The Complexities of Teaching Prime Decomposition and Multiplicative Structure with Tools to Preservice Elementary Teachers.

Preservice elementary teachers often struggle with prime decomposition and other mathematical topics that correlate with number theory. This paper provides a framework for integrating prime factor tiles into their curriculum with a particular emphasis on prime decomposition. Using this framework, preservice teachers explored and evaluated numbers using prime factor tiles. The results of the exploratory inquiry showed that preservice teachers made some progress in their understanding of prime decomposition after exploring with the tools. However, they struggled with problems requiring the application of prime decomposition. More time to delve into this topic is probably needed in order to observe further gains. Introduction Prime decomposition is an important skill for preservice teachers to have as it connects to many different areas of mathematics including Fundamental Theorem of Arithmetic, Modular Arithmetic, Group Theory, Galois Theory and Number Theory among others. Unfortunately, prime decomposition and number theory are often ignored in relation to research (Zazkis, 2000; Zazkis & Campbell, 1996; Zazkis & Liljedahl, 2004). Prime decomposition can be described as the multiplicative decomposition of numbers into their prime number components. Oftentimes, prime decomposition is presumed as difficult (Lenstra, 2000). Although presumed difficult, it is an important concept starting with third through fifth grade Standards. Students in these grades should “recognize equivalent representations for the same number and generate them by decomposing and composing numbers” (NCTM, 2000, p. 148). At the sixth through eighth grade Standards, students should “use factors, multiples, prime factorization, and relatively prime numbers to solve problems” (NCTM, 2000, p. 214). Because of this emphasis, it is imperative that preservice teachers have a robust awareness of prime decomposition and can work interchangeably between factors and the numbers they produce. It is important to help preservice teachers become exposed to some of the richness of prime decomposition and divisibility notions (Brown, 1968). A common method for teaching prime decomposition involves the formation of factor trees (Griffiths, 2010) (see Figure 1). Factor trees help students visualize the multiplicative breakdown of a number but do little else in relation to the development of the concept. Preservice teachers can usually perform the procedure of creating a factor tree. When they have to apply the yielded factor tree in a context or work backward from the factors, difficulties often result (Zazkis, 1999; Zazkis & Liljedahl, 2004). Figure 1. A demonstration of the typical prime factor tree format. To address this, an alternative method to teach prime decomposition was developed (Kurz & Garcia, 2010; 2012). Using tiles, students decompose numbers to their prime factors. These prime numbers can then be used in a number of ways such as discovering the uniqueness of prime decomposition, simplifying fractions, finding the greatest common factor (GCF), finding the least common multiple (LCM), finding all factors and identifying roots. The purpose of this paper is to describe methods for teaching prime decomposition using tools and then present the findings of an inquiry that investigated preservice teachers’ knowledge after exposure to the prime factor tiles. First, an explanation of prime decomposition along with preservice teachers’ difficulties with the topic is described. Then, an alternative method for teaching prime decomposition using tools is explained with supportive lesson ideas. Preservice teachers explored these lessons using the tools. The following question was investigated: How did exposure to prime factor tiles influence preservice teachers’ understanding of prime decomposition? Recommendations for teaching prime decomposition based on the results are provided. Prime Decomposition For preservice teachers, there seems to be real difficulty with understanding and applying prime decomposition in relation to factors (Zazkis, 1999). Several studies have outlined preservice teachers’ difficulties. Zazkis and Liljedahl (2004) found that preservice teachers were able to define (or explain) what makes a number prime. Greater difficulty resulted when preservice teachers were asked to implement their knowledge of prime. Once they had to apply reasoning, the preservice teachers were unsuccessful (for the most part). Their understanding of primes was incomplete or inconsistent and focused primarily on the algorithm. In another study, Zazkis and Campbell (1996) conducted research after the preservice teachers concluded their lessons on number theory. They found that preservice teachers demonstrated misunderstandings and misconceptions about divisibility and factors. They relied on procedural algorithms rather than applying logic and reasoning to determine divisibility. They concluded, “insufficient pedagogical emphasis has been placed on developing an understanding of the most basic and elementary concepts of arithmetic” (p. 562). Bolte (1999) used concept maps to help preservice teachers voice their understanding of 20 terms related to number theory such as factor, prime, composite, prime factorization, multiple, divisible and other terms. In her analysis of two specific cases, one preservice teacher was able to use the concept map to show depth in relation to number theory. The participant was fluent in her connections and properly demonstrated the interconnectedness of the terms. Another participant created a weak map that showed superficial understanding of the interconnectedness of the 20 number theory terms. While Bolte specifically focused on concept maps, the showcased maps and supportive essays provide insight into preservice teachers’ connections and misconceptions regarding number theory. With the documented difficulties preservice teachers have in relation to prime decomposition, it is important that they are provided with opportunities to explore number theory concepts. Alternative methods to teach prime decomposition using tools with an emphasis on discovering the meaning of number theory terms are illustrated. The goal was to implement tools to support preservice teachers’ conceptual understanding of prime decomposition so that they could better understand the multiplicative structure of numbers. Investigative Ideas Prime factor tiles were developed as an alternative method to teach prime decomposition with the goal of implementing the tools. These lessons go beyond the drill aspect of mathematics and instead focus on sense making and reasoning as is recommending in a reform-based curriculum (Cooney, 1999). Prime factor tiles are tiles with a specific prime number written on each tile. In order to build all primes and composites equal or less than 102, the tiles should contain the following numbers: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97 and 101. Other composites beyond 102 are possible. The tiles can be easily made in two ways. The tiles can be printed on cardstock, see Kurz and Garcia (2010) for the template. Using inch-flats, a permanent maker and blank stickers, a more permanent set of factor tiles can be made. Exploring the Fundamental Theorem of Arithmetic The Fundamental Theorem of Arithmetic assures that the prime decomposition of a given number is unique. The instructor gives a number to the class and asks the students to find the prime decomposition using the prime factor tiles. Call on a student and ask what did you get? Call on another student and ask what did you get? Repeat this process for a few more students. All students should have the same result. Repeat this process whole class with several more numbers. Ask, can two students have different prime factor tiles that equal the same number? Can they have tiles that are a little different? Why not? How can we explain what we discovered? What does this mean? Simplifying fractions Prime factor tiles can be used to simplify fractions. A fraction is given to the students, perhaps 24/42. A large fraction bar is drawn. The student decomposes the numerator and denominator into a product of primes using the tiles. Then, the student places the corresponding tiles above and below the fraction bar (see Figure 2). The student continues removing any shared tiles until there are no further common tiles to remove. By multiplying the numbers in the numerator and denominator, the student finds the simplified fraction. What happens when a common tile is removed from both the numerator and the denominator? What does that process do to the fraction? Is the value of the fraction the same once the common tile has been removed? Explain. Can you create a fraction with no common tiles? Can you create a fraction with at least three prime factor tiles for both the numerator and denominator with no common tiles? Can you create a fraction with at least three prime factor tiles located in the numerator and four prime tiles located in the denominator that simplifies yielding a 1 in the numerator? Explain your process.