Reflections on the NCTM Principles and Standards for School Mathematics:

**Six Principles**:

1. Equity- holding high expectations and offering strong support for all students.

2. Curriculum- should be coherent, focused, and well articulated. Math instruction should be constructed around "big ideas" of mathematics. Math should be treated as an integrated whole, not as a list of isolated bits and pieces.

3. Teaching- Teachers should develop an understanding of what their students know and need to learn, and challenge and support them as they learn content *well.* Teachers should have a firm understanding of how children learn math as well as an awareness of each students' development as an individual.

4. Learning- Students actively build knowledge from experience and prior understanding. Students should evaluate their own ideas and those of others and should make and test conjectures. According to Van De Walle, trust must be established with an understanding that it is okay to make mistakes; students will realize that errors are an opportunity for growth.

5. Assessment- should not be something that is done *to* students, but should be done *for* students to guide and enhance learning. Teachers should use feedback to establish and adjust goals.

6. Technology- Is an essential part of contemporary mathematics. Technology influences what is taught and enhances learning.

**Content Standards:**- Number and Operations (numbers and number sense, computation and estimation, counting and writing numbers, interval counting and patterns, comparing greater than and less than, fractions and decimals, rounding)

- Algebra (patterns and functions)

- Geometry

- Measurement

- Data Analysis and probability (statistics)

**Process Standards:**1. Problem Solving- students will build new knowledge, solve problems, apply and adapt a variety of appropriate strategies, and monitor and reflect upon the processes they use. Van De Walle describes problem solving as "the vehicle through which children develop mathematical ideas."

2. Reasoning and Proof- Students recognize reasoning and proof as fundamental elements of doing mathematics, make and investigate their conjectures, develop and evaluate their own arguments and proofs and those of others, and select and use various types of reasoning and proofs. Providing an argument or rationale becomes an integral part of every answer.

3. Communications- Students will organize and consolidate their thinking, communicate thinking coherently and clearly, analyze and evaluate the thinking and strategies of others, and use the language of mathematics to express ideas precisely. Students will talk about, write about, describe, and explain mathematical ideas.

4. Connections - Van De Walle offers that students are practicing reflective thought when they connect related ideas. Students will recognize and use connections among ideas, understand how ideas interconnect and build on one another to produce a coherent whole, and recognize and apply mathematical ideas in other contexts. Students will gain "real world" understandings of the roles of mathematial concepts and integrate mathematical ideas into other content area work.

5. Representation- students will organize, record, and communicate their ideas, select, apply, and translate concepts to solve problems, and use various representations to model and interpret phenomena. Students will gain the understanding that mathematics is a way of communicating ideas.

**Professional Standards:**1. Worthwhile mathematical tasks - knowledge of students' understandings, interests, and experiences and of the range of ways that diverse students learn mathematics. The teacher should pose tasks that promote communication about mathematics and promote the development of all students dispositions to do mathematics. Tasks should stimulate students to make connections and develop a coherent framework for mathematical ideas.

2. Teacher's Role in Discourse- The teaher should ask students to clarify and justiy their ideas orally and in writing; Van De Walle recommends that teachers encourage reflective thought by requiring explanations and justifications in addition to answers. Teachers should decide when and how to atttach mathematical notation and language to students' ideas.

3. Students' Role in Discourse- listen to, respond to, and question the teaher and one another. The teacher should promote discourse in which students try to convince themselves and one another of the validity of particular representations, solutions, conjectures, and answers. Van De Walle suggests that when students try to make sense of the explanations of others, ask questions, and make explanations for or justify their own ideas, they are using reflective thought. Van De Walle points our that Vygotsky's concept of internalization is the transfer of external ideas exchanged in the social environment to those that are internal, personal constructs.

4. Tools for enhancing discourse- the teacher should encourage and accept the use of invented and conventional terms and symbols; computers, calculators, and other technology; metaphors, analogies, and stories; written hypotheses, explanations, and arguments; and oral presentations and dramatizations.

5. Learning Environment- Foster the development of mathematical power by consistently expecting and encouraging students to take intellectual risks by raising questions and formulating conjectures. Van De Walle states that students must come to understand that mathematics makes sense - there is no need for the teaccher or other authority (like the back of the book) to provide judgment for student answers.

6. Analysis of Teaching and Learning

According to Van De Walle, the Professional Standards move us:

1. Toward classrooms as mathematics communities

2. Toward logic and evidencce as verification and away from the teacher as sole authority for right answers.

3. Toward mathematical reasoning and away from mere memorizing procedures.

4. Toward conjecturing, inventing and problem solving and away from an emphasis on mechanistic finding of answers.