ࡱ> SURt @Jbjbj >؝؝x@/BBBBBBBVZZZ8dVv_:   ^^^^^^^$aRc^B0(  0(0(^BB4^@,,,0(BB^,0(^,,WDBB[ @P]Z*X&\._Hv_Yfd+dL[VVBBBB[dB[  (>",#<%,   ^^VVZt,XVVZ TEACHING THE Mathematically Gifted: A PROFESSIONAL DEVELOPMENT COURSE RONNIE KARSENTY AND ALEX FRIEDLANDER Abstract. This paper presents the pilot design of a course, aimed at increasing the pedagogical content knowledge of teachers who work with mathematically gifted students. We point at the need for direct and focused professional development programs that will address specific issues such as fostering mathematical talent and promoting mathematical creativity. We describe and exemplify course activities that relate to this purpose, and also discuss the importance of promoting task design creativity of the participating teachers. Key words: Professional development in gifted education, Mathematics teachers for the gifted, Promoting creativity INTRODUCTION There is a common agreement among researchers in the field of gifted education, that special training for teachers is essential in order to foster academic and creative skills of gifted students. (Baldwin, 1993; Hansen & Feldhusen, 1994; VanTassel-Baska, Johnsen, 2007). Hansen and Feldhusen, for example, reported that trained teachers not only fostered high-level thinking in their classes, but also "placed greater emphasis on creativity and encouragement of creative thinking [] They encouraged fluency, flexibility, originality and elaboration; asked more open-ended questions; and encouraged more risk taking than did untrained teachers in the study" (p. 119). In the USA, the National Council for Accreditation for Teacher Education (NCATE) has recently approved the new set of Teacher Education Standards for the field of gifted education (VanTassel-Baska & Johnsen, 2007). Emphasized in Standard 4 is the prospect that educators of the gifted should "enhance the learning of critical and creative thinking, problem solving, and performance skills in specific domains" (p.195). However, the question of how this goal, among other important ones, should be addressed within domain-specific professional development programs for teachers is an issue scarcely addressed by research. Looking at the domain of mathematics in particular, there is a pressing need for studies which will focus on the preparation of teachers for mathematically gifted students. While there is substantial literature on the unique characteristics of mathematically gifted students (e.g., Krutetskii, 1976;  HYPERLINK "javascript:void(0);" Sowell et al., 1990; Usiskin, 2000), as well as a profusion of classroom activities designed for these students, the issues and dilemmas that teacher educators encounter in the design of effective courses on teaching the mathematically gifted appear to be a neglected aspect of research in mathematics education. We believe that teaching mathematically gifted students is a complex and unique profession that requires (a) robust mathematical content knowledge, and (b) specialized pedagogical content knowledge (as derived from Shulman's framework, 1986). Thus, professional development programs for mathematics teachers of gifted students must specifically train for these requirements, along with training for specialization in gifted education in general, as emphasized in the NCATE new standards, as well as in other sources (Baldwin et al., 2000) In what follows, we describe a set of activities designed for a course on the teaching of mathematically gifted students. We hope that the design and evaluation of this course will shed some light on core issues in the design of similar professional development programs. In this paper, we chose to focus on the following questions: What kind of teacher activities can potentially enhance the development of specialized pedagogical content knowledge for teaching the mathematically gifted? How important is it for a mathematics teacher of the gifted to be creative as a task designer? The teacher course This section presents a brief background, means of data gathering, and a layout of the course activities. Background information The course is presently in its first semester (out of four), and is defined as a "program for promoting excellence in education". It is a two-year inservice program, conducted once in a fortnight for a full day. The participants are 21 science teachers and 12 mathematics teachers from junior high schools all over Israel. The course is designed around three main themes, translated into professional development targets: (a) exposing teachers to theoretical aspects of gifted education in general, and particularly in science and mathematics, through academic lectures and reading of papers; (b) developing leadership qualities, through workshops related to classroom contexts provided by teachers; and (c) increasing teachers' domain-specific pedagogical content knowledge, through intensive workshops conducted separately for science and mathematics teachers. It is the third theme that concerns us here. Data collection In order to assess the effect of course activities on the participant mathematics teachers' pedagogical content knowledge for teaching the mathematically gifted, and to explore the importance of creative task design skills in their professional development, we designed the following instruments of data collection: Teacher questionnaires administered at the beginning and end of the first year of the course, and at the end of the second year. In these questionnaires teachers are requested to describe their teaching practice in classes of gifted and talented students, from various perspectives. For example, teachers are asked to react to given classroom episodes involving creative answers of students. Feedback sheets completed by teachers after workshops on solving and analyzing investigative activities for the gifted. Documentation of teachers' comments and suggestions on the course web forum. Course assignments administered throughout the year (e.g., lesson plans, work sheets and various presentations). Course activities The construction of the course agenda is a dynamic process, influenced by issues and dilemmas that arise along the way. In this report, we describe five types of planned course activities: Characteristics of mathematically gifted students. Teachers get acquainted with cognitive and affective characteristics of students who excel in mathematics, and discuss what teacher behaviors should be derived from recognizing these special qualities. Example: Teachers are required to analyze videotaped or transcribed episodes of lessons or interviews with gifted students, and compare the identified characteristics of gifted students with corresponding research findings (e.g., Hong & Aqui, 2004; Krutetskii, 1976;  HYPERLINK "javascript:void(0);" Sowell et al., 1990) or with their own classroom experiences. Switching viewpoints. Following their work on mathematical activities from the viewpoint of a solver, teachers reflect on employed teaching strategies, lesson structure and implications for the classroom implementation of the same activity. Example: After work on an investigative activity, we examine to what extent the structure of the lesson followed the Launch-Explore-Summarize (LES) instructional model (Connected Mathematics Project, 1995). This three-phase model includes an introductory teacher-led class discussion on the context of the activity, followed by a student-centered investigative group work and closed by a teacher-led summarizing class discussion. The roles of the teacher and of the students are analyzed and illustrated by specific episodes taken from the previously conducted activity. Analysis of student activities. Teachers analyze a wide variety of mathematical tasks for advanced students and discuss their compatibility to gifted students in general and to the characteristics of their particular gifted student population. Following the teachers' work on an investigative activity, we examine the activity's intended purposes (in terms of mathematical content, context, level of openness, representations and sequence of sub-tasks). Design, adaptation or adoption of tasks. The role pf teachers as task developers is an unsettled issue for members of the course team and for participating teachers as well. Opinions vary from limiting the course purposes to promoting skills needed only for a critical-selective adoption of existing student activities, through allotting teachers a limited role of adapting existing activities, to encouraging and preparing teachers to design some of their own tasks. Rather than taking a stand on this issue and structure the relevant course workshops accordingly, we try to cater to a variety of levels of design creativity. However, very few course sessions are dedicated to the design of complete activities. The focus with respect to this issue is placed on workshops that require a lower level of creativity in task design - strategies for adapting routine mathematical tasks to the needs of the gifted, and acquaintance with various sources of student activities. Example: In some of the workshops, the teachers apply the What If Not? problem posing strategy (Brown & Walter, 1990) to adapt routine algebraic word problems to gifted students' higher mathematical abilities (Figure 1). In a similar workshop, the teachers develop routine algebra exercises into tasks that require students to employ higher levels of thinking skills, such as (a) reversed thinking (e.g., finding equations with a given solution, completing missing parts in a solved exercise, and finding the operators in an operation table according to given results); (b) detection of common errors (e.g., differentiating between several correct and incorrect answers in multiple choice items, assessing solutions of "virtual students", and comparing different given methods for solving a certain exercise); and (c) recognition of embedded algebraic expressions i.e., using the answer to a given exercise in order to solve other related exercises, creating categories and classifying a set of exercises (Figure 2). Classroom implementation of activities. Some of the activities presented or developed in workshops are implemented by the teachers in their groups of gifted students. The implementation process is reported and analyzed within sessions allotted specifically for this purpose and also in the course web forum. The purpose of including classroom implementation as a course activity was to raise awareness of differences between intended, enacted and attained curriculum (Clements, 2002), to strengthen the teachers' practice in work with the gifted, and to illustrate the feasibility of principles and ideas presented in the course. Future work As noted earlier, this paper presents the first stage of our study, i.e., the preliminary course design and data collection. In the next stage, we intend to analyze the data gathered throughout the course in an attempt to assess if and to what degree did the participating teachers enhance their specialized pedagogical content knowledge for teaching mathematically gifted students. What If Not? The original (standard) problem Solve the following problem: Two opposite sides of a square are extended by 4 cm. and its other sides are extended by 1 cm. As a result, the area of the new rectangle is increased by 34 cm2. What was the area of the original square? What If Not? - Examples What if it's not a square? What if the area growth is not expressed as a difference between the original and the new areas? The dimensions of a rectangle are integer numbers. Two of its opposite sides are extended by 4 cm each, and the other two by 1 cm. As a result, the area of the new rectangle is increased by 34 cm2. Find all possibilities for the area of the original rectangle. The sides of an equilateral triangle are extended by 5 cm. As a result, the area of the new triangle is 4 times as large as that of the original one. What was the length of the side of the original triangle? What if it's not the sides that get extended? Each diagonal of a square is extended by 4 cm. As a result, the area of the new square is increased by 34 cm2. What was the area of the original square? A square's diagonals are increased by 4 cm, and by 1 cm respectively. As a result, the area of the new rhomb is larger by 34 cm2, as compared to that of the original square. What was the area of the original square? Figure 1. Design of challenging problems by applying the What-If-Not? Strategy Thinking about Algebraic Skills Strategy 1 Reversed Thinking Find different ways to write the expression 900 4x2 as a product. Find different ways to complete each of the following as identities. ( _______ ) ( _______ ) = ___ + ___ + 4 ( _______ ) ( _______ ) = ___ + 3x + ___ ( ___ + ___ )2 = ___ + 3x + ___ Complete operations and possibly parentheses, to make the following identities. 15p ( 3 ( 2p = 22.5 15p ( 3 ( 2p = 2.5 15p ( 3 ( 2p = 3p 15p ( 3 ( 2p = 9p Strategy 2 Detecting Errors Mark the correct factorizations of x4 16. Add two factorizations of your own.  EMBED Equation.3   EMBED Equation.3   EMBED Equation.3   EMBED Equation.3   EMBED Equation.3   EMBED Equation.3  Strategy 3 Recognition of embedded expressions. Solve.  EMBED Equation.3   EMBED Equation.3   EMBED Equation.3   EMBED Equation.3  1 + 2x = 32 1 + 5x = 62 1 + 3x = 42 1 + 6x = 72 1 + 4x = 52 1 + 7x = 82 Figure 2. Design of non-routine exercises. REFERENCES Baldwin, A. Y. (1993). Teachers of the gifted. In K. A. Heller, F. J. Monks, & A. H. Passow (Eds.), International handbook of research and development of giftedness and talent (pp.621-629). Oxford: Pergamon Press. Baldwin, A.Y., Vialle, W. & Clarke, C. (2000). Global professionalism and perceptions of teachers of the gifted. In K.A. Heller, F.J. Mnks, R.J. Sternberg & R.F. Subotnik (Eds.), International handbook of giftedness and talent (2nd ed., pp. 565-572). Oxford: Pergamon. Brown, S. I., & Walter, M. I. (1990). The art of problem posing (2nd ed.). Hillsdale, NJ: Lawrence Erlbaum Associates. Clements, D. H. (2002). Linking research and curriculum development. In L. English (Ed.), Handbook of international research in mathematics education (pp. 599-630). Mahwah, NJ: Lawrence Erlbaum Associates.. Connected Mathematics Project (1995). Getting to know CMP. East Lansing, Michigan: Michigan State University. Hansen, J. B., & Feldhusen, J. F. (1994). Comparison of trained and untrained teachers of gifted students. Gifted Child Quarterly, 38, 115-121. Hong, E., & Aqui, Y. (2004). Cognitive and motivational characteristics of adolescents gifted in mathematics: Comparisons among students with different types of giftedness. Gifted Child Quarterly, 48 (3), 191-201. Krutetskii, V.A. (1976). The psychology of mathematical abilities in school children, Chicago: University of Chicago Press. Shulman, L.S. (1986). Those who understand: Knowledge growth in teaching. Educational Researcher, 15, 4-14. Sowell, E. J., Zeigler, A. J., Bergwall, L., & Cartwright R. M.(1990). Identification and Description of Mathematically Gifted Students: A Review of Empirical Research. Gifted Child Quarterly, 34, 147-154. Usiskin, Z. (2000). HYPERLINK "http://www.eric.ed.gov:80/ERICWebPortal/Home.portal?_nfpb=true&ERICExtSearch_Operator_2=not&searchtype=advanced&ERICExtSearch_SearchType_0=kw&ERICExtSearch_SearchValue_2=primary&ERICExtSearch_SearchValue_1=mathematics&ERICExtSearch_Operator_1=and&eric_displayStartCount=11&ERICExtSearch_SearchType_1=kw&ERICExtSearch_PubDate_To=2007&eric_viewStyle=list&ERICExtSearch_SearchValue_0=talented&ERICExtSearch_PubType=Journal+Articles&ERICExtSearch_SearchType_2=kw&ERICExtSearch_SearchCount=3&ERICExtSearch_PubDate_From=1970&pageSize=10&eric_displayNtriever=false&ERICExtSearch_Operator_3=not&ERICExtSearch_SearchValue_3=elementary&ERICExtSearch_SearchType_3=kw&_pageLabel=RecordDetails&objectId=0900000b8001efd1&accno=EJ606517&_nfls=false%20%20%20%20" The development into the mathematically talented. Journal of Secondary Gifted Education, 11 (3), 152-162. VanTassel-Baska, J., & Johnsen, S. K. (2007). Teacher education standards for the field of gifted education: A vision of coherence for personnel preparation in the 21st century. Gifted Child Quarterly, 51 (2), 182-205. ABOUT THE AUTHORS Ronnie KarsentyAlex FriedlanderWeizmann Institute of Science Rehovot 76100, ISRAELronnie.karsenty@weizmann.ac.ilalex.friedlander@weizmann.ac.il     Teaching the Mathematically Gifted: A Professional Development Course Ronnie Karsenty and Alex Friedlander PAGE 156 DG 9: Promoting Creativity for All Students in Mathematics Education, Section 2 PAGE 157 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 150 DG 9: Promoting Creativity for All Students in Mathematics Education, Section 2 %&GIJopyz ͳui]iOiu@h* 6OJQJ^JmH sH hc<h* 56CJaJhc<h* CJ]aJhc<h* 6CJaJh.h* 6CJaJh.h* 56CJaJh.h* CJaJhSh* 5CJ \aJ h* 5CJ\aJhSh* 5CJ$\aJ$h* 5;CJ$\aJ$hSh* 5;CJ$\aJ$$heKh* CJ$OJQJaJ$mH sH  h>%h* CJOJQJmH sH &IJop   $]a$gd* $d\$]a$gd* $d\$a$gd* $d\$]^a$gd* $d\$a$gd* $d\$]a$gd* d\$gd* xHJ ^       EEFghwx .><N = ""7"8"G"⭟pbbb⭟hSh* 5CJ\aJhSh* 5;CJ\aJhSh* 0J"5CJ\aJhSh* 0J#6CJ]aJhSh* 6CJ]aJ#jhSh* 6CJU]aJhSh* B*CJaJphh* CJaJhSh* CJ]aJhSh* CJaJ#hOzQh* 6OJQJ^JmH sH &  ?0$d\$]a$gd* _$ & F X0xEƀS]^`0a$gd* _$ & F X0xEƀS]^`0a$gd* .>zc$ & F X0xdEƀS\$]^`0a$gd* $d\$]a$gd* ~9c$ & F X0xdEƀS\$]^`0a$gd* c$ & F X0xdEƀS\$]^`0a$gd* <N !w"5X$ & F 0dEƀS.\$]`0a$gd* $d\$]a$gd* c$ & F X0xdEƀS\$]^`0a$gd* G"H"L"v"w"""##%%k''''6(9(u++7/]///1111<3>3k33(4)4ǿ⣗zoaRh,bh* 6CJH*]aJh,bh* 6CJ]aJh,bh* CJaJh,bh* 5CJ\aJhSh* 5;CJ\aJh* 5;CJ\aJhSh* 5CJ\aJhSh* 6CJ]aJh* CJaJh* 6CJ]aJhSh* 0J"5CJ\aJhSh* CJaJ#jhSh* 6CJU]aJ w"h#%k';+7/1111>3j___ $d\$a$gd* X$ & F 0dEƀS.\$]`0a$gd* $ & F 0d\$]`0a$gd* $d\$]a$gd* $ & F 0d\$]`0a$gd* >3K3k33V4n445667h8$ & F hhx$If^ha$gd* $ & F hex$If^e`a$gd* $x$If]^a$gdp$x$Ifa$gdp$d$If\$a$gdp )4V4n455`7a788g8h8i8j8r888888888/90999N:O:R:Ƽ}odXdXdJd jhc<h* CJaJhc<h* CJH*aJhc<h* CJaJhc<h* 5CJ\aJh* h,bh* 56CJ]aJh* 5CJaJh,bh* 5CJaJh,bh* 5>*CJaJhDh* CJhDZh* 5\ hDZh* h,bh* CJH*aJh,bh* CJaJh,bh* 5CJ\aJh,bh* 6CJ]aJh8i8j88888?99qcFF$ & F h$If^`a$gd* $<$Ifa$gdp$xx$Ifa$gdp$a$gd* $xa$gd* kkd$$IfTl J! t0644 laT9999J:^:q: $ & F <<$If^`a$gd* K$ & F <<$If^`gd* K$$ & F hx$If^`a$gd* $ & F $If^`a$gd* R:S:b:c:f:g:v:w:z:{:::::::::;; ;;;;;;;;; ;!;&;';1;3;4;5;缰簚缰q^$jRhc<h* CJEHUZaJ+j"G hc<h* CJUVaJmH sH $j'hc<h* CJEHUZaJ+j#G hc<h* CJUVaJmH sH hc<h* CJZaJ jhc<h* CJUZaJhc<h* CJH*aJhc<h* 5CJ\aJhc<h* CJaJ jhc<h* CJaJ$q:r:::eD $ & F <<$If^`a$gd* K$ & F <<$If^`gd* K$|kdM$IfK$L$l0h6R t0644 lac::::;wiJ$ & F h<$If^`a$gd* $<$Ifa$gdp $$Ifa$gdp|kd$IfK$L$l0h6R t0644 lac;;7;O;P;h;;;Qkd$IfK$L$lFhA    t06    44 lac & F ]<<$If^`gd* K$5;6;7;8;9;>;?;I;K;L;M;N;P;Q;R;W;X;b;d;e;f;g;h;i;j;o;p;z;|;};~;;;;;;;¯ؙp]$jo hc<h* CJEHUZaJ+j!G hc<h* CJUVaJmH sH $j?hc<h* CJEHUZaJ+j#G hc<h* CJUVaJmH sH $jhc<h* CJEHUZaJ+j#G hc<h* CJUVaJmH sH hc<h* CJZaJhc<h* CJaJ jhc<h* CJUZaJ$;;;;;;;;;;;;;;;;;;;;;;;;;<<<<<< < <<<~l[!jhc<h* CJEHUaJ#jJ hc<h* CJUVaJ!jhc<h* CJEHUaJ#jǫJ hc<h* CJUVaJhc<h* 5CJ\aJ jhc<h* CJUZaJ$j hc<h* CJEHUZaJ+jN#G hc<h* CJUVaJmH sH hc<h* CJZaJhc<h* CJaJ";;;;;pdV7$ & F h<$If^`a$gd* $<$Ifa$gdp $$Ifa$gdpkd$IfK$L$lFhA    t06    44 lac;;<<<4<mkd$IfK$L$Tl0 j t V0644 la}$<<$Ifa$gdpK$<<<<<<<#<$<.<0<1<2<3<5<@<A<L<M<X<Y<d<e<p<q<|<}<<<<<<˿˿˿wk_Vh* 5CJaJh* 5;CJ\aJh,bh* 5CJaJh,bh* 5>*CJaJhc<h* CJH*aJ!jVhc<h* CJEHUaJ#jëJ hc<h* CJUVaJhc<h* CJaJhc<h* CJZaJ jhc<h* CJUZaJ!jhc<h* CJEHUaJ#j˫J hc<h* CJUVaJ4<5<6<B<N<Z<s]F0$ <<$Ifa$gdp$ <<$Ifa$gdpK$$ H<<$Ifa$gdp $$Ifa$gdpkd$IfK$L$Tl0 j t V0644 la}Z<f<r<~<<<J:$d$If\$a$gdpkd$IfK$L$lFmC    06    4 lah$ <<$Ifa$gdp$ <<$Ifa$gdpK$<<<<<<<|ddN$0x7$8$H$^`0a$gd* $0d7$8$H$\$^`0a$gd* $d\$a$gd* $xa$gd* mkd$$$IfTl( J! t0644 laT<< =l=F>u>>>>>u??@%@@@@@AAAAABBBBSCiCkCmCCCFFFFFFFFGGGGGGGG货螒h=h* 5CJaJmH sH h`kh* 5CJaJh* 5CJaJhc<h* 0JCJaJjhc<h* CJUaJh* 6CJ]aJhc<h* CJH*aJhc<h* 6CJ]aJhc<h* CJaJh.h* 5CJaJ0<=>??[@@A=BBxCFGGGGgd* $xa$gd* $a$gd* $0^`0a$gd* Q$0Eƀ$Jf^`0a$gd* GGHH6H7HvHwHxHyH{H|H~HHHHHHHHHHHHHHHHII¶}j}]Mh;h k6CJOJQJaJh k6CJOJQJaJ%h8b0JCJOJQJaJmHnHu hAoIh k0JCJOJQJaJ)jhAoIh k0JCJOJQJUaJh kh>%h* CJOJQJaJh8bCJOJQJaJhJRljhJRlU h* h kh,bh* 5CJaJh,bh* 5CJ\aJh=h* 5CJaJmH sH GGHH H6H7H>Dkd$$IfTl! t644 la!T$ 4B$If]a$gdpWkdu$$IfTl0!$ t644 la!T$d$If\$a$gdp7HVHvHwHxHzH{H}H~HHHHHHu$d\$]a$gd* gd* Wkd$$IfTl0!$ t644 la!T$d$If\$a$gdp $$Ifa$gdp HHHHHSITIaIwIxIIIIJ@JAJ$ 9r  ]a$gdN$  ]a$gdN$a$gd+$'&#$+D,a$gdY$hh]h`ha$gd*$y&#$+D,gdY$a$gd* $d\$a$gd* $a$gd* I IGIRITIUI[I\I_I`IaIjIvIxIII?JAJBJHJIJLJMJNJOJUJYJ[JʵtfXfʵF#h;h k0J6CJOJQJaJh-Ch k56CJaJh-Ch k56CJ aJ h;h kCJOJQJaJh k0JCJOJQJaJ%h8b0JCJOJQJaJmHnHu hAoIh k0JCJOJQJaJ)jhAoIh k0JCJOJQJUaJh kh k6CJOJQJaJh;h k6CJOJQJaJ'h;h k6CJOJQJaJmHsHAJNJOJJJJJgd* $a$gd+$ 9r  n!&#$+Dq]a$gd+y&#$+D,gdY[JJJJJJ h* h khJRlh kh k6CJOJQJaJh;h k6CJOJQJaJ< 00&P 1hP:p8b. A!Q"Q#$%QQK$$If!vh5J!#vJ!:Vl t65J!Tk$IfK$L$!vh55R #v#vR :Vl t6,55R ack$IfK$L$!vh55R #v#vR :Vl t6,55R ac+Dd |J  C A? "2 iluHRik`!a iluHRZ` h/xcdd``ed``beV dX,XĐ )KRcgb VP=7T obIFHeA*CXoavcQfbE ri#.VN#8?‡@F0nF!d0g+_Lg M-VK-WMc`fFIo6k`5+̨& y,>eBܤ&\>R ?`u;Gv0y{@I)$5a#\UMDd dlJ  C A? "2hy>5[؍0`!hy>5[؍0h^ QxmQMK@m6 ~x E[ij=#,4Vl@sՃKO<<*GAhȆMM޼yP;(_ ,QInBT,Tf WTAhs4Q*a2.Wⱌ>Yc858vrIW𪫎>j%Ѣm;ޢL;QKgZA6ѫj wBi^q{t͉H9{2S8j;'S-,^G؝= |V@xt_b%Gݤk4]zb##$~F0 /Sh+Dd |J  C A? "2 iluHRi`!a iluHRZ` h/xcdd``ed``beV dX,XĐ )KRcgb VP=7T obIFHeA*CXoavcQfbE ri#.VN#8?‡@F0nF!d0g+_Lg M-VK-WMc`fFIo6k`5+̨& y,>eBܤ&\>R ?`u;Gv0y{@I)$5a#\Us$IfK$L$!vh5 5 5 #v #v :Vl t6,5 5 ac0Dd J  C A? "22Ev;Qu]>n`!f2Ev;Qu]>J (h4xcdd``Ved``beV dX,XĐ  A?d. znĒʂT ~35;a#L ! ~ Ay 9o ŚpM45`|-Xl0p,ĹYg|1&p{VɌ*"1B0^b)53BXI9 LT?=m.h

̅&]fi҄M&k`s) W&00ryp2pAÑ `A``bg##RpeqIj.Ft72ksJ[ Dd [J  C A? "2n=zi9[)w J `!B=zi9[)w `Shxcdd`` @bD"L1JE `xX,56~) M @ k+isC0&dT20Ufz `W018e&@] F\&-, %&P} 1p9AJ ~8$'ZZ"ɌpMքM&k`s) W&00rsFR].hq0 `PllddbR ,.Ie؈Ff:OIs$IfK$L$!vh5 5 5 #v #v :Vl t6,5 5 acDd b  c $A? ?3"`?2ԝu^iH`!ԝu^i  !"#$%&'()*+,-./0123456789:;<=>?@ABCDEFGHIKLMNOPQiTWXY[Z\^]_`acbdefhgpjklmnoqrsvwxyz{|}~Root Entry2 F@s"P]VData J0WordDocument1>ObjectPool4@P]@s"P]_1207902998F@P]@P]Ole CompObjfObjInfo  #$'*-01478:;<=>?@BCDEFGHIJKLMNOPQRSTUVWXYZ[\^ FMicrosoft Equation 3.0 DS Equation Equation.39qLDII (x"2) 2 (x+2) 2 FMicrosoft Equation 3.0 DS EqEquation Native h_1207902744 F@P]@P]Ole CompObj fuation Equation.39q\II x 2 (x 2 "16x 2 ) FMicrosoft Equation 3.0 DS Equation Equation.39qObjInfo Equation Native  x_1207902976 F@P]@P]Ole  CompObj fObjInfoEquation Native l_1207902719F@P]@P]PqI I (x 2 +4)(x 2 "4) FMicrosoft Equation 3.0 DS Equation Equation.39q8II (x 2 "4) 2Ole CompObjfObjInfoEquation Native T_1207903054F@P]@P]Ole CompObjfObjInfo FMicrosoft Equation 3.0 DS Equation Equation.39q,PII (x"2) 4 FMicrosoft Equation 3.0 DS EqEquation Native H_1254730695"'F@P]@P]Ole CompObj fuation Equation.39qtȫ܇ 2x+32+2x+33+2x+34=13 FMicrosoft Equation 3.0 DS Equation Equation.39qObjInfo!!Equation Native "_1254730686,$F@P]@P]Ole %CompObj#%&fObjInfo&(Equation Native )l_1254730699)F@P]@P]P((I x2+x3+x4=13 FMicrosoft Equation 3.0 DS Equation Equation.39q8 x  e@2 Ole +CompObj(*,fObjInfo+.Equation Native /"42+x  e@2 "43+x  e@2 "44=13 FMicrosoft Equation 3.0 DS Equation Equation.39qhT3 x+32+x+33+x+34=1_1254730691.F@P]@P]Ole 2CompObj-/3fObjInfo05Equation Native 61TableudSummaryInformation(39DocumentSummaryInformation8A3Oh+'0 $8 LX t   LABOUT SOME BASIC PRINCIPLES OF THE EXTRACULLAR WORK WITH TALENTED STUDENTSAMI Normal.dot Velikova3Microsoft Word 10.0@G@8 @  xcdd``f 2 ĜL0##0KQ* Wä (d3H1)fY{AP=7T obIFHeA*/,#.#l`0& 焉p\@ʹ `Y2s@ > 1t\> _Jv:~7 !L<< LHB^O+%\ \6T|{6Tu0"#UQK2`LMnHᲖd-ܞlphK9,L>4pS$E@.J@!+KRs@dPdk{> 1v&`*~mpDd b  c $A? ?3"`?2m<Γ-/`!m<Γ-/f  \xcdd``6f 2 ĜL0##0KQ* Wä)d3H1)fY{znĒʂT @I_@YF`b]F"L+,a K&6 & { o &NĠXdpsm^Qc@`܍;b7#MF\ L oC7XAF@`~\~-o ʄ*\>"ŽL@ W&u{a^l #la 6sAc kE:`(6221)W2ePdk{> 1&`<_$IfK$L$!vh55j#v#vj:Vl t V0655ja}Dd Db   c $A? ?3"`?2Rv i``!Rv i`> xcdd``$d@9`,&FF(`Tis A?dbA_fĒʂT @_@ u!f0i09Y@# L,& 7ӂ'fabPp,L7qs\Qc@`|w3݌`7q50y`VedE+߰phT]ꏀ.ˌ>87KaOkbnd{]PPwa W&I9p]/\!2([k o,Ly.hJFHʂ]`(b7221)W2<C `g!0cK|Dd b   c $A ? ?3"`? 2s_42ހ}v8`!s_42ހ}v8@ wxcdd``Nf 2 ĜL0##0KQ* Wä/d3H1)fY{@<~;@P5< %! 8 ĺEV0:Y@ 3Ll% Ll);LL&o &NĠXd/YQc@`܍'m)nFyA44o0p2#A|c8_į?ʊ* U}8̍Lc ?ܤ|pr8? d-?aّYQ'1Cmq.p0"{@I)$5a)ȝ @ ]` bg!X@L h˜$IfK$L$!vh55j#v#vj:Vl t V0655ja}$IfK$L$!vh5 5 5 #v #v :V l065 5 4ahO$$If!vh5J!#vJ!:Vl( t65J!T$$If!!vh5,5;#v,#v;:Vl t0655$a!T$$If!!vh5g #vg :Vl t065a!T$$If!!vh5,5;#v,#v;:Vl t0655$a!T|y@.e8] 6՜.+,D՜.+,x4 hp|  RUu X@ KABOUT SOME BASIC PRINCIPLES OF THE EXTRACULLAR WORK WITH TALENTED STUDENTS TitleX(RZN _PID_HLINKS MTWinEqnsA~2$http://www.eric.ed.gov/ERICWebPortal/Home.portal?_nfpb=true&ERICExtSearch_Operator_2=not&searchtype=advanced&ERICExtSearch_SearchType_0=kw&ERICExtSearch_SearchValue_2=primary&ERICExtSearch_SearchValue_1=mathematics&ERICExtSearch_Operator_1=and&eric_displayStartCount=11&ERICExtSearch_SearchType_1=kw&ERICExtSearch_PubDate_To=2007&eric_viewStyle=list&ERICExtSearch_SearchValue_0=talented&ERICExtSearch_PubType=Journal+Articles&ERICExtSearch_SearchType_2=kw&ERICExtSearch_SearchCount=3&ERICExtSearch_PubDate_Frol`javascript:void(0);l`javascript:void(0);   FMicrosoft Word Document MSWordDocWord.Document.89q$<@< ma NormalCJ_HmH sH tH @@@ Heading 1$$@&a$5\h@h jM Heading 2$<@&,56CJOJQJ\]^JaJmHsHtHDA@D Default Paragraph FontVi@V  Table Normal :V 44 la (k(No List 6U@6 Hyperlink >*B*ph4@4 Header  9r 4 @4 Footer  9r .)@!. Page NumberBB@2B Body Text$a$5CJ\aJROAR jM Char056CJOJQJ\]^J_HaJmHsHtHe@R ma HTML Preformatted7 2( Px 4 #\'*.25@9CJOJQJ^JmHsHtHR^@bR ma Normal (Web)dd[$\$aJmHsHtHHC@rH eKBody Text Indentx^RR@R eKBody Text Indent 2dx^TS@T eKBody Text Indent 3x^CJaJJ>@J eKTitle$a$5CJOJQJ\aJmHsHj@j = Table Grid7:V0=Footnote Text,Footnote -first4$sx5$7$8$9DH$]s^`a$@CJOJQJmHnHuHOH =italic in refs6CJOJQJkHO =reference para6$ TTd$5$7$8$9DH$^T`a$CJOJQJ_HmH sH tH ,O, = highlight0O =hlFV@F =eaFollowedHyperlink >*B* ph*W@!* * Strong5\.X@1. * Emphasis6]B&IJop   .>z~<N  whk;#7'))))>+K+k++V,n,,-../h0i0j00000?11111J2^2q2r2222223373O3P3h33333333444445464B4N4Z4f4r4~4444444445677[889=::x;>?????@@ @6@7@V@v@w@x@z@{@}@~@@@@@@@@@@SATAaAwAxAAAAB@BABNBOBBBB0000000000000000 0 0000000 0 0 0 000 00 00 0 00 00000000000 0 00 0 0000000 0 0 0?1 0?1 0?1 0 0 00 0 0000 0 0 0 00 0 0 0000 0X@0X@0X@0X@0!X@0!X@0X@0#X@0#X@0#X@0#X@0#X@0#X@0#X@0X@0X@0 00x0x0x0x0x000x0x0x0x0000x0x0x0x0x00 0 0 00 0 0 0 0 0X@00X@00X@00X@00@0X@00@0X@00@0X@00 D#@0X@00@0@0X@00@0@0X@00 mB@0@0@0@0@0X@00@0@0y@0X@00 mBX@00 Sy.1 G")4R:5;;<<GI[JJ&).1489<@BFH w">3h89q::;;;4<Z<<<G7HHAJJ'*+,-/023567:;=>?ACDEGJ(E g w 7G3333335373K3M3P3d3f3h3|3~333333333444440424;>>BXX::::::::::Xy1!!!@  @ 0(  B S  ?H0(  1'Dl El F G, H, I J K L Ml N Ol P Q, R, S, T U V, W X Yl Zx [x \x ]x ^x _,x `x alx b,x c,x dx e> f> g ? hL? i? j? z5z55666666677777(8(868@8@8I8O8 : :.: @ @(@/@jAjABB(B0BB      !"$#%&55566666777777748>8>8H8N8Y8Y8*:;:;:'@-@5@5@pApA&B.B.B?BB   !"$#%&>*urn:schemas-microsoft-com:office:smarttags PostalCode9*urn:schemas-microsoft-com:office:smarttagsState8*urn:schemas-microsoft-com:office:smarttagsdateB&*urn:schemas-microsoft-com:office:smarttagscountry-region8"*urn:schemas-microsoft-com:office:smarttagsCity9'*urn:schemas-microsoft-com:office:smarttagsplace=*urn:schemas-microsoft-com:office:smarttags PlaceType=*urn:schemas-microsoft-com:office:smarttags PlaceName 200867DayMonthYear'&'&'"''"'"'"'"'"'"'''"&'&'"&#,apsz3 = ~ z 09 5555556656=666l8u88899=:D:::x;;>>? ???@ @ @'@7@U@V@u@x@x@z@z@{@{@}@~@@@@@@@BB##j0s0004444445556666670727^7`777778[88888889::S;w;>>>?x@x@z@z@{@{@}@~@@@@@AABB33333333333333333333333333d//>0j00011J2233644?@x@x@z@z@{@{@}@~@@@@@@@aAvAB?BBBx@x@z@z@{@{@}@~@@@@@BBd< ,-H7UI^a1x̦we{R}$+]J,'w/h0&1@?3d*B*CJOJQJaJo(phhH^`OJQJ^Jo(hHo^`OJQJo(hH^`OJQJo(hHO O ^O `OJQJ^Jo(hHo^`OJQJo(hH^`OJQJo(hH^`OJQJ^Jo(hHo^`OJQJo(hHh^`OJQJo(hHh^`OJQJ^Jo(hHohpp^p`OJQJo(hHh@ @ ^@ `OJQJo(hHh^`OJQJ^Jo(hHoh^`OJQJo(hHh^`OJQJo(hHh^`OJQJ^Jo(hHohPP^P`OJQJo(hH ^`5CJo(. ^`hH. pLp^p`LhH. @ @ ^@ `hH. ^`hH. L^`LhH. ^`hH. ^`hH. PLP^P`LhH.h^`OJPJQJ^Jo(h^`OJQJ^Jo(hHohpp^p`OJQJo(hHh@ @ ^@ `OJQJo(hHh^`OJQJ^Jo(hHoh^`OJQJo(hHh^`OJQJo(hHh^`OJQJ^Jo(hHohPP^P`OJQJo(hH)/aa^a`56>*B*CJOJQJaJo(phhH.)pp^p`OJQJ^Jo(hHo)@ @ ^@ `OJQJo(hH)^`OJQJo(hH)^`OJQJ^Jo(hHo)^`OJQJo(hH)^`OJQJo(hH)PP^P`OJQJ^Jo(hHo)  ^ `OJQJo(hH L^`Lo(hH. ^`hH. pLp^p`LhH. @ @ ^@ `hH. ^`hH. L^`LhH. ^`hH. ^`hH. PLP^P`LhH.)/aa^a`56>*B*CJOJQJaJo(phhH)pp^p`OJQJ^Jo(hHo)@ @ ^@ `OJQJo(hH)^`OJQJo(hH)^`OJQJ^Jo(hHo)^`OJQJo(hH)^`OJQJo(hH)PP^P`OJQJ^Jo(hHo)  ^ `OJQJo(hHh ^`hH.^`o(.h pLp^p`LhH.h @ @ ^@ `hH.h ^`hH.h L^`LhH.h ^`hH.h ^`hH.h PLP^P`LhH.^`o(. ^`hH. pLp^p`LhH. @ @ ^@ `hH. ^`hH. L^`LhH. ^`hH. ^`hH. PLP^P`LhH.h ^`hH.h ^`hH.h pLp^p`LhH.h @ @ ^@ `hH.h ^`hH.h L^`LhH.h ^`hH.h ^`hH.h PLP^P`LhH.h^`5CJ\aJo(hH.h ^`hH.h pLp^p`LhH.h @ @ ^@ `hH.h ^`hH.h L^`LhH.h ^`hH.h ^`hH.h PLP^P`LhH.hXX^X`B*OJQJo(phhHh((^(`OJQJ^Jo(hHoh^`OJQJo(hHh  ^ `OJQJo(hHh  ^ `OJQJ^Jo(hHohhh^h`OJQJo(hHh88^8`OJQJo(hHh^`OJQJ^Jo(hHoh^`OJQJo(hH]J,w/{?3ZKE&1UIJr= N'@Lu Z8a=\d< K1xw         0        @1F        6|L&       u~y         $nK        0        z:<u~y xʰM       kDDn$FV        &0        t3lBAg5&* ma _%#q]rL:]r *$GV% &C({)O*52H86;~@$EAoIeKjM(YPU=ea6{e"j{^j kJRlxGyzcm{ n{ /8bYW;N|';==1X;.>\MUBGnpP 233=+=&o<))>+h0i000J2^2q2r22223373O3P3h333333444445464N4f4~4444????@@6@7@V@v@w@x@z@}@@@aAABNBOBBB%. ; ;q0q0@V8V8|V V8V8BP@UnknownUserLinda SheffieldGz Times New Roman5Symbol3& z ArialIFMonotype Corsiva?5 : Courier NewY Yu TimesTimes New Roman9Palatino;Wingdings#1hb lfF 6 u 6 uYQ4dX@X@ 2qHP?(YPJABOUT SOME BASIC PRINCIPLES OF THE EXTRACULLAR WORK WITH TALENTED STUDENTSAMIVelikovaL           CompObj]j