ࡱ>  @Jbjbj >؝؝K@8 lvQVlll .  PPPPPPP$RR>UP( ((Pll4P...(@llP.(P..IALlJ ]+lJ)NlP0QJVF-<V4ALVAL "l.\$$% PP.X FROM ONE END TO THE OTHER: RAISING TEACHERS AWARENESS OF MATHEMATICALLY TALENTED STUDENTS IN MIXED ABILITY CLASSES HAGAR GAL, ESTHER LEVENSON, BRURIA SHAYSHON, BERTHA TESLER, TZIPPI EYAL, NAOMI PRUSAK AND SHMUEL BERGER Abstract. "From one end to the other: Instruction of mathematically-talented students" is a new teachers' program aimed at raising teachers' awareness regarding the needs of mathematically-talented students and bridging the gap between theory and practice by combining both in the same program. This paper is divided into three parts: findings from a background study which assessed how teachers relate to talented students in their classes, a description of the "From one end to the other" teachers' program, and initial findings from the research which accompanied the program. BACKGROUND study In order to assess teachers' attitudes and beliefs regarding the teaching of mathematically-talented students, Shayshon & Tesler (2008), members of the program research team, handed out questionnaires to 52 teachers. These questionnaires investigated teachers' self-efficacy regarding the teaching of mathematically-talented students, their beliefs regarding talented students, and the ways in which teachers address the needs of talented students in their classes. Teachers' sense of self-efficacy as found to have a mean score of 4.45, on a scale of 1-6 (SD = 0.63). Teachers believed that mathematically-talented students must be high performers. Less than a quarter of the teachers believed that mathematically-talented may be slow learners or may even have learning disabilities. More than half of the teachers claimed to meet the needs of mathematically-talented students by providing these students with challenging activities. Seven teachers claimed they "try to" deal with talented students. Six teachers claimed not to pay any special attention to these students. Ten teachers related negatively to talented students claiming that these students were often bored and displayed behavioral problems in class. In general, teachers did not actively seek out talented students. Instead, if a student's mathematical talent was obvious, the teacher attempted to meet his needs in various ways, some more appropriate than others. A closer look at some of the teachers' statements (see Table 1) suggested that teachers were not entirely aware of the types of activities recommended for talented students such as: raising new questions, investigating problems, evaluating results (Sheffield, 2008), exploring, and developing mathematics new to the student (Novotna, 2008). In other words, teachers were not necessarily aware of the mathematically-talented students learning in their midst nor were they knowledgeable of the types of activities appropriate for challenging these students in order to promote creativity and curiosity while advancing the students' mathematical knowledge. Table 1: Different approaches to meeting the needs of mathematically-talented students in the classroom students in the classroom At least once a weekAt least once a monthAt least once in half a yearNever Statement52%28%6%14%Enabling self-advancement in the current material7.8%13.7%9.8%68%Permission to be engaged in material other than mathematics 39%39%17%3.9%Giving more complicated activities in the subjects being studied 50%36%4%10%Directing talented students to help a weak student42%25%4.2%29%Giving more exercises at the same level24%42%24.5%8.2%Providing mathematics activities beyond the standard curriculum About the program The project "From one end to the other: Instruction of mathematically-talented students" (an explanation of the name of the project can be found in Gal & Levenson, 2008) which is conducted by the Mathematics and Physics department in the David Yellin College of Education and supported by the Municipality of Jerusalem, is based on the belief that talented students should have their needs met in regular classes (Reed, 2004). The results of the teachers' questionnaires described above supported our intention to establish the following foundations for the program: Raising awareness - The success of teachers' training programs is related to teachers' previous knowledge and beliefs (e.g. Kagan, 1992, Borko & Putnam, 1996). Raising teachers' awareness regarding the needs of mathematically- talented students may be seen as a necessary first step towards incurring a positive change in beliefs. Increasing knowledge Another basic aim was to extend teachers' mathematical as well as pedagogical content knowledge (PCK) specifically related to the instruction of mathematically-talented students Combining theory and practice - By combining both theory and practice in the same program, we hoped to overcome the gap usually developed between these two areas (Aitken & Mildom, 1991). Mastering Participants in the program were presented with tasks designed specifically for talented students learning in heterogeneous classes. These tasks encouraged participants to practice and master new skills in a safe environment. Program structure Three parallel frameworks were created to meet the needs of this program: (1) Teachers' course. (2) Children's after-school enrichment program. (3) Research team. Participants in the teachers' course included 18 pre- and in-service mathematics teachers. The pre-service teachers were in their 3rd or 4th year of a program, training to be either elementary school or junior high school mathematics teachers. The in-service teachers were all experienced elementary or junior high school mathematics teachers. All participants met once a week for two hours. Participants were exposed to both cognitive and social theories regarding mathematically-talented students. This raised a general awareness regarding the needs of such students. More specifically, by contending with activities designed for mathematically-talented students, including original activities written for this program, participants became aware of the possibility of different ways of solving the same problem as well as the possibility of having multiple solutions. In other words, tasks were designed in order to promote creative thinking. Furthermore, observing the children under "lab conditions" would enable participants to track the student's creative thinking, raising the teachers' awareness of the need to supply students with appropriate tasks. The childrens after-school enrichment program was used as the teachers practice field, exposing participants to the creative and often unpredictable ways in which talented students cope with mathematical problems. Teachers joined the children's programs six times a year, observing and interacting with the students as they worked at solving the tasks. Alternatively, they could choose to visit the children's program every week. The children's enrichment program included two after-school work-shops for mathematically-talented students. One group consisted of 3rd graders. The second group consisted of 7th and 8th graders. Tasks were designed in such a way as to be easily adjustable for various learning situations: whole class discussions, small group work, as well as individual study (Prusak & Levenson, 2008). The participant teachers reflected on these observations, tracked changes in their own attitudes, awareness, and coping skills, which in turn we hoped would begin the process of change (Gal, 2005). first results The research team which accompanied the program aimed at describing and assessing the ongoing process of change incurred by participants. Several questionnaires were handed out to participants throughout the first semester of the program. This paper focuses on two important junctions: the mid-semester and end-of-semester questionnaires. As suggested by the names, the first questionnaire was handed approximately mid-way through the first semester of the academic year. The second questionnaire was handed out at the end of this semester. Both questionnaires contained the following three questions: What was the best thing that occurred to you during the previous month's meetings? What, if any, new insight did you gain from the course during this month? Do you plan to apply what you learned in your classroom? How? In addition, the end-of-semester questionnaire asked participants to reflect on any change which they felt occurred during the term under consideration. Using the foundations of the program as a basis for analysis, results of the questionnaires were categorized according to statements which referred to the following three aspects of the program: awareness, knowledge, and mastering. Results are summarized in Table 2. Table 2: Frequency of different aspects noted by participants per questionnaire MasteringKnowledgeAwarenessAspectmathematical tasks repertoiremathematical/ PCK of talented studentsof talented studentsas a teacheras a learner420628Mid-semester 410966End-semester  The findings suggest that mid-semester participants mainly reported on changes in pedagogical content knowledge and pedagogical knowledge related to talented students. It was as if the teachers' newly acquired knowledge regarding talented students was necessary in order to fill a gaping hole in their previous knowledge. By the end of the semester, participants were more attuned to the students in the "field and their observations became more reflective of both the students and themselves as teachers. Changes were also noted in teachers' beliefs and approaches towards teaching mathematics to talented and gifted students. The following are some examples related to each aspect of the program. Awareness: From a learner perspective: "I was made to think!; Ive started to be more creative and open to non-standard situations.; I am now convinced that it is worth checking simple, specific cases which in turn helps me to make generalizations. From a teachers' perspective: Ive acquired self-confidence. Though I myself am not gifted, I can certainly build scaffolding for my students and help them develop the talent within. Awareness of talented students: I was exposed to other ways of thinking, different than mine.; My perspective, while observing the students in their course, was changed. We usually pay attention to those students with learning difficulties and less to students who master the subject matter.; Teaching is not an easy job! The class is heterogeneous and each of the students needs special attention according to his/her level. Knowledge: Changes in mathematical or general pedagogical knowledge regarding talented students I appreciate the importance of (students) co-operative group work, and how students contend with (tasks).; It (the teachers' course) organized my knowledge regarding the traits of talented students. Mastering: I experienced interesting tasks.; The repertoire of tasks appropriate for my (talented) students, has increased." When teachers were asked if they plan to apply their newly acquired knowledge in their classrooms, one participant responded, "Certainly, I already have." Another participant who in the middle of the semester responded that the tasks were not relevant for her classes, responded differently by the end of the semester, "Certainly, I have already used the task with talented 6th graders." Combination of theory and practice: Although no direct question regarding the combination of theory and practice was posed on either questionnaire, some participants did refer to their observations of the children's after-school enrichment program. As one participant noted, I could see, as an outside observer, someone elses lesson, observe the learning processes of talented students as well as their limitations. SUMMING UP and looking ahead This paper reported on the first semester of the "From One End to the Other" program. Asked to reflect on any changes that were noted during this short time period, two teachers responded: "I learned that talented students are happy to face challenges My self confidence grew. I feel that I am able to challenge these students with tasks appropriate for their ability." "I feel a great change. I am aware of the need to plan tasks given to them (the talented students). I make an effort to get into their heads in order to be better prepared for their answers and solutions. All pre-service mathematics teachers should take this course. These teachers are in the process of changing. They are aware of the special needs of mathematically-talented students and have begun themselves to master new skills. As the course continues into its second semester we are focusing on mathematical problems which promote mathematical creativity in the students as well as in the teachers. Teachers then bring these tasks to their classes, providing their talented students with appropriately challenging tasks. Although currently the program is defined as a one-year program, looking ahead, we hope to meet with the participants during the following year in order to assess their work with mathematically-talented students, as well as to provide them with necessary encouragement and support in their endeavors to meet the needs of mathematically-talented students in their class. REFERENCES Aitken, J. & Mildon, D. (1991). The dynamics of personal knowledge and teacher Borko, H. & Putnam R. T. (1996). Learning to teach. In: D. C. Berliner & R. C. Calfee (Eds.) Handbook of Educational Psychology. New York: Macmillan Library Reference. Gal, H. (2005). Identifying Problematic Learning Situations in Geometry Instruction, and Handling Them within the Framework of Teacher Training. Thesis submitted for a Doctor of Philosophy degree. Jerusalem: Hebrew University of Jerusalem (Hebrew). Gal, H. & Levenson, E. (2008). From one end to the other: Instruction of mathematically talented students in mixed-ability classes. In: R. Leikin (Ed.) Proceedings of the 5th International Conference Creativity in Mathematics and the Education of Gifted Students. Haifa, 323-327. Kagan D. M. (1992). Professional growth among preservice and beginning teachers. Review of Educational Research. 62(2), 129-169. Novotna, J. (2008). Activities enhancing gifted children's creativity and reasoning. In: R. Leikin (Ed.) Proceedings of the 5th International Conference Creativity in Mathematics and the Education of Gifted Students. Haifa, 188-192. Prusak, N. & Levenson, E. (2008). Developing mathematical tasks suitable for gifted elementary school students. . In: R. Leikin (Ed.) Proceedings of the 5th International Conference Creativity in Mathematics and the Education of Gifted Students. Haifa, 385-387. Reed, C. (2004). Mathematically gifted in the heterogeneously grouped mathematics classroom: What is a teacher to do? The Journal of Secondary Gifted Education, 15(3), 89-95. Shayshon & Tesler (2008). Does it make a difference? Studying the impact of teachers' program aimed at meeting mathematically talented students' needs. In: R. Leikin (Ed.) Proceedings of the 5th International Conference Creativity in Mathematics and the Education of Gifted Students. Haifa, 217-221. Sheffield, L. (2008). Questioning mathematical creativity questions may be the answer. In: R. Leikin (Ed.) Proceedings of the 5th International Conference Creativity in Mathematics and the Education of Gifted Students. Haifa, 29-34. ABOUT THE AUTHORS Hagar Gal, Ph.D, Head of Mathematics & Physics Department, E-mail:  HYPERLINK "mailto:hagarg@dyellin.ac.il" hagarg@dyellin.ac.il Esther Levenson, M.A., Mathematics & Physics Department,  HYPERLINK "mailto:levensone@gmail.com" levensone@gmail.comE-mail: Bruria Shayshon, Ph.D., Mathematics & Physics Department,  HYPERLINK "mailto:brurya@013.net.il" brurya@013.net.ilE-mail: Bertha Tesler, M.Sc., Mathematics & Physics Department,  HYPERLINK "mailto:tbertha89@gmail.com" tbertha89@gmail.comE-mail: Tzippi Eyal, M.Sc., Mathematics & Physics Department, E-mail:  HYPERLINK "mailto:tzippy@macam.ac.il" tzippy@macam.ac.il Naomi Prusak, M.Sc., Mathematics & Physics Department,  HYPERLINK "mailto:inlrap12@netvision.net.il" inlrap12@netvision.net.ilE-mail: Shmuel Berger, M.Sc., Mathematics & Physics Department,  HYPERLINK "mailto:shmuel@openu.ac.il" shmuel@openu.ac.ilE-mail: David Yellin College of Education, Jerusalem, Israel  PAGE 137 PAGE 137 PAGE 137 PAGE 137 PAGE 137    Hagar Gal, Esther Levenson, Bruria Shayshon, Bertha Tesler, Tzippi Eyal, Naomi Prusak and Shmuel Berger From One End to the Other: Raising Teachers' Awareness of Mathematically- Talented Students in Mixed-Ability Classes PAGE 148 DG 9: Promoting Creativity for All Students in Mathematics Education, Section 2 PAGE 149 ICME 11, Mexico, 2008 Proceedings of the Discussing Group 9 : Promoting Creativity for All Students in Mathematics Education The 11th International Congress on Mathematical Education Monterrey, Mexico, July 6-13, 2008 PAGE 141 ICME 11, Mexico, 2008  ?@fg~y 1 2 zlzlzZ#hk6;OJQJ]^JmH sH hkhk6CJ]aJhkhk6CJaJhkhk56CJaJhkhkCJaJhk5CJ aJ hkhk5CJ aJ hKhK5CJ$aJ$hk5CJ$\]aJ$hK5CJ$\]aJ$hKhK5CJ$\]aJ$ hkhk hkZhkCJZ_H aJo( @g1 2 $d\$a$gdk$Pd\$]P^a$gdk$GdA$\$`Ga$gdk$dA$\$a$gdK$dA$\$a$gdk $A$gdk$dA$\$gdkKHJ2 C  Vl7=>Hd$&`#$/If\$gdk$d$&`#$/If\$a$gdk hd\$^hgdk $d\$a$gdk $d\$a$gdk 2 C `elmHX0@smn9TM^Ǻ㲦㛓}rjrh[ CJaJh~h~CJaJh-h~CJaJh-h-CJaJh-CJaJh~hkCJaJhkhk5CJaJhkCJaJhkhkCJ_H aJhkhkCJZ_H 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