## CAREER: Algebraic Knowledge for Teaching: A Cross-Cultural Perspective

The goal of this CAREER program of research is to identify, from a cross-cultural perspective, essential Algebraic Knowledge for Teaching (AKT) that will enable elementary teachers to better develop students' algebraic thinking. This study explores AKT based on integrated insights of the U.S. and Chinese expert teachers' classroom performance.

What content knowledge is needed for the teaching of mathematics? What practices are more effective for realizing student success? These questions have received considerable attention in the mathematics education community. The goal of this CAREER program of research is to identify, from a cross-cultural perspective, essential Algebraic Knowledge for Teaching (AKT) that will enable elementary teachers to better develop students' algebraic thinking. Focusing on two fundamental mathematical ideas recently emphasized by the Common Core State Standards - inverse relations and properties of operations - this study explores AKT based on integrated insights of the U.S. and Chinese expert teachers' classroom performance. It will be focused on three objectives: (1) identify AKT that facilitates algebraic thinking and develop preliminary findings into teaching materials; (2) refine research-based teaching materials based on the evaluative data; and (3) integrate research with education through course development at Temple University and teacher outreach in Philadelphia.

The model underlying this research program is that improved pedagogy will improve student learning, both directly and indirectly. A design-based research method will be used to accomplish objectives #1 and #2. Cross-cultural videotaped lessons will be first analyzed to identify AKT, focusing on teachers' use of worked examples, representations, and deep questions. This initial set of findings will then be developed into teaching materials. The U.S. and Chinese expert teachers will re-teach the lessons as part of the refinement process. Data sources will include: baseline and updated survey data (control, context, and process variables), observation, documents, videos, and interviews. The statistical techniques will include descriptive and inferential statistics and HLM will to address the hierarchical nature of the data.

This project involves students and teachers at various levels (elementary, undergraduate, and graduate) at Temple University and the School District of Philadelphia (SDP) in the U.S. and Nanjing Normal University and Nantong School District in China. A total of 600 current and future elementary teachers and many of their students will benefit directly or indirectly from this project. Project findings will be disseminated through various venues. Activities of the project will promote school district-university collaboration, a novice-expert teacher network, and cross-disciplinary and international collaboration. It is anticipated that the videos of expert teaching will also be useful future research by cognitive researchers studying ways to improve mathematics learning.

**Publications**^{G} indicates graduate student author; ^{U} indicates undergraduate student author

Journal Articles in English

- Ding, M.,
Chen, W., &^{G}Hassler, R. (2019). Linear quantity models in the US and Chinese elementary mathematics classrooms.^{G}*Mathematical Thinking and Learning, 21,*105-130*.*doi: 10.1080/10986065.2019.1570834 . PDF Barnett, E., & Ding, M. (2019).^{G }*Investigations of Mathematics Learning, 11,*148-166. doi: 10.1080/19477503.2018.1425592 PDF- Ding, M., &
Heffernan, K. (2018). Transferring specialized content knowledge to elementary classrooms: Preservice teachers’ learning to teach the associative property.^{G}*International Journal of Mathematics Educational in Science and Technology, 49,*899-921.doi: 10.1080/0020739X.2018.1426793 PDF - Ding, M. (2018). Modeling with tape diagrams. Te
*aching Children Mathematics*,*25*, 158-165. doi: 10.5951/teacchilmath.25.3.0158 PDF Chen, W., & Ding, M.* (2018). Transitioning from mathematics textbook to classroom instruction: The case of a Chinese expert teacher.^{G}*Frontiers of Education in China, 13,*601-632. doi: 10.1007/s11516-018-0031-z (*Both authors contributed equally). PDF- Ding, M., &
Auxter, A. (2017). Children’s strategies to solving additive inverse problems: A preliminary analysis^{G}*. Mathematics Education Research Journal, 29,*73-92. doi:10.1007/s13394-017-0188-4 PDF - Ding, M. (2016). Developing preservice elementary teachers’ specialized content knowledge for teaching fundamental mathematical ideas: The case of associative property.
*International Journal of STEM Education, 3*(9), 1-19*.*doi: 10.1186/s40594-016-0041-4 PDF - Ding, M. (2016). Opportunities to learn: Inverse operations in U.S. and Chinese elementary mathematics textbooks.
*Mathematical Thinking and Learning, 18,*45-68. doi: 10.1080/10986065.2016.1107819 PDF

Journal Articles in Chinese*Note: *The Chinese journals *Educational Research and Evaluation (Elementary Education and Instruction*) *教育研究与评论 (小学教育教学) *and *Curriculum and Instructional Methods (**课程教材教法)* are both official, core journals in mathematics education field in China.

Chen, W. (2018).^{G }*Curriculum and Instructional Methods.*- Ma, F. ( 2018) – Necessary algebraic knowledge for elementary teachers- an ongoing cross-cultural study.
*Educational Research and Evaluation (Elementary Education and Instruction), 2,*5-7. 小学教师必备的代数学科知识-跨文化研究进行时。*教育研究与评论**(**小学教育教学),**2,*5-7. PDF - Chen, J. (2018) Infusion and development of children’s early algebraic thinking – a comparative study of the US and Chinese elementary mathematics teaching.
*Educational Research and Evaluation (Elementary Education and Instruction), 2,*8-13. 儿童早期代数思维的渗透与培养-中美小学数学教学比较研究。*教育研究与评论**(**小学教育教学)**，2**，*8-13. PDF - Zong, L. (2018). A comparative study on the infusion of inverse relations in the US and Chinese classroom teaching.
*Educational Research and Evaluation (Elementary Education and Instruction), 2,*14-19. 中美逆运算渗透教学对比研究。*教育研究与评论（小学教育教学，**2*，14-19. PDF - Wu, X. (2018). Mathematical representations and development of children’s mathematical thinking: A perspective of US-Chinese comparison.
*Educational Research and Evaluation (Elementary Education and Instruction), 2,*20-24. 数学表征与儿童数学思维发展-基于中美比较视角。*教育研究与评论（小学教育教学，**2,*20-24. PDF

**Dissertations**

- Hassler, R. (2016).
*Mathematical comprehension facilitated by situation models: Learning opportunities for inverse relations in elementary school.*Published dissertation, Temple University, Philadelphia, PA. (Chair: Dr. Meixia Ding) PDF - Chen, W. (2018).
*Elementary mathematics teachers’ professional growth: A perspectives of TPACK*(TPACK 视角下小学数学教师专业发展的研究). Dissertation, Nanjing Normal University. Nanjing, China. PDF

**National Presentations**^{G} indicates graduate student author; ^{U} indicates undergraduate student author

- Ding, M (symposium organizer, 2019, April).
*Enhancing elementary mathematics instruction: A U.S.-China collaboration.*Papers presented at NCTM research conference (Discussant: Jinfa Cai). (*The following three action research papers were written by my NSF project teachers under my guidance).*

- Milewski Moskal, M., & Varano, A. (2019).
*The teaching of worked examples: Chinese approaches in U.S. classrooms*. Paper - Larese, T., Milewski Moskal, M., Ottinger, M., & Varano, A., (2019).
*Introducing Investigations math games in China: Successes and surprises.*Paper - Murray, D., Seidman, J., Blackmon, E., Maimon, G., & Domsky, A. (2019).
*Mathematic instruction across two cultures: A teacher perspective.*

- Ding, M., & Ying Y. (2018, June).
*CAREER: Algebraic knowledge for teaching: A cross-cultural perspective.*Poster presentation at the National Science Foundation (NSF) PI meeting, Washington, DC. Poster - Ding, M., Brynes, J.,
Barnett, E., & Hassler, R. (2018, April)^{G}*. When classroom instruction predicts students’ learning of early algebra: A cross-cultural opportunity-propensity analysis.*Paper presented at 2018 AERA conference. New York, NY. Paper - Ding, M., Li, X., Manfredonia, M., & Luo, W. (2018, April).
*Video as a tool to support teacher learning: A Cross-cultural analysis.*Paper presented at 2018 NCTM conference. Washington, DC. PPT Barnett, E., & Ding, M. (2018, April).^{G}*Teaching the basic properties of arithmetic: A natural classroom investigation of associativity.*Poster presentation at 2018AERA conference, New York, NY. Poster- Hassler, R., & Ding, M. (2018, April).
*The role of deep questions in promoting elementary students’ mathematical comprehension.*Poster presentation at 2018AERA conference, New York, NY. - Ding, M.,
Chen, W.,^{G}Hassler, R., Li, X., &^{G}Barnett, E. (April, 2017).^{G}*Comparisons in the US and Chinese elementary mathematics classrooms.*Poster presentation at AERA 2017 conference (In the session of “Advancing Mathematics Education Through NSF’s DRK-12 Program”). San Antonio, TX. Poster - Ding, M., Li, X.,
Hassler, R., &^{G}Barnett, E. (April, 2017).^{G}*Understanding the basic properties of operations in US and Chinese elementary School.*Paper presented at AERA 2017 conference. San Antonio, TX. Paper - Ding, M.,
Chen, W., &^{G}Hassler, R. (April, 2017).^{G}*Tape diagrams in the US and Chinese elementary mathematics classrooms*. Paper presented at NCTM 2017 conference. San Antonio, TX. Paper - Ding, M., &
Hassler, R. (2016, June).^{G}*CAREER: Algebraic knowledge for teaching in elementary school: A cross-cultural perspective.*Poster presentation at the NSF PI meeting, Washington, DC. Poster - Ding, M. (symposium organizer, 2016, April).
*Early algebraic in elementary school: A cross-cultural perspective.*Proposals presented at 2016 AERA conference, Washington, DC.

- Ding, M. (2016, April).
*A comparative analysis of inverse operations in U.S. and Chinese elementary mathematics textbooks.*Paper Hassler, R. (2016, April).^{G }*Elementary Textbooks to Classroom Teaching: A Situation Model Perspective*. PaperChen, W., & Ding, M. (2016, April).^{G}*Transitioning textbooks into classroom teaching: An action research on Chinese elementary mathematics lessons.*Paper- Li, X.,
Hassler, R., & Ding, M. (2016, April).^{G}*Elementary students’ understanding of inverse relations in the U.S. and China.*Paper - Stull, J., Ding, M.,
Hassler, R., Li, X., &^{G}George, C. (2016, April).^{U}*The impact of algebraic knowledge for teaching on student learning: A Preliminary analysis.*Paper

- Ding, M.,
Hassler, R., Li., X., &^{G}Chen, W. (2016, April).^{G}*Algebraic knowledge for teaching: An analysis of US experts' lessons on inverse relations.*Paper presented at 2016 NCTM conference, San Francisco, CA. Paper Hassler. R., & Ding, M. (2016, April).^{G}*Situation model perspective on mathematics classroom teaching: A case study on inverse relations*. Paper presented at 2016 NCTM conference, San Francisco, CA. Paper- Ding, M., &
Copeland, K. (2015, April).^{G}*Transforming specialized content knowledge: Preservice elementary teachers’ learning to teach the associative property of multiplication*. Paper presented at AERA 2015 conference, Chicago, IL. Paper PPT - Ding, M., &
Auxter, A. (2015, April).^{G}*Children’s strategies to solving additive inverse problems: A preliminary analysis*. Paper presented at AERA 2015 conference, Chicago, IL. Paper