The following principles are philosophical statements that underlie the mathematics content and practice standards and resources in this curriculum framework. They should guide the construction and evaluation of mathematics programs in the schools and the broader community.
These recommended Guiding Principles for Mathematics Programs are adapted from the Massachusetts Curriculum Framework January 2011; http://www.doe.mass.edu/frameworks/current.html
Guiding Principle 1: Learning
Mathematical ideas should be explored in ways that stimulate curiosity, create enjoyment of mathematics, and develop depth of understanding.
Students need to understand mathematics deeply and use it effectively. The standards of mathematical practice describe ways in which students increasingly engage with the subject matter as they grow in mathematical maturity and expertise through the elementary, middle, and high school years.
To achieve mathematical understanding, students should have a balance of mathematical procedures and conceptual understanding. Students should be actively engaged in doing meaningful mathematics, discussing mathematical ideas, and applying mathematics in interesting, thought-provoking situations. Student understanding is further developed through ongoing reflection about cognitively demanding and worthwhile tasks.
Tasks should be designed to challenge students in multiple ways. Short- and long-term investigations that connect procedures and skills with conceptual understanding are integral components of an effective mathematics program. Activities should build upon curiosity and prior knowledge, and enable students to solve progressively deeper, broader, and more sophisticated problems. Mathematical tasks reflecting sound and significant mathematics should generate active classroom talk, promote the development of conjectures, and lead to an understanding of the necessity for mathematical reasoning.
Guiding Principle 2: Teaching
An effective mathematics program is based on a carefully designed set of content standards that are clear and specific, focused, and articulated over time as a coherent sequence.
The sequence of topics and performances should be based on what is known about how students’ mathematical knowledge, skill, and understanding develop over time. What and how students are taught should reflect not only the topics within mathematics but also the key ideas that determine how knowledge is organized and generated within mathematics. Students should be asked to apply their learning and to show their mathematical thinking and understanding by engaging in the first Mathematical Practice, Making sense of problems and persevere in solving them. This requires teachers who have a deep knowledge of mathematics as a discipline.
Mathematical problem solving is the hallmark of an effective mathematics program. Skill in mathematical problem solving requires practice with a variety of mathematical problems as well as a firm grasp of mathematical techniques and their underlying principles. Armed with this deeper knowledge, the student can then use mathematics in a flexible way to attack various problems and devise different ways of solving any particular problem. Mathematical problem solving calls for reflective thinking, persistence, learning from the ideas of others, and going back over one's own work with a critical eye. Students should construct viable arguments and critique the reasoning of others, they analyze situations and justify their conclusions and are able to communicate them to others and respond to the arguments of others. (See Mathematical Practice 3, Construct viable arguments and critique reasoning of others.) Students at all grades can listen or read the arguments of others and decide whether they make sense, and ask questions to clarify or improve the arguments.
Mathematical problem solving provides students with experiences to develop other mathematical practices. Success in solving mathematical problems helps to create an abiding interest in mathematics. Students learn to model with mathematics, they learn to apply the mathematics that they know to solve problems arising in everyday life, society, or the workplace. (See Mathematical Practice 4, Model with mathematics.)
For a mathematics program to be effective, it must also be taught by knowledgeable teachers. According to Liping Ma, “The real mathematical thinking going on in a classroom, in fact, depends heavily on the teacher's understanding of mathematics.”1A landmark study in 1996 found that students with initially comparable academic achievement levels had vastly different academic outcomes when teachers’ knowledge of the subject matter differed.2 The message from the research is clear: having knowledgeable teachers really does matter; teacher expertise in a subject drives student achievement. “Improving teachers’ content subject matter knowledge and improving students’ mathematics education are thus interwoven and interdependent processes that must occur simultaneously.”3
Guiding Principle 3: Technology
Technology is an essential tool that should be used strategically in mathematics education.
Technology enhances the mathematics curriculum in many ways. Tools such as measuring instruments, manipulatives (such as base ten blocks and fraction pieces), scientific and graphing calculators, and computers with appropriate software, if properly used, contribute to a rich learning environment for developing and applying mathematical concepts. However, appropriate use of calculators is essential; calculators should not be used as a replacement for basic understanding and skills. Elementary students should learn how to perform the basic arithmetic operations independent of the use of a calculator.4 Although the use of a graphing calculator can help middle and secondary students to visualize properties of functions and their graphs, graphing calculators should be used to enhance their understanding and skills rather than replace them.
Teachers and students should consider the available tools when presenting or solving a problem. Student should be familiar with tools appropriate for their grade level to be able to make sound decisions about which of these tools would be helpful. (See Mathematical Practice 5, Use appropriate tools strategically.)
Technology enables students to communicate ideas within the classroom or to search for information in external databases such as the Internet, an important supplement to a school’s internal library resources. Technology can be especially helpful in assisting students with special needs in regular and special classrooms, at home, and in the community.
Technology changes what mathematics is to be learned and when and how it is learned. For example, currently available technology provides a dynamic approach to such mathematical concepts as functions, rates of change, geometry, and averages that was not possible in the past. Some mathematics becomes more important because technology requires it, some becomes less important because technology replaces it, and some becomes possible because technology allows it.
Guiding Principle 4: Equity
All students should have a high quality mathematics program that prepares them for college and a career.
All students should have high quality mathematics programs that meet the goals and expectations of these standards and address students’ individual interests and talents. The standards provide clear signposts along the way to the goal of college and career readiness for all students. The standards provide for a broad range of students, from those requiring tutorial support to those with talent in mathematics. To promote achievement of these standards, teachers should encourage classroom talk, reflection, use of multiple problem solving strategies, and a positive disposition toward mathematics. They should have high expectations for all students. At every level of the education system, teachers should act on the belief that every child should learn challenging mathematics. Teachers and guidance personnel should advise students and parents about why it is important to take advanced courses in mathematics and how this will prepare students for success in college and the workplace.
All students should have the benefit of quality instructional materials, good libraries, and adequate technology. All students must have the opportunity to learn and meet the same high standards. In order to meet the needs of the greatest range of students, mathematics programs should provide the necessary intervention and support for those students who are below- or above grade-level expectations. Practice and enrichment should extend beyond the classroom. Tutorial sessions, mathematics clubs, competitions, and apprenticeships are examples of mathematics activities that promote learning.
Because mathematics is the cornerstone of many disciplines, a comprehensive curriculum should include applications to everyday life and modeling activities that demonstrate the connections among disciplines. Schools should also provide opportunities for communicating with experts in applied fields to enhance students’ knowledge of these connections.
An important part of preparing students for college and careers is to ensure that they have the necessary mathematics and problem-solving skills to make sound financial decisions that they face in the world every day, including setting up a bank account; understanding student loans; credit and debit; selecting the best buy when shopping; choosing the most cost effective cell phone plan based on monthly usage; and so on.
Guiding Principle 5: Literacy Across the Content Areas
An effective mathematics program builds upon and develops students’ literacy skills and knowledge.
Reading, writing, and communication skills are necessary elements of learning and engaging in mathematics, as well as in other content areas. Supporting the development of students’ literacy skills will allow them to deepen their understanding of mathematics concepts and help them determine the meaning of symbols, key terms, and mathematics phrases as well as develop reasoning skills that apply across the disciplines. In reading, teachers should consistently support students’ ability to gain and deepen understanding of concepts from written material by acquiring comprehension skills and strategies, as well as specialized vocabulary and symbols. Mathematics classrooms should make use of a variety of text materials and formats, including textbooks, math journals, contextual math problems, and data presented in a variety of media.
In writing, teachers should consistently support students’ ability to reason and deepen understanding of concepts and the ability to express them in a focused, precise, and convincing manner. Mathematics classrooms should incorporate a variety of written assignments ranging from math journals to formal written proofs. In speaking and listening, teachers should provide students with opportunities for mathematical discourse, to use precise language to convey ideas, to communicate a solution, and support an argument.
Guiding Principle 6: Assessment
Assessment of student learning in mathematics should take many forms to inform instruction and learning.
A comprehensive assessment program is an integral component of an instructional program. It provides students with frequent feedback on their performance, teachers with diagnostic tools for gauging students’ depth of understanding of mathematical concepts and skills, parents with information about their children’s performance in the context of program goals, and administrators with a means for measuring student achievement.
Assessments take a variety of forms, require varying amounts of time, and address different aspects of student learning. Having students “think aloud” or talk through their solutions to problems permits identification of gaps in knowledge and errors in reasoning. By observing students as they work, teachers can gain insight into students’ abilities to apply appropriate mathematical concepts and skills, make conjectures, and draw conclusions. Homework, mathematics journals, portfolios, oral performances, and group projects offer additional means for capturing students’ thinking, knowledge of mathematics, facility with the language of mathematics, and ability to communicate what they know to others. Tests and quizzes assess knowledge of mathematical facts, operations, concepts, and skills and their efficient application to problem solving. They can also pinpoint areas in need of more practice or teaching. Taken together, the results of these different forms of assessment provide rich profiles of students’ achievements in mathematics and serve as the basis for identifying curricula and instructional approaches to best develop their talents.
Assessment should also be a major component of the learning process. As students help identify goals for lessons or investigations, they gain greater awareness of what they need to learn and how they will demonstrate that learning. Engaging students in this kind of goal-setting can help them reflect on their own work, understand the standards to which they are held accountable, and take ownership of their learning.
1 Ma, Lipping, Knowing and Teaching Elementary Mathematics, Mahwah, New Jersey: Lawrence Erlbaum Associates, 1999.
2 Milken, Lowell, A Matter of Quality: A Strategy for Answering the High Caliber of America’s Teachers, Santa Monica, California: Milken Family Foundation, 1999.
3 Ma, p. 147.
4 National Center for Education Statistics, Pursuing Excellence: A Study of U.S. Fourth-Grade Mathematics and Science Achievement in International Context. Accessed June 2000.