What We Know About Mathematics Teaching and Learning

By: McREL
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This book supports mathematics education reform and brings the rich world of education research and practice to pre-K-12 educators. Designed for accessibility, each chapter is broken down into important questions. For each question, the authors provide background information from a research perspective, offer implications for improving classroom instruction, and list resources for further reading.

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Chapter 1: Mathematics for All

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All students can learn mathematics, and they deserve the opportunity to do so. The National Council of Teachers of Mathematics’ Principles and Standards for School Mathematics sets forth mathematics literacy expectations for all students and describes what they are expected to learn. However, recognizing the diversity among American children, educators do not expect all students to learn the material in the same manner, with the same resources, or in the same time frame. The Equity Principle in PSSM states:

All students, regardless of their personal characteristics, backgrounds, or physical challenges, must have opportunities to study—and support to learn—mathematics. Equity does not mean that every student should receive identical instruction; instead, it demands that reasonable and appropriate accommodations be made as needed to promote access and attainment for all students. (2000, p. 12)

To achieve “mathematics for all” will take a concerted effort from all stakeholders in our children’s education. We must continue to make progress toward providing rich, well-supported learning environments that respond to the unique educational needs of every student. That is the goal of mathematics education reform.

 

Chapter 2: Teaching Mathematics

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Learning and teaching mathematics are complex, active processes. Teachers constantly make decisions as they facilitate an environment in which students are active learners. They also must undertake long-term planning to connect daily efforts to the broader education of each student. At the same time, teachers share responsibility for their students’ successes with other parts of the educational community, including their colleagues, their institutions, and the policies of the educational system.

The NCTM’s Principles and Standards for School Mathematics outlines theoretical and practical knowledge and understandings about mathematics, how children acquire mathematics content, and mathematics teaching techniques that facilitate each child’s learning. Effective professional development moves teachers toward the goals spelled out in these professional standards for teaching mathematics. Because a teacher’s classroom decisions affect each student’s achievement, teachers need to avail themselves of strategies as varied as their students’ educational needs.

 

Chapter 3: Assessment in Mathematics

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Assessment is a complex, systematic process for collecting and interpreting data and is the primary mechanism for feedback on the attainment of standards to students, teachers, parents, school districts, and communities. The NCTM recommends using multiple assessment methods. Since assessments communicate expectations, providing an operational definition of what is important, the Principles and Standards for School Mathematics promotes the inclusion of authentic assessments—exercises that closely approximate how mathematics is used in the real world.

The PSSM Assessment Principle also recommends measuring both student achievement and opportunity to learn. Interpreted together, this information assists educators and the community at large in assuring that all students can achieve to their potential. Opportunity to Learn measures are important in interpreting both high-stakes individual assessments and international achievement comparisons.

With today’s increased accountability for student learning, educators first think about testing as a way to gather feedback about how students are doing. This U.S. emphasis is solidified by NCLB, which requires states to provide report cards that include a host of statistics about school progress. State and other standardized tests provide data on overall school performance and can be valuable program evaluation tools. New approaches to mathematics teaching have expanded the role of assessment to include monitoring student progress and making instructional decisions. Essentially, assessment now plays four roles: (1) evaluate student achievement, (2) evaluate programs, (3) monitor student progress, and (4) make instructional decisions.

 

Chapter 4: Mathematics Curriculum

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The content standards in the NCTM Principles and Standards for School Mathematics define the content of instruction, outlining what every student should know and be able to do. The district curriculum, however, describes how that content is organized. The curriculum also includes emphases and perspectives placed on the content, creating a map for educators to use in designing classroom experiences for students.

Recognizing that the intent of content standards is to present a goal for all students, teachers must make curriculum decisions that accommodate a wide variety of learning styles, backgrounds, and interests. When educators use multiple means of addressing individual standards, all learners have an opportunity to access common content.

Standards are a set of expectations for what students will learn. Standards have grown increasingly pervasive since the 1990s. Close to 100 percent of all U.S. states and territories are joining the Common Core State Standards initiative. The goal of this initiative is to define standards that are not only based on research and evidence, but are also internationally benchmarked, aligned with college and workforce expectations, and made rigorous in both content and skills.

 

Chapter 5: Instructional Technology in Mathematics

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Instructional technology refers to the tools used to promote classroom learning. In mathematics teaching, instructional technology is often used in problem solving, thereby making the learning experience more learner-centered. Specific technologies include various types of hardware—calculators, handheld data-collection devices, and computers—embedded software, and the Internet.

Instructional technologies such as those found in Using Technology With Classroom Instruction That Works add relevancy and increase student engagement. Further benefits for mathematics instruction include increased accuracy and speed in data collection and graphing; real-time visualization; interactive modeling of ambiguous mathematical processes; ability to collect, compute, and analyze large volumes of data; collaboration for data collection and interpretation; and greater opportunity to vary the presentation of results. Technology can make mathematics more meaningful and standards more attainable for all students. The Technology Principle from the Principles and Standards for School Mathematics states that “technology should be used widely and responsibly, with the goal of enriching students’ learning of mathematics” (2000, p. 25). Technology has become ubiquitous in everyday life, and its use in the classroom enhances real-world connections and relevance.

 

Chapter 6: Learning Mathematics

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What does it mean to learn mathematics? This question is addressed in the Principles and Standards for School Mathematics. Children are natural learners. They are inquisitive about patterns and shapes, recognizing and creating them from a young age. They count, measure, and share objects. For children, mathematics is learned by doing. Their school experience of mathematics learning should include problem solving and reasoning through grade 12, not simply lectures, books, and worksheets.

During the twentieth century, educators’ understanding of the learning process progressed from behavioral observations through cognitive psychology to improved knowledge about neurophysiology. The 1990s were dubbed “the decade of the brain” because of the tremendous increase in understanding of how the brain works. Twenty-first-century educators are improving their classroom practice through application of the newest understandings from neuroscience.

Since the 1990s the general public has become more aware of mathematics education reform than ever before. Publications such as the National Commission on Excellence in Education’s A Nation at Risk, the standards-setting work of the National Council of Teachers of Mathematics, and reports from TIMSS, PISA, and the National Mathematics Advisory Panel highlight the need for reform. Most people agree that mathematical literacy extends beyond knowledge of mathematics concepts and procedures into the ability to create mathematical models of situations, solve the problems represented by the models, and interpret the solutions in terms of societal implications.

 

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