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 LessonSketch platform, that was used to develop and administer those questionnaires. LessonSketch included a variety of tools for multimedia content and questionnaire creation, as well as questionnaire delivery and reporting. LessonSketch 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 Comparing how college mathematics instructors and high-school teachers recognize professional obligations of mathematics teaching when making instructional decisions. Ko I, Herbst P, Shultz M Ko, I., Herbst, P., and Shultz, M. (2024). Comparing how college mathematics instructors and high-school teachers recognize professional obligations of mathematics teaching when making instructional decisions. Journal of Mathematics Teacher Education, 27,1055–1082. https://doi.org/10.1007/s10857-023-09595-2 2024 Are teachers amenable to increasing students’ scope of work in doing proofs? Estimating teachers’ decision-making using a diagnostic classification model. Ko I, Herbst P Ko, I. and Herbst, P. (2020). Are teachers amenable to increasing students’ scope of work in doing proofs? Estimating teachers’ decision-making using a diagnostic classification model. Journal of Educational Research in Mathematics, Special Issue, pp. 169-183. 2020 Presenting proofs in geometry classrooms: A study of what teachers expect from student mathematical communication Dimmel J, Herbst P Dimmel, J., and Herbst, P. (2020). Presenting proofs in geometry classrooms: A study of what teachers expect from student mathematical communication. Educational Studies in Mathematics, 105(1), 71–89. 2020 When what routinely happens conflicts with what ought to be done: A scenario-based assessment of secondary mathematics teaching Erickson A, Herbst P, Ko I, Dimmel J Erickson, A. , Herbst, P., Ko, I., & Dimmel, J. (2021). When what routinely happens conflicts with what ought to be done: A scenario-based assessment of secondary mathematics teaching. Research in Mathematics Education. 23(2), 188-207 2021 Understanding the characteristics of mathematical content knowledge for teaching algebra in high schools and community colleges Ko I, Mesa V, Durancyzk I, Herbst P, Kohli N, Strom A, Watkins L Ko, I., Mesa, V., Duranczyk, I., Herbst, P., Kohli, N., Ström, A., & Watkins, L. (2024). Understanding the characteristics of mathematical content knowledge for teaching algebra in high schools and community colleges. International Journal of Mathematical Education in Science and Technology 55(3), 590-614 2024 Examining the potential of cartoon-based simulations for studying mathematics teachers’ handling of student emotions: A replication study Brown A M, Bardelli E, Herbst P G, Dimmel J K Brown, A. M., Bardelli, E., Herbst, P. G., & Dimmel, J. K. (2023). Examining the potential of cartoon-based simulations for studying mathematics teachers’ handling of student emotions: A replication study. Implementation and Replication Studies in Mathematics Education, 3(2), 243–274 2023 “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 A heuristic approach to assess change in mathematical knowledge for teaching geometry after a practice-based professional learning intervention Herbst P, Ko I, Milewski A Herbst, P., Ko, I., & Milewski, A. (2020). A heuristic approach to assess change in mathematical knowledge for teaching geometry after a practice-based professional learning intervention. Research in Mathematics Education, 22(2), 188-208 2020 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 The Role of Theory Development in Increasing the Subject Specificity of Research on Mathematics Teaching Herbst P, Chazan D Herbst, P. and Chazan, D. (2017). The role of theory development in increasing the subject specificity of research on mathematics teaching. In J. Cai (Ed.), Compendium for Research in Mathematics Education (pp. 102-127). Reston, VA: NCTM. 2017 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
