Guiding Principles

In addition to procedural fluency, study in mathematics involves simultaneously developing skills in problem solving, reasoning, connections, communication, and conceptual understandings. Students should also recognize and apply mathematics in contexts outside the discipline. Students need experience applying mathematics concepts and representations to describe and predict events in almost all academic disciplines, as well as in the workplace as we develop a fully informed citizenry.

Problem Solving

Problem solving means working through a problem for which a solution is unknown in advance. Students must draw on previously acquired knowledge and skills, conducting a process that involves deconstructing a problem, identifying patterns, making deductions and inferences, drawing conclusions, and testing solutions. Solving problems is both the goal and the means of mathematics, which equips students to develop new mathematical understandings. Students should have frequent opportunities to formulate, grapple with, and solve complex problems and reflect on their thinking and process.

Reasoning and Proof

Mathematical reasoning and proof offer powerful ways of developing and expressing insights about a wide range of phenomena. Those who reason and think analytically tend to note patterns, structure, or regularities in both real-world situations and symbolic objects; they ask whether those patterns are accidental or whether they occur for a reason; and they conjecture and prove. Ultimately, a mathematical proof is a formal way of expressing particular kinds of reasoning and justification.

Being able to reason is essential to understanding mathematics. By developing ideas, justifying results, and using mathematical conjectures in all content areas and at all grade levels, students should recognize and expect that mathematics makes sense. Building on the considerable reasoning skills that children bring to school, teachers can help students learn what mathematical reasoning entails.

Communication

Communicating mathematical thinking and reasoning is an essential part of developing understanding. It is a way of sharing and clarifying ideas. Through communication, ideas become objects of reflection, refinement, and discussion and often require adjustments of thinking. The communication process also helps build meaning and permanence for ideas and makes them public. When students are challenged to think and reason about mathematics and communicate the results of their thinking with others, they learn to be clear and convincing in their verbal and written explanations. Listening to others explain gives students opportunities to develop their own understanding. Conversations in which mathematical ideas are explored from multiple perspectives help learners sharpen their ability to reason, conjecture, and make connections.

Connections

Too often individuals perceive mathematics as a set of isolated facts and procedures. Through curricular and everyday experiences, students should recognize and use connections among mathematical ideas. Of great importance are the infinite connections between algebra and geometry. These two strands of mathematics are mutually reinforcing in terms o concept development and the results that form the basis for much advanced work in mathematics as well as in applications. Such connections build mathematical conceptual understanding based on interrelationships across earlier work in what appear to be separate topics.

(Source: National Council of Teachers of Mathematics)