**Ma, L. (1999). Profound understanding of mathematics: When and how is it attained? In A. Schoenfeld (Anniversary ed.), Knowing and teaching elementary mathematics: Teachers’ understanding of fundamental mathematics in China and the United States. (pp. 135–143). New York, NY: Routledge. Available for purchase on Amazon.com.
This chapter illustrates when and how teachers attain a profound understanding of fundamental mathematics (PUFM) —an awareness of the conceptual structure and attitudes towards mathematics combined with the ability to provide a foundation for that structure and instill the attitudes necessary for student mastery of mathematics. The author uses a series of interviews with preservice teachers and ninth grade students in China to examine the teacher education program and the mathematical knowledge a prospective teacher carries upon entering a teacher education program. A closer examination of three teachers identified as having PUFM explores the working conditions that provide support for the growth of their mathematical knowledge and its organization for teaching. The chapter provides insight for how deep understanding of mathematical concepts and attitudes can inform teachers as they instruct students.
Ball, D. L., Hill, H. C., Bass, H. (2005). Knowing mathematics for teaching: Who knows mathematics well enough to teach third grade and how can we decide? American Educator (fall 2005), 14–22, 43–46. Available at http://www.aft.org/sites/default/files/periodicals/BallF05.pdf
This article argues that there is a specialized mathematical knowledge that is unique to teaching and the work that teachers do. Specifically, the authors assert that teachers need fluency with skills such as error analysis and anticipation, modeling, representations, reasoning, example selection, and an understanding of how and when to shift between informal and technical, content-specific language. In their research, the authors assessed the “common” knowledge of mathematics (content knowledge) and “specialized” knowledge (teacher-specific knowledge) of a large sample of teachers. They found that teachers’ specialized knowledge significantly predicts student test scores, even when controlling for student socioeconomic status, student absence rate, teacher credential, teacher experience, and average length of math lessons. The researchers also found that teachers can learn these specialized skills in professional development, especially if the learning opportunities focus on proofs, analysis, exploration, communication, and representation. The article concludes that a focus on developing the specialized mathematical knowledge of teachers could raise student achievement, improve teaching for disadvantaged students, and improve the professional standing of math teachers.
Lewis, C., Perry, R., & Hurd, J. (2004). A deeper look at lesson study. Educational Leadership (February 2004), 18–22. Available at http://www.lessonresearch.net/DeeperLookatLS.pdf
This article identifies key pathways to instructional improvement that emerged from interviews and recent studies of American educators engaged in lesson study—a professional development approach designed to improve instruction that originated in Japan. The authors contend that lesson study’s visible features of planning, observing, and rethinking a lesson combined with the innovative prospects that emerge from key pathways toward improvement can support sustainable and successful instructional improvement. A teacher-led lesson study effort in San Mateo-Foster City School District in California illustrated the ways in which lesson study connects to seven pathways toward instructional improvement: (1) increased knowledge of subject matter, (2) increased knowledge of instruction, (3) increased ability to observe students, (4) stronger collegial networks, (5) stronger connection of daily practice to long-term goals, (6) stronger motivation and sense of efficacy, and (7) improved quality of available lesson plans.
Inoue, N. (2011). Zen and the art of neriage: Facilitating consensus building in mathematics inquiry lessons through lesson study. Journal of Math Teacher Education, 14, 5–23. Available at http://dx.doi.org/10.1007/s10857-010-9150-z
This research article argues for the value of lesson study focused around the development of consensus building discussion—the Japanese concept of neriage—to facilitate learning in the mathematics classroom. The author compares traditional classroom dialogue and consensus building discussion to demonstrate how neriage leads to stronger inquiry-based mathematical learning. The article emphasizes the positive impact of the lesson study experience on teacher growth and subsequent student learning, and a suggestion for how to convene a lesson study cohort.
**This document is considered a priority reading.