OTHMAN N. ALSAWAIE
NUMBER SENSE-BASED STRATEGIES USED BY HIGH-ACHIEVING SIXTH GRADE STUDENTS WHO EXPERIENCED REFORM TEXTBOOKS Received: 24 November 2010; Accepted: 3 August 2011
ABSTRACT. The purpose of this study was to explore strategies used by highachieving 6th grade students in the United Arab Emirates (UAE) to solve basic arithmetic problems involving number sense. The sample for the study consisted of 15 high-achieving boys and 15 high-achieving girls in grade 6 from 2 schools in the Emirate of Abu Dhabi, UAE. Data for the study were collected through individual interviews in which students were presented with 10 basic problems. The results showed that a low percentage of solutions involved aspects of number sense such as appropriate use of benchmarks; using numbers flexibly when mentally computing, estimating, and judging reasonableness of results; understanding relative effect of operations; and decomposing or recomposing numbers to solve problems. It was also found that students were highly dependent on school-taught rules. In many cases, these rules were confused and misused. KEY WORDS: mathematics curriculum, mathematical thinking, number sense, problemsolving strategies, reform textbooks
INTRODUCTION According to Yang (2005), “Number sense refers to a person’s general understanding of numbers and operations and the ability to handle daily life situations which include numbers” (p. 318). The National Council of Teachers of Mathematics (NCTM) has highlighted the importance of number sense in several documents (NCTM, 1989, 2000). Actually, number sense has been considered central to the NCTM’s Number and Operation Standard (NCTM, 2000). Good number sense involves a good understanding of number meanings and relationships in addition to flexibility in thinking about numbers (Schneider & Thompson, 2000). It also involves flexibility in making judgments and developing effective strategies for handling numbers and operations (Reys & Yang, 1998). Therefore, development of number sense should be a central focus throughout the elementary mathematics curriculum (Yang, 2005). In contrast to the belief that computational competence is equal to mathematical ability, research studies have found that high-level skills in International Journal of Science and Mathematics Education (2012) 10: 1071Y1097 # National Science Council, Taiwan 2011
1072
OTHMAN N. ALSAWAIE
computation does not necessarily mean a deep understanding of mathematics (Hiebert & Lindquist, 1990; Kamii, 1989; Yang & Huang, 2004), nor do they reflect a good number sense (Yang, 2005; Stigler, Lee & Stevenson, 1991; McIntosh, Reys & Reys, 1992). The emphasis that instruction places on being rule-based makes students rule followers without—in many cases—understanding the procedures they learn (McIntosh, Reys, Reys, Bana & Farrel, 1997). Many mathematics educators have raised concerns that many students demonstrate little understanding of numerical situations when they have to solve number problems (Burns, 1989). For example, a study revealed that students successfully found that 6 × 6 = 36, but could not use this fact in finding 7 × 6 (Kamii, Lewis & Livingston, 1993). Even when students are able to follow rules, they do not always understand the procedures they have learned (Ghazali, Abdul Rahman, Ismail, Sharifah Norhaidah Idros & Salleh, 2003). A number of mathematics educators seem to attribute the difficulties experienced by students in solving mathematics problems to the lack of number sense (Burns, 1989). Norul Akmar (1997) found that many students represent integer as a point on the number line and use the rule “subtracting a negative number is equal to adding the numbers” by rote, without conceptually understanding the rule. Sackur-Grisvad & Leonard (1985) found that French children considered 3.63 larger than 3.8 because 63 is greater than 8. Clearly, children who make such mistakes lack an understanding of place value. Yang (2005) reported that one third of sixth grade students could not answer a similar question correctly, and G30% of those who answered it correctly could support their answers with number sense-based explanations. Sowder & Wheeler (1987) reported that the majority of students before grade 10 could not correctly compare 5/6 and 5/9. Similarly, Peck & Jencks (1981) reported poor performance on comparing 2/3 and 3/4. In comparing two fractions, students usually compare the numerators if the denominators are equal and the denominators if the numerators are equal (Tsao, 2005; Sowder & Wheeler, 1987; Sowder & Markovits, 1989). Other researchers reported that students do not display number sense for problems involving fractions and decimals (Behr, Harel, Post & Lesh, 1992; Greer, 1992). Interestingly, some students thought that decimals were two different numbers separated by a dot (Threadgill-Sowder, 1984; Hiebert & Wearne, 1986). Kerslake (1986) reported that primary school children considered the denominator and numerator of a fraction to be two separate entities. Poor student understanding is attributed mainly to the organization of mathematics content and mode of instruction (Schmidt, Jakwerth &
NUMBER SENSE STRATEGIES
1073
McKnight, 1998). Traditional mathematics textbooks usually focus on mastering rules and algorithms and do not emphasize thinking, creativity, and number sense, which therefore makes them undemanding (Hiebert, 1999). According to Sheetal Sood & Jitendra (2007), one reason for children's poor understanding of mathematics is the “poorly designed textbooks and educational materials that fail to provide experiences to develop critical mathematical ideas” (p. 145). Attributing lack of students' mathematical understanding to textbooks is highlighted by many other researchers (e.g. Jones, Langrall, Thornton & Nisbet, 2002; Suter, 2000). Researchers suggest that number sense starts to be developed too early in school, and children need a positive start in order to develop confidence, positive beliefs, and attitudes toward mathematics. Experiences that are not mathematically meaningful will lead children to believe that mathematics learning is all about memorizing facts and procedures (NCTM, 1989). Meaningful experiences with mathematics can lead to positive attitudes and beliefs about mathematics (Van de Walle, 2007). Furthermore, learning mathematics with understanding helps children develop more confidence, autonomy, and flexibility in their learning and use of mathematics (NCTM, 2000). Garner (1992) suggested, “Textbooks serve as critical vehicles for knowledge acquisition in school” and can “replace teacher talk as the primary source of information” (p. 53). More recently, speaking about mathematics instruction in the USA, Sheetal Sood & Jitendra (2007) stated, “The increased complexities of a diverse population in the U.S. educational system, along with the less than positive school outcomes in mathematics, clearly call for the need to analyze the adequacy of information presented in textbooks. Textbooks are considered a de facto national curriculum, and are the primary means of imparting new information to students” (p. 146).
ASPECTS
OF
NUMBER SENSE
According to NCTM (2000), number sense is “the ability to decompose numbers naturally, use particular numbers like 100 or 1/2 as referents, use the relationships among arithmetic operations to solve problems, understand the base-ten number system, estimate, make sense of numbers, and recognize the relative and absolute magnitude of numbers” (p. 32). Therefore, in characterizing number sense, mathematics educators focus on its intuitive nature and its gradual development (NCTM, 1989, 2000;
1074
OTHMAN N. ALSAWAIE
Sowder, 1992; Sowder & Schappelle, 1989). Number sense is manifested in many ways, including judging number magnitude; appropriate use of benchmarks; using numbers flexibly when mentally computing, estimating, and judging reasonableness of results; understanding relative effect of operations; and decomposing and recomposing of numbers to solve problems (Yang, 2005; Tsao, 2005; NCTM, 2000; Markovits & Sowder, 1994). Number Magnitude Researchers have considered number magnitude an important component of number sense (Greeno, 1991; Howden, 1989; Markovits, 1989; Greenes, Schulman & Spungin, 1993). Understanding the magnitude of numbers involves understanding the place value concept, the ability to compare and order numbers, and the ability to identify numbers between two given numbers (Sowder & Markovits, 1994). For example, a student who understands the magnitude of numbers should easily recognize that 0.05 is greater than 0.025. Given two numbers such as 2.24 and 2.25, a student with good number sense should be able to see that there are infinite numbers between them (the density of decimals). Understanding number magnitude enables students to flexibly compare fractions such as 7/11 and 7/8; and 18/19 and 15/16. Benchmarking As McIntosh et al. (1992) suggest, “A compass provides a valuable tool for navigation, numerical benchmarks provide essential mental referents for thinking about numbers” (p. 6). For example, using 1/2 as a benchmark, the sum of 2/5 and 9/20 should be more than 1/2 because each of them is close to 1/2. Similarly, using 50% (or half) as a benchmark, 48% of 300 should be about half of 300 or 150. In this regard, Trafton (1989) suggests that benchmarks such as 10% and 50% in particular should be given special attention before teaching calculating exact percents. Depending totally on the rules might not help students in solving a problem like this: place the decimal point correctly in the product: 115.4 × 0.325 = 37505. The rule is to count the number of digits to the right of the decimal point in the two multiplicands. Then you count the same number of digits in the product starting from the right, to place the decimal point. Applying the rule in this problem will give 3.7505. A student with good number sense will not accept this answer simply
NUMBER SENSE STRATEGIES
1075
because the answer should be less than 115.4. Using benchmarking will lead the solver to decide that the correct answer should be 37.505. Estimation, Mental Computation, and Judging Reasonableness An important aspect of number sense is the use of appropriate strategies flexibly to solve problems. The ability to estimate is a useful tool that facilitates problem solving. Mental computation is a powerful means for performing mathematical calculations to solve problems. These two abilities can be used separately or in combination depending on the situation. Combining both can put the problem solver in a better position in dealing comfortably with a difficult situation. The ability to judge the reasonableness of answers is very important in mathematics as it encourages reflection on the results and the processes used to arrive at them (Sowder, 1992). The NCTM Principles and Standards of School Mathematics emphasized the connection between number sense and reasonableness by suggesting that all students should be able to “judge the reasonableness of results” (NCTM, 2000, p. 32). Relative Effect of Operations This aspect of number sense is the feeling of how arithmetic operations affect numbers. It requires understanding the relationship between addition and subtraction as well as the relationship between multiplication and division (Baroody & Coslick, 1998). Experience with operations on whole numbers causes students to feel that multiplication makes “bigger” and division makes “smaller” (Greer, 1992). This understanding of the effect of operations does not help students when operating on decimals or fractions. When working with decimals for example, multiplying by a decimal makes the answer smaller and dividing by a decimal makes it larger. In a problem like this—Which is larger 83 ÷ 0.05 or 83 ÷ 0.025?—the student has to decide on two issues: First, they have to determine which divider is larger, 0.05 or 0.025, and, second, determine whether a smaller or a larger divisor makes the quotient larger. As suggested by McIntosh et al. (1992), “Fully conceptualizing the concept of operation implies understanding the effect of the operation on using numbers including whole and rational numbers” (p. 7). Decomposition/Recomposition of Numbers As part of flexibility in working with numbers, students need to be able to decompose and recompose numbers by expressing them in equivalent
1076
OTHMAN N. ALSAWAIE
forms to facilitate mental computation and problem solving. For example, when dealing with a problem like this—Which is larger 28 × 52 or 30 × 50?—the student has to understand the decomposition and recomposition of numbers, in addition to understanding the effect of operations on numbers. For example, 28 × 52 can be thought of as 28 × (50 + 2), and 30 × 50 can be thought of as (28 + 2) × 50. Now, in comparing 28 × (50 + 2) and (28 + 2) × 50, the student should notice that 28 × 50 exists in both quantities. Therefore, the comparison is really between 28 × 2 and 2 × 50. This procedure, of course, can be done mentally. This type of thinking about numbers and operations can make problem solving more accessible to students.
CONTEXT
AND
RESEARCH QUESTION
This study was conducted to learn about number sense strategies used by highachieving sixth graders in the United Arab Emirates (UAE). More specifically, it aimed at answering the following research question: What strategies do sixth grade high-achieving students use to answer arithmetic problems? The Ministry of Education (MOE) in the UAE adopted a new vision for mathematics education in the country which was finally fulfilled with the publication of the National Document for Mathematics Curriculum in Public Education (NDMCPE; MOE, 2001), which places major emphasis on number sense and learning mathematics for understanding. This document set forth content and process standards identical to those found in NCTM's Principles and Standards for School Mathematics (PSSM; NCTM, 2000). Following the publication of this document, the MOE published new mathematics textbooks for all grade levels in public schools. These textbooks are mostly Arabic versions of textbooks originally published in the USA (Scott Foresman Series). They were translated into Arabic first by a publishing company. Then, committees formed by the MOE worked on verifying their alignment with the NDMCPE and the culture of the UAE. As a result, these textbooks present mathematics in a way that promotes understanding, number sense, problem solving, reasoning, and communication. They also connect mathematical concepts and offer opportunities for a variety of representations of mathematical ideas. The textbooks highly emphasize the inclusion of real-life applications related to the UAE culture as well as international culture. Sixth graders in the UAE, who are the target of this study, have been taught mathematics through these reformed textbooks, starting in grade 1.
NUMBER SENSE STRATEGIES
1077
The following features are in the grade 6 mathematics textbooks in the UAE. The textbooks come in six parts, a student version (two parts), an exercise book, remedial book, enrichment book, teacher’s guide, and assessment book. The last two are for the teacher only. The student version consists of 11 units with one to three chapters in each, with a total of 24 chapters; each includes a number of lessons. There are a team project and a mathematics journal in each unit, and an oral expression activity in each lesson. Each chapter integrates a number of connections (e.g. mathematics, science, geography, history, tourism, and time) and processes (e.g. problem solving with different strategies each time, communication, critical thinking, logical thinking, number sense, procedural sense, estimation, finding patterns, oral expression, and daily journals). Before and during the implementation of these textbooks, experts from the publishing company provided training workshops for teachers to familiarize them with the textbooks and train them in using the books in their daily instruction. The training covers topics such as: opening the lesson, using the teacher’s guide in implementing the lesson and the assessment book for developing student assessment, using the exercise book, the remedial and enrichment books for accommodating student diversity, and using instructional aides to facilitate student learning. In addition, supervisors from educational districts train teachers and observe them in class to give feedback. Now, since many educators have attributed the lack of number sense, at least partially to traditional textbooks, it is worth studying whether the reformed textbooks would provide students with better number sense. However, to the knowledge of the author of this paper, no research on number sense has been done in the UAE. Therefore, it is of high importance to initiate such a type of research. While this research is highly important for the UAE specifically, it is also important for the international mathematics education community as it investigates the number sense of high-achieving students who were taught from the reformed textbooks from grade 1 to grade 6. Not much similar research exists.
METHODOLOGY Sample The sample of this study consisted of 30 high-achieving sixth grade students from two public schools in the same city, in the Emirate of Abu Dhabi, UAE. As schools in the UAE are segregated by gender, 15 boys
1078
OTHMAN N. ALSAWAIE
were selected from one school and 15 girls were selected from another. In each of the schools, the highest five mathematics achieving students in each of three randomly selected sections participated in the study. As each section usually has about 25 students, the participants are the top 20% students in each section. These students had a mathematics score of at least 88 out of 100 in their last academic year. All participants belong to the middle socioeconomic class according to the living standards in the UAE. Instrument All participants were interviewed individually and asked to solve ten mathematics problems. These problems were based on the components of number sense, as indicated in related documents and research reports (NCTM, 2000, 1989; Yang, 2005; MarKovits & Sowder, 1994). In developing the problems, specific attention was given to the instrument developed by Yang (2005) and those of Tsao (2005) and MarKovits & Sowder (1994). These components are: judging number magnitude; appropriate use of benchmarks; using numbers flexibly when mentally computing, estimating, and judging reasonableness of results; understanding relative effect of operations; and decomposing and recomposing numbers to solve problems. The problems focused specifically on whole numbers, decimals, fractions, and percent. The first draft of the problems was given to mathematics education experts to judge the suitability of the problems to the grade 6 mathematics curriculum. The comments of these experts were taken into consideration in order to arrive at the final version of the problems. The researcher (author) interviewed students individually in the last month of school. The problems were given to students in the same order, as presented in Table 1. For each problem, the researcher read the problem statement to the student after giving them the problem on a flash card. When the student gave an answer, the researcher asked them to tell how they had arrived at that answer, or to justify the answer. The interviews were videotaped and later transcribed for the purpose of analysis. Limitations The author would like to shed light on the fact that only 30 students were involved in the study and that all of them were from one city in the UAE. The small sample size is due to the difficulty of interviewing a large
1079
NUMBER SENSE STRATEGIES
TABLE 1 Students’ responses to each of the ten items (N=30) Strategies used Item
Answer
NSB
2
How many different decimal numbers are there between 2.24 and 2.25? Which is larger 7/11 or 7/8?
3
Which is larger 18/19 or 15/16?
4
Please place the decimal point correctly : 115.4×0.325=37505 Is 2/5+9/20 less than or greater than 1/2? Please estimate 48% of 300.
Correct Incorrect Correct Incorrect Correct Incorrect Correct Incorrect Correct Incorrect Correct Incorrect Correct Incorrect Correct Incorrect Correct Incorrect Correct Incorrect Correct Incorrect
1
5 6 7
8 9 10
The estimate of 28×52 is 30×50=1500. Is the exact answer less than, equal to, or greater than 1500? Which is larger 83÷0.05 or 83÷0.025? Is A or B larger? A=1/3×1/4×1/5×1/7, B=1/15×1/28 Is A or B larger? A=8×12×7×16, B=7×6×32×8 Total
Other
NE
1 0 6 0 4 0 4 0 4 0 14 0 1 0
0 20 9 15 18 6 1 25 10 8 0 13 4 20
0 9 0 0 2 0 0 0 0 8 1 2 3 2
8 0 6 0 3 0 51 0
8 13 0 8 0 9 50 137
1 0 4 12 8 10 19 43
NSB number sense-based, Other rule-based or incorrect reasoning, NE no explanation or don’t know
number of students. These factors are to be taken into consideration when interpreting the results of the study.
DATA ANALYSIS The author and another mathematics educator analyzed the data. First of all, students’ responses were coded as either correct or incorrect based on an answer key. Responses of students who answered “I don’t know how to do it” or “I can’t do it without a calculator” were considered incorrect. Second, the transcripts of the interviews were investigated to determine the strategies used to solve each problem. Each response was coded into one of the following three categories (strategies):
1080
OTHMAN N. ALSAWAIE
1. Number sense-based: If the student used one or more of the aspects of number sense (number magnitude, benchmarks, relative effect of operation on numbers, estimation, decomposition/recomposition of numbers, and judging the reasonableness of results) 2. Rule-based or incorrect reasoning: If the student used a correct standard rule or an incorrect rule, or reasoned incorrectly to solve the problem 3. No explanation or don’t know: If the student gave an answer, but provided no explanation, or could not give an answer The two coders examined all of the 300 responses independently and agreed on 282 responses (94%). The remaining responses were re-discussed until an agreement had been reached. RESULTS The numbers of correct and incorrect answers as well as the numbers of strategies for each of the ten questions are presented in Table 1. As shown in Table 1, out of 300 responses, 120 (40%) were correct. Correct answers on the ten items ranged between 1 (3.33%) and 24 (80%), with correct answers to six of the items being G50% (items 1, 4, 5, 7, 9, and 10). Only in 51 responses (17%) were the number sense-based strategies used, and those were all associated with correct answers. None of the 180 incorrect responses involved number sense-based strategies. On the contrary, 187 responses (62%) were based on other strategies (rule-based or incorrect reasoning). Sixty-two students (21%) could not provide explanations to their answers (whether correct or incorrect), or they did not know how to solve the problem. Number sense-based strategies used included: judging number magnitude (13 times on items 1, 3, and 8) using benchmarks (eight times on items 2 and 5), relative effect of operation on numbers (eight times on item 8), estimation (17 times on items 4 and 6), decomposition/ recomposition of numbers (ten times on items 6, 9, and 10), and judging the reasonableness of results (once on item 1). Following are the detailed analyses of the responses on four items (analyses of the remaining items are found in the Appendix). Item (1): How Many Different Decimal Numbers are There Between 2.24 and 2.25? This item was meant to test students’ concept of number density between two decimals. Only 1 student out of 30 (3.33%) answered this item
NUMBER SENSE STRATEGIES
1081
correctly. His explanation reflected deep understanding of number magnitude (density between two decimals). Here is how he explained his answer: R: How many different decimal numbers are there between 2.24 and 2.25? S: Well, many numbers. R: How many? S: Countless numbers. R: How do you know? S: Well. 2.24, 2.241, 2.242, 2.243 and so on. You can also make 2.2411 and you can still make more digits and still you are less than 2.25.
Twenty-nine students gave incorrect answers. Eighteen of them answered “none”, with eight offering no explanation to their answers. Ten students, however, justified their answers by saying that “there are no numbers between 4 and 5.” Here is an example: R: How many different decimal numbers are there between 2.24 and 2.25? S: None. R: Why do you think so? S: Because 5 comes directly after 4. I mean there are no numbers between 4 and 5. R: Can you think of it another way? S: No.
It is clear that these students lacked understanding of number density between two decimals. An understanding of decimal equivalence would help students see that 2.24 = 2.240 = 2.2400 = 2.24000 and so on. When they have this understanding, it would be easy for them to recognize that an infinite number of decimals exist between 2.24 and 2.25. Such responses highlight the weak understanding of place value and number magnitude. Based on this lack of understanding, these students will not be able to compare two decimals successfully (Sowder & Markovits, 1994). Six students answered 9 or 10. All of them used decimal equivalence like this: 2.24 = 2.240, 2.25 = 2.250. So, 2.241, 2.242, …,2.249 are between 2.24 and 2.25. While these students have a better understanding of place value than their counterparts who answered “none,” this understanding is still limited. Five students answered “one.” One of them had no explanation and four had similar explanations. For example: S: One decimal only. R: Can you justify your answer? S: Well. 5 comes after 4 and there is a number in between. R: Can you clarify your idea? S: I think there is a distance between any two numbers, and there is one number in the middle. R: Only one number? S: Yes. R: What is that number? S: I don’t know. Well, maybe four and a half. R: So, you think there is one number between 2.24 and 2.25? S: Yes. R: Can you write that number please? S: Well. No, I don’t know how to write it. R: Can you do the problem another way? S: No.
Students who give such answers with such explanation clearly lack understanding of number magnitude and place value. Even though they
1082
OTHMAN N. ALSAWAIE
learned this in school as part of their curriculum, they still cannot apply it in a novice situation. Item (2): Which is Larger 7/11 or 7/8? Fifteen students (50%) answered this item correctly. Only six of these students used number sense-based strategies to compare the two fractions. Three of these students compared the two fractions with 1 and decided that “7/8 is very close to 1 whereas 7/11 is far from 1.” The other three compared the two fractions with 1/2 and decided that “7/11 is close to 1/2 whereas 7/8 is much greater than a half.” These students successfully used benchmarking to solve the problem. Using 1 or 1/2 as a mental referent enabled them to compare the two fractions based on the distance between each and the referent (McIntosh et al., 1992). Four students used the rule “if numerators are equal, the fraction with the smaller denominator is larger”. Even though these students arrived at the right answer using the correct rule, they did not seem to have the appropriate mental referent. Another four students used cross-multiplication: “7 × 11 = 77, 7 × 8 = 56, 77956, so 7/8 is larger.” The remaining student could not justify her answer, but insisted that 7/8 is larger than 7/11. Fifteen students (50%) answered incorrectly. Eleven of them reversed the rule and answered, “7/11 is larger because 11 is larger than 8”. This shows that depending only on memorizing rules is an irrelevant way of teaching fractions. Four students used other strategies. Two of them multiplied the numerator and denominator of each fraction and compared the product. Therefore, they concluded that “7/11 is larger because 77 is greater than 56.” One student decided, “We can convert the fractions into decimals: “7/11 = 7.11 and 7/8 = 7.8, and since 0.11 is greater than 0.8, 7/11 is greater than 7/8.” One student added the numerator and denominator of each fraction and compared the sums. Therefore, she concluded that 7/11 is greater than 7/8 because 18 is greater than15. These awkward strategies highlight the danger of focusing on memorizing rules routinely when teaching mathematics. Such teaching does not help students build the meaning of what they have learned nor develop a sense of numbers. A student who has a sense of number magnitude would never convert 7/11, which is less than 1 into 7.11 or
NUMBER SENSE STRATEGIES
1083
conclude that 7/11 is larger than 7/8 based on adding the numerator and denominator of each fraction (7 + 1197 + 8). Item (7): The Estimate of 28 × 52 is 30 × 50 = 1,500. Is the Exact Answer Less Than, Equal to, or Greater than 1,500? Eight students (26.7%) answered this item correctly. Only one of these students used a number sense-based strategy to arrive at the solution. The following excerpt of his interview illustrates his strategy: S: 28 × 52 is less than 30 × 50. R: Justify your answer please. S: Here [pointing to 52 in 28 × 52], the 2 is multiplied by 28 and that is 56, but here [pointing to 30 in 30 × 50], the 2 is multiplied by 50 and that is 100. One hundred is greater than 56. R: Where is the 2 here [pointing to 30 in 30 × 50]? S: The 2 in the 30. I mean 30 is 28 plus 2. R: Thank you.
This student successfully used decomposition of numbers. First, he decomposed 52 into 50 + 2, and, second, he decomposed 30 into 28 + 2. Using decomposition, he successfully compared the two quantities. Four students seemed not to fully understand the relative effect of operations as they did not use the two multiplicands to solve the problem. Rather, they focused on one multiplicand as they explained “28 × 52 is less than 30 × 50 because 28 is less than 30.” Here is an example of how a student reasoned about the problem: S: 28 × 52 is less. R: Justify your answer please. S: 28 is less than 30. The answer must be less. R: But at the same time, 50 is less than 52. S: Yes, but 28 is first and 30 is first. R: Is 28 × 52 different from 52 × 28? S: [Pause]. Well, no. They are the same. Multiplication is commutative. R: So, again why do you think that 28 × 52 is less than 30 × 50? S: Well. As I said because 28 is less than 30. R: What about 50 and 52? S: It does not matter. This is less [pointing to 28 × 52] because 28 is less than 30.
Three students had no explanation of why they answered 28 × 52 is less than 30 × 50. The interviewer tried to make them say something about their thinking, but they provided no explanation or justification for their answers. Twenty-two students answered this item incorrectly. Five of them answered 28 × 52 is greater than 30 × 50, with two explaining “estimation makes less” and the other three explaining “because 52 is greater than 50.” Seventeen students answered “equal,” with two of them providing no explanation to their answers. All of the other 15 students based their answers on the fact 30 − 28 = 52 − 50. Clearly, these 15 students not only lacked understanding of the relative effect of operations but also seemed
1084
OTHMAN N. ALSAWAIE
not to fully understand the meaning of multiplication. They tended to use additive reasoning, which is a sign of immaturity in mathematical thinking (Post, Cramer, Behr, Lesh & Harel, 1993). Item (9): Is A or B Larger? A = 1/3 × 1/4 × 1/5 × 1/7, B = 1/15 × 1/28 Ten students (33.3%) answered this item correctly, with four of them providing no explanation. All of the other six students used a decompose/ recompose strategy to solve the problem. They reasoned that “1/3 × 1/5 = 1/15, and 1/4 × 1/7 = 1/28. Therefore, A = B.” This type of reasoning shows that the flexibility of these students in working with numbers enabled them to compare A and B easily by expressing them in equivalent forms. Eight students considered B to be larger because “it contains larger numbers.” Here is an example: R: Justify your answer please. S: B contains larger numbers, 15 and 28.R: So? S: When you multiply large numbers, you get very large products. All numbers in A are small. They don’t give large products. R: Aren’t there any other ways to compare A and B? S: No. I am sure B is larger.
This student not only ignored that 3 × 5 = 15 and 4 × 7 = 28 but also did not realize that having a greater denominator will make the fraction less. If the product of 15 and 28 is larger than that of 3, 5, 7, and 4, then A should be greater than B. The remaining 12 students did not attempt to solve the problem. All of them asked for a calculator to do the problem. Needing a calculator to do simple calculations is a common practice among students in the UAE. That is why these students could not even try the problem without having calculators. Actually, the availability of calculators does not guarantee that students will do well in problem solving. How would the student in the last example have used the calculator if the numbers in the question were different? Most likely, he would have found the products using the calculator and then decided that the fraction with the greater denominator is greater.
DISCUSSION This study aimed at investigating number sense-based strategies used by high-achieving grade 6 students in the UAE. Given the lack of these types of studies in the country, the results presented here are of great
NUMBER SENSE STRATEGIES
1085
importance to all stakeholders. The results of the study revealed important findings. The data indicate that a low percentage of students exhibited number sense when solving arithmetic problems. Even though the participants were all high achievers in mathematics, only 17% of their solutions involved number sense-based strategies. This finding is consistent with the findings of Yang (2005) and those of Markovits & Sowder (1994). Students’ solutions were dominated by rule-based methods. Actually, even pre-service teachers were found to perform poorly on similar problems (Tsao, 2005). Clearly, the overemphasis on written computations in schools was the main reason behind these results. The results of this study add to the accumulated knowledge about number sense in many ways. The results show that the lack of number sense is not exhibited only by low mathematics achievers but also by high achievers. Furthermore, the results show clearly that the existence of the reformed textbooks is not enough to promote number sense among students. The participants of this study have used the reformed textbooks since they entered school up to grade 6. These textbooks presented mathematics in a way consistent with the vision for teaching and learning mathematics set forth by NCTM as presented in the PSSM (NCTM, 2000) and with what was emphasized in the NDMCPS (MOE, 2001). Yet, it is clear that these students were exposed to experiences that did not promote number sense. Rather, written computations seem to have been the focus of mathematics instruction. This overemphasis on written computation using standard algorithms discourages students from developing number sense-based strategies. Actually, it might also hinder their development of thinking and reasoning (Yang, 2005). This is not to say that students should not learn to use standard algorithms. However, these algorithms should be taught after students have had the opportunity to conceptually understand the logic behind these algorithms and to devise their own solution strategies (Baroody & Coslick, 1998), in which case they can make meaning out of these algorithms. Otherwise, students will only memorize the rule-based methods and apply them without understanding why and when they work. That is why students in this study misused and confused rules when confronted with novel problems. The results of this study contribute to the literature on number sense in one more way. Interestingly, in some cases, students gave correct answers by using incorrect rules or inappropriate reasoning. For example, in comparing 15/16 and 18/19, some students answered “18/19 is larger” just because 19 is greater than 16. Some other students added the numerator and denominator of each fraction and compared the sums.
1086
OTHMAN N. ALSAWAIE
Therefore, they concluded that 18/19 is larger than 15/16 because 37 is larger than 31. This finding questions the validity of the results of assessments that simply ask students to do calculations without justifying their answers. These students, for example, would be seen to understand the material while in fact they do not because their answers are falsely true (Baroody & Coslick, 1998). The interview method should be incorporated into classroom assessment as a diagnostic tool that enables teachers to gain a better understanding of the learning and thinking of their students through a careful analysis of their ways in tackling mathematics problems. This assessment method gives teachers the opportunity to understand the strengths and weaknesses of their students’ conceptual understanding of the mathematical content. Consequently, teachers would be better able to make instructional decisions regarding the next steps in their teaching. It is true that this method is time-consuming and requires more effort from the teacher, but it is worthwhile if number sense is considered as an important objective of mathematics teaching. The extensive use of calculators in the classroom seems to have hindered some students’ ability to even try solving problems without calculators. Many students did not attempt to answer some problems (items 9 and 10) because there were no calculators available. While calculators might be useful in solving problems that require complicated calculations, it is not wise to allow students to use them for doing simple calculations that can be done mentally. Calculators are used extensively by students in UAE for doing simple calculations. Through observing student teachers in local schools, the author noticed these students also used calculators for doing calculations as simple as: 40 + 60; 3 × 5; 20 − 8; …etc. Other factors such as parents and private tutors may have played a role in the lack of students’ number sense. Some parents work with their children on doing their homework. Parents who only use rules when computing could have a negative influence on their children’s number sense. Private tutoring is very common in the UAE, and private tutors usually teach students only rules and algorithms, which prevents students from developing good number sense. In conclusion, good textbooks are essential but are not adequate for promoting good student number sense. Based on the results of this study and previous research on number sense, mathematics instruction in UAE and elsewhere should focus on the development of conceptual understanding and meaningful use of mathematical rules and algorithms. Teachers should teach in ways that lead their
NUMBER SENSE STRATEGIES
1087
students to develop a sound, intuitive “feel” for numbers. This implies that instruction should be designed to provide rich opportunities for exploring numbers, number relationships, and number operations and to discover rules and invent algorithms. Actually, unless teachers recognize and use good number sense themselves, which can take a good deal of professional development, it is unlikely that much change will take place. It is essential that mathematics teachers themselves have a deep understanding of the components of number sense and an understanding of the difficulties students encounter in solving numerical problems. Many studies have investigated the mathematics content knowledge of teachers and its influence on their instructional choices. It was found that content knowledge influenced teaching practice, which in turn influenced student success (Hiebert & Stigler, 2000; Hill, Rowan & Ball, 2005; Stigler & Hiebert, 2004; Knuth, 2002). Teachers with adequate mathematic content knowledge are more flexible in their instruction and more likely to involve students in problem solving to promote student thinking and make mathematical connections (Faulkner, 2009). The importance of content knowledge has led educators to design professional development programs that focus on content knowledge. Such programs have been found to be effective (Hill et al., 2005; Garet, Porter, Desimone, Birman & Suk Yoon, 2001).
APPENDIX Item (3): Which is Larger 18/19 or 15/16? Twenty-four students (80%) gave correct answers to this item, but ten of them however, used incorrect rules to solve the problem. Four students used number sense-based strategies. Here is an example. R: Which is larger 15/16 or 18/19? S: 18/19. R: Justify your answer please. S: In the second [18/19], the pieces are smaller, and since in each fraction, one piece is left, this one [18/19] is greater because the piece left is smaller.
Another interesting strategy was used: R: Which is larger 15/16 or 18/19? S: 18/19. R: Justify your answer please. S: It is like drawing a circle, dividing it into 19 parts, and shading 18 parts. 15/16 is the same but with
1088
OTHMAN N. ALSAWAIE
16 parts and you shade 15 parts. Here [pointing to 18/19], more area will be shaded. R: Do you mean more parts? S: Yes, more parts but also more area. R: How do you know that the area will be more? S: Well. Surely, the area is more. You shade everything except 1/19 but in this case [15/16] you shade everything except 1/16. R: So? S: 1/19 is small comparing to 1/16. Therefore, this area [18/19] is more.
Eight students used the school-taught rule, “If the differences between the numerator and the denominator are the same, the greater the denominator the larger the fraction.” Two students could not provide any justification to why they decided that 18/19 is larger than 15/16. Interestingly, ten students, answered correctly by using incorrect rules. Six of them compared the denominators only. Here is an example: R: Which is larger 15/16 or 18/19? S: 18/19. R: Can you justify your answer? S: Because 19 is larger than 16. R: Is it enough to compare the denominators? S: Yes R: What if we change the numerators? S: I don’t look at the numerators. If the denominator is larger, then the fraction is larger. R: Can you do it another way? S: No.
Four students added the numerator and denominator of each fraction and compared the sums. Therefore, they concluded that 18/19 is larger than 15/16 because 37 is larger than 31. Six students answered incorrectly. Two of them decided that 15/16 is larger than 18/19 using the incorrect rule, “If the denominator is smaller, then the faction is larger.” Following is the conversation with one of them: R: Which is larger 15/16 or 18/19? S: 15/16. R: Justify your answer please. S: Because 16 is smaller than 19. R: So? S: When the denominator is less, the fraction is larger. R: What about the numerators? S: It doesn’t matter. For example 1/2 is larger than 1/4 because 2 is smaller than 4. R: Are you sure of your answer? S: Yes.
The other four students decided that 15/16 = 18/19. Two different incorrect strategies were used in arriving at this answer. One student used “cross addition” and got (15 + 19 = 34, 18 + 16 = 34). Therefore, she decided the
NUMBER SENSE STRATEGIES
1089
two fractions are equal. The other three students calculated the difference between the denominator and numerator of each fraction. Therefore, they concluded that the two fractions are equal because 19−18 = 16−15. Item (4): Please Place the Decimal Point in the Product Correctly: 115.4 × 0.325 = 37505 Only five students (16.7%) answered this item correctly. Four of them used number sense-based strategies to solve the problem. Following is an example of these strategies. R: Please place the decimal point correctly: 115.4 × 0.325 = 37505.S: Well. It should be here [3.7505]. No, wrong. R: Ok. Where should it be? S: It can’t be 375 because the answer should be less than 115, and 3.7 is wrong of course. So, the answer is 37.505. R: Why do you think the rule did not work here? S: I think there should be a zero to the right. Yes, because 4 times 5 is 20. There should be zero here [pointing to the right of 5].
This student used the rule first by counting 4 digits from the right but soon realized that the answer was unreasonable. Her behavior reflects good number sense as she did not accept the answer just because she applied the “right rule.” She knew that the answer should be less than 115.4, but not as small as 3.75. Using this type of reasoning, she arrived at the right answer. One student gave a correct answer using an incorrect rule. S: There is one [digit] here [pointing to the 4 in 115.4] and one here pointing to the zero in 0.325]. So we count two digits from the left and put the point here [pointing to the right of 7 in 37505]. R: Are you sure about this rule? S: Yes. R: How do you know your answer is right? S: I don’t know, but that is the rule.
Evidently, this student did not learn this rule in school, but it seems that because of the many rules that students are forced to memorize, they get confused about them or misuse them. Twenty-five students (83.3%) provided incorrect answers. Eleven students used the school-taught rule to solve the problem. All of them counted the number of digits to the right of the decimal point in both multiplicands (four digits). Then, they counted four digits from the right in the product and placed the decimal point. Obviously, these students are
1090
OTHMAN N. ALSAWAIE
not used to judging the reasonableness of results when solving problems because having 3.7505 as an answer should make the student rethink their strategy as the result is unreasonable. They also lack the ability to estimate. They could have just multiplied 100 by 0.3 to get 30, which is close to the correct answer. Seven students showed confusion in applying the school-taught rule. In counting the digits in the multiplicands, they considered one digit in 115.4 (from the right) and one in 0.325 (from the left). Having two digits, they counted two digits from the right in the product to obtain 375.05. Another seven students reversed the school-taught rule and counted four digits from the left to place the decimal point and got 3750.5. Again, obtaining this answer should have alerted students to the fact that there is something wrong in their strategies, but the blind dependence on rules made them accept any answers. This may be a natural result of learning the rules without understanding their conceptual bases. All these answers indicated that the students could not estimate, do mental computation, nor judge the reasonableness of results. Item (5): Is 2/5 + 9/20 Less Than or Greater Than 1/2? Fourteen students (46.7%) answered this item correctly; only four of them used number sense-based strategies in arriving at the answer. Three of them reasoned that “each of the given fractions [2/5 and 9/20] is almost one half and so their sum must be more than one half.” The fourth student, however, reasoned differently: “One fourth plus one fourth equals one half, and each of the two fractions is more than one fourth. So, their sum must be more than one half.” All these students used benchmarks appropriately and based their solutions on familiar referents (1/2 and 1/4). The other ten students made some calculations to answer the question. Six students converted 2/5 into 8/20 and added the two fractions. Two students converted the two fractions into decimals (0.4 and 0.45) and concluded that the sum is greater than one half. The other two students, however, converted the two fractions into percents (40% and 45%). Those who answered the question incorrectly (16 students) could not reason logically or use any appropriate rule. Half of them could not provide any explanation as to why they considered the sum of the two fractions to be less than one half. Five students attempted to add the two fractions by finding a common denominator but they failed to do it
NUMBER SENSE STRATEGIES
1091
correctly. The other three provided peculiar explanations: “Because each is less than one half, the sum must be less than one half”; “2 × 9 divided by 5 × 20 is less than one half. Therefore, the sum must be less than one half”; “2 × 20 = 40 and 5 × 9 = 45, 40≠45. Therefore, the sum must be less than one half.” All of the three students could not justify their reasoning. Here is how one of them responded to the interviewer’s questions: R: Is 2/5 + 9/20 less than or greater than 1/2? S: Let me see. You mean addition? R: Yes. I mean if you add 9/20 to 2/5, will the sum be less than or greater than 1/2? S: Okay. Look. 2 × 9 = 18 and 5 × 20 = 100. So 18/100. R: Okay. So what do you think? Is the sum of 2/5 and 9/20 less than or greater than 1/2?S: 2 × 9 divided by 5 × 20 is less than one half. Therefore, the sum must be less than one half. R: I don’t understand. What does this have to do with the sum of 2/5 and 9/20? S: Well. You want to know if the sum of 2/5 and 9/20 is less than or greater than 1/2. Right? R: Exactly. S: Okay. Numerator times numerator and denominator times denominator. R: Okay. What does that mean? S: I really don’t know. This is how we learned it. R: Can you think of any other way to solve the problem? S: No.
This student had learned in school that working with fractions involves addition, multiplication, and division. But it seems that he does not know when to do what. He is lost in the middle of the rules and operations he was taught in a mechanical way. The same applies to the other two students who used awkward reasoning for solving the same problem. Item (6): Please Estimate 48% of 300. Fifteen students (50%) answered this item correctly. Only one of these students could not provide any explanation for his answer. All other 14 students used number sense-based strategies to answer the item. For example: R : Estimate 48% of 300 please. S : Well. About 150. R : Justify your answer please. S: Okay. 48% is almost 50%. That is half 300.
Fifteen students (50%) answered incorrectly. Six of these students rounded 48–50% and then divided 300 by 50, obtaining 6 as the answer. Here is an example: S : Yes. 48% is about 50%. 300 divided by 50 is 6. R: Why did you divide 300 by 50? S : Because 48% is about 50%. R: Okay. But why did you use division here? S: Because it is
1092
OTHMAN N. ALSAWAIE
a percent. When you have percents you divide. R: Can you think of another way for doing the problem? S: No. It should be division.
Five students rounded 48–50% and multiplied 300 by 2. This is how one responded to the interviewer’s questions: S: Okay. 48% is rounded to 50%. Fifty percent means 50/100. We multiply 300 by 2. Three hundred times two is six hundred. R: Where did you get the two from? S: One hundred over fifty is two. R: You said 50% is 50 over 100 not 100 over 50. S: No. No. When you divide you convert division into multiplication and you switch the numerator and the denominator. R: But, is it reasonable that 50% of 300 is 600. S: Yes. Because when you multiply, the number becomes bigger. R: Can you do it another way? S: No. That is the rule for it.
Two students rounded 48–50% and multiplied 300 by 50. These students claimed that this was a rule that was taught by the teacher. Two other students arbitrarily answered 100 and 200, and could not justify their answers. These strategies or rules used by students reflect a real problem in learning mathematical operations and algorithms. It seems that the meaning of operations and algorithms were missing or confused in the minds of students. Participants in this study were selected as highachieving students based on their achievement in mathematics, and yet they used non-number sense procedures for solving the problems. At least 13 out of 15 students who answered incorrectly did not understand that 50% of 300 means a fraction of 300. With this lack of understanding, some of them divided 300 by 50, and some others multiplied it by 2 or 50. Actually, it seems they did not attach any meaning to 50% of 300. Otherwise, they would have used these meanings in justifying their procedures and answers. Their explanations focused only on rules that they thought to be correct. Item (8): Which is Larger 83 ÷ 0.05 or 83 ÷ 0.025? Seventeen students (56.7%) answered this item correctly. Only eight of them reasoned correctly as they had decided first that dividing by a smaller number gives a larger quotient and second that 0.025 is less than 0.05. Another eight students lacked understanding of place value and number magnitude as they considered 0.025 to be greater than 0.05 and decided that dividing by a larger decimal makes the quotient larger. So
NUMBER SENSE STRATEGIES
1093
the answers of these students were falsely correct as they obtained correct answers by using incorrect rules or reasoning. Some of these students considered 0.025 to be greater than 0.05 because “25 is greater than 5.” Some other students had a different reason: “0.025 represents parts of a thousand but 0.05 represents parts of a hundred. And since 1000 is larger than 100, parts of a thousand are larger than parts of a hundred.” One student could not justify his answer. Thirteen students answered the item incorrectly. Ten of these students reasoned that dividing by a larger number makes the quotient smaller, but they considered 0.025 to be greater than 0.05. On the contrary, three students knew that 0.05 is larger than 0.025, but considered that dividing by a decimal “reverses the rule of division.” That is, “dividing by a larger decimal makes the quotient larger.” The results of this item raise three important points. First, it seems that many students do not understand the conceptual meaning of division, that is, converting the dividend into a number of parts each of which is equal to the divisor. When this meaning is present in students’ minds, it is easy for them to see that a smaller divisor makes the result bigger, without memorizing the rule. Second, comparing decimals seems to be challenging to many students, obviously because of the lack of understanding of the concept of a decimal. Many students treat digits to the right of the decimal point as whole numbers and base their comparison on that understanding. Third, falsely correct answers might mislead teachers and send them the wrong message regarding the understanding of their students. Item (10): Is A or B Larger? A = 8 × 12 × 7 × 16, B = 7 × 6 × 32 × 8 Eleven students (36.7%) answered this item correctly. Only three of them used number sense to solve the problem. All three students started by eliminating 8 and 7 from A and B before they continued solving the problem. Two of them decomposed 12 into 2 × 6 and then recomposed 2 × 16 as 32 to obtain B: “12 × 16 = 2 × 6 × 16 = 6 × 32. Therefore, A = B.” The third student solved the problem using a different approach. After eliminating 7 and 8 from A and B, she reasoned as follows: “12 is twice 6 and 32 is twice 16. Then the two [A and B] are equal.” This student mentally decomposed 12 and 32 into 2 × 6 and 2 × 16, respectively, and used the associative property of multiplication to conclude that A = B. When asked to clarify her thinking, she wrote (2 × 6) × 16 = 6 × (2 × 16) and said: “The 2 here [pointing to the 2 in 2 × 16] can be put here
1094
OTHMAN N. ALSAWAIE
[pointing to the left of 6]. So, they are the same.” The other eight students who gave correct answers could not provide any explanation to their answers. This means they either answered by guessing or they cannot communicate their thinking mathematically. Nineteen students answered the item incorrectly. Following are the types of their responses: 1. B is larger: 7 = 7, 8 = 8, 12−6 = 6, 32−16 = 16, 16−6 = 10. So, B is larger by 10. Six students used this strategy to solve the problem. Instead of thinking about the multiplicative relations between the numbers involved, these students tended to think additively. That is, instead of thinking of 12 as 2 × 6 and of 32 as 2 × 16, they thought of 12−6 and 32−16, respectively. This is a sign of mathematical and logical immaturity. 2. B is larger because it contains 32 and multiplying by a large number such as 32 gives a very large product. Three students thought B is larger just because 32 is a large number compared with the other numbers involved. Again, understanding of the multiplicative domain seems to be lacking. In turn, these students do not have flexibility in seeing numbers as products of other numbers. 3. ”I can’t do it without a calculator.” Ten students did not even try the problem without calculators.
REFERENCES Baroody, A. J. & Coslick, R. T. (1998). Fostering children's mathematical power: An investigative approach to K-8 mathematics instruction. Mahwah, NJ: Erlbaum. Behr, M., Harel, G., Post, T. & Lesh, R. (1992). Rational number, ratio and proportion. In D. Grouws (Ed.), Handbook of research on mathematics teaching and learning (pp. 296–333). New York, NY: Macmillan Publishing. Burns, M. (1989). Teaching for understanding: A focus on multiplication. In P. R. Trafton & A. P. Shulte (Eds.), New directions for elementary school mathematics (pp. 123–134). Reston, VA: NCTM. Faulkner, V. N. (2009). Components of number sense: An instructional model for teachers. Teaching Exceptional Children, 41(5), 24–30. Garet, M. S., Porter, A. C., Desimone, L., Birman, B. F. & Suk Yoon, K. (2001). What makes professional development effective: Results from a national sample of teachers. American Educational Research Journal, 38(4), 915–945. Garner, R. (1992). Learning from school texts. Educational Psychologist, 27, 53–63. Ghazali, M., Abdul Rahman, S., Ismail, Z., Sharifah Norhaidah Idros, S. N., & Salleh, F. (2003). Development of a framework to assess primary students’ number sense in
NUMBER SENSE STRATEGIES
1095
Malaysia. Proceedings of the International Conference, the Decidable and the Undecidable in Mathematics Education, Brno, Czech Republic. The Mathematics Education into the 21st Century Project. Greenes, C., Schulman, L. & Spungin, R. (1993). Developing sense about numbers. Arithmetic Teacher, 40(5), 279–284. Greeno, J. G. (1991). Number sense as situated knowing in a conceptual domain. Journal for Research in Mathematics Education, 22, 170–218. Greer, B. (1992). Multiplication and division as models of situations. In D. A. Grouws (Ed.), Handbook of research on mathematics teaching and learning (pp. 276–295). New York, NY: Macmillan. Hiebert, J. C. (1999). Relationships between research and the NCTM standards. Journal for Research in Mathematics Education, 30, 3–19. Hiebert, J. & Lindquist, M. M. (1990). Developing mathematical knowledge in the young child. In J. N. Payne (Ed.), Mathematics for the young child (pp. 17–36). Reston, VA: National Council of Teachers of Mathematics. Hiebert, J. & Stigler, J. W. (2000). A proposal for improving classroom teaching: Lessons from the TlMSS video study. The Elementary School Journal, 101l, 3–20. Hiebert, J. & Wearne, D. (1986). Procedures over concepts: The acquisition of decimal number knowledge. In J. Hiebert (Ed.), Conceptual and procedural knowledge: The case of mathematics (pp. 199–223). Hillsdale, NJ: Erlbaum. Hill, H. C., Rowan, B. & Ball, D. L. (2005). Effects of teachers' mathematical knowledge for teaching on student achievement. American Educational Research Journal, 42(2), 371–406. Howden, H. (1989). Teaching number sense. Arithmetic Teacher, 36(6), 6–11. Jones, G. A., Langrall, C. W., Thornton, C. A. & Nisbet, S. (2002). Elementary students’ access to powerful mathematical ideas. In L. English, M. B. Bussi, G. A. Jones, R. A. Lesh & D. Tirosh (Eds.), Handbook of international research in mathematics education (pp. 113–141). Mahwah, NJ: Erlbaum. Kamii, C. (1989). Young children reinvent arithmetic: Implications of Piaget's theory. New York, NY: Teachers College Press. Kamii, C., Lewis, B. A. & Livingston, S. J. (1993). Primary arithmetic: Children inventing their own procedures. Arithmetic Teacher, 41(4), 200–203. Kerslake, D. (1986). Fractions: Children's strategies and errors: A report of the Strategies and Errors in Secondary Mathematics Project. Windsor, England: NFERNelson. Knuth, E. (2002). Teachers' conceptions of proof in the context of secondary school mathematics. Journal of Mathematics Teacher Education, 5(1), 61–88. Markovits, Z. (1989). Reactions to the number sense conference. In J. T. Sowder & B. P. Schappelle (Eds.), Establishing foundations for research on number sense and related topics: Report of a conference (pp. 78–81). San Diego, CA: San Diego University, Center for Research in Mathematics and Science Education. Markovits, Z. & Sowder, J. T. (1994). Developing number sense: An intervention study in grade 7. Journal for Research in Mathematics Education, 25(1), 4–29. McIntosh, A., Reys, B. J. & Reys, R. E. (1992). A proposed framework for examining basic number sense. For the Learning of Mathematics, 12, 2–8. McIntosh, A., Reys, B. J., Reys, R. E., Bana, J. & Farrell, B. (1997). Number sense in school mathematics: Student performance in four countries. Perth, Australia: Edith Cowan University.
1096
OTHMAN N. ALSAWAIE
Ministry of Education (2001). The national document for mathematics curriculum in public education. Dubai, UAE: Ministry of Education. National Council of Teachers of Mathematics (1989). Curriculum and evaluation standards for school mathematics. Reston, VA: NCTM. National Council of Teachers of Mathematics (2000). The principles and standards for school mathematics. Reston, VA: NCTM. Norul Akmar, S. (1997). Two students subtraction of integers schemes. Journal of Education, 18, 75–96. Peck, D. M. & Jencks, S. M. (1981). Conceptual issues in the teaching and learning of fractions. Journal for Research in Mathematics Education, 12(5), 339–348. Post, T. R., Cramer, K. A., Behr, M., Lesh, R. & Harel, G. (1993). Curriculum indications from research on the learning, teaching and assessing of rational number concepts: Multiple research perspective. In T. Carpenter & E. Fennema (Eds.), Learning, teaching and assessing rational number concepts: Multiple research perspective. Madison: University of Wisconson. Reys, R. E. & Yang, D. C. (1998). Relationship between computational performance and number sense among sixth- and eighth-grade students in Taiwan. Journal for Research in Mathematics, 29, 225–237. Sackur-Grisvad, C. & Leonard, F. (1985). Intermediate cognitive organizations in the process of learning a mathematical concept: The order of positive decimal numbers. Cognition and Instruction, 2, 157–174. Schmidt, W. H., Jakwerth, P. M. & McKnight, C. C. (1998). Curriculum-sensitive assessment: Content does make a difference. International Journal of Educational Research, 29, 503–527. Schneider, S. B. & Thompson, C. S. (2000). Incredible equations develop incredible number sense. Teaching Children Mathematics, 7(3), 146–148. Sheetal Sood, S. & Jitendra, A. K. (2007). A comparative analysis of number sense instruction in reform-based and traditional mathematics textbooks. The Journal Of Special Education, 41(3), 145–157. Sowder, J. (1992). Estimation and number sense. In D. Grouws (Ed.), Handbook of research on mathematics teaching and learning (pp. 371–389). New York, NY: Macmillan. Sowder, J. T. & Markovits, Z. (1989). Effects of instruction on number magnitude. In C. A. Maher, G. A. Goldin & R. B. Davis (Eds.), Proceedings of Eleventh Annual Meeting: North American Charter of the International Group far the Psychology o f Mathematics Education (pp. 105–110). New Brunswick, NJ: Rutgers University Center for Mathematics, Science, and Computer Education. Sowder, J. T. & Markovits, Z. (1994). Developing number sense: An intervention study in grade 7. Journal for Research in Mathematics Education, 25(1), 4–29. Sowder, J. T. & Schappelle, B. P. (Eds.). (1989). Establishing foundations for research on number sense and related topics: Report of a conference. San Diego, CA: San Diego State University, Center for Research in Mathematics and Science Education. Sowder, J. T. & Wheeler, M. M. (1987). The development of computational estimation and number sense: Two exploratory studies. San Diego, CA: San Diego State University Center for Research in Mathematics and Science Education. Stigler, J. W. & Hiebert, J. (2004). Improving mathematics teaching. Educational Leadership, 61(5), 12–17.
NUMBER SENSE STRATEGIES
1097
Stigler, J. W., Lee, S. Y. & Stevenson, H. W. (1991). Mathematical knowledge of Japanese, Chinese, and American elementary school children. Reston, VA: NCTM. Suter, L. E. (2000). Is student achievement immutable? Evidence from international studies on schooling and student achievement. Review of Educational Research, 70, 529–545. Threadgill-Sowder, J. (1984). Computational estimation procedures of school children. The Journal of Educational Research, 77, 332–336. Trafton, P. R. (1989). Reflections on the number sense conference. In J. T. Sowder & B. P. Schappelle (Eds.), Establishing foundations for research on number sense and related topics: Report of a conference (pp. 74–77). San Diego, CA: San Diego University, Center for Research in Mathematics and Science Education. Tsao, Y. L. (2005). The number sense of preservice elementary school teachers. College Student Journal, 39(4), 647–679. Van de Walle, J. A. (2007). Elementary and middle school mathematics: Teaching developmentally (6th ed.). New York, NY: Longman. Yang, D. C. (2005). Number sense strategies used by sixth grade students in Taiwan. Educational Studies, 31(3), 317–333. Yang, D. C. & Huang, F. Y. (2004). Relationships among computational performance, pictorial representation, symbolic representation and number sense of sixth-grade students in Taiwan. Educational Studies, 30, 373–389.
College of Education United Arab Emirates University P.O. Box 17551 Al-Ain, United Arab Emirates E-mail:
[email protected]