Mathematics Curriculum Framework
Achieving Mathematical Power - January 1996
The Beauty and Power of Mathematics
- The harmony of the world is made manifest in form and number, and the heart and soul and all the poetry of natural philosophy are embodied in the concept of mathematical beauty.
-- Sir D'Arcy Wentworth Thompson
Mathematics is a universal language of numbers that is spoken in all cultures. It is an equalizer. Mathematics is beautiful patterns in nature and art, pleasing proportions in architecture. The form of poetry is mathematics and the shape of music is mathematics. It is systematic, random, and chaotic. Mathematics is about relationships. It brings joy in discovery and satisfaction in mastery. Mathematics is about exploration, intuition, and strategy.
Mathematics has an intrinsic value. It beckons each of us to appreciate and value mathematics for its own sake. When learners progress from intellectual struggle to achieving mathematical knowledge, the aesthetic value of mathematics emerges.
The vision of the Massachusetts Mathematics Curriculum Framework is that all students in the Commonwealth achieve mathematical power through problem solving, communication, reasoning, and connections. This vision of mathematical power exemplifies the principles in the Common Core of Learning. This framework sets forth as a goal that all students become effective problem solvers, master computation in the early elementary grades, and master basic algebraic concepts by the end of eighth grade.
- "Mathematical power includes the ability to explore, conjecture, and reason logically; to solve non-routine problems; to communicate about and through mathematics; and to connect ideas within mathematics and between mathematics and other intellectual activity. Mathematical power also involves the development of personal self-confidence and a disposition to seek, evaluate, and use quantitative and spatial information in solving problems and in making decisions. Students' flexibility, perseverance, interest, curiosity, and inventiveness also affect the realization of mathematical power."
-- National Council of Teachers of Mathematics
Problem solving is the central focus of mathematics education. Whenever we apply our mathematics knowledge, skills, or experiences toward resolving a dilemma or situation that is new or perplexing, we are problem solving. Whether we are adding and subtracting to make change, using proportional reasoning to build a snowman, calculating distance to choose the best route home, or exploring patterns in music to choreograph a dance, we are engaging in mathematical problem solving. To become good problem solvers, students need many opportunities to create and solve problems in both mathematical and real-world contexts. They also need to develop and build upon a strong number sense. As stated in the Massachusetts Common Core of Learning, students "develop, test, and evaluate possible solutions." This involves posing questions, defining problems, considering different strategies, and finding appropriate solutions.
Communication of mathematics reflects mathematical understanding and embraces mathematical power. The Common Core of Learning suggests that all students should ". . . justify and communicate solutions to problems." Students learn mathematics as they talk and write about what they are doing. They become actively engaged in doing mathematics when they are asked to think about their ideas, or talk with and listen to other students, sharing ideas, strategies, and solutions. Writing about mathematics encourages students to reflect on their work and clarify ideas for themselves. Reading what students write is an excellent way for teachers to identify students' understanding and misconceptions.
Mathematical reasoning is necessary if we are to know and do mathematics. The ability to reason enables students to solve problems in their lives, inside and outside of school. Whenever we use reasoning skills to validate our thinking, we enhance our confidence with mathematics and mathematical thinking. Mathematics as a field of study is characterized as much by particular types of reasoning as by particular types of content. The Common Core of Learning states that "all students . . . should make reasoned inferences and construct logical arguments." These reasoning skills suggest an ability to recognize and use deduction and induction, as well as develop and apply these skills in numerical and spatial contexts.
Connections ground mathematics in daily life and provide the synapses through which we relate mathematical topics with one another. Students should understand how mathematics relates to other subject areas--the arts, social studies, health, science and technology, world languages, and English language arts. For example, the Common Core of Learning suggests that "all students . . . should understand concepts such as location and place . . ." This exemplifies one of the interdisciplinary relationships that exists between geography and mathematics. Students appreciate that mathematics is one way of learning about the world, and that it is connected, not isolated, from other ways of learning. Individuals' strengths and interests enhance those subject areas with which they feel less comfortable. For example, a proficient mathematics student may begin to think of himself as a talented arts student when he explores drawing through the study of perspective in mathematics class. Or, a history buff may gain new insight to her potential when she analyzes data from a questionnaire on family histories.
The atmosphere Massachusetts teachers create in their classrooms encourages students to ask questions or propose partial ideas, thereby building students' confidence in their mathematical abilities. Students in elementary grades begin working on complex mathematical ideas in informal ways as they are learning computational skills. This provides contexts for computation as well as laying the ground work for future mathematics learning. Students try uncharted ways of thinking and draw upon a repertoire of problem-solving strategies. They often work in teams, testing and verifying their ideas. No longer do they look only to the teacher for the final validation. Students learn from each others' explanations. They clarify their own understanding as their peers question their findings, enhancing the learning process.
- "Profound thoughts arise only in debate, with a possibility of counterargument, only when there is a possibility of expressing not only correct ideas but also dubious ideas."
-- Andrei Dmitrievich Sakharov
Massachusetts needs all of its citizens to achieve mathematical power. With effective mathematics education today, tomorrow's leaders can call upon more empowered citizens, who will have the resources to make informed decisions, the curiosity to explore new ideas, and the perseverance to face obstacles. They will be mathematically empowered with an adeptness to solve problems, a common language with which to communicate with other citizens and cultures, the self-confidence required for effective reasoning, and an understanding of the connections among mathematical ideas and everyday life.