Three Instructional Principles: IES Recommendations
Institute of Educational Sciences (IES) has published a practice guide that contains seven recommendations for classroom teachers (Pashler et al., 2007). These recommendations are developed from numerous high- quality cognitive research studies and classroom experiments on various subjects. Four of these recommendations (#2, #3, #4, and #7) are directly related to classroom teaching and serve as the instructional principles that governed my NSF project data analysis and the key elements in this book.
Research from a cognitive load perspective shows that learning from worked examples is more effective than asking students to solve problems alone (Gog, Kester, 8c Paas, 2011; Sweller & Cooper, 1985). Successful learning with worked examples enables students to generate a relevant mental schema, which helps decrease students’ cognitive load during problem solving. The effect of using worked examples is called the “worked example effect” (Sweller 8c Cooper, 1985; Zhu 8c Simon, 1987). The IES recommendation also asserts that interleaving worked examples and exercise problems is effective. However, within the confines of a short class period, I argue that the worked example effect (rather than the “interleaving” effect) should take priority. In other words, a classroom lesson should treat worked examples as cases of principles. The example task should be unpacked in depth so that students learn a mathematical method to solve new problems and develop forms of mathematical thinking.
Representations contain both concrete and abstract varieties. Both types of representations have affordances and limitations (Goldstone 8c Son, 2005). For instance, concrete representations (e.g., real-world situations) provide students with familiar contexts that support sense-making (Gravemeijer, 8c Doorman, 1999; Resnick, Cauzinille-Marmeche, 8c Mathieu, 1987); however, concrete representations may contain irrelevant information that could distract students (Kaminski, Sloutsky, 8c Heckler, 2008; Uttal, Liu, 8c Deloache, 1999). In contrast, abstract representations enable transfer of initial learning; yet, abstract representations without sense-making may result in inert knowledge that cannot be retrieved for later usage (Koedinger, Alibali, 8c Nathan, 2008). As such, it is necessary to take advantage of both concrete and abstract representations by making connections between them. Recent research recommends an approach called concreteness fading, in which learning should start with concrete representations that can then be gradually faded into abstract numerical solutions (Fyfe, McNeil, 8c Borias, 2015; McNeil 8c Fyfe, 2012).
Prior studies have found that student explanations can enhance learning in general and learning through worked examples specifically (Chi, 2000; Chi, Bassok, Lewis, Reimann, & Glaser, 1989). However, students may not have the motivation to explain the underlying ideas by themselves. As such, teachers should ask deep questions to elicit students’ deep explanations of the targeted concepts. Deep questions include question stems such as why, what caused X, how did X occur, what if, what-if-not, how does X compare to Y, and so on (Craig, Sullins, Witherspoon, & Gholson, 2006).
Even though these IES recommendations are insightful, it remains largely unknown how these instructional principles can be applied to the teaching of early algebraic concepts such as inverse relations and the basic properties of operations. In fact, teachers who read IES recommendations without outside guidance demonstrated difficulties in putting these principles into action (Ding & Carlson, 2013). This book documents an instructional approach (through the lens of the IES recommendations), TEPS, that is expected to be effective in teaching early algebra and beyond.
TEPS: Teaching through Example-based Problem Solving
This book is the result of my NSF-supported project that involved 17 U.S. and 17 Chinese elementary expert teachers. All participating Chinese teachers specialized in teaching mathematics while the U.S. teachers were generalists who taught all subjects. Despite this difference, all teachers were identified as expert teachers in local or national standards2. The main goal of this project was to identify necessary knowledge for teaching early algebra based on cross-cultural lessons. Cross-cultural studies are extremely powerful in identifying effective instructional insights (Cai, Lew, Morris, Moyer, Ng, & Schmittau, 2005; Cai & Moyer, 2008; Miller & Zhou, 2007; Stigler & Hiebert, 1999).
The overall project was design-based research with the first two years devoted to videotaping teachers’ lessons without researcher intervention. In particular, year 1 focused on inverse relations and year 2 focused on the basic properties of operations. Each teacher taught four lessons on either topic. During year 3 of the project, we analyzed a total of 136 cross- cultural videos and delivered the project intervention through an online- video forum and in-person summer workshops. After the intervention, teachers in each country re-taught their lessons in year 43. In addition to video data, we administered pre- and post-tests on additive and multiplicative inverses, as well as the basic properties of operations, to students from both countries. The main data source in this book is teaching videos along with typical student work, mainly from years 1 and 2 of the project.
We analyzed the cross-cultural videos on inverse relations and basic properties of operations with attention to teachers’ use of worked examples, representations, and deep questions (Pashler et al., 2007). Findings suggest the approach of Teaching through Example-based Problem Solving (TEPS). As described in the Preface, this approach is different from a traditional way of showing the full solution of a worked example to students. It is also different from a more extreme problem-solving approach in which students receive minimum guidance. Rather, this approach aims to establish the worked example effect by engaging students in the problem-solving process. Specifically, the worked example is situated in a concrete story problem context which will be solved to illustrate the targeted mathematical concepts. During this process, concrete representations are gradually faded out (e.g., from word problem context to drawings/manipulatives to number sentences). In addition, deep questions should be asked to elicit students’ understanding of the meaning of operations, representational connections, and comparisons between varied solutions to gauge students’ understanding of the targeted mathematical concepts.
In the following chapters, I will describe the elements of TEPS based on observation from cross-cultural lessons. Roughly, the additive lessons are relevant to grades K-2 curricula while the multiplicative lessons are aligned with content from grades 3-5 in the U.S. For each subtopic, I will report the observed Chinese and U.S. approaches to typical worked examples, identifying insights in representation uses and deep questions during this process. As explained earlier, I will document the Chinese approaches more heavily, supplemented with U.S. insights. However, gaps in the observed cultural differences will be analyzed to highlight possible paths for implementing the TEPS to maximize student learning.