by NCTM President Henry (Hank) Kepner
NCTM Summing Up, October 2009
Mathematics is an integrated field of study with dynamic connections across many perspectives and to a wide range of human endeavors. Although at times we focus our instruction on a narrow area of mathematics to develop our students’ skills and understanding of concepts, I call on you to ensure that students expect to make connections between the mathematics—and the math-related contexts—that they are currently encountering and those that they have already experienced. Students should expect to make connections and capitalize on them, using insights gained in one mathematical context to investigate conjectures in another.
When students connect mathematical ideas, their understanding becomes deeper and more lasting, and learners come to view mathematics as a coherent whole—connected with other subjects and their own interests and experiences. Through instruction that emphasizes the interrelatedness of mathematical ideas, students not only learn mathematics but also discover its utility. What role do connections play in developing your students’ insights about and understanding of mathematics and its use? My challenge to you is to make sure that connections play an essential role in your students’ learning!
The following examples illustrate how we might help our students understand the interconnectedness of mathematical ideas and other aspects of their lives.
The area model is the preeminent model for the multiplication of whole numbers. For children beginning to think about the product of 3 x 4, for example, placing and counting unit squares inside a rectangle with dimensions 3 centimeters by 4 centimeters is foundational. This geometric representation later leads students to understand multi-digit multiplication in the partial products algorithm, which extends to fraction and polynomial multiplication, at least through degree 2. This is a powerful mathematical process for making sense of the often-meaningless FOIL (first, outer, inner, last) multiplication rule in algebra.
Developing a geometric perspective and justification of the Pythagorean theorem through paper folding and other perspectives builds a foundation for the distance formula (in both 2- and 3-D). I challenge you to prepare your students for distance thinking through work with Pythagorean relations. We shouldn’t have our students memorize the often confusing distance formulae but instead understand the concepts. What do you do in your instruction to emphasize the interrelatedness of mathematical ideas and their social and practical value?
A pivotal concept in algebra and calculus is rate of change. In learning about linear relations, students often encounter slope in algebraic formulae such as (y2 – y1)/(x2 – x1), totally missing the geometric representation of slope as conceptualized and justified through similar triangle relationships. What do you do in your instruction to emphasize the interrelatedness of mathematical ideas?
Middle school students might collect and graph data for the circumference (C) and the diameter (d) of a set of different-sized circles. They could extend their previous knowledge of algebra and data analysis by recognizing that the values nearly form a straight line, soC/d is between 3.1 and 3.2—a rough estimation of pi. How do you create classroom experiences that value and build on the connections between mathematics and students’ knowledge, experiences, and interests?
The graphs of functions—particularly graphs created with dynamic graphing utilities—allow students to search for and investigate approximate simultaneous solutions of two functions. Such work with graphs is especially useful in cases where students’ algebraic solution techniques are inadequate (e.g., f(x) = x and g(x) = sin x).
Students should connect mathematical concepts to their daily lives, as well as to applications from the sciences, social sciences, literature, business and the arts. Moreover, rich mathematical problems enable students to recognize the value of mathematics in examining personal, cultural, and social issues.
Have we prepared our students to ask, “Will mathematical analysis of the question that I am studying help me with my response?” Students who understand the usefulness of connections will know that this is a valuable question to ask.
The NCTM Professional Development Focus of the Year for 2009-2010 is “Connections: Linking Concepts and Context.” Resources on the topic of Connections including Reflection Guides for specific journal articles are available on the NCTM web site. Additionally, Learn-Reflect strands on the topic of Connections are offered at all NCTM conferences during 2009-2010, offering participants an opportunity to immerse themselves in the topic and collaborate with other mathematics educators.