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MCS Mathematics Standards
Standard # 1:
Students should be able to solve theoretical and real-world problems which require various approaches to investigate, understand, and apply mathematical concepts.
This "problem solving standard" requires that opportunities be provided for students to encounter diverse types of problems that arise from both real-world and mathematical contexts and to share their thinking with other students and with teachers. Classrooms with a problem-solving orientation are permeated with thought-provoking questions, speculations, investigations, and explorations and with the goal to promote a problem-solving approach for the learning of all mathematical content. For example, when the young child removes 6 counters from a box containing 14 counters and is asked to figure out, without looking inside the box, how many counters are left, he is creating strategies for subtraction; he is also developing, presenting, and following logical reasoning. When the intermediate child participates in a project to design estimation activities for children in other classes, she is also devising her own estimation and problem-solving strategies while, at the same time, she is learning to work with team members from diverse backgrounds to accomplish group goals. And when high school students find the maximum height of the path of a projectile, they are using models to study functions, and they are becoming proficient in locating and organizing different kinds of information to accomplish meaningful tasks.
All Memphis City Schools mathematics students need to be actively and continually engaged in learning ways to represent problems, in developing a variety of problem-solving strategies, and in formulating and explaining problems, processes, and solutions. As they progress toward this standard with its mandate for integrating effective communication, appropriate work habits, and logical reasoning throughout all K-12 mathematical problem-solving experiences, students will be learning knowledge and skills in ways that develop and strengthen their conceptual understanding of mathematics and its applications.
Specific Expectations
1. Use a variety of problem-solving strategies to represent, analyze, solve, and summarize results for theoretical and real-world problems.
2. Apply principles from the mathematics content strands of estimation, number sense/number theory, computation, algebra and functions, geometry, measurement, trigonometry, probability, and statistics in the solving of both theoretical and real-world problems.
3. Represent problem situations with models, and use these models for analysis, prediction, and decision making.
Standard #2:
Students should be able to express ideas and solutions through appropriate mathematical language and symbols.
This "communication standard" focuses on providing opportunities for students to read, write, and discuss ideas so that the use of both the verbal and written language of mathematics becomes natural. As Memphis City Schools mathematics students communicate their ideas with different audiences, through a variety of mediums, to achieve different purposes, they will learn to clarify, refine, and consolidate their thinking.
Children learn language through communication. Memphis City Schools students need to be "reading, writing, and talking" mathematics on a daily basis. In the district's mathematics classes, all students will actively participate in group activities, projects, presentations, game playing, and responding to questions, stones, and demonstrations and will, thereby, construct knowledge, learn other ways to think about ideas, and come to value the thoughts and opinions of others.
All Memphis City Schools mathematics students will engage in listening to, reading about, writing about, speaking about, reflecting on, and demonstrating mathematical ideas. As students participate in individual and small group explorations, they will have multiple opportunities to discuss, question, listen, and summarize. Students will describe their processes and solutions in oral, written, and visual forms and in both formal and informal presentations. They will be continually encouraged to clarify, paraphrase, and elaborate on their understanding, receiving acknowledgment of the merit and importance of their thoughts and enlightenment about their misconceptions. In these ways, students will learn to assess consequences of decisions and actions, perceive issues and circumstances from different points of view, work productively with team members from diverse backgrounds, and deal with conflicts in acceptable ways.
Progress toward this mathematics communication standard will improve the ability of all Memphis City Schools students to participate more fully in academic, service, and social activities. As a result, they will be better prepared for achieving lifelong productivity and success.
Specific Expectations
1. Make, read, or listen to mathematical presentations, and demonstrate understanding by asking questions that clarify and extend, and by using appropriate notation, symbols, and vocabulary.
2. Give results and rationales in written, oral, and visual forms, for answers, solution processes, conjectures, estimation, and predictions.
3. Use formal and symbolic methods of mathematical communication to describe relationships, illustrate ideas, apply mathematical principles, and formulate generalizations.
Standard #3:
Students should be able to use mathematical reasoning to analyze and answer theoretical and real-world questions and problems.
This "reasoning standard" necessitates opportunities for students to make conjectures, gather evidence, and build arguments to support their ideas. Demonstrations of good reasoning should be regarded as highly as correct answers. By applying their reasoning skills in problem discussions, students will learn to explain and justify their thinking and to perceive problem situations as units which include not just an answer but also process details, rationales, predictions, and conclusions.
There are many forms of mathematical reasoning. For example, at the K-4 level, students begin to use proportional reasoning when they determine what combinations of geometric shapes completely cover another shape. Reasoning skills continue to emerge in grades 5-8 as students learn to recognize and apply deductive and inductive reasoning. As they complete projects such as the study of advertising claims, secondary students engage in formal aspects of logic: formulating counterexamples, examining truth tables, judging the validity of arguments, and constructing proofs.
Reasoning about what is happening and why needs to be a constant part of the study of mathematics. All K-12 students can learn to explore, to make conjectures, to validate procedures and results, and to justify their viewpoints and interpretations while working individually and with others on challenging mathematical situations.
Progress toward this mathematics reasoning standard will improve the ability of Memphis City Schools students to use sound reasoning skills and logic, identify and assess risks and consequences of their decisions and actions, perceive issues and circumstances from different viewpoints, develop and present logical reasoning, distinguish fact, opinion, and interpretation, use the scientific method to solve problems and create understanding, and accomplish meaningful tasks. As a result, Memphis City Schools students will be better prepared to meet the demands of an increasingly complex world.
Specific Expectations
1. Apply statistical reasoning and probability theory to gather and organize real-world data, interpret and communicate observations, draw conclusions, and make predictions.
2. Apply principles of logical reasoning to make estimations, test conjectures, evaluate and construct logical arguments, identify patterns, and defend mathematical actions and results.
3. Generalize from patterns discovered in particular cases to develop functional relationships and to extend the generalizations to other situations.
Standard #4:
Students should be able to make connections between mathematics and their daily lives to answer questions, solve problems, and complete authentic projects.
This "connections standard" means that opportunities must be provided for students to learn that mathematics is not a series of isolated topics, and nor is it just a school subject with no connections to other disciplines or the world beyond their classrooms. As they explore and investigate situations with real-world contexts, students will build a foundation for the lifelong application of mathematics to their daily lives. Memphis City Schools mathematics teachers will help their students to understand that when they measure their running times and the heights of their jumps in physical education or when they use proportions and similarity in art, they are connecting mathematics with other disciplines. They are also locating and organizing different kinds of information to accomplish meaningful tasks. Students are connecting mathematics to their environment when they learn about architecture and design while they formulate and solve problems involving geometrical concepts. Analyzing real-world data will allow students to gain insights into problems of social equity and understanding of how political decisions affect their daily lives. Mathematics instruction needs to instill in students an attitude of inquiry and investigation and a sensitivity to the many interrelationships between formal mathematics and the real world. In this way, students will become more knowledgeable of cultural diversity in a global society.
Students also need to engage in making their own connections and in explaining how they have determined them. As students progress toward the mathematical connections standard by learning to apply their mathematical knowledge through meaningful experiences, they will come to use, recognize, and value the varied roles of mathematics in their lives, their culture, and their society.
Specific Expectations
1. Make, discuss, and explain mathematical connections between physical, symbolic, and abstract representations of the same problem situation or of the same mathematical concept.
2. Recognize, interpret, value, explain, and use the connections between different branches of mathematics and between mathematics and other disciplines.
Standard #5:
Students should be able to use appropriate technology to solve problems and to communicate ideas and solutions.
This "technology standard" acknowledges that opportunities and resources need to be routinely provided for students if they are to be adequately prepared for success and productivity in our highly technological world.
The changes in technology and the broadening of areas in which mathematics is applied have occurred simultaneously and have resulted in growth and changes in the discipline of mathematics itself, causing topics such as long division and the factoring of polynomials to receive decreased attention and strands such as statistics and algebra to receive increased attention. As part of the thrust, Memphis City Schools mathematics students, at all grade levels, will use calculators, computers, CD ROMs, and other appropriate technology to broaden the scope of their mathematics education beyond that which is possible with only paper-and-pencil activities.
For example, elementary students will use calculators to learn about skip counting and recognize patterns as they begin their preparation for algebra. Middle school students will make predictions about bacteria growth after using technology to organize and display data. High school students will analyze rates of change and lines of best fit using graphing calculators and computer programs to perform the laborious plotting and computations. In these and other activities, students will not only be using technology to solve problems and create quality products, they will also be developing, presenting, and following logical reasoning as they use technology to extend their thinking about real-world situations and conditions. Progress toward this mathematics technology standard will prepare students for meeting the challenges of their rapidly changing world and for maintaining lifelong productivity and success.
Specific Expectations
1. Employ technological tools to investigate and solve problems, test the validity of results, and interpret and communicate outcomes.
2. Select appropriate technological tools to perform mathematical tasks and solve problems.
3. Explain limitations and difficulties resulting from the use of technology.