The Program for International Student Assessment (PISA) and the Trends in International Mathematics and Science Study (TIMMS) results have revealed that students’ mathematics performance in Taiwan indicated a lack of student interest and a lack of student confidence and that students in Taiwan did not view skills in mathematics as valuable. Although high performance was noted in a substantial minority of the lowest performing students, the government launched many programs out of concern for disadvantaged and underperforming students, and these programs have been implemented for many years. After several years of efforts to conduct supplemental instruction for low-achieving students in mathematics (without satisfactory results), a new approach has emerged and is the fundamental theme of the present study, namely that teachers’ own beliefs about teaching are an important factor that influences the effectiveness of student learning. Empirical research has indicated that those teachers who contribute to effective student learning share similar teaching beliefs. Their teaching beliefs influence the relationship between learning techniques and culture and help students build on prior learning. Additionally, such teachers devise and present curricula that are relevant to students and that are aligned with their learning needs and cultures. 

The aim of this study is twofold. First, although teaching beliefs have been studied in detail in many contexts, this is the first study of aesthetic knowledge in remedial mathematics teaching.This study provides a crucial practical basis for remedial mathematics teaching in middle school classrooms. Second, the findings of this study should motivate future research in this area by providing a basis for action research. The paper is structured as follows. First, we briefly define Vico’s ideals and methods for transferring those ideals into the regular classroom. Second, our conceptual framework highlights how mathematics learning strategies and mathematics learning emotions influence motivational processes within student learning. Third, we present our conclusions resulting from this framework. Questions and implications for future research arising from the analyses then follow. Finally, we discuss how teachers can apply Vico’s ideals and MSE to actual classroom teaching. 

A central issue in this study is how teachers apply Vico’s ideals and MSE to actual classroom teaching. Providing teachers an opportunity to learn how to teach mathematics while engaging in authentic learning activities which combine both key mathematical and pedagogical concepts is one way to determine how teachers apply those concepts to their own teaching. According to recent studies, The New Science is the major work of Italian philosopher Giambattista Vico (1990), who posited that science without aesthetic knowledge results in a world devoid of imagination and creativity. Vico reasoned that without ontological knowledge of reality, the exact science of mathematics has limited insight into the creation of the objects in the world. Based this theory, we applied Vico’s ideals (1990) as key mathematics teaching concepts to guide the core values underlying the teaching of mathematics. Vico’s ideals contain three layers of aesthetic imagery: (1) the development of being to know the self and understand others, (2) the development of superior technical capabilities, and (3) the development of aesthetic performance skills. Vico’s ideals provide a path to transform teaching from the ideal to the realities of the regular classroom through the three S’s: the spiritual, the scientific, and storytelling; these are all conceptual teaching strategies that can support teachers and ensure effective learning for students. 

From 2013 to 2016, our research team constructed an MSE teaching module for supporting low-achieving students who were learning mathematics with the aid of a mathematics learning strategy (MS). Additionally, the module was designed to improve negative mathematics learning emotions and promote positive mathematics learning emotions (MEs). The question of how researchers and teachers can develop interventions in the mathematics classroom to address the learning needs of students . To achieve these purposes, we conducted collaborative action research focused on qualitative research methods and supplemented by quantitative research methods. To summarize, we focused on two classes and Vico’s ideals were applied to the optimization of one class, and MSE was applied to the optimization of the other class. Empirical research material was collected by combining different methods such as questionnaires, recorded interviews, video recordings of lessons in mathematics, and classroom observations. We used the Learning Strategy Scale and the Academic Emotion Scale for before and after testing. Qualitative data collection and analysis. 

The optimization process through Vico’s ideals was accomplished through the three S’s, comprising the spiritual, the scientific, and storytelling. First, a spiritual teaching task provided the opportunity for greater self-awareness through reflection and critical questioning in collaboration with research group members and the observing teacher. In addition, teaching-bydoing and learning-by-doing activities clarified that changes in teachers’ beliefs precede changes in their teaching practices. Second, in the scientific teaching task, the teacher changed the usual textbooks and paper-based evaluations to employ (1) group cooperation with tabletop games (Pagamo) and line and peer teaching to increase communicative opportunities for mathematics learning, (2) evaluation combined online platforms and multiple evaluation methodologies to cater to students with different learning needs. Third, the storytelling task used (1) digital technology to select the plot of a story, create a context, and provide examples of solving problems with mathematics and (2) anime cartoons from YouTube and story solitaire and test questions to increase student opportunities for using their mathematics knowledge. 

In the optimization process for the model of math self-concept, math learning strategies and math academic emotion (MSE), MSE´s infused teaching and teaching images guided teachers in the teaching of mathematics emotions by careful consideration of his or her own emotions before and after class. Students are highly impressionable and are seeking out role models whose actions they will adopt and mirror. Teachers should therefore attempt to lead by example and inspire students to perform at their best. In the MSE optimization class, students drew or wrote emotions notes to increase their awareness of their mathematics learning emotions and then shared these with others. 

The results revealed that, first, this program provided teachers with a variety of learning experiences and with an opportunity to reconsider the underlying assumptions of their approaches to teaching and may serve as a foundation to strengthen and facilitate communication between teachers and students. Awareness and reflection of issues that affected learning and professional development also increased. Second, the MSE program not only enhanced students’ mathematics learning strategies and awareness of mathematics emotions but also improved the personal growth of teachers. Third, collaborative action research helped teachers interpret the process of teacher inquiry and teaching optimization from Vico’s ideals and MSE teaching models and to implement and experience an optimized curriculum. Through Vico’s ideals and the 3 S’s, practical strategies were provided in different teaching contexts, offering teachers and students different learning experiences and allowing them to engage their imaginations and express their creativity. The research recommendations can be used for reference by mathematics teaching or educational psychology researchers and middle school remedial mathematics teachers.