Scope of the Mathematics Curriculum
Conjectures are estimations based on education or experience. A mathematical argument is a connected series of statements (premises) intended to support another statement (conclusion).
This article will provide an overview of the NCTM process and content standards. Educators first studying the standards may feel overwhelmed with the amount of content addressed within each grade-level span. State frameworks that dictate standards for each grade level exacerbate this situation. However, a longitudinal view will show how the same topics are developed over several years in a spiral and interconnected pattern. For example, the concepts of multiplication and division are introduced in the PreK-2 band, but fluency with these operations isn't expected until the 3-5 band. Multiplication and division skills are used in grades 6-8 with problem solving and algebraic equations and in grades 9-12 with vectors, matrices, and other advanced applications. It is critical to keep in mind that deeper study of a few topics is more important for student learning than covering dozens of discrete topics at a surface level. Less is more!
The process standards address ways of acquiring and using knowledge and are developed across the entire mathematics curriculum. They also can be applied across other content areas and real-world problems. These processes are the "verbs" of math. The role of the teacher is to provide settings, models, and guidance for these processes to develop and to assess student skills in using these processes. The process standards are applied at every grade level and across all five content areas.
Select items from one state's mathematics assessment intstrument. Evaluate the process standards required. For example, the following test item requires all five process skills—problem solving, communication, connections, reasoning, and representation:
Which of the following could be the length of the sides of a triangle?
- 1, 2, 1
- 2, 3, 1
- 3, 5, 4
- 7, 15, 7
Problem solving is a major focus of the mathematics curriculum; engaging in mathematics is problem solving. Problem solving is what one does when a solution is not immediate. Students should build mathematical knowledge through problem solving, develop abilities in formulating and representing problems in various ways, apply a wide variety of problem-solving strategies, and monitor their mathematical thinking in solving problems. Problems become the context in which students develop mathematical under-standings, apply skills, and generalize learning. Students frequently solve problems in cooperative groups and even create their own problems.
Students should learn to reason and construct proofs as essential and powerful aspects of understanding and using mathematics. These processes involve making and investigating conjectures, developing and evaluating arguments, and applying various types of reasoning and methods of proof. Reasoning skills are critical for science, social studies, social skills, literature, and most other areas of study.
Communication skills are an integral part of mathematics activities. Students must understand and use the language of mathematics-in listening, speaking, reading, and writing. Mathematics communication involves specialized vocabulary and new symbol systems, and becomes a tool for organization and thinking. More than ever, students and teachers are "talking about math" with each other. Many new mathematics assessments require students to explain their thoughts and processes for solving problems in writing. Some mathematics teachers and mathematicians have tremendous understanding of mathematics concepts, yet have difficulty with communication skills. They can't convey concepts on a level others will understand, or effectively use communication devices such as analogies and examples. Communication must be modeled with a full range of curriculum applications.
Making connections fosters deeper mathematics understanding and assists learning. Students are encouraged to make connections among different mathematics topics, across other content and skill areas, and into the "real" world. When introducing new concepts, it is critical that teachers assist students in making connections with previous, understood concepts. Linking prior knowledge results in more efficient and generalizable learning.
Students are taught to make and apply representations across all mathematics topics. Representations assist with organization, recording, communication, modeling, predicting, and interpreting mathematical ideas and situations. Examples of representations are graphs, diagrams, charts, three-dimensional models, computer-generated models, and symbol systems.