TEPS: The Teaching of Early Algebra
Based on the U.S. and Chinese expert teachers’ early algebra lessons in the project, I have identified four key components of TEPS. These components align with the overarching instructional principles recommended by the IES (Pashler ct ah, 2007). Additionally, they contribute new insights and details into how these recommended principles can be applied to teaching arithmetic in ways that develop elementary students’ algebraic thinking. Among these four components, the first two are related to representation use and the latter two are related to deep questioning (see Figure 6.1). Although these components cannot be completely separated from one another, I will discuss each component individually for clarity:
Situating a Worked Example in a Real-world Context
The first component of the TEPS approach is to situate a worked example in a real-world context. Depending on the grade level, this concrete context could be a combination of pictures and words in various arrangements (e.g., picture only, picture with words, picture with narrative statements, narrative statements only). Regardless of the format, the illustration of the concrete contexts should contain only essential information. In my project, almost all new Chinese lessons situated the worked examples in real- world contexts that were familiar to students (e.g., swimming, making
Figure 6.1 A summary of the key components of TEPS.
flowers, planting trees, jumping rope). For instance, when teaching inverse relations, the worked example was situated within the context of children in a swimming pool; when teaching the associative property of addition, the lesson began with a story context about children jumping rope and kicking shuttlecocks. Although I observed this feature to be common in the Chinese lessons, it was only intermittently applied in U.S. classrooms. Indeed, the literature shows that many U.S. teachers and textbooks consider word problems an application of computational skills. Thus, real- world contexts are often kept at the end of a lesson after students have learned computational strategies (Koedinger & Nathan, 2000). In my project, the feature of situating a worked example in a real-world context to teach the new concepts was quite interesting to our U.S. teacher participants during our cross-cultural lesson exchange (which I will elaborate upon later in this section).
Situating a worked example in a real-world context has at least four benefits. First, a real-world context, in comparison with abstract number sentences, may immediately attract students’ attention at the very beginning of a lesson in ways that can help boost students’ interest in learning a topic. Given that many students have math phobia, especially as their grade level increases, student interest is an important factor to consider. I would like to point out that this is not an attempt to activate student interest by using irrelevant activities to cheer students up. Rather, the presented real-world context is directly related to the information that will be learned and, indeed, should be chosen carefully to accurately reflect the concept being taught.
Second, a real-world context provides opportunities for problem solving and problem posing, both of which are critical mathematical thinking skills (Singer et al., 2015). As seen in many of the observed Chinese lessons in the lower grades, teachers would often project a pictorial context (without solutions) on the board and ask students to verbalize what they noticed about the mathematical information inherent in the picture. Students were then requested to pose a relevant mathematical problem based on what they were observing. Only after the story problem was posed would the class move on to problem solving, which then would often occupy a majority of the class time. In addition, I also observed that some U.S. teachers asked students to create story problems for a given numerical equation. Even though this is a very different process, students did demonstrate great ability and interest in problem posing and solving.
Third, a real-world context can provide a teacher with opportunities to pose deep questions that can help students make sense of the targeted mathematical operations and big ideas that are, in essence, abstract. Recall that in most chapters, Chinese teachers prompted students to explain their ideas by referring back to the story context of the problem. The teachers used this to help students tackle concepts such as the reasonableness of an operation (e.g., Why did you use addition?), the inverse relationships between operations (e.g., How are these two solutions related?), and the basic properties embodied by the solutions (e.g., Why does this generated equation make sense?). These prompts allowed students to explain the abstract ideas using their understanding of real-world contexts as a guide. Likewise, some U.S. teachers also asked deep questions to promote sensemaking based on the real-world contexts (e.g., What does that mean that Teacher Kelly has four times as many apples? Who can tell a story about this [array mode] that will help people understand?)
Last but not least, starting a worked example with real-world context (e.g., word problems) provides opportunities for implementing concreteness fading, a powerful strategy supported by the cognitive learning theory (Bruner, 1966) and recent empirical studies (Fyfe et al., 2015; McNeil & Fyfe, 2012). Given that word problems contain specific real-world contexts, they are arguably more concrete than semi-concrete manipulatives (e.g., cubes or eounters) and diagrams (e.g., tape diagrams, arrays) that do not have speeifie contextual meanings (Ding & Li, 2014). However, as mentioned earlier, studies (e.g., Koedinger & Nathan, 2000) have reported that U.S. teachers and textbooks often perceived word problems as harder than computation tasks and thus tended to arrange word problems at the end after the students had been learned computational strategies. This is not consistent with how students learn because students perform better at solving simple story problems than parallel computation tasks (Koedinger et al., 2008). The Cognitive Guided Instruction project also reported that elementary children have great potential to reason about types of word problems (Carpenter et al., 1999). As such, word problems can serve as an effective context and starting point to discuss a worked example.