Comparing US and Chinese Preservice Teachers’ Mathematical Thinking and Multiple Representation in Teaching Whole-numbers Lessons

• In Event: Comparative Insights on Effective Teaching, Teacher Education, and Digital Literacy

Proposal

Purposes and Rationale

Teaching plays a crucial role in developing students' mathematical thinking (Hanna & Jahnke, 1996) and fostering meaningful learning (Aylar & Sahiner, 2014). Validating mathematical assumptions involves assessing arguments through various thinking schemes (NCTM, 2000), making it essential to integrate mathematical thinking into teaching at all grade levels, especially in elementary school (Ball & Bass, 2003). Teacher preparation programs are expected to equip preservice teachers with foundational mathematical knowledge (Livy et al., 2016; Swars et al.,
2007) and teaching practices focused on mathematical thinking (Stylianou et al., 2006) to support meaningful learning for young students (NCTM, 2000). It is, therefore, important to study how preservice teachers demonstrate mathematical thinking in their teaching. Additionally, teachers' effective use of representations, including single and multiple representations, significantly influences teaching quality (Barton & Neville-Barton, 2003) and students' learning outcomes (Cobb, Yackel, & Wood, 1992). Since representations illustrate mathematical ideas (Dreher, Kuntze, & Lerman, 2016) and help explain complex concepts (Ainsworth, 2006), examining preservice teachers' use of representations in their teaching is critical.

Whole-number concepts and operations represent a basic constituent and a significant part of mathematics (Sbaih, 2022), foundational mathematics topics in the elementary curriculum (Thanheiser, 2009). These concepts and operations help students understand other concepts (Siegler et al., 2011) and develop their problem-solving ability. Thus, examining how US and Chinese preservice teachers can demonstrate mathematics thinking in teaching whole-number concepts and operations using various representations focus using three questions. 1) How do US and Chinese preservice teachers demonstrate mathematic thinking in teaching whole-number concepts and operations differently? 2) How do US and Chinese preservice teachers use different representations in teaching whole-number concepts and operations differently? 3) What is the relationship between mathematical thinking and different types of multiple representations in teaching whole-number lessons?

Theoretical Frameworks

Conception of Mathematical Thinking
The conception suggests that mathematical thinking is an intellectual ability in students’ mathematical development, which includes three kinds (Harel & Sowder, 1998): 1) external conviction proof scheme, 2) the inductive proof scheme, and 3) the deductive proof scheme. This conception guided the development of a coding system to capture the inductive and deductive proof schemes demonstrated in the US and Chinese preservice teachers’ whole-number instruction.

Conception of Different Mathematics Representation
This conception suggests that mathematics ideas can be represented using external actions, images, symbols, and language (Bruner & Kenney, 1965). These representations include (Lesh et al., 1987): 1) Realistic representation, 2) Verbal representations, 3) Pictorial representations, 4) Concrete representations, 5) Symbolic representations. This conception guided the design of interventions and coding systems to capture the representations participants used to teach mathematical concepts and operations.

Methodology

Participants and Context
This study involved five US and five Chinese preservice teachers, all in their student teaching after completing a mathematics methods course. They were selected because they taught number lessons in their practicum classrooms, covering concepts or operations related to addition, subtraction, multiplication, and division.

Data Sources and Analysis
The study analyzed ten videotaped whole-number lessons taught by participants. First, the lessons were transcribed, with Chinese transcripts translated into English. Next, the lessons were coded for types of mathematical thinking and representations successively. The results were then grouped by mathematical thinking and representations for US and Chinese preservice teachers. Finally, similarities and differences between the two groups in their use of mathematical thinking and representations in teaching whole-number concepts and operations were identified.

Findings

Kinds of Mathematical Thinking in Teaching

Both US and Chinese participants used deductive thinking more frequently than inductive thinking when teaching whole-number topics, with US participants using 54.6% and Chinese participants using 55.9% of their total mathematics thinking for deductive approaches.
However, when focusing on specific instructional areas, the study found that US participants preferred inductive thinking (58.6%) over deductive thinking (41.4%) for whole-number concepts, while Chinese participants favored deductive thinking (70.0%) over inductive thinking (30.0%) for the same. For whole-number operations, US participants leaned heavily on deductive thinking (76.3%), whereas Chinese participants balanced deductive and inductive thinking equally (50.0%). See Table 1.

Kinds of Representations in Teaching

Both US and Chinese participants predominantly used multiple representations over single representations when teaching whole-number topics. US participants used 66.7% and Chinese participants 85.5% of their total representations for multiple representations, with single representations accounting for only 33.3% and 14.5%, respectively.

When focusing on multiple representations, US participants most often integrated different single representations simultaneously for both whole-number concepts (80.0%) and operations (28.6%). In contrast, Chinese participants favored sequencing single representations for concepts (52.0%) and using the same single representations repeatedly for operations (16.7%). See Table 2.

Relationship between Mathematical Thinking and Multiple Representations

Both US and Chinese participants frequently using the same single representations repeatedly tended to develop deductive thinking (64.3%) more than inductive thinking (35.7%) when teaching whole-number topics. However, participants using sequencing single representations tended to develop inductive thinking (55.1%) more than deductive thinking (44.9%). And there was no difference in the mathematical thinking tendency when different single representations were simultaneously integrated to teach whole-number topics.

Specifically, when using the same single representations repeatedly, US participants exclusively developed deductive thinking (100%), whereas Chinese participants demonstrated a more balanced approach, developing both deductive (54.5%) and inductive thinking (45.5%) with a smaller difference between the two. When using different single representations simultaneously, both US and Chinese participants tended to develop deductive and inductive thinking in a balanced manner, with each type of thinking demonstrated equally (50% deductive, 50% inductive). When using sequencing single representations, Chinese participants tended to develop inductive thinking (55.1%) rather than deductive thinking (44.9%). While US participants rarely used sequencing single representations to develop mathematical thinking.

Discussions and Implications

This study enhances the empirical understanding of how preservice teachers demonstrate mathematical thinking and develop single and multiple representations in teaching whole-number topics. It also explores the relationship between mathematical thinking and different types of representations in instruction. These insights provide a valuable foundation for shaping teacher education programs and policies aimed at improving preservice teachers' content knowledge and classroom application in mathematics teaching.

Authors

• Xuan He, texas tech University

• Jian Wang, Texas Tech University

• Min Huang, TTU

• Hyunjin Shim

• Raymond Flores, Texas Tech University