Mathematical core attainment evaluation method based on multi-feature fuzzy rank and vector

Abstract

The invention discloses a mathematical core attainment evaluation method based on a multi-feature fuzzy rank and a vector, and belongs to the field of education evaluation and education data mining. The method comprises the following steps: constructing a high school mathematics core accomplishment index system which comprises a plurality of first-level indexes and second-level indexes; multiple examination data of the students are collected, the score rate of each student on each secondary index is calculated, multi-feature data are extracted, and the multi-feature data comprise the minimum value, the mean value and the maximum value of the score rates; performing normalization processing on the multi-feature data to obtain a normalized feature quantity; on the basis of the normalized feature quantity, calculating a fuzzy rank vector of each student under each feature quantity, the fuzzy rank vectors representing membership degrees of the students on different sorting results; and calculating a single-feature fuzzy rank sum vector and a multi-feature fuzzy rank sum vector. According to the method, by constructing the multi-feature data model and introducing the fuzzy rank vector sorting method, the scientificity and accuracy of high school mathematics core attainment evaluation are effectively improved.

Description

Technical Field

[0001] This invention relates to educational assessment and educational data mining, and in particular to a method for assessing mathematical core competencies based on multi-feature fuzzy rank sum vectors. Background Technology

[0002] In the context of current high school education reform, core competencies have become an important direction for curriculum reform and teaching evaluation. The "General High School Mathematics Curriculum Standards (2017 Edition, Revised 2020)" clearly states that mathematics teaching should focus on cultivating students' core mathematical competencies to achieve their comprehensive and individualized development. Therefore, how to scientifically and fairly evaluate students' achievements in core mathematical competencies has become a key issue in teaching reform and educational assessment. Especially against the backdrop of the continuous evolution of big data and educational measurement methods, developing a mathematical core competency evaluation technology that combines objectivity, comprehensiveness, and accuracy is of great significance for promoting high-quality educational development.

[0003] Existing research on core competency assessment mainly focuses on constructing evaluation index systems and using traditional methods such as weighted averages and rank-sum ratios to rank students. These methods are usually based on a student's score in a single dimension, such as the mean score, ignoring the volatility and extreme value characteristics of scores, and failing to comprehensively reflect the stability and potential of students' abilities. Furthermore, most existing ranking methods employ hard ranking (i.e., uniquely determining a rank), which can easily lead to misjudgments when students' scores are similar or their performance is variable. In addition, while some studies have attempted to introduce fuzzy mathematics methods to handle uncertainty, they lack systematic modeling based on the multidimensional characteristics of students' data. Therefore, current methods generally suffer from three shortcomings: first, the evaluation data dimensions are too singular to comprehensively reflect student characteristics; second, the ranking method is rigid and fails to reflect the uncertainty between rankings; and third, there is a lack of mechanisms for the fusion and analysis of multi-feature data. These shortcomings limit the scientific rigor and discriminative power of the evaluation methods. Summary of the Invention

[0004] Purpose of the Invention: The purpose of this invention is to provide a method for evaluating core competencies in high school mathematics based on multi-feature fuzzy rank vectors. This method first constructs six core competencies and their subordinate secondary indicators according to the mathematics curriculum standards, collects data from multiple student exams, and constructs a multi-feature dataset using the minimum, average, and maximum score rates. Then, fuzzy rank vectors are used to represent the students' ranking and membership degrees on the indicators, and the multi-feature fuzzy rank vectors are further integrated to form a comprehensive evaluation result. This method takes into account the stability, baseline, and potential of students' abilities, overcomes the limitations of traditional evaluation methods in ranking expression and information utilization, and has stronger discriminative power and fairness.

[0005] Technical solution: A mathematical core competency assessment method based on multi-feature fuzzy rank sum vectors, including the following steps: Step 1: Construct a core competency indicator system (1) Source of indicators According to the "General High School Mathematics Curriculum Standards (2017 Edition, Revised in 2020)," six core competencies are selected as indicators: mathematical abstraction, logical reasoning, mathematical modeling, intuitive imagination, mathematical operations, and data analysis. These indicators are collectively referred to as primary indicators.

[0006] (2) Detailed indicators Based on the six core competency indicators, this invention further refines each of the six primary indicators into two secondary indicators, such as... Figure 1 As shown. The mathematical symbols for the 6 primary indicators and 12 secondary indicators are as follows: Figure 5 As shown.

[0007] Step 2: Constructing Feature Data (1) Quantification of indicators Collect student answer data through periodic exams (such as monthly exams and midterm exams); establish a "question-indicator" mapping table to assign each question to the corresponding indicator; calculate the student's score rate under each indicator, where the score rate is equal to the student's actual score on the corresponding question for a certain indicator divided by the full score of the corresponding question for that indicator.

[0008] For example, assuming a total of 10 tests, student T1's actual score, full score, and score rate for each test on the "Conceptual Comprehension Ability" indicator are as follows: Figure 6 As shown.

[0009] (2) Selecting feature quantities Descriptive statistical analysis methods are used to calculate the minimum, mean, maximum, and variance of a series of student scores on various indicators. Assume there are n students and m indicators, where the number of secondary indicators involved in this invention is m=12. The d-th characteristic of the j-th student on the i-th secondary indicator is denoted as... Without loss of generality, this invention selects the minimum, mean, maximum, variance, and median of the score rate as the characteristic quantities of students on a certain indicator, respectively, and their corresponding mathematical symbols are: , , , , .

[0010] Step 3: Calculate the fuzzy rank vector of the feature quantities (1) Data normalization The Min-Max normalization method is used to normalize the student's features. (d=1,2,3,4,5) normalized to [0,1].

[0011] Where n represents the number of students; m represents the number of indicators, and in this invention, the number of secondary indicators m=12; d represents the feature number, d=1 corresponds to the minimum value, d=2 corresponds to the mean, d=3 corresponds to the maximum value, d=4 corresponds to the variance, and d=5 corresponds to the median; feature quantity Let d represent the d-th feature of the j-th student on the ith secondary indicator; This represents the normalized characteristic.

[0012] (2) Definition and calculation of fuzzy rank vector Define fuzzy rank vector Traditional rank-sum comparisons perform a "hard" ranking of students' indicator data. For example, if student T1 ranks 3rd in the "conceptual comprehension ability" indicator among 5 students, then student T1's rank in the "conceptual comprehension ability" indicator is 3. However, the ranking (rank) of students in the indicators has a certain degree of uncertainty. Therefore, the "traditional rank" is extended to "fuzzy rank vector".

[0013] This invention believes that student T j Feature quantity The ranking is fuzzy. Let's take the ranking of student T1 on the "conceptual comprehension ability" indicator as an example to explain fuzzy rank vectors. Assume we choose the minimum score as the feature, and there are 5 students. The normalized feature data for these 5 students is as follows: Figure 7 As shown.

[0014] Because there are 5 students, the possible ranking results for each student are 1, 2, 3, ..., 5, and the ideal values ​​corresponding to these 5 ranking results are 1, 0.8, 0.6, 0.4, and 2, respectively. Student T1's feature value for the "conceptual understanding ability" indicator is 0.29. For ranking result 1, the membership degree of student T1 to ranking result 1 on the "conceptual understanding ability" indicator is calculated as 0.29 using min(0.29, 1.00) / max(0.29, 1.00) = 0.29; for ranking result 2, the membership degree of student T1 to ranking result 2 on the "conceptual understanding ability" indicator is calculated as 0.36 using min(0.29, 0.80) / max(0.29, 0.80) = 0.36; for ranking result 3, the membership degree of student T1 to ranking result 2 on the "conceptual understanding ability" indicator is calculated as min(0.29, 0.60) / max(0.29, 0.60) = 0. The membership degree of student T1 in the "conceptual understanding ability" indicator, belonging to ranking result 3, is calculated to be 0.48. For ranking result 4, the membership degree of student T1 in the "conceptual understanding ability" indicator, belonging to ranking result 4, is calculated to be 0.73 using min(0.29,0.40) / max(0.29,0.40)=0.73. For ranking result 5, the membership degree of student T1 in the "conceptual understanding ability" indicator, belonging to ranking result 4, is calculated to be 0.69 using min(0.29,0.20) / max(0.29,0.20)=0.69. Finally, the membership degree vector of student T1 in the "conceptual understanding ability" indicator across the five rankings is (0.29,0.36,0.48,0.73,0.69).

[0015] In the traditional rank-sum ratio, the ranking of student T1 on the "conceptual comprehension ability" indicator is a definite result, namely, a ranking of 4 (rank is 4). This result can be understood as a membership vector of (0,0,0,1,0), meaning that the probability of ranking 4 is 100%, and the probability of other ranking results is 0. Therefore, the traditional rank-sum ratio can be understood as a special case of the fuzzy rank vector proposed in this invention.

[0016] Calculate the fuzzy rank vector Based on the above examples, this invention proposes student T j Feature quantity A method for calculating fuzzy rank vectors.

[0017] Assume there are n students and m indicators, and let k represent the possible ranking results, k = 1, 2, ..., n; Student T j In indicators Features on The membership degree of the sorting result k; using This represents the ideal value of the sorting result k.

[0018] Student T j In indicators By combining and normalizing the k membership degrees on the vector, a fuzzy rank vector is obtained. .

[0019] Step 4: Calculate the fuzzy rank sum vectors for single and multiple features. (1) Calculate the single-feature fuzzy rank sum vector For the d-th feature, the fuzzy rank vector According to weight By performing weighted summation, we obtain a "fuzzy weighted rank sum vector of a single feature". Its formula is: in , .

[0020] (2) Calculate the multi-feature fuzzy rank sum vector Experience weight: The fuzzy weighted rank sum vector of multiple features (d=1,2,3,4,5), according to weights The weighted summation of (d=1,2,3,4,5) yields the "multi-feature fuzzy weighted rank sum vector". Its formula is: in , .

[0021] Entropy weight method dynamic weights: Calculate the weight of each student's normalized feature under the d-th feature: in, Here, n is the normalized characteristic quantity, and n is the number of students. Calculate the information entropy of the d-th feature: like ,but ; Calculate the weight of the d-th feature: satisfy .

[0022] Step 5: Comprehensive ranking based on multi-feature fuzzy rank sum vector Based on the multi-feature fuzzy weighted rank sum vector of n students ( Let D be the set of candidate student IDs, initially D = {1, 2, ..., n}.

[0023] For the student ranked first in the comprehensive ranking, the student ID s1 is calculated using the following formula: Because the values ​​of the multi-feature fuzzy weighted rank sum vector are continuous data, this invention believes that there will not be a situation where the fuzzy weighted rank sum of multiple students is simultaneously at the maximum value.

[0024] Update the set of candidate student IDs: D = D - {s1}.

[0025] For the student ranked 2nd overall, calculate the student ID s2 using the following formula: Update the set of candidate student IDs: D = D - {s2}.

[0026] Repeat the above steps until the nth iteration of the comprehensive sorting is completed.

[0027] Step 6: Ranking assessment based on multi-feature fuzzy rank sum vector The quantile method was used to determine the level thresholds, and students' core competencies were divided into four levels: excellent, good, qualified, and needing improvement. The thresholds were calibrated through large sample data (in this embodiment, the thresholds were optimized based on data from 5 students), making the evaluation results more meaningful for teaching guidance.

[0028] Beneficial Effects: This invention effectively improves the scientific rigor and accuracy of high school mathematics core competency assessment by constructing a multi-feature data model and introducing a fuzzy rank vector ranking method. Compared with traditional assessment methods based on single features and hard ranking, this invention can comprehensively reflect students' baseline abilities, stable performance, and potential development under various indicators, avoiding misjudgments caused by extreme values ​​or data fluctuations. Simultaneously, the fuzzy rank vector quantifies the uncertainty of the ranking results, making the evaluation results more flexible and discriminative, especially suitable for situations where students have similar ability levels. Furthermore, this method has good adaptability and scalability, allowing for flexible adjustment of the indicator system and feature weights according to teaching practice, and is widely applicable to educational assessment, teaching improvement, and personalized tutoring scenarios. Implementation verification shows that this invention outperforms traditional methods in terms of ranking rationality and evaluation fairness, demonstrating application value and promising prospects for promotion. Attached Figure Description

[0029] Figure 1 A diagram of the assessment index system for core competencies in high school mathematics. Figure 2 The distribution chart of the minimum characteristic index data for students T1 and T3; Figure 3 The distribution chart of the mean characteristic index data for students T1 and T3; Figure 4 The distribution chart of the maximum characteristic index data for students T1 and T3; Figure 5 A diagram showing the mathematical symbol representation of the indicator; Figure 6 The graph shows the actual scores and full marks of Student T1 in the "Conceptual Comprehension Ability" indicator; Figure 7 Normalized characteristic graphs of five students on the "conceptual comprehension ability" indicator; Figure 8 A graph showing the minimum characteristic values ​​of the core competencies indicators in high school mathematics. Figure 9 A graph showing the mean characteristics of the core competencies indicators in high school mathematics. Figure 10 A graph showing the maximum value characteristic of the core competencies indicators in high school mathematics. Figure 11 A variance characteristic graph of the core competencies indicators in high school mathematics; Figure 12 A graph showing the median characteristic of the core competencies in high school mathematics. Figure 13 Normalized data plot for minimum feature quantity; Figure 14 A normalized data plot for mean characteristic quantities; Figure 15 A normalized data plot of the maximum feature value; Figure 16 A normalized data plot of variance features; Figure 17 Normalized data plot for median characteristic; Figure 18 A fuzzy rank vector diagram of the minimum value feature quantity; Figure 19 A fuzzy rank vector diagram of the mean feature quantity; Figure 20 A fuzzy rank vector diagram of the maximum feature quantity; Figure 21 A fuzzy rank vector diagram of variance features; Figure 22 A fuzzy rank vector diagram of the median feature; Figure 23A fuzzy rank-sum vector graph of the minimum value feature quantity; Figure 24 A fuzzy rank-sum vector diagram of mean feature quantities; Figure 25 A fuzzy rank-sum vector graph of the maximum feature quantity; Figure 26 A fuzzy rank-sum vector diagram of variance features; Figure 27 A fuzzy rank-sum vector diagram of the median feature; Figure 28 A multi-feature fuzzy rank-sum vector graph (empirical weights); Figure 29 A multi-feature fuzzy rank-sum vector graph (dynamic weighting using entropy weighting method); Figure 30 This is a comprehensive sorting chart; Figure 31 The graph shows the results of the traditional rank-sum ratio calculation. Detailed Implementation

[0030] To make the technical solution of the present invention clearer, the present invention will be further described in detail below with reference to the accompanying drawings and specific embodiments.

[0031] Example Step 1: Construct a core competency indicator system The core competency indicator system of this embodiment is shown in Table 1.

[0032] Step 2: Constructing multi-feature data In this embodiment, the minimum, mean, maximum, variance, and median of the score rate are selected as the characteristic quantities of students on the indicator, as shown in the data below. Figures 8 to 12 As shown.

[0033] Step 3: Calculate the fuzzy rank vector of the feature quantities (1) Data normalization Figures 8 to 12 After data normalization, as follows Figures 13 to 17 As shown.

[0034] (2) Calculate the fuzzy rank vector in accordance with Figures 13 to 17 The feature quantity, the calculated fuzzy rank vector is as follows: Figures 18 to 22 As shown.

[0035] Step 4: Calculate the fuzzy rank sum vectors for single and multiple features. (1) Calculate the single-feature fuzzy rank sum vector Based on experience, the weights of the 12 indicators are all selected as 1 / 12, and the calculated fuzzy rank sum vectors are as follows: Figures 23 to 27 As shown.

[0036] (2) Calculate the multi-feature fuzzy rank sum vector The weights are calculated using two methods: empirical weights and dynamic weights based on entropy weights. ① Based on experience, the weights of the minimum, mean, maximum, variance, and median features are selected as 0.15, 0.4, 0.15, 0.15, and 0.15, respectively. The calculated multi-feature fuzzy rank sum vector is as follows: Figure 28 As shown.

[0037] ② Dynamic weight calculation using the entropy weight method ( =0.18、 =0.42、 =0.16、 =0.10、 =0.14), the calculated multi-feature fuzzy rank sum vector is as follows: Figure 29 As shown.

[0038] by Taking students as an example, the calculation process of multi-feature fuzzy rank sum vector under dynamic weighting of entropy weight method is as follows: Minimum feature contribution: (0.19, 0.21, 0.21, 0.20, 0.18) × 0.18 = (0.034, 0.038, 0.038, 0.036, 0.032) Contribution to mean characteristics: (0.20, 0.23, 0.23, 0.20, 0.14) × 0.42 = (0.084, 0.097, 0.097, 0.084, 0.059) Maximum feature contribution: (0.21, 0.24, 0.24, 0.19, 0.12) × 0.16 = (0.034, 0.038, 0.038, 0.030, 0.019) Variance feature contribution: (0.14, 0.17, 0.22, 0.26, 0.21) × 0.10 = (0.014, 0.017, 0.022, 0.026, 0.021) Median feature contribution: (0.21, 0.26, 0.23, 0.17, 0.13) × 0.14 = (0.029, 0.036, 0.032, 0.024, 0.018) Summing these values ​​gives (0.19, 0.23, 0.22, 0.20, 0.16), which is consistent with... Figure 29 middle The student data is consistent.

[0039] Step 5: Comprehensive ranking based on multi-feature fuzzy rank sum vector according to Figure 29 The multi-feature fuzzy rank sum vectors shown are sorted as follows: Figure 30 Because the traditional rank-sum ratio uses single-feature data as input, we select [the appropriate feature here]. Figure 14 The data is used as input for the traditional rank-sum ratio. The result is calculated using the traditional rank-sum ratio method as follows: Figure 31 As shown.

[0040] The difference between the multi-feature fuzzy rank sum vector method proposed in this invention and the traditional rank sum ratio in this case is as follows: the method of this invention results in student T3 ranking first and student T1 ranking second; while the traditional rank sum ratio results in student T1 ranking first and student T3 ranking second.

[0041] from Figure 2 , Figure 3 and Figure 4 The differences in score distribution between students T1 and T3 across the minimum, mean, and maximum feature dimensions are clearly visible. Traditional rank-sum ratio methods, primarily based on simple summation and ranking, may overestimate overall performance due to high scores in individual dimensions. In contrast, the multi-feature fuzzy rank-sum vector method employed in this invention comprehensively considers data stability and extreme value characteristics at the minimum, mean, and maximum levels, thereby enhancing the scientific rigor and discriminative power of the ranking.

[0042] exist Figure 2 In the (minimum distribution), T3 has a greater advantage over T1 in several indicators, especially in u11 and u12, where the scores are significantly higher than T1, reflecting its strong bottom-line ability. Figure 3 The mean distribution shows that T3's overall score was consistently higher than T1's from u9 to u12, indicating that its overall performance was more stable. Figure 4 In the (maximum value distribution), T3 achieved full marks in multiple indicators (such as u8 to u12), demonstrating extremely strong upper limit capabilities. In contrast, although T1 scored highly in some indicators, its minimum value distribution was low and highly volatile, and its mean was not as stable as T3.

[0043] Meanwhile, the teachers of these five students reported that student T3's core mathematical competencies were better than student T1's.

[0044] In summary, the method of this invention identifies T3 as the number 1 more reasonably than the traditional method, taking into account both extreme value ability and overall stability level, and is more fair and accurate in comprehensive evaluation.

[0045] The embodiments described above are merely illustrative of several implementations of the present invention, and while the descriptions are specific and detailed, they should not be construed as limiting the scope of the present invention. It should be noted that those skilled in the art can make various modifications and improvements without departing from the concept of the present invention, and these modifications and improvements all fall within the scope of protection of the present invention. Therefore, the scope of protection of this patent should be determined by the appended claims.

Claims

1. A mathematical core competency assessment method based on multi-feature fuzzy rank-sum vectors, characterized in that, Includes the following steps: S1. Construct a core competency indicator system for high school mathematics, including multiple primary and secondary indicators; S2. Collect students' test data from multiple exams, calculate each student's score rate on each secondary indicator, and extract multi-feature data, including the minimum, mean, maximum, variance, and median of the score rate. S3. Normalize the multi-feature data to obtain normalized feature quantities; S4. Based on the normalized feature quantity, calculate the fuzzy rank vector of each student under each feature quantity, where the fuzzy rank vector represents the membership degree of the student in different ranking results. S5. Calculate the single-feature fuzzy rank sum vector and the multi-feature fuzzy rank sum vector, and fuse the multi-feature data by weighted summation. The weights are adopted using empirical weights or dynamic weights using the entropy weight method. S6. Student comprehensive ranking based on multi-feature fuzzy rank sum vector; S7. Based on the comprehensive ranking results and the numerical distribution of the multi-feature fuzzy rank sum vector, conduct a core competency level assessment and output the evaluation results and level.

2. The mathematical core competency assessment method based on multi-feature fuzzy rank sum vectors according to claim 1, characterized in that, In step S1, the indicator system is based on the "General High School Mathematics Curriculum Standards" and includes six primary indicators: mathematical abstraction, logical reasoning, mathematical modeling, intuitive imagination, mathematical operation and data analysis. Each primary indicator is further subdivided into two secondary indicators.

3. The mathematical core competency assessment method based on multi-feature fuzzy rank sum vectors according to claim 1, characterized in that, In step S2, the score rate is calculated by dividing the student's actual score on the corresponding question of a certain indicator by the full score of the corresponding question, and the minimum, mean and maximum feature values ​​are extracted based on multiple test data. The variance characteristic is the square of the standard deviation of the multiple test scores; the median characteristic is the median of the multiple test scores arranged in ascending order, and when the sample size is even, the average of the two median values ​​is taken.

4. The mathematical core competency assessment method based on multi-feature fuzzy rank sum vectors according to claim 1, characterized in that, In step S3, the normalization process uses the Min-Max method to normalize the feature values ​​to the [0,1] interval. The calculation formula is as follows: in, Represents the original feature quantity. Let i represent the normalized feature quantity, j be the index number, j be the student label, and d be the feature quantity number.

5. The mathematical core competency assessment method based on multi-feature fuzzy rank sum vectors according to claim 1, characterized in that, In step S4, the calculation of the fuzzy rank vector includes: for the possible results k in the sorting, calculating the membership degree: in, Indicates the ideal value for sorting; The fuzzy rank vector is represented as: 。 6. The mathematical core competency assessment method based on multi-feature fuzzy rank sum vectors according to claim 1, characterized in that, In step S5, the single-feature fuzzy rank sum vector The calculation formula is: in, As the indicator weight, satisfying ; Multi-feature fuzzy rank sum vector The calculation formula is: in, For feature weights, satisfying .

7. The mathematical core competency assessment method based on multi-feature fuzzy rank sum vectors according to claim 6, characterized in that, Weight settings include two methods: (1) Experience weight: weight Set to equal weights, i.e. Where m is the number of indicators; weight Based on experience, the weights of the features of minimum, mean, maximum, variance, and median are set to 0.15, 0.4, 0.15, 0.15, and 0.15, respectively. (2) Entropy weight method dynamic weight: The weight is calculated based on the information entropy of each feature quantity to achieve objective weight allocation.

8. The mathematical core competency assessment method based on multi-feature fuzzy rank sum vectors according to claim 1, characterized in that, In step S6, the comprehensive ranking is achieved by iteratively selecting the student corresponding to the maximum value in the multi-feature fuzzy rank sum vector, specifically including: Initialize the candidate student set D = {1, 2, ..., n}; For sorting positions s (s=1 to n), calculate ,in These are the components of a multi-feature fuzzy rank sum vector; Update D = D - {s k Continue until all students have been sorted.

9. The mathematical core competency assessment method based on multi-feature fuzzy rank sum vectors according to claim 7, characterized in that, The calculation steps for dynamic weights using the entropy weight method include: (1) Calculate the weight of each student's normalized feature under the d-th feature: in, Here, n is the normalized characteristic quantity, and n is the number of students. (2) Calculate the information entropy of the d-th feature: like ,but ; (3) Calculate the weight of the d-th feature: satisfy .

10. The mathematical core competency assessment method based on multi-feature fuzzy rank sum vectors according to claim 1, characterized in that, In step S7, the quantile method is used for grading, and the maximum value component of the multi-feature fuzzy rank sum vector is divided into grades according to the following thresholds: Excellent: Maximum value component ≥ 0.22; Good: 0.18 ≤ maximum value component < 0.22; Acceptable: 0.14 ≤ maximum value component < 0.18; Needs improvement: Maximum value component < 0.14.

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