# ThEMaT II

Thought Experiments in Mathematics Teaching IIIn order to test a theory of mathematics teacher decision making developed in the context of a previous project (ThEMaT I), in ThEMaT II (a.k.a., ThEMaT Online), we developed survey measures of constructs that we hypothesized influence the instructional decisions made by mathematics teachers in particular recurrent instructional situations in high school geometry and algebra. Those constructs include high school mathematics teachers’ recognition of, and attitude towards, both norms of those instructional situations and professional obligations of mathematics teaching, as well as the mathematical knowledge needed to teach high school geometry and algebra. Those instructional situations included doing proofs and geometric calculations in geometry, solving equations and doing word problems in algebra.

Those measures included both traditional text-based questionnaires as well as scenario-based questionnaires, which confronted participants with storyboard representations of classroom scenarios or images of mathematical problems. The development of both those representations of practice and the questionnaires that contained them involved several stages, including reviews by our team of mathematics educators as well as by experienced teachers, during focus groups. The questionnaires were also piloted with several small samples of teachers and revised again before eventually being administered to a national sample of high school mathematics teachers.

Using latent variable models, we explored the dimensionality of several constructs and evaluated the validity and reliability of the determined scales. We were also able to estimate the relationships between various constructs (using both structural equation models and more-traditional regression techniques). For example, we provided evidence that the mathematical knowledge needed to teach geometry, as well as the mathematical knowledge needed to teach algebra, depend on the task of teaching being completed (e.g., whether the teacher is choosing the givens for a problem or interpreting student work) and the instructional situation (e.g., whether the class is composing a geometric proof or solving an equation). We also demonstrated that one’s mathematical knowledge for teaching a given course is positively associated with their experience teaching that course. Similarly, we demonstrated that a teacher’s attitudes towards breaching a given norm are positively correlated with both their experience teaching and their mathematical knowledge for teaching (in particular, when that norm is breached in a way that allows the teacher to attend to one of their professional obligations as mathematics teachers). We also demonstrated that a teacher’s preference for following a norm of an instructional situation is associated with their recognition of that norm.

The project also supported the development of a Flash-based web application, the Lesson*Sketch* platform, that was used to develop and administer those questionnaires. Lesson*Sketch* included a variety of tools for multimedia content and questionnaire creation, as well as questionnaire delivery and reporting. Lesson*Sketch* was also used and made available for others to use to develop and deliver content for teacher education. These teacher education uses included a series of modules for inservice teachers to learn the Common Core Standards for Mathematical Practice, which reused content developed during ThEMaT I (e.g., the animations). The project tested the role the tools could support in teacher education using various online experiences and forums for teachers.

Thus the project contributed to three of the five components of NSF’s cycle of innovation: synthesizing and theorizing notions of teacher knowledge and practical rationality; hypothesizing and clarifying how those participate in decision-making in specific situations in algebra and geometry instruction; and designing, developing, and testing a “virtual setting” for practice-based teacher education.

Below is a list of all publications produced with NSF grant funding from DRL–0918425.

Title | Author(s) | Citation | Date |
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“It Depends …”: Using Ambiguities to Better Understand Mathematics Teachers’ Decision-making | Milewski A, Erickson A, & Herbst P | Milewski, A., Erickson, A. & Herbst, P. (2021). “It Depends …”: Using Ambiguities to Better Understand Mathematics Teachers’ Decision-making. Can. J. Sci. Math. Techn. Educ. 21, 123–144. | 2021 |

Subject Matter Knowledge of Geometry Needed in Tasks of Teaching: Relationship to Prior Geometry Teaching Experience | Ko I, & Herbst P | Ko, I., & Herbst, P. (2020). Subject Matter Knowledge of Geometry Needed in Tasks of Teaching: Relationship to Prior Geometry Teaching Experience, Journal for Research in Mathematics Education, 51(5), 600-630. | 2020 |

Representations of mathematics teaching and their use in teacher education: What do we need in a pedagogy for the 21st century? | Herbst P, Bieda K, Chazan D, González G | Herbst, P., Bieda, K., Chazan, D., and González, G. (2010). Representations of mathematics teaching and their use in teacher education: What do we need in a pedagogy for the 21st century? Proceedings of the 2010 Annual PME-NA conference. Columbus, OH: Ohio State University | 2010 |

Where’s the proof? Proof in U.S. high school geometry content standards | Kosko K, Herbst P | Kosko, K.^ and Herbst, P. (2011). Where’s the proof? Proof in U.S. high school geometry content standards. Proceedings of the 2011 Annual PME-NA Conference. Reno, NV | 2011 |

Representations of mathematics teaching and their use in transforming teacher education: Contributions to a pedagogical framework | Herbst P, Aaron W, Bieda K, González G, Chazan D | Herbst, P., Aaron, W., Bieda, K., González, G., and Chazan, D. (2011). Representations of mathematics teaching and their use in transforming teacher education: Contributions to a pedagogical framework. Discussion document for the working group ‘representations of mathematics teaching’. Proceedings of the 2011 Annual PME-NA Conference. Reno, NV | 2011 |

Representations of mathematics teaching and their use in transforming teacher education: The role of approximations of practice | Herbst P, Aaron W, Bieda K, Moore-Russo D | Herbst, P., Aaron, W., Bieda, K., and Moore-Russo, D. (2012). Representations of mathematics teaching and their use in transforming teacher education: The role of approximations of practice. Proceedings of the 34th Annual Meeting of the North American Chapter of the International Group for the Psychology of Mathematics Education. Kalamazoo, MI. Available on Deep Blue at The University of Michigan | 2012 |

Expanding students’ role when doing proofs in high school geometry | Herbst P, Shultz M, Ko I, Boileau N, Erickson A | Herbst, P., Shultz, M., Ko, I., Boileau, N., and Erickson, A. (2018). 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 | 2018 |

Does the medium matter? A comparison of secondary mathematics preservice teachers’ representations of practice created in text and storyboarding media | Rougée A, Herbst P | Rougée, A., & Herbst, P. (2018). Does the medium matter? A comparison of secondary mathematics preservice teachers’ representations of practice created in text and storyboarding media. In R. Zazkis and P. Herbst (Eds.), Scripting approaches in mathematics education: Mathematical dialogues in research and practice (pp. 265-292). Cham, Switzerland: Springer. | 2018 |

LessonSketch: An environment for teachers to examine mathematical practice and learn about its standards | Herbst P, Aaron W, Chieu V M | Herbst, P., Aaron, W., & Chieu, V. M. (2013). LessonSketch: An environment for teachers to examine mathematical practice and learn about its standards. In D. Polly (Ed.), Common core mathematics standards and implementing digital technologies (pp. 281-294). Hershey, PA: IGI Global. | 2013 |

Solving equations: Exploring instructional exchanges as lenses to understand teaching and its resistance to reform | Buchbinder O, Chazan D I, Capozzoli M | Buchbinder, O., Chazan, D. I., & Capozzoli, M. (2019). Solving equations: Exploring instructional exchanges as lenses to understand teaching and its resistance to reform. Journal for Research in Mathematics Education, 50(1), 51-83. | 2019 |

Constructing plausible, but uncommon stories: Gaining subversive insight into the school mathematics tradition. | Chazan D, Gilead S, Cochran K | Chazan, D., Gilead, S., & Cochran, K. (2018). Constructing plausible, but uncommon stories: Gaining subversive insight into the school mathematics tradition. In R. Zazkis & P. Herbst (Eds.), Scripting approaches in mathematics education (pp. 53-72). Cham, Switzerland: Springer. | 2018 |

Technology-Mediated Mathematics Teacher Development: Research on Digital Pedagogies of Practice | Herbst P, Chazan D, Chieu V M, Milewski A, Kosko K, Aaron W, | Herbst, P., Chazan, D., Chieu, V. M., Milewski, A., Kosko, K., and Aaron, W. (2016). Technology-Mediated Mathematics Teacher Development: Research on Digital Pedagogies of Practice. In M. Niess, K. Hollebrands, & S. Driskell (Eds.), Handbook of Research on Transforming Mathematics Teacher Education in the Digital Age (pp. 78-106). Hershey, PA: IGI Global. | 2016 |

Research on the Teaching of Mathematics: A Call to Theorize the Role of Society and Schooling in Mathematics | Chazan D, Herbst P, Clark L, | Chazan, D., Herbst, P., and Clark, L. (2016). Research on the Teaching of Mathematics: A Call to Theorize the Role of Society and Schooling in Mathematics. In D. Gitomer and C. Bell (Eds.), Handbook of research on teaching (5th ed., pp. 1039-1097). Washington, DC: AERA. | 2016 |

A study of the quality of interaction among participants in online animation-based conversations about mathematics teaching | Chieu VM, Herbst P, | Chieu, V. M., & Herbst, P. (2016). A study of the quality of interaction among participants in online animation-based conversations about mathematics teaching., Teaching and Teacher Education, (57, pp. 139-149). | 2016 |

What Details Do Teachers Expect From Student Proofs? A Study of Proof Checking in Geometry | Dimmel J, Herbst P, | Dimmel, J. K., & Herbst, P. G. (2018). What Details Do Teachers Expect From Student Proofs? A Study of Proof Checking in Geometry, Journal for Research in Mathematics Education JRME, 49(3), 261-291. | 2018 |

Secondary mathematics teachers’ attitudes toward alternative communication practices when doing proofs in geometry | Dimmel J, Herbst P, | Dimmel, J and Herbst, P. (2017). Secondary mathematics teachers? expectations of student communication practices when doing proofs in geometry., Teaching and Teacher Education, (69, pp. 151). | 2017 |

Mathematics Teachers’ Recognition of an Obligation to the Discipline and Its Role in the Justification of Instructional Actions | Herbst P, Dimmel J, Ericson A, Ko I, Kosko K, | Herbst, P., Dimmel, J., Erickson, A., Ko, I., & Kosko, K. (2014). Mathematics teachers? recognition of an obligation to the discipline and its role in the justification of instructional actions, Proceedings of the 2014 annual meeting of the International Group for the Psychology of Mathematics Education, (3, pp. 273). | 2014 |

Mathematical Knowledge for Teaching High School Geometry | Herbst P, Kosko KW, | Herbst, P. and Kosko, K. (2012). Mathematical Knowledge for Teaching High School Geometry, Proceedings of the 34th Annual Meeting of the North American Chapter of the International Group for the Psychology of Mathematics Education, (pp. 438). | 2012 |

Measuring Recognition of the Professional Obligations of Mathematics Teaching: The Prob Surveys | Herbst P, Ko I, | Herbst, P. and Ko, I. (2017). "Measuring recognition of the professional obligations of mathematics teaching, Proceedings of the 39th annual meeting of the North American Chapter of the International Group for the Psychology of Mathematics Education., (pp. 1242). | 2017 |

Teoría y métodos para la investigación de la racionalidad de la práctica en la enseñanza de las matemáticas | Herbst P, | herbst, P. (2018). Teoría y métodos para la investigación de la racionalidad de la práctica en la enseñanza de las matemáticas (Theory and methods for research on the practical rationality of mathematics teaching), Educación Matemática (Mexico), (30, pp. 11). | 2018 |

Teachers’ Recognition of the Diagrammatic Register and its Relationship with their Mathematical Knowledge for Teaching | Boileau N, Dimmel J, Herbst P, | Boileau, N., Dimmel, J.K., & Herbst, P. G. (2016). Teachers? recognition of the diagrammatic register and its relationship with their mathematical knowledge for teaching., Proceedings of the Thirty-Eighth Annual Meeting of the North American Chapter of the International Group for the Psychology of Mathematics Education, (pp. 266). | 2016 |

Teachers’ Expectation about Geometric Calculations in High School Geometry | Boileau N, Herbst P, | Boileau, N., & Herbst, P. G. (2015). Teachers' expectations about geometric calculations in high school geometry, Proceedings of the Thirty-Seventh Annual Meeting of the North American Chapter of the International Group for the Psychology of Mathematics Education, (pp. 269). | 2015 |

An Instrumental Co-Genesis Approach to Developing an Online Practice-based Environment for Teacher | Chieu VM, Boileau N, Herbst P, | Chieu, V. M., Boileau, N., & Herbst, P. (2015). An Instrumental Co-Genesis Approach to Developing an Online Practice-based Environment for Teacher Education., Society for Information Technology & Teacher Education International Conference, (1, pp. 2763). | 2015 |

Evaluating Teachers’ Decisions in Posing a Proof Problem | Kosko K, Herbst P, | Kosko, K. and Herbst, P. (2012). Evaluating Teachers? Decisions in Posing a Proof Problem, Proceedings of the 34th Annual Meeting of the North American Chapter of the International Group for the Psychology of Mathematics Education., (pp. 813). | 2012 |

How Are Geometric Proof Problems Presented? Conceptualizing and Measuring Teachers’ Recognition of the Diagrammatic Register | Herbst P, Kosko K, Dimmel J, | Herbst, P., Kosko, K., and Dimmel, J. (2013). How are geometric proof problems presented? Conceptualizing and measuring teachers? recognition of the diagrammatic register., Proceedings of the 35th annual meeting of the North American Chapter of the International Group for the Psychology of Mathematics Education, (pp. 179). | 2013 |

Insights into the School Mathematics Tradition from Solving Linear Equations | Buchbinder O, Chazan D, Fleming E, | Buchbinder, O., Chazan, D., and Fleming, E. (2015). Insights into the school mathematics tradition from solving linear equations, For the Learning of Mathematics, (35, pp. 2). | 2015 |

When mathematics teachers consider acting on behalf of the discipline, what assumptions do they make? | Milewski A, Erickson A, Herbst P, Dimmel J, | Milewski, A., Erickson, A., Herbst, P., & Dimmel, J. (2015). When mathematics teachers consider acting on behalf of the discipline, what assumptions do they make?, Proceedings of the 37th Annual Meeting of the North American Chapter of the International Group for the Psychology of Mathematics Education, (pp. 1130). | 2015 |

Using representations of practice to elicit mathematics teachers’ tacit knowledge of practice: a comparison of responses to animations and videos | Herbst P, Kosko KW, | Herbst, P. and Kosko, K. (2014). Using representations of practice to elicit mathematics teachers? tacit knowledge of practice: a comparison of responses to animations and videos, Journal of Mathematics Teacher Education, (17, pp. 515). | 2014 |

Using multimedia questionnaires to study influences on the decisions mathematics teachers make in instructional situations | Herbst P, Chazan D, Kosko K, Dimmel J, Erickson A, | Herbst, P, Chazan, D., Kosko, K., Dimmel, J., and Erickson, A. (2016). Using multimedia questionnaires to study influences on the decisions mathematics teachers make in instructional situations., ZDM-The International Journal of Mathematics Education, (48, pp. 167). | 2016 |

Will Teachers Create Opportunities for Discussion when Teaching Proof in a Geometry Classroom? | Erickson A, Herbst P, | Erickson, A. and Herbst, P. (2018). Will teachers create opportunities for discussion when teaching proof in a geometry classroom?, International Journal of Mathematics and Science Education, (16, pp. 167). | 2018 |

Investigating Secondary Mathematics Teachers’ Attitudes toward Alternative Communication Practices While Doing Proofs in Geometry | Dimmel J, Herbst P, | Dimmel, J. and Herbst, P. (2015). Investigating secondary mathematics teachers? attitudes toward alternative communication practices while doing proofs in geometry, Proceedings of the 37th Annual Meeting of the North American Chapter of the International Group for the Psychology of Mathematics Education, (pp. 277). | 2015 |

Designing Reference Points in Animated Classroom Stories to Support Teacher Learners’ Online Discussions | Chieu VM, Herbst P, | Chieu, V. and Herbst, P. (2013). Designing reference points in animated classroom stories to support teacher learners? online discussions., 10th International conference on Computer Supported Collaborative Learning, (1, pp. 89). | 2013 |

LessonSketch: A Rich-Media Scenario-Based Learning Environment for Teacher Development | Chieu VM, Herbst P, | Chieu, V. and Herbst, P. (2012). LessonSketch: A Rich-Media Scenario based learning environment for teacher development, Proceedings of Society for Information Technology & Teacher Education International Conference, (pp. 968). | 2012 |

Approximating the Practice of Mathematics Teaching: What Learning Can Web-based, Multimedia Storyboarding Software Enable? | Herbst P, Chieu VM, Rougee A, | Herbst, P., Chieu, V.M., and Rougee, A. (2014). Approximating the Practice of Mathematics Teaching: What Learning Can Web-based, Multimedia Storyboarding Software Enable?, Contemporary Issues in Technology and Teacher Education, CITE journal, (14) | 2014 |

Using multimedia scenarios delivered online to study professional knowledge use in practice | Herbst P, Chazan D, | Herbst, P. and Chazan, D. (2015). Using Multimedia Scenarios Delivered Online to Study Professional Knowledge Use in Practice., International Journal of Research and Method in Education, (38). | 2015 |

An Analysis of Evaluative Comments in Teachers’ Online Discussions of Representations of Practice | Chieu VM, Kosko K, Herbst P, | Chieu, V. M., Kosko, K., and Herbst P. (2015). An analysis of evaluative comments in teachers? online discussions of representations of practice., Journal of Teacher Education, (66). | 2015 |

“You Are Learning Well My Dear”: Shifts in Novice Teachers’ Talk About Teaching During Their Internship | Bieda K, Sela H, Chazan D, | Bieda, K., Sela, H., and Chazan, D. (2015). "You are learning well my dear": How student teaching influences intern teachers talk' about teaching during their internship. Journal of Teacher Education, (66). | 2015 |

What actions do teachers envision when asked to facilitate mathematical argumentation in the classroom? | Kosko K, Rougee A, Herbst P, | Kosko, K., Rougee, A., and Herbst, P. (2014). What actions do teachers envision when asked to facilitate mathematical argumentation in the classroom?, Mathematics Education Research Journal, (26, pp. 459). | 2014 |

Research on Practical Rationality: Studying the justification of actions in mathematics teaching | Herbst P, Chazan D, | Herbst, P., and Chazan, D. (2011). Research on Practical Rationality: Studying the Justification of Actions in Mathematics Teaching., The Mathematics Enthusiast, (8, pp. 405). | 2011 |

On the instructional triangle and sources of justification for actions in mathematics teaching | Herbst P, Chazan D, | Herbst, P and Chazan, D. (2012). On the instructional triangle and sources of justification for actions in mathematics teaching. ZDM The International Journal of Mathematics Education (44th ed., pp. 601-612). | 2012 |

On creating and using representations of mathematics teaching in research and teacher development | Herbst P, Chazan D, | Herbst, P., and Chazan, D. (2011). On creating and using representations of mathematics teaching in research and teacher development: Introduction to this issue., ZDM Mathematics Education, (43, pp. 1). | 2011 |

A deeper look at how teachers say what they say: A quantitative modality analysis of teacher-to-teacher talk | Kosko K, Herbst P, | Kosko, K. and Herbst, P. (2011). A deeper look at how teachers say what they say: A quantitative modality analysis of teacher-to-teacher talk., Teaching and Teacher Education, (28, pp. 589). | 2011 |

Using comics-based representations of teaching, and technology, to bring practice to teacher education courses | Herbst P, Chazan D, Chen C, Chieu VM, Weiss M | Herbst, P., Chazan, D., Chen, C., Chieu, V., and Weiss, M. (2011). Using comics-based representations of teaching, and technology, to bring practice to university 'methods' courses., ZDM-The international journal of mathematics education. | 2011 |

Mathematical Knowledge for Teaching and its Specificity to High School Geometry Instruction | Herbst P, Kosko KW, | Herbst, P., & Kosko, K. (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. 23-45). New York, NY: Springer. | 2014 |

Studying Decision Making in Instructional Situations: The Affordances of Multimedia Questionnaires | Herbst P, Chazan D, Kosko K, Dimmel J, Erickson A, | Herbst, P., Chazan, D., Kosko, K., Dimmel, J, and Erickson, A. (2015). Studying Decision Making in Instructional Situations: The Affordances of Multimedia Questionnaires, ZDM-The International Journal of Mathematics Education, | 2015 |