Research on Organization of Mathematics Knowledge in Good Mathematical Cognitive Structure

This research adopted the method of network data analysis, chose 31 senior high school first-grade outstanding students as participants and 44 concepts about trigonometric function as materials, analyzed the organization of mathematics knowledge in good mathematical cognitive structure (GMCS) with the help of software Ucinet. The results indicated: (1) The connections between the concepts in GMCS were more extensive, especially those connections with higher tightness degree. (2) Most connections in GMCS were direct. (3) There were more abstract concepts as accumulation points connecting with others. (4) There were a number of concrete and frequently used concepts connecting with others directly. Therefore, the mathematics teachers should help students to construct extensive and direct connections between mathematics knowledge in their mind. These new findings expanded and deepened the current research about mathematical cognitive structure(MCS), pointed out the direction and target for educators helping their students form GMCS. Meanwhile, this research demonstrated the network data analysis method was feasible and valuable to analyze mathematics psychological issues.


INTRODUCTION
Mathematical cognitive structure (MCS) was a specific concept presented by mathematical education researchers in the late 19th century based on the concept of cognitive structure in psychology (Cao and Cai, 1989;Wu and Guo, 1997;Tu, 2003;Yu, 2011;Alison et al., 2013).It was an internal hypothetical structure and used to represent the storage and the organization of mathematics knowledge (Bruner, 1989;Cao and Cai, 1989;Li, 2001;Tsai, 2001;He, 2002;Yu, 2004).Much mathematical education research had demonstrated the MCS formed in the process of mathematics learning and played a vital role in individual mathematical activities (Ausubel, 1963;Ausubel, 1994;Zhang, 2003;Yu, 2004;Sofia et al., 2017).It could affect individual not only to understand and master mathematics knowledge but also to apply mathematics knowledge, etc. (Skemp, 1971;Yang, 1993;Dixon, 2005;Zhang, 2007;Zhang, 2012).Therefore, in the process of mathematical teaching, almost all teachers expected to help the students to form a good mathematical cognitive structure (GMCS) (Wilkerson-Jerde & Wilensky, 2011).To achieve this goal, numerous related research was conducted in recent years and obtained fruitful results.
Regarding the content of GMCS, Guan Peng and He Xiaoya thought GMCS should contain massive and varied basic mathematics knowledge, especially the knowledge about the production and the problem-solving strategy (Guan, 1998;He, 2002).The so-called production was a string of knowledge made up of the conditions and the related action.It was a program that some kinds of behaviors happen when certain conditions were given (He, 2002;Kong and Zeng, 2009).The problem-solving strategy referred to the general guideline and idea which an individual could use to understand conditions, select methods and determine the steps in the process of problemsolving (Li, 1998).Wang Wenjing and Zheng Yanping thought GMCS should include the representation system of relations and the representation system of ideas.The relations representation was awareness of relations of mathematical knowledge, and the idea representation referred to the understanding of the cause of the relations between knowledge (Wang and Zheng, 2008).Mi Meng and Hou Wansheng proposed that GMCS should include much meta-cognitive knowledge (Mi and Hou, 2011).
As for the characteristics of knowledge in GMCS, Jin Qifeng, Wang Guangming, Han Bing and Wang Yue thought it should be reasonable, clear, individualistic, interconnected and flexible (Jin, 2002;Wang and Wang, 2004;Han and Wang, 2005).Wang Wenjing, Zheng Yanping, Jin Xiaofeng, Yan Zhengxiang, Huang Decheng, Han Bing and Wang Yanwen held the idea it should be stable and firm (Wang and Zheng, 2008;Jin, 2011;Yan and Huang, 2005;Han and Wang, 2005).Mi Meng, Hou Wansheng, Yan Zhengxiang and Huang Decheng thought it should be given a high-level "internalization" and deep understanding (Mi and Hou, 2011;Yan and Huang, 2005).Wang Guangming and Wang Yue thought the correlation density of knowledge in GMCS was larger; Different representation forms of the same mathematical knowledge were interlinked (Wang and Wang, 2004).Chen Jie thought it had the following three characteristics: (1) It could absorb new knowledge quickly; (2) It could provide the knowledge timely; (3) It could create new knowledge (Chen, 2003).
When it came to the organization of knowledge in GMCS, many scholars considered that the contents were organized in the form of network (Papert, 1993;Guan, 1998;Li, 2001;He, 2002;Wilkerson-Jerde & Wilensky, 2011).Besides, Yan Zhengxiang and Huang Decheng thought that GMCS was ordered and well organized (Yan and Huang, 2005).Zhao Chunxiang and Jin Xiaofeng believed that the knowledge in GMCS could be divided into two parts, the first part and the second part.The knowledge belonging to the first part was general, and those belonging to the second part was the concrete (Zhao, 2013;Jin, 2011).Mi Meng and Hou Wansheng believed that the knowledge which had the highest degree of the inclusive level were in central position (Mi and Hou, 2011).Wang Wenjing and ZhengYanping thought that the organization of GMCS was a pattern of "standard pyramid" formed according to the level of the abstraction of mathematical knowledge.The most abstract and general knowledge was at the top of the "standard pyramid" while the subordinate knowledge was at the bottom (Wang and Zheng, 2008).Yu Ping proposed that mathematics knowledge in GMCS should be organized together based on the knowledge field and knowledge system.The so-called knowledge field was a knowledge network, in which the knowledge connected with each other according to their equivalence relations; the so-called knowledge system was a series of knowledge which was formed according to their relations about abstraction or deduction (Yu et al., 2011).
Reviewing these research, it could be seen many aspects of GMCS had been discussed and obtained fruitful results.However, there were still some aspects of GMCS need to be explored in depth, such as its density, subgroups, and centrality.Therefore, instead of using the qualitative method which was mostly adopted by the previous research, the further research which is conducted with the quantitative method which can make the research about GMCS to be more clear and precise is necessary and valuable (Lu and Yu, 2010;Sun and Yang, 2015).This research explores the organization of mathematics knowledge in GMCS with the quantitative method, focuses on the research question "what is organizational characteristics of the mathematics knowledge in GMCS?"

THEORETICAL CONSIDERATIONS
The MCS was an internal structure about mathematics knowledge (Bruner, 1989;Cao and Cai, 1989;Li, 2001;Tsai, 2001;He, 2002;Tu, 2003;Yu, 2004).It could be described as a network consisted of many nodes and their connections.The nodes were the mathematics concepts, signs, figure, formula, axiom, theorem and their properties (Papert, 1993;Wilkerson-Jerde & Wilensky, 2011).Their connections were the relations between nodes (Guan, 1998;He, 2002).The MCS could be divided into three basic forms: linear structure, tree structure and network structure.These three basic structures could combine each other and construct a Three-dimensional and synthetical structure (Li, 2001).In which the quantity of nodes and the connections and the tightness of connections and the way of connections varied from person to person.The quantity of nodes and connections contained in mathematics gifted students' MCS was larger (Wang and Wang, 2004).In which the distribution of nodes was also uneven, that was the MCS could be divided into many respectively centralized substructures which formed around one or several

Contribution of this paper to the literature
• With the help of network data analysis method, this research analyzed the organization characteristics of mathematics knowledge in GMCS quantitatively.
• The results indicated the connections of mathematics knowledge in GMCS were more extensive and most of them was direct.There were more abstract knowledge connecting with others and a number of concrete and frequently used concepts connecting with others directly in GMCS.
• Mathematics teachers should help students to construct extensive and direct connections between mathematics knowledge in their mind, attach importance to the frequently and widely used mathematics knowledge and help students to construct the direct connections between this kind of mathematics knowledge and others.important mathematics concepts or theorems (Yu, 2004).As for the organization of these nodes, some researchers referenced the perspectives of cognition psychologist about the cognitive structure and believed they could be classified many different layers (Wo, 2000;Wang and Zheng, 2008).The abstract mathematics knowledge was on the top, and the concrete mathematics knowledge was on the bottom.However, the other researchers did not believe it (Zhang and Chen, 2000).
As for the method to probe the MCS, the concept map method was considered as the most convenient and effective commonly at present (Novak, 1998;Liu, 1994;Ruiz-Primo & Shavelson, 1997), because it could show clearly not only the quantity of knowledge and their connections, but also the integral organization of all knowledge (Preece, 1976;Zhang and Chen, 2000;Tsai, 2001;Ifenthaler et al., 2001).However, it was obviously inconvenient to conduct quantitative analysis (Zhang and Chen, 2000;Ifenthaler et al., 2001).So the researchers developed some new methods, such as flow-map method (Anderson and Demetrius, 1993) , the technique of "establishment by the subject" (Liu, 1994;Enger, 1996;Ruiz-Primo & Shavelson, 1997;Ruiz-Primo et al., 2001;Lavigne, 2005;Shavelson et al., 2005;Muller et al., 2008;Lindstrom & Sharma, 2011), the method of "Pathfinder Associative Networks" (Casas et al., 2011;Geeslin & Shavelson, 1975;Jonssen et al., 1993;Fenker, 1975;Preece, 1976;Wainer & Kaye, 1974), and the method with HIMATT (Ifenthaler, 2011).After verifying by practice, it could be found these new methods were feasible, they promoted the quantitative research to cognitive structure and made the understanding of MCS to be more distinct and profound.
Recently, a new method, that was network data analysis method, was introduced in research of psychology to help the related quantitative analysis (Xu et al., 2011).This method used to analyze social network characteristics based on the calculation of data relation, such as density, centrality, and subgroups (Liu, 2009;Luo, 2010).The reason why it was introduced into research about psychology was that lots of psychological relation networks were similar with the social relation networks.Indeed, excepting the actual meaning of content involved in the psychological networks and social networks, the abstract connections which need to be discussed in two different networks were same (Ma et al., 2011).Additionally, it had been verified to be feasible and effective to research psychological problems (Hou et al., 2014).Therefore, when discussing the organization of mathematical knowledge of MCS, this method should be feasible and valuable, and it will bring us more new discoveries.

Participants
We chose 31 first-grade senior high school outstanding students in mathematics learning respectively coming from two key senior high schools in Jinan city and two key senior high schools in Zibo city of Shandong province in China as participants.The reason why we chose them as participants were the primary students did not learn any knowledge about trigonometric function.When they were in junior high school, they were just arranged to learn some elementary concepts of trigonometric function, such as sine and cosine (Ministry of Education of the people's Republic of China, 2012).Only the senior high school students learn the 44 concepts of trigonometric function (Ministry of Education of the people's Republic of China, 2003).The reason why we chose outstanding students was their MCS were usually thought as the good (Tsai, 2001, Wang andWang, 2004).
The reason why we chose students from the above four schools was the key senior high school in Jinan city and Zibo city had admittedly a higher level of mathematical education in China.The outstanding students in mathematics learning chosen from the above four schools had great representation.These students included 19 female students and 12 male students.Their average age was 16.12 ( = 0.11).
The standards we adopted for selecting students were the most popular one in mathematical education currently, which focused on the students' performances in mathematical learning and their daily achievements.The outstanding students' mathematical learning was usually positive, stability, effective and efficient, their daily achievements were usually high (Maker, 1981;National Council of Teachers of Mathematics, 2000;Johnson, 2000).
The process that the outstanding students were selected was as follows: (1) selected a class randomly in each school mentioned above firstly.(2) selected the top 15 of students of that class according to the scores of the end examination of last semester secondly (15 was approximately the one-third of the number of all students in a class in Chinese senior high school.Thus, 60 students were selected totally).(3) analyzed these students' rankings among all students in their school, deleted the students whose ranking was not in top 10% of all students in their school.(13 students were deleted.)(4) analyzed the rest students' rankings at each chapter examination to identify their stability of learning.When the students started the chapter of trigonometric function, they had learned 7 chapters of the textbook and had 7 chapter examinations totally.After that, we deleted the students who were not always on top 15.(12 students were deleted).( 5) analyzed the each student's daily performance of the rest students, deleted the students without enthusiasm for mathematics learning.(4 students were deleted.)

Materials
We chose all 44 mathematics concepts related to trigonometric function from the most popular senior high school mathematics textbook in China published by people's education press as research materials.The reason why we chose these concepts was their relations between each other were abundant, which could help us to find the organizational characteristics of MCS not only easily but also clearly.The reason why we chose a particular group of mathematics concepts as the material was that it can make the research more careful and in depth (Sofia et al., 2017).

Data Collection
This research was carried out after the students had finished the chapter of trigonometric function and learned all 44 concepts above.The process of data collection was as follows: (1) encouraged the students to recall all concepts about trigonometric function at first; (2) presented above 44 mathematical concepts to students to help them have a clear understanding of these concepts; (3) let each student draw a map of relations between these concepts according to his or her own understanding; (4) asked students to indicate the connection tightness degrees of each relation respectively with integer numbers 1 to 5 from the weakest to the strongest.
The reason why we encouraged students to recall at first and then provided students with all 44 concepts in this period was the focus of this research was the organization of knowledge in GMCS, we expected to know all relations of these concepts, so that we could find out their organizational characteristics in students' MCS instead of their storage characteristics.
In order to collect the data expediently and effectively, with the agreement of all students' parents and the help of their mathematics teachers, we took the way of cluster sampling then.There were totally 213 first-grade students in four classes, including 96 male students and 117 female students, 31 outstanding students, 127 medium students, and 55 general students.The general students were usually ranked behind 15 in chapter examinations, its selection process was similar with that of the outstanding students.Excepting the outstanding students and general students, the rest was regarded as medium students.
We received 213 concept relation maps at last.After rejecting 7 unclear and disorder sets (they all were general students' maps), finally we got 206 concept relation maps that were suitable for further analysis.The reason why the maps of medium and general students were also reserved was that we planned to compare their maps with the outstanding students' maps to find more characteristics of GMCS.
What needs to be declared is this research was carried out in accordance with the recommendations of "The guidelines of the International Committee of Medical Journal Editors and the Adolescent Mental Health Specialized Committee of Chinese Mental Health Association" with written informed consent from all parents of participants.All parents of participants gave written informed consent in accordance with the Declaration of Helsinki.Meanwhile, the ethics and academics committee of the School of Psychology at Shandong Normal University approved this research.

Data Analysis
To identify the characteristics of GMCS, we extracted the number representing the connection tightness degrees of concepts from the 206 concept relation maps mentioned above at first, transformed them into one-mode multivalued relations matrices, and calculated the average value matrices of three different level students respectively.Thereout, we analyzed the overall density, subgroups, ego density and ego centrality of their MCS with the help of network analysis software Ucinet6.0.We believe these four aspects were necessary and valuable to access to characteristics of GMCS.

The Overall Density of MCS
The overall density of a network was the ratio of numbers of actual relations to numbers of theoretical relations between members in the network.Its calculation formula was /( • ( − 1)/2) ( was the number of actual relations,  was the number of theoretical relations).So it could help researchers to find out the relative quantity of relations in a network (Liu, 2009;Luo, 2010).
To find out the differences precisely of the overall density of the networks of MCS of different level students, we picked up the maximum values respectively from three average matrices and calculated their average value firstly, that was 1.77.With 0, 1.77 and the quartile of 1.77, namely 0.4475, 0.875 and 1.3225 as equidistant critical values, we input the three average value matrices into Unciet6.0and calculated overall density values in above 5 cases of three average value matrices by executing the command "Network→Cohesion→Density overall".
When the critical value was 0, all relations would be considered.When the critical value was 0.4475, all relations which connections tightness degree were less than 0.4475 were ignored.When the critical value was 0.875, all relations which connections tightness degree were less than 0.875 were ignored.When the critical value was 1.3225, all relations which connections tightness degree were less than 1.3225 were ignored.When the critical value was 1.77, all relations which connections tightness degree were less than 1.77 were ignored.After calculating, the results were as shown in Table 1.
From Table 1, it could be seen that no matter which the critical value was, the overall density value of the network of outstanding students' MCS was greater than that of the network of medium students and general students' MCS.When the critical value was 0.4475, the density value of the network of outstanding students' MCS was 2 times as big as that of medium students' MCS, the density value of the network of medium students' MCS was 1.5 times as big as that of general students' MCS.When the critical value was 1.3225, the density value of the network of outstanding students' MCS was 30 times as big as that of medium students' MCS, the density value of the network of medium students' MCS was 2 times as big as that of general students' MCS.So the connections between above 44 concepts in the outstanding students' MCS should be more than that in the medium students' MCS and general students' MCS, especially the connections with higher tightness degree.

The Cohesive Subgroups of MCS
The cohesive subgroup of a network was a subset formed by some members of the network by connecting with each other directly.It was a maximal complete sub-graph and what it reflected was the uniformity and local grouping characteristics of the network (Liu, 2009;Luo, 2010).To understand the situation of cohesive subgroups in detail, we input three average value matrices into Ucniet6.0and calculated three times respectively according to the critical value 1, 2, 3 by executing the command "Network→Regions→Components→Simple graphs".The reason why we chose these three numbers as critical values is the maximum value of all tightness degree was 3.35, the minimum value of that was 0. When the critical value was 1, all relations which connections tightness degree were less than 1 were ignored.When the critical value was 2, all relations which connections tightness degree were less than 2 were ignored.When the critical value was 3, all relations which connections tightness degree were less than 3 were ignored.After calculating, the numbers of cohesive subgroups found and the numbers of subgroups whose number of concepts were more than 3 were as shown in Table 2.It could be seen from Table 2, the numbers of cohesive subgroups in MCS of outstanding students were always less than or equal to the numbers of cohesive subgroups in MCS of other two kind students.The cohesive subgroups whose number of concepts was more than 3 always existed in MCS of outstanding students, while in MCS of medium and general students it did not.The MCS of outstanding students always could be divided into several bigger subgroups and some small subgroups.While the MCS of medium and general students did not.When the critical value was 1, a cohesive subgroup whose number of concepts was 32 appeared in in MCS of outstanding students, it contained 72.7% of all concepts.The largest cohesive subgroups in MCS of medium students only contained 6 concepts, no cohesive subgroup whose number of concepts was more than 3 existed in MCS of general students.It seemed that most concepts in MCS of outstanding students were connected each other directly.While in MCS of other students, the concepts connected each other directly not much.

The Ego Density of Members in MCS
The ego density of members in a network referred to the number of relations that formed by this member connecting others of the network.What it revealed was who was the accumulation point in the network (Liu, 2009;Luo, 2010).We input three average value matrices into Unciet6.0and calculated the ego density of members in MCS of three different level students by executing the command "Network→Ego-networks→Egonet basic measures".After calculating, all values obtained of ego density were as shown in Table 3.
From Table 3, it could be seen that the values of ego density of concepts in MCS of outstanding students were generally bigger than that of concepts in MCS of medium students, and that of concepts in MCS of medium students was bigger than that of concepts in MCS of general students.Indeed the average value of the values of ego density of concepts in MCS of outstanding students was 20.545, while the average values of values of ego density of concepts in MCS of medium and general students were respectively 18.409 and 11.091.There was a significant difference between the values of ego density of concepts in MCS of outstanding and medium students and that of ego density of concepts in MCS of general students (t=8.227, p=0.000; t=5.931 p=0.000).This should indicate the connections of concepts in MCS of outstanding students were more extensive than that in MCS of other students again.
Additionally, it could be found that the concept which had the biggest value of ego density in MCS of all students was always Trigonometric function.Just the biggest values of ego density in MCS of three level students were different, they were respectively 38, 43 and 28.That meant the concept of Trigonometric function was the prominent accumulation point in MCS, it should be in the central place of whole MCS.Considering the Trigonometric function was abstract and of the highest generalization level concept among 44 concepts above, it seemed that the abstract concept with the highest generalization level was the most important in MCS, it connected with other concepts extensively.
It was noteworthy there were additional three concepts whose values of ego density were close to the value of Trigonometric function in MCS of outstanding students, they were Tangent function, Cosine function, Sine function, their values of ego density differed from others' values significantly (t=-12.878, p=0.000; t=-11.754, p=0.000; t=-11.754, p=0.000).While there were additional two concepts whose values of ego density were close to the value of the Trigonometric function in MCS of medium students, they were Periodic transformation and Trigonometric function line.Their values of ego density differed from others' value significantly (t=-12.636, p=0.000; t=-11.632, p=0.000).There were no similar concepts in MCS of general students.So it seemed that there were some additional remarkable accumulation points in MCS of outstanding students beside the prominent accumulation point.

The Ego Centrality of MCS
The ego centrality of a member in a network referred to the numbers of relations that formed by this member connecting with others directly.It was different from the ego density of a member, what it revealed was the influence power or force of this member in the network (Liu, 2009;Luo, 2010).Inputting three average value matrices into Unciet 6.0, we calculated the ego centrality of the concepts in MCS of three different level students by executing the command "Network→Centrality→degree".After that, all values obtained of ego centrality of 44 concepts were as shown in Table 4.
From the Table 4, it could be seen that the values of ego centrality of concepts in MCS of outstanding students were generally greater than that in MCS of medium students and general students.Indeed, the average value of values of ego centrality of concepts in MCS of outstanding students was 14.1.The average value of values of ego centrality of concepts in MCS of medium students was 6.5842.The average value of values of ego centrality of concepts in MCS of general students was 3.9968.There was the significant difference between the values of ego centrality of concepts in MCS of outstanding students and that of ego centrality of concepts in MCS of medium and general students significantly (t=5.895, p=0.000; t=8.296, p=0.000).So, the concept in MCS of outstanding students should have more direct connections with other concepts.
It could be found that the concept of the biggest value of ego centrality in MCS of outstanding students was the Tangent function, and its value of ego centrality was 39.8.The concept with the biggest value of ego centrality in MCS of medium level students was Cosine function, and its value of ego centrality was 14.33.The concept with the biggest value of ego centrality in MCS of general students was the Trigonometric function, and its value of ego centrality was 8.5.Hence, the concepts who had the most direct connections with other concepts in MCS of outstanding and medium students were different.Comparing these three concepts with others, it could be seen, the concept of Tangent function and the concept of Cosine function were relatively concrete concepts and used frequently in student's mathematics learning, especially in students' mathematics problem solving, and their generalization level was not high.While the concept of Trigonometric function was abstract and of the highest generalization level and used seldom in students' mathematics problem-solving.So it seemed that the concrete concepts whose generalization level was not high and used frequently were the most influential in MCS of outstanding students, namely they had more direct connections with other concepts.
Additionally, the concept of the Cosine function and Sine Function in MCS of outstanding students had the higher value of ego centrality too, and the two values were close to the value of ego centrality of Tangent function in MCS of outstanding students.Their values of ego centrality differed from others ' value significantly (t=-20.331, p=0.000; t=-17.936, p=0.000).The concept of Trigonometric function, Sine Function, Functional image and Tangent function had higher values of ego centrality in MCS of medium students, the four values were close to the value of ego centrality of Cosine function in MCS of medium students.Their values of ego centrality differed from others' value significantly (t=-16.804, p=0.000; t=-14.372, p=0.000; t=--14.453, p=0.000; t=-12.02, p=0.000).The concept of the Cosine function and Tangent function in MCS of general students had the higher value of ego centrality too, the two values were close to the value of ego centrality of Tangent function in MCS of general students.Their values of ego centrality differed from others' value significantly (t=-13.875, p=0.000; t=-14.154, p=0.000).So it seemed there were a number of influential concepts in MCS of outstanding students, they all were concrete and used frequently and of slightly lower generalization level.

DISCUSSIONS
The MCS played a vital role in individual mathematical activities, almost all teachers expected to help their students form a GMCS (Skemp, 1971;Ausubel, 1994;Zhang, 2003;Dixon, 2005;Wilkerson-Jerde & Wilensky, 2011;Sofia et al., 2017).To achieve this goal, much research with various method had been conducted.Reviewing these research, it could be seen there some aspects about GMCS were still not be explored in depth (Sun and Yang, 2015).Therefore, this research chose the concepts about trigonometric function as materials, adopted the network data analysis method, analyzed the overall density, subgroups, ego density, and ego centrality of MCS of outstanding students.
From the data analyzing above, it could be seen that the overall density of MCS of outstanding students was greater than that of MCS of medium and general students, especially when only the bigger value of tightness degree of connections was considered, this situation became more clear.So the connections between above 44 concepts in the outstanding students' MCS should be more than that in the medium level students' MCS and general students' MCS, especially those connections with higher tightness degree.Because the overall density was a ratio of numbers of actual relations to numbers of theoretical relations between members in the network, what it reflected was the relative quantity of relations in a network.This situation could be found from another aspect too, that was ego density.From the analyzing above, it could be found that the values of ego density of concepts in MCS of outstanding students were generally and significantly greater than that of concepts in MCS of medium and general students.While the ego density of a member was the number of relations that formed by this member connecting with others of the network.So the connections formed by each concept with others in MCS of outstanding students should be more than the connections formed by each concept with others in MCS of medium and general students.Hence, overall, the connections in MCS of outstanding students must be more than that in MCS of medium and general students.This result obviously supported some former views (Wang and Wang, 2004;Ifenthaler, 2011).However, they were still a little different, because the former research did not explain the situation of connections with higher tightness degree.
The connections could be classified into two types, namely direct and indirect connections.What type do the connections in MCS of outstanding students belong to?From the analyzing above, it could be seen the numbers of cohesive subgroups were less and the cohesive subgroups whose number of concepts was more than 3 always existed in MCS of outstanding students.When the critical value was 1, a cohesive subgroup whose number of concepts was 32 appeared in MCS of outstanding students, it contained 72.7% of all concepts.While the cohesive subgroup of a network was formed by some members of the network by connecting each other directly and tightly.So most concepts in MCS of outstanding students should be connected each other directly, or most connections in MCS of outstanding students should be direct.This situation should be found from the ego centrality.From the analyzing above, it could be found that the values of ego centrality of concepts in MCS of outstanding students were generally greater than that in MCS of medium students and general students.While the ego centrality was the number of relations formed by this member connecting with others directly.This result obviously was different from the views of "standard pyramid" about MCS (Wang and Zheng, 2008).
Additionally, from the analyzing about ego density of concepts, it could also be found that the concept of Trigonometric function had the biggest value of ego density in MCS of all students.So it should be the concept with the most connections and was the prominent accumulation point in all MCS.Besides, it could be found there are 3 and 2 other concepts having the higher value of ego density respectively in MCS of outstanding students and in MCS of medium students, however, there was no similar concept in MCS of general students.So the MCS of outstanding students should have more abstract concepts as accumulation points to connect with other concepts.
Even the concept of Trigonometric function was the prominent accumulation point in MCS of outstanding students, it was not the most influential.Because it could be found from the analyzing about ego centrality of concepts, that the concept of the biggest value of ego centrality in MCS of outstanding students was the Tangent function.Besides, since the concept of the Cosine function and Sine function had the bigger value of ego centrality which was close to that of the Tangent function.So there were a number of influential concepts in MCS of outstanding students.Meanwhile, considering the Tangent function, the Cosine function and Sine function all were concrete and used frequently concepts, it could be seen there were a number of concrete and frequently used concepts in MCS of outstanding students connecting with other concepts directly.
The outstanding students generally were deemed to hold GMCS (Tsai, 2001, Wang andWang, 2004).Thus, the GMCS should have following characteristics: (1) The connections between the concepts in GMCS were more extensive, especially those connections with higher tightness degree.(2) Most connections in GMCS were direct.(3) There were more abstract concepts as accumulation points connecting with other concepts.(4) There were a number of concrete and frequently used concepts connecting with other concepts directly.
Thus, the mathematics teachers should help students to construct extensive connections between mathematics knowledge in their mind in the process of teaching, especially the direct connections, so as to make students form a compact and smart MCS.Meanwhile, mathematics teachers should pay attention to all abstract mathematics knowledge with higher generalization level and instruct students construct extensive connections between this kind of mathematics knowledge and others so as to form more mathematics knowledge accumulation points in their MCS.Moreover, mathematics teachers should attach importance to the frequently and widely used mathematics knowledge and help students to construct the direct connections between this kind of mathematics knowledge and others.
These new findings enriched the existed research results, expanded the research field about MCS, pointed out the direction and target for educators helping their students form GMCS.Moreover, this research demonstrated the network data analysis method was feasible and effective when it was applied in research about MCS, it could undoubtedly be used in related researches.So this research promoted researches about MCS, especially those researches about the organization of knowledge in cognitive structure.
However, this research had one limitation at least objectively, that was the participants we chose was just students in Shandong province, China, and the number of participants was only 31.A small number of participants coming from the same district would definitely affect the accuracy of the results.So we will choose more appropriate participants from different districts to do more deep and extensive research.And then we will compare and analyze all results and give a more accurate conclusion about organizational characteristics of mathematics knowledge in GMCS.
They are Periodic transformation, Acute angle, Trigonometric function value, Trigonometric function line, Tangent line, Cosine line, Sine line, Directed line segment, Tangent function, Cosine function, Sine function, Central angle, Radian, Angle system, Trigonometric function, Round angle, Quadrant angle, Terminal side, Initial line, Zero angle, Radian system, Arbitrary angle, Sinusoid, Phase, Initial phase, Cosine, Frequency, Amplitude, Definition domain, Function domain, Interval, Decreasing function, Increasing function, Even function, Odd function, Minimal positive period, Period, Periodic function, Functional image, Induced formula, Tangent function value, Cosine function value, Obtuse angle, Unit circle.

Table 1 .
The overall density values of MCS

Table 2 .
The numbers of cohesive subgroups of MCS

Table 3 .
The ego density of all concepts

Table 4 .
The ego centrality of all concepts