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Mathematics Curriculum Framework
Achieving Mathematical Power  January 1996
Mathematics Content
The core concept of this Massachusetts Mathematics Framework is that students develop mathematical power through problem solving, communication, reasoning, and connections. Consider how incorporating problem solving in a mathematics curriculum provides the context in which students work together and communicate with one another. Or, imagine ways students communicate mathematically, providing opportunities to explain their reasoning, and to listen to and understand the reasoning of others. Think of a student's explaining his reasoning about fractals to a friend, then discovering that his solution is based on a diagram while his friend's solution is grounded in number patternsthis signifies connections between geometry and algebra.
 Mathematical power is conceived as consisting of mathematical abilities (conceptual understanding, procedural knowledge, and problem solving) within a broader context of reasoning and with connections across the broad scope of mathematical content and thinking. Communication is viewed as both a unifying thread and a way for students to provide meaningful responses to tasks.
 Mathematics Framework for the 1996 National Assessment of Educational Progress, U.S. Department of Education
An effective mathematics curriculum is one in which the skills and knowledge of problem solving, communication, reasoning, and connections are subsumed. The pages that immediately follow present models of what teachers, administrators, and districts are working toward in mathematics education, which is contrasted with what they are moving away from.
The standards set forth by the National Council of Teachers of Mathematics (NCTM) and The Massachusetts Adult Basic Education Math Standards that derive from the core concept of this framework are also presented. Each of these standards should be an integral part of each district's mathematics curriculum.
The four content strands in this framework are Number Sense; Patterns, Relations, and Functions; Geometry and Measurement; and Statistics and Probability. Each content strand represents a grouping of the NCTM standards. A chart showing the groupings is found in the appendix. It is important to understand that the four components of the core concept (problem solving, communication, reasoning, and connections ) are embedded and crosscutting within and among content strands. That is why the components of the core concept are found here, in the mathematics content chapter. They cannot be separated from the conceptual understanding and procedural knowledge that are also a part of mathematics education.
Following the discussions of problem solving, communication, reasoning, and connections are the Learning Standards, which identify what students should know and be able to do across grade spans. Examples of Student Learning and How It Looks in the Classroom contextualize some of the learning standards for the reader. They are not, however, intended as prescriptive or isolated activities.
 "Problem solving is not a distinct topic, but a process that should permeate the entire program and provide the context in which concepts and skills can be learned."
 National Council of Teachers of Mathematics
Here is a model of the paradigm shift that summarizes mathematics education reform:
...solving routine word problems
...problems that are computational exercises embedded in a simple context.
...problem solving as doing problems following introduction and practice of a specific procedure.
... assuming that complex problems are appropriate for some students, and too difficult for other students.
...problem solving as the focus of mathematics programs.
...recognizing that all students are capable of solving problems, and are thereby given many opportunities to solve problems.
...students' developing and using a variety of approaches and strategies when solving problems.
...students' using multiple strategies to solve the same problem.
...students' realizing that there are multiple pathways to solutions as well as multiple solutions to some problems.
elementary students explore area and perimeter by creating solar panel roof designs by guessing, estimating, and checking with the aid of a geoboard, making a drawing, modeling with geometric tiles, or using a computer program.
Below are listed the standards that exemplify mathematics as problem solving.
All students will...
 Use multiple approaches to investigate and understand mathematical content.
 Formulate problems from everyday and mathematical situations.
 Develop and apply strategies to solve a wide variety of problems, including multistep and nonroutine problems.
 Verify and interpret results with respect to the original problem.
 Generalize solutions and strategies to new problem situations.
 Acquire confidence in using mathematics meaningfully.
 Recognize and formulate problems from situations within and outside mathematics.
 Apply the process of mathematical modeling to realworld problem situations.
All students will...
 Explore and use multiple strategies for solving problems.
 Determine, collect, and analyze appropriate data with respect to the original problem or in new
 problemsolving situations.
 Access and use appropriate problemsolving tools, including calculators, computers, and measurement devices.
 Generalize problemsolving strategies to a wide range of adult oriented, realworld situations.
 "All students should develop and present conclusions through speaking, writing, artistic, and other means of expression."
 Massachusetts Common Core of Learning
Here is a model of the paradigm shift that summarizes mathematics education reform.
...the primary means of communicating mathematics as onewayfrom the teacher (or textbook) to the students.
...students' solely communicating their knowledge of mathematics by using taught procedures in response to questions on tests.
...an emphasis on students' memorizing mathematical definitions as the method of internalization.
...students learning by means of reflecting on ideas and communicating their thoughts with others.
...students interacting with other students to solve problems, share strategies, and seek solutions to problems.
...students explaining and justifying their mathematical thinking in writing.
...students exploring mathematical terms by comparing and contrasting examples with other students before presenting a formal definition.
high school students make designs in coordinate planes and challenge partners to describe them mathematically, using systems of linear inequalities; and descriptively, using English or another language.
Below are listed the standards that exemplify mathematics as communication.
All students will...
 Relate physical materials, pictures, and diagrams to mathematical ideas.
 Reflect on and clarify thinking about mathematical ideas and situations.
 Relate everyday language to mathematical language and symbols.
 Use the skills of reading, listening, and viewing to interpret and evaluate mathematical ideas.
 Model situations by using oral, written, concrete, pictorial, graphical, and algebraic methods.
 Develop mathematical ideas, formulate mathematical definitions, and express generalizations discovered through investigations.
 Ask clarifying and extending questions related to mathematics students have read or heard about.
 Appreciate the economy, power, and elegance of mathematical notation and its role in the development of mathematical ideas.
All students will...
 Develop the appropriate reading, writing, listening, and speaking skills necessary for communicating mathematically in a variety of settings.
 Discuss mathematics with others, reflecting and clarifying individual thinking about mathematical outcomes.
 Make convincing arguments and informed decisions.
 Define everyday, workrelated, or testrelated mathematical situations by using concrete, pictorial, graphic, or algebraic methods.
 Appreciate the value of mathematical language and notation in relation to mathematical ideas.
 "If we would guide by the light of reason, we must let our minds be bold."
 Louis Dembitz Brandeis
Here is a model of the paradigm shift that summarizes mathematics education reform.
...some students investigating and applying mathematical reasoning, while others learn mathematics solely by memorization of rules and procedures.
...developing mathematical understanding through teacher explanations, and only elementary students use manipulatives as tools for learning.
...the notion that basics must be mastered before proceeding to higher level mathematics.
...the belief that most students are not mature enough to do complex or abstract reasoning.
...all students' reasoning about mathematics, responding to the reasoning of others, and communicating their reasoning to others.
...students' making conjectures, thinking about and selecting sensible ways to solve problems, and justifying their solutions.
...students' recognizing different types of reasoning as they use them.
...students' understanding the degrees of certainty associated with deductive, inductive, analogic, and statistical reasoning.
students find examples of misguiding statistics in the media.
Below are listed the standards that exemplify mathematics as reasoning.
All students will...
 Draw logical conclusions about mathematics.
 Use models, known facts, properties, and relationships to explain mathematical thinking.
 Justify solutions and explain solution processes.
 Use patterns and relationships to analyze mathematical situations.
 Believe that mathematics makes sense.
 Recognize and apply deductive and inductive reasoning.
 Make and evaluate mathematical conjectures and arguments.
 Make and test conjectures.
 Follow logical arguments.
 Judge the validity of arguments.
 Appreciate the pervasive use and power of reasoning as a part of mathematics.
All students will...
 Draw logical conclusions from mathematical situations, using concrete models and verbal skills.
 Understand and apply deductive, inductive and proportional reasoning, with special attention to spatial and visual reasoning with proportions and graphs.
 Pose mathematical questions and evaluate arguments.
 Validate individual thinking and intuition.
 See how mathematics makes sense.
 All students should ...explore the relationship of mathematics to other areas of knowledge.
 Massachusetts Common Core of Learning
Here is a model of the paradigm shift that summarizes mathematics education reform.
...mathematics content that is broken down into several large strands, such as arithmetic and geometry.
...mathematics strands that are organized into a series of small, sequential objectives, which are taught separately, without exploring the commonalties of reasoning and skills in different strands.
...students learning new content by memorization of facts and formulas.
...students believing that mathematics is a collection of many isolated topics, rules, and procedures.
...mathematics content that is organized into separate standards, wherein students make connections among mathematical topics and domains.
...students making connections between mathematical ideas and other disciplines.
...students fostering meaningful connections between mathematical ideas and students' experiences in their daily lives.
students connect their understanding of patterns to measurement when they make open boxes to explore the relationships of area to surface area or to volume, and surface area to volume.
Below are listed the standards that exemplify mathematics as connections.
All students will...
 Link conceptual and procedural knowledge.
 Relate various representations of concepts or procedures to one another.
 Recognize and value the relationships among different topics in mathematics.
 Use mathematics in other curriculum areas and in daily living.
 Explore problems and describe results by using graphical, numerical, physical, algebraic, and verbal mathematical models or representations.
 Apply mathematical thinking and modeling to solve problems that arise in other disciplines.
 Recognize equivalent representations of the same concept.
 Relate procedures in one representation to procedures in an equivalent representation.
All students will...
 View mathematics as an integrated whole, which is connected to past learning, the real world, adult life skills, and workrelated settings.
 Explore problems by using appropriate technology, and describe results by using a variety of mathematical models and representations.
 Apply mathematical thinking and modeling to solve problems that arise in other disciplines and in the real world, including workrelated settings.
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Last Updated: January 1, 1996
