Explicitness in Creating Access for all Students to Complex Mathematics

• In Event: 33.040 - Revisiting Explicit Instruction in English and Mathematics

Abstract

Purpose and Theoretical Framework
This paper explores what is involved in making mathematical content and practices explicit for students. Our research takes a perspective situated between two views of instruction that have divided mathematics education: (1) a tradition of “direct instruction” in which teachers show students what to do and students follow teachers’ direction (e.g., Good, Grouws, & Ebmeier, 1983), and (2) a perspective on learning rooted in constructivism, which prioritizes students’ opportunities to figure things out themselves (e.g., von Glasersfeld, 1995). On one hand, for students to have equitable access to challenging mathematical work, the teacher’s role in explaining, modeling, and highlighting key ideas and strategies is crucial. The goals promoted in the Common Core entail significant levels of reasoning, understanding, and problem solving, disciplinary work that is not entirely natural and for which students need support. On the other hand, if students are to develop understanding of core mathematical ideas, their opportunities to practice mathematical work, to struggle and persevere, and to engage firsthand in mathematical problems and discourse are central. Too much instructional guidance could shortchange students’ learning by degrading the cognitive demand of the tasks and discourse (Stein, Grover, and Hennigsen, 1996), but too little structure can leave students without adequate support.

Data Sources/Methods
Our analyses are part of a program of research over the last 25 years, which has engaged in the design and study of equitable and mathematically rigorous instruction. Our central goal has been to understand what is involved in making complex mathematical work learnable by students who have often not had successful experiences with mathematics (Author, 2008) . We have identified three commitments important to explicitness in teaching mathematics: creating access, scaffolding students’ learning, and the enabling authentic mathematical work by all students. Attending to these commitments requires balance and can create dilemmas of what to make explicit and what to leave open. The paper focuses on explicitness by unpacking what is involved in acting on these commitments in the context of three instructional practices: (a) choosing and using mathematical tasks; (b) asking questions and (c) public recording.

Results
The analyses suggest that explicitness is not an abstract quality of teaching independent of context. Instead, what to make explicit and in what ways demands sensitivity to the specific mathematics at hand and to the experiences that one’s particular students bring to their mathematical encounters. In addition to what to make explicit, the specific techniques of making content explicit that we describe and analyze in this paper include highlighting, naming and labeling, co-sponsoring, and reflecting.

Significance
“Explicitness” in teaching is not an uncontroversial idea in mathematics education. Some advocates promote it as a lever for attending to equity in access and learning opportunity; others criticize it for reducing the complexity of instruction and thereby curtailing access and opportunity. Our investigation of making mathematical content and practice “explicit” shows that “explicitness” itself is an undetermined idea. Our analyses highlight key considerations and techniques for making mathematics explicit to advance equity and complex learning.

Authors

• Deborah Loewenberg Ball, University of Michigan - Ann Arbor

• Lindsey Mann, University of Michigan

• Meghan M. Shaughnessy, University of Michigan