Defining Connections
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Let's look at several examples of connections both within and outside mathematics.

The Connections Standard states that "instructional programs should enable students to understand how mathematical ideas interconnect and build on one another to produce a coherent whole." Connections are not apparent to many students until they are explicitly brought to the forefront for discussion and examination. "Teachers should help students explore and describe mathematical connections and ensure that they see mathematical ideas in a variety of contexts and models" (NCTM, 2000, p. 203). Curricular materials may also draw attention to connections through questioning and through the choice of examples and explanations.

Consider the following problem and its various connections:

 Here is a dartboard. Assume that a dart has an equal chance of landing on any spot on the board. Which prize do you have the best chance of winning? Explain your answer. (Answer: The blue area that covers 4/10 of the entire square is the largest. Thus, you have the best chance of winning a key chain.)

Students sometimes learn the skill of naming equal fractions and decimals without conceptually understanding that these numbers are the exact same amount. Working with models, such as rectangular areas, number lines, and rulers, can give learners a visual verification that the amounts truly are the same. For this problem, students can draw on their prior experience with naming shaded parts of regions that are subdivided into equal parts. They can decide to draw more lines to show 20 equal parts and use that as a common denominator to name the part for each color. But they can also think of the region as 100 hundredths, or 1.00, and recognize 0.1, 0.4, and 0.25 as names for various colored parts.

Through careful questioning and guiding discussion, an upper elementary teacher can help students see that a geometric model can be connected to probability. At the same time, students can extend their understanding of the numbers used to express probability and of fractions and decimals in general.

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