Defining Connections
 Introduction | Varieties of Connections | Other Contexts | The Teacher's Role | Your Journal

Connections between mathematics topics, between mathematics and other subjects, and between mathematics and everyday life all contribute to making mathematics understandable and meaningful. Let's look at some classroom techniques that help illustrate what we mean by "connections" and can contribute to students' ability to make and talk about connections in their work.

The Interconnectedness of Mathematics

"In grades 9-12, students not only learn to expect connections but they learn to take advantage of them, using insights gained in one context to solve problems in another . . . [S]tudents should routinely ask themselves, 'How is this problem or mathematical topic like things I have studied before?' From the perspective of connections, new ideas are seen as extensions of previously learned mathematics. Students learn to use what they already know to address new situations." (NCTM, 2000, p. 65)

Students should be given frequent opportunities to see connections within the mathematics they are learning and that they have encountered in earlier grades. One example of this is relating algebraic expressions for phenomena to graphical representations -- for instance, students might explore how changes in one parameter (such as the interest rate or time period in the Interest Calculator problem) affect both the graphic and algebraic representation of a situation.

 Compound interest situations, which are first encountered in middle school, can be related to the mathematics of exponential growth situations that may be discussed in later grades, for example, population growth. In both cases, the expected quantity is found by using time as an exponent with a base that represents 100% plus the percent increase for a time period. If students can understand why t = number of years in the equation Future Value = f(t) = (1 + r)t • P, when P = \$500, and r = 3.75%/year, they have a grounding that will enable them to approach a similar use of t in a bacteria growth problem with doubling in each time period: Population = f(t) = (1 + r)t • P, when r = 100%, and P = the initial population, f(t) = (2)t • P.

Noting similarities to and differences from prior work can help deepen students' understanding. For instance, when shown a periodic graph of a situation, students should speculate that a circular function may be involved and continue investigating that conjecture. Mathematical similarities that appear in seemingly unrelated instances help reinforce the interconnected nature of mathematics and encourage a sense of wonder and inquisitiveness. As an example, consider the many ways that Pascal's Triangle can be applied -- in combinatorial problems or in study of the binomial theorem. Similarly, triangular numbers appear in various visual pattern-finding settings, such as the Staircase problem in Session 3, along with the related question, "Which whole numbers can be expressed as a sum of consecutive whole numbers?"

Preparing students to work with connections to data and statistics is also key. For example, students may collect quantitative data in a science lab, enter them as a table of values in a graphing calculator or spreadsheet program, create a scatter plot, and establish a line or curve of best fit. Through this process, they may be able to link the original situation to an algebraic equation.

Watch the video segment (duration 0:22) in the viewer box on the upper left to hear a reflection from William Masalski, a high school educator.

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