Examining Pictorial Models and Virtual Manipulatives for Third-Grade Fraction Instruction

The purpose of this study was to examine pictorial representations, whether in static or dynamic modalities, and their impact on student learning in a classroom with low-achieving students. The investigation emerged from a classroom teacher’s action research project. During a three-week fraction unit, nineteen third-grade low-achieving students participated in two groups – a Dynamic Virtual Manipulatives (DVM) group using virtual manipulatives, and a Static Pictorial Models (SPM) group using pictorial models. Students in both the DVM and SPM groups showed significant improvements between the preand post-tests of fraction concepts. Students’ visualization skills increased while using pictorial models, in both the static and dynamic modalities. The Common Core State Standards emphasize Standards for Mathematical Practice. Standard #4 “model with mathematics” (http://www.corestandards.org/the-standards) emphasizes students’ use of models to communicate their mathematical thinking. The National Council of Teachers of Mathematics (NCTM, 2000) Principles and Standards for School Mathematics (PSSM) encourage teachers and students to use multiple representations during mathematics instruction. The PSSM states that all students should “create and use representations to organize, record, and communicate mathematical ideas; select, apply, and translate among mathematical representations to solve problems; [and] use representations to model and interpret physical, social, and mathematical phenomena” (p. 67). This means that visual or pictorial representations should be used in school mathematics. Mathematics education research has demonstrated the benefits of mathematics instruction that employs a variety of representational forms (Cuoco & Curcio, 2001). Representations may include manipulatives, pictures, written symbols, spoken language, real-life situations (Lesh, Post, & Behr, 1987) or virtual manipulatives (Moyer, Bolyard, & Spikell, 2002). Evidence suggests that the use of pictorial representations in mathematics can be an important way for children to express their mathematical thinking (Woleck, 2001). The use of pictorial representations as an option for children to think about mathematics concepts is further supported by research on multiple intelligences and students’ different learning styles (e.g., Gardner, 1997, 2002; Marzano, 2010). Although the use of Journal of Interactive Online Learning Moyer-Packenham, Ulmer, and Anderson 104 concrete (or physical) manipulatives is well situated in the mainstream of school mathematics as a teaching tool (Kamii, Lewis, Kirkland, 2001; Martin, Svihla, Smith, 2012; Moyer, 2001a; Moyer, 2001b; Moyer & Bolyard, 2002; Moyer & Jones, 2004; Raphael & Wahlstrom, 1989; Sowell, 1989; Suydam, 1985; Uribe-Florez & Wilkins, 2010), less attention has been given to the use of pictorial representations as a part of daily mathematics instruction (Larkin & Simon, 1987). The purpose of this study was to examine pictorial representations and their impact on student learning in a classroom with low-achieving students. Of particular interest in this study were the pictorial representations that students used for fraction learning, whether those pictorial representations were in static or dynamic forms, and the influence on learning for a group of low-achieving students. Research reports that high-achieving students are quite capable at visualization, in comparison with low-achieving students, who have much more difficulty with this mathematical practice (Moreno & Mayer, 1999). Because this study was a teaching experiment conducted by a classroom teacher for her Master’s Degree action research project, the purpose of the study was not to generalize the results, but to inform the classroom teacher’s own instruction when teaching mathematics using pictorial representations. Visual (or Pictorial) Representations Numerous studies have been conducted on the use of manipulatives for classroom instruction (Kamii, Lewis, & Kirkland, 2001; Martin, Svihla, & Smith, 2012; Moyer, 2001; Moyer & Jones, 2004; Raphael & Wahlstrom, 1989; Sowell, 1989; Suydam, 1985; Uribe-Florez & Wilkins, 2010), but fewer studies focus specifically on teachers using visual (or pictorial) representations (Larkin & Simon, 1987). Static or dynamic pictorial models support students’ visualization of mathematical ideas. Arcavi (2003) defines visualization in mathematics as the ability to create, interpret, use, and reflect on images in the mind, on paper, or with technological tools. Visualization provides some students with the opportunity to see physical objects and pictorial representations with meaning, and to connect these to abstract concepts in the mathematics curricula. Van Garderen (2006) highlighted the importance of mathematics visualization in a study that examined students of varying achievement levels and use of visual imagery and spatial visualization. The findings indicated that high achieving students displayed the highest levels of spatial visualization. Based on this study, students’ ability to visualize has a significant correlation with the ability to understand mathematics. Virtual manipulatives have been defined as “computer based renditions of common mathematics manipulatives and tools” (Dorward, 2002, p. 329) and “an interactive, web-based visual representation of a dynamic object that presents opportunities for constructing mathematical knowledge” (Moyer, Bolyard, & Spikell, 2002, p. 373). Virtual manipulatives can develop students’ visualization skills by connecting words, pictures, and symbols simultaneously. This simultaneous presentation can assist students in developing a solid understanding of mathematical concepts (Paivio, 2007). The effective use of virtual manipulatives in mathematics classrooms relies greatly on the visual images presented to students and students’ ability to visualize mathematics concepts beyond the use of the virtual manipulatives. A recent meta-analysis compared the effects of virtual manipulatives on student achievement with other instructional treatments. Findings from 29 research reports, containing 79 effect size scores, yielded a moderate averaged effect size of 0.44 for virtual manipulatives as compared with other instructional treatments (Moyer-Packenham, Westenskow, & Salkind, 2012). Among these articles, four specifically focused on the instruction of fractions (with a moderate effect size, 0.66), and four others reported the effect of virtual manipulatives with students of different achievement levels. The majority of studies focusing on fraction instruction reported significant differences in achievement in favor of the virtual manipulatives (Lin, 2010; Moyer, Niezgoda, & Stanley, 2005; Reimer & Moyer, 2005; Suh, 2005). For example, Reimer and Moyer (2005) studied 19 third-grade Journal of Interactive Online Learning Moyer-Packenham, Ulmer, and Anderson 105 students during a two-week classroom investigation of students’ use of virtual fraction applets. Preand post-test scores on conceptual knowledge and procedural computation indicated that students showed significant improvement on a conceptual knowledge post-test. Interviews and attitude surveys collected by Reimer and Moyer showed that improvements on the conceptual knowledge post-test could be attributed, in part, to the tutorial nature of the virtual fraction applets and the specific feedback provided. Other studies report similar findings related to learning fractions (Lin, 2010; Moyer, Niezgoda, & Stanley, 2005; Suh, 2005). For example, Lin (2010) attributed greater gains in pre-service teachers’ understanding of fractions using fraction applets to the interactive and learner-centered nature of the virtual manipulatives. Lin based these findings on three factors: (a) the virtual manipulatives captured the students’ attention, focusing their attention on learner-active mastery rather than on teacher presentation; (b) the dynamic and animated nature of the virtual manipulatives provided tools for improving students’ visual and conceptual abilities and permitted students to explore the concepts; and (c) the virtual manipulatives provided immediate feedback and required that students obtain the correct answers. Lin’s assessment gathered data on the procedural and conceptual understanding of fraction operations of addition, subtraction, multiplication and division. Reimer and Moyer (2005) and Lin (2010) also commented on the potential of virtual fraction applets to positively impact students’ ability to visualize mathematics concepts. Virtual manipulatives offer students many opportunities to practice using a visual model. The active manipulation of this model may contribute to increasing students’ growth in conceptual understanding (Reimer & Moyer, 2005). Many virtual manipulatives incorporate animations that “permit students to explore complex and dynamic relationships in the subject matter” (Lin, 2010, p. 67). For example, through the use of dynamic virtual models of fractions, students can identify the need for equivalent fractions during fraction addition and represent them accordingly. These tutorial types of virtual manipulatives aid students in visualizing the division of fractions (e.g., in the problem 1/3 ÷ 1/6, students visualize how many 1/6 pieces it takes to make 1/3) – a difficult concept for most students to comprehend. The four studies on virtual manipulatives that described outcomes with students of varying achievement levels reported mixed findings (Drickey, 2000; Kim, 1993; Moreno & Mayer, 1999; Moyer-Packenham & Suh, 2012). Drickey (2000) and Kim (1993) each indicated no statistically significant difference in learning gains between lowand high-achieving groups. Although the lowachieving students made numerically greater gains than other groups on tests of conceptual knowledge, these gains were not statistically significant. Moreno and Mayer (1999) examined students’ work with multi-representational tools and reported that high-achieving students and those students with higher spatial ability (or visualization skills) significantly outperformed students of lower achievement and ability. The researchers attributed these differences to the working memory of individual students (Clark, Nguyen, & Sweller, 2006). Those in the high-spatial or high-achieving groups did not become overloaded in their working memory, and thus performed more successfully on assessment tasks. In contrast, other research indicates that tutorial features built into virtual manipulatives significantly influence students of varying achievement levels (Moyer-Packenham & Suh, 2012). Moyer-Packenham and Suh (2012) found statistically significant gains for low-achieving students using virtual manipulatives, but only numerical gains for averageand high-achieving students. Additionally, they observed that high-achieving students quickly recognized patterns and transitioned to symbolic representations. Low-achieving students in the study relied more explicitly on pictorial representations as they worked with mathematical symbols. These mixed findings suggest that virtual manipulatives have different learning effects for students of varying achievement levels. Journal of Interactive Online Learning Moyer-Packenham, Ulmer, and Anderson 106 Action Research in the Classroom Teachers engage in classroom action research to better understand the effects of their own instructional practices and what they can do to improve the educational experience of their students (Beckett, McIntosh, Byrd, & McKinney, 2011; Bonner, 2006; Herbel-Eisenmann, Drake, & Cirillo, 2009). In 1946, Kurt Lewin originally described action research as a three-step spiral process involving: (a) planning, (b) taking actions, and (c) systematically collecting information on the results of the actions. For the classroom teacher, this means: (a) making observations of what is happening in his or her classroom, (b) doing something to improve outcomes, and (c) seeing if what was done made a difference in what or how students learned. More recently, Mills (2003) defined action research in terms of the school context: Action research is any systematic inquiry conducted by teacher researchers...to gather information about how their particular schools operate, how they teach, and how well students learn. This information is gathered with the goals of gaining insight, developing reflective practice, effecting positive changes in the school environment (and on education practices in general), and improving student outcomes and the lives of those involved (p. 5). Classroom action research, as discussed in this study, refers to one teacher systematically addressing a perceived need in her own classroom. The research questions naturally emerged from the teacher’s desire to improve her instruction. The teacher conducting the action research in this project believed that pictorial models help some students to visualize mathematics concepts. The teacher was interested in how static and dynamic representations may benefit learning with different groups of students in different contexts providing an opportunity for greater learning and understanding of mathematical content. Her classroom teaching experiment sought to contribute to the research on mathematics representations by examining how pictorial images, in teachers’ drawings or pictures (static) and in virtual manipulatives (dynamic), influenced low-achieving students’ learning in a fraction unit. The Current Classroom Project This investigation emerged from a classroom teacher’s (one of the authors of this paper) action research project in a course taught by a professor at the university where she was earning her Master’s Degree. The design of the investigation was a classroom teaching experiment in which the classroom teacher wanted to address the needs of her low-achieving students through the use of pictorial models and virtual manipulatives. The teacher conjectured that greater use of pictorial representations would help her students make connections between pictures, concepts, and symbols. The study focused on the following research question: How do static and dynamic pictorial representations influence learning for low-achieving third-grade students during a fraction unit? School Context The elementary school that served as the context for this study was in a suburban city and was built in the 1950s, with permanent trailers surrounding the school to serve as additional classroom space. There were four or five classes in each grade level, K-6. Almost 600 students attended the school with 48% eligible for free/reduced lunch. The ethnicities of the students in the school included Hispanic (45.61%), Asian (25.84%), White (20.44%), and Black (1.69%). English Speakers of Other Languages (ESOL) comprised 40.37% of the student population with students speaking Spanish, Vietnamese, Chinese, Portuguese, and Urdu. Many of the students’ parents did not speak English. Journal of Interactive Online Learning Moyer-Packenham, Ulmer, and Anderson 107 Participants At the beginning of the study, a fraction pre-test was administered to all 78 of the students in the five third-grade classrooms to identify the students in the lowest one-third of the third grade class prior to the instructional unit. Teachers regularly used these pre-tests to group students in the third grade according to their needs and performance. One-third of the third-grade class (i.e., 26 students) were identified and invited to participate in the study. Nineteen of the 26 students in the lowest onethird of the third-grade class agreed to participate. The teacher divided the low achieving students into two groups—a Dynamic Virtual Manipulatives (DVM) group and a Static Pictorial Models (SPM) group—and matched the students based on similar characteristics (e.g., gender, ability, special needs status). She matched the students in the two groups to keep them as similar as possible for the teaching experiment. Nine students (five girls and four boys) were placed in the DVM group and ten students (six girls and four boys) were placed in the SPM group. Both groups had similar backgrounds and learning abilities. The DVM group contained two students under review for learning disabilities and one ESOL student. The SPM group contained one special education student and one ESOL student. The ethnicities of the DVM group included Hispanic, African American, and Native American. Seven students were of Hispanic background and spoke both English and Spanish at home. The other two students in the DVM group spoke only English (one African American and one Native American). Hispanic and Caucasian students comprised the SPM group. Five of the students were of Hispanic background and spoke both English and Spanish at home. Five of the students were Caucasian and spoke only English. The teacher’s original action research question emerged from a desire to know if the use of static or dynamic pictorial representations made a difference in students’ learning. Based on her own experience and study of the research literature, she believed the use of pictorial representations to be more effective than traditional algorithmic-only instructional techniques (Arcavi, 2003; van Garderen, 2006). Therefore, she valued the use of pictorial representations for all students, and did not wish to use a control group receiving only algorithmic instruction. This decision aligns with the nature of action research, in that research questions arise from classroom situations and the purpose is to improve instruction for that particular group of children (Mills, 2003). Teachers The classroom teacher who conducted the action research in this study was in her fifth year of teaching third grade. This teacher taught the DVM group. She had a Bachelor’s Degree in Education, a K-6 Elementary Teaching Certificate, and she was completing her final project before earning her Master’s Degree in Education at the time of this study. The teacher had professional leadership experiences in mathematics that demonstrated a high level of expertise in this area. Experiences that exemplified her level of expertise included: (a) a refereed journal publication based on teaching activities in her classroom, which she co-authored, published in the National Council of Teachers of Mathematics (NCTM) journal, Teaching Children Mathematics; and (b) a presentation titled Impacting Mathematical Learning at Multiple Levels through School and University Collaboration, which she presented with a team of colleagues at the Annual Meeting of the American Association of Colleges for Teacher Education (AACTE) in Washington, DC. The teacher had learned about the use of virtual manipulatives and visual representations in her teacher licensing program and had used various instructional modalities in her teaching of mathematics. The classroom teacher who taught the SPM group was in her eleventh year of teaching, and she had taught fourth grade for four years and third grade for seven years. She had a Bachelor’s Degree in Education, a K-6 Elementary Teaching License, and a Master’s Degree in Elementary Education. During the study, she had been serving as the third-grade team leader for the past three years. The teacher participated in continuing professional development for mathematics teaching after earning her Master’s Degree. Journal of Interactive Online Learning Moyer-Packenham, Ulmer, and Anderson 108 Classroom Environments The DVM group received instruction in the classroom and in the computer lab. Student centers located around the perimeter of the classroom served as mathematics rotation centers. In addition, five student tables in the center of the classroom housed four student workspaces. Physical manipulatives were readily available for student use at these workspaces. The computer lab had two printers and 30 computers allowing students to work individually with the computers. In addition to written work on the chalkboard, student and teacher interactions were facilitated through whole group discussions as the teacher used an LCD projector to demonstrate the virtual manipulatives. The SPM group received instruction in a separate third-grade classroom. This classroom was set up in much the same way as the DVM classroom with physical manipulatives readily available for student use and student centers for mathematics around the perimeter of the classroom. The teachers of the SPM and DVM groups shared the belief that teachers should incorporate learning centers into their daily routines. The SPM classroom and its resources mirrored the DVM classroom.