Counting: The Art of Enumerative Combinatorics

Abstract

Counting : the art of enumerative combinatorics / George E. Martin.p. cm. -(Undergraduate texts in mathematics) lncludes bibliographical references and index.

Student connections among counting problems: an exploration using actor-oriented transfer

“…This is an important illustration for a researcher, as it exemplifies a detail in the problem that a student emphasized, but that an expert/ observer may not. Each of the two cases of the Over Three Million problem is called an arrangement with limited repetition problem by some (e.g., Martin, 2001), in which objects are arranged, some of which are indistinguishable from one another. For a researcher, in order to categorize a problem as an arrangement with limited repetition problem, details such as the actual number of repeats are not typically relevant.…”

“…As Martin (2001) points out, one reason that counting is difficult for students is because "there are few formulas and each problem seems to be different" (p. 1). The fact that student difficulties remain (in spite of clearly defined structural similarities for experts) suggests that students may not attend to the same kinds of categorizations that experts do, or at least they struggle to do so (e.g., English, 2005;Batanero et al, 1997).…”

“…The primary distinctions tend to involve notions of order and constraints on repetition in specifying these objects. Many texts (e.g., Cameron, 1994;Martin, 2001;Roberts, 1984;Tucker, 2002) offer the following two-by-two grid for selecting r objects from n distinguishable objects (table taken from Martin, 2001). Tables such as these 1 clearly suggest the existence of pre-existing, structural similarities by which experts delineate problem types (Table 1).…”

“…The students were thus instructed to work freely and were given relatively little external input. The problems 3 ranged in difficulty, and they were chosen to represent common types of problems as categorized by combinatorics textbooks (e.g., Martin, 2001;Tucker, 2002) and studies in the combinatorics education literature (e.g., Batanero, et al, 1997). These included both basic and complex examples of arrangement and selection problems, featuring a variety of constraints.…”

“…It is "advanced" because it follows a discrete mathematics course, which is a juniorlevel course that offers an introduction to combinatorial topics.3 The list of problems used in the study was compiled from a variety of sources includingTucker (2002) andMartin (2001). The Test Questions problem and the Over Three Million problem come from Tucker.…”