Sources we love…

Ball, D. & Bass, H. (2000). Making believe: The collective construction of public mathematical knowledge in the elementary classroom. In D.C. Philips (Ed.) Constructivism in Education: Opinions and second opinions on controversial issues. Chicago: University of Chicago Press.

This work explores the construction of mathematical knowledge in classroom teaching and learning, focusing on the mathematical reasoning and constructed knowledge within a community of third grade students. Ball and Bass state that publicly shared knowledge and mathematical language work together to determine the granularity of and the medium in which claims are developed within mathematical reasoning, respectively. The authors underscore three fundamental concepts that link the construction of mathematical knowledge to the work of teaching: (a) the role of reasoning/justification in the construction of mathematical knowledge; (b) what it means to support the identification, use, and ongoing development of publicly shared knowledge base; and (c) the significance of mathematical language in reasoning. The authors challenge educators to work toward bridging the gap between theory and teaching practice.

Bransford, J. D., Brown, A. L., & Cocking, R. R. (Eds.). (2000). How people learn: Brain, mind, experience, and school. Washington, D.C.: National Academy Press.

 
This book presents findings of research on the brain and the process of learning with implications for what and how we teach. Specific examples and general recommendations are provided.

Boaler, J., & Humphreys, C. (2005). Connecting mathematical ideas: Middle school video cases to support teaching and learning. Portsmith, NH: Heinemann.

This book is accompanied by a CD containing video clips to match the reading.  The book is a chance for a teacher (Humphreys) to accurately reflect on her teaching practice.  Each chapter addresses a certain lesson (and matching video clip).  The chapter begins with the teacher introducing the lesson and her goals for the lesson.  Readers are then prompted to watch the clip and then read along with her analysis of her discussion.  Finally, a professor of mathematics education (Boaler) will analyze what she noticed in the clip.  The book makes a great case for the use of video case studies as well as focuses on some major educational topics (highlighted by Boaler’s analysis).

Cohen, E. G. (1994). Designing groupwork (2nd ed.). New York: Teachers College Press.

Properly designed groupwork has evolved into a powerful tool for teaching all students, especially those from diverse backgrounds. The time has come to bring teachers up-to-date on the latest developments in the field. Designing Groupwork combines easy-to-follow theory with examples and teaching strategies that are adaptable to any situation. The advantages and dilemmas of groupwork are discussed, as well as it’s use in multiability and bilingual classrooms, and step-by-step approaches to successful planning, implementation, and evaluation of groupwork activities. The 2^nd edition includes new material on skill-building for more advanced students, on the development of roles for older and younger students, on how to use multiple ability treatments and how to avoid common pitfalls, on cooperation and anti-social behavior, and on a new treatment for status problems (From the back cover).

Cohen, E. G. (1994). Status treatments for the classroom [VHS]. New York: Teachers College Press.

In a cooperative learning situation, how can a teacher keep a popular student from taking over a small group? How can a student with lower social status be kept from fading into the background? This video, designed to accompany the book " Designing Groupwork: Strategies for the Heterogeneous Classroom " by Elizabeth G. Cohen, outlines techniques for promoting social equity in the classroom. Narrated by Cohen.

Findell, B., Kilpatrick, J., & Swafford, J. (2001). Adding It Up: Helping Children Learn Mathematics. Center for Education (CFE).

This report from the National Research Council discusses the state of math education in the early grades (Grades K – 8).  The report suggests reform methods for both teachers and teacher educators to improve the understanding of elementary and middle school mathematics students.  The report shares research on what mathematical knowledge students “bring to the table” in kindergarten as well as how students refine and improve their mathematical knowledge.  The later stages of the report focus specifically on what teachers can do to improve their teaching practices.

Henningsen, M.A., & Stein, M.K. (2002). Supporting students' high-level thinking, reasoning, and communication in mathematics. Lessons learned from research (pp. 27-35). Reston: National Council of Teachers of Mathematics.

The chapter in this book covered two factors that support students' high-level thinking. One factor involved teachers as an important link to the way students engage in high-level thinking. It described the role of the teacher as being two-fold in that the teacher must select mathematical tasks that are appropriate to a high cognitive level and also intentionally and consistently provide students with support and scaffolding without interrupting the complexity and cognitive demands of the task. Another factor involved a description of the framework and language used to label classroom events that would otherwise be difficult to describe effectively and clearly. In doing so, classroom-based factors were organized into five categories: task conditions and appropriateness; quality of communication of mathematical ideas; sustaining thought over time; tailored assistance, and classroom norms.

Kazemi, E. (1998). Discourse that promotes conceptual understanding. Teaching Children Mathematics, 411-415.

 
This article focuses on the types of questions and teacher interactions that produce high-press conceptual thinking. Low-press classrooms are characterized by superficial thinking, questioning that elicits responses about process rather than conceptual understanding, and prioritizing social norms over sociomathematical norms. High-press classrooms are characterized by four main components: (a) an explanation consisting of a mathematical argument (not just a description of procedure); (b) an understanding of the relationships among differing mathematical strategies; (c) errors are valued, providing opportunities for deeper conceptual mathematical understanding; and (d) collaborative work involves individual accountability and reaching consensus or agreement through mathematical argumentation. The teacher makes a profound difference on whether or not his/her classroom will be high-press or low-press; progress requires teacher questioning that encourages students to look deeply at why their conjectures make sense.

Lampert, M. (1990). When the problem is not the question and the solution is not the answer: Mathematical knowing and teaching. American Educational Research Journal, 27(1), 29-63.

This essay is a description of a research and development project designed to examine whether or not it is possible to bring the practice of “knowing mathematics in school closer to what it means to know mathematics within the discipline.” The essay describes the activity of a class of fifth graders whom the author believes to have learned to do mathematics together in a way that is consistent with Lakatos and Polya’s assertions about what doing and knowing mathematics involves. That is, these students participate in mathematical arguments with both courage and modesty, and are able to learn about the establishment of mathematical assertions through discourse as they “zig-zag” between their own observations and generalizations, proofs and refutations – revealing and testing their own ideas as they go along. In addition to illustrating what students can learn about how to participate in the doing and learning of mathematics, this case study also shows how the teacher can act to create and maintain the culture in which such discourse and learning can occur.

Stein, M. K., Smith, M. S., Henningsen, M. A., & Silver, E. A. (2000). Implementing standards-based mathematics instruction: A casebook for professional development. New York: Teachers College Press.

The authors examine mathematics instructional tasks in terms of their cognitive demand, which indicates the “kind and level of thinking require of student in order to successfully engage with and solve the task” (p11). Tasks are classified according to their level of cognitive demand: memorization & procedures without connections (lower level demand); and procedures with connections & doing mathematics (higher level demand). The authors use a Mathematical Task Framework (Tasks as they appear in curriculum -> Tasks as set up by teachers -> Tasks as implemented by students -> Student Learning) to guide lesson analysis, and comment that teachers like to use it as a lens for reflection on their practices, since decline of demand often decreases as lessons are implemented. The remainder of the book illustrates these ideas vis-à-vis classroom cases.

Van de Walle, J. A. (2004). Elementary and Middle School Mathematics: Teaching Developmentally (5^th ed.). Pearson Education Inc.

This text is an excellent reference for all mathematics teachers. The first eight chapters explore learning mathematics and teaching practices from a constructivist perspective.

Additional topics of study include assessment, lesson planning, students with special needs, and the use of technology in the classroom. Chapters nine through twenty-four explore specific mathematical topics, how students learn this content and problem-based activities that support learning. Additional features that will strengthen teachers’ understanding of the content include a focus on “big ideas”, NCTM standards correlations, assessment notes, technology notes and opportunities for teacher reflections.

Yackel, E. & Cobb, P. (1996). Sociomathematical norms, argumentation, and autonomy in mathematics. Journal for Research in Mathematics Education, 27(4), 458-477.

This purpose of this paper is to show how students and teachers develop specific mathematical beliefs and values within the classroom discourse. Yackel & Cobb establish the differences between social norms and sociomathematical norms. The authors further explain that mathematical activity is an interactive between transmitting knowledge and understanding how and when to apply that knowledge. They describe three parts in the development of math discourse in their study: (a) The process by which sociomathematical norms are established in the classrooms they observed; (b) how these sociomathematical norms regulate math argumentation and learning for students and teachers; and (c) how students’ and teachers’ justification of acceptable mathematical discourse evolves over time as the math community and understanding develops. The following quotes seemed particularly interesting: “…the teacher plays a central role in establishing the mathematical quality of the classroom environments and in establishing norms for mathematical aspects of students’ activity. It further highlights the significance of the teacher’s own personal beliefs and values and their own mathematical knowledge and understanding.” ; “The understanding of learning and teaching mathematics... supports a model of participating in a culture rather than a model of transmitting knowledge…” (Bauersfield)

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Boaler, J. (2002). Experiencing School Mathematics: Revised and Expanded Edition. NJ: Lawrence Erlbaum Associates, Inc.

Cochran-Smith, M. (2004). Walking the Road: Race, Diversity, and Social Justice in Teacher Education. New York: Teachers College Press.

Dichter, A., McDonald, E. C., McDonald, J. P., & Mohr, N. (2003). The Power of Protocols: An Educator’s Guide to Better Practice. New York: Teacher College press.

Donovan, M. S., & Bransford, J. D. (2005). Introduction. In M. S. Donovan & J. D. Bransford (Eds.), How students learn: History, math, and science in the classroom (pp. 1-30). Washington, D.C.: National Academy Press.

Erickson, T. (1989). Get It Together: Math Problems for Group Grades 4-12. CA: Equals Lawrence Hall of Science.

Lortie, D. C. (1975). Schoolteacher. Chicago: University of Chicago Press.

Ma, Liping. (1999). Knowing and teaching elementary school mathematics. NJ: Lawrence Erlbaum

Tyack, D., & Cuban, L. (2003). Tinkering toward utopia. Cambridge, Massachusetts: Harvard University Press.

http://archive.cabinetoffice.gov.uk/servicefirst/2000/learninglabs/Framework/staff_practical.htm