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Preceding Page
- Looking Ahead
- Part A: Beliefs About Best Practices for Teaching and Learning Mathematics and Science
Current Page
- Part B: Beyond Beliefs: The Guiding Principles
Following Pages
- Guiding Principles
- 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
The seven
Guiding Principles are the heart of Maine's Curriculum
Framework for Mathematics and Science. They incorporate the
thinking of Framework developers and the national standards
for science and mathematics. The guiding principles are intended
to help establish clear, coherent expectations about what all
Maine students should know and be able to do in K-12 mathematics
and science.
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.
The guiding principles must also be connected with
content areas outside of mathematics and science. Just as the
guiding principles cannot stand alone, neither can individual
content areas. Connecting content areas helps students to apply
their knowledge and skills in an integrated fashion that reflects
the real world and relates learning to life outside the classroom.
The Guiding Principles
are the central ideas of Maine's Curriculum Framework for Mathematics
and Science. After each guiding principle is stated, its practical
meaning is explored in more detail through presentation of Content
Standards which describe important strands and themes suggested
by that principle.
A discussion of Instructional Implications describes useful
teaching strategies and other ideas helpful in working toward
each principle. These illustrate the overall effect that use of
the Guiding Principle may have on a classroom learning environment.
Snapshots offer a practical picture of the kinds of instructional
activities or assessments that might promote the principle.
Finally, each content standard is explored in depth. Discussion
of these standards includes particular Performance Indicators,
which state specifically the behaviors which knowledgeable students
will demonstrate within each of four grade clusters: Primary
(Grades Pre-K-2), Intermediate (Grades 3-5), Middle
(Grades 6-8) and Secondary (Grades 9-12). Accomplishment
of these indicators within each grade cluster demonstrates that
the student is moving toward achievement of the guiding principles.
The sum of these parts, enacted through the Best Practices,
can result in a rich, challenging mathematics and science curriculum
for all Maine students.
The organization used for discussion of each of the Guiding Principles
is illustrated in Figure 2.
Content Standard(s)Important strands and themes within the Guiding Principle.
A brief description of the guiding principle follows.
Instructional Implications Broad descriptions of teaching strategies and ideas which are important for this Guiding Principle.
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Specific classroom scenarios that give a "picture" of how this Guiding Principle (or Content Standard) might look in the classroom. These snapshots may suggest classroom activities or assessments.
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Content Standard(s)Each Content Standard for the Guiding Principle is restated and briefly described here.
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Performance Indicators All Levels
Specific descriptions of what students should know and be able to do at each of four grade clusters in order to move toward achievement of the content standard and guiding principle.
Primary (Pre-K-2)
Intermediate (Grades 3-5)
Middle (Grades 6-8)
Secondary (Grades 9-12)
Content Standards A. Students use scientific inquiry to provide insight into and comprehension of the world around them.
B. Students use mathematical inquiry to develop conjectures and work to prove or disprove them within a mathematical system.
C. Students use models to understand the world around them.Science and mathematics are creative, systematic human enterprises that help us understand the universe. Scientific understanding - the exploration of the workings of the universe - is not the same as the accumulation of facts. Similarly, mathematical understanding - the study of patterns and logic to describe the world - is more than the mastery of computation skills. The application of scientific knowledge, together with the investigation of interesting questions and the use of mathematics to analyze data and model phenomena, leads to understanding.
Mathematics and science as enterprises share many values and features: belief in order, ideals of honesty and openness, the usefulness of colleagues' critical judgment, and the essential role played by imagination. Developments in science often stimulate innovations in mathematics by presenting new kinds of problems to be solved. Likewise, developments in mathematics often stimulate new innovations in science.
Students need ample opportunities to do science and mathematics. At the earliest levels, students can begin learning and practicing the process skills of science and mathematics. Observing, questioning, classifying, representing and communicating should be part of the youngest students' instruction. As students get older, instruction in those skills becomes more complex. As students gain experience, they will be able to formulate hypotheses, create models, analyze data and interpret results. It is only in doing science and mathematics that students can appreciate their true value.
Instructional Implications To understand how mathematicians and scientists work, and to appreciate both the utility and beauty of these enterprises, students need experiences that allow them to construct their own knowledge and to see connections within their world. Integrating science and mathematics with each other and with other subjects helps students develop an appreciation for the power and utility of the disciplines.
Teachers can encourage and support this understanding and appreciation by developing students' sense of wonder for science and mathematics, by presenting situations which promote student exploration, by providing students with tools which help them construct their own understanding, by seeking opportunities for students to work as and interact with scientists and mathematicians, and by providing learners with opportunities to present a variety of ideas and projects. In addition to teacher-chosen topics, students need opportunities to decide what to explore and how to design those explorations. The skills which students must practice from a young age include observing, questioning, formulating hypotheses, creating mathematical models, making predictions, testing and experimenting, analyzing data, interpreting and justifying results, creating and communicating explanations, and applying knowledge to new situations.
Observing involves using the five senses to perceive qualities of an object (although vision, tasting, touching and smelling may be restricted due to the nature of the object being observed). Sometimes these qualities can be observed directly, and sometimes it is beneficial to use a tool. Students learn that good observations should be quantitative and may involve the use of sense extending devices, such as microscopes and thermometers.
Formulating hypotheses involves proposing a tentative explanation based on previous observations or prior knowledge. Many learners believe that a hypothesis is a guess, but it is more. Students need to learn that good hypotheses must be testable and that scientific hypotheses are not proved or disproved, but merely supported or not. As additional support for a hypothesis is gathered by a wider community of scientists, the hypothesis can become a theory or even a natural law. Mathematical hypotheses, however, can usually be proved true though mathematical inquiry and logic arising from a common set of basic assumptions.
Models are theoretical or physical representations of phenomena. Formulating models means creating objects (physical models) or conjectures (conceptual models) to describe something unfamiliar in terms of something that is more readily known or formulating an equation or function that describes a situation mathematically (theoretical model). Models can be tested under controlled conditions. Effective models can be used to predict the outcome of a process even when conditions are different from those under which the process was studied.
Predicting involves the forecasting of future events on the basis of observed patterns. Good predicting requires students to think about occurrences and to extend what they have noticed. Taken further, predicting skills are necessary to making good hypotheses or models.
Experimenting involves the process of defining a problem, planning and carrying out a fair test of a hypothesis, and using the recorded results to pose a possible solution. Students should be given opportunities to experiment, ranging from the simple to the more complex, in order to gain an understanding of all that is involved in the process. They need to learn that in a properly designed experiment, all factors which may affect results must be held constant except the factor being tested. Students can first be given opportunities to identify variables by determining if tests are "fair," and later, to control variables through opportunities to design and carry out their own investigations.
Analyzing and interpreting data involves determining the meaning of the results in order to learn more about the system under study. Collected results are used to answer questions, solve problems or test hypotheses. This may include drawing generalizations or conclusions, extending patterns, and extracting relationships using equations and graphs. Analyzing data may involve looking at and thinking about either a single measurement or the relationship of several variables.
Creating and communicating explanations includes two process skills: inference and communication. Inference involves making a judgment based on evidence from observations. Students should be taught the difference between observations and inference and have opportunities to practice both. Thus they can learn that inference involves making decisions about observations based on prior knowledge and experience, as well as on the data they have collected. Communicating can be defined as describing objects, events or findings so that others can understand.
Communicating can be done verbally or visually and can include drawings, photographs and presentations of data in the form of charts, tables and graphs. Accurate communication is a fundamental skill in mathematics and science, since it is important that others be able to replicate and interpret one's results. Students should be given opportunities to communicate in varied contexts and for varied purposes.
Teachers must also look for opportunities to involve students in activities that make connections to the usefulness of science and mathematics. For example, students can study discoveries that have been made by scientists and mathematicians of different cultures. They can gain an understanding of the predictive capabilities of science and mathematics. In addition, young people should become aware that while career opportunities in both subjects can enhance their own lives, mathematics and science are useful in all careers._____________________________________________
SNAPSHOTS Primary students explore patterns of numbers by discovering similarities among objects grouped into even-numbered and odd-numbered sets. The teacher asks for the students' observations. Some note that even-numbered sets can be arranged in a line and that no object is in the middle of the line, but that odd numbered sets have an object in the middle. Other students show that objects in even sets can be paired, while objects in odd sets have one remaining when paired. Other activities include exploring patterns and relationships among shapes, colors, direction, orientation and size.
~~~~~ Third-graders doing a free exploration using magnets with various materials are talking with others as they work. After a class discussion on their observations, groups of students are told to choose at least one problem to explore and report to the class. Their report, a demonstration including both spoken and visual elements, must state the problem being researched, findings, implications and usefulness of these findings for others. The visual element might be a chart, table, picture, graph or magnet demonstration. All group members take part in the report and answer questions from the class on their problem. Groups select such problems as determining which items are attracted to magnets, how many paper clips in touching chains each magnet type can hold, and the thickness of materials through which magnets' attraction power can still be felt. On subsequent days, groups verify others' findings.
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Content StandardA. Students use scientific inquiry to provide insight into and comprehension of the world around them.
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Figure 3.
Performance Indicators
At each level, students moving toward achievement of this standard will: Primary
1. Make accurate observations using appropriate toosl and units of measure. (S-J1)
2. Ask questions and propose strategies and materials to use in seeknig answers to questions (S-J2)
Intermediate
1. Make accurate observations using appropriate tools and units of measure. (S-J1)
2. Conduct scientific investigations: make observations, collect and analyze data, and do experiments. (S-J2)
3. Explain how differences in time, place, or experimenter can lead to different data. (S-J5)
4. Explain how different conclusions can be derived from the same data. (S-J6)
5. Explain the importance of repeated trials.
Middle
1. Make accurate observations using appropriate tools and units of measure. (S-J1)
2. Design and conduct scientific investigations which include controlled experiments and systematic observations. Collect and analyze data, and draw conclusions fairly. (S-J2)
3. Compare and contrast the processes of scientific inquiry and the technological method. (S-J4)
4. Explain how personal bias can affect observations. (S-J5)
Secondary
1. Make accurate observations using appropriate toosl and units of measure.(S-J5)
2. Demonstrate the ability to use scientific inquiry and the technological method with short-term and long-term investigations, recognizing that there is more than one way to solve a problem. Demonstrate knowledge of when to try different strategies. (S-J3)__________________________________________
SNAPSHOT
Middle school students have been studying the physical properties of matter. On a given day the students choose unknown substances (from a teacher generated selection) and are allowed to investigate in their own fashion to determine as many of the substances' physical properties as possible. The experimental process and the inquiry method allow the students then to attempt some classification schemes using a computer database to sort substances by physical properties. By cross-referencing their database with the characteristics of known substances, the students try to identify the unknown substances. Based on their results, student groups assess the effectiveness of their own classification scheme.
Students then design a classification system for examining plastics: types of plastic, uses of various plastics, and recyclable or non-recyclable properties of different types. The students visit a plastics recycling center to see the process. They talk with recycling technicians and plastics engineers about their work and investigate different careers.
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Content Standard B. Students use mathematical inquiry to develop conjectures and to prove or disprove them within a mathematical system.
__________________________________________________Mathematical inquiry consists of a cycle of investigation leading to the development of valid mathematics ideas. The cycle consists of three stages: representation, manipulation and validation. Like scientific inquiry, mathematical inquiry is not a rigid process-one may begin at any step.
Students need many chances to make conjectures about mathematical ideas and try to prove or disprove these statements. Making conjectures leads students to making connections within mathematical systems. For example, the idea that what works for one set of numbers might work for another set may lead students to an understanding of real numbers by studying integers. They may begin to "prove" this by checking a few examples, but as their mathematical sophistication grows, they should quickly move to generalization and formal mathematical proof.
Performance Indicators Primary
1.Demonstrate that performing the same operation with the same operant on the same numbers will yield the same result every time.
2.Use physical examples to justify math facts.
Intermediate
1. Demonstrate that some mathematical rules hold for all numbers.
Middle
1. Provide reasons to support mathematical conclusions and verify that numerous examples are not necessarily sufficient to prove a statement.
2. Recognize that within a mathematical system, some mathematical statements are always true, and that these can be used to demonstrate or prove that other statements are always true as well.Secondary
1. Differentiate between different mathematical systems and recognize that rules which hold in one system may not hold in others.
2. Construct formal mathematical arguments to prove or disprove conjectures.
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High school students have been studying chaos theory and meteorology. Beginning work at computer terminals about two weeks before graduation, they analyze up-to-date weather data transmitted by satellite, using the information to prepare a number of long-range graduation day forecasts, making small changes in the initial conditions and deciding on the most likely forecast. Students collect data daily and make new predictions and refinements as graduation day nears. At the conclusion of the exercise, students are asked to reflect on the difficulty of weather prediction and the application of chaos theory to meteorology.
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Content Standard C. Students develop models to understand the world around them.
___________________________________________________Scientists and mathematicians use models to make predictions based on hypotheses, to simulate phenomena and to generate data. This information is then used to form and justify explanations for the situation being modeled.
Performance Indicators Primary
1. Identify shapes can be found both in nature and in human designed objects.
2. Represent and describe mathematical relationships.
Intermediate
1. Use variables and open sentences to express relationships.
2. Identify an instance when answers that may be right theoretically may not appropriately answer a practical problem.
3. Represent mathematical ideas concretely, graphically and symbolically.
Middle
1. Use different methods to solve problems.
2. Describe and represent relationships with tables, graphs and rules.
3. Use patterns and functions to represent and solve problems.
4. Describe how computer models are used to simulate phenomena too dangerous, too expensive or too difficult to observe (e.g., hydrogen bombs, weather models).
Secondary
1. Make predictions using statistics, probability and functions.
2. Form logical arguments to justify explanations.
3. Represent and analyze relationships using tables, verbal rules, equations and graphs.
4. Translate among tabular, symbolic and graphical representations of functions.
5. Demonstrate an understanding that while mathematical statements may be true, it does not mean that models based on these statements are always valid; models must be tested against reality just as scientific theories are tested.
