*Principles and Standards in School Mathematics,*a revised document that makes more explicit the basic underlying assumptions of the original Standards documents. The central messages about high-quality mathematics teaching and learning remain unchanged. We refer to these documents collectively as "the Standards" or "the NCTM Standards."

## New Views of Mathematics Learning

- Mathematical literacy is essential if students are to become informed and competent citizens.
- All students can and should become mathematically literate, not just students who have traditionally performed well in mathematics classes.
- Literacy involves understanding mathematical principles; developing mathematical ways of thinking; and acquiring fluency with number, geometry, and data.
- Students develop this literacy by actively doing mathematics, using their skills and knowledge to solve problems and to investigate mathematical ideas.

## Evaluating Standards-Based Curriculums

## Mathematical Content

## Mathematical Processes

*Problem solving*. Students use mathematically productive ways to approach problems, which include hypothesizing, building a variety of representations, abstracting, and making generalizations.

*Reasoning and proof*. Students think systematically and critically about mathematics by making observations, proposing and investigating conjectures, and developing mathematical arguments and proofs.

*Communication*. Students effectively organize and articulate their thinking, consider the ideas of others, and develop facility with the language of mathematics.

*Connections*. Students recognize the coherence of mathematics as a discipline by seeing interrelations among ideas and by understanding the power of mathematics through connections with outside disciplines and contexts.

*Representation.*Students develop and effectively use a repertory of representations to organize thinking and to model and interpret mathematical situations. (NCTM has added this process standard to its forthcoming

*Principles and Standards.*)

#### Figure

## Mathematics for All

**Figure 1. Multiple Ways to Solve a Problem**

**Crossing the River**

#### A group of 8 adults and 2 children needs to cross a river. A small boat is available that can hold 1 adult or 1 child or 2 children. Everyone can row the boat.

#### How many one-way trips does it take for them all to cross the river?

#### How many trips would it take for 2 children and 100 adults?

*Student 1*: It takes four trips to get 1 adult across the river. And one additional trip for the children to get across the last time. For 100 adults to get across the river, including 2 kids, it would take 401 total one-way trips.

*Student 2*: Begin with 3 trips to get 1 adult over. Each additional adult is four trips. For part B, subtract 1 from the total number of adults. Multiply that answer by 4. Add 3 to the total of that. Add 2 to get the kids across and there's your answer! (A − 1) x 4 + 3 + 2

*Student 3*: See diagram.

*The first student describes a rule for counting the total number of trips. The second uses a traditional algebraic representation, and the third student describes the solution with a diagram and a written explanation.*

## A Different Kind of Mathematics Curriculum

- Interact with a range of materials for representing problem situations, such as manipulatives, calculators, computers, diagrams, tables, and charts;
- Work collaboratively as well as individually;
- Discuss mathematical ideas; and
- Focus on making sense of the mathematics they are studying as well as on learning to achieve accurate and efficient solutions to problems.

- Standards-based materials take an integrated approach to topics from the earliest grades, with several areas of mathematics appearing at each grade level and developing connections to one another. Skill acquisition and practice are often embedded in a larger activity or are presented in a context other than pure drill. For example, in the elementary Everyday Mathematics program (UCSMP, 1998), games such as "Addition and Subtraction Top-It" allow students to practice their number skills and arithmetic facts (fig. 2).
- Mathematical ideas reappear at different grade levels in increasingly sophisticated forms. For example, investigations of patterns at the elementary level lay a foundation for the study of algebraic relationships in the upper grades.
- Mathematical knowledge is developed within both practical and conceptual contexts, with less emphasis on rote symbol or number manipulation. For example, the 9th grade unit "Testing 1, 2, 3" in the high school curriculum Mathematics: Modeling Our World (COMAP, 1998) uses the context of steroid and medical testing as the context for investigating the mathematical ideas of probability, optimization, modeling, and area.
- Many problems are complex, involving a number of mathematical ideas and skills and requiring more time and thought to solve than the problems of the past.
- The programs emphasize different kinds of representations, such as charts, tables, graphs, diagrams, and formal notation, for exploring, describing, and testing problem situations
- Lessons employ less direct instruction and more student collaboration, conjecture, exploration, and discussion of mathematical ideas. Lessons extend over several days and involve student activity followed by class discussion.

#### Figure 2. A Skills Practice Game

**Addition Top-It**

#### Each player turns over two cards and calls out the sum. The player with the higher sum wins the round and takes all the cards. Play ends when not enough cards are left for each player to have another turn. The player with more cards wins.

*Assessment tip:* You might observe whether children rely on counting dots or using fingers to add the numbers. Children who have command of the addition facts will automatically call out the sum.

#### Note: Reprinted with permission from *Everyday Mathematics: Second-Grade Teacher's Manual and Lesson Guide*, Vol. A (Chicago: Everyday Learning Corporation, 1998).