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Preservice Elementary Teachers’ Reflective Insights from Teaching Mathematics during an Authentic Early Practicum Experience

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Preservice Elementary Teachers’ Reflective Insights from Teaching Mathematics during an Authentic Early Practicum Experience
Series Title:
Journal of Authentic Learning
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Rule, Audrey C. ( author )
Arthur, Scott C. ( author )
Dunham, Eric ( author )
Miller, Ryan ( author )
Stoker, Jonn ( author )
Thibado, Nichole ( author )
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English

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preservice teachers
early practicum experience
mathematics education
reflection
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This content analysis examined 1,710 statements made in post-lesson reflections of elementary education preservice teachers (N=120) after their first and second unassisted lessons during an early practicum experience that accompanied a mathematics methods course. The activities constituted authentic learning experiences in which preservice teachers planned and executed two appropriate mathematics lessons featuring hands-on materials and open-ended problem solving for elementary students. Predominant reflection themes focused on elementary student motivation, student learning, improvement of lessons, student skill levels, student behaviors and feelings. The authentic learning format allowed preservice teachers to develop professionally, taking responsibility for their own learning with the support of their mathematics methods instructor, host teacher, and peers. Reflections showed that preservice teachers had become aware of the complexity of teaching, analyzing problems from many viewpoints.
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Submitted by Brian McDonald (bmcdonal@oswego.edu) on 2007-06-13.
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Made available in DSpace on 2007-06-13T14:26:12Z (GMT).

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SUNY Oswego
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SUNY Oswego
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1558-7320 ( issn )

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http://hdl.handle.net/1951/41488

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Journal of Authentic Learning

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Preservice Elementary Teachers Reflect ive Insights from Te aching Mathematics 43 Preservice Elementary Teachers Reflective Insights from Teaching Mathematics during an Auth entic Early Practicum Experience Audrey C. Rule Scott C. Arthur Eric Dunham Ryan Miller Jonn Stoker Nichole Thibado State University of New York at Oswego This content analysis examined 1,710 statemen ts made in post-lesson reflections of elementary education preservice teachers (N=120) after their first and second unassisted lessons during an early practicum experie nce that accompanied a mathematics methods course. The activities constituted authentic learning experiences in which preservice teachers planned and executed two appropriate mathematics lessons featuring hands-on materials and open-ended problem solving for elementary students. Predom inant reflection themes focused on elementary student motivation, student learning, improvemen t of lessons, student skill levels, student behaviors and feelings. The authentic learning format allowed preservice teachers to develop professionally, taking responsibility for their own learning with the support of their mathematics methods instructor, host teacher, and peers. Re flections showed that preservice teachers had become aware of the complexity of teaching, analyzing problems from many viewpoints. Key Words: Preservice teachers, Early practicum experience, Mathematic s education, Reflection ______________________________________________________________________________ Extensive hours of field experience have been required by school of education accreditation agencies like the National Council for the Accreditation of Teacher Education in recent years (Vergari & Hess, 2002). However, limited research evidence is available that shows what preservice teachers gain from such early practicum experiences, most of it focusing on student teaching rather than earlier experiences (e.g., Chepyator-Thomson & Liu, 2003; Dobbins, 1996), and with only a few studies (Mewborn, 1999a, 1999b; Moyer & Husman, 2006) focusing on mathematics instruction. Therefore, this analysis of preservice teacher reflections on initial mathematics lessons contributes to a needed discourse in this area. In the following sections, we review the theoretical underpinnings of reflective practice and pertinent literature on preservice teacher reflective learning during practicum experiences in general and with respect to mathematics. Next we show how the activities discussed here constitute an authentic learning experience. Then we describe our study anal yzing the reflections of preservice teachers following the planning and teaching of their first lessons in the field to determine the reflective insights they glean from their authentic public school-based teaching experiences. Learning from Practicum Experiences through Reflection Dewey (1964, p. 211) found reflection to go beyond being merely a teaching tool to being an aim of education: reflective thinking enables us to know what we are about when we act, it converts action that is merely appetitive, blind, and impulsive into intel ligent action. Dewey (1933) outlined the steps of reflection: (1) Journal of Authentic Learning, Volu me 4, Number 1, Pages 43-64, June 2007

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Preservice Elementary Teachers Reflect ive Insights from Te aching Mathematics 44 feeling confused or doubtful about the situation in which one finds oneself; (2) speculation and tentative interpretation of the situation and its possible consequences; (3) observation and examination of the problem with which one is confronted and analysis of possible factors and considerations that will help define and clarify the problem; (4) Elaboration of hypotheses; (5) formulating a plan of action and then trying to implement a desired result. Dewey also acknowledged that attitudes, qualities and character traits (such as open-mindedness, sincerity, and responsibility) were im portant partners to skilled methods. Additionally, Dewey differentiated the contexts of reflection: retrospective reflection on past events, anticipatory reflection on future experiences, and contemporaneous reflection on inprogress experiences. The latter context of reflection is more difficult for new practitioners to achieve and is what Schn (1983, p. 54) calls thinking on your feet or Having your wits about you. Van Manen (1995) suggests that pedagogical tact is needed during on-the-s pot refection. Such tact includes sensitivity to the feelings, attitudes, and understa ndings of students, appropriate interpretation of the significance of these perceived feelings, a sense of standards, limits and balance that guide the teacher in entering or distancing from students situations, and moral intuitiveness of sensing what is right and doing it. These skills require a lot of reflective practice for development. Many researchers have investigated student teachers use of reflection. Dobbins (1996) conducted a four-year study of preservice elementary student teachers, finding that when they consciously reflected on their practicum experiences they had an enhanced learning experience, but were more tired and emotionally drained. Host teachers found that student teachers who reflected on the experience exhibited improved classroom practices with a better effect for students. Dobbins facilitated reflection by setting aside time for preservice teachers to reflect, requiring journal entries, involvement in school-wide experiences, and discussion during group meetings. Garca, Snchez, & Escudero (2006) who anchor their ideas concerning thoughtful reflection in situated learning (Collins, Brown, & Newman, 1989) and communities of practice (Wenger, 1998), noted that reflectiononaction, interpretations of past classroom events, helped teachers and preservice teachers define their future actions. They suggested that mathematics educators help preservice teachers connect theory and practice through reflection. Reflection, although taxing, is an important endeavor that leads to professional development. Hascher, Cocard, and Moser (2004) after a longitudinal analysis of preservice teachers through three phases of practicum experiences, raised important questions about learning outcomes from the field. They made a key distinction between two possible scenarios: preservice teachers experiencing the practicum as an apprenticeship in which they view their host teachers as experts and learn to act like them through social learning and imitation, as compared to experiencing the practicum as professional development in which preservice teacher learning is supported by host teacher mentors. This latter situation was more desirable for avoiding the pitfalls of preservice teachers developing an aversion to learning theory, which is necessary for generalizing and extending situations, and adopting the teaching practices to which they are exposed without reflection. Parsons and Stephenson (2005) emphasized the importance of reflection in Journal of Authentic Learning, Volu me 4, Number 1, Pages 43-64, June 2007

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Preservice Elementary Teachers Reflect ive Insights from Te aching Mathematics 45 deepening the practicum experience for preservice teachers. They defined students engaged in reflective practice as aware of and able to monitor their own thinking, understanding and knowledge about teaching (p. 97). Such reflective preservice teachers identify problems in practice, situations where there is some doubt about how to proceed, or instances where something could be changed or improved. Parsons and Stephenson suggested discussion an d collaboration with peers as a beneficial way to promote deeper reflection. This is echoed by Walsh and Elmslie (2005) who described a successful program in which preservice teachers were paired for their practicum experiences. Parkinson (2005) employed an unusual technique in eliciting reflections from preservice teachers in a practicum that preceded student teaching that of asking students to write a friendly letter in which they completed these prompts: (1) a new belief I have due to my practicum is; (2) I wonder; (3) I feel worried about; and (4) I wish She found that student reflections centered on five major themes: the realities of teaching; pedagogy and content; parents roles; elementary students needs, and preparedness for teaching. She concluded that pres ervice teachers reflectiveness is developmental and must be nurtured in many ways throughout their teacher preparation experience. The foregoing studies show that reflection on experiences assists preservice teachers in growing professionally and independently making sense of their experiences rather than merely acting as apprentices. In the next section we review the literature on pres ervice teachers using reflection during field placements related to mathematics. Reflection on Mathematics during Practicum Experiences In a study of four preservice teachers during a field experience connected to a mathematics methods course (1999a), Mewborn identified four areas of concern for preservice teachers (listed in the order they were addressed): 1) matters of classroom organization and management apart from mathematics; 2) mathematics pedagogy; 3) childrens mathematical thinking; and 4) mathematics content. When confronted with a new situation, preservice teachers tended to start again with the first area of concern and progress through the four identified stages. Mewborn found that the preservice teachers in her study engaged in all stages of Deweys (1933) five stages of reflective thinki ng during the field experience, but only when they felt they were in authority (in charge of the lesson and reflection). Otherwise, they became passive consumers of reflection ideas generated by the host teacher or methods course instructor. Mewborn (1999b) asked educators to pay more attention to the ecology of field experiences the structure and content, the characteristics of placement sites, and the relationships between preservice teachers, host teachers, and supervisors. She identified three characteristics crucial to shifting the locus of authority to preservice teachers. The first component is an inquiry approach involving analysis of and reflection upon what ha ppens in the field. Inquiry is supported when preservice teachers trust those which whom they discuss field experiences, and when they have the opportunity to discuss specific problematic situations and generate possible solutions. This requires that a significant amount of time be devoted to reflection and discussion of the field experiences. The second component identified by Mewborn Journal of Authentic Learning, Volu me 4, Number 1, Pages 43-64, June 2007

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Preservice Elementary Teachers Reflect ive Insights from Te aching Mathematics 46 was a cohort peer group sharing experiences and developing a sense of community. The final important component of Mewborn was the nature of the school-university collaboration. The field experience is optimal when the methods course and field experiences are interwoven so that preservice teachers see examples of the mathematical ideas being discussed in the college classroom in the field and problems arising in the practicum are discussed in the methods classroom. Moyer and Husman (2006) compared the reflections of preservice elementary teachers collected during the semester before student teaching under two conditions: 1) preservice teachers enrolled in a mathematics methods course taught at an elementary school with practicum experiences at that scho ol, and 2) preservice teachers enrolled in a mathematics methods course on the college campus with practicum field placements at elementary schools. They iden tified four main categories in the data: relationships with children, relationships with the school, lesson performances, and future goals. Similar to Mewborn (1999a), they found that preservice teachers first addressed general classroom organization and management issues, then narrowed their focus to mathematics pedagogy, sometimes considered childrens thinking, and very rarely reflected on mathematics content. However, they found that the group of preservice teachers immersed in the elementary school setting for their methods courses focused more on developing skills that would support them in their future careers as elementary teachers than those preservice teachers on campus who maintained a college-student focus. Those preservice teachers at the elementary school setting also had more opportunity for interactions in the school culture and for higher quality interac tions. These impacted their discussions of teaching mathematics and facilitated their un derstandings of how mathematics fits into the role of a teacher and how their practice lessons fit into the social and political context of schooling. Moyer and Husman found that campusbased preservice teachers focused more on classroom management and mathematics pedagogy issues, while the school-based preservice teachers advanced to also discuss childrens thinking. Our study adds to the current literature by examining written reflections of a much larger group of preservice teachers after teaching their first two lessons. Additionally, these reflective insights relate specifically to their experience of teaching their first lessons, rather than more generally related to their overall practicum experiences. Although the mathematics methods course was campus based and the field experiences took place mostly at urban schools in a nearby large city, the set-up of the current study otherwise supported Mewborns (1999b) three components of ecology for effective field experiences. The methods course instructor integrated discussions of the field experiences with the course, facilitating the integration of the field placement with the course as much as possible. Cohorts of students carpooled to the field placements and students discussed field placement issues in small and large groups during the course, providing a sense of community. Preservice teachers assumed the position of authority, planning and executing two open-ended problem-solving mathematics lessons with elementary students and reflecting in writing upon these experiences, thereby su pporting inquiry into the teaching-learning process. We analyzed these reflections to discover the types of reflective insights they had after teaching their first lessons in an early practicum experience. In the next section we explain Journal of Authentic Learning, Volu me 4, Number 1, Pages 43-64, June 2007

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Preservice Elementary Teachers Reflect ive Insights from Te aching Mathematics 47 how the lesson experiences described here constitute authentic learning experiences. An Authentic Learning Experience Rule (2006) found four themes that recur in authentic learning experiences: 1. Students conduct inquiry and use thinking skills and metacognition; 2. Students are involved in the work of professionals and make connections to the real world; 3. Students are part of a community of learners and engage in meaningful discourse; and 4. Students are empowered through choice in some aspects of the learning experience. In the following sections, we show how designing and teaching two mathematics lessons during a practicum experience at a public school qualifies as authentic learning for preservice elementary teachers. 1. Students Are Involved in Inquiry Preservice teachers needed to consider many different factors in choosing/designing their lessons: constraints such as supporting the state standards and being approved by the host teacher, the needs and interests of the elementary students with whom they worked, their own interests and mathematical strengths, resources for producing the hands-on mathematics lesson materials, and time constraints. These variables allowed many different solutions to the problem of choosing appropriate lessons. Preservice teachers designed an identical pretestposttest instrument related to lesson content that they administered to elementary students before and after each lesson. This information about student performance helped them as they reflected on the first lesson and prepared for the second one a few weeks later. Preservice teachers each wrote a one-page analysis of each lesson experience, describing what happened during the lesson, what aspects they might continue or change, and what they learned from the experience. These reflections formed the raw data for this study. Hands-on lesson materials were presented during class with peers scoring the materials with a rubric, noting strengths and suggestions for improvement. The assignment involved authentic assessment of preservice teacher performance. The instructor graded the lesson plan projects on the correspondence of lesson plans and materials to effective teaching of mathematics, using the LaunchExplore-Summarize teaching model as recommended by research following the Third International Mathematics and Science Study (Annenberg Media, 2000). Host teachers evaluated preservice teachers on professional dispositions and efficacy in working with students. 2. The Work of Professionals in the Real World The mathematics methods course promoted authentic, learner-centered mathematics with hands-on representations of concepts presented through a problemsolving approach. Teaching two lessons to elementary students during the practicum experience was a natural part of the course in which preservice teachers implemented instructional methods they were learning. Besides learning mathematics content and lesson pl anning, preservice teachers discovered how classrooms operate and how to interact wi th other professionals and elementary students, understandings not available in college classrooms (Brown & Kysilka, 2002). To be effective, today's teachers must "be students of human behavior, social events and their causes, and the characteristics of the citizens they serve" (Blair & Jones, 1998, p. 77). Preservice teachers needed to develop both professional (e. g., classroom discipline, pupils, curriculum, school culture) and cultural Journal of Authentic Learning, Volu me 4, Number 1, Pages 43-64, June 2007

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Preservice Elementary Teachers Reflect ive Insights from Te aching Mathematics 48 understandings (e.g., pupil's living conditions, cross-cultural communication, historical understanding) to see the multiple realities of a classroom and its community setting (Stachowski & Frey, 2003). Therefore, preservice elementary teachers were preparing for their future careers as teachers as they conducted these first lessons during practicum experiences. 3. Discourse among a Community of Learners Preservice teachers interacted cooperatively with the host teacher, with elementary students, wi th peers, and with the instructor to successfully complete the problem. Observations of elementary students and discussions with host teachers allowed preservice teachers to design effective lessons. Preservice teachers worked together in small groups during the methods course helping each other understand mathematics concepts, planning lessons, and discussing their practicum experiences. The mathematics instructor (first author) scaffo lded learning throughout the course by addressing mathematics topics in developmental or der, representing concepts concretely with manipulatives, and teaching through a problem-solving approach. The two le ssons taught by preservice teachers occurred near the end of the semester-long course, when they were well prepared to tackle this complex assignment. Preservice teachers had the opportunities, both during class discussions and informally before and after class to discuss their field observations and experiences with peers. Additionally, in most situations, severa l preservice teachers were assigned to the same classroom and many students carpooled for the forty-five minute ride to the school, discussing ideas during the trip. 4. Empowerment through Choice Preservice teachers were enrolled concurrently in a mathematics methods course and the practicum experience. They chose the two lessons that they planned and executed (generally for a small group of learners) with the provision that these lessons supported the state curriculum, were agreeable to the host teacher, and involved problem-solving with hands-on materials. Preservice teachers were encouraged to challenge themselves by choosing less familiar mathematical concepts that might require more preparation. Method Participants One hundred twenty preservice teachers (105 F, 15M; 116W, 4B) at a public mid-sized college in central New York State enrolled concurrently in one of five sections of a mathematics methods course taught by the same instructor and a practicum field experience at an area elementary school participated in the study. This course occurred during the se cond semester of the junior year for most students, and was the second of four practicum experiences that culminated in student teaching, but the first in which a formal lesson was taught to elementary students. Procedure The study was a mixed-methods design involving a content analysis of preservice teacher reflections, including frequency counts of categories of ideas, and triangulation with ot her qualitative data from another study (Rule and Harrell, 2006) of the preservice teachers from two of the five sections of the course. Each preservice teacher planned and implemented for elementary students two mathematics lessons that involved hands-on materials and open-ended problem-solving. Most lessons were conducted with a small Journal of Authentic Learning, Volu me 4, Number 1, Pages 43-64, June 2007

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Preservice Elementary Teachers Reflect ive Insights from Te aching Mathematics 49 group of students, but a few preservice teachers were asked to engage the entire class. Preservice teachers designed one-page quizzes to assess student learning before and after the lesson on the instructed concepts. The results from the assessment and events during the lesson were described and analyzed in a one-page reflection written by each preservice teacher for each of his/her lessons that was attached to the lesson plan. Instructions for the reflection were to provide a one-page word-processed reflection that gave the five or six most important events that occurred during the lesson and then a paragraph of reflection about what was learned from teaching the lesson. No example reflections were provided, nor were additional instructions or suggestions provided. This ensured that the preservice teachers decided the content and directions of their reflections. During the content analysis, each reflection was divided into a series of idea statements which were usually one sentence long, but varied betwee n a half-sentence and two sentences. These idea statements were transcribed word for word onto separate lines of a spreadshee t. There were 1710 statements in all. These were read and reread until several statem ents were found that relayed the same or similar ideas. In the second column of the spreadsheet, a short phrase was entered that captured the essence of the idea. After all statements had been categorized in this way, these short phrases in the second column were grouped into more general ideas, which were written in the next column. This process was continued until four levels of classification were achieved and major themes of the reflection statements were identified. This inductive method of initial open coding resulted in categories grounded in the research data (Creswell, 2005). The frequency of statements in each category was determined. Triangulation of results was achieved by comparing themes from reflections to the results of a study by Rule and Harrell (2006). The participants in Rule and Harrells study were 52 of the same participants in the cu rrent study (two of the five sections of the course). In Rule and Harrells study, at the beginning and end of the course, preservice teachers drew images representing three events that shaped their current attitudes toward mathematics. They also listed associated emotions and a wrote short self-analysis of what the images mean with regard to teaching mathematics to elementary students in their future careers. Although only data from 52 preservice teachers were included in Rule and Harrells study, all preservice teachers in the current study participated in these activities. Additionally, the reflection data were analyzed to compare male versus female responses to discern any trends, although the number of males in the study was small. Preservice teacher reflections on the first and second lessons taught were also compared to determine growth in the reflective process. Results and Discussion The content analysis of reflections revealed six main categories of response: student motivation, student learning, student skills, student behavior, teaching/ revising the lesson, and feelings of both students and preservice teachers. Each of these six major categories is discussed in the following sections with example quotes from the reflections given to illustrate the ideas. The six main categories identified in this study reveal pres ervice teachers' focus on students during the practicum. College methods classes tend to focus on pedagogy of teaching the subject area content. Although instructors do spend time talking Journal of Authentic Learning, Volu me 4, Number 1, Pages 43-64, June 2007

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Preservice Elementary Teachers Reflect ive Insights from Te aching Mathematics 50 about the attitudes, sk ills, and learning of elementary students, it is in the practicum that preservice teachers actively observe these factors. Naturally, then, elementary students, what motivates them, how they learn, what skills they have, and how they behave, are at the center of their thoughts. Another category, teaching/ revising the lesson, results from preservice teacher's main task during the practicum: to deliver two effective mathematics lessons. The emotional aspects of the entire experience comprise the final category. Similar to the preservice teachers in Moyer and Husmans study (2006), the preservice teachers in our study rarely discussed the mathematics content in their reflections. However, because our preservice teachers reflections were directly related to mathematics lessons they taught, our students paid less attention to general classroom organization and management issues, focusing more on mathematics pedagogy and management related directly to their lessons. Also in contrast to Moyer and Husman, our students considered student thinking about mathematics more frequently, possibly because they had pretest/posttest results to consider. Student Motivation Subcategories of statements related to student achievement are displayed in Table 1. Comments related to student motivation were the most numerous of all the statements made by preservice teachers, possibly because they openly reflected on their own past experien ces and feelings as elementary students during the course as various elementary topics were addressed. Table 1 Categories of Statements Related to Student Motivation General Category N Subcategory N More Detailed Subcategory N Students excited, enjoyed manipulatives 90 Next time will add or modify manipulatives 41 Students wanted to extend their time/ lessons with manipulatives 15 Students fascinated with tactile aspects of manipulatives 9 Observation that manipulative lessons were different than those traditionally taught in classroom 8 Students were overexcited about the manipulatives 8 Excitement Related to Manipulatives 177 Students were engaged during manipulative lessons 6 Enthusiasm and motivation displayed 43 Interest affects task engagement 19 Students were motivated because they felt special 17 Students wanted to continue working on math activities 16 Students' pride in understanding/ accomplishment 13 Other students wanted to participate 9 Novel activity or game sparked interest 9 Lack of interest or boredom shown 8 Choice of favorites or ownership motivated students 8 Frustration does not motivate 6 Considering adding reward to motivate 4 Factors Affecting Motivation 154 Safety of situation pr omotes participation 2 Students attracted to colorful materials 18 Change materials to make more colorful 17 Student Motivation 372 Beauty Appreciation 41 Change materials to make more durable 6 Journal of Authentic Learning, Volu me 4, Number 1, Pages 43-64, June 2007

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Preservice Elementary Teachers Reflect ive Insights from Te aching Mathematics Factors affecting student motivation. Preservice teachers not ed the enthusiasm displayed by students fo r their activities. Example statements included: "Being able to work with a student who was so eager to hear what I had to say was a big encouragement and it is the kind of teaching atmosphere that I think all teachers crave;" and "The student I worked with was always behind, but he hurried to do this project." Preservice teachers saw that a student's interest in the activity strongly affected motivation to participate, as indicated by, "I learned that involving topics of interest when doing word problems makes a big difference," and "My host teacher thought that students were more involved because of the cake theme of the activity. Two participants made the following important insights, "I have realized that anything can be fun if presented correctly," and "I was able to change their attitudes and to get them excited about a subject that even I did not like when I was young." The study participants observed with delight how students enjoyed working with them. They marveled at students' drive to continue working after the activity had been completed. Many preservice teachers acknowledged the importance of student's pride in work and sense of accomplishment. "Students were excited to take the posttest and show what they knew." "My student did all the cards. I though t it would have been overwhelming and repetitive, but she did it as a challenge." Additionally, they identified the novelty of an activit y as inspiration for students and the importance of student choice of favorites and ownership to motivation. Preservice teachers also identified times at which students showed a lack of motivation because of disinterest, boredom, or frustration. "This time I had a student who didn't want to participate and it was difficult to persuade him." An interesting insight in this category was, "For the next lesson, I will reorganize the quiz so the last question is the hardest and students are not discouraged early on." Finally, with regard to motivation, a few preservice teachers considered offering rewards, shown by statements such as, "If I do this activity again where student buy donuts with nickels, I'll provide real donuts at the end so students can eat them." Others affirmed to make the situation more comfortable for students, "In the future, I will create a more comfortable environment where the child can take risks," and, "The student started out shy but warmed up with my support." Excitement related to manipulatives. The subcategory with the most statements was that of student interest in, excitement related to, or enjoyment of manipulatives used during the mathematics lessons, confirming the motivating effects of involving students with concrete representations of mathematics. Ninety statements were recorded that expressed these sentiments; examples follow. "Students love manipul atives; it makes the lesson more interesting and fun for them." "Children called the lesson a game because they liked the manipulatives so much." "My student went back to the classroom to tell his friends how cool the stuff he worked with was." Students were often so enthralled with the manipulative lessons that they wanted to extend them. "Students asked if they could stay and play with the cards when the lesson was over." "One student asked to take the game home; I said, Ill bring it to class again.'" Excitement about manipulatives extended beyond the lesson, "During free time, one student drew bead bars." Because of the effects of manipulatives on motivation and on understanding (discussed in the next section), many preservice teachers reflected

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Preservice Elementary Teachers Reflect ive Insights from Te aching Mathematics 52 on how they would improve or add to their use of manipulatives in future lessons. "Next time I will make pretend wallets to go along with the purses with money clues." "All the class should learn with the teen numeral cards and bead bars so they will understand the teen numbers. I will have to make more sets." Sometimes this enthusiasm became over-excitement, as shown by the following comments. "My student wanted to skip the review and get on with using the manipulatives." I found that the students were overexcited about the manipulatives in the lesson and not focused on my explanations." This over-excitement may be related to the novelty of using hands-on materials. Students and preservice teachers observed that manipulatives were not generally used in ma ny classrooms. "All the kids loved the flat marbles for dynamic addition. They do not usually get to work with this stuff and so they enjoyed it." Many students enjoyed the tactile aspect of the manipulat ives. "Students held the bead bars rather than placing them on the table because the beads were so cool." Several preservice teachers noticed that students were more engaged in lessons that included interesting manipulatives: "I had their full attention during the lesson because of the materials," and One student who might have an attention deficithyperactivity disorder did focus when given the manipulatives." Beauty appreciation. Forty-one statements addressed appreciation of the beauty of materials used in teaching mathematics. Preservice teachers noted children's attraction to colorful materials. Childrens and preservice teachers perceptions of beau ty of mathematics materials affect mathematical performance (Rule, Sobierajski, & Schell, 2005). Children complimented the preservice teachers on making the attractive materials and one preservice teacher observed, "The children were very careful with my pompoms and baskets, showing their respect for my hard work." Preservice teachers reflected on changes they would make to their materials to make them more attractive or durable. Triangulation with other data. In the study by Rule and Harrell (2006) preservice teachers (actually, 52 of the same preservice teachers involved in the current study) drew three images (and also listed associated emotions) representing significant events that contributed to their attitudes toward mathematics at the beginning and end of the course. In the beginning of the course, almost two-thirds of the images and associated emotions were negative, whereas at the end of the course, about 70% of the images and emotions were positive. Preservice teachers were also asked to analyze their images and interpret them considering that they were preparing to be mathematics teachers. These statements were compiled and analyzed for major themes. The most frequent themes on at the end of the course were that manipulatives are necessary for effective mathematics, and mathematics must be made interesting and engaging to students. The most frequently drawn positive images were of manipulatives used during the course, followed by smiling faces, and images of preservice teachers delivering practicum lessons. Certainly the practicum experiences had a positive effect on attitude and motivation of preservice teachers. Therefore, preservice teachers were very much aware of the attitude and mo tivation of elementary students during their practicum lessons, particularly noting successes in motivation students in their reflections. Teaching mathematics presents challenges not encountered to the same extent in other subject areas. The main challenges are 1) the often poor preparation Journal of Authentic Learning, Volu me 4, Number 1, Pages 43-64, June 2007

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Preservice Elementary Teachers Reflect ive Insights from Te aching Mathematics 53 of preservice teachers to understand mathematics because of the traditional, rulebound, procedural way mathematics has been taught in the past; and 2) the unfamiliar new pedagogies of problem-solving and inquiry with ample use of manipulatives for concrete representation of concepts and frequent mathematical discourse. Consequently, many preservice teachers initially react to mathematics methods courses with fear of re vealing their levels of mathematical understanding and resistance to learning new ways of teaching mathematics that seem foreign to them. Changes in attitude must occur during a methods course for preservice teachers to become successful teachers themselves and it is therefore fitting th at preservice teachers focused on attitude and motivation most frequently during their reflections. Student Learning The second-largest category of statements dealt with student learning. This is not unexpected, as preservice teachers were required to pretest and posttest their students on lesson concepts, therefore providing data for reflection. Subcategories of student learning are shown in Table 2. Table 2 Subcategories of Statements Related to Student Learning General Category N Subcategory N More Detailed Subcategory N Students improved on posttest 48 Students showed improvement in understanding and skills during lesson 38 Surprised by students' speed of learning 8 Evidence of Learning 96 Students showed they did not understand 2 Student understanding increased with use of manipulatives 44 Need to have more manipulatives so more students may be involved 23 Manipulatives help learning 82 Need different manipulatives to help student learn concept 15 Students lacking needed mathematics vocabulary 31 Wording of exercises difficul t or confusing for students 14 Students experienced di fficulty in reading 5 Students lacked everyday background vocabulary 4 Language 57 Students confused by multiple ways of saying something 3 Students made personal connections to math lesson 18 Connection to a real life use of math made 8 Students applied new technique to familiar circumstance 5 Other subject area connected to math 4 Real world connections 38 Need for more real world connections noted 3 Activity designed so all were involved 9 Noticed that students need to have own sets of materials 9 Students enjoyed posing their own problems 8 Involvement 30 Students should be independent and self-check work 4 Organized activities help structure and fa cilitate learning 4 Student Learning 310 Order and organization 7 Neatness is important 3 Evidence of student learning. Fortyeight statements addressed increases in pretest to posttest scores, indicating that students had gained knowledge from the lessons. Overall, 947 elementary students were taught by these preservice teachers with a mean pretest score of 6.1 out of 10 (standard deviation = 2.8) and a mean posttest score of 8.5 out of 10 (standard deviation = 2.0). Only 6 children showed a decrease in performance from pretest to posttest, while 193 had the same pretest and posttest scores, with over half (118) of those achieving ten out of ten on both the pretest and posttest (no room to show improvement). An analysis of variance Journal of Authentic Learning, Volu me 4, Number 1, Pages 43-64, June 2007

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Preservice Elementary Teachers Reflect ive Insights from Te aching Mathematics 54 indicated that there was a significant difference between elementary student scores from pretest to posttest ( alpha = 0.5; F critical = 3.84; degrees of freedom = 1/1854; p < 0.001). This indicates that in general, elementary students displayed substantial performance gains after the lessons. Preservice teachers did grade the pretests/posttests, but included these with their lessons so that the instructor could examine the grading. Additionally, elementary student scores did not play a role in the preservice teachers grades for the assignment and the instructor emphasized the need for preservice teachers to use the information reflectively rather than viewing it as a scoring contest. This gain in understanding is reflected in statements made by preservice teachers. "I reviewed place value during my second lesson and found that students remembered a lot from my first lesson." "The practicum teacher expressed to me how she liked the activ ity and how much my student had progressed!" "Overall the child I worked with learned a lot about making maps and schedules." Preservice teachers expressed surprise at how quick ly students caught on to lesson ideas. "I was shocked at how quickly the kindergarteners understood the lesson and were counting by tens." Only two preservice teachers noted student difficulties in understanding: "A student asked me a question that made me r ealize that the he did not understand what I was explaining," and "The students' score did not change after the lesson." Helpfulness of manipulatives to student understanding. Preservice teachers wrote forty-four statements expressing the usefulness of manipulatives in helping children understand mathematics. "All but one child stated that having coins made it easier to figure out the coins in the coin purses from the lists of clues." "As I was completing the activity related to teen numbers, students told me that they liked this way of learning because they can see how one bead bar represents ten." Vocabulary and language. Participants noted that many children lacked the necessary language skills or vocabulary to succeed in their mathematics lessons. They found that many students were unfamiliar with coin names, inequality symbols, skip counting, multiples, or terms such as "addend," "sum," "dynamic," and "static." Students had difficulty with some everyday vocabulary along with listening and reading comprehension. One preservice teacher wrote that sometimes it was hard to word the questions so that students would understand. Another observed that third graders had difficulty reading number words. Connections to the real world. Many participants made connections to other subjects and everyday life as they taught their lessons, as in this excerpt from a reflection, "Students enjoyed the book I wrote about time and thought it was cool to write a book in math class." Another preservice teacher noted that during her lesson, "Students found examples of teen numbers inside and outside the classroom." During a lesson on coins, another participant observed, "As I reviewed, students made connections to their previous experiences with money." In the Principles and Standards for School Mathematics the National Council of Teachers of Mathematics (2000, p. 15-16) states that a mathematics curriculum should focus on important mathematics: "the curriculum should offer experiences that allow students to see that mathematics has powerful uses in mode ling and predicting real-world phenomena." After an exercise is using a zoo map to make a time schedule for visiting animals, a student remarked that he will be able to read a map next time he goes Journal of Authentic Learning, Volu me 4, Number 1, Pages 43-64, June 2007

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Preservice Elementary Teachers Reflect ive Insights from Te aching Mathematics 55 Triangulation with other data. In Rule and Harrells study, preservice teachers noted that they had learned important pedagogy of teaching mathematics, had a more positive attitude about teaching mathematics as a result of the course and were very much looking forward to teaching mathematics to elementary students. on vacation. An insight from another preservice teacher about a different lesson expressed these sentiments, "I'm glad that she was able to make personal connections to my lesson because I think it will help her to remember the information." Active involvement. The NCTM (2000, p. 18) acknowledges, "Teachers establish and nurture an environment conducive to learning mathematics through the decisions they make, the conversations they orchestrate, and the physical setting they create." Active involvement of students in mathematics is essential to their learning. Thirty reflection statements recognized student involvement in mathematics with such comments as, "Students enjoyed the problem posing aspects," and "They had fun creating different examples to make the expression true." Insights such as, "Next time I'll make more manipulatives so all students can be involved," show preservice teachers recognized the value in active involvement. Teaching and Revising Lessons Preservice teachers considered the components of effectiv e teaching in their reflections, addressing timing (Next time allow more work time so they can problem solve.); review of con cepts (I reviewed the greater than less than symbols and that was good because they didn't remember them); preparation (Next time be more prepared and not leave items out in my car.); challenge (Students enjoyed sorting clues and determining the coins. It was challenging); explanations (A student shared a mnemonic device for recognizing a penny as different color than other coins); practice (Students needed more examples and practice before they were able to move on.); simplifying (Next time, use only phrases with pennies as the other coins were too hard.); and expectations (I thought some students would have problems but I was wrong. They surpassed my expectations and wanted to count way past 100). More details of these catego ries are provided in Table 3. Order and organization. Seven statements revealed appreciation of order and organization, two concepts intimately tied to mathematics. A well-structured mathematics curriculum is more than a collection of activities; it must be coherent to effectively organize and integrate important mathematical ideas. Similarly, a single lesson must be focused and orderly. Several participants noted that wellorganized activities facilitated student learning. Others noted the importance of neatness in mathematical work. Journal of Authentic Learning, Volu me 4, Number 1, Pages 43-64, June 2007

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Preservice Elementary Teachers Reflect ive Insights from Te aching Mathematics 56 Table 3 Subcategories of Statements Relate d to Teaching/ Revising Lessons General Category N Subcategory N More Detailed Subcategory N Needed more time to complete planned lesson 22 Rushed through the lesson because there wasn't time 10 Host teacher cut off preservice teacher's lesson 6 Lacked preparation and so timing was poor 4 Desired to extend the lesson, but no time 4 Pretest/posttest took longer than expected 4 Student questions took a lot of time 2 Timing 53 Pacing of reading pretest/posttest questions difficult 1 Next time will review concepts prior to lesson 23 In the future, take more time to review more ideas 7 Initiated a review in response to student questions 7 Need to find ways to improve the review of skills 5 Began lesson with a review of background skills 5 Review concepts 51 Students caught on quickly after the review 4 Will change the lesson set-up to remedy problems 19 Will improve the pretest/posttest 8 Was not prepared for unanticipated student questions 5 Next time will practice lesson first to improve it 4 Preparation 40 Need to make an additional reference chart or model 4 Pretest/posttest was too easy or too hard for students 9 Needed to make lesson more challenging for students 8 Make the activity more complex to challenge students 7 Students need support when the activity is challenging 5 Preservice teachers were challenged by creating lesson and pretest/posttest 6 Many students appreciate challenging tasks 4 Independence makes the lesson more challenging 4 Challenge 46 Need to find harder questions to ask 3 Model and give more examples to improve explanation 18 Student reactions indicated inadequate explanations 12 Thought of hints to help students 5 Mnemonic devices given helped students 3 Difficult to provide the best level of explanation 2 Explanations 42 Initial confusion in problem solving is normal but students are not used to this. 2 In the future, provide more practice for students 15 Student performance / understanding improved with practice 11 Practice 28 Desired to leave activity for students to practice later 2 Start with easier or more limited problems 10 Simplify 14 Break the activity into smaller steps 4 Underestimation of student skills/ knowledge 9 Teaching/ Revising the Lesson 286 Expectations 12 Overestimation of student skills/ knowledge 3 Journal of Authentic Learning, Volu me 4, Number 1, Pages 43-64, June 2007

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Preservice Elementary Teachers Reflect ive Insights from Te aching Mathematics 57 Table 4 Subcategories of Statements Related to Student Skills General Category N Subcategory N More Detailed Subcategory N Noticed that numeration and pl ace value skills were poor 36 Stronger foundation in ma th and basic skills needed 25 Clocks and time concepts difficult for students 25 Students had adequate skills for lesson 19 Underestimated the ab ilities of students 14 Overestimated the abilities of students 13 Inequalities were a difficult concept for students 12 Pretest performance was poor 9 Lesson needed to be adjusted for poor performers 7 Students' reading skills were poor 6 Skill level 169 Noticed that students had poor memory skills 3 Noticed that students performed at different skill levels 32 Additional assistance was needed by some students 17 Desire future work with a more diverse group of students 11 Recognized the need for more advanced work for high achievers 10 Some learners needed to repeat the lesson to grasp the concepts 4 Diversity of ability 77 Gender differences in performance were noted 3 The complex, applied activity challenged students 17 Students enjoyed acting like detectives 7 Students were puzzled and lacked confidence at first 6 Students had difficulty understanding complex directions 5 Student Skills 285 Problem solving newness 39 Multiple correct solutions were surprising to students 4 Student Skills This aspect of practicum reflections, shown in Table 4, re vealed preservice teachers' surprise in observed skills of students, the diversity of abilities indicated by performance, and the outdated approaches being used in many contemporary classrooms. Skill levels. The skill level subcategory was the second largest subcategory of responses, second only to statements about manipulative excitement. This is understandable because, after all, lessons are designed to improve student knowledge and skills. The mathematics methods course in which the preservice teachers were enrolled emphasized place value as one of the big ideas of mathematics. Therefore, it is not surprising that 36 statements expressed preservice teachers' dissatisfaction in the place value and numeration skills observed in students in the field, noting student difficulty with one-toone correspondence in counting, confusion of odd and even numbers by upper elementary students, difficulty in regrouping, and lack of various place value skills. Participants also found that students struggled with elapsed time concepts and inequalities. However, almost equal numbers of preservice teachers wrote that they overor under-esti mated the abilities of the students for whom they designed lessons. Many made adjustments for low performers during the lesson. Besides mathematics skills being discussed, several reflections mentioned student problems with reading and memory. Diversity of ability Many participants obser ved that students performed at different skill levels during the lessons, noticing that, "one student got the concept right away while the others struggled." Another saw that, "Students took different amounts of time to think and respond." Several realized that some students needed individualized lessons: "I think you should make sure to have extra work or practice for high achievers," and, similarly, "I had difficulty keeping more Journal of Authentic Learning, Volu me 4, Number 1, Pages 43-64, June 2007

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Preservice Elementary Teachers Reflect ive Insights from Te aching Mathematics 58 proficient students busy while slower students worked." The importance of review and repetition, along with observed gender differences were noted. Problem solving newness. The lessons taught by preservice teachers in this study were open-ended, somewhat complex problems that integrated several background skills. Most of the classrooms in which preservice teachers were placed practiced traditional methods of mathematics instruction. Students were at first confused by problems-solving. Students felt challenged, but enjoyed their role as detectives in figuring out the solution. Example statements for this category include: Students were really challenged by the inequalities, and At first they were discouraged by all the different answers until I explained how each was correct. Table 5 Subcategories of Statements Related to Student Behavior General Category N Subcategory N More Detailed Subcategory N Challenging to keep students on task 24 Encountered jealousy and other emotional issues 14 Important to assign tasks for order and fairness 14 Exercised a teacher's author ity to maintain order 9 Next time will work with fewer students for control 8 Witnessed misbehavior and acting out of students 7 Need rules of raising hands and responding 5 Witnessed students cheating 5 Discipline 90 Lack of space and overcrow ding caused problems 4 Peer tutoring occurred during lesson 25 Students worked cooperatively on the activity 17 Students showed care/respect for materials or others 14 Competition between students was observed 13 Students encouraged their peers 4 Some students dominated or were very assertive 3 Peer interaction 79 Gender issues noticed: boys pr eferred working with boys 3 Interruptions by teachers, students and events 21 Other lessons being conducted distracted students 16 Materials distracted students from instruction 16 Noise distracted students 8 Student Behavior 236 Distraction 67 Students not chosen for lesson commented or asked questions and distracted students in lesson 6 Student Behavior A study of preservice physical education teachers reflections on student teaching (Chepyator-Thomson & Liu, 2003) revealed that they focused mostly on techniques of class management and discipline, with secondary student teachers emphasizing these components (71.1% of reflective comments) more than elementary teachers (55.4% of reflective comments). Reflections on student be haviors were also a major component of the reflections analyzed in this study, in which preservice teachers discussed discipline, pe er interactions, and lesson distractions. Table 5 shows details of subcategories of comm ents about student behavior. Discipline. First and foremost, preservice teachers noted the difficulty of keeping many students on task. Usually, these practicum lessons were taught in small groups of three to six students, but a few of the preservice teachers were requested by their host teachers to engage the entire class. Some preservice teachers remarked that they would work with fewer students next time for better control. G oodman (1991) states, Reflection suggests much more than taking Journal of Authentic Learning, Volu me 4, Number 1, Pages 43-64, June 2007

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Preservice Elementary Teachers Reflect ive Insights from Te aching Mathematics 59 a few minutes to think about how to keep pupils quiet and on task Reflection implies a dynamic way of being in the classroom (p. 60). Parsons and Stephenson (2005) stress the importance of preservice teachers recognizing their ideas and beliefs leading to their actions and considering if these underlying values are appropriate, then reconstructing their pro cedures if necessary. This deeper reflection occurred for many of the participants in our study as they witnessed misbehavior of students and considered what to do about it. Here are some examples: When I was setting up, all three wanted to help I should have had something for them to do, None of them could wait for an answer and when you answer one first, all of them get mad that they were not first to be answered I will have to explain to them that we will take turns, and They don't think they need to give me the respect they do a teacher, but I simply explained to them that I was the teacher at the moment and they needed to act appropriately." Peer Interaction. The majority of peer interactions report ed were positive with preservice teachers noting how students engaged in peer tutoring without prompting, worked cooperatively, and showed care and respect. A student who understood buddied with a student who struggled and helped him, "The thing that really got me was when everyone clapped at the answers. The students would cheer on each other and clap for those that got the answers right." Competition was also recognized and the fact that some students dominated the lessons: Students raced each other to see who could match the clocks fastest, and One boy made every part of the lesson into a competition, dominating the lesson. Distraction. Preservice teachers noted the many distractions in the classroom and pondered how to deal with them. Extra ESL children were present in the classroom and their teacher conducted a different lesson with them next time, I will include the ESL students so there is only one lesson going on and not so much distraction, and The phone kept ringing during the lessons, so next time Ill have the lesson in a quieter environment outside the classroom. Feelings Preservice elementary teachers have the highest levels of mathematics anxiety of all college majors (Hembree, 1990). Rule and Harrell (2006) found that participation in a mathematics methods course changed preservice teacher attitudes toward mathematics from predominantly negative to positive. The practicum experience was credited with some of the change as preservice teachers r ealized the importance of attitude, deep understanding of concepts, and the exciting sense of accomplishment in teaching others. Feelings, therefore, are an important part of teaching mathematics. The overall effect and feeling like a teacher. Preservice teachers had overwhelmingly positive experiences in teaching lessons during their field placements; there were seventy-two statements in reflections saying that the activities went smoothly and they felt positive about the lesson. Many expressed that the experience allowed them to feel like a teacher. They felt pride in teaching others successfully and accomplishment in seeing eager students motivated by their lessons. Several mentioned that they grew professionally and/or learned mathematical concepts by teaching others. Acknowledgment of their contributions from the host teacher added to their feeling of being a professional. Si x preservice teachers mentioned that the experience of teaching students during the practicum confirmed their decisions to enter the teaching profession. Journal of Authentic Learning, Volu me 4, Number 1, Pages 43-64, June 2007

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Preservice Elementary Teachers Reflect ive Insights from Te aching Mathematics 60 Table 6 Subcategories of Statements Related to Feelings General Category N Subcategory N More Detailed Subcategory N Overall effect 72 Activities went smoothly; felt good about lesson 72 Felt pride in successful teaching of students 14 Student enjoyment of lesson motivated preservice teacher 13 Grew professionally and mathematically as a result of the lesson 12 Felt respected as a teacher by host teacher 7 Felt like teacher 52 Confirmed career decision to be a teacher 6 General student apprehension concerning pretest 18 Non-graded aspect relieved some student tension 6 Student concern about taking pretest without preparation 5 Fear that teacher would see poor pretest performance 4 Observed students working hard on pretest 2 Assessment anxiety of students 37 Students irritated that the posttest was the same as pretest 2 Preservice teacher gained self-confidence 11 Students gained self-confidence with support 9 Confidence 28 Students were afraid to take risks for new lesson 8 Practice helps ease nerves 8 Preservice teacher fear of disappointing others 4 Preservice teacher relaxed as lesson unfolded 3 Nervous 18 Positive student reactions eased nerves 3 Effect on schools 9 Teachers adopted preservice teachers' activities for their classrooms 9 Feelings 221 Complexity of teaching 5 Many different tasks must occur simultaneously during a lesson 5 Assessment anxiety of students. Many students expressed concern about taking a pretest before they had been taught the lesson material. Many were afraid that if they performed poorly on the pretest, the host teacher would find out. "There seemed to be some uneasiness among the students about the pretest because they didn't know the answers and didn't like having to guess." Finding out that the pretest did not count toward their grades eased tension somewhat, but students were accustomed to having scores on quizzes carry consequences. On the second lesson, preservice teachers were more careful to explain that the pretest was merely a way for them to ascertain what students already knew. Confidence and nervousness. Developing a professional identity is one of the most difficult issues at hand when a preservice teacher starts the first lesson; this evolving process can be one of stress and strain on the preservice teacher (Pittard, 2003). Nervousness or general anxiety was expressed in both th e first and second lessons taught. A preservice teacher composing a first lesson reflection commented, I got kind of nervous and was ready to give up. I think sometimes children intimidate me. Another writing a second lesson reflection noted, It seems like I am always very nervous when it comes to presenting my own lessons, so I should practice more before I present. Knowing that other novices have the same feelings can help to restore confidence. Much of the anxiety precipitates from a desire to not fail the students and to appear as a professional in front of the host teacher. Many preservice teachers reported gains in self-confidence as a result of their experiences: I am slowly gaining the confidence that I feel is necessary for being in the teacher position, and I was pleasantly surprised that my lesson plan was as effective as I hoped it would be. They Journal of Authentic Learning, Volu me 4, Number 1, Pages 43-64, June 2007

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Preservice Elementary Teachers Reflect ive Insights from Te aching Mathematics 61 reported that students al so made strides in confidence, as shown in this example comment, The student learned from her mistakes and had more confidence in doing subtraction at the end of the lesson. However, the newness of the hands-on lesson formats intimidated some students, as shown in these statements, A student pretended not to know so as not to make a mistake, and At the start of the lesson, the students did not want to do the activity because it was new to them. This was an eye-opener to me as a preservice teacher as I did not realize that sometimes you have to coax even your best students to partake in an activity." Effect on schools. The learning that occurs during practicum experiences occurs for host teachers, too. Nine preservice teachers reported that their host teachers adopted some of their activities for use in their classrooms, as shown by these examples: "My host teacher was excited when I told her I had this assignment and she was eager to see new materials she could use in her class with students," and The teachers in my class and next door said how this checkerboard would be great as a permanent station in the classroom. Complexity of teaching. Five preservice teachers noted the complexity of teaching, in which many different tasks occur simultaneously during a lesson. One remarked, "I also observed how very difficult it was to plan a lesson for kindergartners. I though t that kindergarten teachers got to play all day, but this is definitely not the case." Comparison of Female to Male Reflections and First to Second Lessons The sample population contained only 15 male preservice teachers or 12.5%. Therefore, one cannot generalize too far from this small subset of the population. However, a few trends are worth noting. Female preservice teachers wrote an average of 14.5 idea statements per preservice teacher, combining both reflections form lessons 1 and 2; male preservice teachers recorded somewhat fewer ideas with an average of 12.2 statements per preservice teacher. This may mirror the more verbal, chatty nature of women compared to men. Table 7 presents data comparing female and male preservice teacher reflections. Female and male preservice teachers offered similar percentages of ideas related to student learning, student skills, student behavior, and feelings. However, female students provided more comments related to student motivation than males, while male preservice teachers gave more statements related to teaching and revising lessons than females. This may also reflect gender differences of females being more attuned to feelings, while males focus on executing and refining required tasks (Gray, 2004). An increase in reflective observations occurred for both sexes from lesson 1 to lesson 2, probably because of the increased familiarity with writing a reflection and the larger store of experience. Second lessons tended to focus more on teaching and revising the lessons, particularly for male s, and on analyzing student learning. The re flections of second lessons devoted fewer statements to student skills, student motivation, and student behavior, probably because these were more general observations of the students that were already discussed in the first lesson reflection. The statements related to feelings remained fairly constant for females, but decreased somewhat for males. Mewborns (1999a) ideas that matters of classroom organization and management apart from mathematics will be addressed first seem applicable to our results. Journal of Authentic Learning, Volu me 4, Number 1, Pages 43-64, June 2007

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Preservice Elementary Teachers Reflect ive Insights from Te aching Mathematics 62 Table 7 Comparison of Numbers and Percents of Statemen ts in the Six Categories by Sex and Lesson Student Motivation Student Learning Teaching / Revising the Lesson Student Skills Student Behavior Feelings Totals Group Lesson Number of Statements Percent of Total for Row Number of Statements Percent of Total for Row Number of Statements Percent of Total for Row Number of Statements Percent of Total for Row Number of Statements Percent of Total for Row Number of Statements Percent of Total for Row Number of Statements Percent of Total for Row Both Lessons 372 21% 310 18% 286 17% 285 17% 236 14% 221 13% 1710 100 Lesson 1 193 23% 123 15% 108 13% 156 19% 137 17% 109 13% 826 100 All Preservice Teachers Lesson 2 179 20% 187 21% 178 20% 129 15% 99 11% 112 13% 884 100 Both Lessons 348 23% 285 19% 243 16% 240 16% 209 14% 202 13% 1527 100 Lesson 1 180 24% 113 15% 93 13% 133 18% 122 16% 99 13% 740 100 Female Preservice Teachers Lesson 2 168 21% 172 22% 150 19% 107 14% 87 11% 103 13% 787 100 Both Lessons 24 13% 25 14% 43 23% 45 15% 27 15% 19 10% 183 100 Lesson 1 13 15% 10 12% 15 17% 23 17% 15 17% 10 12% 86 100 Male Preservice Teachers Lesson 2 11 11% 15 15% 28 29% 22 13% 12 12% 9 9% 97 100 Summary and Conclusion The foregoing analysis of preservice teachers reflections after teaching their first and second lessons in an early practicum placement shows that they learned much about the essentials of teaching from this experience. Their focus was mostly on elementary students, what motivates them, how they learn, how to improve lessons to teach them better, along with observations of student skills, student behaviors, and feelings. The preservice teachers in our study exhibited evidence of progressing through Deweys reflective stages. During class discussions preservice teachers expressed confusion and doubt over how to go about teaching lessons. During lesson planning, they attempted to prepare for students with special learning needs a nd to build in extra activities for those students who excelled or needed more practice. During their reflections, preservice teachers identified major events during their teaching and problems that had occurred. They attempted to explain why these events happened and to determine ways to improve their lessons for the future. Because they taught and reflected on two lessons, many were able to formulate a plan of action and attempt to implement a desired result. The authentic nature of the experience in which the preservice teachers collaborated with peers and host teachers to determine a suitable mathematics lesson, executed the lesson, and reflected on it alone and through discussions with others assisted them in extracting esse ntial aspects of the Journal of Authentic Learning, Volu me 4, Number 1, Pages 43-64, June 2007

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Preservice Elementary Teachers Reflect ive Insights from Te aching Mathematics 63 experience and achieving professional growth rather than merely emulating the host teacher as previously discussed by Hascher, Cocard, and Moser (2004). Many of the reflective statements expressed inferences about cause and effect, underlying social issues, and moral responsibilities, indi cating the professional growth experienced by the participants. Similar to Moyer and Husmans findings, our campus-based preservice teachers focused mostly on classroom management and mathematics pedagogy issues, especially student motivation to study mathematics. However, many preservice teachers advanced to also discuss childrens mathematical thinking, which is Mewborns (1999a) third level of preservice teachers concerns. This was particularly evident when they discussed the utility of manipulatives, language and vocabulary issues, real world conne ctions that students made between the activity and everyday life, challenge of concepts, and the newness of the problem-solving approach. However, because all preservice teachers in this study were campus-based rather than participating in methods classes at an elementary school, we are not able to di scern how a change of venue would affect the results as suggested by Moyer and Husman (2006). Additionally, although the instructor did not ask preservice teachers to give an example of student thinking and analyze it, this would probably facilitate preservice teachers growth and should be imp lemented in future reflection assignments. The fact that many preservice teachers were able to engage in this sort of reflective analysis spontaneously is encouraging. This analysis confirms the value of early field experiences in helping preservice teachers transition to the demanding position of teacher through authentic learning of planning and teaching of two problemsolving lessons to elementary students. References Annenberg Media. (2000). Planning a math unit: Launch-explore-summarize teaching model. In The missing link: Essential concepts for middle school math teachers (pp. 152-156). Washington, DC: A-Plus Communications and Levine Production Group. Blair, T. R., & Jones, D. L. (1998). Preparing for student teaching in a pluralistic classroom Boston: Allyn & Bacon. Brown, S. C., & Kysilka, M. L. (2002). Applying multicultural and global concepts in the classroom and beyond Boston: Allyn & Bacon. Chepyator-Thomson, J. R., & Liu, W. (2003). Preservice teachers reflections on student teaching experiences: Lessons learned and suggestions for reform in PETE programs. Physical Educator, 60(2), 2-12. Collins, A., Brown, J., & Newman, S. (1989). Cognitive Apprenticeship: Teaching the crafts of reading, writing, and mathematics, in L. Resnick (ed.), Knowing, learning, and instruction. Essays in Honour of Robert Glaser, Hillsdale, NJ:LEA Publishers. Creswell, J. W. (2005). Educational research: Planning, conducting, and evaluating quantitative and qualitative research (2nd ed.). Upper Saddle River, NJ: Pearson Merrill Prentice Hall. Dewey, J. (1933). How we think. New York: Heath. Dewey. J. (1964). John Dewey selected writings New York: Modern Library. Dobbins, R. (1996). The challenge of developing a reflective practicum. Asia-Pacific Journal of Teacher Education, 24 (3), 269-280. Garca, M., Snchez, V., & Escudero, I. (2006). Learning through reflection in mathematics teacher education. Educational Studies in Mathematics, 64, 1-17. Goodman, J. (1991). Using a methods course to promote reflection and inquiry among preservice teachers, in: R. Tabachnick & Zeichner, K. (Eds) Issues and practices Journal of Authentic Learning, Volu me 4, Number 1, Pages 43-64, June 2007

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Preservice Elementary Teachers Reflect ive Insights from Te aching Mathematics 64 in inquiry-oriented teacher education. London: Falmer Press. Gray, J. (2004). Men are from Mars, women are from Venus: The classic guide to understanding the opposite sex. New York: HarperCollins. Hascher, T, Cocard, Y., & Moder, P. (2004). Forget about theory practice is all? Student teachers learning in practicum. Teachers and Teaching, 10 (6), 623-637. Hembree, R. (1990). The nature, effects, and relief of mathematics anxiety. Journal for Research in Mathematics Education, 21, 33-46. Mewborn, D. S. (1999a). Reflective thinking among preservice elementary mathematics teachers. Journal for Research in Mathematics Education, 30(3), 316-341. Mewborn, D. S. (1999b). Learning to teach elementary mathematics: Ecological elements of a field experience. Journal of Mathematics Teacher Education, 3, 27-46. Moyer, P. S., & Husman, J. (2006). Integrating coursework and field placements: Elementary mathematics teachers connections to teaching. Teacher Education Quarterly, 33(1), 37-56. National Council for the Accreditation of Teacher Education (NCATE). National Council of Teachers of Mathematics (2000). Principles and standards for school mathematics. Reston, VA: Author. Parkinson, D. D. (2005). Unexpected student reflections from an underused genre. College Teaching, 53 (4), 147-151. Parsons, M., & Stephenson, M. (2005). Developing reflective practice in student teachers: Collaboration and critical partnerships. Teachers and Teaching: Theory and Practice, 11(1), 95-116. Pittard, M. (2003). Developing identity: The transition from student to teacher Paper presented at the Annual Meeting of the American Educational Research Association (Chicago, IL, April 21-25, 2003). (ERIC Document Reproduction Service No. ED481729) Rule, A. C. (2006). The components of authentic learning. Journal of Authentic Learning, 3 (1), 1-10. Rule, A. C., & Harrell, M. H. (2006). Symbolic drawings reveal changes in preservice teacher mathematics attitudes after a mathematics methods course. School Science and Mathematics, 106 (5), 241258. Rule, A.C., Sobierajski, M. J., and Schell, R. (2005). The effect of beauty and other perceived qualities of curriculum materials on mathematical performance. Journal of Authentic Learning 2 (2), 26-41. Schn, D. A. (1983). The Reflective Practitioner: How professionals think in action. New York: Basic Books. Stachowski, L. L., & Frey, C. J. (2003). Lessons learned in Navajoland: Student teachers reflect on professional and cultural learning in reservation schools and communities. Action in Teacher Education, 25(3), 38-47. van Manen, M. (1995). On the epistemology of reflective practice. Teachers and Teaching: Theory and Practice, 1 (1), 33-50. Vergari, S., & Hess, F. M. (2002). The accreditation game. Education Next, 2 (3), 48-57. Walsh, K., & Elmslie, L. (2005). Practicum pairs: An alternative for first field experience in early childhood teacher education. Asia-Pacific Journal of Teacher Education, 33 (1), 5-21. Wenger, E. (1998). Communities of practice: Learning, meaning, and identity. Cambridge, UK: Cambridge University Press. Journal of Authentic Learning, Volu me 4, Number 1, Pages 43-64, June 2007