Who Builds Understanding? Teachers’ and Students’ Roles in Developing Conceptual Understanding in Chinese and U.S. Preservice Whole Number Lessons

• In Event: Teacher Education in Comparative Perspective: Instructional Understanding, Relational Agency, and Equity

Proposal

Purposes and Rationale
Conceptual understanding (CU) is a core component of mathematical understanding (Hiebert & Lefevre, 1986) and represents essential aspects of mathematical proficiency necessary for successful learning (National Research Council, 2001).
Additionally, creating connections between math ideas is a key factor in mathematical understanding (Hiebert & Carpenter, 1992; National Research Council, 2001). Therefore, it is important to examine how teachers offer opportunities for students to make mathematical connections to build mathematical understanding. In particular, the source of CU—whether generated by the teacher or constructed by students—may shape the depth of students’ mathematical understanding.
The whole number topics are foundational in the elementary curriculum (Thanheiser, 2012)
help students understand other concepts (Siegler et al., 2011) and develop their problem-solving ability. In recent international assessments, Chinese students have consistently demonstrated outstanding mathematics performance in both PISA and TIMSS (OECD, 2023; IEA, 2024). Thus, this study aims to investigate how Chinese and U.S. preservice teachers support students’ CU in whole number instruction using three questions.
1）How do Chinese and U.S. preservice teachers and students demonstrate CU during whole number instruction?
2）To what extent do students exhibit mathematical connections during whole number instruction in the two countries?
3）What is the relationship between teacher- and student-enacted CU and the mathematical connections demonstrated by students?
Theoretical Frameworks
Conception of Conceptual Understanding (CU)
Conceptual understanding refers to an integrated and functional grasp of mathematical ideas. A primary indicator of conceptual understanding is the ability to represent situations in different ways and recognize how different representations are useful for different purposes (National Research Council, 2001). This conception guided the development of a coding system to capture teachers' and students' CU demonstrated during whole number instruction
Conception of Mathematical Connections
A foundational assumption of this study is that mathematical understanding is constructed through making connections among ideas, representations, and procedures (Hiebert & Carpenter, 1992). There are several kinds of connections that learners construct to create in the classroom. 1) Similarities and differences connections; (2) Inclusion connections (Hiebert & Carpenter, 1992). 3)Representation connection (Lesh et al., 1987).4) Strategy connections (Rittle-Johnson & Star, 2009). 5) Experience connections (Bransford et al., 2000). 6) Conceptual and procedural connections, which is constructed in this study. This conception guided the development of a coding system to capture the mathematical connections that students exhibit during whole-number instruction.
Methodology
Participants and Context
This study involved five Chinese and five U.S. preservice teachers. They were selected from different mathematics methods courses at a Chinese university and a U.S. university's elementary program, based on whether they taught whole numbers lessons during their practicum experiences.
Data Sources and Analysis
The data sources for the study included ten videotaped whole-number lessons.
To answer the first question, we coded each whole-number lesson for the CU of preservice teachers and students. Next, calculate the median of CU separately for the Chinese and U.S. groups. Finally, conduct a Mann-Whitney U test to examine whether there are significant differences between the two groups.
To answer the second question, we coded each whole-number lesson for mathematical connection. Then, calculate the median of each of the mathematical connections students demonstrated separately for the Chinese and U.S. groups. Finally, conduct a Mann-Whitney U test to examine whether there are significant differences between the two groups.
To answer the third question, we conducted Spearman’s rank correlation to explore the relationship between teacher- and student-enacted CU and the mathematical connections demonstrated by students.
Findings and Discussions
Teachers’ and Students' Demonstration of CU
A Mann–Whitney U test was conducted to compare the frequency of CU demonstrated by Chinese and U.S. preservice teachers in whole-number lessons. There was no significant difference in CU between the Chinese (Md = 14.0, n = 5) and U.S. (Md = 16.0, n = 5) groups, U = 8.50, z = −0.84, p = .402, with a small effect size (r = .26).
However, Chinese students showed significantly more CU (Md = 25.0, n = 5) than U.S. students (Md = 2.0, n = 5), U = 2.50, z = -2.1, p = .036, with a large effect size (r = .66).
Comparison of Students’ Mathematical Connections Across Countries
A Mann–Whitney U test was conducted to compare students’ mathematical connections in lessons taught by Chinese and U.S. preservice teachers. Although Chinese students showed higher levels of mathematical connections (Md = 31.0, n = 5) than U.S. students (Md = 9.0, n = 5), the difference was not statistically significant, U = 10.00, z = -0.522, p = .602, with a small effect size (r = .17).
Among the six types of mathematical connections, only the experience connections showed a statistically significant difference between groups. Chinese students demonstrated more frequent experience-based connections (Md = 3.0, n = 5) than U.S. students (Md = 0.0, n = 5), U = 5, z = -2.60, p = .008, with a large effect size (r = .82).
Associations Between Mathematical Connection and Conceptual Understanding
For Chinese participants, there are no significant relationships were found between teacher-enacted CU and any connection type. However, representation connections were positively correlated with student-enacted CU, rₛ = .87, n=5, p = .054, showing a strong positive trend but not reaching significance at the .05 level. All other connection types were not significantly related to student-enacted CU (all p > .05).
For U.S. participants, there are no significant relationships were found between teacher-enacted CU and any connection type. However, the conceptual and procedural connections were strongly and significantly correlated with student-enacted CU, rₛ =1.00, n=5, p < .001. All other connection types were not significantly related to student-enacted CU (all p > .05).
Significance
This study contributes to understanding how preservice teachers from different educational cultures support students' mathematical understanding. By distinguishing between teacher- and student-generated CU, it highlights what is taught, but who constructs the understanding. This study also contributes to investigating students’ different types of mathematical connections as in-class evidence of deep understanding.

Authors

• Xuan He, Texas Tech University

• Jian Wang, Texas Tech University

• Raymond Flores, Texas Tech University

• Min Huang, TTU

• Robert Joseph Lynn