Learning Environment

The Professional Standards for Teaching Mathematics presents six standards for the teaching of mathematics organized under four headings: tasks, discourse, environment, and analysis—major arenas of teachers' work that are logically central to shaping what goes on in mathematics classrooms. These four arenas are interwoven and interdependent.

• TASKS are the projects, questions, problems, constructions, applications, and exercises in which students engage.
• DISCOURSE refers to the way of representing, thinking, talking, and agreeing and disagreeing that teachers and students use to engage in these tasks.
• ENVIRONMENT represents the setting for learning.
• ANALYSIS is the systematic reflection in which teachers engage.

Source: National Council of Teachers of Mathematics. (1991). Professional standards for teaching mathematics, pp. 25, 35, 45, 52, 57, 63. Reston, VA: NCTM.

Tasks

Worthwhile Mathematics Tasks

The teacher of mathematics should pose tasks that are based on—

• sound and significant mathematics;
• knowledge of students' understandings, interests, and experiences;
• knowledge of the range of ways that diverse students learn mathematics

and that

• engage students' intellect;
• develop students' mathematical understanding and skills;
• stimulate students to make connections and develop a coherent framework for mathematical reasoning;
• promote communication about mathematics;
• represent mathematics as an ongoing human activity;
• display sensitivity to, and draw on, students' diverse background experiences and dispositions;
• promote the development of all students' dispositions to do mathematics.

Discourse

Teacher's Role in Discourse

The teacher of mathematics should orchestrate discourse by—

• posing questions and tasks that elicit, engage, and challenge each student's thinking;
• listening carefully to students' ideas;
• asking students to clarify and justify their ideas orally and in writing;
• deciding what to pursue in depth from among the ideas that students bring up during a discussion;
• deciding when and how to attach mathematical notation and language to students' ideas;
• deciding when to provide information, when to clarify an issue, when to model, when to lead, and when to let a student struggle with a difficulty;
• monitoring students' participation in discussions and deciding when and how to encourage each student to participate.

Student's Role in Discourse

The teacher of mathematics should promote classroom discourse in which students—

• listen to, respond to, and question the teacher and one another;
• use a variety of tools to reason, make connections, solve problems, and communicate;
• initiate problems and questions;
• make conjectures and present solutions;
• explore examples and counterexamples to investigate a conjecture;
• try to convince themselves and one another of the validity of particular representations, solutions, conjectures, and answers;
• rely on mathematical evidence and argument to determine validity.

Tools for Enhancing Discourse

The teacher of mathematics, in order to enhance discourse, should encourage and accept the use of—

• computers, calculators, and other technology;
• concrete materials used as models;
• pictures, diagrams, tables, and graphs;
• invented and conventional terms and symbols;
• metaphors, analogies, and stories;
• written hypotheses, explanations, and arguments;
• oral presentations and dramatizations.

Environment

Learning Environment

The teacher of mathematics should create a learning environment that fosters the development of each student's mathematical power by—

• providing and structuring the time necessary to explore sound mathematics and grapple with significant ideas and problems;
• using the physical space and materials in ways that facilitate the learning of mathematics;
• providing a context that encourages the development of mathematical skill and proficiency;
• respecting and valuing students' ideas, ways of thinking, and mathematical dispositions.

Woven into the fabric of the Professional Standards for Teaching Mathematics are five major shifts in the environment of mathematics classrooms from current practice to teaching for the empowerment of students. We need to shift—

• toward classrooms as mathematical communities—away from classrooms as simply a collection of individuals;
• toward logic and mathematical evidence as verification—away from the teacher as the sole authority for right answers;
• toward mathematical reasoning—away from merely memorizing procedures;
• toward conjecturing, inventing, and problem solving—away from an emphasis on mechanistic answer-finding;
• toward connecting mathematics, its ideas, and its applications—away from mathematics as a body of isolated concepts and procedures.

Analysis

Analysis of Teaching and Learning

The teacher of mathematics should engage in ongoing analysis of teaching and learning by—

• observing, listening to, and gathering other information about students to assess what they are learning;
• examining effects of the task, discourse, and learning environment on students' mathematical knowledge, skills, and dispositions;

in order to

• ensure that every student is learning sound and significant mathematics and is developing a positive disposition toward mathematics;
• challenge and extend students' ideas;
• adapt or change activities while teaching;
• make plans, both short- and long-range;
• describe and comment on each student's learning to parents and administrators, as well as to the students themselves.