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2 What Should WeTeach?
We live in a time of extraordinary and accelerating change. New knowledge, tools, and ways of doing and communicating mathematics continue to emerge and evolve…. The need to understand and be able to use mathematics in everyday life and in the workplace has never been greater and will continue to increase. (Principles and Standards for School Mathematics, p. 4)
The mathematical knowledge necessary to succeed in this changing world is tied to what is taught in schools. The core of a mathematics program is its curriculum—what is taught to whom and when. Careful sequencing of mathematical ideas can build understanding and sense making about important mathematical topics. To help educators consider their curriculum, we offer three questions, then suggest a resource generated at the national level that can provide advice for states, districts, and schools formulating their own plans in response to these questions:
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What mathematics should all students know and be able to do?
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What mathematics should be taught at what grade levels?
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What mathematics is important for students in the 21st century?
RESOURCE AVAILABLE
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Principles and Standards for School Mathematics, developed by the National Council of Teachers of Mathematics, 2000.
OVERVIEW OF THE RESOURCE
The National Council of Teacher of Mathematics (NCTM) intends its Principles and Standards for School Mathematics to be a “resource and guide
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for all who make decisions that affect the mathematics education of students in pre-kindergarten through grade 12” (p. ix). NCTM is an organization of over 110,000 mathematics educators concerned with pre-K–12 mathematics education. This update of the NCTM's three previously developed sets of standards for curriculum, teaching, and assessment is intended to establish a curriculum framework to bring focus and coherence to K–12 mathematics. The document was developed through an extensive and inclusive process that engaged a wide spectrum of experts on issues concerning mathematics education. As such, Principles and Standards represents a negotiated position about appropriate content for school mathematics to which educators should give careful consideration.
The developers offer the standards as a guide for ensuring quality, developing goals, and promoting change by suggesting common language, examples, and recommendations to engage people at state, provincial and local levels in conversations about mathematics education. The document is intended to (p. 6):
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Set forth a comprehensive and coherent set of goals for mathematics for all students that will orient curricular, teaching, and assessment efforts.
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Serve as a resource for teachers, education leaders, and policymakers to use in examining and improving the quality of mathematics instructional programs.
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Guide the development of curricular frameworks, assessments, and instructional materials.
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Stimulate ideas and ongoing conversations about how best to help students gain a deep understanding of important mathematics.
Principles and Standards is built on the following vision (p. 5):
In this changing world, those who understand and can do mathematics will have significantly enhanced opportunities and options for shaping their futures. Mathematical competence opens doors to productive futures. A lack of mathematical competence keeps those doors closed. NCTM challenges the assumption that mathematics is only for the select few. On the contrary, everyone needs to understand mathematics. All students should have the opportunity and the support necessary to learn significant mathematics with depth and understanding. There is no conflict between equity and excellence.
To fulfill this vision, the document describes what mathematics in pre-K–12 school programs should look like including how mathematical ideas should be developed across five content areas and five process domains. The standards present a deeper look at the mathematics within each of four grade-level bands, pre-K–2, 3–5, 6–8, and 9–12; they also suggest how mathematics should grow
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across the grades. The examples were chosen to illustrate these ideas and to portray teaching practices that will support all students in learning such mathematics.
RECOMMENDATIONS MADE IN THE REPORT
Essentially the recommendations made in Principles and Standards are the principles and standards themselves. The document offers six overarching themes—the “principles for schools mathematics” —to provide a lens for considering decisions about the content and character of school mathematics (p. 11):
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Equity. Excellence in mathematics education requires equity—high expectations and strong support for all students.
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Curriculum. A curriculum is more than a collection of activities: it must be coherent, focused on important mathematics, and well articulated across the grades.
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Teaching. Effective mathematics teaching requires understanding what students know and need to learn and then challenging and supporting them to learn it well.
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Learning. Students must learn mathematics with understanding, actively building new knowledge from experience and prior knowledge.
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Assessment. Assessment should support the learning of important mathematics and furnish useful information to both teachers and students.
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Technology. Technology is essential in teaching and learning mathematics: It influences the mathematics that is taught and enhances students' learning.
The mathematical content of the curriculum is organized into five major content strands for school mathematics (numbers and operations, algebra, geometry, measurement, and data analysis and probability) and five major process domains (problem solving, reasoning and proof, communication, connections, and representations). The five strands describe the content students should learn, and the five domains describe the processes through which students learn mathematics and demonstrate the mathematics they have learned. The heart of Principles and Standards elaborates on the mathematical expectations, instructional strategies, and assessment practices that guide the effective and coherent implementation of these standards at the four grade bands.
Highlighted recommendations include the following:
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“Teachers must help students be confident, engaged mathematics learners.” (p. 374)
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“Teacher-leaders support on a day-to-day basis can be crucial to a teacher's work life.” (p. 375)
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“Administrators and policymakers must carefully consider the impact of high-stakes assessments on the instructional climate in schools.” (p. 377)
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“Families become advocates for education standards when they understand the importance of a high-quality mathematics education for their children.” (p. 378)
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“Conceptual understanding is an important component of proficiency.” (p. 20)
Putting this vision into action requires thoughtful and ongoing consideration of the following questions:
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“How can all students have access to high-quality mathematics education?” (p. 368)
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“Are good instructional materials chosen, used, and accepted?” (p. 369)
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“How can teachers learn what they need to know?” (p. 370)
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“Do all students have time and the opportunity to learn?” (p. 371)
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“Are assessments aligned with instructional goals?” (p. 372)
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“Is technology supporting learning?” (p. 372)
ACTIONS EDUCATORS MIGHT CONSIDER
As a resource for high-quality mathematics programs, Principles and Standards can be used to guide analysis and decision making about current programs and evolving program components. To consider what mathematics all students should know and at what grade bands it should be taught, educators and policymakers might do the following:
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Compare their curricular guidelines and frameworks with the standards and expectations delineated in Principles and Standards and examine the differences.
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Compare the content and format of classroom, district, and state assessments in current use with the vision of curriculum described in these documents and consider the implications of mismatches.
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Disaggregate enrollment and achievement data to analyze patterns of equity of opportunity and equity of outcome and develop plans and programs that respond to any differences that are found.
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Develop and implement professional development opportunities to ensure that all teachers of mathematics have the mathematical and pedagogical knowledge and skill needed to implement a curriculum that will meet the needs of their students.
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Develop programs and policies to ensure that school, district, and state administrators who are responsible for shaping instructional programs provide the human and material resources necessary for their implementation at the school, district, and state levels.
Develop programs and policies to ensure that all those involved in preservice preparation of teachers and responsible for in-service or professional development programs have a sufficient understanding of content, curriculum, teaching, and learning.
Discussion about these suggestions can be found in some of the documents discussed in later chapters of this report. For example, both Before It's Too Late and Adding It Up provide recommendations about teacher preparation.
