mathematical competency in the fourth‐grade mathematics test in TIMSS‐2011 through cognitive diagnosis analysis?" and (iii) "How to construct the learning path of students' mathematical competency according to the evaluation data?"</p> <hd id="AN0169726498-5">METHODS</hd> <p></p> <hd id="AN0169726498-6">Participants</hd> <p>This study takes the 67 released mathematical items for the fourth grade in TIMSS‐2011 as assessment items since the items of TIMSS‐2011 are in the public domain and can be downloaded from the website of the IEA. The items of TIMSS‐2011 are equipped with high reliability and validity and are still applied to recently published research (T. Hu et al., [<reflink idref="bib16" id="ref45">16</reflink>]). The participants of the present study comprised three primary schools of 573 fourth graders from Shanghai, China. These three schools are at a medium level in Shanghai, which can represent the average situation of Shanghai students to a certain extent. The test was divided into three parts, and each part needed to be completed within 40 min. All three parts of the test were carried out in April 2021. The testing process was all strictly supervised by the mathematics teachers of each class to ensure high quality. A total of 573 students participated in all three parts of the test. The scoring and data entry work was carried out by 13 postgraduate students majoring in mathematics education. One point for correct answers, and 0 points for wrong answers to each item.</p> <hd id="AN0169726498-7">Cognitive attribute</hd> <p>The cognitive attribute is the core concept of CDA. There is no universal definition of attributes. Previous studies defined attributes as processing skills and knowledge structures needed to complete a task (Leighton & Gierl, [<reflink idref="bib28" id="ref46">28</reflink>]), posited knowledge and thinking skill (Tatsuoka et al., [<reflink idref="bib47" id="ref47">47</reflink>]), or descriptions of the procedures, skills, processes, strategies, and knowledge a student must possess to solve an item (Dogan & Tatsuoka, [<reflink idref="bib9" id="ref48">9</reflink>]). Generally, in an educational context, we can take the attributes as the knowledge and skills that students need to apply in problem‐solving. The construction of cognitive attributes is a key link for understanding children's cognitive structure in a specific learning field (Wu, Wu, et al., [<reflink idref="bib54" id="ref49">54</reflink>]). Cognitive diagnosis is exactly the diagnosis of students' mastery of cognitive attributes, through which students' performance can be inferred.</p> <p>To understand the cognitive structure of students in solving mathematical problems, it is necessary to construct mathematical cognitive attributes first. TIMSS is one of the most comprehensive international large‐scale studies that investigate students' mathematical competency. The assessment framework of TIMSS‐2011 not only focuses on mathematical content domains (algebra, geometry, etc.) but also includes cognitive domains that encompass a range of cognitive processes involved in working mathematically and solving problems. The cognitive dimension is specified by three aspects, that is, knowing, applying, and reasoning and each aspect contain different cognitive attributes. Cognitive attributes describe the sets of behaviors expected of students as they respond to the items. The cognitive framework of TIMSS‐2011 is formed based on the mathematical curriculum standards of various countries and the mathematical competency of students, with high reliability and validity. Therefore, this research draws on the attribute division in the cognitive framework of TIMSS‐2011 and makes adjustments based on the Chinese mathematics curriculum, initially forming a cognitive framework containing seven cognitive attributes. Then, according to the expert opinions of three professors of mathematics education, the connotation of attributes was further revised and clarified, and finally, a cognitive framework of mathematical competency containing seven cognitive attributes was formed, as shown in Table 1.</p> <p>1 Table Cognitive attribute and definition of mathematical competency.</p> <p> <ephtml> <table><thead valign="bottom"><tr valign="bottom"><th align="left">Code</th><th align="left">Attribute</th><th align="left">Definition</th></tr></thead><tbody valign="top"><tr><td>A1</td><td>Recall/Recognize</td><td>Recall definitions; terminology; unit rate; algorithm; geometric properties; and formulas (e.g., rectangular area formula S = ab). Recognize mathematical objects, for example, shapes, numbers, expressions, and quantities. Recognize mathematical entities that are mathematically equivalent (e.g., equivalent familiar fractions, decimals, and percents; different orientations of simple geometric figures).</td></tr><tr><td>A2</td><td>Classify/Order</td><td>Classify objects, shapes, numbers, and expressions according to common properties; order numbers and objects by attributes.</td></tr><tr><td>A3</td><td>Compute/Measure</td><td>Carry out algorithmic procedures for +, −, ×, ÷, or a combination of these with whole numbers, fractions, decimals, and integers. Approximate numbers to estimate computations. Carry out routine algebraic procedures. Use measuring instruments; carry out measurement activities; choose appropriate units of measurement.</td></tr><tr><td>A4</td><td>Retrieve/Solve Routine Problems</td><td>Retrieve information from graphs, tables, or other sources; read simple scales. Select an efficient/appropriate operation, method, or strategy for solving problems where there is a known procedure, algorithm, or method of solution; implement instructions or strategies to solve standard problems similar to those encountered in class.</td></tr><tr><td>A5</td><td>Represent/Model</td><td>Display mathematical information and data in diagrams, tables, charts, or graphs, and generate equivalent representations for a given mathematical entity or relationship. Generate an appropriate model, such as an equation, geometric figure, or diagram for solving a routine problem.</td></tr><tr><td>A6</td><td>Analyze/Integrate</td><td>Determine and describe the relationship between variables or objects in mathematical situations. Make connections between different elements of knowledge and related representations, and make linkages between related mathematical ideas. Evaluate alternative strategies and solutions for solving problems.</td></tr><tr><td>A7</td><td>Generalize/Justify</td><td>Make valid inferences from the given information. Extend the domain to which the result of mathematical thinking and problem‐solving is applicable in more general and more widely applicable terms. Provide a justification by reference to known mathematical results or properties.</td></tr></tbody></table> </ephtml> </p> <hd id="AN0169726498-8">Items and Q‐matrix</hd> <p>The calibration of cognitive attributes is based on the determination of cognitive components involved in the cognitive process of students' problem‐solving. Q‐matrix plays a key role in ensuring the accuracy of the evaluation. It determines the internal structure of the test and correlates items with cognitive attributes, which can clearly reflect the important characteristics of items (Wu, Xu, et al., [<reflink idref="bib56" id="ref50">56</reflink>]). Through the item answers, we can identify students' mastery of attributes, so as to obtain diagnostic information about students and determine their knowledge state. To effectively ensure the quality of the test, 67 released mathematical items for the fourth grade in TIMSS‐2011 were used as assessment tools in this study (IEA, [<reflink idref="bib18" id="ref51">18</reflink>]). The cognitive attributes for each item were independently calibrated through questionnaires by 33 primary school mathematics teachers, 15 master and doctoral students majoring in mathematics education, and 4 mathematics education experts. Attributes calibrated by more than 50% of the people were taken as the attributes of the item (Wu, [<reflink idref="bib52" id="ref52">52</reflink>]). After that, researchers and the three professors of mathematics education who participated in the definition of attribute connotation further discussed and modified the item attributes based on the results of the questionnaire, and the final Q‐matrix of 67 item attributes was determined. Three example items and corresponding attributes are shown in Table 2.</p> <p>2 Table Example items and corresponding attributes.</p> <p> <ephtml> <table><thead valign="bottom"><tr valign="bottom"><th align="left">Item ID</th><th align="left">Item</th><th align="left">Attribute</th></tr><tr valign="bottom"><th align="left">A1</th><th align="left">A2</th><th align="left">A3</th><th align="left">A4</th><th align="left">A5</th><th align="left">A6</th><th align="left">A7</th></tr></thead><tbody valign="top"><tr><td align="left">M031187</td><td align="left">Stands for the number of pencils Pete had. Kim gave Pete 3 more pencils. How many pencils does Pete have now?</td><td align="left">0</td><td align="left">0</td><td align="left">0</td><td align="left">1</td><td align="left">0</td><td align="left">1</td><td align="left">0</td></tr><tr><td align="left"><graphic href="" /></td></tr><tr><td align="left">M031210</td><td align="left">Which of these fractions is larger than <p><math altimg="urn:x-wiley:00333085:media:pits22916:pits22916-math-0001" xmlns="http://www.w3.org/1998/Math/MathML"><semantics xmlns="">12<annotation encoding="application/x-tex"> $\frac{1}{2}$</annotation></semantics></math></p>?</td><td align="left">1</td><td align="left">1</td><td align="left">0</td><td align="left">0</td><td align="left">0</td><td align="left">0</td><td align="left">0</td></tr><tr><td align="left"><graphic href="" /></td></tr><tr><td align="left">M051601</td><td align="left">Cooney has to form figures 1–4 with matches.</td><td align="left">0</td><td align="left">0</td><td align="left">0</td><td align="left">1</td><td align="left">0</td><td align="left">1</td><td align="left">1</td></tr><tr><td align="left">Figures 1‐3 are shown below.</td></tr><tr><td align="left">He needs 4 matches to form figure 1, 7 matches to form figure 2, and 10 matches to form figure 3.</td></tr><tr><td align="left">He uses the same rule each time to make the next figure in the pattern.</td></tr><tr><td align="left"><graphic href="" /></td></tr><tr><td align="left">How many matches will he need to form figure 4?</td></tr></tbody></table> </ephtml> </p> <hd id="AN0169726498-9">MODEL SELECTION AND INSTRUMENT ANALYSIS</hd> <p></p> <hd id="AN0169726498-10">Deterministic inputs, noisy, "and" gate (DINA) model and its applicability analysis</hd> <p>Appropriate model selection is an important prerequisite for accurate diagnosis or classification of examinees in cognitive diagnosis assessment. The DINA model is a simple and interpretable model that requires only two parameters for each item and has been shown to provide a good model fit (Junker & Sijtsma, [<reflink idref="bib19" id="ref53">19</reflink>]). As an effective part of the DINA model, Q‐matrix can explicitly show the cognitive specification for each item. The model is (de la Torre, [<reflink idref="bib49" id="ref54">49</reflink>]): <ephtml> <math altimg="urn:x-wiley:00333085:media:pits22916:pits22916-math-0002" display="block" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>P</mi><mi>j</mi></msub><mrow><mo>(</mo><msub><mi>α</mi><mi>i</mi></msub><mo>)</mo></mrow><mo>=</mo><mi>P</mi><mrow><mo>(</mo><msub><mi>X</mi><mi mathvariant="italic">ij</mi></msub><mo>=</mo><mn>1</mn><mo>|</mo><msub><mi>α</mi><mi>i</mi></msub><mo>)</mo></mrow><mo>=</mo><msubsup><mi>g</mi><mi>j</mi><mrow><mn>1</mn><mo>−</mo><msub><mi>η</mi><mi mathvariant="italic">ij</mi></msub></mrow></msubsup><msup><mrow><mo>(</mo><mn>1</mn><mo>−</mo><msub><mi>s</mi><mi>j</mi></msub><mo>)</mo></mrow><msub><mi>η</mi><mi mathvariant="italic">ij</mi></msub></msup><mo>,</mo></mrow><annotation encoding="application/x-tex"> ${P}_{j}({\alpha }_{i})=P({X}_{{ij}}=1|{\alpha }_{i})={g}_{j}^{1-{\eta }_{{ij}}}{(1-{s}_{j})}^{{\eta }_{{ij}}},$</annotation></semantics></math> </ephtml> where <ephtml> <math altimg="urn:x-wiley:00333085:media:pits22916:pits22916-math-0003" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>P</mi><mi>j</mi></msub><mrow><mo>(</mo><msub><mi>α</mi><mi>i</mi></msub><mo>)</mo></mrow></mrow><annotation encoding="application/x-tex"> ${P}_{j}({\alpha }_{i})$</annotation></semantics></math> </ephtml> is the probability of examinee <emph>i</emph> with the attributes vector <ephtml> <math altimg="urn:x-wiley:00333085:media:pits22916:pits22916-math-0004" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>α</mi><mi>i</mi></msub></mrow><annotation encoding="application/x-tex"> ${\alpha }_{i}$</annotation></semantics></math> </ephtml> answering item <emph>j</emph> correctly. <ephtml> <math altimg="urn:x-wiley:00333085:media:pits22916:pits22916-math-0005" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>η</mi><mi mathvariant="italic">ij</mi></msub></mrow><annotation encoding="application/x-tex"> ${\eta }_{{ij}}$</annotation></semantics></math> </ephtml> is a latent response vector produced by an examinee's attributes vector and the Q‐matrix. <ephtml> <math altimg="urn:x-wiley:00333085:media:pits22916:pits22916-math-0006" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>η</mi><mi mathvariant="italic">ij</mi></msub></mrow><annotation encoding="application/x-tex"> ${\eta }_{{ij}}$</annotation></semantics></math> </ephtml>  = 1 if examinee <emph>i</emph> has all the attributes of item <emph>j</emph> and <ephtml> <math altimg="urn:x-wiley:00333085:media:pits22916:pits22916-math-0007" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>η</mi><mi mathvariant="italic">ij</mi></msub></mrow><annotation encoding="application/x-tex"> ${\eta }_{{ij}}$</annotation></semantics></math> </ephtml>  = 0 otherwise. <ephtml> <math altimg="urn:x-wiley:00333085:media:pits22916:pits22916-math-0008" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>g</mi><mi>j</mi></msub></mrow><annotation encoding="application/x-tex"> ${g}_{j}$</annotation></semantics></math> </ephtml> and <ephtml> <math altimg="urn:x-wiley:00333085:media:pits22916:pits22916-math-0009" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>s</mi><mi>j</mi></msub></mrow><annotation encoding="application/x-tex"> ${s}_{j}$</annotation></semantics></math> </ephtml> are the slip and guessing parameters of item <emph>j</emph>. The slip parameter refers to the probability that the examinee who has mastered all the attributes of the item will answer the item incorrectly, and the guessing parameter refers to the probability that the examinee who lacks at least one attribute of the item will answer the item correctly. The DINA model is suitable for dealing with multidimensional binary latent traits.</p> <p>DINA model assumes that to respond correctly to an item, the examinee must master all the attributes of the item, and the failure to master any attribute of the item will reduce the probability of a correct response. It is a completely noncompensatory cognitive diagnostic model (Junker & Sijtsma, [<reflink idref="bib19" id="ref55">19</reflink>]). According to the characteristics of mathematical learning, it is generally believed that mathematical competencies are interlinked, and it is difficult for the attributes of mathematical competency to replace and compensate for each other. Therefore, mathematical cognitive attributes tend to be non‐compensatory. As a noncompensatory model, the DINA model is more in line with the evaluation of mathematical discipline traits and has been widely used in cognitive diagnostic studies involving mathematics learning (Akbay et al., [<reflink idref="bib1" id="ref56">1</reflink>]; Choi et al., [<reflink idref="bib7" id="ref57">7</reflink>]; Lee et al., [<reflink idref="bib26" id="ref58">26</reflink>]).</p> <hd id="AN0169726498-11">Absolute model fit indices</hd> <p>Absolute model fit is based on the fit of a model to the observed response data without comparison to other models. The absolute fitting index in this study is Limited information of the root mean square error of approximation (RMSEA<subs>2</subs>) (Houts & Cai, [<reflink idref="bib14" id="ref59">14</reflink>]). RMSEA<subs>2</subs> is an absolute fitting index commonly used in cognitive diagnosis models. In the construction of the index model, RMSEA<subs>2</subs> is different from RMSEA because it only uses two moments: univariate and bivariate interaction (Maydeu‐Olivares & Joe, [<reflink idref="bib35" id="ref60">35</reflink>]). It is suggested that RMSEA<subs>2</subs> < 0.05 is a conservative criterion for model fit in cognitive diagnosis models (J. Hu et al., [<reflink idref="bib15" id="ref61">15</reflink>]). By using the generalized DINA (G‐DINA) package in R software, the overall model fit index RMSEA<subs>2</subs> = 0.0201 < 0.05, which indicates an acceptable model fit.</p> <hd id="AN0169726498-12">Item fit analysis</hd> <p>In the analysis of the cognitive diagnosis model, in addition to examining the overall fit of the model, the fit effect of each test item should also be analyzed. The fitting degree of data and items directly determines the accuracy of the model diagnosis. In CDA, residual statistics (RMSEA) can be used to measure the fitting effect of items. RMSEA mainly compares the square root errors of observed and predicted responses under different potential classifications. The RMSEA formula for item <emph>j</emph> is (Kunina‐Habenicht et al., [<reflink idref="bib24" id="ref62">24</reflink>]): <ephtml> <math altimg="urn:x-wiley:00333085:media:pits22916:pits22916-math-0010" display="block" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>RMSEA</mi><mi>j</mi></msub><mo>=</mo><msqrt><munderover><mo>∑</mo><mrow><mi>c</mi><mo>=</mo><mn>1</mn></mrow><msup><mn>2</mn><mi>K</mi></msup></munderover><mrow><mi>p</mi><mrow><mo>(</mo><msub><mi mathvariant="bold-italic">α</mi><mi mathvariant="bold-italic">c</mi></msub><mo>)</mo></mrow><mo>[</mo><msub><mi>P</mi><mi>expected</mi></msub><mo>(</mo><msub><mi>X</mi><mi>j</mi></msub><mo>=</mo><mn>1</mn><mrow><mo>|</mo><mrow><msub><mi mathvariant="bold-italic">α</mi><mi mathvariant="bold-italic">c</mi></msub><mo>)</mo><mo>−</mo></mrow></mrow><msub><mi>P</mi><mi>observed</mi></msub><msup><mrow><mo>(</mo><msub><mi>X</mi><mi>j</mi></msub><mo>=</mo><mn>1</mn><mrow><mo>|</mo><mrow><msub><mi mathvariant="bold-italic">α</mi><mi mathvariant="bold-italic">c</mi></msub><mo>)</mo></mrow></mrow><mo>]</mo></mrow><mn>2</mn></msup></mrow></msqrt><mo>,</mo></mrow><annotation encoding="application/x-tex"> ${\mathrm{RMSEA}}_{j}=\sqrt{\sum _{c=1}^{{2}^{K}}p({{\boldsymbol{\alpha }}}_{{\boldsymbol{c}}})[{P}_{\mathrm{expected}}({X}_{j}=1|{{\boldsymbol{\alpha }}}_{{\boldsymbol{c}}})-{P}_{\mathrm{observed}}{({X}_{j}=1|{{\boldsymbol{\alpha }}}_{{\boldsymbol{c}}})]}^{2}},$</annotation></semantics></math> </ephtml> where <ephtml> <math altimg="urn:x-wiley:00333085:media:pits22916:pits22916-math-0011" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi mathvariant="bold-italic">α</mi><mi mathvariant="bold-italic">c</mi></msub><mo>=</mo><mo>(</mo><msub><mi>α</mi><mn>1</mn></msub><mo>,</mo><msub><mi>α</mi><mn>2</mn></msub><mo>,</mo><mtext>⋯</mtext><mo>,</mo><msub><mi>α</mi><mi>K</mi></msub><mo>)</mo></mrow><annotation encoding="application/x-tex"> ${{\boldsymbol{\alpha }}}_{{\boldsymbol{c}}}=({\alpha }_{1},{\alpha }_{2},\text{\unicode{x022EF}},{\alpha }_{K})$</annotation></semantics></math> </ephtml> represents the attribute vector for latent class <ephtml> <math altimg="urn:x-wiley:00333085:media:pits22916:pits22916-math-0012" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>c</mi><mo>=</mo><mn>1,2</mn><mo>,</mo><mtext>⋯</mtext><mo>,</mo><msup><mn>2</mn><mi>K</mi></msup></mrow><annotation encoding="application/x-tex"> $c=\mathrm{1,2},\text{\unicode{x022EF}},{2}^{K}$</annotation></semantics></math> </ephtml> and <ephtml> <math altimg="urn:x-wiley:00333085:media:pits22916:pits22916-math-0013" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>P</mi><mo>(</mo><msub><mi mathvariant="bold-italic">α</mi><mi mathvariant="bold-italic">c</mi></msub><mo>)</mo></mrow><annotation encoding="application/x-tex"> $P({{\boldsymbol{\alpha }}}_{{\boldsymbol{c}}})$</annotation></semantics></math> </ephtml> is the marginal proportion of respondents in latent class <emph>c</emph>, whose estimation is based on the respondents' most likely latent class membership, namely the maximum a posterior probability. <emph>K</emph> is the number of attributes. In our study, <emph>K</emph> = 7. The residual information of mathematical competency test items based on the DINA model is obtained as shown in Table 3.</p> <p>3 Table Residual information of items.</p> <p> <ephtml> <table><tbody valign="top"><tr><td align="left">Item1</td><td align="left">0.0893</td><td align="left">Item18</td><td align="left">0.0424</td><td align="left">Item35</td><td align="left">0.0115</td><td align="left">Item52</td><td align="left">0.0598</td></tr><tr><td align="left">Item2</td><td align="left">0.1174</td><td align="left">Item19</td><td align="left">0.0939</td><td align="left">Item36</td><td align="left">0.0157</td><td align="left">Item53</td><td align="left">0.0247</td></tr><tr><td align="left">Item3</td><td align="left">0.0320</td><td align="left">Item20</td><td align="left">0.1462</td><td align="left">Item37</td><td align="left">0.0095</td><td align="left">Item54</td><td align="left">0.0929</td></tr><tr><td align="left">Item4</td><td align="left">0.0987</td><td align="left">Item21</td><td align="left">0.0211</td><td align="left">Item38</td><td align="left">0.0076</td><td align="left">Item55</td><td align="left">0.0263</td></tr><tr><td align="left">Item5</td><td align="left">0.0412</td><td align="left">Item22</td><td align="left">0.0529</td><td align="left">Item39</td><td align="left">0.0990</td><td align="left">Item56</td><td align="left">0.0329</td></tr><tr><td align="left">Item6</td><td align="left">0.0592</td><td align="left">Item23</td><td align="left">0.0946</td><td align="left">Item40</td><td align="left">0.0109</td><td align="left">Item57</td><td align="left">0.0581</td></tr><tr><td align="left">Item7</td><td align="left">0.0757</td><td align="left">Item24</td><td align="left">0.0331</td><td align="left">Item41</td><td align="left">0.0955</td><td align="left">Item58</td><td align="left">0.0080</td></tr><tr><td align="left">Item8</td><td align="left">0.0999</td><td align="left">Item25</td><td align="left">0.0528</td><td align="left">Item42</td><td align="left">0.0494</td><td align="left">Item59</td><td align="left">0.0000</td></tr><tr><td align="left">Item9</td><td align="left">0.0221</td><td align="left">Item26</td><td align="left">0.0457</td><td align="left">Item43</td><td align="left">0.1279</td><td align="left">Item60</td><td align="left">0.0098</td></tr><tr><td align="left">Item10</td><td align="left">0.0073</td><td align="left">Item27</td><td align="left">0.0011</td><td align="left">Item44</td><td align="left">0.0979</td><td align="left">Item61</td><td align="left">0.0344</td></tr><tr><td align="left">Item11</td><td align="left">0.0026</td><td align="left">Item28</td><td align="left">0.0254</td><td align="left">Item45</td><td align="left">0.0333</td><td align="left">Item62</td><td align="left">0.0802</td></tr><tr><td align="left">Item12</td><td align="left">0.0417</td><td align="left">Item29</td><td align="left">0.0077</td><td align="left">Item46</td><td align="left">0.0683</td><td align="left">Item63</td><td align="left">0.0255</td></tr><tr><td align="left">Item13</td><td align="left">0.0010</td><td align="left">Item30</td><td align="left">0.0337</td><td align="left">Item47</td><td align="left">0.1003</td><td align="left">Item64</td><td align="left">0.0241</td></tr><tr><td align="left">Item14</td><td align="left">0.0843</td><td align="left">Item31</td><td align="left">0.0008</td><td align="left">Item48</td><td align="left">0.1083</td><td align="left">Item65</td><td align="left">0.0065</td></tr><tr><td align="left">Item15</td><td align="left">0.0443</td><td align="left">Item32</td><td align="left">0.0103</td><td align="left">Item49</td><td align="left">0.0314</td><td align="left">Item66</td><td align="left">0.0460</td></tr><tr><td align="left">Item16</td><td align="left">0.0887</td><td align="left">Item33</td><td align="left">0.0329</td><td align="left">Item50</td><td align="left">0.0404</td><td align="left">Item67</td><td align="left">0.0223</td></tr><tr><td align="left">Item17</td><td align="left">0.0147</td><td align="left">Item34</td><td align="left">0.1704</td><td align="left">Item51</td><td align="left">0.0171</td><td /><td /></tr></tbody></table> </ephtml> </p> <p>The closer the RMSEA value is to 0, the smaller the deviation is, and the better the fitting effect is. In Oliveri and Von Davier's ([<reflink idref="bib41" id="ref63">41</reflink>]) study, the critical value of RMSEA is set to 0.1. When RMSEA > 0.1, it indicates that the items fit is poor. According to this criterion, the RMSEA value of almost every item in current research is less than 0.1, and only the RMSEA value of items 20, 34, 43, 47, 48 is slightly greater than 0.1.</p> <hd id="AN0169726498-13">Reliability</hd> <p>The reliability of cognitive diagnostic models can be measured in two ways. One is to calculate Cronbach's <emph>α</emph> coefficient based on the CTT. In this study, <emph>α</emph> = .849 > 0.7, which means high reliability. In addition, as a new generation of evaluation theory, the reliability of the cognitive diagnostic model can also be assessed by calculating the retest consistency of the attributes. Assuming that the probability of attributes mastered by examinees is constant, repeated testing occasions are simulated through repeated draws from an examinee's posterior distribution, and the model reliability measure is obtained by estimating the correlation of the examinee's attribute mastery probability in the two tests (Templin & Bradshaw, [<reflink idref="bib48" id="ref64">48</reflink>]). The reliability measures of the seven attributes in this study are 0.9451, 0.9346, 0.9472, 0.9599, 0.7408, 0.8913, 0.8519, and the average value is 0.8958, indicating high reliability.</p> <hd id="AN0169726498-14">RESULTS</hd> <p></p> <hd id="AN0169726498-15">Attribute mastery probability</hd> <p>Based on the Q‐matrix, the probability of each student mastering each attribute was obtained by evaluating students' responses by the DINA model using the G‐DINA package in R software. Then, according to the probability of each student mastering each attribute, we computed the mean value of the probability that each student mastered the attribute, that is, the probability of all students mastering the attribute. According to the probability, the results were plotted into a histogram, as shown in Figure 1.</p> <p> <img src="https://imageserver.ebscohost.com/img/embimages/rdk/PIS/01sep23/pits22916-fig-0001.jpg?ephost1=dGJyMNXb4kSepq84yOvqOLCmsE6epq5Srqa4SK6WxWXS" alt="pits22916-fig-0001.jpg" title="1 Attribute mastery probability." /> </p> <p></p> <p>According to the probability of mastering the attributes, we can analyze which attributes of A1–A7 students may master well or not as a whole, so as to provide information for teaching. Figure 1 is the histogram of attribute mastery probability. The horizontal axis represents seven attributes, and the vertical axis represents the probability of all students mastering the attribute. As we can see, the mastery of each attribute is relatively balanced on the whole, basically between 70% and 80%. On the whole, the mastery probability of attributes A3, A4, and A5 in the middle is lower than that of attributes A1, A2, A6, and A7 at both ends. Among them, the probability of mastering attribute A7 Generalize/Justify is the highest, there is an 80% probability that all students will master this attribute, while the probability of mastering attribute A5 Represent/Model is relatively the lowest.</p> <hd id="AN0169726498-17">Learning path analysis</hd> <p>The construction of a learning path based on CDA takes the individual differences among groups into account. It assumed that the learning sequence, along with the cognitive structure and cognitive order of the learning process, is the result of the internalization of the group's learning consciousness into individual learning behaviors (Wu, Wu, et al., [<reflink idref="bib55" id="ref65">55</reflink>]). When individuals in a group are at different stages of development, they can be connected in a certain cognitive order to form a learning path. The learning path focuses on the cognitive development process of students' learning, which reflects the different order of acquiring knowledge or skill among individuals (Wu, Zhang, et al., [<reflink idref="bib58" id="ref66">58</reflink>]). Under the theory of cognitive diagnosis, it is the basis for constructing the learning path to clarify the knowledge state of students which can be obtained by evaluating the response data of students through the cognitive diagnosis model. Based on the analysis of students' knowledge states, groups with the same knowledge state are clustered into the same category. Then, by searching the inclusion relationship between different knowledge states, a hypothetical learning path is constructed for this group.</p> <p>The basic assumption of learning path construction is that the mastery of learning knowledge or skills has a certain order, which can be reflected by the knowledge state. Therefore, it is necessary to obtain the knowledge state of each student before constructing the learning path. According to the student's responses, the cognitive diagnostic model can evaluate each student's mastery status of attributes (1 indicates that the attribute has been mastered, and 0 indicates that the attribute has not been mastered). The mastery status of each student's attributes constitutes a multidimensional vector composed of elements 0 and 1, known as the knowledge state. Table 4 shows the distribution of the number of people in different knowledge states. It can be seen that the number of people who have mastered all seven attributes is the largest, more than 60%, indicating that most students have mastered the attributes well. The second largest number of students are those who haven't mastered all the seven attributes, more than 16%, while the number of students in the intermediate state is obviously small.</p> <p>4 Table Distribution of students in different knowledge states.</p> <p> <ephtml> <table><thead valign="bottom"><tr valign="bottom"><th align="left">Knowledge state</th><th align="left">Students number</th><th align="left">Knowledge state</th><th align="left">Students number</th><th align="left">Knowledge state</th><th align="left">Students number</th><th align="left">Knowledge state</th><th align="left">Students number</th></tr></thead><tbody valign="top"><tr><td align="left">0000000</td><td align="char" char=".">93</td><td align="left">0100010</td><td align="left">1</td><td align="left">1010001</td><td align="left">1</td><td align="left">1101011</td><td align="char" char=".">1</td></tr><tr><td align="left">0000001</td><td align="char" char=".">1</td><td align="left">0100111</td><td align="left">1</td><td align="left">1010101</td><td align="left">2</td><td align="left">1101111</td><td align="char" char=".">2</td></tr><tr><td align="left">0001110</td><td align="char" char=".">1</td><td align="left">0101110</td><td align="left">6</td><td align="left">1010111</td><td align="left">5</td><td align="left">1110011</td><td align="char" char=".">1</td></tr><tr><td align="left">0001111</td><td align="char" char=".">1</td><td align="left">0110000</td><td align="left">1</td><td align="left">1011011</td><td align="left">1</td><td align="left">1110101</td><td align="char" char=".">1</td></tr><tr><td align="left">0010000</td><td align="char" char=".">1</td><td align="left">0110001</td><td align="left">4</td><td align="left">1100011</td><td align="left">2</td><td align="left">1110111</td><td align="char" char=".">3</td></tr><tr><td align="left">0010001</td><td align="char" char=".">2</td><td align="left">0111100</td><td align="left">5</td><td align="left">1100101</td><td align="left">1</td><td align="left">1111011</td><td align="char" char=".">35</td></tr><tr><td align="left">0010011</td><td align="char" char=".">1</td><td align="left">1000001</td><td align="left">1</td><td align="left">1101000</td><td align="left">1</td><td align="left">1111111</td><td align="char" char=".">361</td></tr><tr><td align="left">0011011</td><td align="char" char=".">6</td><td align="left">1001111</td><td align="left">3</td><td align="left">1101001</td><td align="left">4</td><td /><td /></tr><tr><td align="left">0100000</td><td align="char" char=".">3</td><td align="left">1010000</td><td align="left">2</td><td align="left">1101010</td><td align="left">5</td><td /><td /></tr></tbody></table> </ephtml> </p> <p>After estimating the knowledge state of each student, the same knowledge state can be classified, and then stratified according to the number of mastered attributes in the knowledge state. The knowledge state (0000000) is marked as level 0, and the knowledge state that represents mastering one attribute as level 1, for example (0100000). There are total 8 levels from the knowledge state (0000000) at level 0 to the knowledge state (1111111) at level 7. The learning path is characterized by the inclusion relationship of knowledge states between different levels. In this study, not only the first‐order path, that is, the path between adjacent levels were considered but also the multilevel path, which means that it is possible for students to obtain multiple attributes at a time. For example, as shown in Figure 2, some learning paths are leapfrog. From the knowledge state (0000000), one might get one attribute to the knowledge state (0100000) at the first level, two attributes to the knowledge state (0010001) at the second level, or three attributes to the knowledge state (1001001) at the third level. From the knowledge state (0000000) to (1101111), there are both paths with four levels from (0000000) → (0100000) → (0101110) → (1101111), and paths with five levels from (0000000) → (0100000) → (1100011) → (1101111).</p> <p> <img src="https://imageserver.ebscohost.com/img/embimages/rdk/PIS/01sep23/pits22916-fig-0002.jpg?ephost1=dGJyMNXb4kSepq84yOvqOLCmsE6epq5Srqa4SK6WxWXS" alt="pits22916-fig-0002.jpg" title="2 Learning path of mathematical competency." /> </p> <p></p> <hd id="AN0169726498-19">Knowledge structure analysis</hd> <p>The notable feature of CDA that differs from other measurement theories consists of the personalized diagnosis and analysis of students' knowledge structure. In the cognitive diagnosis assessment, after evaluating the probability of each student's attribute mastery, the student's knowledge structure can be formed, which is a refined measurement model with the attribute as the basic analysis unit. From Figure 3, we can see three students with the same total score but have different knowledge structures. Although their total scores are the same, there are certain differences in the mastery probability of different attributes, that is, the knowledge structure is different. For example, the mastery probability of subject 342 on attributes 1, 3, 4, 6, and 7 is close to 1, the mastery probability on attribute 2 is slightly lower than 0.9, and the mastery probability on attribute 5 is lower, less than 0.8; the mastery probability of subject 490 on attributes 2, 3, and 4 reached 1, the mastery probability of attributes 1 and 7 was close to 1, and the mastery probability of attributes 5 and 6 was about 0.9; the mastery probability of subject 466 on attribute 1, 2, 3, 4, 6, and 7 basically reached 1, while the mastery probability of attribute 5 is about 0.7. Therefore, even if students have the same total score and the same knowledge state, there may be differences in their probability of attribute mastery. Through cognitive diagnostic analysis, we can carry out personalized diagnostic analysis for each student, and find out the weak points, and provide targeted learning feedback for students and teaching reference information for teachers.</p> <p> <img src="https://imageserver.ebscohost.com/img/embimages/rdk/PIS/01sep23/pits22916-fig-0003.jpg?ephost1=dGJyMNXb4kSepq84yOvqOLCmsE6epq5Srqa4SK6WxWXS" alt="pits22916-fig-0003.jpg" title="3 Comparison of different knowledge structures with the same total score." /> </p> <p></p> <hd id="AN0169726498-21">DISCUSSION</hd> <p>The development of mathematical competency is not only of great significance to students' learning and life in school but also has an important impact on student's future career development. The purpose of this study is to understand the mathematical competency of Shanghai students in the context of TIMSS and to conduct cognitive diagnostic analysis. First of all, based on the attribute division in TIMSS's cognitive framework, adjustments were made according to the Chinese mathematics curriculum, combined with the results of expert interviews, a cognitive framework of mathematical competency containing seven cognitive attributes was constructed. Second, 67 mathematical items for the fourth grade in TIMSS‐2011 were used as evaluation items to test 573 fourth‐grade students from Shanghai, China, and the cognitive attributes of each item were calibrated based on the constructed cognitive framework, and the Q‐matrix of 67 item attributes was obtained. The effectiveness of the assessment tool was also verified based on the DINA model. Finally, the cognitive diagnosis of the student's mathematical competency was carried out to identify students' mastery of various attributes, draw the students' learning path, and illustrate the personalized analysis of different knowledge structures of students with the same score.</p> <p>The results of this study show that students' performance in mastering attributes of mathematical cognition is excellent on the whole which indicates that the sample of Shanghai fourth‐grade students have good mathematical competency and is consistent with the excellent performance of Shanghai students in other international assessments (Mervis, [<reflink idref="bib36" id="ref67">36</reflink>]; OECD, [<reflink idref="bib40" id="ref68">40</reflink>]). Compared with the mastery of other attributes, the performance in attribute Represent/Model is relatively poor, indicating that if students encounter obstacles in the development of mathematical competency, it is likely that there are problems in representation or modeling. Representation and modeling are both vital skills in solving math‐related problems in different situations. The conversion between different representations will lead to information gain or information loss. Focusing on such gains and losses is a significant element of this competency (Niss & Højgaard, [<reflink idref="bib39" id="ref69">39</reflink>]). More than 60% of the students have mastered all the attributes, while more than 16% of the students have not mastered any of the attributes, which is also a very high proportion, indicating that some students in the sample have lagged behind in the development of mathematical competency. Teachers may need to pay attention to this part of students in the follow‐up teaching, adjust teaching strategies and make timely remedies. Learning path analysis can provide information for students' mathematical thinking and learning. Through learning path analysis, it is convenient for students in different learning paths to choose appropriate learning methods and achieve different learning goals. Different from previous studies on learning paths (Wu, Wu, et al., [<reflink idref="bib55" id="ref70">55</reflink>]; Wu, Zhang, et al., [<reflink idref="bib57" id="ref71">57</reflink>]), we can see from the results of learning path analysis in this study that some learning paths are leapfrog, which indicates that students can acquire multiple attributes at the same time, rather than only one attribute at a time. Learning path analysis emphasizes the development of students' cognitive order and clarifies the importance of learners in guiding future teaching, curriculum, and evaluation (Confrey, [<reflink idref="bib8" id="ref72">8</reflink>]). Through the cognitive diagnosis analysis, a personalized formative assessment report has also been generated for each student, which can provide a detailed understanding of each student's performance in various cognitive attributes. Based on this, students can obtain accurate feedback, and teachers can adjust teaching in time for students' weaknesses. As we all know, even the students obtain the same score on a test, there might be great differences in their skill mastery profiles. This study shows that the CDA approach can provide students with personalized and refined diagnostic information from the aspects of attribute mastery, learning path, and knowledge structure, which cannot be assessed through traditional measurement methods. CDA can enrich the judgment of test scores by and for students, teachers, curriculum developers, and policymakers and assist teachers to tailor remedial instruction for each student (Kabiri et al., [<reflink idref="bib20" id="ref73">20</reflink>]). The application of CDA has far‐reaching implications for promoting students' development and school education.</p> <p>Based on the TIMSS cognitive framework, this study constructs a cognitive assessment framework for mathematical competency and carried out an analysis on 573 fourth graders in Shanghai, China. Cognitive diagnosis provides unique diagnostic information for the development of students' mathematical competency and also provides a new perspective for the evaluation of student's competency. However, there are still some limitations in this study. For example, only the fourth‐grade students in Shanghai were recruited in this study, leading to an insufficient representation of the sample. For future research, if conditions permit, the sample scope can be further expanded for large‐scale evaluation to identify the general situation of students in Mainland China. In addition, longitudinal data can be collected to verify the learning path constructed in the study. It can also be considered in future research how to link cognitive diagnosis analysis with adaptive learning, form students' personalized assessment reports through cognitive diagnosis, and then recommend targeted resources.</p> <hd id="AN0169726498-22">ACKNOWLEDGMENTS</hd> <p>This work was supported by Yuanhui Youth Development Program: A study on the appropriateness of cognitive diagnostic assessment in Mathematics; Teacher Education "JIEBANGLINGTI" Project of Northeast Normal University: Learning Progression Construction and Learning Path Analysis Based on Cognitive Diagnosis (JSJY20220305).</p> <hd id="AN0169726498-23">CONFLICT OF INTEREST STATEMENT</hd> <p>The authors declare no conflict of interest.</p> <hd id="AN0169726498-24">DATA AVAILABILITY STATEMENT</hd> <p>The datasets generated during and/or analyzed during the current study are available from the corresponding author upon reasonable request.</p> <hd id="AN0169726498-25">ETHICS STATEMENT</hd> <p>Ethical review and approval were not required for the study on human participants in accordance with the local legislation and institutional requirements. 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Items – Name: Title
  Label: Title
  Group: Ti
  Data: Cognitive Diagnostic Analysis of Students' Mathematical Competency Based on the DINA Model
– Name: Language
  Label: Language
  Group: Lang
  Data: English
– Name: Author
  Label: Authors
  Group: Au
  Data: <searchLink fieldCode="AR" term="%22Xu%2C+Tianshu%22">Xu, Tianshu</searchLink><br /><searchLink fieldCode="AR" term="%22Wu%2C+Xiaopeng%22">Wu, Xiaopeng</searchLink> (ORCID <externalLink term="http://orcid.org/0000-0003-0147-8228">0000-0003-0147-8228</externalLink>)<br /><searchLink fieldCode="AR" term="%22Sun%2C+Siyu%22">Sun, Siyu</searchLink><br /><searchLink fieldCode="AR" term="%22Kong%2C+Qiping%22">Kong, Qiping</searchLink>
– Name: TitleSource
  Label: Source
  Group: Src
  Data: <searchLink fieldCode="SO" term="%22Psychology+in+the+Schools%22"><i>Psychology in the Schools</i></searchLink>. Sep 2023 60(9):3135-3150.
– Name: Avail
  Label: Availability
  Group: Avail
  Data: Wiley. Available from: John Wiley & Sons, Inc. 111 River Street, Hoboken, NJ 07030. Tel: 800-835-6770; e-mail: cs-journals@wiley.com; Web site: https://www.wiley.com/en-us
– Name: PeerReviewed
  Label: Peer Reviewed
  Group: SrcInfo
  Data: Y
– Name: Pages
  Label: Page Count
  Group: Src
  Data: 16
– Name: DatePubCY
  Label: Publication Date
  Group: Date
  Data: 2023
– Name: TypeDocument
  Label: Document Type
  Group: TypDoc
  Data: Journal Articles<br />Reports - Research
– Name: Audience
  Label: Education Level
  Group: Audnce
  Data: <searchLink fieldCode="EL" term="%22Elementary+Education%22">Elementary Education</searchLink><br /><searchLink fieldCode="EL" term="%22Grade+4%22">Grade 4</searchLink><br /><searchLink fieldCode="EL" term="%22Intermediate+Grades%22">Intermediate Grades</searchLink>
– Name: Subject
  Label: Descriptors
  Group: Su
  Data: <searchLink fieldCode="DE" term="%22Cognitive+Processes%22">Cognitive Processes</searchLink><br /><searchLink fieldCode="DE" term="%22Mathematics+Skills%22">Mathematics Skills</searchLink><br /><searchLink fieldCode="DE" term="%22Competence%22">Competence</searchLink><br /><searchLink fieldCode="DE" term="%22Grade+4%22">Grade 4</searchLink><br /><searchLink fieldCode="DE" term="%22Models%22">Models</searchLink><br /><searchLink fieldCode="DE" term="%22Student+Evaluation%22">Student Evaluation</searchLink><br /><searchLink fieldCode="DE" term="%22Learning+Processes%22">Learning Processes</searchLink><br /><searchLink fieldCode="DE" term="%22Knowledge+Level%22">Knowledge Level</searchLink><br /><searchLink fieldCode="DE" term="%22Foreign+Countries%22">Foreign Countries</searchLink><br /><searchLink fieldCode="DE" term="%22Evaluation+Methods%22">Evaluation Methods</searchLink><br /><searchLink fieldCode="DE" term="%22Cognitive+Measurement%22">Cognitive Measurement</searchLink>
– Name: Subject
  Label: Geographic Terms
  Group: Su
  Data: <searchLink fieldCode="DE" term="%22China+%28Shanghai%29%22">China (Shanghai)</searchLink>
– Name: DOI
  Label: DOI
  Group: ID
  Data: 10.1002/pits.22916
– Name: ISSN
  Label: ISSN
  Group: ISSN
  Data: 0033-3085<br />1520-6807
– Name: Abstract
  Label: Abstract
  Group: Ab
  Data: Considering the importance of mathematics in modern society, it is crucial to understand the cognitive processes involved in the acquisition of complex mathematical competency. As a new generation of evaluation theory, cognitive diagnosis has its unique advantages in personalized evaluation. Based on the mathematical cognitive framework of Trends in International Mathematics and Science Study (TIMSS)-2011 and the Chinese mathematics curriculum, this research has formed a mathematical competency model composed of seven cognitive attributes. Sixty-seven released mathematical items for the fourth grade in TIMSS-2011 were used as assessment tools in this study. The deterministic inputs, noisy, "and" gate model was selected as the cognitive diagnosis model in this research, and the parameters of the model were evaluated according to the response data, based on which the effectiveness of the assessment tool was further verified, forming the evaluation framework of students' mathematical competency. This framework was used to analyze the data of 573 students' mathematical competency from Shanghai, China, specifically from three aspects: attribute mastery probability, learning path, and knowledge structure. Results show that students' performance in mastering attributes of mathematical cognition is excellent on the whole; some students' learning paths are leapfrog; there are certain differences in students' knowledge structure despite that they have the same total score. This research is performed as a systematic case study of the evaluation of students' mathematical competency and also provides a new perspective in assessing other knowledge and skills.
– Name: AbstractInfo
  Label: Abstractor
  Group: Ab
  Data: As Provided
– Name: DateEntry
  Label: Entry Date
  Group: Date
  Data: 2023
– Name: AN
  Label: Accession Number
  Group: ID
  Data: EJ1386792
PLink https://search.ebscohost.com/login.aspx?direct=true&site=eds-live&db=eric&AN=EJ1386792
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  BibEntity:
    Identifiers:
      – Type: doi
        Value: 10.1002/pits.22916
    Languages:
      – Text: English
    PhysicalDescription:
      Pagination:
        PageCount: 16
        StartPage: 3135
    Subjects:
      – SubjectFull: Cognitive Processes
        Type: general
      – SubjectFull: Mathematics Skills
        Type: general
      – SubjectFull: Competence
        Type: general
      – SubjectFull: Grade 4
        Type: general
      – SubjectFull: Models
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      – SubjectFull: Student Evaluation
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      – SubjectFull: Learning Processes
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      – SubjectFull: Knowledge Level
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      – SubjectFull: Foreign Countries
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      – SubjectFull: Evaluation Methods
        Type: general
      – SubjectFull: Cognitive Measurement
        Type: general
      – SubjectFull: China (Shanghai)
        Type: general
    Titles:
      – TitleFull: Cognitive Diagnostic Analysis of Students' Mathematical Competency Based on the DINA Model
        Type: main
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            NameFull: Xu, Tianshu
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            NameFull: Wu, Xiaopeng
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            NameFull: Sun, Siyu
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            NameFull: Kong, Qiping
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          Dates:
            – D: 01
              M: 09
              Type: published
              Y: 2023
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              Value: 60
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              Value: 9
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            – TitleFull: Psychology in the Schools
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