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This study examined teachers‘ knowledge for teaching algebra, with a particular focus on teaching the concept of function and quadratic relations in China and the United States. An embedded mixed methods design was adapted, a design in which the main data set consists of written answers to a questionnaire, while the supportive data set is comprised of the written answers to open-ended questions and follow-up interviews. A structural equation model was adopted to analyze the status and structure of teacher knowledge for teaching algebra in China and the U.S. A qualitative analysis of the answers to the open-ended questions and follow-up interviews is aimed to further illustrate and interpret the quantitative findings. Three hundred and seventy six Chinese and 115 U.S. prospective middle and high school mathematics teachers participated in this survey. Based on an extensively quantitative and qualitative data analysis, the following conclusions were made. First, the Chinese participants demonstrated a stronger knowledge for teaching algebra when compared with their U.S. counterparts. Second, the structure of knowledge for teaching algebra of the Chinese participants is much more interconnected than that of their U.S. counterparts. Third, the Chinese participants showed flexibility in choosing appropriate perspectives of function concept and in selecting multiple representations in contrast to their U.S. counterparts. Fourth, this flexibility is found to be closely related to school math and teaching math. Finally, the number of college math and math education courses taken impacts teachers‘ knowledge for teaching algebra. The findings of this study hold several implications for mathematics teacher preparation in general and studies on mathematics teachers‘ knowledge in particular. Theoretically, the complexity of understanding and measuring mathematics teachers‘ knowledge for teaching was examined and discussed. This study also enriches the understanding of mathematics teachers‘ knowledge for teaching at middle and high schools in China and the United States. Specifically, the Chinese practice of developing teachers‘ basic knowledge, skills, and flexibility provides an alternative for U.S. mathematics teacher educators to reflect on their practice. Practically, what we can learn from this study to improve mathematics teacher preparation in China and the U.S. is discussed. Finally, the limitations of this study are discussed and further studies are suggested.

For students with learning disabilities, positive academic achievement outcomes are a chief area of concern for educators across the country. This achievement emphasis has become particularly important over the last several years because of the No Child Left Behind legislation. The content area of mathematics, especially in the higher order thinking arena of algebra, has been of particular concern for student progress. While most educational research in algebra has been targeted towards remedial efforts at the high school level, early intervention in the foundational skills of algebraic thinking at the elementary level needs consideration for students who would benefit from early exposure to algebraic ideas. A key aspect of students' instruction with algebraic concepts at any level is the degree and type of preparation their teachers have received with this content. Using a mixed methods design, the current researcher investigated the usage of the Developing Algebraic Literacy (DAL) framework with preservice special education teacher candidates in an integrated practicum and coursework experience. Multiple survey measures were given at pre-, mid-, and post- junctures to assess teacher candidates' attitudes about mathematics, feelings of efficacy when teaching mathematics, and content knowledge surrounding mathematics. An instructional knowledge exam and fidelity checks were completed to evaluate teacher candidates' acquisition and application of algebraic instructional skills. Focus groups, case studies, and final project analyses were used to discern descriptive information about teacher candidates' experience while engaging in work with the DAL framework. Results indicated an increase in preservice teachers' attitudes towards mathematics instruction, feelings of efficacy in teaching mathematics, and in the content knowledge surrounding mathematics instruction. Instructional knowledge also increased across preservice teacher candidates, but abilities to apply this knowledge varied across teacher candidates', based on their number of sessions working with students within their practicum site. Further findings indicate the desire of preservice teachers to increase the length and number of student sessions within the DAL experience, as well as the need for increased levels of instructional support to enhance their own experience. This study provides preliminary support for utilizing the DAL instructional framework within preservice teacher preparation experiences for future special educators.