The Korea Society Of Educational Studies In Mathematics

Current Issue

Journal of Educational Research in Mathematics - Vol. 30

[ Special ]
Journal of Educational Research in Mathematics - Vol. 30, No. SP1, pp.169-183
Abbreviation: JERM
ISSN: 2288-7733 (Print) 2288-8357 (Online)
Print publication date 31 Aug 2020

Are Teachers Amenable to Increasing Students’ Scope of Work in Doing Proofs? Estimating Teachers’ Decision-Making Using a Diagnostic Classification Model
Inah Ko* ; Patricio Herbst**,
*Postdoctoral Research Fellow, University of Michigan, USA
**Professor, University of Michigan, USA

Correspondence to : Email:,

Funding Information ▼

Please cite this article as: Ko, I. & Herbst, P. Are teachers amenable to increasing students’ scope of work in doing proofs? Estimating teachers’ decision-making using a diagnostic classification model.


We investigate teachers’ decision making in contexts where they could choose to provide students more authentic experiences with proving. Specifically, we investigate their preferences to depart from norms about what proof problems to assign to students. Scenario-based instruments consisting of two sets of items reflecting two hypothesized norms in doing geometric proofs, the given and prove norm and the diagrammatic register norm, were used to operationalize teachers’ preference to depart from instructional norms in order to increase students’ share of labor. By applying a diagnostic classification model (DCM) to classify teachers with respect to their depart from two norms, this study shows that teachers’ decisions depend on the norm at issue. To examine individual factors associated with preference profiles, we use scores of teachers’ mathematical knowledge for teaching, beliefs on the importance of student autonomy, and confidence in mathematics teaching. This study also illustrates methodological benefits of a DCM model in estimating a binary construct (i.e., teachers’ preference), which has more than one sub-construct, with a small number of items.

Keywords: Doing proofs, Instructional situations, Scenario-based assessments, High school geometry, Teacher decision making, Instructional norm, Diagnostic classification model


A preliminary version of this paper was presented at the 2019 Annual Meeting of the American Educational Research Association in Toronto, Canada. The instrument development, data collection, and initial data analysis were done with resources from NSF grant DRL-0918425 to P. Herbst. All opinions are those of the authors and do not necessarily represent the views of the National Science Foundation.

1. Aaron, W. & Herbst, P. (2019). The teacher’s perspective on the separation between and proving in high school geometry classrooms. Journal of Mathematics Teacher Education, 22(3), 231-256.
2. Bieda, K. N. (2010). Enacting proof-related tasks in middle school mathematics: Challenges and opportunities. Journal for Research in Mathematics Education, 41(4), 351-382.
3. Boileau, N., Dimmel, J. K., & Herbst, P. G. (2016, November). Teachers’ recognition of the diagrammatic register and its relationship with their mathematical knowledge for teaching. In M.B. Wood, E.E. Turner, M. Civil, & J.A. Eli (Eds.), Proceedings of the Thirty-Eighth Annual Meeting of the North American Chapter of the International Group for the Psychology of Mathematics Education (pp. 266-269). Tucson, Arizona.
4. Bradshaw, L., Izsák, A., Templin, J., & Jacobson, E. (2014). Diagnosing teachers’ understandings of rational numbers: Building a multidimensional test within the diagnostic classification framework. Educational Measurement: Issues and practice, 33(1), 2-14.
5. Brown, L., & Coles, A. (2011). Developing expertise: How enactivism re-frames mathematics teacher development. ZDM Mathematics Education, 43(6-7), 861–873.
6. Cirillo, M. & Herbst, P. (2012). Moving toward more authentic proof practices in geometry. The Mathematics Educator, 21(2), 11-33.
7. De Villiers, M. (1990). The role and function of proof in mathematics. Pythagoras, 24, 17-2
8. Dunekacke, S., Jenßen, L., & Blömeke, S. (2015). Effects of mathematics content knowledge on pre-school teachers’ performance: A video-based assessment of perception and planning abilities in informal learning situations. International Journal of Science and Mathematics Education, 13(2), 267-286.
9. Erickson A. W, Dimmel, J. K, Herbst, P. G, & Ko, I. (2015). When what routinely happens conflicts with what ought to be done: a scenario-based assessment. Paper presented at the Annual Meeting of the American Educational Research Association in Washington, DC.
10. Henson, R. A., Templin, J. L., & Willse, J. T. (2009). Defining a family of cognitive diagnosis models using log-linear models with latent variables. Psychometrika, 74(2), 191-210.
11. Herbst, P. G. (2002). Establishing a custom of proving in American school geometry: Evolution of the two-column proof in the early twentieth century. Educational Studies in Mathematics, 49(3), 283-312.
12. Herbst, P., & Chazan, D. (2012). On the instructional triangle and sources of justification for actions in mathematics teaching. ZDM Mathematics Education, 44(5), 601–612.
13. Herbst, P. Chen, C., Weiss, M., González, G., Nachlieli, T., Hamlin, M., & Brach, C. (2009). “Doing proofs” in geometry classrooms. In M. Blanton, D. Stylianou, and E. Knuth (Eds.), Teaching and learning of proof across the grades: A K-16 perspective (pp. 250-268). New York: Routledge.
14. Herbst, P., Aaron, W., Dimmel, J., & Erickson, A. (2013, April). Expanding students' involvement in proof problems: Are geometry teachers willing to depart from the norm? Paper presented at the 2013 meeting of the American Educational Research Association. Deep Blue at the University of Michigan.
15. Herbst, P., Kosko, K., & Dimmel, J. (2013). How are geometric proof problems presented? Conceptualizing and measuring teachers’ recognition of the diagrammatic register. In Martinez, M. & Castro Superfine, A (Eds.). Proceedings of the 35th annual meeting of the North American Chapter of the International Group for the Psychology of Mathematics Education (pp. 179-186). Chicago, IL: University of Illinois at Chicago.
16. Herbst, P., & Kosko, K. W. (2014). Mathematical knowledge for teaching and its specificity to high school geometry instruction. In J. Lo, K. R. Leatham, & L. R. Van Zoest (Eds.), Research trends in mathematics teacher education (pp. 23-46). New York: Springer.
17. Herbst, P., Chazan, D., Kosko, K., Dimmel, J., & Erickson, A. (2016). Using multimedia questionnaires to study influences on the decisions mathematics teachers make in instructional situations. ZDM Mathematics Education, 48, 167-183.
18. Herbst, P., Dimmel, J., & Erickson, A. (2016, April). High school mathematics teachers’ recognition of the diagrammatic register in proof problems. Paper presented at the Annual Meeting of the American Educational Research Association, Washington DC.
19. Herbst, P., Shultz, M., Ko, I., Boileau, N., & Erickson, A. (2018, October). Expanding students’ role when doing proofs in geometry. In T. Hodges, G. Roy, & A. Tyminski (Eds.), Proceedings of the 40th Annual Meeting of the North American Chapter of the International Group for the Psychology of Mathematics Education. Greenvile, SC: University of South Carolina.
20. Jurich, D. P., & Bradshaw, L. P. (2014). An illustration of diagnostic classification modeling in student learning outcomes assessment. International Journal of Testing, 14(1), 49-72.
21. Ko, I. (2019). Investigating the dimensionality of teachers' mathematical knowledge for teaching secondary mathematics using item factor analyses and diagnostic classification models. Unpublished doctoral dissertation. University of Michigan, Ann Arbor.
22. Ko, I., & Herbst, P. (in press). Subject matter knowledge of geometry needed in tasks of teaching: relationship to prior geometry teaching experience. Journal for Research in Mathematics Education.
23. Lievens, F., Peeters, H. & Schollaert, E. (2007). Situational judgement tests: A review of recent research. Personnel Review, 37(4), 426–441.
24. Manders, K. (2008b). The Euclidean Diagram (1995). In Mancosu, P. (Eds.), The philosophy of mathematical practice (pp. 80-133). Oxford: Oxford University Press.
25. Mason, J. (1989). Mathematical abstraction as the result of a delicate shift of attention. For the learning of mathematics, 9(2), 2-8.
26. Muthén, L. K. & Muthén, B. O. (1998-2015). Mplus User’s Guide. Seventh Edition. Los Angeles, CA: Muthén & Muthén
27. O'Halloran, K. L. (1998). Classroom discourse in mathematics: A multisemiotic analysis. Linguistics and Education, 10(3), 359-388.
28. Rupp, A. A., Templin, J., & Henson, R. A. (2010). Diagnostic measurement: Theory, methods, and applications. New York: Guilford Press.
29. Schoenfeld, A. H. (2008). On modeling teachers’ in-the-moment decision making. Journal for Research in Mathematics Education. Monograph, 14, 45–96.
30. Schoenfeld, A. H. (2010). How we think: A theory of goal-oriented decision making and its educational applications. New York: Routledge.
31. Stahnke, R., Schueler, S., & Roesken-Winter, B. (2016). Teachers’ perception, interpretation, and decision-making: a systematic review of empirical mathematics education research. ZDM Mathematics Education, 48(1-2), 1-27.
32. Stipek, D. J., Givvin, K. B., Salmon, J. M., & MacGyvers, V. L. (2001). Teachers beliefs and practices related to mathematics instruction. Teaching and Teacher Education, 17, 213-226
33. Tversky, A., & Kahneman, D. (1981). The framing of decisions and the psychology of choice. Science, 211(4481), 453-458