# Beyond Teaching Early Algebra: More Implications

Even though the TEPS components were gleaned from cross-cultural early algebra lessons, I believe that this approach can be applied to teach mathematics topics beyond early algebra in elementary school. The components align with the selected IES recommendations (Pashler et al., 2007), which have general applicability to enhance classroom teaching. Additionally, this approach draws heavily from insights taken from Chinese lessons that cover topics beyond early algebra. Recall that many videotaped lessons in this project involved informal teaching of the targeted topics and indeed focused on a variety of other concepts (e.g., computational algorithms, problem solving). As such, TEPS has the potential for a broader applicability' than teaching early algebra alone.

Consider, for example, the topic of fraction multiplication with a worked example of — x 12. A common computational procedure is to first simplify 3 and 12 to obtain 4 and then use 2 x 4 to obtain the answer of 8. To help students make sense of this computational procedure, a teacher can situate this example in a real-world context such as “Mary picked 12 flowers. She shared 2/3 of them with her friends. How many flowers did she share?” After students agree that this problem needs to be solved by 2

— x 12, the teacher can use 12 counters to help students understand the computational method based on modeling. According to the meaning of a fraction, 2/3 of 12 flowers means that one should divide the total flowers (n = 12) into three equal parts and consider two of these parts. Since each part would be 12 -r 3 = 4 (flowers), two parts would be 2 x 4 = 8 (flowers). As such, 2/3 x 12 = 8 (flowers). The above modeling and reasoning process with counters explains the computational procedure described above. Of course, to promote understanding, the teacher should also ask deep questions that elicit student explanations of the processes (e.g., Why did we simplify 3 and 12 to obtain 4? What does this “4” mean in the counter model or in the story situation?). Such questions likely promote connection-making between the concrete and abstract representations and reveal the undergirding ideas behind the example task.

Components of TEPS have been piloted in my project. As noted earlier, the videotaped lessons shared in this book were mainly collected during the first two years of the project. These videos were then analyzed to identify the cross-cultural instructional insights that arc discussed in this book. The main insights (e.g., concreteness fading, deep questioning) were shared with all teachers in both countries as part of the project intervention. I have reported in detail what teachers seem to have learned from the cross-cultural videos (Ding, Manfredonia & Luo, 2018, April). After this project intervention, teachers in both countries re-taught their lessons to implement what they had learned. An analysis of these post-intervention videos and corresponding teacher interviews indicate that teachers intentionally implemented the above components, although perhaps to different degrees (Ding, Spiro, & Mochaourab, 2021, April). Many of the U.S. teachers expressed interest in the concreteness fading approach and some shared their successes and challenges with employing this approach at the research conference of the National Council of Teachers of Mathematics (Milewski & Varano, 2019). For instance, they had to adapt their textbook materials due to a lack of available concrete contexts and images. To do so, they created their own pictures or found relevant pictures from the Internet. Some even took photos of their own students doing activities (e.g., reading books, making cookies, playing in the playground) and used those to support the TEPS activities. This effort was noted repeatedly even after the project videotaping had ended. The above observations indicate the promise and feasibility of implementing TEPS for teaching elementary mathematics in U.S. classrooms.

Teachers’ reported challenges, such as the lack of textbook support, call for the attention of textbook designers. In fact, as frequently shown by our findings, many U.S. teachers in this project followed their textbook presentations, which do not necessarily encourage concreteness fading and deep questioning. For instance, the textbooks often present the definition of a property at the beginning of a lesson, followed by an example task to illustrate that conclusive statement. Some textbook lessons do not contain a real-world situation or deep questions. These findings are consistent with prior comparative textbook studies (Ding & Li, 2010, 2014; Ding, 2016). In addition, textbooks sometimes separate the teaching of the same concept across different lessons with a focus on different representations. One lesson teaches how to use manipulatives (breaking apart an array to multiply), another abstract calculation (e.g., breaking apart a number to multiply), and the third lesson solving word problems. This tendency to present related pieces of knowledge as separate skills is then passed down to instructors who may not feel comfortable adjusting the textbook in ways that merge these lessons. I suggest textbook designers streamline the lessons and incorporate the TEPS approach during their book revision (e.g., adding relevant story contexts for worked examples, displaying a definition after the worked example, adding deep questions such as “what property of operations did you use in your strategy?”).

Of course, the proposed TEPS approach is not the sole possible method that could work to promote learning in elementary mathematics classrooms. Rather, it is an alternate path to teach based on insights gleaned from the Chinese and U.S. expert teachers’ practices that has been aligned with the IES-reeommended instructional principles. In addition, I have observed and catalogued variations between teachers’ instructional approaches that might serve to enrich and inform the components this book has suggested. For instance, Chinese teachers often facilitate a folding back from abstract to concrete to strengthen meaning-making while some U.S. teachers ask students to ereate a story problem for a given number sentence. These choices should not be seen as a “violation” of concreteness-fading, but rather methods of recontextualizing a flexible tool. Derivations on these themes can work in conjunction with our overarching components of the TEPS approach in a way that serves student needs as they arise.