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DER-CHING YANG, ROBERT E. REYS AND BARBARA J. REYS NUMBER SENSE STRATEGIES USED BY PRE-SERVICETEACHERS IN TAIWAN Received: 12 August 2007; Accepted: 24 December 2007ABSTRACT. This study examined number sense strategies and misconceptions of 280Taiwanese pre-service elementary teachers who responded to a series of real-life problems. About one-fifth of the pre-service teachers applied number sense-basedstrategies (such as using benchmarks appropriately or recognizing the number magnitude)while a majority of pre-service teachers relied on rule-based methods. This finding isconsistent with earlier studies in Taiwan that fifth, sixth, and eighth grade students tended torely heavily on written methods rather than using number sense-based strategies. This studydocuments that the performance of pre-service elementary teachers on number sense is low.If we want to improve elementary students ’ knowledge and use of number sense, then actionshould be taken to improve the level of their future teachers ’ number sense.KEY WORDS: number sense, pre-service teachers, Taiwan R ATIONALE AND P URPOSE The importance of number sense in school mathematics has beenhighlighted by many national reports (Australian Education Council,1991; Cockcroft, 1982; Japanese Ministry of Education, 1989; NCTM, 1989, 2000; National Research Council, 1989). However, research has shown that many students in the middle grades are poor in number sense(Markovits & Sowder, 1994; McIntosh, Reys, Reys, Bana & Farrel, 1997; Reys, Reys, McIntosh, Emanuelsson, Johansson & Yang, 1999; Reys &Yang, 1998; Van den Heuvel-Paanhuizen, 1996, 2001; Yang, 2005; Yoshikawa, 1994; Verschaffel, Greer & DeCorte, 2007). Many factors, such as lack of attention to number sense in mathematics textbookstogether with heavy emphasis on rules associated with written computa-tion may account for this situation. The number sense knowledge that teachers have and the value they place on its importance are other factors.Thus children ’ s lack of number sense may be partly due to their teachers ’ lack of number sense as well as not knowing how to help studentsdevelop number sense.Reys (1994) advocates that “ teachers play an important role in buildingnumber sense in the type of classroom environment they create, in theteaching practices they employ, and in the activities they select. ” (p. 116). International Journal of Science and Mathematics Education (2009) 7: 383 Y 403 # National Science Council, Taiwan (2008) She goes on to argue that number sense becomes meaningful and valuableto students when teachers believe that developing number sense is moreimportant than mastering the rules associated with written computation.Research has demonstrated that teachers play a key role in helpingchildren develop number sense through creating a good learningenvironment that encourages children to explore numbers, operations,and their relationships freely and meaningfully (McIntosh, 2004; Siegler & Booth, 2005; Yang & Reys, 2001a , b). While research has demonstrated that teachers play a powerful role inhelping students develop number sense, there has been little research on pre-service teachers ’ number sense. Therefore, the purpose of this studywas to investigate the strategies used by pre-service elementary teachersin Taiwan when responding to number sense-related questions.B ACKGROUND What is number sense? Number sense refers to a person ’ s generalunderstanding of numbers and operations and the ability to handle daily-life situations that include numbers. This ability is used to develop flexibleand efficient strategies (including mental computation and estimation) tohandlenumericalproblems(Howden,1989; McIntosh, Reys & Reys, 1992; Reys, 1994). Number sense components? Although number sense is a relatively newterm in the Taiwanese mathematics curriculum, the emphasis on meaningfullearning and understanding has been extensively discussed, and is widelyaccepted in mathematics education (Kilpatrick, Swafford & Findell, 2001). Number sense is a complex process involving many different componentsof numbers, operations, and their relationships, and it has been the focus of research and discussions among mathematics educators, educational psychologists, researchers, and curricula developers. As a result, different psychological perspectives have been provided (Case & Sowder, 1990);theoretical frameworks of number sense proposed (Greeno, 1991;McIntosh, Reys & Reys, 1992); characteristics of number sense described(Howden, 1989; Reys, 1994) and essential components of number sense have been enumerated (Sowder, 1992; Yang, Hsu & Huang, 2004). Based on a review of the number sense literature, this study is limitingits focus on number sense to include:(1). Using benchmarks in recognizing the magnitude of numbers: It implies that children use benchmarks, such as 1, ½, 0, when dealing DER-CHING YANG ET AL. 384 with fractions and 1, 0.5, 0 with decimals to compare the magnitudeof numbers. Additionally, children do not need to rely on thestandard written methods (such as finding the least commondenominator or changing fractions to decimals). Rather, they candraw meaningful interpretations by using benchmarks or examiningattributes such as the same numerator, same denominator, andresiduals to compare fractions.(2). Knowing the relative effects of an operation on various numbers:This means that an individual should recognize how the four basicoperations interact with various numbers to affect the results. For example, when children are asked to determine the result 2435 919 ,they know 2435 is less than 1 and 919 is less than 12 , so they canconclude that the result will be less than 12 . Thus students must beable to make sense of the meaning of the operations and understandthat multiplication does not always produce larger numbers as products. The Importance of Mathematics Teachers ’ Subject Matter Knowledge Several studies report the importance of mathematics teachers ’ subject matter knowledge and assert that the mathematical knowledge teachers possess has a profound impact on what and how they teach (Ball, 1996;Ma, 1999; Schifter, 1999; Shulman, 1987; Bobis, 2004). For example, Alajmi (2004) argues that if teachers are to promote and teach students torecognize reasonable answers, the teachers must then respect and value theimportance of reasonable answers. Schifter (1999) says mathematicsteachers need to “ understand the big ideas of mathematics and be able torepresent mathematics as a coherent and connected enterprise. ” Further-more, the Principles and Standards for School Mathematics (NCTM, 2000)emphasizes that “ effective teaching requires knowing and understandingmathematics. ” (p. 17). These statements significantly underscore theimportance of mathematics teacher ’ s subject matter knowledge.Teachers play a key role in teaching number sense, and lead children tolearn and value the importance of number sense. Research in Kuwait examined one aspect of number sense, namely reasonableness of answers.This research reported low performance of eighth graders on questionsrelated to determining the reasonableness of answers to a range of mathematics problems (Alajmi & Reys, 2007). It also reported that Kuwaiti teachers valued exact answers and did not consider determiningwhether answers were reasonable an important instructional goal. Several NUMBER SENSE STRATEGIES USED BY PRE-SERVICE TEACHERS 385 studies in Taiwan have demonstrated that teachers need to have a profound understanding on number sense if they are to capitalize onclassroom events and help students develop number sense (Yang & Reys,2001a , b). Teachers need to create a learning environment that provides opportunities for students to explore numbers, ask questions, and handleunexpected student responses that arise as operations and their relation-ships with numbers are encountered. If teachers don ’ t understand themathematics and have a solid knowledge of number sense, it is unlikelythey will be able to promote number sense in their students. Investigating pre-service teachers ’ understanding of number sense will provide baselinedata that may be useful in reshaping future experiences in mathematicseducation courses for pre-service elementary teachers.M ETHOD Participants A total of 280 pre-service teachers from one Taiwanese university participated. Although all were preparing for teaching in elementaryschools, they represented six different majors, including ElementaryEducation 1 (36 pre-service teachers), Social Education (32), LanguageEducation (71), Mathematics education (19), Science Education (36), andElementary Education 2 (86). The pre-service teachers majoring inElementary Education 1, Social Education, and Language Education hadtaken BM(3) [Basic Mathematics — 3 credit hours] and MTC(2) [Math-ematics Teaching Course — 2 credit hours]. The pre-service teachers inScience Education and Elementary Education 2 also had taken MTC(2).The pre-service teachers majoring in mathematics education hadcompleted many mathematics courses (such as Calculus, Linear Algebra,and Abstract Algebra) and MTC(2). About two-thirds of the participantshad achieved senior status with ages typically ranging from 21 to 22 yearsold. Even though the pre-service teachers were from only one university,all of the students had passed a highly competitive entrance examinationin order to enter the university. Thus this group is likely to berepresentative of other pre-service teachers preparing to be elementaryteachers in Taiwan.Teachers in the elementary schools (Grade 1 to 6) in Taiwan need toteach language, mathematics, and social science, whereas science, physicor music courses are taught by specialists. Hence, no matter what their major, most participants will teach mathematics when they teach in theelementary schools. DER-CHING YANG ET AL. 386 Test Instrument The test instrument contained 12 items designed to investigate twocomponents of number sense, namely using benchmarks, and estimationin recognizing the relative effect of different operations on the magnitudeof numbers. Three mathematics educators reviewed the test items for clarity and their alignment with the number sense components. A pilot study with ten pre-service teachers who are comparable in generalacademic background to the participants in the larger study was done toensure that items were clear, appropriate, and that the test time allocatedwas appropriate. Procedure Each participant was given a testing book. Each page included one itemand ample space was provided to allow students to record the reasons for their answers. The test took about 40 min. Before the test, the researcher read the rules that were to be followed during the test. Specifically, thefollowing directions were given: 1. Participants were told to estimate or mentally compute and not to carry out a written algorithm to find an exact answer on each item; 2. Participants were asked to write an answer to thequestion and then to briefly explain how they arrived at their answer; 3.Participants were told the time on each item was controlled (3 mins), sothey should not turn to the next page without permission. Controlling thetime ensured that all students would have an opportunity to respond toeach question. The researcher monitored the test to control the pace andalso to validate that the directions, such as not executing a writtenalgorithm, were followed. Analysis The participants ’ answers and explanations were examined and classified by two raters independently. These initial reviews produced categoriza-tion agreement on over 90% of the pre-service teachers ’ responses. Theremaining contested responses were reexamined and discussed by bothraters until complete categorization agreement was reached.R ESULTS Five items and student responses on them are shown in Table I. These itemsare representative of all the open-ended questions, and the performance patterns of this subset are similar to the results on all of the items. NUMBER SENSE STRATEGIES USED BY PRE-SERVICE TEACHERS 387
