ࡱ>  @bjbj >b؝؝y"*******>QQQ8:Q\Q>$nT^T`U`U`UVlW,0W$ܤR.*bVVbb**`U`U4rvvvb*`U*`UvbvvK**`UbT pTQqKCL0c\ot<0>>*****HWZv]_OHWHWHW>>LQuX>>Q RESEARCH ON THE CHARACTERISTICS OF MATHEMATICALLY GIFTED STUDENTS IN KOREA YOUNGGI CHOI, JONGHOON DO Abstract: The purpose of this paper is to make an analysis of characteristics of the mathematically gifted students in Korea through a mathematics aptitude test, a problem solving test, and the Figural Torrence Test of Creative Thinking. Key words: Creativity, Intelligence Characteristics of the gifted `!. INTRODUCTION Mathematically gifted students refer to students who have talent or potential to get creative achievement in mathematics and strong desire for mathematical exploration. Their mathematical ability and potential are valuable social assets and necessary for developing and maintaining leadership in the technical scientific society. Education and research of mathematically gifted students are needed in a social viewpoint as well as in a personal and educational one related to the self-realization of individual students (Clark,1983; House,1987). We need to perform a concrete and precise analysis on characteristics of mathematically gifted students to educate them so that they can realize their potential to the utmost. Especially, we need a long-term, empirical research to establish and administer educational policy optimized for the Korean context. There have been various researches on definition and components of mathematical giftedness by Terman, Getzels, Jackson, Renzulli, Krutetskii, etc. Especially, Krutetskii did a 12-year research from 1955 to 1966, which involved multiple methods like observation, analysis of students' problem solving process and survey to mathematicians and scientists. His research cast a lot of implications on mathematically gifted students' thinking characteristics. Lately, we are witnessing different researches on the traits of mathematically gifted students in Korea. However, it still has some limitations in its history and scope. It is urgent that we have empirical analysis and research on Korean mathematically gifted students. Education for the gifted students should be provided based on empirical research data. Mathematics division of the Seoul National University Gifted Education Center has(SNUGEC here after) been providing education for mathematically gifted students for 8 years since 1998. Students are selected through 3 steps of recommendation of his/her parents, teacher, and principal, test on mathematical problem solving ability, and interview. They are top 0.1% students in terms of mathematical ability and it has been verified that they have considerable mathematical talent. In this article we analyze intellectual, affective, and creative characteristics of mathematically gifted students with subjects attending the SNUGEC. The research questions are as follows: What intellectual characteristics do mathematically gifted students have? Are mathematically gifted students different from scientifically gifted and ordinary students in their mathematical disposition? What are characteristics of mathematically gifted students' social coping? Are mathematically gifted students different from ordinary students in general creativity? In the long term, the results of empirical research on characteristics of mathematically gifted students, including this, will be basis for decision making in the selection and education of mathematically gifted students. This paper is a translation of an article published in Korean in the Journal of Korea Society of Educational Studies in Mathematics < School Mathematics> 6(4) , 2004. a!. METHODOLOGY 1. Subject We analyzed characteristics of 8th grade mathematically gifted students who have attended or are attending the SNUGEC. The comparative groups include scientifically gifted students in SNUGEC (Physics, Chemistry, Earth Science, Biology, and Information division) and 8th graders from 4 middle schools in Seoul. 2. Data To analyze the intellectual characteristics of mathematically gifted students, we categorized their proofs into three types and three cases representing each type were analyzed. And we administered problem solving test which evaluates their intuition and ability to overcome fallacies derived from false intuition, to 17 mathematically gifted students and 57 scientifically gifted students. To analyze the affective characteristics of mathematically gifted students, we used the self-questionnaire on mathematical disposition, which is made up of three main components of confidence, preference, and task commitment on mathematics, to 109 mathematically gifted students, 64 scientifically gifted students, and 142 ordinary ones (for details of the questionnaire, see Appendix). To analyze the difference among three groups, one way ANOVA was used. In addition, we let students fill in the questionnaire on social coping, made up of personality traits, interpersonal relationship, satisfaction and adaptation to school life. To look into the general creativity of the mathematically gifted students, we carried out the Figural Torrance Test of Creative Thinking to 28 mathematically gifted and 28 ordinary students. Five values-fluency, originality, abstractness, and elaboration and their arithmetic mean were compared. To analyze the difference between two groups, one-way ANOVA was used. b!. RESULT 1. Intellectual Characteristics V.Krutetskii (1969, 1976) reported that mathematically gifted students show excellence in grasping structures and tend to abridge reasoning or problem solving process. Rigorous proof and intuition of mathematical structure underlying mathematical objects are two main aspects of mathematical discovery. V.Krutetskii (1976) reports that mathematically gifted students Intuition, one the typical representations of such thinking characteristics, refers to a cognitive process directly realized without explicit justification or interpretation. 1.1. Mathematical proposition and proof Van Hiele (1986) defined a student's thinking ability to rigorously prove a proposition as Level 3. For example, if a student can prove that three medians of a triangle meet at one point, he or she can be said to have reached the Level 3 in Van Hiele's Model of Geometric Thought. Through the different types and levels, we can estimate intellectual characteristics of mathematically gifted students. Twenty seven mathematically gifted students were asked to provide as many proofs as possible to the proposition that the sum of interior angles of a triangle is 180 degree. The proofs have been categorized into 3 types. The first type is the most ordinary method that makes use of the characteristics of parallel lines and corresponding angle(alternate interior angle), which students have learned before. Every student provided a proof identical or similar to Fig. 1.  INCLUDEPICTURE "C:\\DOCUME~1\\ADMINI~1\\LOCALS~1\\Temp\\UNI00000c3c0002.gif" \* MERGEFORMATINET  Figure 1. First type of proof The second type (Figure 2) corresponds to the 3rd grade level of the middle school (9th grade) but is not ordinary in that it is not similar to the methods students have learned. This is different from Type 1 in that it introduces a circle, not parallel lines, to prove the proposition on the sum of internal angles of a triangle. In fact, the proposition "the sum of internal angles of a triangle is 180 degree." is equivalent to the proposition "The circle is unique which passes three points that are not co-linear." Therefore, it is natural to prove the proposition using a circle which circumscribes a triangle. We cannot say that students have learned or explicitly grasped the relationship between two propositions. However, mathematically gifted students' intuition is noteworthy in that twenty three of them have provided a proof using a circle.  INCLUDEPICTURE "C:\\DOCUME~1\\ADMINI~1\\LOCALS~1\\Temp\\UNI00000c3c0005.gif" \* MERGEFORMATINET  Figure 2. Second type of proof Third type is not distinct from Type 1 and 2. It cannot be learned through the acceleration programs and is very creative in that it penetrates mathematical structure underlying the given proposition and nature of mathematical proof. This can be said to correspond approximately to Level 4 in terms of Van Hiele's Geomatric Model (Figure 3). Two of the mathematically gifted students have provided the proof as this.  INCLUDEPICTURE "C:\\DOCUME~1\\ADMINI~1\\LOCALS~1\\Temp\\UNI00000c3c0008.gif" \* MERGEFORMATINET  Figure 3. Third type of proof The proof illustrated in Figure 3 is made up of 3 steps as follows. Proves that the sums of interior angles of two triangles, which share a side, are same. Proves the sums of interior angles of any two similar triangles are the same. Given (i) and (ii). Proves it is 180 degree if the sums of interior angles of triangles are the same. This proof is unique from Type 1 and 2 in step (p!) and (q!). Students who submitted Type 1(Figure1) and 2(Figure 2) proofs paid attention to the value of 180 degree and the phenomenal fact that the sum of internal angles of a triangle is 180 degree, which corresponds to Step 3 of Type 3 proof. On the other hand, the student who submitted the Type 3 proof first raised a fundamental question "Is the sum of interior angles of a triangle constant?" underlying a phenomenal question "Is the sum of internal angles of a triangle 180 degree?" and settled it through Step 1 and 2 of the proof. In effect, the proposition that the sum of interior angles of all triangles is constant is proved through Step 1 and 2. That is, given any two triangles ABC and DEF, two similar triangles ABC and GBF have the same sum of interior angles by (q!) and two triangles GBF and DEF has the same sum of internal angles by (p!), when we overlap point B and E and place side BC and side EF on a line. Therefore, the sum of interior angles of triangle ABC and DEF is the same (Figure 4).  INCLUDEPICTURE "C:\\DOCUME~1\\ADMINI~1\\LOCALS~1\\Temp\\UNI00000c3c000b.gif" \* MERGEFORMATINET  Figure 4. The proof has a small defect but it is not a fundamental one. The whole structure and process of the proof show students' superb thinking ability to understand the mathematical structure and underlying nature, which clearly reveals the intellectual characteristics of mathematically gifted students. 1.2. Intuition and Problem Solving by Overcoming Errors by Intuition We explained the roles and importance of intuition in mathematical discovery and problem solving with some examples and analyzed students' responses to the following 2 questions. [Question 1] Some day a straight airway line from airport Seoul to airport Pyongyang is opened to traffic. A plane flies in the airway from airport Seoul to airport Pyongyang, then back in the same airway from Pyongyang to Seoul. It travels with a constant engine speed and there is no wind. Will its travel time for the same round trip be greater, less or the same if, throughout both flights, at the same engine speed, a constant wind blows from Seoul to Pyongyang? [Question 2] A watermelon 10kg in weight of which 99% are water was laid on a sandy beach. Water in this watermelon evaporated to be 98% of the watermelon. How much does this water-evaporated watermelon weigh? These two questions have been given in two turns. At the first turn, students were asked to provide the answer the question using their intuition without any operation involving formulas. Later, they were asked to fill out the answer paper using calculation using formulas. These questions can discriminate students' intuitive power to settle mathematical problems, level of intuitive errors and their ability to overcome the errors. It is very probable that students may make fault answers to both of the questions when they use their intuition based on superficial phenomenon without understanding the whole relationship and nature of given problems. Question 1Question 2l first turnsecond turnfirst turnsecond turna percentage of correct answers gifted students in mathematics8/17 (47%)11/17 (65%)9/17 (52%)9/17 (52%)`gifted students in biology0/27 (0%)0/27 (0%)2/27 (7%)5/27 (19%)`gifted students in earth science4/30 (14%)3/30 (10%)0/30 (0%)5/30 (17%) For Question 1, we can expect fault answers that state the flight time is the same when it blows and when it does not. This comes from intuition that the time spared when it blows in one direction offsets the time delayed when it does not blow. If students answered correctly without setting any formula, it means that students' thinking process, which formulates the given problem and discovers the algebraic relationship in it, took place in a second. For Question 2, we can expect fault answers that state the watermelon would be 9.9kg or 9.8 kg or around these two weights. This comes from students' naive guess that the weight of the watermelon will go through a small change which focuses on the non-fundamental phenomenon of the change of water from 99% to 98%. However, if a student gave a correct answer, he or she can be said to focus on the conservation of the mass throughout the evaporation process, not on the given change in value of water. Mathematically gifted students show approximately 50% of the ratio of correct answers at the first and second turn. The ratio of correct answers of gifted students in biology and earth science is comparatively low. Especially, biologically gifted students showed 0% of correct answers at both turns. Moreover, only 3 students set valid formulas at the second turn. Three students didn't understand the question at all and took the question for a real situation only to recognize it as a non-mathematical problem. Students admitted to the SNU Science Education Center for Gifted Students, irrespective of their division, have a great learning ability in general and also for mathematics. However, the significant difference in the results of two questions implies that mathematical thinking characteristics of mathematically gifted students are different from those of scientifically gifted students. Some argues that scientific gift is not different from mathematical gift and we must select and educate mathematically and scientifically gifted students simultaneously. However, the results of this research show that there exists a great difference in thinking characteristics of mathematically and scientifically gifted students. This will be a substantial ground on which mathematically gifted students should be differentiated from scientifically gifted students. 2. AFFECTIVE CHARACTERISTICS Mathematically gifted students hold affective characteristics like concentration and task focus in problem solving process, positive attitude towards math(confidence and preference) as well as intellectual ability. Kepler's unbelievable perseverance and focus on tasks in the process of discovering the laws of motion of planets revolving around the sun shows his genius. In addition, we need to pay close attention to mathematically gifted students' affective as well as intellectual development taking the fact that historically great mathematicians held extraordinary interest in mathematics as well as distinct mathematical ability into account. Maintaining and protecting mathematically gifted students' affective and intellectual characteristics is an essential aspect of gifted education. We classified mathematically gifted students' affective characteristics into math-related characteristics (mathematical disposition) and social characteristics (social coping). Mathematical disposition has been divided into two factors of attitude towards mathematics and task commitment. Again, the former was divided into confidence and preference. We have divided these two because one can have different thoughts on two factors because of the unique cultural characteristics of South Korea. For example, one may have interest in mathematics but lack confidence owing to excessive stress on good grades, comparison and competition with peers, etc. The vice versa also can happen. Social coping has been divided into adaptation to school life, interpersonal relationship, and personal traits related to social life. 2.1. Mathematical disposition The results of test on mathematical disposition show that mathematically gifted students, scientifically gifted students, and ordinary ones are statistically different in confidence, preference, and task commitment (See Table 1). The post-hoc analysis(Scheffe) revealed that three factors are statistically different between any two groups, that is, mathematically and scientifically gifted students, scientifically gifted and ordinary students, and mathematically gifted and ordinary students. We can see that mathematically gifted students show higher scores than scientifically gifted students, scientifically gifted students than ordinary students in mathematical disposition which is divided into confidence, preference, and task commitment. The analysis of group means show that mathematically and scientifically gifted students have as much difference in mathematical disposition as scientifically gifted and ordinary students. FactorGroupN of studentsMstdF value (Sig. level)ConfidenceM1093.718.69353.105* (.00)ConfidenceS643.308.77053.105* (.00)ConfidenceO1422.720.82053.105* (.00)PreferenceM1094.250.60381.119* (.00)PreferenceS643.453.77081.119* (.00)PreferenceO1422.8251.03381.119* (.00)Task CommitmentM1094.131.58366.129* (.00)Task CommitmentS643.519.79566.129* (.00)Task CommitmentO1422.935.96766.129* (.00)*p < .05M=Mathematically gifted students S=Scientifically gifted students O = Ordinary students Considering the uniqueness of mathematically gifted students, it is not surprising that they show high scores in mathematical disposition. However, to look more closely, 21 students (20%) scored less than 3 points in confidence and the average in confidence is far lower than the average in preference or task commitment. One-way ANOVA showed that scores of three factors are statistically different(see Table 2). Post-hoc analysis using Tukey revealed that preference and task commitment are not statistically different(p=.34>.05) while confidence and preference, confidence and task commitment are statistically different(p=.00<.05). Students involved in this research have been selected to represent their schools and then admitted to the Center through very challenging problem solving tasks and interview. Given that the point of time when students filled in this questionnaire is between their admission and first class and thus there was little possibility that students might show inferiority in the group, we need further research on which factors caused their lack of confidence. FactorMstdF value (Sig. level)Confidence3.718.69321.473* (.00)Preference4.250.60321.473* (.00)Task Commitment4.131.58321.473* (.00)`*p < .05 2.2. Social Adaptation Table 4 summarizes the results of survey on mathematically gifted students' characteristics in social adaptation. The response 'Dissatisfied' covers the average score below 2.5, 'Normal' from 2.5 to 3.5, 'Satisfied' above 3.5. Values in each parenthesis show the percentage of students who belong to the respective responses. More than 90% of the students showed scores above normal in Personality(Sociality) and Companionship and more than 60% showed overall satisfaction in their school life. The result, contrary to some arguments that mathematically gifted students are idiosyncratic and distinct from ordinary students, shows that they are generally stable in their social adaptation, which are in harmony with Gallagher(1975) and Goldberg(1965). FactorN of studentsunsatisfaction (%)satisfaction (%)very satisfaction (%)Personality (Sociality)626(9.7)26(41.9)30(48.4)Companionship626(9.7)19(30.6)37(59.7)School life6223(37.1)19(30.6)20(32.3)Total35(18.8)64(34.4)87(46.8) We need to note that6 students (approximately 10 percentage), though small in percentage, showed unstableness in Personality(Sociality) and Companionship and as many as 23 students (approximately 37.1 percentage) expressed dissatisfaction in their school life. We need to pay close attention to our students lest they fall in affective disparity or personality problems considering that mathematically gifted students' ability can contribute greatly or do harm to the technology and science-centered society depending on their ethics and values. 3. General Creativity: the Figural Torrance Test of Creative Thinking Creativity has long been regarded as one of the main components which characterizes giftedness. Many scholars have studied characteristics of the construct and how to measure it. Especially, divergent thinking has been identified with creativity for a long time (Getzels & Jackson, 1962; Guilford, 1967; Torrance, 1990,1998). Guilford(1967) divided human thinking into convergent and divergent thinking and considered creative outcomes as a consequence of divergent thinking process on a specific problem. Test of divergent thinking ability demands testees to present as many answers to a given question as possible. Basically, the score depends on the number of answers, their diversity, and the level of uniqueness of responses. The Figural Torrence Test of Creative Thinking is one of tests on outcomes produced by divergent thinking. Torrance(1984,1990,1998) has defined creativity as 5 factors of originality, abstractness, fluency, elaboration, and resistance to hasty conclusion and 13 other creative strengths. According to Torrence, originality refers to the ability to escape from routine things and produce new and unique ideas, abstractness refers to ability to derive and express abstract ideas, and fluency refers to produce as many ideas as possible. Elaboration is the ability to develop unrefined established ideas into more elaborated ones. The results of the Figural Torrence Test of Creative Thinking are summarized in Table 5. Two groups of mathematically gifted students and ordinary students showed statistically difference in uniqueness (t=2.48, p=.02<.05) and abstractness (t=2.67, p=.01<.05) while fluency(t=.03, p=.97>.05), elaboration(t=1.38, p=.17>.05), and creativity index (t=2.018, p=.05) were not statistically different. The difference of 2.5 between two groups in creativity index comes mainly from originality and abstractness while fluency and elaboration bridge the gap. We defined the average of four factors as the creativity index. This is a faithful reflection of the scoring method for the Figural Torrence Test of Creative Thinking. Considering this result, the fact that there is little difference in mathematically gifted and ordinary students' creativity index means that mathematical giftedness are not so closely correlated with general creativity. However, we can see that the two factor of uniqueness and abstractness have significant correlation with mathematical giftedness. The question whether creativity is domain-general or domain-specific is one of the main concerns in creativity research and this is directly connected with how to define mathematical creativity in discriminating mathematical giftedness. Therefore, we need to further explore what factors cause the difference between mathematically talented and ordinary students in each score. GroupN of studentsMstdF value (Sig. level)uniquenessmathematically talented2823.507.152.48* (.02)ordinary students2818.795.022.48* (.02)abstractnessmathematically talented289.074.252.67* (.01)ordinary students285.144.612.67* (.01)fluencymathematically talented2826.007.20.03 (.97)ordinary students2825.936.19.03 (.97)elaborationmathematically talented287.793.291.38 (.17)ordinary students286.542.461.38 (.17)creativitymathematically talented2816.604.582.018 (.05)ordinary students2814.103.312.018 (.05)*p < .05 IV. CONCLUSION It is necessary that we perform empirical research on characteristics of mathematically gifted students and analyze the data. In this research, mathematically gifted students' cognitive and affective characteristics have been compared with those of ordinary students using case analysis of their proof examples, test on mathematical disposition, survey on social adaptation, and the Figural Torrence Test of Creative Thinking. The results of this research about can be empirical basis on which we select and educate mathematically gifted students. Gifted education of mathematics should be planned and executed on this kind of empirical data. Analysis of proof examples illustrated that mathematically gifted students have ability to provide extraordinary proofs, which they have not dealt with in the precedent learning and some of them have intuition to understand the nature of mathematical structure and proof. Gifted education of mathematics must maintain and develop these types of talents and intuition, which can be done by 'doing mathematics.' However, it is a problem that we depend a great portion of gifted education on private education biased towards advanced learning and excessive problem solving practices. If we really want to help them develop mathematical experiences and maximize their potential, the people involved must plan and do the work through in-depth research. In terms of mathematical disposition, mathematically gifted students show positive characteristics when compared with scientifically gifted or ordinary students. They were also generally stable in social adaptation. However, it should not be overlooked, in spite of generally positive characteristics, that there exist some mathematically gifted students with lack of confidence in mathematics or weakness and unstableness in social adaptation such as adjustment to school life. It turned out that mathematically gifted students and ordinary students were statistically different in uniqueness and abstractness while their fluency, elaboration, and creativity index were not statistically different. It is needed we further study the fundamental factors which cause group differences in each factor to define the concept of creativity and explore creative characteristics of mathematically gifted students. The research results have been acquired by analysis of accumulated records about characteristics of mathematically gifted students for years. They include information as detailed as possible such as their future vision, award history, intelligence quotient, social economic status, etc and are kept up-to-date by reflecting students' recent development. The records can be empirical data for mathematically gifted students' various characteristics and foundation and procedures in which they are discriminated from the long-term perspective on educational curriculum. In addition, we can utilize the records as a ground for providing students with appropriate and individually tailored education by analyzing data about their strengths and weaknesses. We expect that the sustained accumulation and analysis of students data for scores of years will strengthen the ground for gifted education of mathematics in South Korea. Therefore it is necessary each gifted education institute keep piling and analyzing students' characteristics while conducting gifted education program. REFERENCES Clark, B.(1983). Growing up gifted : Developing the Potential of Children at Home and at School. Prentice Hall. Cosgrave, J.B.(1999). An Introduction to Number Theory with Talented Youth. School science and mathematics 99(6), 48-53. Getzels, J. & Jackson, P.(1962). Creativity and Intelligence. Wiley : New York. Gallagher, J.J.(1975). Characteristics of gifted children: a research summary, In Barbe, W.B. & Renzulli, J.S.(1975), Psychology and Education of the Gifted. NewYork: Irvington Publishers. Goldberg, M.L.(1965). Research on the talented. Teachers College, Columbia University. Guilford, J.P.(1967). The Nature of Human Intelligence. McGraw Hill : New York. Heid, M.K.(1983). Characteristics and special needs of the gifted students in mathematics. Mathematics teacher 76(4), 221-227. House, P.(1987). Providing Opportunities for the Mathematically Gifted K-12. NCTM Krutetskii, V. (1976). The psychology of mathematical abilities in schoolchildren. University of Chicago Press. Swiatek, M. (2001) Social coping among gifted high school students and its relationship to self-concept. Jour. of Youth and Adolescence 30(1), 19-39. Torrance, E.B. & Ball, O.E.(1984). Torrance Tests of Creative Thinking : Streamlined Scoring Book Figural A. Scholastic Tetsting Service. Torrance, E.B.(1990,1998). Torrance Tests of Creative Thinking : Norms-Technical Manual-figural forms A&B. Scholastic Testing Service Inc : Bensenvill, Illinois. Van Hiele, P.M.(1986). Structure and insight. Academic press, Inc. ABOUT THE AUTHORS Younggi Choi,Ph.D. Department of Mathematics Education, Seoul National University, Seoul 151-748, South Korea -mails:  HYPERLINK "mailto:yochoi@snu.ac.kr" yochoi@snu.ac.kr Jonghoon Do, Ph.D. Department of Mathematics Education, Seowon University, Cheongju, Chungbuk, 361-742, South Korea $%MNZ[\hir) W Z d   ʼyfyfy[K@hj6h5fCJaJhj6h5f5CJQJ\aJhsh5fCJaJ$hj6h5f6B*CJPJaJphhj6h5f6CJaJhj6h5f56CJaJhsh5f5CJ\aJhj6h5f5@xCJ\aJh5f5CJ\aJhj6h5f5CJ\aJh*h5f5CJ\h5f5CJ$\aJ$hnph5f5CJ$\aJ$ hsh5f5B*CJaJph%MNhiZ 2 0zG$ & F 0x`0a$gd5f $[$a$gd5f $d\$a$gd5f$d\$]^a$gd5f$[$]^a$gd5f $d\$a$gd5f 1 2 (4PX  T X l n Y"["e"f"H#p#y#z###&&&&='>'ҢҔҢҌҌҌҔ҄ҌҌn]n!hj6h5fB*CJPJaJph*jhj6h5fB*CJPJUaJphhzCJaJh5fCJaJhj6h5f5CJ\aJhj6h5f5CJQJ\aJhj6h5fB*CJaJph hj6h5fB*CJ\aJphhj6h5fCJaJ!hj6h5fB*CJPJaJph!hj6h5fB*CJPJaJph )*4PXY T Y"H#p##%$dVD\$^`a$gd5f$xVD^`a$gd5f $d\$a$gd5f $d\$a$gd5f $xa$gd5f$a$gd5f $WDd`a$gd5f$a$gd5f%%&A'_'*+;+,C-a---.d/5A6K6 $[$a$gd5f$ & F 0x`0a$gd5f 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