, 43:547 | Cite as

Beliefs and engagement structures: behind the affective dimension of mathematical learning

  • Gerald A. GoldinEmail author
  • Yakov M. Epstein
  • Roberta Y. Schorr
  • Lisa B. Warner
Original Article


Beliefs influencing students’ mathematical learning and problem solving are structured and intertwined with larger affective and cognitive structures. This theoretical article explores a psychological concept we term an engagement structure, with which beliefs are intertwined. Engagement structures are idealized, hypothetical constructs, analogous in many ways to cognitive structures. They describe complex “in the moment” affective and social interactions as students work on conceptually challenging mathematics. We present engagement structures in a self-contained way, paying special attention to their theoretical justification and relation to other constructs. We suggest how beliefs are characteristically woven into their fabric and influence their activation. The research is based on continuing studies of middle school students in inner-city classrooms in the USA.


Goal Orientation Engagement Structure Motivational Orientation Behavioral Engagement Belief Structure 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.



The authors thank Robert Capraro, Mary Margaret Capraro, and Nicole Shechtman for stimulating discussions. This research was partly supported by the US National Science Foundation (NSF), Grants 0138806, ESI-0333753, and REESE-1008770. Any opinions, findings, conclusions, or recommendations are the authors’ and do not necessarily reflect the views of the NSF.


  1. Abelson, R. P. (1981). Psychological status of the script concept. American Psychologist, 36(7), 715–729.CrossRefGoogle Scholar
  2. Alston, A., Goldin, G. A., Jones, J., McCulloch, A., Rossman, C., & Schmeelk, S. (2007). The complexity of affect in an urban mathematics classroom. In T. Lamberg & L. R. Wiest (Eds.), Exploring mathematics education in context: Proceedings of the 29th annual meeting of PME-NA, Lake Tahoe, NV (pp. 326–333). Reno: University of Nevada.Google Scholar
  3. Anderman, E. M., & Wolters, C. A. (2006). Goals, values, and affect: Influences on student motivation. In P. Alexander & P. Winne (Eds.), Handbook of educational psychology (pp. 369–389). Mahwah, NJ: Erlbaum.Google Scholar
  4. Anderson, E. (1999). Code of the street. NY: W. W. Norton.Google Scholar
  5. Beck, A. T. (1976). Cognitive therapy and the emotional disorders. NY: New American Library.Google Scholar
  6. Capraro, M. M., Capraro, R. M., & Henson, R. K. (2001). Measurement error of scores on the mathematics anxiety rating scale across studies. Educational and Psychological Measurement, 61, 373–386.CrossRefGoogle Scholar
  7. Cattell, R. B., & Scheier, I. H. (1961). The meaning and measurement of neuroticism and anxiety. NY: Ronald Press.Google Scholar
  8. Csikszentmihalyi, M. (1990). Flow: The psychology of optimal experience. NY: Harper & Row.Google Scholar
  9. Dai, D. Y., & Sternberg, R. J. (Eds.). (2004). Motivation, emotion, and cognition: Integrative perspectives on intellectual functioning and development. Mahwah, NJ: Erlbaum.Google Scholar
  10. Dance, L. J. (2002). Tough fronts: The impact of street culture on schooling. NY: Routledge-Falmer.Google Scholar
  11. Davis, R. B. (1984). Learning mathematics: The cognitive science approach to mathematics education. Norwood, NJ: Ablex.Google Scholar
  12. DeBellis, V. A., & Goldin, G. A. (2006). Affect and meta-affect in mathematical problem solving: A representational perspective. Educational Studies in Mathematics, 63(2), 131–147.CrossRefGoogle Scholar
  13. Dweck, C. S. (2000). Self-theories: Their role in motivation, personality, and development. Philadelphia, PA: Taylor & Francis.Google Scholar
  14. Epstein, Y. M., Goldin, G. A., Schorr, R. Y., Capraro, R., Capraro, M. M., & Warner, L. B. (2010). Measuring engagement structures in middle-grades urban mathematics classrooms. Paper presented at the 2010 annual meeting of AERA, Denver, CO.Google Scholar
  15. Epstein, Y., Schorr, R. Y., Goldin, G. A., Warner, L., Arias, C., Sanchez, L., et al. (2007). Studying the affective/social dimension of an inner-city mathematics class. In T. Lamberg & L. R. Wiest (Eds.), Exploring mathematics education in context: Proceedings of the 29th annual meeting of PME-NA, Lake Tahoe, NV (pp. 649–656). Reno: University of Nevada.Google Scholar
  16. Evans, J., Morgan, C., & Tsatsaroni, A. (2006). Discursive positioning and emotion in school mathematics practices. Educational Studies in Mathematics, 63(2), 209–226.CrossRefGoogle Scholar
  17. Fennema, E., & Sherman, J. A. (1976). Fennema-Sherman Mathematics Attitudes Scales: Instruments designed to measure attitudes toward the learning of mathematics by females and males. Journal for Research in Mathematics Education, 7, 324–326.CrossRefGoogle Scholar
  18. Finn, J. D. (1993). School engagement and students at risk. Washington, DC: U. S. Department of Education, National Center for Educational Statistics (ERIC Document Reproduction Service No. ED 362 322).Google Scholar
  19. Flavell, J. H. (1979). Metacognition and cognitive monitoring: A new area of cognitive-developmental inquiry. American Psychologist, 34(10), 906–911.CrossRefGoogle Scholar
  20. Fredricks, J. A., Blumenfeld, P. C., & Paris, A. H. (2004). School engagement: Potential of the concept, state of the evidence. Review of Educational Research, 74(1), 59–109.CrossRefGoogle Scholar
  21. Furinghetti, F., & Pehkonen, E. (2002). Rethinking characterizations of beliefs. In G. Leder, E. Pehkonen, & G. Törner (Eds.), Beliefs: A hidden variable in mathematics education? (pp. 39–57). Dordrecht: Kluwer.Google Scholar
  22. Goffman, E. (1974). Frame analysis. NY: Harper Colophon.Google Scholar
  23. Goldin, G. A. (2000). Affective pathways and representation in mathematical problem solving. Mathematics Thinking and Learning, 2, 209–219.CrossRefGoogle Scholar
  24. Goldin, G. A. (2002). Affect, meta-affect, and mathematical belief structures. In G. Leder, E. Pehkonen, & G. Tőrner (Eds.), Beliefs: A hidden variable in mathematics education? (pp. 59–72). Dordrecht: Kluwer.Google Scholar
  25. Goldin, G. A., Epstein, Y. M., & Schorr, R. Y. (2007). Affective pathways and structures in urban students’ mathematics learning. In D. K. Pugalee, A. Rogerson, & A. Schinck (Eds.), Mathematics education in a global community: Proceedings of the 9th international conference of the mathematics education into the 21st century project (pp. 260–265). Charlotte, NC: University of North Carolina.Google Scholar
  26. Goldin, G. A., Rösken, B., & Törner, G. (2009). Beliefs: No longer a hidden variable in mathematics teaching and learning processes. In J. Maass & W. Schlöglmann (Eds.), Beliefs and attitudes in mathematics education: new research results (pp. 1–18). Rotterdam: Sense.Google Scholar
  27. Gomez-Chacon, I. M. (2000). Affective influences in the knowledge of mathematics. Educational Studies in Mathematics, 43, 149–168.CrossRefGoogle Scholar
  28. Green, T. F. (1971). The activities of teaching. Tokyo: McGraw-Hill.Google Scholar
  29. Greeno, J. G., & Goldman, S. V. (1988). Thinking practices in mathematics and science learning. Mahwah, NJ: Erlbaum.Google Scholar
  30. Greenwood, C. R. (1991). Longitudinal analysis of time, engagement, and achievement in at-risk versus non at-risk students. Exceptional Children, 57, 521–535.Google Scholar
  31. Grouws, D. A., & Lembke, L. O. (1996). Influential factors in student motivation to learn mathematics: The teacher and classroom culture. In M. Carr (Ed.), Motivation in mathematics (pp. 39–62). Cresskill, NJ: Hampton Press.Google Scholar
  32. Hannula, M. S. (2002). Attitude towards mathematics: Emotions, expectations and values. Educational Studies in Mathematics, 49, 25–46.CrossRefGoogle Scholar
  33. Jost, J. T., & Kay, A. C. (2010). Social justice: History, theory, and research. In S. T. Fiske, D. T. Gilbert, & G. Lindzey (Eds.), Handbook of social psychology (5th ed., Vol. 2, pp. 1122–1165). Hoboken, NJ: Wiley.Google Scholar
  34. Kloosterman, P. (1996). Students’ beliefs about knowing and learning mathematics: Implications for motivation. In M. Carr (Ed.), Motivation in mathematics (pp. 131–156). Cresskill, NJ: Hampton Press.Google Scholar
  35. Laurent, J., Catanzaro, S. J., Joiner, T. E., Jr., Rudolph, K. D., Potter, K. I., Lambert, S., et al. (1999). A measure of positive and negative affect for children: Scale development and preliminary validation. Psychological Assessment, 11(3), 326–338.CrossRefGoogle Scholar
  36. Leder, G. C., Pehkonen, E., & Törner, G. (Eds.). (2002). Beliefs: A hidden variable in mathematics education? Dordrecht: Kluwer.Google Scholar
  37. Lesh, R., Hamilton, E., & Kaput, J. J. (Eds.). (2007). Foundations for the future in mathematics education. Mahwah, NJ: Erlbaum.Google Scholar
  38. Linnenbrink, E. A. (2007). The role of affect in student learning: A multi-dimensional approach to considering the interaction of affect, motivation, and engagement. In P. A. Schutz & R. Pekrun (Eds.), Emotion in education (pp. 107–124). London: Elsevier.Google Scholar
  39. Linnenbrink, E. A., & Pintrich, P. R. (2000). Multiple pathways to learning and achievement: The role of goal orientation in fostering adaptive motivation, affect, and cognition. In C. Sansone & J. M. Harackiewicz (Eds.), Intrinsic and extrinsic motivation: The search for optimal motivation and performance (pp. 195–227). San Diego, CA: Academic Press; cited in Anderman & Wolters (2006).Google Scholar
  40. Linnenbrink, E. A. & Pintrich, P. R (2002). Achievement goal theory and affect: An asymmetrical bidirectional model. Educational Psychologist, 37, 69–78; cited in Anderman & Wolters (2006).Google Scholar
  41. Maass, J., & Schlöglmann, W. (Eds.). (2009). Beliefs and attitudes in mathematics education: New research results. Rotterdam: Sense.Google Scholar
  42. Malmivuori, M.-L. (2006). Affect and self-regulation. Educational Studies in Mathematics, 63, 149–164.CrossRefGoogle Scholar
  43. Marks, H. M. (2000). Student engagement in instructional activity: Patterns in the elementary, middle, and high school years. American Educational Research Journal, 37, 153–184.Google Scholar
  44. McLeod, D. B. (1994). Research on affect and mathematics learning. Journal for Research in Mathematics Education, 25, 637–647.CrossRefGoogle Scholar
  45. McLeod, D. B., & McLeod, S. H. (2002). Synthesis—beliefs and mathematics education: Implications for learning, teaching, and research. In G. C. Leder, E. Pehkonen, & G. Törner (Eds.), Beliefs: A hidden variable in mathematics education? (pp. 115–123). Dordrecht: Kluwer.Google Scholar
  46. Middleton, J. A., & Spanias, P. A. (1999). Motivation for achievement in mathematics: Findings, generalizations, and criticisms of the research. Journal for Research in Mathematics Education, 30(1), 65–88.CrossRefGoogle Scholar
  47. Murray, H. A. (2008). 70th Anniversary edition, explorations in personality. NY: Oxford University Press.Google Scholar
  48. Op ‘t Eynde, P., De Corte, E., & Verschaffel, L. (2002). Framing students’ mathematics-related beliefs. In G. C. Leder, E. Pehkonen, & G. Törner (Eds.), Beliefs: A hidden variable in mathematics education? (pp. 13–37). Kluwer: Dordrecht.Google Scholar
  49. Op ‘t Eynde, P., De Corte, E., & Verschaffel, L. (2007). Students’ emotions: A key component of self-regulated learning? In P. Schutz & R. Pekrun (Eds.), Emotion in education (pp. 185–204). Burlington, MA: Academic Press.Google Scholar
  50. Pajares, M. F. (1992). Teachers’ beliefs and educational research: Cleaning up a messy construct. Review of Educational Research, 62, 307–332.Google Scholar
  51. Pajares, M. F., & Graham, L. (1999). Self-efficacy, motivation constructs, and mathematics performance of entering middle school students. Contemporary Educational Psychology, 24, 124–139.CrossRefGoogle Scholar
  52. Park, S. (2005). Student engagement and classroom variables in improving mathematics achievement. Asia Pacific Education Review, 6(1), 87–97. doi: 10.1007/BF03024970.CrossRefGoogle Scholar
  53. Pekrun, R., Frenzel, A. C., Goetz, T., & Perry, R. P. (2007). The control-value theory of achievement emotions: An integrative approach to emotions in education. In P. Schutz & R. Pekrun (Eds.), Emotion in education (pp. 13–36). Burlington, MA: Academic Press.Google Scholar
  54. Pintrich, P. R. (2000). Multiple goals, multiple pathways: The role of goal orientation in learning and achievement. Journal of Educational Psychology, 92, 544–555.CrossRefGoogle Scholar
  55. Richardson, F., & Suinn, R. (1972). The mathematics anxiety rating scale: Psychometric data. Journal of Counseling Psychology, 19, 551–554.CrossRefGoogle Scholar
  56. Rogers, T. B. (1983). Emotion, imagery, and verbal codes: A closer look at an increasingly complex interaction. In J. Yuille (Ed.), Imagery, memory, and cognition: Essays in honor of Allan Paivio (pp. 295–305). Mahwah, NJ: Erlbaum.Google Scholar
  57. Rokeach, M. (1968). Beliefs, attitudes, and values: A theory of organization and change. San Francisco: Jossey-Bass.Google Scholar
  58. Rokeach, M. (1973). The nature of human values. NY: The Free Press.Google Scholar
  59. Ross, L., & Nisbett, R. E. (1991). The person and the situation: Perspectives of social psychology. NY: McGraw-Hill.Google Scholar
  60. Schoenfeld, A. H. (1985). Mathematical problem solving. NY: Academic Press.Google Scholar
  61. Schoenfeld, A. H. (2010). How we think: A theory of goal-oriented decision making and its educational applications. NY: Routledge.Google Scholar
  62. Schorr, R. Y., Epstein, Y. M., Warner, L. B., & Arias, C. C. (2010a). Don’t disrespect me: Affect in an urban math class. In R. Lesh, P. L. Galbraith, C. R. Haines, & A. Hurford (Eds.), Modeling students’ mathematical modeling competencies: ICTMA 13 (pp. 313–325). NY: Springer.CrossRefGoogle Scholar
  63. Schorr, R. Y., Epstein, Y. M., Warner, L. B., & Arias, C. C. (2010b). Mathematical truth and social consequences: The intersection of affect and cognition in a middle school classroom. Mediterranean Journal for Research in Mathematics Education, 9, 107–134.Google Scholar
  64. Schorr, R. Y., & Goldin, G. A. (2008). Students’ expression of affect in an inner-city SimCalc classroom. Educational Studies in Mathematics, 68, 131–148.CrossRefGoogle Scholar
  65. Schunk, D. H. (2001). Social-cognitive theory and self-regulated learning. In B. Zimmerman & D. Schunk (Eds.), Self-regulated learning and academic achievement: Theoretical perspectives (2nd ed., pp. 125–151). Mahwah, NJ: Erlbaum; cited in Anderman & Wolters (2006).Google Scholar
  66. Schunk, D. H., & Zimmerman, B. J. (Eds.). (2008). Motivation and self-regulated learning: Theory, research, and applications. NY: Routledge.Google Scholar
  67. Schutz, P., & Pekrun, R. (Eds.). (2007). Emotion in education. Burlington, MA: Academic Press (Elsevier).Google Scholar
  68. Skinner, E., Kindermann, T., & Furrer, C. (2009). A motivational perspective on engagement and disaffection: Conceptualization and assessment of children’s behavioral and emotional participation in academic activities in the classroom. Educational and Psychological Measurement, 69, 493–525.CrossRefGoogle Scholar
  69. Törner, G. (2002). Mathematical beliefs—a search for a common ground. In G. C. Leder, E. Pehkonen, & G. Törner (Eds.), Beliefs: A hidden variable in mathematics education? (pp. 73–94). Dordrecht: Kluwer.Google Scholar
  70. Turner, J. C., & Meyer, D. K. (2009). Understanding motivation in mathematics: What is happening in classrooms? In K. R. Wentzel & A. Wigfield (Eds.), Handbook of motivation at school (pp. 527–552). NY: Routledge.Google Scholar
  71. van de Sande, C., & Greeno, J. G. (2010). A framing of instructional explanations: Let us explain with you. In M. K. Stein & L. Kucan (Eds.), Instructional explanations in the disciplines (pp. 69–82). NY: Springer.CrossRefGoogle Scholar
  72. Wellborn, J. G. (1991). Engaged and disaffected action: The conceptualization and measurement of motivation in the academic domain. Unpublished doctoral dissertation, University of Rochester, NY; cited in Skinner et al. (2009).Google Scholar
  73. Wigfield, A., & Eccles, J. S. (1992). The development of achievement task values: A theoretical analysis. Developmental Review, 12, 265–310.CrossRefGoogle Scholar
  74. Wigfield, A., & Eccles, J. S. (2000). Expectancy-value theory of achievement motivation. Contemporary Educational Psychology, 25, 68–81.CrossRefGoogle Scholar
  75. Wigfield, A., & Eccles, J. S. (2002). The development of competence beliefs, expectancies for success, and achievement values from childhood through adolescence. In A. Wigfield & J. S. Eccles (Eds.), Development of achievement motivation (pp. 91–120). San Diego, CA: Academic Press.CrossRefGoogle Scholar
  76. Zajonc, R. B. (1980). Feeling and thinking: Preferences need no inferences. American Psychologist, 35, 151–175.CrossRefGoogle Scholar
  77. Zan, R., Brown, L., Evans, J., & Hannula, M. S. (2006). Affect in mathematics education: An introduction. Educational Studies in Mathematics, 63(2), 113–121.CrossRefGoogle Scholar
  78. Zimmerman, B. J., & Schunk, D. H. (2008). Motivation: An essential dimension of self-regulated learning. In D. H. Schunk & B. J. Zimmerman (Eds.), Motivation and self-regulated learning: Theory, research, and applications (pp. 1–30). NY: Routledge.Google Scholar

Copyright information

© FIZ Karlsruhe 2011

Authors and Affiliations

  • Gerald A. Goldin
    • 1
    Email author
  • Yakov M. Epstein
    • 1
  • Roberta Y. Schorr
    • 2
  • Lisa B. Warner
    • 3
  1. 1.Rutgers UniversityNew BrunswickUSA
  2. 2.Rutgers UniversityNewarkUSA
  3. 3.William Paterson UniversityWayneUSA

Personalised recommendations