Conceptual and procedural angle knowledge: do gender and grade level make a difference?

  • Utkun Aydın MEF University


The study examined differences in students’ conceptual and procedural knowledge of angles among two grades and gender. Participants were 382 sixth and 376 seventh graders from a metropolitan city in [Country]. [Nation] students’ conceptual and procedural knowledge of angles declined from sixth to seventh grade. Gender differences were found for procedural knowledge, but not for conceptual knowledge. Since conceptual and procedural knowledge of angles may have significant influences on the essential subsequent topics in geometry, we need to seriously consider the implications of these gender- and grade-related differences and pay attention particularly to males in Grade 7. The patterns of [Nation] students’ conceptual and procedural angle knowledge were discussed, and educational implications were offered.

Author Biography

Utkun Aydın, MEF University
Department of Mathematics Education


1. AERA, APA, & NCME. (1999). Standards for educational and psychological testing. Washington DC: Author.
2. Baroody, A. J. (2003). The development of adaptive expertise and flexibility: The integration of conceptual and procedural knowledge. In A. J. Baroody & A. Dowker (Eds.), The development of arithmetic concepts and skills: Constructing adaptive expertise studies. Mahwah, N.J. : Lawrence Erlbaum Associates, Inc.
3. Baroody, A. J., & Gannon, K. E. (1984). The development of the commutativity principle and economical addition strategies. Cognition and Instruction, 1(3), 321−339.
4. Bartlett, M. S. (1954). A note on the multiplying factors for various chisquare approximations. Journal of the Royal Statistical Society, 16(Series B), 296–298.
5. Bartolini Bussi, M. G., & Baccaglini-Frank, A. (2015). Geometry in early years: sowing seeds for a mathematical definition of squares and rectangles. ZDM, 47(3), 391-405.
6. Battista, M. T. (1990). Spatial visualization and gender differences in high school geometry. Journal for Research in Mathematics Education, 21(1), 47-60.
7. Battista, M. T. (2007). The development of geometric and spatial thinking. In F. Lester (Ed.), Second handbook of research on mathematics teaching and learning (pp. 843-908). Charlotte, NC: Information Age Publishing.
8. Benbow, C. P., & Stanley, J. C. (1982). Consequences in high school and college of sex differences in mathematical reasoning ability: A longitudinal perspective. American Educational Research Journal, 19(4), 598-622.
9. Bielinski, J., & Davison, M. L. (1998). Gender differences by item difficulty interactions in multiple-choice mathematics items. American Educational Research Journal, 35(3), 455-476.
10. Bisanz, J., & LeFevre, J. A. (1990). Strategic and nonstrategic processing in the development of mathematical cognition. In D. F. Bjorklund (Ed.), Children’s strategies: Contemporary views of cognitive development (pp. 213–244). Hillsdale, NJ: Lawrence Erlbaum Associates, Inc.
11. Bransford, J. D., Brown, A. L., & Cocking, R. R. (2001). How people learn: Brain, mind, experience, and school. Washington, DC: National Academy Press.
12. Byrnes, J. P., & Wasik, B. A. (1991). Role of conceptual knowledge in mathematical procedural learning. Developmental Psychology, 27(5), 777−786.
13. Canobi, K. H. (2005). Children’s profiles of addition and subtraction understanding. Journal of Experimental Child Psychology, 92(3), 220-246.
14. Canobi, K. H., Reeve, R. A., & Pattison, P. E. (2003). Patterns of knowledge in children's addition. Developmental Psychology, 39(3), 521.
15. Casas-García, L. M., & Luengo-González, R. (2013). The study of the pupil’s cognitive structure: the concept of angle. European Journal of Psychology of Education, 28(2), 373-398.
16. Casey, M., Nuttall, R., & Pezaris, E. (2001). Spatial-mechanical reasoning skills versus mathematics self-confidence as mediators of gender differences on mathematics subtests using cross-national gender-based items. Journal for Research in Mathematics Education, 32(1), 28–57.
17. Cattell, R. B. (1978). The scientific use of factor analysis. New York: Plenum Press.

18. Chazan, D. (1993). High school geometry students' justification for their views of empirical evidence and mathematical proof. Educational Studies in Mathematics, 24(4), 359-387.
19. Clements, D. H., & Battista, M. T. (1992). Geometry and spatial reasoning. In D. A. Grouws (Ed.), Handbook of research on mathematics teaching and learning (pp. 420-464). New York: Macmillan.
20. Clements, D. H., Battista, M. T., & Sarama, J. (1998). Development of geometric and measurement ideas. In R. Lehrer & D. Chazan (Eds.), Designing learning environments for developing understanding of geometry and space (pp. 201–225). Mahwah, NJ: Erlbaum.
21. Clements, D. H., & Burns, B. A. (2000). Students' development of strategies for turn and angle measure. Educational Studies in Mathematics, 41(1), 31-45.
22. Cohen, J. (1998). Statistical power analysis for the behavioral sciences. Hillsdale, NJ: Erlbaum.
23. Couto, A., & Vale, I. (2014). Pre-service teachers' knowledge on elementary geometry concepts. Journal of the European Teacher Education Network, 9, 57-73.
24. De Jong, T., & Ferguson-Hessler, M. G. M. (1996). Types and qualities of knowledge. Educational Psychologist, 31(2), 105-113.
25. Devichi, C., & Munier, V. (2013). About the concept of angle in elementary school: Misconceptions and teaching sequences. The Journal of Mathematical Behavior, 32(1), 1-19.
26. Doyle, W. (1983). Academic work. Review of Educational Research, 53(2), 159-199.
27. Duatepe-Paksu, A., & Ubuz, B. (2009). Effects of drama-based geometry instruction on student achievement, attitudes, and thinking levels. The Journal of Educational Research, 102(4), 272-286.
28. Else-Quest, N. M., Hyde, J. S., & Linn, M. C. (2010). Cross-national patterns of gender differences in mathematics: a meta-analysis. Psychological Bulletin, 136(1), 103.
29. Freudenthal, H. (1983). Didactical phenomenology of mathematical structures. Dordrecht: Reidel.
30. Friedman, L. (1989). Mathematics and the gender gap: A met-analysis of recent studies on sex differences in mathematical tasks. Review of Educational Research, 59(2), 185-213.
31. Gelman, R., Meck, E., & Merkin, S. (1986). Young children's numerical competence. Cognitive Development, 1(1), 1−29.
32. Gilmore, C. K., & Bryant, P. (2006). Individual differences in children's understanding of inversion and arithmetical skill. British Journal of Educational Psychology, 76(2), 309-331.
33. Hair, J. F., Black, W. C., Babin, B. J., Anderson, R. E., & Tatham, R. L. (2006). Multivariate data analysis. Upper Saddle River, NJ: Pearson Education.
34. Hallett, D., Nunes, T., & Bryant, P. (2010). Individual differences in conceptual and procedural knowledge when learning fractions. Journal of Educational Psychology, 102(2), 395.
35. Hallett, D., Nunes, T., Bryant, P., & Thorpe, C. M. (2012). Individual differences in conceptual and procedural fraction understanding: The role of abilities and school experience. Journal of Experimental Child Psychology, 113(4), 469-486.
36. Halpern, D. F. (2000). Sex differences in cognitive abilities. Mahwah, NJ: Lawrence Erlbaum Associates.
37. Halpern, D. F., Benbow, C. P., Geary, D. C., Gur, R., Hyde, J. S., & Gernsbacher, M. A. (2007). The science of sex differences in science and mathematics. Psychological Science in the Public Interest, 8, 1–51.
38. Henderson, D. W., & Taimina, D. (2005). Experiencing geometry. Euclidean and non-Euclidean with history. New York: Cornell University.
39. Herbst, P., & Brach, C. (2006). Proving and doing proofs in high school geometry classes: What is it that is going on for students?. Cognition and Instruction, 24(1), 73-122.
40. Hiebert, J., & Carpenter, T. P. (1992). Learning and teaching with understanding. In D. A. Grouws (Ed.), Handbook of research on mathematics teaching and learning (pp. 65-97). New York: Macmillan.
41. Hiebert, J., & Lefevre, J. (1986). Conceptual and procedural knowledge in mathematics: An introductory analysis. In J. Hiebert (Ed.), Conceptual and procedural knowledge: The case of mathematics (pp. 1–27). Hillsdale, NJ: Lawrence Erlbaum Associates, Inc.
42. Hiebert, J., & Wearne, D. (1996). Instruction, understanding, and skill in multidigit addition and subtraction. Cognition and Instruction, 14(3), 251−283.
43. Hollebrands, K. F. (2007). The role of a dynamic software program for geometry in the strategies high school mathematics students employ. Journal for Research in Mathematics Education, 38(2), 164–192.
44. Hyde, J., Fennema, E., & Lamon, S. (1990). Gender differences in mathematics performance: A meta-analysis. Psychological Bulletin, 107(2), 139–155.
45. Jacobs, J. E., Lanza, S., Osgood, D. W., Eccles, J. S., & Wigfield, A. (2002). Changes in children’s self-competence and values: Gender and domain differences across grades one through twelve. Child Development, 73(2), 509–527.
46. Jones, K. (2000). Providing a foundation for deductive reasoning: Students' interpretations when using dynamic geometry software and their evolving mathematical explanations. Educational Studies in Mathematics, 44(1-2), 55-85.
47. Jöreskog, K., & Sörbom, D. (1993). Structural equation modeling with the SIMPLIS command language. Hillsdale, NJ: Lawrence Erlbaum Associates.
48. Jöreskog, K. G., & Sörbom, D. (2012). LISREL 9.1 [Computer software]. Lincolnwood, IL: Scientific Software International.
49. Kaiser, H. F. (1974). An index of factorial simplicity. Psychometrika, 39(1), 31-36.
50. Keiser, J. M. (2004). Struggles with developing the concept of angle: Comparing sixth-grade students' discourse to the history of the angle concept. Mathematical Thinking and Learning, 6(3), 285-306.
51. Kimball, M. M. (1989). A new perspective on women's math achievement. Psychological Bulletin, 105(2), 198.
52. Kim, H., Plake, B. S., Wise, S. L., & Novak, C. D. (1990). A longitudinal study of sex-related item bias in mathematics subtests of the California Achievement Test. Applied Measurement in Education, 3(3), 275-284.
53. Leder, G. C., Pehkonen, E., & Törner, G. (2006). Beliefs: A hidden variable in mathematics education? (Vol. 31). Springer Science & Business Media.
54. Lehrer, R., Jenkins, M., & Osana, H. (1998). Longitudinal study of children's reasoning about space and geometry. In R. Lehrer & D. Chazan (Eds.), Designing learning environments for developing understanding of geometry and space (pp. 137-167). Mahwah, NJ: Erlbaum.
55. Levine, S. C., Huttenlocher, J., Taylor, A., & Langrock, A. (1999). Early sex differences in spatial skill. Developmental Psychology, 35(4), 940.
56. Linn, M. C., & Petersen, A. C. (1985). Emergence and characterization of sex differences in spatial ability: A meta-analysis. Child Development, 56(6), 138–151.
57. Little, R. J. (1988). A test of missing completely at random for multivariate data with missing values. Journal of the American Statistical Association, 83(404), 1198-1202.
58. Lowrie, T., & Diezmann, C. M. (2011). Solving graphics tasks: Gender differences in middle-school students. Learning and Instruction, 21(1), 109-125.
59. Mabbott, D. J., & Bisanz, J. (2003). Developmental change and individual differences in children's multiplication. Child Development, 74(4), 1091-1107.
60. Mayberry, J. (1983). The van Hiele levels of geometric thought in undergraduate preservice teachers. Journal for Research in Mathematics Education,14(1), 58–69.
61. Ministry of National Education (MoNE) (2013).
62. Ministry of National Education Archives (2010–2014).
63. Mitchelmore, M. C. (1997). Children's informal knowledge of physical angle situations. Learning and Instruction, 7(1), 1-19.
64. Mitchelmore, M. C. (1998). Young students’ concepts of turning and angle. Cognition and Instruction, 16(3), 265–284.
65. Mitchelmore, M., & White, P. (1998). Development of angle concepts: A framework for research. Mathematics Education Research Journal, 10(3), 4-27.
66. Mitchelmore, M. C., & White, P. (2000). Development of angle concepts by progressive abstraction and generalisation. Educational Studies in Mathematics, 41(3), 209-238.
67. Moore, K. C. (2013). Making sense by measuring arcs: A teaching experiment in angle measure. Educational Studies in Mathematics, 83(2), 225-245.
68. Munier, V., Devichi, C., & Merle, H. (2008). A physical situation as a way to teach angle. Teaching Children Mathematics, 14(7), 402-407.
69. National Council of Teachers of Mathematics. (2000). Principles and standards for school mathematics. Reston, VA: Author.
70. Owens, K., & Outhred, L. (2006). The complexity of learning geometry and measurement. In A.Gutierrez & P.Boero (Eds.), Handbook of Research on the Psychology of Mathematics Education: Past, Present and Future (pp. 83-116). Sense Publishers.
71. Pallant, J. (2013). A step by step guide to data analysis using IBM SPSS survival manual. Berkshire: McGraw-Hill.
72. Rittle-Johnson, B., & Koedinger, K. R. (2005). Designing knowledge scaffolds to support mathematical problem solving. Cognition and Instruction, 23(3), 313–349.
73. Rittle-Johnson, B., & Schneider, M. (in press). Developing conceptual and procedural knowledge of mathematics. In R. Kadosh, & A. Dowker (Eds.), The Oxford handbook of numerical cognition. Oxford, UK: Oxford University Press.
74. Rittle-Johnson, B., Schneider, M., & Star, J. R. (2015). Not a one-way street: Bidirectional relations between procedural and conceptual knowledge of mathematics. Educational Psychology Review, 27(4), 587-597.
75. Rittle-Johnson, B., Siegler, R. S., & Alibali, M. W. (2001). Developing conceptual understanding and procedural skill in mathematics: An iterative process. Journal of Educational Psychology, 93(2), 346.
76. Rittle-Johnson, B., & Star, J. R. (2007). Does comparing solution methods facilitate conceptual and procedural knowledge? An experimental study on learning to solve equations. Journal of Educational Psychology, 99(3), 561.
77. Sabers, D., Cushing, K., & Sabers, D. (1987). Sex differences in reading and mathematics achievement for middle school students. The Journal of Early Adolescence, 7(1), 117-128.
78. Schneider, M., Rittle-Johnson, B., & Star, J. R. (2011). Relations among conceptual knowledge, procedural knowledge, and procedural flexibility in two samples differing in prior knowledge. Developmental Psychology, 47(6), 1525.
79. Schreiber, J. B., Stage, F. K., King, J., Nora, A., & Barlow, E. A. (2006). Reporting structural
80. equation modeling and confirmatory factor analysis results: A review. The Journal of
81. Educational Research, 99(6), 323−337.
82. Senk, S., & Usiskin, Z. (1983). Geometry proof writing: A new view of sex differences in mathematics ability. American Journal of Education, 91(2), 187-201.
83. Simmons, M., & Cope, P. (1990). Fragile knowledge of angle in turtle geometry. Educational Studies in Mathematics, 21(4), 375-382.
84. Smith, C. P., King, B., & Hoyte, J. (2014). Learning angles through movement: Critical actions for developing understanding in an embodied activity. The Journal of Mathematical Behavior, 36, 95-108.
85. Soury-Lavergne, S., & Maschietto, M. (2015). Articulation of spatial and geometrical knowledge in problem solving with technology at primary school. ZDM, 47(3), 435-449.
86. Spelke, E. S. (2005). Sex differences in intrinsic aptitude for mathematics and science?: A critical review. American Psychologist, 60(9), 950.
87. SPSS (2012). IBM SPSS Statistics for Windows, Version 21. Boston, Mass: International Business Machines Corporation.
88. Star, J. R. (2005). Re-conceptualizing procedural knowledge. Journal for Research in Mathematics Education, 36(5), 404-411.
89. Star, J. R. (2007). Foregrounding procedural knowledge. Journal for Research in Mathematics Education, 38(2), 132-135.
90. Star, J. R., & Stylianides, G. J. (2013). Procedural and conceptual knowledge: exploring the gap between knowledge type and knowledge quality. Canadian Journal of Science, Mathematics and Technology Education, 13(2), 169-181.
91. Swafford, O. J., Jones, G. A., & Thornton, C. A. (1997). Increased knowledge in geometry and instructional practice. Journal for Research in Mathematics Education, 28(4), 467-483.
92. Tabachnick, B. G., & Fidell, L. S. (2007). Using multivariate statistics. Boston: Pearson Education.
93. Tsamir, P., Tirosh, D., Levenson, E., Barkai, R., & Tabach, M. (2016). Developing a mathematically rich environment for 3-year-old Children: The case of geometry. In T. Meaney, O. Helenius, M. L. Johasson, T. Lange, A. Wernberg (Eds.) Mathematics Education in the Early Years (pp. 325-340). Springer International Publishing.
94. Usiskin, Z. (1982). Van Hiele levels and achievement in secondary school geometry (Final report of the Cognitive Development and Achievement in Secondary School Geometry Project). Chicago: University of Chicago, Department of Education. (ERIC Document Reproduction Service No. ED 220 288).
95. Vasilyeva, M., & Bowers, E. (2006). Children’s use of geometric information in mapping tasks. Journal of Experimental Child Psychology, 95(4), 255-277.
96. Vasilyeva, M., Casey, B. M., Dearing, E., & Ganley, C. M. (2009). Measurement skills in low-income elementary school students: Exploring the nature of gender differences. Cognition and Instruction, 27(4), 401-428.
97. White, P., & Mitchelmore, M. C. (2010). Teaching for abstraction: A model. Mathematical Thinking and Learning, 12(3), 205-226.
98. Wigfield, A., & Eccles, J. S. (2000). Expectancy-value theory of achievement motivation. Contemporary Educational Psychology, 25(1), 68–81.
99. Yeung, A. S. (2011). Student self-concept and effort: Gender and grade differences. Educational Psychology, 31(6), 749-772.