“Knowledge of students” is a core domain of professional knowledge. According to the Professional Standards for Teaching Mathematics (NCTM 1991), both initial and continuing professional education should offer teachers opportunities to develop understandings about these topics:
Deborah Loewenberg Ball
This article reports an analysis of 19 prospective elementary and secondary teachers' understanding of division. Interview questions probed the prospective teachers' understanding of division in three contexts. Although many of the teacher candidates could produce correct answers, several could not, and few were able to give mathematical explanations for the underlying principles and meanings. The prospective teachers' knowledge was generally fragmented, and each case of division was held as a separate bit of knowledge.
Mark Hoover Thames and Deborah Loewenberg Ball
Effectively helping others learn is demanding work that necessitates sensibility as well as specialized knowledge and skill.
Deborah Loewenberg Ball and Thomas L. Schroeder
Deborah Loewenberg Ball and Susan N. Friel
In March of this year, NCTM published the Professional Standards for Teaching Mathematics (Professional Teaching Standards) (1991), a companion to the earlier Curriculum and Evaluation Standards for School Mathematics (Curriculum and Evaluation Standards) (1989). Whereas the earlier document focuses on curriculum, the new document addresses teaching. It elaborates the Curriculum and Evaluation Standards's vision of teaching, in which mathematical reasoning, problem solving, communication, and connections are central. It addresses such questions as, What are classrooms like in which students are able to encounter, develop, and use mathematical ideas and skills in the context of genuine problem and situations? What role might a teacher play in helping students learn to use a variety of resources and tools, such as calculators and computers, and concrete and pictorial models? What is meant by engaging students in mathematical reasoning—in making conjectures, presenting arguments, constructing proofs—at various grade levels? How can adequate mathematical skill be developed in concert with mathematical reasoning? The list of questions can be extended indefinitely, for what we are trying to create is quite different from what we experienced when we were in school and even quite different from much of what we are doing now as teachers.
Deborah Loewenberg Ball
Edited by Susan N. Friel
Despite its title, the Professional Standards for Teaching Mathematics (NCTM 1991) should not be read as a set of prescriptions about how to teach. The document will not deliver on such expectations, not because it fails but because no document can prescribe good teaching. No set of standards can be expected to stipulate what teachers should do. The potential of the Professional Teaching Standards rests instead in its use as a set of tools with which to construct productive conversations about teaching. It should be viewed as a resource with which to build teaching rather than as a measuring stick by which to judge teaching. With new ideas about things to pay attention to in our classrooms, to ask ourselves, to wonder about, we would have increased power to analyze and improve our teaching — alone and as members of a wider community of educators. In this article I explore possible outcomes of using the Professional Teaching Standards in such ways.
Heather C. Hill and Deborah Loewenberg Ball
Widespread agreement exists that U.S. teachers need improved mathematics knowledge for teaching. Over the past decade, policymakers have funded a range of professional development efforts designed to address this need. However, there has been little success in determining whether and when teachers develop mathematical knowledge from professional development, and if so, what features of professional development contribute to such teacher learning. This was due, in part, to a lack of measures of teachers' content knowledge for teaching mathematics. This article attempts to fill these gaps. In it we describe an effort to evaluate California's Mathematics Professional Development Institutes (MPDIs) using novel measures of knowledge for teaching mathematics. Our analyses showed that teachers participating in the MPDIs improved their performance on these measures during the extended summer workshop portion of their experience. This analysis also suggests that program length as measured in days in the summer workshop and workshop focus on mathematical analysis, reasoning, and communication predicted teachers' learning.
Heather C. Hill, Deborah Loewenberg Ball and Steven G. Schilling
There is widespread agreement that effective teachers have unique knowledge of students' mathematical ideas and thinking. However, few scholars have focused on conceptualizing this domain, and even fewer have focused on measuring this knowledge. In this article, we describe an effort to conceptualize and develop measures of teachers' combined knowledge of content and students by writing, piloting, and analyzing results from multiple-choice items. Our results suggest partial success in measuring this domain among practicing teachers but also identify key areas around which the field must achieve conceptual and empirical clarity. Although this is ongoing work, we believe that the lessons learned from our efforts shed light on teachers' knowledge in this domain and can inform future attempts to develop measures.