THE USE OF MANIPULATIVES IN MATHEMATICS CLASSROOMS: AN EXAMINATION OF TEACHER PERCEPTIONS by Amanda Curtis A project submitted in partial fulfillment of the requirements for the degree of MASTER OF EDUCATION IN CURRICULUM AND INSTRUCTION WEBER STATE UNIVERSITY Ogden, Utah October 3, 2020 2 THE USE OF MANIPULATIVES IN MATHEMATICS CLASSROOMS Acknowledgements This project was propelled by many great people. Dr. Rachel Bachman has played a significant role in my studies and project and has become my greatest mentor and teacher. Rachel brought the field of mathematics to life and pushed me to be more than I could have on my own. She is a master teacher. I would also like to acknowledge the tireless efforts of Dr. Caitlin Byrne, my committee chair and cheerleader. She guided me through this process and made each draft better than the last. I would also like to thank Dr. Peggy Saunders, whose expertise and experience were invaluable. Throughout my graduate courses my mind was opened to many new ideas as I learned from all of my professors, especially C. David Walters, thank you for teaching me how to persevere. I could not have accomplished anything without the love and support of my husband. His constancy and encouragement pushed me forward. He never wavered as he was subjected to hours of mathematical talk. My four wonderful children sacrificed and grew throughout this process— learning how to cook, clean, and care for one another as we united as a family. I hope they know there is never an end to learning and they too, can do hard things. Many thanks to my mother, the first to attend college in her family, and instilled in me a love for learning. She was always willing to care for me and my family. Thank you to my father who made it to every game, taught me the value of hard work, and was always willing to drive me to school. I am so thankful to them for believing in me. 3 THE USE OF MANIPULATIVES IN MATHEMATICS CLASSROOMS Table of Contents NATURE OF THE PROBLEM .…………………………………………………………………7 Literature Review…………………………………………………………………………9 Manipulatives…………………………………………………………………….11 Benefits…………………………………………………………………………..15 Student Engagement……………………………………………………..15 Diverse Learners………………………………………………………....16 Knowledge Retention……………………………………………………16 Barriers to Implementation……………………………………………................17 Teacher Experience………………………………………………………17 Classroom Management………………………………………………….19 Ongoing Teacher Support…………………………………………….….21 Secondary School Use…………………………………………………………...23 Summary………………………………………………………………………………...24 PURPOSE……………………………………………………………………………………….25 METHOD……………………………………………………………………………………….26 Setting…………………………………………………………………………………..26 Participants…………………………………………………………………………..…27 Instrument………………………………………………………………………………27 Procedures………………………………………………………………………………28 FINDINGS AND DISCUSSION……………………………………………………………….30 Teachers’ Perceptions………………………………………………………………..…31 Outcomes by Grade Band………………………………………………………32 Elementary Grade Band Outcomes..……………………………………32 Secondary Grade Band Outcomes..….…………………………………34 4 THE USE OF MANIPULATIVES IN MATHEMATICS CLASSROOMS Individual Shifts in Perception….….……………………………………35 Manipulatives Use Frequency……………………..…………………………………….40 Barriers to Manipulatives Use …………………….…………………………………….40 Classroom Management…………………………………………………………40 Teacher Support………………………………………………………………….42 Conclusion……………………………………………………………………………….44 Limitations and Recommendations………………………………………………45 Final Reflection………………………………………………………………….47 REFERENCES…………………………………………………………………………………..48 APPENDIX A……………………………………………………………………………………52 APPENDIX B…………………………………………………………………………………....57 5 THE USE OF MANIPULATIVES IN MATHEMATICS CLASSROOMS List of Figures Figure 1. Teachers’ Perceptions of Manipulatives Before and After Training.………………….32 Figure 2. Elementary Teachers’ Perception Before and After Training…………………………..33 Figure 3. Secondary Teachers’ Perception Before and After Training…………………………...34 Figure 4. Frequency of Manipulative Use Before Training ……………………………….……….37 Figure 5. Change of Frequency of Manipulative Use in Elementary and Secondary After Training……………………………………………………………………………………..………….…38 Figure 6. Frequency of Behavior Problems…………………………………………………………42 Figure 7. Side By Side Comparison of Teachers’ Perceptions of Manipulatives in Elementary and Secondary Grade Bands………………………………………………………………………………..44 6 THE USE OF MANIPULATIVES IN MATHEMATICS CLASSROOMS Abstract Students in the United States continue to perform poorly on standardized tests in mathematics. Students struggling with the depth of understanding needed for secondary mathematics are unprepared for collegiate mathematics and either revert to rudimentary mathematics courses or forgo college. Helping students to develop a deeper understanding of mathematical concepts in order to be more successful mathematicians necessitates changing traditional instructional methodology. Incorporating mathematical manipulatives into instruction supports the idea of allowing students multiple representations of mathematical concepts which facilitates deep mathematical understanding. This descriptive survey-based study sought to understand teacher perception of manipulatives, the frequency of manipulative use, and barriers to implementing manipulatives. Participants included teachers attending a training specific to manipulative implementation. The data supported a change in teacher perception of manipulatives after receiving instruction. The post-training results showed teachers found manipulatives not only to be engaging for students but necessary for student learning. Despite barriers such as time, teacher knowledge of manipulative use, and behavioral issues, teacher comments supported implementation. While additional research is needed to further understand the implications of this change in teacher perception, the research is encouraging in that teachers are willing to change their teaching practices to incorporate manipulatives into their classrooms and work to aid their students in achieving a deeper understanding of mathematics. 7 THE USE OF MANIPULATIVES IN MATHEMATICS CLASSROOMS NATURE OF THE PROBLEM Since the enactment of No Child Left Behind in 2001, increased numbers of American eighth-grade students score at or above proficiency in mathematics (Lim & Sireci, 2017). However, the increased mathematics scores in eighth grade do not correlate with U.S. high school students’ test scores, which have shown a flat line of progression since 1973 (NCES, 2009). According to 2019 SAT and ACT scores, only 44% of high school graduates were prepared for collegiate mathematics (ACT, 2020). Low-level mathematics proficiency is cause for concern because of the implications on future achievement for unprepared students. Compared to other nations, the US had a much smaller increase in students achieving at or above the advanced level of mathematics from 2003 to 2011 (Lim & Sireci, 2017). These results were supported by the Programme for International Student Assessment (PISA), which showed that U.S. students performed low-level tasks well, but lacked the ability to interpret real-world situations using mathematical reasoning (OECD, 2013). Furthermore, in the US only 16% of high school seniors were proficient in mathematics and desired to pursue STEM (science, technology, engineering, and mathematics) careers (U.S. Department of Education, 2014). As the number of mathematics-proficient high school graduates drops, the need for remedial courses at the college level rises, increasing the expense and time for college graduation (Gningue, Menil, & Fuchs, 2014). How teachers approach mathematical instruction is one possible cause for poor mathematics achievement. Generally, mathematics teachers favor frontal lecture, a traditional style of lecture using primarily algorithms [i.e., symbolic representations of a mathematics question] yet, frontal lecture is the least effective teaching style in producing conceptual understanding and retention (Gningue et al., 2014). Where frontal lecture falls short, research 8 THE USE OF MANIPULATIVES IN MATHEMATICS CLASSROOMS suggests all students may benefit from visualizing and reasoning mathematics equations through mathematics manipulatives (Belenky & Nokes, 2009; Carbonneau, Marley, & Selig, 2013; Mueller, Yankelewitz, & Maher, 2014; Satsangi, Bouck, Taber-Doughty, Bofferding, & Roberts, 2016; Sinatra, 1990). Other benefits of using manipulatives include greater knowledge retention (Carbonneau et al., 2013), student engagement (Belenky & Nokes, 2009; Golafshani, 2013; Vinson, 2001), and the ability to reach diverse learners (Gningue et al., 2014; Satsangi et al., 2016; Sinatra, 1990). In an attempt to bolster students’ conceptual mathematical understanding, the National Council of Teachers of Mathematics (2014) recommended eight mathematical teaching practices. Included in these practices is the need for teachers to use and connect different representations for their students. Using manipulatives is one of the representational methods recommended. While manipulative use has benefited elementary students, secondary teachers hesitate to implement them due to the assumption that they are solely for the benefit of those in primary grades or those with learning disabilities (Satsangi et al., 2016). Other concerns in implementing manipulatives include: teacher knowledge of manipulative use and mathematics reasoning (Alat & Dedeoglu, 2013; Puchner, Taylor, O’Donnell, & Fick, 2008; Vinson, 2001), time (Golafshani, 2013; Puchner et al., 2008), and classroom management challenges (Golafshani, 2013; Satsangi et al., 2016). Due to these hurdles, teachers need support from administrators in funding materials, time off for mathematics workshops, and opportunities to collaborate with other teachers in order to successfully implement a different type of mathematics instruction (Puchner et al., 2008). Without these supports, teachers may continue relying on direct instruction to the detriment of their students. 9 THE USE OF MANIPULATIVES IN MATHEMATICS CLASSROOMS Literature Review In the United States, students have consistently scored below proficiency in mathematics and science from 4th through 8th grade, with even fewer students showing proficiency from 8th through 12th grade (NCTM, 2014). In 2018, U.S. students scored below average on the Programme for International Student Assessment (PISA), a worldwide survey administered to 15-year-old students to assess essential skills and knowledge acquired by adolescents (OECD, 2018). Students lacking proficiency in pre-algebra, an eighth-grade skill, have less than a 12% chance of graduating from college (Gningue et. al., 2014). Due to the lack of foundational mathematical knowledge U.S. colleges have been forced to offer remedial courses to bring students up to proficiency before advancing their education (Gningue et. al., 2014). These courses add additional time and expense to traditional four-year programs. Not only did U.S. student test scores demonstrate a lack of proficiency in mathematics, test scores showed U.S. students only performed at a moderate level of procedural fluency, scoring even lower in conceptual knowledge (Vinson, 2001). The NCTM (2014) defines conceptual understanding as “the comprehension and connection of concepts, operations, and relations” (NCTM, 2014, p. 4). Current teaching practices emphasize using fact recall and memorization to solve mathematical problems (NCTM, 2014; Satsangi et al., 2016). If students are unable to relate the problem to these memorized steps, the lack of conceptual knowledge impedes their ability to problem solve. Lacking problem-solving skills, high school graduates are unprepared for college and careers (NCTM, 2014). In addition, the lack of mathematical knowledge means fewer students are prepared to contribute to ever-growing STEM careers (NCTM, 2014). Addressing this gap between procedural and conceptual knowledge may be key in improving student performance. 10 THE USE OF MANIPULATIVES IN MATHEMATICS CLASSROOMS Tripathi (2008) related the complexity of mathematical concepts to the parable of five blind men encountering an elephant for the first time. Each man experienced a different part of the elephant (trunk, ear, leg, tail, or stomach), and each concluded the animal was very different. Due to the limited view each had of the elephant, their perception of the animal as a whole was inaccurate. Likewise, giving students only one view of a mathematical concept (a formula, procedures, symbols) limits their ability to make connections and gain deeper understanding of the problem as a whole (Tripathi, 2008). Students need support in viewing a problem as more than just a list of procedures. They need to understand deeply what is being asked, what numbers represent, and different strategies to approach problems. Bridging the gap between procedural and conceptual understanding and changing the outcome for U.S. students may begin with adapting teaching techniques. Mathematics is traditionally taught using frontal lecture, the least effective teaching method for retaining knowledge and conceptual understanding (Gningue et al., 2014; Satsangi et al., 2016). In this practice, the teacher is the primary, or in many cases only speaker, while the students simply observe and replicate teacher actions. Replicating actions without providing conceptual meaning for students was found to be ineffective (Mitchell, Charalambous, & Hill, 2013; Protheroe, 2007). Puchner et al. (2008) stated, “mathematical understanding involves the process of developing internal representations of mathematical ideas” (p. 314). The idea that students need to internalize mathematics ideas to create conceptual knowledge contradicts traditional mathematics classroom protocol. Mathematics educators’ perceptions of what mathematics instruction should look like directs their pedagogical practices (Moyer & Jones, 2018). Thus, changing perceptions of mathematics instruction in general is imperative to changing instructional practices. Teachers 11 THE USE OF MANIPULATIVES IN MATHEMATICS CLASSROOMS cannot be successful in a method they do not believe will be effective (Hull, Hinerman, Ferguson, Chen, & Naslund-Hadley, 2018). Protheroe (2007) recommended several mathematical teaching practices, which differ greatly from the frontal lecture approach and included active engagement, solving challenging problems, sharing of student ideas, pair and group work, mathematical communication, and manipulative and other tool use. Similarly, the NCTM (2014) has recommended the implementation of eight standard teaching practices. These practices include establishing goals, implementing mathematical tasks, using and connecting mathematical representations, facilitating discourse, questioning, fluency through conceptual understanding, supporting productive struggle, and eliciting evidence of student thinking. While each of these practices is essential for well-rounded mathematical instruction, the remainder of this literature review focuses on using and connecting mathematical representations using manipulatives. Manipulatives Manipulatives are concrete or virtual objects used to model mathematical problems (Belenky & Nokes, 2009). The concept of using manipulatives is not new. In southern Africa, the bone of a baboon was discovered containing 29 deliberate markings. Dating back to 35000 B.C., the markings appearing on the bone are believed to be tallies (Joseph, 2000). Other historical evidence of manipulatives are apparent: fingers and hands used in reckoning to emulate pairing (Leslie, 1817); in Babylonian mathematics, the number 10 is believed to be represented by two hands pressed together (Cajori, 1896); and in Rome, noble youth toted the abacus— a tool they borrowed from the Greeks (Leslie, 1817). Other manipulatives include bean counters that farmers used to account for produce and grain and nuts, or knots on a string that were used to number armies (Leslie, 1817). As counts grew too large for one-to-one 12 THE USE OF MANIPULATIVES IN MATHEMATICS CLASSROOMS representation, patterns and groupings emerged (Leslie, 1817). Later, the convenience of writing in symbols became more prevalent (Cajori, 1896). Symbolic representations eased communication in mathematics but did not explain the mathematical applications. The practice of reckoning numbers without relation to geometry, as had previously been done, began to emerge (Cajori, 1896). Without many advances during the dark ages, the Renaissance era brought a return to mathematical exploration. With the advancement of printed material and the field of mathematics being established in universities, symbolic knowledge of mathematics became more commonplace (Cajori, 1896). Manipulatives were reintroduced in the 1950s, during which Piaget emphasized the need for mathematical experiences in education (Golafshani, 2013). During this time, manipulatives began to take on a new role, being used as a learning tool rather than as objects to communicate quantities. Using manipulatives allowed students to experience mathematics in different ways creating richer and deeper connections (Tripathi, 2008). In the 1960s the “New Math” initiative was introduced to change mathematics teaching. This method encouraged learning through mathematical discovery rather than through taught techniques (Malkevitch, 2011). Later with the “New Math” initiative failing, further mathematical teaching reform in the 1980s urged more “real world” problems (Malkevitch, 2011). Today the NCTM promotes the use of multiple representations in teaching mathematics, along with seven other standards for teaching (NCTM, 2014). Concrete mathematical representations can be used in the form of mathematical manipulatives. Tools such as protractors, calculators, and rulers are not considered manipulatives since they are simply instruments used for precision and not representative of a mathematical concept (Carbonneau et al., 2012; Golafshani, 2013). Manipulatives include, but are not limited 13 THE USE OF MANIPULATIVES IN MATHEMATICS CLASSROOMS to, fraction strips, fingers, pattern blocks, base-ten blocks, scales, counting chips, and virtual manipulatives (Golafshani, 2013; Satsangi et al., 2016). Virtual manipulatives are computer programs that demonstrate a math concept using pictorial examples that can be manipulated (Gningue et al., 2014). These concrete or virtual representations help students connect concepts and understand mathematical relationships (Gningue et al., 2014; NCTM, 2014). While manipulative use has increased in the classroom, lecture-style teaching is still the predominant format for teaching mathematics (Carbonneau et al., 2013; Sitsangi et al., 2016). One challenge in incorporating manipulatives into instruction is choosing the correct manipulative. First, the manipulative must accurately demonstrate the concept (Belenky & Nokes, 2009). As with the parable of the blind men and the elephant (Tripathi, 2008), any one manipulative may leave out some of the concept. Thus, multiple representations may be needed. Another consideration in choosing a manipulative is the structure. Picking an object that is too detailed or realistic may cause students to focus on the manipulative itself rather than the concept it represents (Belenky & Nokes, 2009). In a study of manipulative use by college students, Belenky and Nokes (2009) found that such distractions led to difficulty in transferring conceptual knowledge to different scenarios. Therefore, teachers need to carefully choose the manipulative they use to build conceptual understanding. Another consideration in choosing a manipulative is whether to use one that is virtual or concrete. In one study, three secondary algebra students with learning disabilities used balance scales to solve algebraic equations [e.g., 4x + 1 = 9] (Satsangi et al., 2016). The students were taught using a physical scale with concrete objects to manipulate and a virtual scale in which students used a mouse to manipulate virtual objects. The task using concrete manipulatives involved loading multiple chips onto a scale, one at a time, to balance the equation. This task was 14 THE USE OF MANIPULATIVES IN MATHEMATICS CLASSROOMS time consuming and required greater instruction from the teacher compared to the virtual task. Satsangi et al. (2016) found that the virtual manipulative, involving the same action but using a computer mouse, allowed for greater student independence and took less time to solve. Due to the greater independence associated with virtual manipulatives, they may have a greater potential for secondary grade levels (Satsangi et al., 2016). While all students demonstrated mastery at a 90% or higher accuracy using either manipulative, two of the three students showed greater understanding using the concrete over the virtual manipulative. One student commented, “With the concrete manipulative, I can see where I went wrong” (Satsangi et al., 2016, p. 250). Given these findings both virtual and concrete methods have value in secondary mathematics classrooms. The NCTM recommended students have access to manipulatives to increase mathematical understanding (Carbonneau et al., 2013). While both virtual and concrete manipulatives can be effective, simply providing access to manipulatives did not lead to mathematical understanding (Belenky & Nokes, 2009; Carbonneau et al., 2013). For example, Moyer & Jones (2018) found that when a middle school teacher equipped the classroom with manipulatives but failed to demonstrate or relate the content using the manipulatives, the manipulatives were not effective. Students did not use the manipulatives, and the teacher concluded students in her class found them unnecessary. Thus, students needed instruction that facilitated the use of manipulatives before they were able to use them successfully (Mueller et al., 2014; Vinson, 2001). Manipulatives were most successful when combined with strategic questioning (Belenky & Nokes, 2009; Puchner et al., 2008). Instruction that facilitates manipulative use includes strategic prompts and thoughtful questioning leading to student discovery of concepts (Mueller et 15 THE USE OF MANIPULATIVES IN MATHEMATICS CLASSROOMS al., 2014). Researchers found that more instruction was needed to initiate a task, but once students were engaged, experimentation and learning followed (Belenky & Nokes, 2009; Carbonneau et al., 2013). When students disengaged, more reflective, deeper questioning was used to create interest and a greater cognitive result (Belenky & Nokes, 2009). Benefits Manipulative-based mathematics instruction has many benefits. Manipulatives are effective in engaging students, teaching diverse student populations, and increasing the rate of knowledge retention (Carbonneau et al., 2013; Gningue et al., 2014; Sinatra, 1990; Vinson, 2001). In a meta-analysis study, Carbonneau et al. (2013) found that manipulatives improved student learning over using abstract symbols alone. Student engagement. Keeping students engaged is a struggle in any discipline, but particularly in mathematics, as negative attitudes and anxiety may serve as stumbling blocks for student engagement (Vinson, 2001). Mathematical anxiety is manifested in uneasiness, avoidance, physical illness, and/or poor test performance. Vinson (2001) found that this anxiety led to negative attitudes and perceptions of mathematics, making students disengage before lessons began. Effective teaching practices, including using manipulatives, may reduce or eliminate mathematical anxiety (Vinson, 2001). Specifically, manipulative use lowered anxiety levels through emphasizing understanding rather than the solution. Eliminating mathematical anxiety can help students engage in more mindful mathematics. Belenky and Nokes’s (2009) data suggested that the best way to engage students using manipulatives was to monitor their interest. In their study, a group of 90 university students were aided in manipulative use through prompts and questioning. Once students were actively working, it was best to allow them the independence to pursue the self-guided task. When 16 THE USE OF MANIPULATIVES IN MATHEMATICS CLASSROOMS students were not engaged, the researchers recommended instructors seek more difficult questions to elicit deeper thinking. Once engaged, students were best supported through minimal interruptions (Belenky & Nokes, 2009). Teachers also reported that the use of manipulatives had a great impact on the classroom environment (Mueller et al., 2014). Manipulative use encouraged group work and discussion, which led to classroom communities in which students were able to share ideas openly and construct concepts (Golafshani, 2013; Mueller et al., 2014). Such communities were effective in engaging students and encouraging them to build arguments to support solutions (Mueller et al., 2014). These learning communities embody many of the standards set by NCTM (2014) for improved mathematics instruction. Diverse learners. Mathematics manipulatives can be used to support diverse learners. For example, Satsangi et al. (2016) found that effective use of virtual and concrete manipulatives improved outcomes for students with disabilities. They stated, “struggles faced by students with a learning disability in mathematics may be attributed to the instructional manner in which students are conventionally taught” (Satsangi et al., p. 240). Thus, integrating manipulatives into instruction is one way to beneficially alter mathematical instruction. Teachers noted the benefits of using manipulatives for struggling students, and these benefits extended to those with visual or abstract learning styles (Golafshani, 2013). The opportunity to experience mathematics in a different way allows more students access to the content. Knowledge retention. Another benefit of using manipulatives was the increased ability students had to retain knowledge (Belenky & Nokes, 2009; Carbonneau et al., 2013). Through manipulatives, students connected previous knowledge to new knowledge, which contributed to better recall amongst students. Using self-performed tasks rather than teacher lecture during 17 THE USE OF MANIPULATIVES IN MATHEMATICS CLASSROOMS instruction enhanced knowledge retention (Carbonneau et al., 2013). These tasks may include breaking up a pie into fraction pieces, regrouping base ten blocks for addition and subtraction, or using a scale to balance an algebraic equation. In explaining the increased retention, Belenky and Nokes (2009) suggested concrete materials may lighten memory load. Carbonneau et al. (2013) attributed the increased recall following manipulative use to the dual coding theory. The dual coding theory states that verbal and nonverbal information are stored in different, though connected, areas of long-term memory. Information is stored both as an image and as a verbal cue. Connecting to either the image or verbal cue stimulates the long-term memory. Students engaged in self-performed tasks use nonverbal techniques to obtain knowledge. When the students are then prompted to verbalize their knowledge, they create a verbal connection to stored material. Purportedly, activating one form of representation also activates the other, thus, increasing the rate of recall. This increased rate of recall is not based on memorization of procedural details, but conceptual connections made through teacher-guided, self-performed actions (Carbonneau et al., 2013). Barriers to Implementation Despite the known benefits of manipulative use, Golafshani (2013) found that teachers were still reluctant to integrate them into instruction. Factors leading to this reluctance included teacher experience (Golafshani, 2013; Moyer & Jones, 2018), classroom management (Mueller et al., 2014), and ongoing teacher support (Golafshani, 2013; Puchner et al., 2008). Understanding and addressing these factors may help overcome them. Teacher experience. Moyer and Jones (2018) and Golafshani (2013) examined the connection between teacher experience and manipulative use. Teachers who expressed reluctance to use manipulatives also had the least experience in using them, and these 18 THE USE OF MANIPULATIVES IN MATHEMATICS CLASSROOMS inexperienced teachers exhibited a lack of confidence in using manipulatives (Moyer & Jones, 2018). In contrast, Golafshani (2013) found teachers educated in manipulative use in the classroom reported greater self-confidence resulting in more effective teaching. Specifically, a 20-week training focused on manipulative use in the applied mathematics classroom shifted ninth grade mathematics’ teachers’ perceptions of manipulatives (Golafshani, 2013). This finding is supported by Vinson (2001), who asserted that mathematics educators needed opportunities to gain experience and confidence in using manipulatives themselves before they felt confident enough to use them in the classroom. The way teachers perceived manipulatives influenced if, and how often, they incorporated them into instruction (Moyer & Jones, 2018; Puchner et al., 2008). One study found that teachers perceived manipulatives as a reward activity, rather than a tool for conceptual understanding (Moyer & Jones, 2018). Using manipulatives as a reward meant students merely played with the manipulatives but did not understand the deeper concept. This misconception led to time wasting and manipulatives being used procedurally instead of conceptually, which did not improve student understanding (Moyer & Jones, 2018). Thus, manipulatives need to be used strategically to benefit students. Teacher perception was not the only inhibitor of using manipulatives. The NCTM stated that teachers need “a profound understanding of fundamental mathematics” in order to teach well (Puchner et al., 2008, p. 313). However, most teachers do not possess this deep understanding; many continue to struggle with mathematical content and pedagogical strategies (Hull et al., 2018; Puchner et al., 2008). Having a deeper understanding of concepts allows teachers to help students make connections and build conceptual understanding. Effective teachers need to be 19 THE USE OF MANIPULATIVES IN MATHEMATICS CLASSROOMS able to map out where discussions may go during student think-time and choose pedagogical strategies to support their students (Puchner et al., 2008). This lack of deep mathematical understanding is evident among pre-service teachers, and may be perpetuated by the programs that train them (Alat & Dedeoglu, 2013). Data compiled to assess early education classes and in-service programs at an early childhood program showed most students felt preservice mathematics education courses were ineffective in preparing them for teaching. One student commented “All we did throughout the semester was making teaching materials. Nothing else. It was no different than the material development course. No mention of how to teach mathematics. I learned nothing in that course that I can use now” (Alat & Dedeoglu, 2013, p. 2141). Another student stated, “I did not get anything at college that I could apply in my practice now” (p. 2141). Students also expressed frustration in student teaching. They felt they were not mentored but left to teach lessons and manage the classroom on their own. Without proper instruction and guidance on how to use manipulatives, teachers used them ineffectively (Alat & Dedeoglu, 2013). To be comfortable and confident with manipulatives, teachers needed to learn how to teach with them, beginning in preservice programs. Classroom management. The ability to make conceptual connections is not the only potential deterrent to teachers implementing manipulatives. Teachers also showed significant concern regarding the investment of time involved in using manipulatives (Puchner et al., 2008). As with any new instructional practice, initially there is a greater investment of time required for planning, practicing, and carrying out each lesson. The allotment of class time to allow for student exploration and concept development also played a significant role in teachers choosing to use manipulatives (Golafshani, 2013). Suggested instructional time for manipulatives has yet to be established, though research has suggested the need for instruction over an extended period 20 THE USE OF MANIPULATIVES IN MATHEMATICS CLASSROOMS (Carbonneau et al., 2013). Setting up and assembling manipulatives may also be time consuming. For example, Satsangi et al., (2016) pointed out that some manipulatives come with several parts and require assembly. Virtual manipulatives are an alternative that relieve some of this burden and allow for greater student independence but necessitate electronic devices for each student (Satsangi et al., 2016). Other classroom management concerns teachers face when using manipulatives include distributing materials, mishandling of manipulatives, and storage space (Golafshani, 2013; Satsangi et al., 2016). Golafshani (2013) found that teachers experienced in the use of manipulatives still anticipated some classroom disruption but felt the benefits of the manipulatives outweighed the inconvenience. In a yearlong study involving ten middle school classrooms, teachers were challenged to change how manipulative materials were managed (Moyer & Jones, 2018). During the first half of the year, manipulatives were controlled solely by teachers and distributed only on occasion. During the second half of the year, students were granted free access to manipulatives. Teachers expressed their apprehension at the ability they would have to maintain control in the class and keep track of the manipulatives. During the first week some students did misbehave with the manipulatives. However, teachers concluded that, even with free access, manipulatives were not a disruption to the class (Moyer & Jones, 2018). As with any new technique, it took time to establish a routine and expectations. Teachers may employ management techniques to ensure positive experiences with manipulatives. Techniques include: setting rules and expectations (i.e. students were not able to use the manipulatives during teacher instruction), assigning group leaders to count and return materials at the end of each class, keeping manipulatives in the middle of each table, or setting aside an accessible area within the classroom for manipulatives (Moyer & Jones, 2018). They 21 THE USE OF MANIPULATIVES IN MATHEMATICS CLASSROOMS found that students who had direct access to manipulatives on their tables used them more than students who had to retrieve them. Further, teachers reported that students with manipulatives at their tables explored and played mathematical games with the manipulatives, even during free time. While this approach disrupted the traditional role of mathematics educators, it was an effective approach in encouraging conceptual construction. The shift from procedural education, using only symbols, to conceptual education involved teachers relinquishing their role as the expert and empowered students to become the experts (Moyer & Jones, 2018). Giving students choice in their use of manipulatives transferred some of the responsibility of teaching to the student and gave them control to make decisions, construct solutions, and support their thinking. Given free access, students used manipulatives as an aid in explaining concepts to peers, and only on occasion picked inappropriate manipulatives to solve equations (Moyer & Jones, 2018). Classrooms effectively integrating manipulatives deviated from chalk and talk lectures to more of a classroom community (Mueller et al., 2014). An effective community involves problems that challenge students; teachers who help students develop strategies; open-ended tasks, student collaboration, and a setting that welcomes a variety of ideas and strategies for solving. Mueller et al. (2014) found that this type of community was conducive for manipulative use and conceptual reasoning. In addition, this community reinforces the teaching practices recommended by NCTM (2014). Ongoing teacher support. In order for teachers to overcome such barriers as experience and classroom management, they need support from their administrators and coworkers. One form of support that teachers need is access to manipulatives, as noted by Hull et al. (2018) who found that the expense and availability of manipulatives was a concern for some teachers. While manipulatives can be costly, Hull et al. found that introducing manipulatives was less expensive 22 THE USE OF MANIPULATIVES IN MATHEMATICS CLASSROOMS and more effective than reducing class size had been in a similar study. Using manipulatives that are inexpensive or constructed by students and teachers can be cost effective (Hull et al., 2018). Manipulatives do not have to be extravagant; they can be as easy as folding papers. In Think3d!, a spatial training program for education, paper is the only manipulative used (Burte, Gardony, Hutton, & Taylor, 2017). Think3d! encouraged students to manipulate paper into 2D and 3D objects to better reason through mathematical problems. Using this strategy, students involved in the program demonstrated improved understanding in real-world, spatial, and visual problems (Burte et al., 2017). Obtaining manipulatives is not the only support teachers needed for implementation. Alat and Dedeoglu (2013) stated, “teacher professional development is to be far from a simple provision of a training to bring out the natural instinct of an individual to teach, on the contrary, it is to be seen as a complex process in which many elements and factors come to play in interaction” (p. 2140). All teachers, especially new teachers, need pedagogical strategies to anticipate misconceptions and questions students may present (NCTM, 2014). These strategies can be effectively obtained through teacher collaboration (Golafshani, 2013). In addition, different approaches and classroom management techniques can be taught through mentor teachers (Livy, Muir, & Sullivan, 2018). Golafshani (2013) found that mathematics teachers needed mentors, collaborative in-school teams, and professional groups to share and learn. Successful incorporation of manipulatives is dependent not only on the support received from other teachers but administrators as well. Golafshani (2013) found that administrative support to invest time and money in professional development was necessary to be successful in teaching using concrete materials. Specifically, teachers need support through professional development to prepare lesson plans, map-out student thinking related to conceptual 23 THE USE OF MANIPULATIVES IN MATHEMATICS CLASSROOMS understanding, and become familiar with how and when to use manipulatives (Puchner et al., 2008). Following professional development, lesson observations and critique are needed to fully implement better instruction (Puchner et al., 2008). Secondary School Use The lack of manipulative use in secondary education may be attributed to the lack of availability, time, or lack of teacher knowledge (Alat & Dedeoglu, 2013; Belenky & Nokes, 2009; Golafshani, 2013; Puchner et al., 2008; Satsangi et al., 2016). In addition to these hurdles, secondary educators face the additional barrier of students’ negative perceptions of mathematics manipulatives. Gningue et al. (2014) found that students in secondary and collegiate grades perceived concrete manipulatives as juvenile and that virtual manipulatives were preferred. A study involving college-aged students found they did not consider manipulatives useful and instead preferred to rely on internal representations (Belenky & Nokes, 2009). These findings may be due to students’ experience and expectations of traditional mathematics classes. Despite their negative perceptions of manipulatives, research shows that secondary students do benefit from their use. Satsangi et al. (2016) examined how manipulatives supported students learning algebra. Algebra-related learning barriers included deficits in working and long-term memory as well as a lack of organization skills. Secondary students lacking these skills benefited from using manipulatives (Satsangi et al., 2016). Golafshani (2013) stated that manipulatives “bring mathematics to life and make the invisible mathematics concepts visible” (p. 139). Because algebraic skills are a key indicator of future success, aiding secondary students in mastery is imperative (Gningue et. al., 2014). Where manipulatives can improve mental processing skills that are fundamental to learning algebra, it is worth challenging students’ perceptions of manipulatives and engaging students in their use. 24 THE USE OF MANIPULATIVES IN MATHEMATICS CLASSROOMS Changing the perception of traditional mathematics classes, especially in secondary settings, may ease the transition to manipulative implementation. However, it may be necessary to change teacher perception first. If teaching strategies go unaltered, mathematics educators will continue in their reluctance to incorporate manipulatives and the number of U.S. students showing advanced achievement in mathematics may continue to drop (Lim & Sireci, 2017). Summary Compared to other developing countries, students in the United States continue to flounder on the mathematics portion of the PISA test (NCTM, 2014). Low mathematics performance may be attributed to current prevailing pedagogical practices used in mathematics instruction today. Mathematics teachers traditionally teach in a lecture-style environment that encourages procedural rather than conceptual understanding (Gningue et al., 2014; Puchner et al., 2008; Satsangi et al., 2016). The NCTM (2014) has recommended that teachers integrate multiple representations into their teaching repertoire, as students need to be able to view mathematical concepts from diverse angles in order to deepen their understanding (Tripathi, 2008). Using manipulatives engages students, increases the rate of retention, and has a positive effect on diverse learners (Belenky & Nokes, 2009; Carbonneau et al., 2013, Golafshani, 2013; Satsangi et al., 2016; Vinson, 2001). Although there are many benefits to using manipulatives many teachers still hesitate to use them. Secondary educators in particular are less prone to use manipulatives (Alat & Dedeoglu, 2013; Belenky & Nokes, 2009). Teacher perception, knowledge, and ongoing support are key factors that may impact manipulative use (Alat & Dedeoglu, 2013; Belenky & Nokes, 2009; Golafshani, 2013; Puchner et al., 2008; Satsangi et al., 2016). 25 THE USE OF MANIPULATIVES IN MATHEMATICS CLASSROOMS PURPOSE Students in the United States continue to score below proficiency on standardized mathematics exams (ACT, 2020; NCTM, 2014). Traditional mathematics classrooms using frontal lecture as the basis for student instruction have become insufficient in improving scores (NCTM, 2014). Adapting teaching methods to include multiple representations, such as manipulatives, helps students to reason mathematically (Golafshani, 2013; NCTM, 2014; Satsangi et al., 2016). Some teachers, especially those beyond elementary grades, are hesitant to implement non-traditional mathematical teaching practices due to their perceptions of traditional mathematics (Alat & Dedeoglu, 2013; Moyer & Jones, 2018). Teacher perception of how mathematics is taught needs to change before teachers will be willing to implement new techniques. The purpose of this study was to examine teachers’ perceptions and use of manipulatives in the classroom prior to and after participating in teacher education specific to mathematical manipulative implementation. Understanding these factors provides insight into the barriers that teachers face in implementing manipulatives after receiving instruction. This study addressed the following research questions: a. How do mathematics teachers perceive manipulatives prior to, and following professional development? b. How frequently do teachers use manipulatives as part of their mathematics instruction prior to, and following professional development? c. What barriers, if any, do mathematics teachers encounter when incorporating manipulatives into instruction? 26 THE USE OF MANIPULATIVES IN MATHEMATICS CLASSROOMS METHOD This descriptive research study used a survey to gain insight into teachers’ perceptions and use of mathematical manipulatives prior to and following professional development. Specifically, this study examined teachers’ perceptions of manipulatives, the frequency with which they used manipulatives, and any barriers teachers experienced in using manipulatives. Setting The survey was distributed to mathematics teachers who had participated in a professional learning project called the 8x8 (eight by eight) Teacher Leader Project. The 8x8 project is led by educators from Weber State University in Ogden Utah and is based on the eight mathematics teaching practices outlined in Principles to Actions: Ensuring Mathematical Success for All (NCTM, 2014). The researcher served for two consecutive years as a student assistant for the 8x8 project. Extending across three school years, the training consisted of a series of full-day meetings in which teachers learned about each of the eight practices with a focus on implementing them. The participants, teachers of kindergarten through 12th grade, met together as a large group to facilitate vertical alignment of practices but also had times they split into their designated grade bands (Grades K-4, 5-8, 9-12) to discuss specific ways to implement new practices. A full day of training was dedicated to each of the eight teaching practices during which teachers received research-based instruction. Following the instructional day, teachers were expected to create a video of themselves implementing the practice in their own classroom. The video was then presented within the appropriate grade band with the intent of receiving feedback for further refinement of the focused practice. For this study, the survey was sent to teachers after they had taken part in training on using and connecting mathematical representations. While the practice included visual, 27 THE USE OF MANIPULATIVES IN MATHEMATICS CLASSROOMS contextual, verbal, symbolic, and physical representations, the survey focused on the implementation of physical representations in the form of manipulatives. The survey was distributed approximately three months after the training. The first two months following instruction, teachers taught in their classrooms as usual. However, during the third month, teachers were mandated to teach remotely as all schools in the area were temporarily closed due to COVID-19. Participants Participants in the study included 25 mathematics teachers in grades K-12 from local school districts participating in the 8x8 Teacher Leader Project. Of those who participated, 84% (n = 21) had at least 4 years of teaching experience. Of the participants 92% (n = 23) taught general education courses. In addition to the general education courses, 44% (n = 11) reported teaching either honors/gifted classes or remedial/special education (SPED) courses. Based on personal knowledge of the area’s schools, this percentage most likely reflects differentiation within the classroom rather than teaching separate remedial, gifted, and general education courses. However, the survey did not include a question differentiating between these two possibilities. Only two participants reported teaching solely remedial/SPED classes, both of whom taught in the secondary grade band. Instrument The survey instrument was constructed by the researcher and distributed via email. The survey was self-administered through Google Forms and contained two sections: the consent form and the survey. In addition to collecting demographic information, the researcher developed survey items based on relevant constructs from the literature. Specifically, where the literature suggested that teacher perception, among other barriers, effects the frequency of manipulative 28 THE USE OF MANIPULATIVES IN MATHEMATICS CLASSROOMS use within the classroom (Golafshani, 2013; Moyer & Jones, 2018; Puchner et al., 2008)the survey instrument included items about frequency of manipulative use, perception of manipulatives, and barriers (if any) to manipulative implementation. Questions in the survey included short answer, Likert-scaled, and multiple-choice items. Given the shift from traditional in-person instruction to virtual/online learning due to COVID-19, the survey was designed to distinguish between these time frames. Procedures The researcher obtained permission to conduct this study through Weber State University’s institutional review board (IRB). Once approved, the survey instrument (see Appendix A) was distributed via email to participants. The survey instrument contained a disclosure giving consent to use data once the survey was completed and submitted. Participants were given the opportunity to discontinue the survey or skip over any question at any time. The survey was conducted using Google Forms, which allowed responses to be submitted anonymously. The survey was open for a period of three weeks. A reminder email was sent after the second week in hopes of increasing participation. Only the researcher and committee chair had access to the raw data of the survey. During the three-week survey window, 25 survey submissions were recorded, 10 from the secondary grade band and 15 from the elementary grade band. Thus, the survey had a 28.7% response rate. In social research, a response rate of 50% is generally deemed as acceptable (Richardson, 2005). However, Nulty (2008) argued that online surveys receive closer to a 33% response rate and that, for class evaluation purposes, any response rate may be useful depending upon the purpose of the survey. Thus, a response rate of 28.7%, while not ideal, may still be 29 THE USE OF MANIPULATIVES IN MATHEMATICS CLASSROOMS useful for understanding teachers’ perceptions of mathematical manipulatives before and after receiving a training. Low participation in this survey may have been due to several factors. The survey was distributed the week after school dismissed for the summer, following an especially challenging time for most teachers. The COVID-19 pandemic forced school closures, and the teachers in this study were required to teach remotely for the final quarter of the school year. Following this challenging time, teachers may have been less inclined to take part in an optional survey. The survey responses and all results were kept in a password protected drive until the completion of the study, at which time the files will be deleted. Once the survey data was obtained the researcher analyzed the data. Specifically, the researcher used descriptive statistics to analyze the selected-response items and thematic coding to analyze the open-ended response items. 30 THE USE OF MANIPULATIVES IN MATHEMATICS CLASSROOMS FINDINGS AND DISCUSSION The purpose of this survey-based study was to explore teachers’ perceptions and use of manipulatives after taking part in training specific to the incorporation of manipulatives into mathematical curricula. Key considerations included the differences in teacher perception, frequency of use, and barriers to manipulative use. Following data collection, the researcher explored different patterns within the data to develop possible connections between frequency of use and barriers teachers experience incorporating mathematical manipulatives. The data was separated into two grade bands to contrast elementary (kindergarten through sixth grade) and secondary (Grades 7-12) responses. While the local school districts represented in this study divide secondary grades into two groups: junior high (Grades 7-9) and high school (Grades 10-12), there were not enough respondents at the junior high level to divide the groups as such in this study. Thus, the data were split into the aforementioned grade bands, with 15 elementary participants and 10 secondary participants respectively. The data collected consisted of numerical values and short answers. The numerical data were entered into spreadsheets to analyze. The data were analyzed as a whole, as elementary and secondary grade bands, and as individual surveys. The researcher systematically looked for connections within the numerical data and used descriptive statistics to make comparisons. The survey concluded with several short response questions. These short answers were analyzed using thematic coding. Key words were used to group comments into categories in which certain themes emerged. These themes were considered in relation to the descriptive statistics to gain a deeper understanding of the research questions. 31 THE USE OF MANIPULATIVES IN MATHEMATICS CLASSROOMS Teachers’ Perceptions One of the primary purposes of this research was to explore teachers’ perceptions of mathematical manipulatives both before and after receiving training. Prior to receiving training, teachers held both positive and negative perceptions of manipulatives. Negative perceptions included 12% (n = 3) of participants thinking manipulatives were unnecessary, 12% (n =3) thought they were a waste of time, only 44% (n = 11) found them necessary, and 40% (n = 10) thought of them as a distraction (see Figure 1). Positive perceptions prior to training included 72% (n = 18) of participants thinking manipulatives were engaging for students and 64% (n = 16) thinking they enhanced learning. While not necessarily a negative perception, 64% (n = 16) of participants thought of manipulatives as an aid for remedial students. Manipulatives are helpful for remedial students but to think of them as only for remedial students would inhibit the idea that manipulatives are beneficial for all learners and would have negative implications. However, noticing that manipulatives can be used as a remedial aid in addition to regular classroom use has positive implications. Thus, a clarifying question asking the intent of the participants’ response on whether they considered mathematical manipulatives a remedial aid was needed to draw any further conclusions regarding teachers perceiving manipulatives as a remedial aid In contrast, following the training, positive perceptions included 0% of participants considering manipulatives to be unnecessary or a waste of time, 92% (n = 23) thought they were engaging, 88% (n = 22) thought they were necessary, and 100% (n = 25) thought they enhanced learning (see Figure 1). Despite the trend toward positive perception, some negative perceptions remained with 16% (n = 4) of participants thinking of manipulatives as a distraction. The post-training results also showed 8% (n = 2) of participants thought they were used to entertain 32 THE USE OF MANIPULATIVES IN MATHEMATICS CLASSROOMS students, and 84% (n = 21) thought they were an aid for remedial students. Perceiving manipulatives as entertainment for students can be positive, if the manipulatives are being used to drive conceptual connections. However, when manipulatives are viewed for their entertainment value students tend to use them for play rather than conceptual connections (Jones & Moyer, 2004). Figure 1. Teachers’ perceptions of manipulatives before and after training. Outcomes by Grade Band Where the literature established differences in the perception and frequency of use of manipulatives among grade bands (Alat & Dedeoglu, 2013; Belenky & Nokes, 2009; Golafshani, 2013; Puchner et al., 2008; Satsangi et al., 2016), it is helpful to consider the survey outcomes for each respective group. Thus, the elementary and secondary grade band outcomes are discussed separately here. Elementary grade band outcomes. After training, it was evident that participants in the elementary grade band, consisting of 15 survey responses, perceived manipulatives in a more positive light. Following the training, 0% considered manipulatives to be a waste of time, which 0 20 40 60 80 100 Entertain Entertain EntertainEntertain Entertain DistractionDistractionDistractionDistraction Distraction Distraction Distraction Enhance EnhanceEnhance EnhanceEnhance AideAideAideAide NecessaryNecessaryNecessaryNecessaryNecessaryNecessaryNecessary Necessary Engaging Engaging EngagingEngaging Waste of Time Waste of Time Waste of Time Waste of Time UnnecessaryUnnecessaryUnnecessaryUnnecessaryUnnecessaryUnnecessaryUnnecessaryUnnecessaryUnnecessary Unnecessary Percentage of TeachersPercentage of TeachersPercentage of TeachersPercentage of TeachersPercentage of TeachersPercentage of TeachersPercentage of Teachers Percentage of TeachersPercentage of TeachersPercentage of Teachers Percentage of TeachersPercentage of Teachers Percentage of TeachersPercentage of TeachersPercentage of TeachersPercentage of Teachers Perception of ManipulativesPerception of ManipulativesPerception of ManipulativesPerception of ManipulativesPerception of ManipulativesPerception of ManipulativesPerception of ManipulativesPerception of ManipulativesPerception of ManipulativesPerception of ManipulativesPerception of ManipulativesPerception of ManipulativesPerception of ManipulativesPerception of ManipulativesPerception of ManipulativesPerception of ManipulativesPerception of ManipulativesPerception of ManipulativesPerception of ManipulativesPerception of Manipulatives AfterAfter After Before Before Before33 THE USE OF MANIPULATIVES IN MATHEMATICS CLASSROOMS is down from 6.67% (n = 1) prior to the training (see Figure 2). Prior to the training, 73.33% (n = 11) of participants found manipulatives to be engaging for students; in the post-training responses, 93.33% (n = 14) of participants found them engaging. The number of participants who considered manipulatives necessary rose from 53.33% (n = 8) to 86.67% (n = 13). The perception that manipulatives are an aid for remedial users went from 60% (n = 9) prior to the training to 80% (n = 12) following the training. Perceiving manipulatives as solely an aid for remedial students is detrimental in the implementation in the use of manipulatives in the general education classrooms; however, acknowledging them as necessary for all students in addition to being a remedial aid is a positive shift in perception. While 60% (n = 9) of participants felt manipulatives enhanced learning prior to the training, following the training 100% (n = 15) indicated manipulatives enhanced learning. After the training, fewer teachers (20%, n = 3) considered manipulatives to be a distraction, compared to 53.33% (n = 8) prior to training. Figure 2. Elementary teachers’ perceptions before and after training. Thus, the training elicited a positive shift in the perception of manipulatives for participants in the elementary grade band. Of special note are the decline in the perception that manipulatives are a distraction and the increase in the belief that manipulatives enhanced learning for students. 0 20 40 60 80 100 EntertainEntertain EntertainEntertain EntertainEntertainEntertain DistractionDistractionDistractionDistractionDistraction DistractionDistractionDistractionDistraction EnhanceEnhance Enhance Enhance AideAideAide NecessaryNecessaryNecessaryNecessaryNecessaryNecessaryNecessaryNecessary EngagingEngaging Engaging Engaging Waste of TimeWaste of TimeWaste of TimeWaste of TimeWaste of TimeWaste of Time Waste of TimeWaste of TimeWaste of Time UnnecessaryUnnecessary UnnecessaryUnnecessaryUnnecessaryUnnecessaryUnnecessaryUnnecessary Percentage of TeachersPercentage of TeachersPercentage of Teachers Percentage of TeachersPercentage of Teachers Percentage of TeachersPercentage of TeachersPercentage of TeachersPercentage of TeachersPercentage of Teachers Percentage of TeachersPercentage of TeachersPercentage of TeachersPercentage of TeachersPercentage of Teachers Percentage of Teachers Perception of ManipulativesPerception of ManipulativesPerception of ManipulativesPerception of ManipulativesPerception of ManipulativesPerception of ManipulativesPerception of ManipulativesPerception of ManipulativesPerception of ManipulativesPerception of ManipulativesPerception of ManipulativesPerception of ManipulativesPerception of ManipulativesPerception of ManipulativesPerception of ManipulativesPerception of Manipulatives AfterAfter AfterAfter Before Before34 THE USE OF MANIPULATIVES IN MATHEMATICS CLASSROOMS This change in perception could be a result of teachers receiving better strategies through training to implement manipulatives which led to fewer distractions in the classroom (Golafshani, 2013). Secondary grade band outcomes. In the secondary grade band, consisting of 10 survey responses, teacher perceptions also showed a more positive trend following the training. Initially, 30% (n = 3) of secondary participants found manipulatives to be unnecessary and 20% (n = 2) found them to be a waste of time (see Figure 3). However, both numbers dropped to 0% following the training. Prior to the training, 70% (n = 7) of the secondary participants found manipulatives to be engaging for students; following the training, this rose to 90% (n = 9). Prior to the training, 70% (n = 7) of secondary participants considered manipulatives an aid for remedial students; following the training, 90% (n = 9) considered them an aid for remedial students. The greatest increase was evident in the number of secondary teachers who perceived manipulatives as being necessary. Statistically, this perception increased from 40% (n = 4) prior to the training to 100% (n = 10) post training. Figure 3. Secondary teachers’ perceptions before and after training. 0 20 40 60 80 100 EntertainEntertain EntertainEntertain EntertainEntertainEntertain DistractionDistractionDistractionDistractionDistraction DistractionDistractionDistractionDistraction EnhanceEnhance Enhance Enhance AideAideAide NecessaryNecessaryNecessaryNecessaryNecessaryNecessaryNecessaryNecessary EngagingEngaging Engaging Engaging Waste of TimeWaste of TimeWaste of TimeWaste of TimeWaste of TimeWaste of Time Waste of TimeWaste of TimeWaste of Time UnnecessaryUnnecessary UnnecessaryUnnecessaryUnnecessaryUnnecessaryUnnecessaryUnnecessary Percentage of TeachersPercentage of TeachersPercentage of Teachers Percentage of TeachersPercentage of Teachers Percentage of TeachersPercentage of TeachersPercentage of TeachersPercentage of TeachersPercentage of Teachers Percentage of TeachersPercentage of TeachersPercentage of TeachersPercentage of TeachersPercentage of Teachers Percentage of Teachers Perception of ManipulativesPerception of ManipulativesPerception of ManipulativesPerception of ManipulativesPerception of ManipulativesPerception of ManipulativesPerception of ManipulativesPerception of ManipulativesPerception of ManipulativesPerception of ManipulativesPerception of ManipulativesPerception of ManipulativesPerception of ManipulativesPerception of ManipulativesPerception of ManipulativesPerception of ManipulativesPerception of Manipulatives AfterAfter AfterAfter Before Before35 THE USE OF MANIPULATIVES IN MATHEMATICS CLASSROOMS Post-training, teachers in the secondary grade band no longer considered manipulatives a distraction or entertainment for students. Because manipulative implementation happens less in secondary grades, this positive shift in perception is a significant finding. Given training, teachers in the secondary grade band found manipulatives not only engaging for students but a necessary part of their curriculum. This result speaks to the effectiveness of the 8X8 teacher training program. Individual Shifts in Perception While the summary of the population as a whole and by grade band has great implications indicating the effects a training may have on teacher perceptions of mathematical manipulatives, of further importance are the findings shown on individual surveys. For example, Participant A’s survey is significant in that it shows the degree to which a teacher’s perception may change after training. Participant A was a general education mathematics teacher in the secondary grade band with four to six years of teaching experience. Prior to training, the teacher perceived manipulatives as being unnecessary, a waste of time, and an aid for remedial students. The data reinforced the literature, in that manipulatives being perceived as an aid for remedial students is typical across all grade bands (Gningue et al., 2014). Following training, this same teacher perceived manipulatives as being engaging, necessary, and used to enhance learning. The teacher also left the following concluding remarks: Teachers need training on them [manipulatives], especially at the high school level. Manipulatives are generally seen as beneficial for elementary school, but too basic or little kid like for high school. I didn’t see the use for them before participating in the 8x8. 36 THE USE OF MANIPULATIVES IN MATHEMATICS CLASSROOMS Not only did this teacher’s perception change, the teacher was aware of the change and the possible cause of their previous perception. This is especially poignant coming from a secondary teacher, since the frequency of manipulative use was generally low in the secondary grade band. Another example of how impactful the 8x8 training was for individuals can be seen in Participant B’s survey. This survey was submitted by a general education elementary school teacher with four to six years of teaching experience. Prior to training, the teacher reported perceiving the use of manipulatives as a waste of time and a distraction. In contrast, following the training, the same teacher reported manipulatives to be engaging, necessary, an aid for remedial students, and a way to enhance learning. This teacher concluded the survey by adding, “manipulatives allow students to explore mathematical concepts in a physical way. They can create a deeper understanding of the concepts and engage students in their learning.” Participant B concluded that not only are manipulatives engaging, but they are necessary. This participant emphasized the importance of creating a conceptual connection when using manipulatives, rather than memorizing arithmetic steps. Though this teacher had access to six different sets of manipulatives, Participant B was using them less than once each month prior to the training. Following the training, the participant said they used manipulatives more frequently and felt they were somewhat successful in implementation. This further supports the literature in that demonstrating a change in teacher perception may be a precursor to changing their practice (Moyer & Jones, 2018; Puchner et al., 2008). Though participant B found they frequently encountered behavioral problems when using manipulatives, they selected knowledge of manipulative use as the greatest barrier for their use. This result implies the need for further trainings and collaboration to aid teachers as they 37 THE USE OF MANIPULATIVES IN MATHEMATICS CLASSROOMS transition their practice to include manipulatives. Without additional support, teachers may abandon manipulatives and revert to their previous instructional methods. Manipulative Use Frequency In addition to examining teachers’ perceptions of manipulatives, the survey included questions about how frequently teachers used manipulatives in their instruction before and after the 8x8 training. Reflecting on manipulative use frequency prior to training, 8% (n = 2) of participants reported never using manipulatives. All 8% (n = 2) were in the secondary grade band, which aligns with the literature illustrating the lack of manipulatives in the secondary grade bands (Alat & Dedeoglu, 2013; Belenky & Nokes, 2009; Golafshani, 2013; Puchner et al., 2008; Satsangi et al., 2016). In addition, 56.5% (n = 13) of participants who used manipulatives, used them once a month or less. Comparatively, 28% (n = 7) of participants reported using manipulatives at least once each week. Of that 28%, the majority (86%, n = 6) were in the elementary grade band. Figure 4. Frequency of manipulative use before training. 0 10 20 30 40 50 60 More than Once/MonthMore than Once/MonthMore than Once/MonthMore than Once/MonthMore than Once/MonthMore than Once/MonthMore than Once/MonthMore than Once/MonthMore than Once/MonthMore than Once/MonthMore than Once/MonthMore than Once/MonthMore than Once/Month More than Once/MonthMore than Once/MonthMore than Once/MonthMore than Once/MonthMore than Once/Month Once/WeekOnce/WeekOnce/Week Once/WeekOnce/Week Once/WeekOnce/Week Once/2 WeeksOnce/2 WeeksOnce/2 Weeks Once/2 WeeksOnce/2 WeeksOnce/2 WeeksOnce/2 Weeks Once/2 WeeksOnce/2 WeeksOnce/2 Weeks Once/monthOnce/monthOnce/month Once/monthOnce/month Once/monthOnce/month Less than Once/Week Less than Once/WeekLess than Once/WeekLess than Once/WeekLess than Once/Week Less than Once/WeekLess than Once/WeekLess than Once/WeekLess than Once/WeekLess than Once/WeekLess than Once/Week Less than Once/WeekLess than Once/Week Less than Once/WeekLess than Once/Week Never NeverNeverNever Percentage of TeachersPercentage of TeachersPercentage of Teachers Percentage of TeachersPercentage of Teachers Percentage of TeachersPercentage of TeachersPercentage of TeachersPercentage of TeachersPercentage of Teachers Percentage of TeachersPercentage of TeachersPercentage of TeachersPercentage of TeachersPercentage of Teachers Percentage of Teachers Frequency of Manipulative UseFrequency of Manipulative UseFrequency of Manipulative UseFrequency of Manipulative UseFrequency of Manipulative UseFrequency of Manipulative UseFrequency of Manipulative UseFrequency of Manipulative UseFrequency of Manipulative UseFrequency of Manipulative UseFrequency of Manipulative UseFrequency of Manipulative UseFrequency of Manipulative UseFrequency of Manipulative UseFrequency of Manipulative UseFrequency of Manipulative UseFrequency of Manipulative UseFrequency of Manipulative UseFrequency of Manipulative Use Frequency Before TrainingFrequency Before Training Frequency Before Training Frequency Before Training Frequency Before TrainingFrequency Before TrainingFrequency Before TrainingFrequency Before TrainingFrequency Before Training Frequency Before TrainingFrequency Before TrainingFrequency Before Training Frequency Before TrainingFrequency Before Training Frequency Before Training SecondarySecondarySecondary SecondarySecondarySecondarySecondarySecondary Elementary ElementaryElementary ElementaryElementary ElementaryElementary38 THE USE OF MANIPULATIVES IN MATHEMATICS CLASSROOMS After the training, 88% (n = 22) of participants said they used manipulatives more often, 12% (n = 3) reported no change in their use frequency, and 0% of participants reported using manipulatives less often. This increase in use frequency could be due to the experience teachers received during the training. This finding is supported by Moyer & Jones (2018) who observed a lack of teacher experience in using manipulatives affects the frequency of manipulative use in the classroom. Thus, the 8x8 training increased teachers’ experience with manipulatives, which prompted them to incorporate manipulatives in their instruction more often following the training. It is interesting to note that even though elementary teachers reported more frequent use of manipulatives in comparison to secondary teachers before the training, more than 90% (n = 14) of elementary teachers said they increased the use of manipulatives after the training compared to only 80% (n = 8) of secondary teachers (see Figure 5) Figure 5. Change of frequency of manipulative use in elementary and secondary after training. All participants reported increased confidence in using manipulatives after the training. Following the training, all participants reported that manipulatives were at least somewhat 0 20 40 60 80 100 Less LessLess No Change No ChangeNo Change No ChangeNo ChangeNo ChangeNo Change MoreMoreMoreMore Percentage of TeachersPercentage of TeachersPercentage of Teachers Percentage of TeachersPercentage of Teachers Percentage of TeachersPercentage of TeachersPercentage of TeachersPercentage of TeachersPercentage of Teachers Percentage of TeachersPercentage of TeachersPercentage of TeachersPercentage of TeachersPercentage of Teachers Percentage of Teachers Frequency of Manipulative UseFrequency of Manipulative UseFrequency of Manipulative UseFrequency of Manipulative UseFrequency of Manipulative UseFrequency of Manipulative UseFrequency of Manipulative UseFrequency of Manipulative UseFrequency of Manipulative UseFrequency of Manipulative UseFrequency of Manipulative UseFrequency of Manipulative UseFrequency of Manipulative UseFrequency of Manipulative UseFrequency of Manipulative UseFrequency of Manipulative UseFrequency of Manipulative UseFrequency of Manipulative UseFrequency of Manipulative Use Change in Frequency Change in Frequency Change in FrequencyChange in Frequency Change in Frequency Change in Frequency Change in Frequency Change in Frequency Change in Frequency SecondarySecondarySecondary SecondarySecondarySecondarySecondarySecondary Elementary ElementaryElementary ElementaryElementary ElementaryElementary39 THE USE OF MANIPULATIVES IN MATHEMATICS CLASSROOMS successful and had at least a somewhat positive effect on conceptual understanding in their classrooms. These findings support those of Golafshani (2013), who reported teachers educated in manipulative use showed greater confidence in teaching with them and were more effective. Perhaps due to increased confidence, 28% (n = 7) of teachers used manipulatives during the time of remote instruction. Of the 28%, the majority (57.1%, n = 4) reported using manipulatives once a month or less prior to the training, possibly demonstrating the change in perception and seeing manipulatives as necessary to mathematics. This finding is especially encouraging considering teachers were unexpectedly pitched into remote learning. A number of considerations impact the frequency with which teachers implement manipulatives. Participants were asked the greatest consideration impacting the frequency of manipulative use, 44% (n = 11) responded that being able to create a conceptual connection using manipulatives was their greatest consideration, while 28% (n = 7) of participants said knowledge of manipulatives use was the biggest consideration impacting the frequency of manipulative use. While it is encouraging that the ability to make conceptual connections is a primary motivator for teachers’ using manipulatives, it may also explain the frequency in which they are used. Both Hull et al. (2018) and Puchner et al. (2008) stated that many teachers lack a deep understanding of mathematics. Teachers lacking this deep understanding may not be able to connect the manipulatives to conceptual content, decreasing the likelihood of manipulatives being incorporated. One teacher’s comments regarding barriers supports this idea. The teacher said that one barrier to manipulative use was “my knowledge of how to use them [manipulatives] and my students’ ability to understand what the manipulatives represent.” Thus, providing teachers with more training on manipulatives may help them build the conceptual connections necessary to confidently implement manipulatives into teaching practices. 40 THE USE OF MANIPULATIVES IN MATHEMATICS CLASSROOMS Based on the responses of participants in the elementary grade band, there appears to be some correlation between the frequency of manipulative use and behavior problems. Teachers who reported encountering behavior problems least often also reported using manipulatives most often (at least once each week). In contrast, 75% (n =3) of those who reported frequent behavioral problems during manipulative use implemented them less often (once each month or less). Further exploration is needed to determine if there is a connection between more frequent use of mathematical manipulatives and the drop in behavioral problems. This pattern was not evident in the secondary grade band data, which may be due to the generally low frequency of use that was reported in the secondary grade band. Only 20% (n = 2) of participants in the secondary grade band reported using manipulatives at least once every two weeks. The literature suggests effective use of manipulatives can create collaborative student communities within the classroom (Golafshani, 2013; Mueller et al., 2014). Without definitive suggestions for the frequency of use in the literature, this opens the opportunity to investigate the optimal frequency of manipulatives use to create such communities. Barriers to Manipulative Use In order to facilitate manipulative use, it was beneficial to know the barriers teachers faced as they left the training to implement this practice. To identify these barriers, participants were given a multiple response question relating to barriers identified in the literature such as time, access, knowledge, and conceptual connection, as well as a short answer response question where participants could identify other barriers and provide context for their responses. Classroom Management Another factor that impacts manipulative use is classroom management, which includes elements such as class time, student behavior, and manipulative access. When asked about 41 THE USE OF MANIPULATIVES IN MATHEMATICS CLASSROOMS barriers to using manipulatives, 44% (n = 11) of participants stated that time was a barrier. Comments included concern for the lack of class time to help students create a conceptual connection with the manipulatives, class time to distribute and clean up materials, as well as the time in professional learning communities (PLCs) to help other teachers understand the benefits of using manipulatives. Other barriers included lack of knowledge or a need for training (40%, n = 10), student behavior (12%, n = 3) and the availability of manipulatives (12%, n = 3). Although only 20% (n = 5) of teachers reported student behavior as a barrier to manipulative implementation, it was a recurring theme throughout the data (see Figure 4). All participants in the K-6 grade band reported experiencing behavioral problems while using manipulatives; 80% (n = 12) reported behavior problems sometimes to frequently. It does not appear to matter if teachers distributed manipulatives or allowed students free access to manipulatives, there were an equal amount of behavioral disruptions in both settings. In contrast, no participants in the secondary grade band reported frequent behavior problems, and 20% (n = 3) reported never having behavioral problems during manipulative use. All participants in the secondary grade band kept manipulatives in a cupboard accessed by the teacher. The behavioral issues for the younger grades could be due to the age and maturity of the students. Regarding behavioral issues one K-6 teacher stated, No matter the number of times I explain “expectations” there are students who see them as toys. I’m sure there is something I can do about that, but it is silly to just think that it’s all about explaining the expectations and [sic] viola! every student uses them as intended. This participant’s comment demonstrates the need for further training on classroom management during manipulative use. More specific instruction on how to navigate disruptive behavior when 42 THE USE OF MANIPULATIVES IN MATHEMATICS CLASSROOMS using manipulatives may be beneficial in future trainings, especially in the elementary grade band where behavioral problems were a more pronounced issue. Figure 6. Frequency of behavior problems. This comment also demonstrates the ongoing need for teacher collaboration when implementing manipulatives. Collaboration with other teachers may illuminate beneficial teacher practices to help deter behavioral issues when using manipulatives (Alat & Dedeoglu, 2013). While these disruptions may be attributed to poor behavior, there is a possibility that the teacher is simply not accustomed to a classroom culture centered on student exploration through manipulatives. This assertion is supported by Jones & Moyer (2004) who found that teachers needed to change their instructional behaviors and turn some of the classroom control over to students for successful implementation. Teacher Support Some of the barriers that teachers experience may be alleviated through co-worker and administrative support. Specifically, providing access to manipulatives and opportunities for 0 20 40 60 80 FrequentlyFrequently Frequently Frequently FrequentlyFrequently SometimesSometimes SometimesSometimesSometimesSometimesSometimesSometimes SeldomSeldomSeldomSeldom NeverNeverNever Never Percentage of TeachersPercentage of TeachersPercentage of Teachers Percentage of TeachersPercentage of Teachers Percentage of TeachersPercentage of TeachersPercentage of TeachersPercentage of TeachersPercentage of Teachers Percentage of TeachersPercentage of TeachersPercentage of TeachersPercentage of TeachersPercentage of Teachers Percentage of Teachers Frequency of Behavior ProblemsFrequency of Behavior ProblemsFrequency of Behavior ProblemsFrequency of Behavior ProblemsFrequency of Behavior ProblemsFrequency of Behavior ProblemsFrequency of Behavior ProblemsFrequency of Behavior ProblemsFrequency of Behavior ProblemsFrequency of Behavior ProblemsFrequency of Behavior ProblemsFrequency of Behavior ProblemsFrequency of Behavior ProblemsFrequency of Behavior ProblemsFrequency of Behavior ProblemsFrequency of Behavior ProblemsFrequency of Behavior Problems SecondarySecondarySecondarySecondary Secondary ElementaryElementaryElementaryElementaryElementaryElementary ElementaryElementary43 THE USE OF MANIPULATIVES IN MATHEMATICS CLASSROOMS collaboration and training are supports they may increase teachers’ manipulative use in the classroom. In this study, only 12% (n = 3) of teachers reported the availability of manipulatives as a barrier. All teachers reported having access to at least two sets of manipulatives within their school. In the elementary grade band, 93% (n = 14) of participants had access to six or more types of manipulatives. However, 40% (n = 6) of participants in the elementary grade band had used personal money to purchase manipulatives in comparison to only 10% (n = 1) of teachers in the secondary grade band. This result could be due to the higher percentage of secondary participants viewing manipulatives as unnecessary before the training or the perception that elementary teachers use manipulatives more often than secondary teachers. The need for manipulative access was apparently in one elementary teacher’s response regarding barriers of manipulative implementation. Articulating one of the barriers, the teacher stated, “time and organization because I don't have a classroom set of most manipulatives. Too large of groups using a singular manipulative usually means that it is a management nightmare.” This problem may be especially apparent in Utah, where the study took place, due to the large class sizes. While many manipulatives come in sets of 20-25, the average class size in Utah is greater than that. As of 2017, the average class size in Utah was 24 students for elementary schools and 29 students for secondary schools (Utah State Board of Education, 2017). Thus, even though teachers may have a variety of manipulatives to choose from, not having enough for a class set could also create management issues. Such issues may be addressed by encouraging added funding from the administration for large enough sets of manipulatives. In addition to funding, administrative support is needed to implement time for PLCs. PLCs can be a time for collaboration and mentoring (Alat & Dedeoglu, 2013). The responses 44 THE USE OF MANIPULATIVES IN MATHEMATICS CLASSROOMS addressing collaboration on manipulative use during PLCs were mixed. In the elementary grade band, 40% (n = 6) of teachers had never addressed manipulative use in their PLCs, and 33.3% (n = 5) discussed them more than once each week. In the secondary grade band, 40% (n = 4) stated they addressed the topic once each month and 20% (n = 2) stated they had never collaborated in their PLCs on how to use manipulatives. Due to the great variation in response, it is difficult to draw any clear conclusions on the consistency of teachers using PLCs to discuss the incorporation of manipulatives and how that impacts classroom use. This could be explored in future research. Conclusion This study explored teacher perceptions, frequency of use, and barriers to implementing mathematical manipulatives prior to and following teacher training. After the training, changes in the perceptions of manipulative use from teachers in both the elementary and secondary grade bands were evident (figure 7). Figure 7. Side by side comparison of teacher perception of manipulatives in elementary and secondary grade bands. When compared side by side, before receiving instruction the secondary grade band in particular had many negative perceptions of manipulatives. However, after the training the perceptions of 45 THE USE OF MANIPULATIVES IN MATHEMATICS CLASSROOMS secondary teachers shifted to more closely resemble those in the elementary grade band. This shift in perception points to the malleability of teacher perceptions through training. Teachers in the elementary grade band found manipulatives to be less of a distraction and more of an enhancement for learning. Teachers in the secondary grade band came to find them necessary. More importantly, individual teacher perceptions changed drastically for some, demonstrating the positive impact that teacher training can have on practice. In addition, 88% (n = 22) of participants reported an increased use of manipulatives in the months following the training. While time, classroom management, and knowledge were reported as barriers, the overall positive perception of manipulative use is encouraging. One teacher stated, “even though they can be distracting they are just too valuable not to allow access to students who choose to use them.” This participant found that the benefits of manipulatives outweighed the barrier of distraction. Providing teachers with training, materials, and ongoing support can break down some of the barriers to manipulative implementation. Where teachers’ perceptions of manipulatives impact their willingness to use manipulatives in the classroom, training designed to improve teachers’ perceptions of manipulatives is essential (Moyer & Jones, 2018; Puchner et al., 2008). Limitations and Recommendations While this study illuminated the ways that training can impact teachers’ perceptions and use of manipulatives, there were some limitations. For example, this study is limited by its sample size. A small response rate limits the implications of the study findings. In addition, some questions in the survey could have benefited from follow-up questions. As mentioned in the findings and discussion, the data depicting teacher perception of manipulatives as a remedial aid could be considered positive or negative depending on the intended meaning participants 46 THE USE OF MANIPULATIVES IN MATHEMATICS CLASSROOMS ascribed to the phrase “manipulatives as a remedial aid”. If participants meant that manipulatives are only perceived as a remedial aid, that would be negative, however, if they perceive manipulatives as an aid in addition to using them in a traditional classroom setting this would be considered a positive finding. An interview or follow up question would have clarified the intent of the response. This is also the case for perceiving manipulatives as entertainment. If the teacher perceived manipulatives as entertaining for students and that was the cumulative value of manipulatives, that would be considered a negative perception. However, if the teacher perceived manipulatives as entertaining for students because manipulatives engaged the students in learning, that would be a positive implication. Where these terms were not specifically defined for participants and no follow-up questions were asked to better understand how participants interpreted the terminology, this represents a limitation of the study. The significant change in individual perception noted in the findings create an opportunity for further research. For example, a follow-up qualitative study examining the driving force behind the perception shift may illuminate significant turning points for teachers. Based on these points of impact, strategies may be developed and implemented in teacher trainings to shift teacher perception and urge implementation of new practices. Because 8X8 is an ongoing teacher training program, continued research could be done to build upon the current findings. Sending out the survey prior to the end of the school year may increase participation. In addition, the response rate in future research may increase as result of not be impacted by the pandemic conditions that potentially limited the participation in this study. Another possible adjustment to future research would be to add more background questions on previous teacher training experience. Insight into preservice and in-service 47 THE USE OF MANIPULATIVES IN MATHEMATICS CLASSROOMS experiences with manipulatives may have illuminated further correlation to the frequency with which teachers use manipulatives. Future research could investigate the correlation between manipulative use frequency and behavioral problems. Exploring the participants’ continued efforts to implement manipulatives and possible future changes in teacher perspectives regarding manipulative use would be another research possibility. Of further interest would be to examine different types of student access to manipulatives and the possible effects on behavior during manipulative use. Research might also compare outcomes of teacher trainings that require implementations, as 8X8 does, in contrast to those that require attendance only. Exploration on the effectiveness of PLCs and their uniformity from school to school may also deepen understandings regarding manipulative implementation. Furthermore, this idea of shifting teacher perceptions through training creates possibilities for future research. For example, a follow-up qualitative study investigating the turning points for teachers, whether through their own efforts of implementation or from impactful moments during the training could provide further insight into how teachers’ perceptions and practices shift in relation to training. Final Reflection The findings of this study highlight the power and promise of quality professional development and its impact on teachers. Specifically, the findings demonstrate the malleability of teachers’ perceptions through training. That is, effective training can alter teachers’ perceptions and motivate them to implement research-based techniques into their instruction. This study demonstrated drastic individual shifts in teachers’ perceptions of manipulatives following participation in the 8x8 training. 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Science, Technology, Engineering and Math: Education for Global Leadership. Washington, D.C.: ED, 2014. http://www.ed.gov/ Vinson, B. M. (2001). A comparison of preservice teachers’ mathematics anxiety before and after a methods class emphasizing manipulatives. Early Childhood Education Journal, 29(2), 89–94. 52 THE USE OF MANIPULATIVES IN MATHEMATICS CLASSROOMS APPENDIX A Google Forms Survey Questions For the purpose of this study Mathematics Manipulatives refer to physical objects used to demonstrate mathematical concepts. Not included in this definition are mathematics tools: rulers, compasses, calculators, protractors, etc. As you answer the survey items, please consider your usage of physical/concrete manipulatives only, not virtual/computer-based manipulatives. 1. Which grade(s) do you teach? K-6 7-9 10-12 2. In the 8X8 Teacher Leadership project which cohort did you attend? K-4 5-8 9-12 3. How long have you taught mathematics? 1-3 years 4-6 years 7-9 years more than 9 years Perception 4. Which classes/students do you use manipulatives with? select all that apply) Honors /Gifted General education None Remedial/Special Education 5. BEFORE being involved in the 8X8 Mathematics Teacher Leader Project this year I considered mathematical manipulatives to be (check all that apply) Unnecessary An aid for remedial students 53 THE USE OF MANIPULATIVES IN MATHEMATICS CLASSROOMS A waste of time Used to enhance learning Engaging for students A distraction for students Necessary Used to entertain students 6. AFTER being involved in the 8X8 Mathematics Teacher Leader Project this year I considered mathematical manipulatives to be (check all that apply) Unnecessary An aid for remedial students A waste of time Used to enhance learning Engaging for students A distraction for students Necessary Used to entertain students 7. How would you rate the use of manipulatives in your class BEFORE being involved in the 8x8 Mathematics Teacher Leader Project this year? Did not use manipulatives Neither successful or unsuccessful Unsuccessful Somewhat successful Somewhat unsuccessful Successful 8. How would you rate the use of manipulatives in your class AFTER being involved in the 8x8 Mathematics Teacher Leader Project this year? Did not use manipulatives Neither successful or unsuccessful Unsuccessful Somewhat successful Somewhat unsuccessful Successful 9. Overall, how effective do you feel manipulatives have been at increasing conceptual understanding in your classroom? I don’t use manipulatives at all in my classroom Negative effect 54 THE USE OF MANIPULATIVES IN MATHEMATICS CLASSROOMS Somewhat negative effect No effect Somewhat positive effect Positive effect 10. Currently, how confident are you in using manipulatives to connect mathematical concepts? Unconfident Somewhat confident Somewhat unconfident Very Confident Neither unconfident or confident 11. How has your confidence in using mathematical manipulatives changed since being involved in the 8X8 Mathematics Teacher Leader Project this year? Less confident Somewhat more confident Somewhat less confident More confident Neither less or more confident 12. BEFORE being involved in the 8X8 Mathematics Teacher Leader Project this year how often did you incorporate manipulatives in your class? More than once a week Once each month Once each week Less than once each month Once every two weeks Never 13. How has the frequency of mathematical manipulative use changed in your classroom since receiving instruction on multiple representations in January? (NOT including the time of remote teaching due to COVID-19) Use less often 55 THE USE OF MANIPULATIVES IN MATHEMATICS CLASSROOMS No change Use more often 14. During this time of remote instruction have you incorporated physical manipulatives into your online instruction? Yes No 15. If yes, how? If no, why not? (please explain) 16. What is your biggest consideration in the frequency with which you use manipulatives? Time Conceptual connection Access Knowledge of use 17. How do students access manipulatives in your classroom? When you distribute them Available in a closed cupboard Available at their table/desk I do not use manipulatives in my classroom 18. How often have you encountered behavioral problems when using manipulatives? I do not use manipulatives Sometimes Never Frequently Seldom Always 19. How many different manipulatives are available for your use at your school? 1 7 2 8 3 9 4 10 56 THE USE OF MANIPULATIVES IN MATHEMATICS CLASSROOMS 5 11 6 12 or more 20. How were the sets of manipulatives purchased? School funded Do not have manipulatives Personal money Do not know 21. If you and your colleagues were eligible for the perfect attendance resource funds, what have you already purchased or plan to purchase for your school? Not eligible Math Tools Manipulatives Undecided 22. How often do you collaborate within your PLC (Professional Learning Community) on how to use manipulatives? Never Once each month Once each year Once each week or more Once each quarter 23. What do you consider the barriers (if any) to using manipulatives? (please explain) 24. Is there anything you would like to add that you think will add to the research regarding teachers’ perception of manipulatives after instruction? (please explain) 57 THE USE OF MANIPULATIVES IN MATHEMATICS CLASSROOMS Appendix B