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The number sense of preservice elementary school teachers.

The purpose of this study was to explore cognitive process used by selected preservice elementary teachers on basic types of problems involving number sense. The sample was selected from students in six intact entry-level mathematics sections of a course populated by preservice elementary school teachers. In the beginning of the semester, the instrument used to collect data was the Number Sense Test (NST). The score from the NST was analyzed. Six participants were randomly selected for individual interviews from students who scored in the top 10% and six participants whose scores ranked in the bottom 10% on the NST to be interviewed. All students interviewed in this study do so voluntarily and completed data collection tasks during the Spring 2002 semester for the study. According to the interview data, the data indicated that items including fractions were more difficult for low ability students than decimals or whole numbers. The low ability students tended to use rule-based methods more often than the high ability students. Furthermore, the low ability students relied on standard written algorithms more than reflecting number-sense-based methods. High ability students applied number sense twice as often as low ability students.

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Schneider and Thompson (2000) note that a student who has good number sense has a good understanding of number meanings and numerical relationships and is flexible in thinking about numbers. Number sense also includes the ability and inclination to use this understanding in flexible ways to make mathematical judgments and develop useful strategies for handling numbers and operations. (Reys & Yang, 1998; McIntosh al., 1999). The development of number sense is important in mathematics education. The National Council of Teachers of Mathematics, in their Principles and Standards for School Mathematics, note that number sense is one of the foundational ideas in mathematics in that students (1) Understand number, ways of representing numbers, relationships among numbers, and number system; (2) Understand meanings of operations and how they related to one another; (3) Compute fluently and make reasonable estimates. (NCTM, 2000, p.32).

The few studies that have investigated the mathematical understanding of elementary teachers and preservice elementary teachers indicate that many exhibit weakness in mathematics, may misapply mathematical rules, do not understand true meanings of mathematical concepts, and that they are, generally, not prepared to teach the mathematical subject matter entrusted to them (Cuff, 1993; Hungerford, 1994). In a the study of pre-service teachers' understanding of the operation of division, Ball (1990) found that their understanding relied on rules and was unrelated to other mathematical operations. The data reveals that preservice teachers apply well-ingrained whole number rules, instead of weakly understood fraction and decimal concepts, to draw false conclusion about rational number representations, such as 0.45 is greater than 0.5 because 45 is greater than 5 (Ball, 1990).

Johnson (1998) conducted a study yielding more evidence that prospective teachers' general number sense and rationale number concept knowledge are inadequately developed. These students, resist looking at mathematics in creative, non-algorithmic ways. The participants were asked to generate solutions using mental arithmetic and then explain why their answers were correct without resorting to algorithmic procedures. Further analysis of responses identified common misconceptions held by prospective elementary majors which included:

1. The belief that the fraction having the larger denominator is always large;

2. The belief that two fractions that are almost equal are equivalent;

3. The confusion about decimal place value;

4. The use of flawed algorithms, such as multiplying fractions by using a common denominator and multiplying numerators; and

5. The belief that area models must be rectangular or regular in order to find a fractional portion.

In that elementary teachers provide the first formal mathematical training children receive, it is reasonable that the educators responsible for preparing them to teach should know what skills they possess and what skills they lack in order to design their curriculum. It is also reasonable to assume that, if the perceptions and misconceptions teachers possess are addressed during preservice training, or even during inservice training, their teaching performance will be strong.

The Purpose of the Study

The purpose of this study was to study what cognitive processes do preservice elementary school teachers use when asked to solve problems involving number sense. The research question requires a qualitative approach. The problem-centered interview was applied to answer the research question. The categorization of student strategies, and field notes from each interview, were used to describe each student and each problem. Each student's strategies were investigated in an attempt to explain the approach used in number sense problems by that student, and each of the interview problems were matched with the strategies used by all the interviewees in order to determine if there were certain strategies that were preferred for certain problems.

Review of Literature

The literature reviewed for this study will describe characteristics of number sense, researches related to number sense, and the misconception of number sense.

Characteristics of Number Sense

Characteristics of number sense has been discussed by researchers (Case, 1989; Greenes, Schulman, & Spungin, 1993; Greeno, 1991; McIntosh, Reys, & Reys, 1992; NCTM, 1989; Resnicks, 1989; Reys, B., 1994; Reys et al., 1991; Schoen, 1989; Sowder, 1989, 1992a, 1992b, Reys, Resys, Nohda & Emori, 1995; Reys & Yang, 1998; McIntosh, B. Reys, & R. Reys, 1992; Sowder & McIntosh, 1994; Yang, 1997). The theoretical framework of number sense for this study is adopted from research by mathematicians and educators. To summarize, the characteristics of number sense included: well-understood number and symbol meanings, the ability to decompose/recompose numbers, recognition of the relative and absolute magnitude of numbers, having the ability to use benchmarks, and flexibility while applying knowledge of numbers and operations to computational situations (including mental computation and computational estimation).

Well-Understood Number and Symbol Meanings

The most important single element of number sense is an understanding of numbers (Sowder and Schappelle, 1994). One cannot make sense of numbers without attaching meaning to them. Heibert (1984) believes that a meaningful understanding of the Hindu-Arabic number system and its structure are essential. The NCTM Standards (2000) states that children must understand number meanings if they are to make sense of the ways that numbers are used in real-life situations. Hiebert (1989) has noted that children are often surprisingly proficient at solving mathematical problems outside of school because in these settings the context provides a means for understanding what they are doing. These same children when given more formal school-related tasks are often unable to complete the task because they cannot relate the formal or written symbolism to their own informal conceptualizations (Hiebert, 1989).

Mack (1990) examined the development of students' understanding of fractions by looking at ways students are able to use their informal knowledge to give meaning to symbols and procedures associated with fractions. Her study "suggests that knowledge of rote procedures interferes with students' attempts to construct meaning algorithms " (p. 30). The degree of a student's understanding is determined by the number, accuracy, and strength of connections. A concept is well understood if it has many links to other aspects of knowledge that are accurate and strong (Hiebert and Carpenter, 1992).

Decomposition/Recomposition Numbers

The ability to decompose and recompose numbers is a behavior which demonstrates the presence of number sense (Greeno, 1991; Resinck, 1983; Sowder, 1990; Reys, Reys, Nohda & Emori, 1995; McIntosh, B. Reys, & R. Reys, 1992; McIntosh & Sowder, 1994; Kaminski, 1997, Yang 1997).

"Decomposition/recomposition of numbers involves expressing a number in an equivalent form as a result of recognizing how this equivalent form facilitates operating on the recomposed numbers" (McIntosh, Reys, & Reys, 1992, p. 6). It also includes the ability to flexibly use different representations to display numbers and to appropriately choose useful representations to execute the computation. For example when considering 240 x 0.5, 1/2 can be substituted for 0.5 or it can be decomposed in the following manner: 24 x 5 to 6 x 4 x 5, or 6 x 20 in order to facilitate mental computation.

Resnick (1983) noted that the notion of unique partitioning occurs during the initial stage of learning multi-digit numbers when children are presented a number as the composition of a tens values and a ones value in standard form. Later, multiple partitioning of multi-digit numbers occurs when children use the nonstandard form such as 57=4 tens 17 ones or 3 ten and 27 ones. Resnick (1986) posits that many powerful mental thinking strategies build on compositions of smaller numbers. She believes that a great number of informal strategies invented by students utilize decomposing a number to the sum of smaller numbers.

Ross (1989) stated "Pupils need to engage in problem-solving tasks that challenge them to think about useful ways to partition and compose numbers " (p. 50). She found such activity having a reciprocal effect on developing understanding of place value and the numeration system. The ability to view 7 tens and 21 ones enables a child to decompose the dividend in a useful way to calculate 91 as 7 tens and 21 ones divided by 7, or 1 ten and 3 ones, or 13.

Carraher, Carraher, & Schliemann (1987) found in their study that students used the method of decomposition in the study of oral mathematics. For example, when asking students to compute 252-57, one subject decomposed the 252 into 200 + 52, and decomposed 57 into 52 + 5. Both 52 were deleted, and there remained another five to be taken away from 200 (p. 92). They found that "the decomposition heuristic was used at least once by each child both in addition and subtraction", and 85% of them were correct.

Recognizing the Relative and Absolute Magnitude of Numbers

Number magnitude has been identified frequently by many researchers and reported (Case, 1989; Greene et al., 1993; Greeno, 1991; Hope, 1989; Howden, 1989; NCTM, 1989; Markovits, 1989; McIntosh, Reys, & Reys, 1992; Resnick, 1989; Reys, B. 1994; Reys et al., 1991; Schoen, 1989; Silver, 1994; Sowder, 1989, 1992a; Markovits & Sowder, 1994; Threadgill-Sowder, 1988; Van De Walle, & Watkins, 1993, Yang, 1997, McIntosh et al., 1999) as an important component of number sense. Number magnitude can be divided into two areas: relative magnitude of numbers and absolute magnitude of numbers.

Understanding a number as a quantity of specific magnitude and being able to judge how it compares to another number is basic to number sense (Sowder, 1988). A sense of the size of a number including the ability to compare and order numbers is a corner stone of number sense. For example, students know that place value of 5 is more than place value of 3, or 1359 is less than 1500. This characteristic also includes a general sense of large numbers, often using benchmarks to reveal their size. Markovits and Sowder (1994) state that "understanding number magnitude within a number domain encompasses the abilities to compare numbers, to identify which of two numbers is closer to a third number, to order numbers, and to find or identify numbers between two given numbers" (p. 6).

Sowder and Markovits (1989) believe that a mathematics curriculum devoted to more work on number magnitude can increase the understanding of numbers and the number system. For example, research has indicated that "percent" is a difficult topic in the middle grades' mathematics curriculum (McGivney & Nitschke, 1988). Sowder and Markovits believe that meaningful understanding of the size of fraction and decimal numbers can help students in developing number sense in general.

Sowder and Schappelle (1994) state that "knowledge of relative and absolute number size is essential to judging the reasonableness of computation" (p. 343). For example, students who have difficulty ignoring the 5 in estimating 15394 + 9581 + 2568 + 5 show an inadequate understanding of the relative magnitude of numbers; or students who find it difficult to place the decimal point in the product by using a mental estimation of 50000 x 0.05 display an inadequate knowledge of number magnitude. These examples may give students a sense of the orderliness of numbers and the regularity of the number system (McIntosh, Reys, & Reys, 1992) and also promote a discussion of relative and absolute size.

Relative magnitude of numbers is the ability to compare and to order numbers (Sowder, 1992b). Several research studies have shown that, particularly for rational numbers, children have an inadequate ability or understanding to determine relative size (Peck & Jencks, 1981; Hiebert & Weame, 1986; Sackur-Grisvard & Leonard, 1985). Sowder and Wheeler (1987) found that most students before tenth grade were unable to correctly compare 5/6 and 5/9. The study results of Peck and Jencks (1981) demonstrate similar poor performance for comparing fractions such as 2/3 and 3/4. In fact, some students thought the two fractions were equal, because "there are the same number of pieces left over" (p. 344). A frequent strategy cited by students to compare fractions is to compare numerators if the denominators are equal, and denominators if the numerators are equal (Behr, Wachsmuth, Post, & Lesh, 1984; Sowder & Markovits, 1989; Sowder & Wheeler, 1987 ; Reys , Reys, Nohda & Emori, 1995). Sowder and Markovits (1989) found that instruction focusing on the concept of fraction number size could improve students' understanding of fractions. Behr, Wachsmuch, Post, & Lesh (1984) and Behr, Wachsmuth, & Post (1985) also found that with appropriate instructions through an extended period of time, most children have the ability to develop suitable strategies to compare and order fractions.

Sowder (1992a) believed that if children want to compare decimal numbers accurately, then they must have a good understanding of whole number place value. Hiebert and Weame (1986) found that nearly half of sixth and seventh graders chose 0.1814 as the largest decimal among 0.09, 0.385, 0.3, and 0.1814 by using the method of more digits makes bigger. This finding supports the research of Sackur-Grisvard & Leonard (1985) that children choose 3.63 as the larger when giving 3.63 and 3.8, because 63 is larger than 8. Children incorrectly based their judgement of the relative magnitude of numbers on the number and value of the places in the decimal position of the number, paying little attention to the digits in these places or to the overall meaning of the number (Resnick, 1987).

The absolute magnitude of numbers is the ability to make sense of the general magnitude of a given number (McIntosh, Reys, & Reys, 1992). Children at lower grade levels often have little understanding of the meaning of very large numbers, such as 1 million for students in grades 3 or 4. In the study of McIntosh, Reys, & Reys (1997), children in grades 2, 4, and 6 were asked "About how many days have you lived?" and given the choices "A. 300 B. 3000 C. 30000 D. 300000." They had to select one answer that made sense to them. In grade 2, forty-two percent of the children chose 300, and thirty-two percent of the children chose 30000 or 300000. In grade 4, there were forty percent and twenty-two percent of children who chose 30000 and 300000, respectively. Even in grade 6, forty-three percent and twenty one percent of the children chose 30000 and 300000, respectively. The finding of this study indicated that a very high percentage of children could not make sense of absolute number size in real life situations (McIntosh, Reys, & Reys 1997).

Sowder (1992a) posits that it is difficult for children to make sense of large numbers if they do not have opportunities in an instructional setting to explore big numbers. Saxe (1988) found that unschooled Brazilian children could manage large numbers with the Brazilian money system. Zaslavsky (2001) stated that students can begin to appreciate the Hindu-Arabic numeration system better as they learn about, and experience, the many ways that people have counted and recorded numbers throughout the development of society over a period of thousands of years. Students not only gained in their understanding of other mathematical systems but also developed greater appreciation of the fact that all people count Zaslavsky (2001).

The Use of Benchmarks

Benchmarks are described by the phrase, "a compass provides a valuable tool for navigation, numerical benchmarks provides essential mental referents for thinking about numbers" (McIntosh, Reys, & Reys, 1992, p. 6), For example, using 1 as a benchmark, the sum of 6/7 and 14/15 should be near 2 and less than 2, because both of the fractions are a little less than 1. McIntosh, Reys, & Reys (1992) state that "benchmarks are often used to judge the size of an answer or to round a number so that it is easier to mentally process" (p. 6).

Trafton (1989) believes that percentages should be taught with an early emphasis on the meaning of percent using estimation and mental computation as vehicles for developing the concept of percents. In particular, Trafton (1989) posits that particular attention should be given to establishing key benchmarks such as 10% and 50%, performing mentally computations with these benchmarks, estimating percent using benchmarks, and finally calculating exact percents.

Resnick (1989) believes that the use of benchmarks is to use well-known number facts to figure out facts of which one is not sure. In the study of skilled and less skilled estimators' strategies for estimating discrete quantities, Crites (1992) found that successful estimators tended to use decomposition/recomposition and multiple benchmark strategies, whereas less successful estimators generally used perceptually based strategies.

Flexibility in the Application of Number Senses and Operations to Computational Situations

McIntosh, Reys, & Reys (1992) stated that "solving real world problems which require reasoning with numbers involves making a variety of decisions including : determining what type of answer is appropriate (exact or approximate), determining what computational tool is efficient and/or accessible(calculator, mental computation, computation estimation, etc.) choosing a strategy, applying a strategy, reviewing the data, and result for reasonableness". The researchers see the use of mental computation and estimation as necessary.

Teaching basic facts has always been a part of any successful mathematics program and is very important in developing mental math skills and flexibly applying estimation skills. (Leutzinger, 1999). Leutzinger (1999) addressed that too much time is spent on repetitive practice instead of exploratory experiences, which give students the opportunity to develop thinking strategies. Calvert (1999) agrees that practicing procedures without thought or understanding enforces a dependence on standard paper and pencil algorithms. Mental math strategies and flexibly applying the knowledge of number senses and operations to computational situations, need to be developed from conceptual understanding rather than memorized rules (NCTM, 1995).

McIntosh, Reys, & Reys (1992) state that "fully conceptualizing (the concept of) operation implies understanding the effect of the operation on using numbers including whole and rational numbers" (p. 7). The NCTM Standards (1989 a) highlight that "operation sense also involves acquiring insight and intuition about the effects of operations on two numbers" (p. 43). For example, children should sense that the sum of two numbers, each of which is greater than 50, must be greater than 100.

Behr (1989) characterized the effects of operations on number in two respects. One is to understand how to compensate when one or more operations is changed in an computational question; for example, if 248-189 is 59, then what is 258-189. The other is to understand when the result of a computation remains the same after changing the original numbers computed; for example, 252-194 = 58 can be used to find 247-189. In the study conducted Reys, Bestgen, Rybolt, and Wyatt (1982), the process of reformulation and compensation were identified by the researchers as common processes used by good estimators.

Misconceptions of Number Sense

Understanding one's own knowledge of number sense indicates the depth of acquisition of the concept. A number of studies have focused on the children's understanding of number sense (Mack, 1990; Markovits & Sowder, 1991; Reys, Resys, Nohda & Emori, 1995; Reys & Yang, 1998; Yang, 1997), but few focus on the teacher's understanding of rational numbers. Simon and Blume (1992) have targeted learning about the teaching of the how and why of rational number concepts. However, teachers need to understand how each piece of rational number knowledge fits into the larger mathematical meaning system children have already developed, they need to know how previously learned procedures may block the learning of later material, and to clearly discriminate new elements from those that are similar to old elements during the student's instruction and evaluation (Behr, Lesh & Post, 1981; Mack, 1990).

Children's and adults' misconceptions are often difficult to uncover. They are often hidden behind computationally correct answers, but answers that might be given for one of several wrong reasons. Teachers also fail to analyze why particular mistakes were made and thus miss information critical to appropriate remediation (Galbraith, 1995).

Teachers with the awareness of potential problem areas can design assessment that reveals the presence of the known misconceptions. Teachers, however, are hardly aware of such analyses and generally are unable to use such knowledge in planning, instruction, and evaluation of student performance (Mack, 1990; Fuller, 1996). Different curricular sequences produce different patterns of rule invention and different misconceptions. Misconceptions arise from the students' effort to integrate new material into previously established knowledge structures (Resnick et al., 1989). Different difficulties occur when students use fractions and when they use decimals. Some common difficulties associated with fractions follow.

1. Students have difficulty with the meaning of operations. In whole number arithmetic, division is understood to mean making smaller, and multiplication is understood to mean making larger. The rules for addition, subtraction, multiplication, and division of fractions are easily confused and misapplied. (Ball 1990; Mack 1995).

2. Students have difficulty interpreting mixed numerals correctly. They may not interpret the number 4(1/2) to mean 4+1/2, but rather as 5/2 (Ball, 1990).

3. Students have difficulty associating meaning with rational numbers expressed as fractions. Students may not be able to recognize that the units compared must be the same size, and that the numerator and denominator do not refer to distinct regions of a closed figure. (Nik Pa, 1989).

4. Students have difficulty translating one rational number model, whether verbal, pictorial, or symbolic, into another. Fractions as symbols representing rational numbers may not be associated with real world situations and, therefore, may not have practical implications than merely computation. (Johnson, 1998).

5. Rational number meaning may not allow judgement of reasonable results. For example, when fair sharing or equally dividing 7 yards of ribbon between 4 kids, many students do not recognize 4/7 yards as an unreasonable answer, whereas 7/4 yards is the length of ribbon each would receive. (Johnson, 1998).

Teaching for meaning takes longer than teaching that produces correct answers. However, teaching for meaning is more efficient in the long run in that it avoids the student's construction of erroneous rules (Ball, 1990, Nik Pa, 1989, Mack, 1995)

Johnson (1998) found that preservice elementary teachers have a gap in their rational number understanding and that they rely on the use of algorithms when approaching non-standard problems. The misconceptions they exhibit tend to be similar across different representations of rational numbers. The findings of Rasch (1992) and Hungerford (1994) suggest that preservice elementary school teachers exhibit difficulties with rational numbers that may be indicative of a lack of intuitive conceptual understanding of the meaning and properties of the number system. Thus, the scope of number sense was restricted to the understandings that could be derived mentally, without resorting to computation, rules, or algorithms. Believing that understanding the level of number sense should play an important role in pre-service teaching programs, the motive for conducting this study rises from a deep concern for the development of number sense for preservice teachers.

Methodology

Population and Sample

The population of this study consists of preservice elementary school teachers at a mid-sized, four-year, state university in a mid-sized town in the Rocky Mountain region. The sample was composed of students in six intact entry-level mathematics sections of a course populated by preservice elementary school teachers. In the beginning of the semester, the instrument used to collect data was the Number Sense Test (NST). The score from the NST was analyzed. Six participants were randomly selected for individual interviews from students who scored in the top 10% and six participants whose scores ranked in the bottom 10% on the NST to be interviewed. All students interviewed in this study do so voluntarily and completed data collection tasks during the Spring 2002 semester for the study.

Data Collection Procedures

At the beginning of the semester, the Number Sense Test (NST) was given to all classes. Calculator use was allowed. The score from the NST was analyzed. Six participants were randomly selected for individual interviews from students who scored in the top 10% and six participants whose scores ranked in the bottom 10% on the NST to be interviewed.

Number-Sense Test

During the first week, a 25-item Number Sense Test (NST) was given to the students. Students were given a copy of the NST and instructed not to begin work until told to do so by the researcher. The researcher and instructors were provided with general instructions and answer questions from students. Students were asked to obey the rules of this test: timing per item is about 45 seconds and students were told not to spend too much time on any one question. The Number Sense Test (NST) was developed by Yang (1997) for grade 6 and 8 students in Taiwan. The 25 item NST includes whole number, fraction, and decimal items as well as the four basic operations. According to Yang, the split-half reliability of the NST is over 0.80 for both 6th and 8th grade of students. Table 1 shows two items.

The NST included items representing three number dimensions: Whole Numbers, Decimals, and Fraction numbers. Table 2 displays the percents of correct responses and standard deviations on the NST by number domains for the one hundred fifty-five participants. The pretest data show that the number domain of Fraction percent of the Number Sense Test was relatively low when compared with Whole Number (42.5%) and Decimal (43.81%).

Number Sense Interview

The Number Sense Interview Instrument (for example see Table 3) includes 14 items. The researcher presented one item at a time. Probing and other types follow-up questions were asked in an effort to gain a clearer understanding of the students' thinking. The interviews were audio taped and recorded. The interviewer were also recorded the responses and explanations made by the students, recording limited written notes. After each interview, the researcher documented field notes with respect to the subject's response during the interview. In order to insure consistency of measuring the characteristics of "number sense", the researcher recoded each taped response. The interview procedure displayed a series of number sense items as a means of drawing out responses built on the characteristics of number sense. The interview was conducted in the mixed-method form and the questions were not change that based on the response of the subjects. However, subjects were be encouraged to talk aloud and explain their answers.

Number Sense Interview Instrument

The interview provided additional perspective of number sense possessed by high- and low-performance students. The interview explored five characteristics of number sense which have been identified by Yang (1997). They include: having the ability to decompose/recompose numbers; recognizing the relative and absolute magnitude of numbers; having the ability to use a benchmark; understanding the relative effects of operations on numbers; and flexibly applying the knowledge of numbers and operations to computational situations (including mental computation and computational estimation). The Number Sense Interview Instrument (NSII) was developed by Yang in 1997. The NSII was individually administered and will include 14 questions. Table 4 shows the framework of items by number domain and characteristics of number sense.

Analyzing Data

The categorization of student strategies, and field notes from each interview, were used to describe each student and each problem. Each student's strategies were investigated in an attempt to explain the approach used in number sense problems by that student, and each of the interview problems were matched with the strategies used by all the interviewees in order to determine if there were certain strategies that were preferred for certain problems. The problem-centered interview was characterized by three central criteria: problem centering, i.e. the research orientation to relevant problems; object orientation, i.e. that methods are developed or modified with respect to an object of research; and finally process orientation in the research process and in the understanding of the object of research (Witzel, 1985).As suggested by Strass and Corbin (1998), the research looked for key issues and recurrent themes and events in the data and form categories of focus. The research viewed data analysis in this study as an "ongoing cyclical process integrated into all phases of qualitative research ... a systematic process of selecting, categorizing, comparing, synthesizing, and interpreting" (McMillan & Schumacher, 1989, p. 414).

The researcher reviewed the interview tapes in order to make extensive comments about the use of characteristics of number sense possessed by the subjects. When listening to each tape, the researcher transcribed and reviewed the transcripts to document characteristics of number sense displayed by those interviewees. The researcher randomly chose one high and one low ability students' responses to recode. In Yang's (1997) study, reliability between the two time recoding was 99%.

Results

The interview results will be presented in the following order: subjects' responses, the difference of each characteristic used between subject of high and low abilities.

Subjects' Responses

The following number sense interview items were designed to detect several characteristics of number sense: Number magnitude, using benchmarks, decomposition/recomposition of numbers, relative effect of operations on numbers, and flexibility with numbers and operations to computational situations. The correctness of each answer was analyzed and classified into one of the following categories: (a) explanation reflecting one or more of the characteristics of number sense, (b) rule-based explanations, (c) correct answers, but the explanations were incorrect or does not have any explanation and (d) incorrect answers and explanations.

Score on group administered was used to categorize students into either high or low ability, depending on whether they score within the top 10% or bottom 10% respectively, on the Number Sense Test. Fourteen interview questions were used to understand the students' thinking process in the use of characteristics of number sense. The first four questions examined number magnitude; questions five to seven examined use of the benchmark; questions eight to ten examined the flexibility of numbers and operations in computations; questions eleven and fourteen examined the relative effect of operations on numbers; and questions twelve and thirteen focused on the decompositive/recomposition of numbers. Each item was to reflect a certain characteristic of number sense; each characteristic is reported in a single table.

Number Magnitude

Number magnitude items are related to number size, comparing and ordering numbers, and understanding the density of rational numbers. Table 5 shows the results of students' responses to the number magnitude items between the high and low group students. Students' responses to number magnitude items were judged according to their answer and explanations.

Item 1: How many different decimals are there between 1.42 and 1.43?

All high ability students responded correctly. Their responses all reflected the infinite concept. For example, "Decimals can be infinitely extended, for example, 1.421, 1.4211, 1.42111, ... Number can be added infinitely after 2 of 1.42. These decimals can be extended to many different decimals between 1.42 and 1.43..... 1.421, 1.422, ... 1.4211, are between 1.42 and 1.43.", "Infinite. Because decimals have tenth, hundredth, thousandth, ... and so on digits, these numbers are infinite. For example, 1.421, 1.422, ... 1.4211, 1.4212, ..." These types of explanations illustrated the correct answer. Their explanations were classified as the use of number magnitude. If responses were correct, the explanations were classified as the use of number magnitude and typically provided two different types of explanations.

All students in the low ability group gave incorrect responses. Two different answers were given. Three students of low ability believed that there were only nine possibilities. For example, "There are nine. 1.421, 1.422, ... and 1.429 are between 1.42 and 1.43.", "The number following 1.42 is 1.43., there was no decimal numbers between 1.42 and 1.43.". The responses differed significantly between the high and low abilities in answering item 1. The low ability students had difficulty in understanding that there is an infinite number of decimal numbers between 1.42 and 1.43 and were unable to support their response with at least one example. Three other students in the low ability grouping believed there were no decimals between 1.42 and 1.43.

Item 2. How many fractions are there between 2/5 and 3/5?

Three of six students of high ability and two students of low ability answered this item correctly. Those responding correctly stated that an infinite number of fractions exist between 2/5 and 3/5. Explanations included "2/5 = 20/50 = 200/500, 3/5 = 30/50 = 300/500 are fractions between 2/5 and 3/5". I can multiply the same number to the denominators and the numerator. Hence there are many fractions between 2/5 and 3/5. The two students in the low ability group who responded correctly gave answers that were similar to those given by students of high ability. For example, "Infinite. Since I changed 2/5 and 3/5 to be 0.4 and 0.6 so there are a lot of decimals between 0.4 and 0.6." The students who responded incorrectly were unable to provide accurate explanations and gave responses such as "no fractions between 2/5 and 3/5" or "I do not know".

Item 3. Order 0.595, 61%, 3/5, 5/8 and 0.3562.

Only one-fourth of the students (two from high ability, one from low ability) responded correctly. One student provided this explanation: "I would like to change these numbers to be decimals, and compared them. 61% is equal to 0.61, 5/8 = 0.625, and 3/5 = 0.6, then 5/8 > 61% > 3/5 > 0.595 > 0.3652."

Another student in the high ability group responded: "I changed them to fractions with common denominators, 0.595 = 595/1000 = 5950/10000, 0.3562 = 3562/10000, 5/8 = 0.625 = 625/1000 = 6250/10000, 3/5 = 0.6 = 6000/10000, 61% = 6100/10000, so now I can compare all of their numerators. 0.3562 < 0.595 < 3/5 < 61% < 5/8. The remaining three-fourth of the students responded incorrectly.

Five of the students in the low ability group naturally separated fractions from decimals when they ordered these numbers. This indicated that these students considered fractions and decimals as different entities and they did not necessarily make any connections between them. The proportion of correct responses between high ability and low ability is two to one, this revealed that fraction problems were more difficult for the low ability students. This supports the results of the NST. Approximately 83% students in the low ability group could not connect their understanding of decimal numbers with fractions and could not make sense of the relationship between fractions and decimals.

However, comparison of number sets which included fractions and decimals proved more difficult for the students. Only two students in the high ability group, and one in the low ability group, were able to convert fractions to decimal forms or used strategies in which common denominators were used to compare the fractions and decimals. The researcher also found two students in the high ability group and four students in the low ability group were not able to do long division for 5 / 8.

Item 4a: Which letter in the number line names a fraction where the numerator is slightly more than the denominator?

[ILLUSTRATIONS OMITTED]

Four of the six students in the high ability group and three of the six students in the low ability group responded correctly. The typical correct response from students from both ability groups was to estimate the fraction as a rationale: "D is about 1 1/4 which is equal to 5/4 and 5 is slightly more than 4. Hence, D is the answer." Another type explanation given by a high ability student was "Since the numerator must be greater than the denominator a little, this fraction is larger than 1 and near 1. D is bigger than 1 and near 1, hence D is the answer." Some students who responded inaccurately were unable to provide a rationale and frequently answered with "I don't know."

The high ability students appeared to have developed a stronger understanding of fractional number size than those students in the low ability group. Additionally, the data indicates that fractional items which include the use of relative fractional number size, seemed to be more difficult for the low ability students than for those students in the high ability group.

Item 4b: Which letter in the number line names a fraction where the numerator is nearly twice the denominator?

Five of the high ability students and two of the low ability students were able to correctly answer this question. Several different explanations were provided. Students from both ability groups stated that "the numerator is nearly twice the denominator. If the denominator is 5 then the numerator is about 10. F is about 2 1/4 = 9/4, and 11 is about twice of the denominator 5, hence F is the answer." A student from the low ability group responded, "F is about 2.1 and 2.1 is nearly twice of 1, hence F is the answer." Finally, another high ability group student said, " is the answer. The reason is similar to the Item 4 (a). F is near two, then the numerator is nearly twice of the denominator."

Four in the low ability group could not answer this question. All of these students answered "I don't know" or were unable to provide an appropriate reason for their guess.

The Use of a Benchmark

The use of a benchmark focuses on the use of or basic facts, multiples of powers of 10 a unit number, such as ten, one or one-half as a referent to make sense of the results. Table 6 shows the results on items designed to identify the use of a benchmark.

Item 5: Without calculating an exact answer, 72 x 0.46 is more than 36, or less than 36?

Five of the six students in the high ability group and two of the six students in the low ability group used 1/2 or 0.5 as a benchmark to decide the answer. Three different explanations were observed. High ability students provided an explanation involving the use of fractions such as "Less than 36. Because 72 x 1/2 = 36, 0.46 is less than 1/2, then 72 x 0.46 is less than 36." and "Because one half of 72 is 36. Since 0.46 is less than a half, then 72 x 0.46 is less than 36." Secondly, both high and low students were able to use decimals as part of their explanation. For example, "Since 72 x 0.5 = 36 and 0.46 is less than 0.5, therefore 72 x 0.46 is less than 36."

However, two in the low ability group misunderstood the concept believing that multiplication makes an answer larger. This student's conceptualization is consistent with the previous studies of Graeber & Tirosh (1990) and Greer (1992, 1994). For example, "Greater than 36, 72 is much greater than 36 and it is multiplied by the other number. Since a number is multiplied by another number, the result is larger. Therefore, 72 x 0.46 is greater than 36." Five-sixth high ability and one- third of the low ability students could use the benchmark 1/2 or 0.5 when determining that 72 x 0.46 is less than 36.

Item 6:6 2/5 / 15/16 is:

A. more than 6 2/5

B. less than 6 2/5

C. equal to 6 2/5

Two students in the high ability group and two of the six students in the low ability group gave incorrect explanations. For example, "6 2/5 / 15/16 can be changed to a format like 6 2/5 x 16/15 and 16/15 is larger than 1, therefore the result is greater than 6 2/5."

Students in both high and low ability groups gave, examples such as "The division should be changed to be multiplication. 6 2/5 / 15/16 = 6 2/5 x 16/15 after the computation by paper-and-pencil, the answer should be greater than 6 2/5." Explanations were classified as rule-based because the students did not use a benchmark, but rather employed either the standard computational rule or a memorized method.

In both high and low abilities student stated, "When a number is divided by another number, the result should be less than the original number. Therefore, 6 2/5 divided by 15/16 is less than 6 2/5." This was judged as an incorrect answer.

Only two of the six students in the high ability provided correct explanations. However, only two students in high ability showed use of a benchmark. "If a number is divided by a divisor which is less than 1, then the computational result is greater than the original number. When a number is divided by a divisor which is larger than 1, the answer is less than the original number. Since 15/16 is less than 1, then 6 2/5 / 15/16 is greater than 6 2/5." The results indicate that high ability students are better at using a benchmark than the low ability students. However, both groups of students frequently tended to use standard written algorithms to answer this question.

Item 7 a) : Without calculating an exact answer, the sum 5/11 + 3/7 is :

A. more than 1/2

B. less than 1/2

Two of the six students in the high ability group and none of the students in the low ability group used a benchmark to answer this item. The following explanations reflected the use of a benchmark: "Both 5/11 and 3/7 are less than 1/2 but it is close to 1/2. So I think their sum is larger than 1/2." One of the six students in the high ability group and two out of six students in the low ability group responded: "5/11 + 3/7. I need to find the common denominator. It is to equal 77, then 35 + 33/77 = 68/77. So sum of these two fractions is more than 1/2. " This thinking process produced a correct answer but the justification was judged as rule-based. Some students in both groups answered: "Both 5/11 and 3/7 are less than 1/2. Therefore their sum is less than 1/2." The answer 4 + 1/2, but rather as 5/2 (Ball, 1990).

3. Students have difficulty associating meaning with rational numbers expressed as fractions. Students may not be able to recognize that the units compared must be the same size, and that the numerator and denominator do not refer to distinct regions of a closed figure.(Nik Pa, 1989).

4. Students have difficulty translating one rational number model, whether verbal, pictorial, or symbolic, into another. Fractions as symbols representing rational numbers may not be associated with real world situations and, therefore, may not have practical implications than merely computation. (Johnson, 1998).

5. Rational number meaning may not allow judgement of reasonable results. For example, when fair sharing or equally dividing 7 yards of ribbon between 4 kids, many students do not recognize 4/7 yards as an unreasonable answer, whereas 7/4 yards is the length of ribbon each would receive. (Johnson, 1998).

Teaching for meaning takes longer than teaching that produces correct answers. However, teaching for meaning is more efficient in the long run in that it avoids the student's construction of erroneous rules (Ball, 1990, Nik Pa, 1989, Mack, 1995)

Johnson (1998) found that preservice elementary teachers have a gap in their rational number understanding and that they rely on the use of algorithms when approaching non-standard problems. The misconceptions they exhibit tend to be similar across different representations of rational numbers. The findings of Rasch (1992) and Hungerford (1994) suggest that preservice elementary school teachers exhibit difficulties with rational numbers that may be indicative of a lack of intuitive conceptual understanding of the meaning and properties of the number system. Thus, the scope of number sense was restricted to the understandings that could be derived mentally, without resorting to computation, rules, or algorithms. Believing that understanding the level of number sense should play an important role in preservice teaching programs, the motive for conducting this study rises from a deep concern for the development of number sense for preservice teachers.

Methodology

Population and Sample

The population of this study consists of preservice elementary school teachers at a mid-sized, four-year, state university in a mid-sized town in the Rocky Mountain region. The sample was composed of students in six intact entry-level mathematics sections of a course populated by preservice elementary school teachers. In the beginning of the semester, the instrument used to collect data was the Number Sense Test (NST). The score from the NST was analyzed. Six participants were randomly selected for individual interviews from students who scored in the top 10% and six participants whose scores ranked

Item 8: Closest estimate of 38 x 86 is:

A. 40 x 90

B. 40 x 86

C. 38 x 90

Two students in the high ability group responded correctly to this item. All of them provided a similar rationale for their choice. For example, "because the difference of 86 and 90 is 4, then 38 x 90 is 38 x 4 more than 38 x 86. The difference between -38 and 40 is 2, so 40 x 86 is 2 x 86 more than 38 x 86. Since 4 x 38 is equal to 152 which is less than 2 x 86 = 172, 38 x 90 is the closest estimate." This type of justification illustrates an appropriate explanation. This response shows flexibility with numbers and operations in computational situations, an understanding of multiplication, and understanding the number magnitude of error when mental calculating or estimating products by rounding was done.

The responses for the rest of the students in both abilities did not reflect this flexibility with numbers and operations. This typical explanation was given by both types of students: "In 40 x 90, both 40 and 90 increased, 40 is 2 more from 38 and 90 is 4 more from 86, so their difference is 6. In 40 x 86 and 38 x 86, both equations including 86 are equal, and the difference of 40 and 38 is 2. However, in 38 x 90 and 38 x 86, both including 38 are equal, but the difference of 86 and 90 is 4. Therefore, 40 x 86 is the answer."

The current data reveals that one-third of the students in the high ability group could use the "multiplicative strategy; that is, they considered the effect of the rounding on the final product (Markovits & Sowder, 1994, p. 19)." These high ability students considered that the change of multiplier or multiplicand would affect the result in the final product. No students in the low ability could make sense of this problem. Students in the low ability grouping used a strategy in which they looked only at the differences between the original numbers and the numbers in the choices presented. The data also indicates that the determination of an exact answer in which the result would be more or less than the given estimate was a difficult task for all the students, but particularly challenging for those in the low ability group.

Item 9: A school bus holds 45 students. If 915 students are being bused to a museum, how many buses are needed?

All of the students knew they needed to use division to work this problem. However, one-half of the high ability students were able to use a mental method to correctly answer this problem. Typical correct responses from both high and low ability group included: "Because 915 divided by 45 is about 20, and the remainder is 15. Fifteen students also need a bus. Hence, 21 buses are needed."

"Because 45 time 2 is equal to 90 and 90 x 10 is equal to 900, then we need 2 x 10 = 20 buses. The remaining 15 students also need a bus. Therefore, we need 21 buses."

A typical incorrect response given by students in both groups was, "Because 915 divided by 45 is about 20. Hence, 20 buses are needed." This explanation was judged as incorrect. Results indicate that all but one student could make sense of their computational results regarding the problem posed by this interview item. This student did not understand that the remaining students also needed a bus to bring them to the museum. The current result is consistent with Silver's findings (1994) "The failure to solve such problems successfully in school contexts appears to be directly related to students' failure to make sense of their computational result with respect to the problem situation."

Item 10: Without calculating an exact answer, circle the best estimate for 9965 + 8972 + 8138 + 8090 and explain your answer.

A. 24000

B. 30000

C. 36000

D. 42000

All students in the high ability group and five of the six students in the low ability group correctly answered this item. Several types of different explanations were observed among these students. High ability students were able to group compatible numbers such as : "8972 plus 8138 is about 17000, and 9965 plus 8090 is about 18000. Then 17000 plus 18000 is about 35000. Therefore, 36000 is the best answer." Also a couple of high ability students utilized the front-end method (left to-right addition but involves only the leading digit of each number) with adjustment reformulation (change of numbers in an item to a mentally manageable form). For example, "9 + 8 + 8 + 8 is 33, and the remaining nine hundred plus nine hundred is about 1800. Then the sum of 33 thousand and one thousand eight hundred is 34800. Therefore, the sum is near 36000." Another method utilized by high ability students is translation (change the mathematics structure of an item). "There are three eight thousands, then the sum is 3 x 8000 = 24000. The 24000 + 9000 = 33000. Then the sum should be over 33000. C is the best answer."

Both high and low ability students applied the rounding method (rounding a number to a certain place). A typical response was: "9965 is about 10000, 8972 is about 9000, and 8138 and 8090 are near 8000. Then 10000 + 9000 + 8000 + 8000 is equal to 35000. Hence, 36000 is the closest estimate."

Only one student answered incorrectly. This particular student, in the low ability group, responded: "Because the sum is over 30000 a little. Hence, I think B is the answer." The student added up the sum, but the answer was wrong.

Interview data reports few differences in the responses to this interview item between high and low ability students. However, some different key processes in response to this interview item were consistent with those found in the study of Reys, Bestgen, Rybolt, and Wyatt (1982). Three students in the high ability group and five students in the low ability group used the front-end method.

Five of the six of the low ability students and nearly half of the high ability students appeared to rely heavily on the straight rounding method (rounding a number to a certain place). The results indicated that there were no differences in the use of estimation strategies and explanations for this item between the high and low groups. However, high ability students showed evidence of better number sense by flexibly ignoring smaller numbers (one-digit or two-digit numbers). They proficiently grouped compatible numbers and more consistently arranged numbers throughout the estimation procedure to accommodate their rounding strategies better than students in the low ability group. High ability students seemed to be more capable in dealing with numbers than the low ability students. These students were more flexible in using different strategies often such as reformulation, grouping by compatible number, or translation. The low ability students tended to primarily use the rounding method, which was taught in the mathematics course they were enrolled in during this study.

Relative Effect of Operations on Numbers

Items designed to demonstrate knowledge of the relative effect of operations on numbers are shown in Table 8. Table 8 also shows results related to relative effect of operations on numbers items observed in the interviews.

Item 11: Which of the following is correct:

A. [96.sup.2] = (100 + 4)(100 - 4) - [4.sup.2]

B. 38 x 42 = 40 x 40 + 6

C. 48 = (4 x 2) + (4 x 2)

D. 2 1/7 x 5 = (2x5) + (1/7 x 5)

Three students in the high ability group and two students in the low ability group provided correct responses similar to this : "By using the distribution property because 2 1/7 x 5 is equal to (2 + 1/7) x 5, and then equal to (2 x 5) + (1/7 + 5), hence D is the answer." One student in the high ability group used a standard written algorithm and rule-based approach : "D is the answer, because 2 1/7 x 5 is equal is to 15/7 x 5 = 75/7 = 10 + 5/7. Then it is equal to 2 x 5 + 1/7 x 5."

Problems requiring the identification of which equation is correct by using the knowledge of the relative effect of operations and numbers were more difficult for the low ability students than for those in the high ability grouping. One of the six students in the high ability group and three of the six students in the low ability responded, "I don't know." or gave incorrect answers without explanations.

This situation also occurred with the high ability students. Even though five students in the high ability group determined D was correct and C was incorrect, they tried to recall the formula which they learned from a mathematics class. However, they still were unable to explain why A and B were incorrect. Results also indicated that comparing 38 x 42 and 40 x 40 + 6 was difficult for all of the students. Students were able to apply a standard written algorithm, but they did not try to use more efficient mental strategies in order to determine an answer. Clearly the evidence shows that standard written algorithms directly affect the mathematical thinking strategies for these students. The determination of whether the exact answer would be more than or less than the approximation was also difficult for all of the students.

Item 14: Fill in the blanks with appropriate integers so that the computational result is equal to 100. That is ( ) + ( ) ( ) - ( ) x ( ) = 100

Three of the six students in the high ability group and two of the six students in the low ability group gave a correct response. Three typical explanations demonstrate the order of operations rules.

* "0 + 108 x 1 - 16 / 2 = 100. Multiplication and division should be calculated earlier than addition and subtraction in an equation. Hence, 11 x 9 and 25 - 5 should be calculated first." (High ability group)

* "100 + 2 x 50 - 200 / 2. First, I start 2 x 50 = 100 and 200 / 2 = 100, then decided the number in the first blank. Because multiplication and division must be calculated earlier than addition and subtraction." (High ability group)

* "22 + 108 - 18 / 9 = 22 + 80 - 2 = 100. If an equation includes four basic operations, then multiplication and division should be calculated first. Hence, I decided the numbers in the first three blanks ( ) + ( ) x ( ) and then decided - ( ) / ( ) to make the answer is 100. For example, I decided 22 + 10 x 8 = 22 + 80 first, then decided 18 / 9 = 2." (Low ability group).

Three of the six students in the high ability group and four of the six students in the low ability group had solutions judged as incorrect. Typical explanations included stating "I don't know", or giving incorrect examples such as "50 + 50 x 2 - 0 / 2 = 100."

Results indicate that the high ability students had a better conceptual understanding of the order of the four basic operations when dealing with number problems. Low ability students tended to do the computation from left to right, not considering which operations should be calculated in advance. Results also indicate that the high ability students more flexibly use their knowledge of numbers and operations to build their answers and deal with numbers than low ability students.

Decomposition/Recomposition of Numbers

Items designed to demonstrate decomposition/recomposition were used to measure student ability in flexibly decomposing and recomposing numbers, or expressing a number in an equivalent form in concert with the computation. Table 9 displays the results of these two items. A student's response was classified as using the decomposition/recomposition method when she or he used an equivalent form to display a number or chose to decompose/recompose numbers when looking for an answer.

Item 12: Is A or B larger? Why?

A. 14 x 17 x 21 x 12,

B. 12 x 17 x 7 x 42.

Three students in the high ability group and one student in the low ability group gave correct answers. Correct explanations from both high and low ability student included: "Both A and B have 17 and 12, so we don't need to consider 17 and 12. In A the remaining numbers are 14 x 21. In B the remaining numbers are 7 x 42. 14 in A is double that of 7 in B, and 21 in A is a half of 42 in B. Then, the product of double and half is equal to 1. Hence, they are equal."

Two high ability students presented well articulated correct responses:

"Since 17 and 12 are equal, so they can be cancelled. 7 x 42 is equal to 7 x 2 x 21 which is equal to 14 x 21. Then, A and B are equal", or decomposed" 14 x 21 to 7 x 2 x 21 " and then recompose it to "7 x 42."

"Both A and B all have 17 and 12, so we don't need to consider 17 and 12. In A, the remaining numbers are 14 x 21. In B, the remaining numbers are 7 x 42. 14 of A is double that of 7 in B, and 21 of A is half of 42 in B. Then, 2 x 2 is equal to 1. Hence, they are equal."

Three of the high ability students and five of the low ability students gave incorrect responses.

High and low ability students' responses included:

* "B is larger. Since 17 and 12 are equal, and 42 is much more than the other numbers in A. Hence, B is larger."

* "A is larger. Because 14 is larger than 12, and the difference is 2. 21 is larger than 7, and the difference is 14. Hence, A is larger."

* "I can't compare if I can not use a calculator."

Item 13: Is A or B larger?

A. 1/5 x 1/6 x 1/9 x 1/7?

B. 1/45 x 1/42?

Three students in the high ability group and one student in the low ability group demonstrated the use of decomposition/recomposition. A typical response from high and low ability students was: "Because the numerator of both A and B are 1, and 5 x 9 = 45 and 6 x 7 = 42. Then, 1/5 x 1/9 = 1/45 and 1/6 x 1/7 = 1/42. Hence, A and B are equal." Three students in the high ability group and five students in the low ability group were unable to apply a decomposition/recomposition method. A typical explanation was:

"A is larger. The denominators of A are less than the denominators of B. Hence, A is larger."

Results revealed that the use of decomposition and then the recomposition method (14 x 21 = 7 x 2 x 21 = 7 x 42, or 7 x 42 = 7 x 2 x 21 = 14 x 21) seemed more difficult than the use of the recomposition method only (1/5 x 1/9 = 1/45 and 1/6 x 1/7 = 1/42, then A and B are equal), for the low ability students than for the high ability students. High ability students were able to successfully use the decomposition and then recomposition strategy more frequently than the low ability students.

Discussion

The data indicates that the high ability students were more successful on each type of number sense item than the low ability students. Table 10 shows the response differences for item category cluster for the high and low ability interviewees. Furthermore, for four of these item clusters (number magnitude, use of a benchmark, flexibility with number and operations and decomposition/recomposition of number) the high group achieved twice the frequency of correct responses over the low ability group.

Five cognitive processes (number magnitude, use of benchmarks, decomposition/recomposition of number relative effect of operations on numbers, and flexibility with numbers and operations to computational situations) were observed in varying degrees during the interviews with the preservice elementary school teachers. The low ability students tended to use the rule-based method more frequently when answering interview items than high ability students. The low ability students also preferred the use of standard written computation algorithms rather than the use of number sense based strategies. The high ability students tended to use of benchmarks, to apply knowledge of the relative of operations on numbers and decomposition/recomposition of operation on number, to reflect knowledge of number magnitude, and to response flexibility with number operations when answering interview items. Overall, the low ability of group students demonstrated less frequent use of characteristics of number sense than the high ability students.

Interview results indicate that items including fractions were more difficult than whole number and decimal items for both groups of students, which is consistent with the Number Sense Test results. Rubenstein (1985) similarly found that eighth-grade students had more difficulties with decimals than with whole numbers. Interview data also reveal that most low ability students had not established connections or important understandings between decimals and fractions.

Some problems involving long division are difficult for low ability students. This supports Yang (1997) and Hiebert's (1984) findings that most students do not connect their understanding of fractions with decimal symbols. Data are also consistent with the statement of Markovits and Sowder (1994) that students initially separate fractions from decimal numbers when ordering, indicating that they regard two types of numbers as separate entities, and these are not necessarily related to one another. Gay and Aichele (1997), in their investigation of difficulties students had in mastering the cases of percent problems, found similar results. They note that students had difficulty applying knowledge about fractions and decimals when solving percent problems. Students in the low ability group of the present study misunderstood the concept that multiplication does not always make an answer larger. This misunderstanding was consistent with the previous studies of Graeber & Tirosh (1990).

During the interview, many commented that they could "do better" if they had pencil-and-paper or a calculator. Apparently, most participants were uncomfortable providing estimates and were especially nervous about being unable to calculate using an instrument. Students' strong aversion to estimating is not unusual. R.E. Reys et al. (1991) and Yang (1997) found that many students in both Japan and Taiwan, respectively, were more comfortable solving computational problems exactly than estimating a solution. These researchers indicate that students resist giving estimates because they either did not understand the meaning of estimation or were reluctant to accept error. In the current study, the preservice elementary school teachers who were reluctant to estimate either appeared confused about why anyone would want an estimate instead of an exact answer or did not understand the meaning of estimation and how to apply it.

Interviews also indicate that some students in the high ability group and most students in the low ability group displayed little number sense. Data indicates that approximately 35% of the high ability students and 75% of low ability students tended to use the standard written computation methods when answering number sense related interview items. This tendency to use traditional paper-and-pencil methods appears to interfere with their use of number sense. The interviews indicate that success in calculating the written computation items using paper-and-pencil methods does not imply good number sense. This provides further evidence that number sense does not necessarily develop from instruction focused on standard written computation algorithms. In fact, the current data indicates that tradition paper-and-pencil computation algorithms may interfere with student's development and number sense. This provides additional evidence to support previous by research Yang (1997) and Kamii & Lewis (991) as well as finding of Reys et. al (1991) that the over learning of written algorithms may inhibit a number of important factors that contribute to success at estimation, namely, flexible use of numbers, tolerance for error, use of multiple strategies, and adjustment techniques.

Implications and Recommendations

This study found that many preservice elementary teacher subject of research are not ready to be immersed into a curriculum that reflects the vision of less emphasis on paper-and-pencil computation and more emphasis on number sense and mental arithmetic stated in the NCTM Standards. Based on the findings of this study and the review of the literature, the following Elementary Education Teacher program recommendations are made:

Number sense is a major theme of the NCTM Principles and Standards for School Mathematics (2000). Colleges and universities must help students develop number sense ideas. In particular, the ability to recognize the relative magnitude of numbers, ability to deal with the absolute magnitude of numbers, ability to link numeration, operation and, relation symbols in meaningful ways, ability to understand the effect of operations, ability to perform mental computation through "invented" strategies that take advantage of numerical and operational properties, ability to use numbers flexibly to estimate numerical answers to computations and to recognize when estimate is appropriate, and a disposition towards making sense of numbers. If preservice teachers have a good number sense, they may be likely to select appropriate computational methods and they may be more confident about how they work with numbers.

The interview process should be incorporated into content or method courses. The interview can become a diagnostic tool that enables instructors to gain a better understanding of, and insight into, the learning and thinking of students through the careful analysis of the problems confronted by the subject. We may think the students are proficient in mathematics by their grades, but conceptually some students may be very weak. By relying only on paper-and pencil assessment, the true nature of a preservice teacher's ability to make appropriate computational choices and execute them will remain unknown. Continued research should develop a broader range of entries for the number sense item bank. The individual interviews included in this study address a few selected items from the research instrument and were conducted in the limited time. Interviews should include more items and multiple interviews to provide a complete profile of the "number sense" characteristics possessed by students.

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YEA-LING TSAO

Department of Math Computer Science Education

Taipei Municipal Teachers College
Table 1. Sample of Number Sense Test Items

1. Without calculating        A. More than 6 2/5
an exact answer, circle the   B. Less than 6 2/5
best estimate for:            C. Equal to 6 2/5
                              D. Impossible to tell
6 2/5/15/16                   without working it out

2. Without calculating        A. 1
an exact answer, circle the   B. 2
best estimate for:            C. 19
                              D. 21
12/13 + 7/8                   E. I don't know

Table 2. Mean and Percent of Correct Responses on Number Domain
Items for NST

                  Number    Possible                       Deviation
Number Domain    of Items    Scores     Mean   Standard     Percent

NST Whole           6          12       5.10     2.67       42.50
NST Decimal         8          16       7.01     3.59       43.81
NST Fraction        11         22       8.04     4.15       36.50

Table 3. Sample of Number Sense Interview Instrument

1. How many decimals are there between 1.42 and 1.43?
   Please show me some examples.

2. How many fractions are there between 2/5 and3/ 5?
   Please show me some examples.

3. Please order 0.595, 61%, 5/8, 0.3562. Can you tell me how
   do you compare these numbers?

Table 4. The Framework of Items Characteristics of Number Sense
and Number Domain

             Number      Benchmark    Relative
            Magnitude                 Effect Of
                                      Operation

Whole                       8,            14
Numbers
Decimals        1            5
Fraction      2.3.4         6.7

            Decompose/   The Number
            Recompose    Knowledge of
                         Application

Whole         11,12      9,10
Numbers
Decimals
Fraction        13

Table 5. Number of Responses to Number Magnitude Items of High
and Low Ability Students

Interview
items       Response                     High N=6   Low N=6

1           Correct
              Correct explanation           6          0
                Incorrect explanation       0          0
            Incorrect                       0          6
                Incorrect explanation       0          0
                  No explanation            0          6

2           Correct
              Correct explanation           3          2
              Incorrect explanation         2          0
            Incorrect                       1          4
              Incorrect explanation         1          2
              No explanation                0          2

3           Correct
              Correct explanation           2          1
              Incorrect explanation         1          0
            Incorrect                       3          5
              Incorrect explanation         3          5
              No explanation                0          0

4(a).       Correct
              Correct explanation           4          1
              Incorrect explanation         0          2
            Incorrect                       2          3
              Incorrect explanation         1          2
              No explanation                1          1

4(b).       Correct
              Correct explanation           5          2
              Incorrect explanation         1          0
            Incorrect                       0          4
              Incorrect explanation         0          0
              No explanation                0          4

Table 6. Number of Responses to Use of a Benchmark Item of High
and Low Ability Students

Interview items   Response                     High N=6   Low N=6

5                 Correct
                    Correct explanation        5          2
                      Incorrect explanation    0          0
                  Incorrect                    1          4
                    Incorrect explanation      1          4
                      No explanation           0          0

6                 Correct
                    Correct explanation        2          0
                    Incorrect explanation      0          0
                    Ruled-based                2          2
                  Incorrect                    4          4
                    Incorrect explanation      4          4
                    No explanation             0          0

7(a)              Correct
                    Correct explanation        2          0
                      Incorrect explanation    0          0
                      Ruled-based              1          2
                  Incorrect                    3          4
                      Incorrect explanation    3          4
                      No explanation           0          0

7(b)              Correct
                    Correct explanation        2          0
                      Incorrect explanation    0          3
                      Ruled-based              3          2
                  Incorrect                    1          1
                      Incorrect explanation    1          1
                      No explanation           0          0

Table 8. Number of Responses to Relative Effect of Operations
on Number Item of High and Low Ability Students

Interview items    Response                    High N=6   Low N=6

11                 Correct
                     Correct explanation       3          2
                       Incorrect explanation   0          0
                       Ruled-based             2          1
                   Incorrect                   1          3
                       Incorrect explanation   0          0
                       No explanation          1          3

14                 Correct
                     Correct explanation       3          2
                       Incorrect explanation   0          0
                   Incorrect                   3          4
                       Incorrect explanation   2          3
                       No explanation          1          1

Table 9. Number of Responses to the decomposition/recomposition
Item of High and Low Ability Students

Interview items    Response                     High N=6    Low N=6

12                 Correct explanation          3           1
                     Incorrect explanation      0           0
                       Incorrect                3           5
                   Incorrect explanation        3           3
                       No explanation           0           2
13                 Correct
                     Correct explanation        3           1
                       Incorrect explanation    0           0
                   Incorrect                    3           5
                       Incorrect explanation    3           5
                       No explanation           0           0

Table 10. Percent of response differences for item category cluster
for high and low ability interviewees

Interview Items Category               High Ability
                                   Correct   Incorrect

Number Magnitude                   67.67%     33.33%
(#1, 2, 3, 4a, 4b)
Use of Benchmark(#5, 6, 7a, 7b)    58.33%     41.67%
Flexibility with Number and        61.11%      8.89%
Operations (#8, 9,10)

Use of Relative Effect of          66.67%     33.33%
Operations on Number(411, 14)
Decomposition/Recomposition of     50.00%     50.00%
Number(# 12,13)

Interview Items Category                Low Ability
                                   Correct    Incorrect

Number Magnitude                   26.67%     73.33%
(#1, 2, 3, 4a, 4b)
Use of Benchmark(#5, 6, 7a, 7b)    25.00%     75.00%
Flexibility with Number and        38.89%     61.11%
Operations (#8, 9,10)

Use of Relative Effect of          41.67%     58.33%
Operations on Number(411, 14)
Decomposition/Recomposition of     16.66%     83.34%
Number(# 12,13)
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Date:Dec 1, 2005
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