THE FIRST FOUR STANDARDS
STANDARD 3 - MATHEMATICAL CONNECTIONS
Making connections enables students to see relationships between different topics and to draw on those relationships in future study. This applies within mathematics, so that students can translate readily between fractions and decimals, or between algebra and geometry; to other content areas, so that students understand how mathematics is used in the sciences, the social sciences, and the arts; and to the everyday world, so that students can connect school mathematics to daily life.
Meaning and Importance
Although it is often necessary to teach specific concepts and procedures, mathematics must be approached as a whole; concepts, procedures, and intellectual processes are interrelated. More generally, although students need to learn different content areas, they also should come to see all learning as interwoven. In a very real sense, the whole is greater than the sum of its parts. Thus, the curriculum should include deliberate efforts, through specific instructional activities, to connect ideas and procedures both among different mathematical topics and with other content areas.
K-12 Development and Emphases
One important focus of this standard is that of unifying mathematical ideas - major mathematical themes which are relevant in several different strands. They emerge when a higher-level view of content is taken. They tie together individual mathematical topics, revealing general principles at work in several different strands and showing how they are related. Unifying ideas also set priorities, and the curriculum should be designed so that students develop depth of understanding in each of the unifying ideas at their grade level. Since it often takes years to achieve understanding of these unifying ideas, students will encounter them repeatedly in many different contexts.
An example of a unifying idea is the concept of proportional relationships. Proportional relationships play a key role in a wide variety of important topics, such as ratios, proportions, rates, percent, scale, similar geometric figures, slope, linear functions, parts of a whole, probability and odds, frequency distributions and statistics, motion at constant speed, simple interest, and comparison. Awareness of the commonprinciple operating in all these topics is an important part of mathematical understanding. Unifying ideas for grades K-4 include quantification (how much? how many?), patterns (finding, making, and describing), and representing quantities and shapes. Unifying ideas for the middle grades include proportional relationships, multiple representations, and patterns and generalization. For the high school, the unifying ideas are mathematical modeling, functions and variation (how is a change in one thing associated with a change in another?), algorithmic thinking (developing, interpreting, and analyzing mathematical procedures), mathematical argumentation, and a continued focus on multiple representations.
Another focus of this standard is mathematical modeling, developing mathematical descriptions of real-world situations and (usually) predicting outcomes based on that model. Even very young children use mathematics to model real situations, counting candies or cookies or matching cups and saucers on a one-to-one basis. Affirming the ways things make sense outside of school and connecting them to things in school is important for students. Older students use mathematical modeling as they develop the concepts of function and variable, extending ideas they already have about growth, motion, or cause and effect. At various grade levels, students can use their experiences waiting in lines to investigate the general relationships among waiting time, the number of people in the line, the position at the end of the line, and the length of time each person takes to buy a ticket or a lunch. Older students can further develop a mathematical model that can help them predict waiting time and formulate and evaluate their suggestions for solving a real problem - getting everyone through the school lunch line more quickly.
Throughout the grades, students need to develop their understanding of the relationship of mathematics to other disciplines. Mathematics is frequently used as a tool in other disciplines. In science, students measure quantities and analyze data. In social studies, they collect and analyze data and make choices using discrete mathematics. In art, they generate designs and show perspective. Other disciplines can also provide interesting contexts to learn about new mathematical ideas. For example, students might learn about symmetry by generating symmetric designs with paint, or they might explore exponential functions by modeling a dying population by repeatedly scattering M&Ms, at each step removing those that have the M showing, so that about half of the population "dies" each time.
Every person views the process of making connections between disciplines differently, and everyone works in different circumstances in which often-competing goals must be balanced. Nevertheless, it is important to establish some means of making explicit the connections between mathematics and other disciplines. There are several ways in which the presentation of content can be approached, ranging from "fragmented" to "integrated" (adapted from Fogarty, 1991):
While the first two approaches do not provide for the establishment of appropriate connections between subjects, the remaining six do provide some degree of interaction. Teachers must consider carefully which of these is most appropriate and feasible for their own situation.
This discussion addresses the connections between mathematics and other disciplines. Of special significance because of their many commonalities is the relationship of mathematics to science. Berlin and White (1993) have identified six areas in which mathematics and sciences share concepts or skills: ways of learning, ways of knowing, process and thinking skills, conceptual knowledge, attitudes and perceptions, and teaching strategies.
Ways of Learning - The two disciplines have a common perspective on how students experience, organize, and think about science and mathematics. Both disciplines endorse active, exploratory learning with opportunities for students to share and discuss ideas. Students must do science and mathematics in order to learn science and mathematics.
Ways of Knowing - Both science and mathematics use patterns to help students develop understanding. Even very young children seek to make sense of patterns in order to make sense of their world. They make generalizations based on what they have observed and apply these generalizations to new situations. Sometimes their guess works, reinforcing the generalization, and sometimes it doesn't, requiring the child to revise the generalization.
An example from chemistry demonstrates the similar ways of knowing used in mathematics and science. Chemist Dmitri Mendeleev proposed a periodic table of the elements based on increasing atomic weights. Sometimes he left open spaces in the table, where he reasoned that unknown elements should go. In 1869, when he arranged his table, the element gallium was unknown; however, Mendeleev predicted its existence. He based his predictions on the properties of aluminum (which appeared directly above gallium in the table). Mendeleev even went so far as to predict the melting point, boiling point, and atomic weight of the thenunknown gallium, which he called ekaaluminum. Six years later, while analyzing zinc ore, the French chemist Lecoq de Boisdaudran discovered the element gallium. Its properties were almost identical to those Mendeleev had predicted.
Process and Thinking Skills - Central to both disciplines are process skills. Mathematics focuses primarily on the following four process skills: problem solving, reasoning, communication, and connections. Basic process skills in science (Tobin and Capie, 1980) include observing, inferring, measuring, communicating, classifying, formulating hypotheses, experimenting, interpreting data, and formulating models.
Conceptual Knowledge - There is considerable overlapping of content between science and mathematics. By examining the concepts, principles, and theories of science and mathematics, those ideas that are unique to one subject and those which overlap both disciplines can be identified. Some of the "big ideas"which are common to both include conservation (of number, volume, etc.), equilibrium, measurement, models (including both concrete and symbolic), patterns (including trends, cycles, and chaos), probability and statistics, reflection, scale (including size, duration, and speed), symmetry, systems, variables, and vectors. An example of a way to interrelate science and mathematics contents is to link population dynamics and genetics in science with sampling and probability in mathematics.
Attitudes and Perceptions - Mathematics and science share certain values, attitudes, and ways of thinking: accepting the changing nature of science and mathematics, basing decisions and actions on data, exhibiting a desire for knowledge, having a healthy degree of skepticism, relying on logical reasoning, being willing to consider other explanations, respecting reason, viewing information in an objective and unbiased manner, and working together cooperatively to achieve better understanding. Both disciplines also value flexibility, initiative, risktaking, curiosity, leadership, honesty, originality, inventiveness, creativity, persistence, and resourcefulness, as well as being thorough, careful, organized, selfconfident, selfdirected, and introspective, and valuing science and mathematics (Science for All Americans). Students' engagement in personal and social issues and interests may also help to encourage, support, and nurture their confidence in their ability to do science and mathematics.
Teaching Strategies - The shared goals of mathematics and science instructions are, according to Science for All Americans, to have students acquire scientific and mathematical knowledge of the world as well as scientific and mathematical habits of mind. Both disciplines support teaching strategies which foster inquiry and problem-solving, promote discourse among students, challenge students to take responsibility for their own learning and to work collaboratively, encourage all students to participate fully, and nurture a community of learners (National Science Education Standards).
In summary, making connections within mathematics and between mathematics and other subjects not only helps students understand the mathematical ideas more clearly, it also captures their interest and demonstrates how mathematics is used in the real world. Important connections that need to be established include working with unifying mathematical themes, using mathematical modeling, and relating mathematics to other disciplines and to the real world.
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