Research on Organization of Mathematics Knowledge in Good Mathematical Cognitive Structure
Zezhong Yang 1  
Ming Zhu 1
Zhaohua Qu 1
More details
Hide details
Shandong Normal University, CHINA
Zezhong Yang   

Shandong Normal University, No.88, Wenhuadong Rd., Lixia District, Ji’nan City, Shandong Province, 250014 Ji'nan, China
Online publish date: 2017-10-05
Publish date: 2017-11-06
EURASIA J. Math., Sci Tech. Ed 2018;14(1):291–302
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.
Alison P., Markus, G., & Alan, S. (2013). Developments in Research on Mathematical Practice and Cognition. Topics in Cognitive Science, 5(2), 224-230.
Anderson O. R., & Demetrius O. J. (1993). A Flow-Map Method of Representing Cognitive Structure Based on Respondents’ Narrative Using Science Content. Journal of Research in Science Teaching, 30(8), 953-969.
Ausubel D. P. (1963). The Psychology of Meaningful Verbal Learning. New York: Grune and Stratton.
Ausubel D. P. (1994). Educational Psychology: a Cognitive View. Beijing: The people education Press.
Bruner J. S. (1974). Beyond the Information Given. London: Allen and Unwin.
Bruner J. S. (1989). A Selection of Bruner’s Education Works. Translated by Shao, R. Zh. Beijing: People’s Education Press.
Cao, C. H., & Cai J. F. (1989). Introduction to Mathematics Education. Nanjing: Jiangsu Education Publishing House.
Casas, L., Luengo, R., & Godinho, V. (2011). Software GOLUCA: Knowledge Representation in Mental Calculation. US-China Education Review, 4, 592-600.
Chen, J. (2003). The Study of Optimizing and Perfecting Mathematics Cognition Structures of Middle School Class Teaching (Master Thesis). Jinan: Shandong Normal University.
Dixon, J. A. (2005). Mathematical Problem Solving: the Roles of Exemplar, Schema, and Relational Representations. In J. I. D. Campbell (Ed.), Handbook of Mathematical Cognition. NewYork: Psychology Press.
Enger, S. K. (1996). Concept Mapping: Visualizing Students Understanding. Paper Presentado en el Anual Meeting of Mid-South American Educational Research Association, Tuscalossa (ERIC Document Reproduction Service No ED 406413).
Fenker, R. M. (1975). The Organization of Conceptual Materials: A Methodology for Measuring Ideal and Actual Cognitive Structures. Instructional Science, 4, 33-57.
Geeslin, W. E., & Shavelson, R.J. (1975). Comparison of Content Structure and Cognitive Structure in High School Students’ Learning of Probability. Journal of Research in Mathematics Education, 12, 109-120.
Guan, P. (1998). Principles of Cognitive Psychology on Forming Good Mathematical Cognitive Structure. Theory and Practice of Education, 18(2), 40-45.
Han, B, & Wang, G. M. (2005). Role of Cognitive Structure and Reflections on It in the Process of Problem Solving. Journal of Middle School Mathematics, (6), 5-7.
He, X. Y. (2002). Instructional Strategies on Constructing well Mathematical Cognitive Structure. Journal of Mathematics Education, 11(1), 24-27.
Hou, K., Zou, H., Liu, Y., Jin, C. C., & Jiang, S. (2014). Peer Group Influence on Adolescents’ Problem Behavior: a Social Network Analysis. Psychological Development and Education, (03), 259-267.
Ifenthaler, D. (2011). Identifying Cross-domain Distinguishing Features of Cognitive Structure. Education Technology Research Development, 59, 817-840.
Ifenthaler, D., Masduki, I., & Steel, N. M. (2001). The Mystery of Cognitive Structure and How We Can Detect It: Tracking the Development of Cognitive Structures over Time. Instructor Science, (39), 41-61.
Jin, Q. F. (2002). Research on Mathematics Cognitive Structure of Students in Elementary School and Teaching Management in Classroom (Master Thesis). Changchun: Northeast Normal University.
Jin, X. F. (2011). Talk about the Establishment of Good Mathematical Cognitive Structure again. New Curriculum Research, 209, 179-181.
Johnson, D. T. (2000). Teaching Mathematics to Gifted Students in a Mixed-Ability Classroom. (Eric DIGESTE594).
Jonassen, D., Beissner, K., & Yacci, M. (1993). Structural Knowledge: Techniques for Representing, Conveying and Acquiring Structural Knowledge. Hillsdale, NJ: Laurence Erlbaum Associates.
Kong, F. Z., & Zeng, Z. (2009). Psychology of Mathematics Learning. Beijing: Peking University press.
Lavigne, N. C. (2005). Mutually Informative Measures of Knowledge: Concept maps Plus Problem Sorts in Statistics. Educational Assessment, 10(1), 39-71.
Li, M. Z. (1998). A Study on Mathematics Problem Solving Strategies and Its Training. Journal of Guizhou Normal University (Natural Science), 2, 72-76.
Li, Sh. Q. (2001). PME: Psychology of Mathematics Education. Shanghai: East China Normal University Press.
Lindstrom, C., & Sharma, M. D. (2011). Teaching Physics Novices at University: A Case For Stronger Scaffolding. Physical Review Special Topics-Physics Education Research, 7(1), 010109.
Liu, J. (2009). Lecture Notes on Whole Network Analysis. Shanghai: Truth & Wisdom Press.
Liu, X. (1994). The Validity and Reliability of Concept Mapping as an Alternative Science Assessment When Item Response Theory is Used for Scoring. Paper Presented at the Annual Meeting of the American Educational Research Association, New Orleans (ED 370992).
Lu, J., & Yu, P. (2010). Reflection of Research about the Chinese Mathematics Cognitive Structure. Journal of Mathematics Education, 19(2), 19-22.
Luo, J. D. (2010). Lecture notes on Social Network Analysis. Beijing: Social Sciences Academic Press.
Ma, Sh. Q., Jiao, C., & Zhang, M. Q. (2011). Application of Social Network Analysis in Psychology. Advances in Psychological Science, (05), 755-764.
Maker, C. J. (1981). Curriculum Development for the Gifted (Hakluyt Society third series). Aspen Publications.
Mi,M., & Hou, W. S. (2011). Cognitive Structure Affect Solving Mathematics Problem. Journal of Yan’an Vocational & Technical Institute, 1, 68-69.
Ministry of Education of the people’s Republic of China. (2003). The Standards for Senior High School Mathematics Curriculum. Beijing: People’s Education Press.
Ministry of Education of the people’s Republic of China. (2012). The Standards for Compulsory education Mathematics Curriculum. Beijing: Beijing Normal University Press.
Muller, D. A., Sharma, M. D., & Reimann, P. (2008). Raising Cognitive Load with Linear Multimédia to Promote Conceptual Change. Science Education, 92(2), 278-296.
National Council of Teachers of Mathematics. (2000). Principles and Standard for School Mathematics. Reston: Author.
Novak, J. D. (1998). Learning, Creating, and Using Knowledge: Concept Maps as Facilitative Tools in School and Corporations Mahwah. New Jersey: Lawrence Erlbaum Assoc.
Papert, S. (1993). The Children’s Machine: Rethinking School in the Age of the Computer. New York: Basic Books.
Preece, P. F. W. W. (1976). Mapping Cognitive Structure: A Comparison of Models. Journal of Educational Psychology, 68 (1), 1-8.
Ruiz-Primo, M. A., & Shavelson, R. J. (1997). Concept-map Based Assessment: On Possible Sources of Sampling Variability. U.S. Department of Education, Office of Educational Research and Improvement, Educational Resources Information Center.
Ruiz-Primo, M. A., Schultz, S. E., Li, M., & Shavelson, R. J. (2001). Comparison of the Reliability and Validity of Scores from Two Concept-mapping Techniques. Journal of Research in Science Teaching, 38(2), 260-278.
Shavelson, R. J., Ruiz-Primo, M. A., & Wiley, E. W. (2005). Windows into the Mind. Higher Education: The International Journal of Higher Education and Educational Planning, 49(4), 413‐430.
Skemp, R. (1971). The Psychology of Learning Mathematics. London: Penguin Books.
Sofia, M. V. C., Vitor, G. L., Luis, M. C. G., & Ricardo, L. G. (2017). Evaluation of Changes in Cognitive Structures after the Learning Process in Mathematics. International Journal of Innovation in Science and Mathematics Education, 25(2), 17-33.
Sun, D. D., & Yang, Z. Zh. (2015). Research on Good Mathematical Cognitive Structure in Mainland China. Advances in Social and Behavioral Sciences: Proceeding of 2015 3rd Asian Conference on the Social Sciences, 15, 216-219.
Sun, D. D., & Yang, Z. Zh. (2015). Study on Content and Organization of Mathematical Cognitive Structure in Mainland China. Advances in Intelligent Systems Research, 283.
Tsai, C. C. (2001). Probing Students’ Cognitive Structures in Science: The Use of a Flow Map Method Coupled with a Meta-Listening Technique. Studies in Educational Evaluation, (27), 257-268.
Tu, R. B. (2003). The cognition of Mathematical Teaching. Nanjing: Nanjing Normal University.
Wainer, H., & Kaye, K. (1974). Multidimensional Scaling of Concept Learning in an Introductory Course. Journal of Educational Psychology, 66, 591-598.
Wang, G. M., & Wang, Y. (2004). The Comparison of Top and Ordinary Students’ Mathematical Cognitive Structures in Senior High School, the Possible Reasons for difference, and Teaching Suggestions. Reference for Middle School Mathematical Education, (12), 1-4.
Wang, W. J., & Zheng, Y. P. (2008). The Characteristic of Good Mathematical Cognitive Structure and it’s Instructional Enlightment. Friend of Science Amateurs, 23, 113-114.
Wang, Y. W. (2013). On Cultivating Students’ Good Cognitive Structure in Advanced Mathematics Teaching. Journal of Chendu Normal University, 29(11), 119-121.
Wilkerson-Jerde, M., & Wilensky, U. (2011). How do Mathematicians Learn Math?: Resources and Acts for Constructing and Understanding Mathematics. Educational Studies in Mathematics, 78, 21-43.
Wo, J. Zh. (2000). Cognitive Structure and Information Processing. Journal of Beijing Normal University, (1), 80-86.
Wu, X. F., & Guo X. H. (1997). Mathematics Education. Wuhan: Huazhong Normal University Press.
Xu, W., Chen, G. H, Zeng, Y., & Zhang, W. X. (2011). A New Approach of Relation Research: Social Network Analysis. Journal of Psychological Science, 34(2), 499 -504.
Yan, Z. X., & Huang, D. C. (2005). How to Construct the Students’ Good Mathematical Cognitive Structure. Journal of Shenyang Agricultural University (Social Science Edition), 2, 190-191.
Yang, Q. (1993). On the Influence of Cognitive Structure on Mathematics Learning - One of the Explores about Psychological Factors that Have Influence on Mathematics Learning. Journal of Mathematics Education, (1), 66-70.
Yu, P. (2004). Influence of Self-controlling Ability and CPFS Structure on the Mathematical Achievement in High School Students. Journal of Mathematics Education, 13(1), 23-26.
Yu, P. (2004). Psychology of Mathematics Education, Nanning: Guangxi Education Publishing House.
Yu, P., Lian, S. Q., & Wo, X. H. (2011). Review on Research about Psychology of Mathematics Education for 30 Years in Mainland China. Beijing: Science Press.
Zhang, Ch. W. (2003). The Psychological Meaning of Students’ Mathematical Cognitive Structure in Mathematics Teaching. Research of Mathematics Teaching-Learning, (12), 2-4.
Zhang, J. W., & Chen, Q. (2000). Test Method of Cognitive Structure. Journal of Psychological Science, (06), 750-751.
Zhang, M. L. (2007). How the Knowledge and Cognitive Structure Promote Each Other? China Education Daily, (5), 11-23.
Zhang, Y., & Wu, J. H. (2012). The Function of Good Mathematical Cognitive Structure. Proceedings of Conference on Psychology and Social Harmony.
Zhao, C. X. (2013). Optimize and Improve the Mathematics Cognitive Structure of Students in the Inquiry Learning. Educational Practice and Research, 5, 53-55.