| Based on Item Response Theory(IRT)and Cognitive Diagnosis(CD)theory,we investigated the problem of plane vector problems solving at different stages of senior high school students,which included the first stage students who just finished the plane vector knowledge in first grade of high school,the second stage students who had finished plane vector knowledge for a term in the first grade of high school and the third stage students who were about to enter the stage to general review in the third grade in high school.On the basis of IRT and the G-DINA model(generalized deterministic inputs,noisy and gate model),we analyzed and found as follows:1.The probability of every cognitive attribute that the test samples mastered: The probability of the whole participants mastered plane vector “concept”,“operation” and “use” 3 attributes(0.42,0.5 and 0.21 probability)was not high enough.Especially the “use” attribute mastery probability(0.21)was the lowest.It was more difficult for subjects to master “operation” attribute than the “concept” and “use” attribute.That reflected the compensatory G-DINA model can deal with multi-level and multidimensional characteristics of cognitive attribute mastering in the comprehensive test of the plane vector problem solving,which also illustrated that plane vector “concept” and “operation” attribute played basic and premise roles for the “use” attribute.2.Subjects cognition latent classification and the probability distribution.The interrelated and interdependent relationship among the cognitive attributes of plane vector could be reflected constantly under the data analysis of G-DINA model,and the comprehensiveness and abstractness of the cognitive attributes of plane vector could be reflected constantly by the compensate and saturated characteristics of G-DINA model.“100”,“110”,“010” and“111” 4 types of potential distributions had greater probabilities to combine,which showed 3 cognitive attributes of plane vector more likely combined like these 4 combinatorial forms.And the “110” and “111” type both contained “concept” attribute,which illustrated that “concept” was the basic attribute and had interdependencies with the 2 back high level attributes.The probability distribution of “100” and “010” model was relatively large under mastering a singer attribute.The reason why “000” model had large probability(0.3)distribution was that it didn't contain any plane vector attribute and it would not be restricted by the relationships among attributes.At the same time that the master still needed to continue to strengthen in the cognitive attribute of plane vector in high school students.At last,other types of latent distribution(“011”,“101”)ratios were relatively lower.3.The subjects overall scores and cognitive model of distributions:(1)cognitive attribute mastery pattern distribution based on all kinds of scores on formal questionnaire: The same cognitive attribute master pattern subjects had different scores,the same scores subjects had different cognitive attribute master patterns.That was regarded as a very different kind of subjects evaluation method with the traditional evaluation method whose test only to give a total score.(2)The subjects overall score and cognitive attributes mastery probability distribution: The higher the scores of the subjects and the higher levels of the probability of the property.Subjects who hadthe high leveld of single attribute mastery probability may get medium scores.The same scores subjects had different cognitive attribute master patterns: Low score subjects mostly mastered a single or two attribute.Medium-scores subjects were mainly to master the single or 2 attributes.High score subjects were mainly to master the 2 attributes or master all attributes.The subjects mastered more numbers of attributes,higher scores he or she got.The subjects mastered less numbers of attributes,lower scores he or she got.These showed that analysis results of cognitive attribute master patterns based on the cognitive diagnostic model G-DINA had certain conformity with the reality of learning in mathematics.This not only fitted in with the practical experience of mathematics learning,but also satisfieded the regular patterns of cognitive development in mathematics learning.(3)All mastery patterns,average scores and the average ability value: Subjects who had the highest score not always had the highest ability value.This evaluation method was different with the traditional evaluation methods that the highest scoring subjects were regarded as the highest ability value subjects and regarded total scores as the only evaluation criteria.Mathematics abilities of “111” mastery pattern senior high school students needed to be further improved.(4)Subjects mastery patterns reaction probability and probability of items correct answer based on formal test questions(G-DINA model): Various cognitive attributes of questions had different contribution rates to items correct answer probability,and the subjects who mastered part of plane vector cognition attributes also had some probability of items correct answer.Meanwhile,all mastery patterns reaction probability reflected the G-DINA model not only contained all the single attribute parameters,but also contained the interaction parameters among multiple attributes.This was also consistent with the theoretical basis of the G-DINA cognitive model.The compensatory and saturation characteristics of the G-DINA model could also be reflected in the interactive attributes parameters of the test questions: The probability of the response of each model and the probability to answer to the question correctly could reflect the dependence and complementarity between the attributes;Under the condition of individual questions the probabilities of the trial reaction of the subjects who all mastered the attributes were lower than the subjects who belonged to single attribute mastery models;Individual item,master all the attribute pattern test reaction probability was higher than single master attributes mastery probability model,but lower than the master 2 properties under the condition of individual questions;Under the condition of individual questions guess parameters were high(individual questions of “000” master mode the participants' reaction probability was high).4.Plane vector data analysis at each stage:(1)The probabilities of mastering each attribute at each stage decreased in turn: The plane vector A1,A3 cognitive attributes of the overall average mastery probability decreased from the first stage students who just finished the plane vector knowledge in first grade of high school--to the second stage students who had finished plane vector knowledge for a term in the first grade of high school--and kept decreasing to the third stage students who were about to enter the stage to general review in the third grade in high school.The probability of mastering the knowledge attributes of the plane vector decreased with the progress of the learning stage.It also consisted with law of mathematics learning that it was more and more difficult to master the 3 attributes “concept”,“operation” and “application” and mastering the former knowledge attribute was the base of mastering the latter one.Especially A3 attribute mastery probability level decreased significantly from the first stage students who just finished the plane vector knowledge in first grade of high school to the second stage students who had finished plane vector knowledge for a term in the first grade of high school.A2 attribute mastery probability level rose firstly and then dropped,and it decreased significantly from the second stage students who had finished plane vector knowledge for a term in the first grade of high school to the third stage students who were about to enter the stage to general review in the third grade in high school.(2)The significance of probability difference in each attribute of the subjects at each stage: Most of the attributes were not significantly different to master at each stage,but A2 attribute mastery probability level had significant differences between stage 2 with stage 3(The P value is 0.022<0.05).In other words,“operation” attribute mastery probability level the third stage students who were about to enter the stage to general review in the third grade in high school was higher significantly than in the second stage students who had finished plane vector knowledge for a term in the first grade of high school.(3)Cognitive latent classification and proportion distribution of samples at various stages: From stage 1(the first stage students who just finished the plane vector knowledge in first grade of high school)to stage 2(the second stage students who had finished plane vector knowledge for a term in the first grade of high school)and to stage 3(the third stage students who were about to enter the stage to general review in the third grade in high school)that “101”,“011” and “111” attribute master pattern were tested respectively in the proportion distribution reduced gradually.From the data,it was also shown that with the promotion of the learning stage before the review of the college entrance examination,numbers of subjects that mastered all 3 attributes of plane vector “111” the proportion decreased obviously,most subjects mastered 2 attributes also reduced.The “000” and “100” master pattern subjects proportions decreased first and then increased.The subjects proportions of the “000” mastery mode greatly increased from stage 2 to stage 3.There was a slight rise first and a large drop of the average ability value proportion of the “000” mastery mode subjects from stage 1 to stage 2 to stage 3.The average ability level of the plane vector problem solving of senior high school students in senior high school before college entrance examination had been greatly reduced.(4)The significant difference in the overall ability levels of the subjects at each stage: The overall ability level differences between stage 2 and stage 3 were very significant(p=0.003<0.05),that was subjects ability level in stage 2(the average ability level value of this stage subjects was 0.202)was higher significantly than stage 3(the average ability level value of this stage subjects was-0.217).(5)The significant difference in ability levels of the subjects that in various mastery patterns at each stage: the differences of ability levels between the various phases of each latent classification type were not significant.(6)The latent classification and the proportion distribution in the subjects of of the liberal arts and science students: “000”(44.44%),“010”(16.67%),“100”(12.96%)and “111”(9.26%)master mode proportion distributed more among the subjects of the liberal arts.The largest proportion of which was “000” mastery pattern,the smallest proportion was “101”(1.85%)and “011”(1.85%)mastery pattern.That was most cognitive attribute three plane vector have not mastered.The proportions of simultaneously mastered the first and second attributes and simultaneously mastered the second and third attributes were both small.Among science subjects,there were more distribution proportions in “000”(23.40%),“100”(12.77%),“010”(21.28%)and “110”(24.47%)mastery patterns,whose “110” had the largest distribution proportions and “101” had the least distribution proportions.(7)The significance differences conditions of the mastery patterns and ability levels between subjects of liberal arts and science subjects: There were significance differences in mastering A1 attributes between subjects of liberal arts and science subjects in stage 3(p=0.000<0.05 and p=0.000<0.05),and there were significance differences in mastering A2 attributes between subjects of liberal arts and science subjects in stage 3(p=0.000<0.05 and p=0.000<0.05),what's more,the ability levels between subjects of liberal arts and science subjects at stage 3 also reflected significance differences(p=0.001<0.05).By comparing and analyzing data of subjects of liberal arts at stage 2 and stage 3,there were significance differences both in mastering A1 attribute and A2 attribute between 2 stages(p=0.000<0.05 and p=0.000<0.05),and subjects ability levels had significant differences between 2 stages,too.(p=0.001<0.05).By comparing and analyzing data of science subjects at stage 2 and stage 3,there were significance differences in mastering A3 attribute in both 2 stages(p=0.048<0.05).By comparing and analyzing data of subjects at stage 2 and subjects at stage 3,there were significant ability levels stage differences between 2 stages subjects.At stage 3 when the third stage students who were about to enter the stage to general review in the third grade in high school,and there were significantly higher mastery levels in mastering A1 attribute and mastering A2 attribute of science subjects than subjects of liberal arts,and there were significantly higher ability levels of science subjects than subjects of liberal arts.However,A1 and A2 attributes mastery possibility of subjects of liberal arts at stage 2 was significantly higher than A1 and A2 attributes mastery possibility of subjects of liberal arts at stage 3,and ability levels of subjects of liberal arts at stage 2 was significantly higher than ability levels of subjects of liberal arts at stage 3.Based on the above data information,we could conclude that:1.The difference in the contribution of different cognitive attributes to the same problem was significant:(1)Attribute dependency relation was obvious.(2)Students who mastered only part attributes also have some probability to answer questions correctly.(3)The probability to master attribute was not high.2.The relationship between mastering knowledge attributes and problem solving ability was the nonlinear.3.Differences in mastering attributes of students in various stages were existed,and these were existed between science students and students of liberal arts.4.The subjects of the same total score or the same ability had the difference in the mastery patterns.Further put forward the thinking of teaching: 1.We should pay attention to knowledge dependent learning.2.Strengthen the cultivation of comprehensive problem solving ability.3.Perfecting intensively the knowledge structure of the students.4.Pay attention to the teaching based on the zones of proximal development in different kinds of high school students:(1)Strengthen the mastery levels of the “000” mastery patterns subjects.(2)Remedy knowledge pertinently for subjects with only mastered part of the knowledge attributes.(3)We should attach importance to the cultivation of the strict and flexible mathematical thinking of high school students with all attributes. |
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