Linear quantity models in US and Chinese elementary mathematics classrooms

Rethinking the role of “quality instruction” in predicting algebraic learning within an opportunity-propensity framework: An exploratory cross-cultural study.

“…We considered teaching quality based on the aforementioned three-aspect construct with each aspect further divided into two subcategories: worked examples (example, practice), representations (concrete, abstract), and deep questions (question, explanation). Table 2 illustrates a coding framework, which was developed and validated through prior studies on teachers’ mathematics lesson planning (Ding & Carlson, 2013) and classroom teaching (Ding et al, 2019). Overall, a total of 64 videotaped lessons were collected from both countries (16 teachers, each of whom taught four lessons).…”

“…In this study, we strove to conduct a cross-cultural study in which the cultures and educational systems were quite different to obtain a larger variation in teaching. We chose to compare China and the United States because international research has consistently shown large gaps between the two countries in terms of students’ mathematical achievements (e.g., Cai, 2004; Ding et al, 2019; Li et al, 2008; OECD, 2014, 2018, 2019; Trends in International Mathematics & Science Study [TIMSS], 2011, 2015, 2019), textbook presentations (Ding, 2016; Ding & Li, 2010, 2014; Li et al, 2008), teacher knowledge (An et al, 2004; Cai, 2005; Ma, 1999), and actual classroom teaching (Ding, 2021; Ding et al, 2019).…”

“…Particularly relevant to our defined “teaching quality” in this study, prior research has also shown cross-cultural differences in U.S. and Chinese teachers’ use of worked examples, representations, and deep questions. For instance, with regard to the use of worked examples, U.S. teachers tended to either provide minimum guidance on worked examples (Kirschner et al, 2006; Stigler & Hiebert, 1999) or to rapidly go through several repetitive examples (Ding et al, 2019, 2021; Ding & Carlson, 2013), which contrasts with Chinese counterparts, who spent a significant amount of time unpacking one in-depth example (Ding, 2021; Ding et al, 2019). In terms of representation uses, U.S. teachers believed in using manipulatives and drawing to find answers, while their Chinese counterparts viewed these approaches only as tools to understand concepts and quantitative relationships (Cai, 2005; Ding et al, 2019).…”

“…For instance, with regard to the use of worked examples, U.S. teachers tended to either provide minimum guidance on worked examples (Kirschner et al, 2006; Stigler & Hiebert, 1999) or to rapidly go through several repetitive examples (Ding et al, 2019, 2021; Ding & Carlson, 2013), which contrasts with Chinese counterparts, who spent a significant amount of time unpacking one in-depth example (Ding, 2021; Ding et al, 2019). In terms of representation uses, U.S. teachers believed in using manipulatives and drawing to find answers, while their Chinese counterparts viewed these approaches only as tools to understand concepts and quantitative relationships (Cai, 2005; Ding et al, 2019). Chinese teachers also tended to use the concreteness fading method (Fyfe et al, 2015), starting with a real-world context that was modeled with diagrams and eventually faded into abstract equations (Ding, 2021).…”

Comparación de resoluciones algebraicas con y sin tablas de problemas de edades

“…Otras propuestas usan representaciones intermedias menos abstractas como paso intermedio, arguyendo que una progresión suave hacia las ecuaciones reporta mejores resultados (Ding y Li, 2014;McNeil y Fyfe, 2012). De hecho, el uso de representaciones intermedias se enseña como método de resolución en varios países (Ding et al, 2019;Ng y Lee, 2009). Sin embargo, el uso de estas representaciones no siempre se asocia con mejores resultados (Ainsworht, 2006;Rellensmann et al, 2017), y centrarse en características visuales de la representación puede inhibir estrategias más generales de resolución e inducir a errores (Bieda y Nathan, 2009).…”