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Current Page
- Looking Ahead
- Part A: Beliefs About Best Practices for Teaching and Learning Mathematics and Science
Following Pages
- Part B: Beyond Beliefs: The Guiding Principles
- Structure for the Discussion of Guiding Principles
- Guiding Principles
- 1. Students understand the nature of mathematics and science.
- 2. Students communicate effectively in mathematics and science.
- 3. Students reason effectively in mathematics and science.
- 4. Students are problem-solvers in mathematics and science.
- 5. Students understand their roles in the natural world.
- 6. Students understand historical and societal implications of mathematics and science.
- 7. Students attain and apply essential knowledge and skills of mathematics and science.
- References
This section outlines the components of an exemplary mathematics and science curriculum for Maine schools. Maine's Curriculum Framework authors have translated broad-based national research and guidelines for science and mathematics into a form useful to teachers, curriculum developers, and other educators across the state.
The purpose of these curriculum standards is not to dictate what will be taught in each school system or classroom; rather, it is to establish statewide agreement about what students should know and be able to do in mathematics and science, while allowing local educators the flexibility to determine how these standards can be reached.
This Framework can undergird significant, lasting improvement in Maine's mathematics and science education -- but only with support from all essential participants: parents, students, administrators, business and community members, researchers, and all faculty from pre-kindergarten through graduate school, including university-based scientists and mathematicians as well as teacher educators.
While Maine's Curriculum Framework evolved through a consensual process which invited the personal and professional expertise of many individuals, the writing teams also drew heavily on four documents which together represent the best current thinking available in mathematics and science education. These works are the primary sources for ideas presented in Section II: Maine's Common Core of Learning; National Council of Teachers of Mathematics (NCTM), Curriculum and Evaluation Standards for School Mathematics; American Association for the Advancement of Science (AAAS), Benchmarks for Science Literacy; and National Research Council, National Science Education Standards.
Appendix A synthesizes the principal ideas of these seminal documents and shows their relationship to Maine's Curriculum Framework for Mathematics and Science. Local curriculum development work will be greatly enhanced by consultation with these vital sources.
PART A: Beliefs about Best Practices
Section II begins with a discussion of Framework authors' Beliefs about Best Practices in mathematics and science education. These beliefs form the basis for Framework discussions about how students can most effectively learn the mathematics and science they need for their future. Some Beliefs about Best Practices are illustrated by a description of a classroom scene which shows the beliefs in action, followed by a brief analysis of the scene.
At the close of Beliefs about Best Practices is a set of questions, "Measuring Up," intended to help educators assess the use of these practices in classroom settings.
PART B: Beyond Beliefs: The Guiding Principles
The section continues with a description of seven Guiding Principles which broadly describe what all Maine students should know and be able to do in mathematics and science. They are:1. Students understand the nature of mathematics and science.
2. Students communicate effectively in mathematics and science.
3. Students reason effectively in mathematics and science.
4. Students are problem solvers in mathematics and science.
5. Students understand their roles in the natural world.
6. Students understand historical and societal implications of mathematics and science.
7. Students attain and apply knowledge and skills of mathematics and science.
These principles, which reflect the national standards in science and mathematics, represent a consensus in thinking of Framework developers. The Guiding Principles are not only the key ideas forming the infrastructure of Section II, they are the central statements of Maine's Curriculum Framework for Mathematics and Science.
The dynamic interplay among the student, the teacher and the curriculum
is at the heart of school mathematics and science learning. The
quality of these interactions depends not only on the teaching/learning
triad itself, but on the nurturing context provided by the rest
of the school community, including parents, administrators, school
boards, other local and state policymakers, businesses, institutions
of higher learning and community residents.
The developers of Maine's Curriculum Framework for Mathematics
and Science believe that the following educational practices
most effectively promote mathematics and science learning in schools.
While these "best practices" focus on what teachers
and students do, all other members of school communities must
help to create the conditions under which such work can be accomplished.
The energetic support of the wider community is essential for
student success.
From Madawaska to Kittery, Bethel to Eastport, Maine's community of learners includes many individuals who learn in diverse ways, but they all must learn essential science and mathematics.
In the Classroom
A third-grade class is investigating multiplication using Cuisenaire Rods(tm) (see note) or similar objects. The students, working in groups of three, have been charged with developing multiplication families for twelve and twenty-four. Previous work has included the underlying concepts of multiplication and use of the rods to model basic multiplication facts.
Before this lesson, the teacher has met with three parent volunteers to explain the activity. As students work, the teacher and parent volunteers circulate among groups, guiding them, answering their questions, and posing new questions that help the group clarify their thinking.
The classroom is a beehive of activity: students are discussing the task in small groups, while parent volunteers and the teacher work with the groups. The students are actively involved in learning: they are posing questions, exploring relationships, helping each other without fear of embarrassment, and recording their findings in a "What I Notice" journal.
At the conclusion of the activity, everyone gathers in a circle. All are familiar with the process: the teacher poses a question -- "What mathematical patterns did you notice?" -- and each member of the group offers a short response. The expectations are simple: all participate, and at this point, lengthy discussion is not expected. The benefits are quickly realized, as all voices are heard and every comment is valued, no matter how brief. Trust and mutual respect are strengthened within this learning community.Classroom Comment
Students in this class feel valued because their teacher has created a supportive,respectful classroom environment where students can question, investigate and offer ideas without fear of embarrassment at being wrong. If a child's conclusions are seriously flawed, the teacher, volunteers, and other students encouragingly guide that student to a more accurate conclusion without attacking the original idea.
The teacher expects to hear every child, drawing out the soft voices that are not easily heard. All listeners consider divergent viewpoints. The adults model respectful and productive learning behaviors. Community members are welcome because teacher and students perceive the community as their extended classroom, an available and important learning environment.In Another Classroom
A seventh-grade teacher asks her class to create a timeline of important scientific discoveries. The classroom computers contain a database including thousands of relevant facts, from the development of Euclidean geometry in ancient Greece, through 17th-century discoveries about gravity, to the modern creation of the computer.
From among these landmark scientific developments, students choose the discoveries they believe to be most important to contemporary society. They take turns working at computers to extract database information for use on their timelines. The computers, modified through use of peripherals, are accessible to all, including physically disabled students.
Students organize their timelines around different themes: discoveries made by women or men, for example, or discoveries regarded as purely science or purely mathematics. The timelines may chart a chain of discoveries leading to a modern innovation. The teacher expects students to justify their selections in a descriptive paragraph displayed with each timeline.
The class continues to work on this project for some time daily over a week, after which the projects are examined and discussed. Individual timelines are evaluated for accurate information, clear presentation, and adequate justification.
After the class has examined all of the work to date, students collectively determine the historical scale that would include all their timelines. Then they create a new timeline that includes all individual efforts and gives an accurate scale distance between discoveries. Students examine this class timeline for any new patterns or relationships it might reveal.Classroom Comment
This seventh-grade teacher has employed a variety of teaching strategies and technologies, each of which addresses different student needs. These methods provide rich learning situations for all students, while maintaining flexibility in instruction, assignments, completion time and assessment.
This teacher's strategies address a variety of learning styles -- visual, verbal, kinesthetic and others. Her classroom is equipped to meet the needs of all learners, including those with disabilities.
The assignment is sufficiently open-ended so that all students can be successful in showing what they have learned about the history of mathematics and science. Students are able to "personalize" the assignment -- to pursue themes that are particularly meaningful to them and which allow them to see themselves connected to the work. The assignment requires that students form and justify positions about ideas. Additionally, students must use measurement and computation skills to create their scaled timelines.
Every student participates in the assignment and is asked to meet high expectations; the teacher has no preconceived notions about who can or cannot learn mathematics and science. This teacher has modeled positive attitudes and expectations of success by encouraging and supporting all students.
In student-centered classrooms, learning is accessible, challenging, relevant and engaging for all. Here teachers know that they can learn from students as well as guide them.
In the Classroom
Fourth-graders are exploring estuarine soils and plants. After a field trip to the estuary to collect soil samples, the students examine the various soils, noting characteristics of each one (texture and amount of organic matter).
The class discusses student observations and matches soil descriptions (peat, mud orsand) with the soils. To learn which characteristics enable plants to succeed in particular soil types, groups of children check the soils for water percolation rates.
Next, students synthesize what they have learned through investigation, research and discussion by "designing" a plant that could live in each of the soil types. Finally, the groups present their plant designs to the rest of the class. The plant may be fictitious or real, but students must explain how the plant is able to get the appropriate amounts of nutrients, gases and moisture from the soil. The group agrees to investigate further by growing some real plants in each of the different soil types.Classroom Comment
In this classroom, students are actively engaged in learning about the world around them. Collecting three soil samples from the field, children analyze their samples in the laboratory. From prior knowledge and new information about estuarine plants, they are able to determine characteristics of plants that might live in these soils.
The students have been acting as scientists, constructing knowledge by observing, exploring relationships and examining underlying concepts, questioning, collecting data, forming and testing hypotheses, analyzing, justifying findings, forming generalizations, applying knowledge to new situations, and developing new research questions. Activities which provide this variety of experiences can engage students' interests, address their areas of strength and inspire investigative learning.
Thinking across the disciplines of study can enhance student's understanding of mathematics and science. When students forge connections -- not only among the various disciplines, but also between these disciplines and their lives -- they develop deeper and more functional understandings.
One way teachers can incorporate this practice is through employing themes to integrate mathematics and science with each other and with other content areas. For instance, the use of change as a theme enables teachers and students to explore a number ofmathematical and scientific ideas concurrently. It also provides an opportunity to connect with literature, the arts, history and other content areas.In The Classroom
Change is being used as an interdisciplinary theme for the year in a sixth grade classroom. In September, students brainstorm a list of changes they have observed in the world around them. Then, after interviewing a family adult about memories of changes in the world since their childhood, students write a short essay documenting this oral history. Later in the year children will study the history of their community, reflecting on the oral history project and trying to identify important factors in the community's development.
A weekly study of changes in the shadow cast by the school's flagpole is conducted throughout the year. At three different times of day, students measure the shadow's length and use a compass to determine the shadow's direction. They record the data and discuss apparent patterns in the changes they notice. These observations lead to hypotheses about the relationship between the sun and the Earth, as children study the changing seasons and the relative movements of the Earth, sun and moon.
In mathematics students work with tiles to determine how the area of a rectangle changes when the perimeter is kept constant but the length of the sides is changed. Later they will examine the relationship of changes in the area of a circle to its diameter and radius. Throughout the year, the class studies a number of other topics through the lens of the change theme. As a final project, students will make a presentation on a change they have chosen to research independently.Classroom Comment
The teacher has selected change as the theme because it offers rich connections for exploration among various mathematics and science concepts and skills. It also suggests easy bridges to other disciplines and to interesting problems and phenomena in the world beyond school.
During the year, students will be looking at a variety of changes, the causes and effects of change, the role of variables and constants, and the effect of holding a variable constant. In many cases, the teacher determines the changes to be studied. However, students themselves also identify and study changes of particular interest to them, working individually or in small groups.
Three important questions must be continuously answered in mathematics and science classrooms: What will be taught?, How will it be taught? and How will learning be assessed? These three questions represent the three major components of mathematics and science teaching: curriculum, instruction and assessment. These three components must be meaningfully connected for effective teaching and learning.
In the Classroom
Secondary students are researching the possible effects of building a dam on a local river. After studying current land use policy and the record of local decisions on relevant issues, students visit the proposed dam site to collect a variety of water quality data, including pH, turbidity and dissolved oxygen content.
Prior to the site visits the teacher has assessed the knowledge and skills of each student to assure that they are prepared to complete the data collection procedures safely and effectively. The teacher and students have also discussed the culminating projects which will comprise assessment of student learning, including a model, a report, a presentation, and a portfolio of work collected throughout the study. Together they have developed criteria for good projects and set clear, high expectations for student achievement. Learners understand the goals of the study and are ready to work with data from the field.
Students work together to interpret the collected data, develop hypotheses, and build a physical model of the dam to test their hypotheses and draw conclusions. Students compare their findings with a professional study of the proposed dam project, developing explanations for any discrepancies and refining their hypotheses for further testing. At the conclusion of the unit, students complete a self-assessment based on the criteria developed earlier and meet with their teacher to get feedback from her assessment of their work.Classroom Comment
As a working definition of curriculum evolves to include process skills in addition to student mastery of essential knowledge, our understanding of the scope of the terms instruction and assessment must also expand. Educators who wish to nurture more complex abilities use instructional practices and assessments which invite students to demonstrate and practice those skills. Instruction includes opportunities to create, perform and apply knowledge and skills. Assessments can take various forms: student presentations of research findings in letters to the editor, debates, video presentations, skits shared with the community or other inventive demonstrations.
Criteria and standards for each type of assessment can be developed collaboratively with students and clearly stated at the beginning of a project. A rubric or detailed description of the characteristics of a quality performance or product is useful in establishing and maintaining high expectations. Through instruction and curriculum, the teacher helps students achieve those high levels of performance. When assessment methods match the goals of curriculum and instruction, assessment becomes a tool to promote and enhance learning.
An excellent assessment program promotes student learning rather than merely measuring it. It informs instructional decisions and program evaluation as well as providing documentation of student learning. Multiple forms of assessment, such as student journal entries, teacher observations, student self-evaluation, presentations, performance tasks, open-response tasks and traditional tests provide members of the school community with a wide variety of data to use in evaluating student progress. When expectations are clear, feedback is focused and learners are given several opportunities to demonstrate their understanding. They continuously improve by capitalizing on their strengths and improving areas of weakness.
Self-confidence is the belief in one's ability to succeed. In a confidence-nurturing environment, adults encourage and support student expression and risk-taking. The classroom is a safe place to question and explore. Every student is encouraged to defend viewpoints, conjecture, design, and experiment individually and in groups. Students practice skills thatempower them to engage effectively in the challenges of new mathematics and science situations. They work on problems that matter.
Teachers use a variety of techniques to enhance all students' opportunities for success. In light of the historical inequities in delivery of mathematics and science instruction, it is especially important to engage all students and to provide them with the chance to demonstrate their knowledge in a variety of ways.In the Classroom
Sixth-grade students work in mixed-gender teams representing a range of abilities in the class. Their task is to insulate a soda can full of hot water, in order to retain the most heat during a given time.
After assigning individual responsibilities in the group, students collaborate in designing, building and testing their insulated cans. Groups report to the class about the nature of their design, their process of design development and the construction of their model. Other students respond by asking questions about designs, materials and results and by pointing out strengths in the group's work.
A classroom discussion follows. The teacher probes for reasons why some materials apparently held heat better than others. The class listens to all responses and all students are encouraged to offer ideas.
A later discussion focuses on the factors that determine a material's ability to slow the transfer of heat energy. The teacher encourages students to support their ideas using facts they learned earlier regarding heat convection and conduction. The original ideas students had offered about their cans are analyzed for their consistency with scientific principles. Students then refine and test their designs, collecting and comparing data about the effectiveness of the insulation.
The teacher later records the day's events in a journal and makes note of the fact that a student who entered the class in September saying, "I really stink in math and science," has been actively engaged in these heat-related studies. No longer does the student passively agree only to take on the recorder role in a group. This learner now freely volunteers ideas and takes a hands-on role in trying out ideas with the group.Classroom Comment
Students of any age who are confident in their mathematical and scientific abilities can explain their understanding of mathematics and science concepts and gain the ability to apply those concepts in new situations. They can learn to use a variety of tools and resources to construct knowledge independently and collaboratively.
Confident students freely share their ideas and listen to the ideas and perspectives of others. They can learn to give and respond to ideas and suggestions in a nonaggressive, constructive manner. They view mistakes as opportunities to analyze, question and learn. Confident students persevere to see a challenge through to completion. They can accept a variety of roles and responsibilities in the classroom.
Confident learners are free to follow their curiosity. They are willing to raise questions about the world around them and seek answers, not only in school but throughout their lives.
Some past schooling practices have presented students with two apparently contradictory maxims: "Two heads are better than one" and "Always do your own work." While the need for independent work skills will never disappear, contemporary workplace and societal challenges demand that workers and citizens communicate and collaborate more effectively. Learning how to be interdependent is becoming more important.
The various classroom activities described in Beliefs about Best Practices show students collaborating in a variety of groups and communicating their knowledge during the learning process. When students collaborate, they develop necessary interpersonal skills, especially an awareness of their responsibilities to the group. These skills include offering ideas, listening, completing tasks, and knowing when to lead and when to follow. Collaboration also helps students develop their communication skills.
The expectations of communication and collaboration add valuable dimensions to learning by providing students opportunities to discuss or write about underlying concepts. When a task demands a single product from a group or team, students learn how to come to consensus in order to reach a joint accomplishment. The informal discussion in theconsensus- building process is as important as communicating more formally in writing or speaking. Students should also, however, practice formal communication skills by writing papers, conducting research, and making oral presentations to various audiences.
Students in our classroom pictures have been given a variety of problems to solve, some which have invited them to learn about the mathematical and scientific contributions of past thinkers, and some which have engaged them in relevant, timely issues that both challenge and interest well-educated adults in their own communities. Such activities invite students to explore in a disciplined fashion, with room for creativity and initiative. These investigations provide rich motivation because students can see the practical consequences of knowing or not knowing essential mathematics and science. Whether constructing mathematical concepts or growing in understanding of the natural world, students solve such problems by building on prior knowledge and connecting new knowledge to related ideas.
Learners need to understand the relationship between technology and scientific investigation. Science is concerned with understanding both the "how" and "why" of natural phenomena. Technology uses that information to develop new products, tools and inventions meant to improve the human condition.
Consider, for example, the Wright Brothers -- two bicycle mechanics who used the scientific inquiry process to develop one of the most significant inventions of the twentieth century: powered flight. After a few years of designing, testing, failing, redesigning, retesting and failing again, they came up with a machine that flew 120 feet under its own power along the shore at Kitty Hawk, North Carolina. People had built flying machines before, but the Wrights used new techniques and materials to make their airplane lighter and more powerful. Thus they made their design work.
The Wrights probably never envisioned the massive changes their work would bring about in the world. Now, with supersonic flight, no place on Earth is more than one day'sflight time from any other, allowing exchange of ideas, materials and people with increasing ease.
Often, as in this case, technology (the invention of the airplane) leads to new science (development of the study of aerodynamics), which leads to new technologies (the development of the jet engine), and so on. Scientists and engineers play pivotal roles in this spiraling process of developing understanding.
Knowing how to choose and use technology and its tools appropriately is becoming more important than ever, as rapid technological change continues to affect daily living and the educational process. In education, the term "technology" is sometimes used as a synonym for calculators and computers -- tools that, when used appropriately, can enhance learning. There actually exists a wide range of technology with the potential to change not only how something is taught (e.g., multimedia presentations instead of slide shows) but also what is taught (e.g., design and construction of robots).
Students should be helped to understand technology in its own right. Preparing the young to be technologically competent does not mean simply teaching them to use a calculator, fax machine or computer. It also means promoting their understanding of the design process and of the relationship between technology and science. Further, it means cultivating their ability to evaluate the implications of technology.
In order to enhance the learning of mathematics and science for all students, classrooms at all levels must be equipped with a variety of educational tools, from library books to CD-ROM encyclopedias, from mathematical manipulatives to laboratory apparatus, from computers and graphing calculators to video microscopes, and from skill practice programs to tools and software for collecting and interpreting data. Students and teachers need to learn how to use these tools and how to choose the appropriate tool for a particular application.
Use of technology is not a substitute for knowledge and inquiry. Spreadsheets and databases can analyze large quantities of data efficiently and provide fast access to data, but students need to experiment, collect and analyze data themselves.
Equity requires special attention in the area of technology: all Maine students and teachers should have access to the tools essential for preparing children and adolescents to besuccessful in a changing world.
The array of instructional resources available today (including people, physical materials such as artifacts and specimens, equipment, instrumentation, print media and electronic media) is staggering in its variety and sometimes in its complexity. This variety will likely continue to expand, as the information highway promises to give us the world at our fingertips. Students and teachers need to know how to select wisely from this wealth of information.
In the Classroom
Students in a chemistry class are researching chemistry in the world around them: after each student has chosen a particular household cleaner, the class has been given the task of determining each cleaner's active ingredients.
Since the chemistry text holds little relevant information, the teacher has made arrangements with the school librarian for some additional research assistance. The library contains reference books, databases on CD- ROM, and on- line connections to the local university campus. The librarian is knowledgeable in the use of these resources and can help the students find the chemical constitution of their cleaners' active ingredients.
In another case a team of teachers wants information about the use of Cuisenaire RodsTM to teach mathematical concepts. They gather their information by talking to others in their own school, calling the nearby College of Education, contacting their mathematics professional organization, searching through electronic databases for magazine articles or books written on the subject, and connecting on- line to chat with teachers from around the country about instructional practices using Cuisenaire RodsTM .Classroom Comment
The benefits of these educational tools and resources, as well as the knowledge of how to use them effectively, must be shared with all Maine teachers and learners, regardless of geographic, economic, physical or cultural differences.
- Does the planned activity address all types of learning styles?
- Is the planned activity accessible to all students?
- Are students actively learning and solving problems?
- Does the environment encourage risk-taking?
- Can every student expect to be listened to in this classroom?
- Do students collaborate effectively?
- Do students have opportunities to practice communicating their new knowledge? .Do assessment practices match curriculum and instructional methods?
- Are appropriate tools and technology for learning being used?
- What additional resources could enhance the learning environment?
_______________________________
Note: While specific products are used in some examples,
no endorsement of the products is intended.
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