2003 Teaching and Learning Number Sense –

DER-CHING YANG

TEACHING AND LEARNING NUMBER SENSE –AN INTERVENTION STUDY OF FIFTH GRADE STUDENTS

IN TAIWAN

ABSTRACT. Two classes (one experimental and one control) in a public elementaryschool located in southern Taiwan participated in this study. Number sense activities wereconducted in the experimental class as supplementary teaching materials, while the controlclass followed the standard mathematics curriculum. Data indicate that there are statisti-cally significant differences between pretest and posttest (pretest and retention-test) scoresfor the experimental and control classes at the 0.01 level. The scores for the experimentalclass increased 44% after instruction (the mean score went from 12.35 to 17.81), while thescores for the control class increased only 10% after instruction (the mean score went from11.29 to 12.42). Compared to the control class, the experimental class made much moreprogress on number sense tests. Results indicate that students in the teaching class (notincluding the students in the low level) advanced in their use of number sense strategieswhen responding to interview questions. The data demonstrate that the teaching of numbersense activities, executed in the experimental class, is effective in developing children’snumber sense. Furthermore, the results of retention demonstrate that the students’ learningwas meaningful and significant.

KEY WORDS: benchmarks, control class, estimation, experimental class, number sense

RATIONALE AND PURPOSE

The teaching and learning of number sense is considered to be a majortopic in international mathematics curricula (Anghileri, 2000; AustralianEducation Council, 1991; Cockcroft, 1982; Japanese Ministry of Educa-tion, 1989; Markovits & Sowder, 1994; McIntosh, Reys, Reys, Bana & Far-rel, 1997; National Council of Teachers of Mathematics (NCTM), 1989,2000; National Research Council, 1989). The Curriculum and EvaluationStandards for School Mathematics highlighted that the teaching of numbersense is an essential goal of a school’s mathematics curriculum. Further-more, the Number and Operations Standard of Principles and Standardsfor School Mathematics (PSSM) (NCTM, 2000) states that “central to thisStandard is the development of number sense” (p. 32).

Over the past decade, several research studies in Taiwan have focusedon fractional concepts and children’s conceptions of numbers (Lin, 1989;

International Journal of Science and Mathematics Education 1: 115–134, 2003.© 2003 National Science Council, Taiwan. Printed in the Netherlands.

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Ning, 1992), but research related to number sense has received little at-tention. During the past two years, due to its emphasis in the NCTMStandards, number sense has stimulated a growing amount of attentionand research in Taiwan. There is currently an important mathematics edu-cational revolution in Taiwan. The guidelines for a Nine-Year Joint Math-ematics Curricula Plan (Ministry of Education in Taiwan, 2000) stressesthat the learning and teaching of mathematics should highlight meaning-ful connections with real life. The new guidelines for the mathematicscurricula plan and number sense both focus on meaningful learning. Ifimportant reforms of mathematics education in Taiwan are to be made, thedevelopment of number sense should be integrated into the mathematicscurriculum.

The purpose of this study was to report the results of an interventionstudy conducted on Taiwanese fifth graders. The related research questionis:

Can number sense be fostered by appropriate teaching?

BACKGROUND

What Is Number Sense?

Number sense refers to a person’s general understanding of numbers andoperations and the ability to handle daily-life situations that include num-bers. This includes the ability to develop useful, flexible, and efficientstrategies (i.e., mental computation or estimation) for handling numeri-cal problems (Howden, 1989; McIntosh, Reys & Reys, 1992; Reys, 1994;Reys & Yang, 1998; Sowder, 1992a, 1992b; Treffers, 1991; Yang, 2002a,2002b).

What Are the Number Sense Components?

Since number sense has been an important topic in mathematics educa-tion, it has produced much research and discussion among mathematicseducators, cognitive psychologists, researchers, teachers, and mathematicscurricula developers (Howden, 1989; Greeno, 1991; Markovits & Sowder,1994; McIntosh et al., 1992; NCTM, 1989, 2000; Reys, 1994; Reys &Yang, 1998; Sowder, 1992a, 1992b; Yang, 2002a, 2002b). The researchergeneralized the above research reports and defined the number sense com-ponents as following:

(1) Understanding the basic number meanings: This implies making senseof numbers and developing a conceptual understanding of numbers.

TEACHING AND LEARNING NUMBER SENSE 117

(2) Recognizing the magnitude of numbers: It includes the ability to com-pare numbers (whole numbers, fractions, decimals, and so on), toorder numbers correctly, and to recognize the density of numbers.

(3) Using benchmarks appropriately: It includes the ability to develop andflexible use the benchmarks, such as 1, 1/2, 100, and so on, in differentsituations.

(4) Understanding the relative effect of operations on numbers: It includesthe ability to identify how the different operations affect the result ofnumerical problems.

(5) Developing different strategies appropriately and assessing the rea-sonableness of an answer: This implies developing different strategies(i.e., estimation, mental computation) to solve problems appropriatelyand knowing that the result is reasonable.

The above five number sense components constitute the basis for thisstudy.

The Role of Number Sense in the Taiwanese Mathematics Curriculum

A major reform of Taiwanese mathematics curriculum standards fromgrade 1 to 9 has been underway since 2001. In 2000, the Guidelines fora 9-Year Joint Curricula Plan (Guidelines (Ministry of Education in Tai-wan, 2000)) was released and covered all subjects, including mathematics.Number sense has been considered an important topic in school mathemat-ics curriculum (Anghileri, 2000; Japanese Ministry of Education, 1989;NCTM, 1989, 2000; National Research Council, 1989). However, the topicof “number sense” has not been integrated into the new mathematics cur-ricula. If significant improvements in mathematics are to be made in Tai-wan, the development of number sense should be considered as an impor-tant topic in the new curricula.

The new Guidelines publication does not provide a standard nationalpolicy for the teaching and learning of number sense. However, the newstandards do emphasize that mathematics teaching and learning shouldfocus on making connections between mathematics and daily life situa-tions. Significantly, the goals of the Guidelines for mathematics curriculaand the spirit of the development of number sense are consistent. Eventhough it lacks a clear policy on the teaching and learning of number sense,the Guidelines emphasizes many of the fundamental principles of numbersense.

Research Studies Related to Teaching and Learning of Number Sense

Several research projects ( Cobb, Wood, Yackel, Nicholls, Wheatly, Tri-gatti & Perlwitz, 1991; Treffers, 1991; Warrington & Kamii, 1998) have

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demonstrated that students who participate in well-designed activities aremore likely to develop number concepts than students who receive in-struction focusing on the development of standard written algorithms andcomputational proficiency. Several studies (Anghileri, 2000; Markovits &Sowder, 1994; Yang, 2002b; Yang & Reys, 2001a, 2001b) have demon-strated the effectiveness of instruction designed to promote number sense.The study of Cramer, Post & delMas (2002) showed that students in theRNP can develop fractional number sense because “they spent their timeinteracting with fraction ideas in multiple ways and were provided ex-tended periods of time to develop an understanding of the meaning ofsymbols” (p. 140). Yang’s study (2002a, 2002b) further demonstrated thatsixth graders’ number sense can be developed through process-orientedteaching models.

METHOD

Sample

A public school located in a city in southern Taiwan was selected to partic-ipate in this study. Two classes (an experimental class and a control class)were studied. Number sense activities were conducted in the experimentalclass as supplementary teaching materials, while the control class followedthe standard mathematics curriculum. The school, which comprised about3000 students, serves children from diverse areas. The students in the studycome from families with a wide range of occupations, incomes, and edu-cational levels. The experimental class consisted of 37 students (20 boysand 17 girls), and the control class consisted of 38 students (20 boys and18 girls).

Based on the students’ performance on the pretest, students in eachclass were divided into the following three levels: High – top 10%; Middle– 40–60%; Low – bottom 10%. Two students were randomly selected fromeach level and interviewed before instruction, after instruction, and fourmonths after the study. Therefore, the sample consisted of 12 students, in-cluding the low- (TL12, CL12), middle- (TM12, CM12), and high- (TH12,CH12) level students.

Instructional Activities

This study defined the five number sense components based on the existingnumber sense related research reports and documents as described earlier.The instructional activities were designed according to the above numbersense framework. Five units of number sense activities were designed for


the use of the teacher. Each unit included 5 lessons. For example, one ofthe estimation lesson was as following:

1. (a) Please estimate the area and perimeter of the playground in ourschool.Explain how you reached your answer.

(b) Estimate how many students can be contained in our school’s play-ground. Explain how you reached your answer.

Assessment Instruments

Assessment instruments included paper-and-pencil tests and interviews.

Paper-and-Pencil Tests included Pretest, Posttest, and Retention test, allof which used the “Number Sense Rating Scale (NSRS)” designed by Hsuet al. (2001). The test items in the NSRS based on the related researchstudies McIntosh et al. (1992, 1997). The NSRS was given in multiplechoice format and included 37 items. The reliability test of internal consis-tency for the NSRS was 0.889. The test-retest reliability coefficient for theNSRS was 0.894 at the 0.01 level. The NSRS also has high content validity,specialist validity, and construct validity (p. 368). The test was given fora period of about 35 min. There are clear explanation about NSRS in thestudy of Hsu, Yang & Li (2001).

Interview Instruments. Pre-interview, Post-interview, and Retention-in-terview questions were designed by the researcher. All of the three testsused the same items. Each interview included 15 questions designed toinvestigate whether or not students could effectively use the five numbersense components as described in the number sense framework. Studentswere given as much time as they needed to respond to each question,and each interview lasted about 60 min. All interviews were video-tapedand later transcribed. The schedule for instructional units and assessmentinstruments is presented in Table I.

Analysis

The NSRS included 37 items. Each item was given 1 point if correct,no point if incorrect. No partial credit was awarded. Therefore, the totalpossible score for the test is 37 points.

Interview. The students’ responses were examined and sorted. In aneffort to identify the different strategies used by students, each responsewas coded (as correct or incorrect) according to one of the following threecategories:

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TABLE I

Schedule for instructions and assessments

Date

02/01/01–08/30/01 Review the related papers and design the number sense project

08/30/01–09/08/01 Pretest and pre-interview for both Exp- and Con-class

09/10/01–12/31/01 Number sense activities integrated into Exp-class

01/01/02–01/18/02 Posttest and post-interview for both Exp- and Con-class

01/19/02–02/17/02 Winter break

02/18/02–06/01/02 No number sense activities were integrated into classes

06/01/02–06/12/02 Retention-test and retention interview for Exp- and Con-class

• Number sense based – strategies that utilized one or more componentsof number sense (i.e., benchmarks, number magnitude, relative effectof operations on numbers).

• Rule based – applied the rules of standard written algorithms but wasunable to go beyond the direct application rule in the explanation.

• Could not explain – despite probes and queries by the interviewer,clear explanations could not be obtained.

The researcher and the teacher used the transcripts to categorize re-sponses to each correct or incorrect answer. Another teacher served assecond coder by using the same transcripts and categorizing the responsesindependently. These initial reviews produced categorization agreement onmore than 90% of the students’ responses. The remaining responses werereexamined and discussed by both coders until a consensus was reached.

HOW WERE THE NUMBER SENSE ACTIVITIES INTEGRATED INTO

THE MATHEMATICS CLASS?

Reys (1994) advocated providing a class with process-oriented activitiesand establishing a classroom environment that encourages meaningful dis-cussion, exploration, thinking, and reasoning is the best way for studentsto develop number sense. Several research studies (Anghileri, 2000; Reys,1994; Fraivillig, 2001) further supported the idea that the effective teachingof mathematics should focus on the learning process. This process not onlyhighlights small-group problem solving and whole class discussions to de-velop mathematical thinking and reasoning, but also encourages studentsto defend, query, and prove their ideas. In order to effectively help children


Figure 1. Process-oriented teaching model.

develop number sense, the researcher designed the process-oriented teach-ing model (Figure 1) to help the teacher effectively integrate number senseactivities into mathematics class.

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RESULT

Table II reports the results of mean scores (correct percent), standard devi-ations, and t-test results for the experimental class and the control class.

The data indicate that there are statistically significant differences be-tween pretest and posttest (pretest and retention-test) scores for bothclasses. However, the scores for the experimental class increased 44% afterinstruction (the mean score went from 12.35 to 17.81), while the scoresfor the control class increased only 10% after instruction (the mean scorewent from 11.29 to 12.42). Compared to the control class, the experimentalclass made much more progress on number sense tests. This indicates thatthe teaching of number sense is highly effective. The retention test resultsfurther indicated that the teaching experiment was meaningful.

Tables III and IV report the results of students’ responses in the experi-mental class and the control class on pre-, post-, and retention-interviews.

The results indicate that students in the teaching class (except for thestudents in the low level) progressed much more in their use of numbersense strategies when responding to interview questions. The results of theretention-interview further show that students’ learning was meaningfuland therefore long-lasting. However, students’ responses in the controlclass did not advance in the use of number sense strategies after the in-struction. In order to highlight the differences in the development and useof number sense strategies between the experimental class and the controlclass, several interview questions were presented before instruction, afterinstruction, and four month after the study.

TABLE II

The mean scores (correct percent), standard deviations, and t-tests results ofnumber sense pretest, posttest, and retention tests for both experimental andcontrol classes

Class Test Mean score (Correct%) SD T-value

Pretest 12.35 (33.4%) 4.83 −10.949∗Exp-classes Posttest 17.81 (48.1%) 5.29(N = 37) Pretest 12.35 (33.4%) 4.83 −13.218∗

Retention 19.08 (51.2%) 5.46

Pretest 11.29 (30.5%) 4.09 −3.775∗Con-classes Posttest 12.42 (33.6%) 3.94(N = 38) Pretest 11.29 (30.5%) 4.09 −4.734∗

Retention 13.23 (35.9%) 4.35

Note. ∗P < 0.01.


TABLE III

The results of students’ responses in the experimental class (and control class) on pre-,post-, and retention-interviews

TH1 TH2 TM1 TM2 TL1 TL2(CH1) (CH2) (CM1) (CM2) (CL1) (CL2)

Pre-interviewCorrect

NS-based 6 (9) 4 (5) 4 (4) 2 (3) 2 (3) 2 (3)Rule-based 2 (4) 5 (3) 2 (0) 3 (1) 0 (0) 0 (1)Couldn’t Ex 1 (0) 2 (1) 2 (2) 2 (2) 4 (2) 3 (2)

IncorrectNS-based 0 (0) 0 (0) 0 (0) 0 (0) 0 (0) 0 (0)Rule-based 0 (2) 0 (3) 0 (4) 2 (4) 0 (3) 0 (3)Couldn’t Ex 6 (0) 4 (3) 7 (5) 6 (5) 9 (7) 10 (6)

Post-interviewCorrect



Retention-interviewCorrect



Notes. 1. Each interview included 15 questions.2. NS- (rule)-based represents number sense- (rule)-based method; Couldn’t Ex

means couldn’t give clear explanation.

Table IV presents a summary of results from the fifth grader’s responsesin the Exp- and Con-class on the pre-, post-, and retention-interview aboutquestion 1. This question focused on investigating the students’ conceptualunderstanding of fractions. Data indicated that only one student could use

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TABLE IV

Results from fifth grade students’ responses to the following problem:

1. Two same sized watermelons were shared by five students equally. How manywatermelons can each student have? Why?

(1) Greater than a half of a watermelon

(2) Equal to 15 of a watermelon

(3) Equal to 25 of a watermelon

(4) Greater than 2.5 watermelons

Exp-A Con-B

Pre- Post- Retention Pre- Post- Retention

CorrectNS-based 1 (17%) 6 (100%) 6 (100%) 2 (33%) 3 (50%) 3 (50%)Rule-based 2 (33%) 0 (0%) 0 (0%) 0 (0%) 1 (17%) 1 (17%)Couldn’t explain 1 (17%) 0 (0%) 0 (0%) 1 (17%) 1 (17%) 1 (17%)

IncorrectNS-based 0 (0%) 0 (0%) 0 (0%) 0 (0%) 0 (0%) 0 (0%)Rule-based 0 (0%) 0 (0%) 0 (0%) 1 (17%) 0 (0%) 0 (0%)Couldn’t explain 2 (33%) 0 (0%) 0 (0%) 2 (33%) 1 (17%) 1 (17%)

Notes. 1. Percentages may not add to 100 because of rounding.2. Each class had six students who were interviewed.

the number sense-based method to answer question 1. For example, (Pre-interview of TH2):

R: Can you tell me the answer and your reasons about question 1?

TH2: One watermelon was shared by five persons, so each one got 15 wa-

termelon. Since there are two watermelons, each person can get 25

watermelon.

TH2 gave the correct answer and reasonable explanations, so this responsewas coded as Number sense-based method. Two (33%) students gave thecorrect answer, by using the rule-based method. For example, (Pre-inter-view of TM2):

TM2: 2 ÷ 5 = 25 , so the answer is 2 ÷ 5 = 2

5 .

R: Can you tell me why the answer is 25?

TM2: Because 2 ÷ 5 = 25 , so the answer is 2 ÷ 5 = 2

5 .

R: Can you explain it in a different way?

TM2: I don’t know.


Since TM2 used the rule “2 ÷ 5 = 25 ”, but could not give an explanation,

this response was coded as “ruled-based”. Another student produced thecorrect answer but also couldn’t explain it (Pre-interview of TH1):

TH1: The answer is 25 .

R: Can your tell me your reasons?TH1: I don’t know how to explain it.

This was coded as “correct and could not explain.”Two students gave incorrect answers and also could not explain them

(Pre-interview of TL1):

TL1: The answer is 15 .

R: Can your tell me your reasons?TL1: I don’t know.

During the post- and retention-interview, all six students in the exper-imental class could use the number sense-based method to explain theirreasoning. In addition, four of the six students could support their answerswith pictorial representations. For example, (Post-interview of TM2):

R: Can you tell me your answer and reasons about question 1?TM2: As you can see the picture, each watermelon can be divided into 5

parts. Each person can have 15 of one watermelon. Since there are

two watermelons, each one can get 25 watermelons.

In Con-class, the students’ responses showed not much difference on thepre-, post-, and retention-interviews. Two (33%) students used the numbersense-based method to answer question 1 on the pre-interview, however,only three (50%) students could apply the number sense strategy to explaintheir reasons on the post- and retention-interviews. One student was ableto give the correct answer, yet could not explain why on the pre-, post-, andretention-interviews. For example, (pre-, post-, and retention-interviews ofCM2):

R: Can you tell me your answer and reasons about question 1?CM2: 2

5 . I don’t know how to explain it.

One student gave an incorrect answer on pre-, post-, and retention-inter-views. CL2 answered: “1

5 ” and “I don’t know how to explain it.”The data indicate that students in the Exp-class more frequently used

number sense strategies during the post and retention-interviews (100% vs.17%), and were more successful than students in the control class (100%vs. 50%).

Table V presents a summary of the results from fifth grader’s responsesfor question 6. This question focused on investigating whether or not stu-dents could use benchmarks. It is a more challenging question than the

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TABLE V

Results from fifth grade students’ responses to the following problem:

6. Without calculating, select the best estimate for 45 + 6

7 ? Why?

(1) 12 (2) 10 (3) 2 (4) 1 (5) Without calculating can’t find the answer

Exp-A and B Con-A and B


Correct

NS-based 0 (0%) 3 (50%) 4 (66%) 0 (0%) 0 (0%) 0 (0%)

Rule-based 1 (17%) 0 (0%) 0 (0%) 1 (17%) 2 (33%) 2 (33%)

Couldn’t explain 0 (0%) 1 (17%) 0 (0%) 1 (17%) 1 (17%) 1 (17%)

Incorrect

NS-based 0 (0%) 0 (0%) 0 (0%) 0 (0%) 0 (0%) 0 (0%)

Rule-based 3 (50%) 1 (17%) 1 (17%) 2 (33%) 1 (17%) 1 (17%)

Couldn’t explain 2 (33%) 1 (17%) 1 (17%) 2 (33%) 2 (33%) 2 (33%)

above question. The data indicate that no student in either class could usethe benchmark on the pre-interview. One student in each class applied therule-based method. For example, (pre-interview of TH2):

R: Can you tell me the answer for question 6 and your reasoning?TH2: I think the answer is 2.

R: Can you tell me your reasons?TH2: Because

4

5+ 6

7= 28

35+ 30

35= 58

35= 1

23

35,

the answer is close to 2.R: Can you do it by another way?

TH2: (Hesitate for a while!) I don’t know.R: How did you know to find the sum of two fractions with different

denominators?TH2: I learned it at private school.

This kind of question (find the sum of two fractions with different de-nominators) is not taught until the seventh grade in the Taiwanese newmathematics curricula. However, TH2 had learned this way at a privateschool.

Three (50%) students in the Exp-class and two (33%) students in theCon-class gave incorrect answers based on rule-methods. For example,(Pre-interview of TH1):


TH1: The answer is 10.R: Why? Please tell me your method.

TH1: Because 6 + 4 is equal to 10.R: Can you do it another way?

TH1: (Hesitating for a while!) I don’t know.

Or (Pre-interview of CM1)

CM1: 1.R: Why? Can you tell me your reasons?

CM1: 45 + 6

7 = 1012 , hence the answer is close 1.

R: Can you do it another way?CM1: (Shaking her head!) I don’t know.

These methods were coded as incorrect and rule-based. Two students ineach class gave incorrect answers and no explanations. For example, (Pre-interview of TL1):

TL1: The answer is about 1.R: Can you tell me your reasons?

TL1: I guess.

Or (Pre-interview of CL2)

CL2: I don’t know.

Three (50%) students in the post-interview and four (66%) students in theretention-interview used benchmarks to solve this question. For example,(Post-interview of TH1):

TH1: 2.R: Can you tell me your reasons?

TH1: Because 45 is less than 1 and near 1, 6

7 is also less than 1 and near 1.Therefore, the sum is about 2.

Or (Post-interview of TH2)

TH2: The answer is 2.R: Why? Can you tell me your reasons?

TH2: Since 45 is near 1 and 6

7 is also near 1, the sum is near 2.R: Can you do it by different way?

TH2: I can find the common denominator and compute it.R: Can you tell me what’s the difference between these methods.

TH2: I can save time, so I don’t need to spend a lot of time to do complexcomputation. It is also more meaningful and useful to me.

However, no student in the control class could use 1 as a benchmark. In theCon-class, two (33%) students in the post- and retention-interview applied

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TABLE VI

Results from fifth grade students’ responses to the following problem

8. Without calculating, which one 710 or 7

12 is larger? Why?

(1) 710 (2) 7

12 (3) same (4) Without calculating can’t find the answer

Exp-A Con-B


Correct

NS-based 1 (17%) 4 (66%) 4 (66%) 0 (0%) 0 (0%) 0 (0%)

Rule-based 1 (17%) 0 (0%) 0 (0%) 2 (33%) 4 (66%) 4 (66%)

Couldn’t explain 1 (17%) 0 (0%) 0 (0%) 2 (33%) 1 (17%) 1 (17%)

Incorrect

NS-based 0 (0%) 0 (0%) 0 (0%) 0 (0%) 0 (0%) 0 (0%)

Rule-based 0 (0%) 0 (0%) 0 (0%) 0 (0%) 0 (0%) 0 (0%)

Couldn’t explain 3 (50%) 2 (33%) 2 (33%) 2 (33%) 1 (17%) 1 (17%)

written algorithms. They found the common denominator first and got theexact answer, then decided on the answer. Two (33%) students continuedto use the same method before and after instruction. For example, (Post-interview and retention-interview of CL2):


As shown in Table VI, students in the Exp-class were not only moresuccessful on the post- and retention-interviews than the pre-interview inapplying benchmarks (50% vs. 0%) and (66% vs. 0%), but they also usedbenchmarks more frequently than students in the Con-class on the post-and retention-interviews (50% vs. 0%) and (66% vs. 0%).

Table VI presents data from the fifth grader’ responses for the Exp-and Con-class on comparing fractional size: without calculating, is 7

10 or7

12 is larger? The fifth graders’ mathematics textbooks in Taiwan do notteach children to find a least common denominator or cross-product proce-dure for comparing fractions. The instruction for the Exp-class focused ondeveloping conceptual understanding through pictorial and verbal repre-sentations. The textbooks used in the Con-class taught children to comparefractions by using concrete materials. For example, when comparing 3

8 and5

12 , the teacher gave students 24 pieces of chips as a unit and taught themto find the pieces of 3

8 and 512 . 24 × 3

8 = 9; 24 × 512 = 10, therefore, 5

12 islarger because it equals 10 pieces of chips.

On the pre-interview, one student (17%) in the Exp-class responded:“A pizza was cut into 10 pieces and the other pizza was cut into 12 pieces.


Each piece in the 10 pieces group is larger than in the 12 piece group, so7

10 is larger than 712 .” This was coded as number sense-based. No student

in the Con-class could apply a number sense-based method. One studentin the Con-class memorized the rule but was unable to explain his reason.For example, (Pre-interview of CH1):

R: Can you tell me your answer and reasoning for question 8?CH1: Since the numerators are the same, and one of the denominators is

smaller, then that fraction is larger.R: Why is the fraction larger, if the denominator is smaller?

CH1: I don’t know. I memorized the rule I learned in mathematics class.

One student in each class gave the correct answer but they could not pro-vide an explanation. This was coded as correct and could not explain.Three (50%) students in the Exp- and two (33%) students in the Con-class gave incorrect answers and incorrect explanations. For example, (Pre-interview of TL2):

TL1: 712 > 7

10 .R: Why? Can you tell me your reason?

TL1: Because the numerators are same, I compared the denominators.The 12 is larger than 10, hence 7

12 > 710 .

Or (Pre-interview of CL2)


On the post- and retention-interviews, four (66%) students in the Exp-class applied the number sense methods. One used a meaningful way andthree used pictorial representations to help solve the problem. For example,(Post-interview of TH1 and TM2):

R: Can you tell me your answer and reasoning for question 8?TH1: A pizza was cut into 10 pieces and another was cut into 12 pieces.

Each piece in the 10 piece group is larger than in the 12 piece group,so 7

10 is larger than 712 .

Or

TM2: You see the shaded area of 710 is larger than the shaded area of 7

12 ,then 7

10 > 712 .

However, no student in the Con-class could compare 710 and 7

12 in ameaningful way. Four (66%) students in the Con-class applied rule-basedmethods. For example, (Post-interview of CH1 and CH2):

CH1: Since the numerators are the same, I only need to compare denom-inators. The fraction that has the smaller denominator is the largerfraction.

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R: Why? Can you tell me your reasons?CH1: I don’t know. I memorized the rule I learned in mathematics class.

Two (33%) students in the Exp-class and one (17%) student in the Con-class still could not compare fractions with the same numerators. For ex-ample, (Post-interview of CL2):


Or (Post-interview of TL1)

TL1: 712 > 7

10 , because 12 is larger than 10.

These responses were coded as incorrect and could not explain.In examining the summary of students’ responses, we found that stu-

dents in the Exp-class advanced in their use of number sense strategieson the post- and retention-interviews as compared to the pre-interview(66% vs. 0%). The data also indicated that students in the Exp-class weremore successful in using number sense-based methods than students in theCon-class on the post- and retention-interviews (66% vs. 0%).

In summary, the data indicates that there was an apparent change inthe Exp-class after instruction and that students’ number sense ability im-proved greatly after they were provided with opportunities to explore num-bers, operations, and their relationships, and to communicate their ideasin a conducive learning environment. The post-interviews revealed thatstudents’ responses reflected number sense strategies. Furthermore, theresults of retention-interviews indicated that students’ learning was mean-ingful and significant. Students’ responses in the Con-class in using num-ber sense components reflected little change after instruction.

CONCLUSION

Although this teaching experiment was implemented over only one se-mester (about 4 and a half months) and limited to two classes from oneschool, the improvement made by the fifth graders in the experimentalclass is readily apparent. Though the generalizations of this study are lim-ited due to the small sample size, the results do provide some importantand interesting findings, as follows:

1. Students’ performance on number sense group tests in both the Exp-class and the Con-class indicated that there were statistically significantdifferences for the scores on the post-tests and retention-tests as comparedwith the pretests at the α = 0.01 level. However, the scores for the Exp-class increased 44% after instruction (the mean score went from 12.35 to


17.81), while the scores for the control class increased only 10% after in-struction (the mean score went from 11.29 to 12.42). This indicated that theExp-class made much more progress on number sense tests as comparedwith the Con-class. These data demonstrate that the teaching of numbersense activities is effective and helpful in the development of students’number sense.

2. In examining the individual interviews of students in the Exp-classand Con-class on pre-, post-, and retention-interviews, the changes andprogress made by students in the Exp-class is apparent. Students in theExp-class were not only able to effectively use number sense strategiesafter instruction, but they were also more successful than students in theCon-class in using number sense strategies to solve problems on the post-and retention-interviews. The interview results showed that these numbersense activities were effective in helping children develop their numbersense abilities. Furthermore, the results of the retention-interviews furtherdemonstrated that students’ learning in the Exp-class was meaningful andtherefore long-lasting.

The data from the interviews provide opportunities not only to bet-ter understand students’ improvement in using number sense strategies,but also to explore students’ thinking and comprehension of numbers andoperations.

3. There is a major difference between number sense activities designedfor this study and the mathematics textbooks used in Taiwanese classes.The mathematics textbooks used for the fifth grade level do not teach stu-dents to find a least common denominator or the cross-product procedurefor comparing fractions. The textbooks teach students to use a unit quantityto order fractions. For example, when comparing 1

3 and 38 , the teacher gave

the students 24 pieces of chips as a unit and taught them to find how manypieces are equivalent to 1

3 and 38 . The textbooks show that 24 × 1

3 = 8;24 × 3

8 = 9, therefore, 38 is larger due to having 9 pieces of chips. In the

context of this situation, students knew how to order fractions when theunit was given. However, they were unable to solve the problem if the unitquantity was not given. On the contrary, the instruction for the Exp-classfocused on the development of conceptual understanding through pictorialrepresentations. The teacher in the Exp-class encouraged children to drawdiagrams to help them understand fractions and also encourage students torepresent fractions by verbal language. This result is consistent with thestudy of Cramer et al. (2002).

4. The use of benchmarks for students in the Exp-class was much moreprevalent after instruction as compared with students in the control class.The use of benchmarks is not introduced in the Taiwanese textbooks. The-

132 D.-C. YANG

refore, the progress made by students in the experimental class is apparent.For example, when students were asked to estimate: 4

5 + 67 , students in

the Exp-class knew to select the 1 as a benchmark to decide the answer,but students in the Con-class were unable to use benchmarks to find theanswer.

In summary, this study demonstrates that students’ number sense can beeffectively developed through establishing a classroom environment thatencourages communication, exploration, discussion, thinking, and reason-ing. The results of this research study confirm the earlier studies (Markovits& Sowder, 1994; Yang, 2002a, 2002b; Yang & Reys, 2001a, 2001b) thatchildren’s number sense can be fostered through appropriate teaching. Fur-thermore, the process-oriented teaching model practically applied in thisstudy not only supports the NCTM process standards and the guidelines fora Nine-Year Joint Mathematics Curricula Plan, but also can be used as aguide to help teachers implement the mathematics curriculum. In fact, thisteaching model provides an artifact that could go a long way toward help-ing teachers actually embed these processes in their practice. The teachingof number sense focuses on conceptual understanding. It directs childrento pursue meaningful learning. This further confirms the statement in theGuidelines that students’ learning of mathematics should be meaningful.From this research study, one can gather that well-designed instructionallessons not only help children develop number sense, but actually promotethe development of critical thinking and reasoning about numbers andoperations.

ACKNOWLEDGEMENTS

This paper is a part of a research project supported by the National ScienceCouncil in Taiwan with grant no. NSC 90-2521-S-415-001. Any opinionsexpressed in here are those of the author and do not necessarily reflect theviews of the National Science Council in Taiwan.

The author gratefully acknowledges the help of teacher Miss Lin, a Tai-wanese 5th grade teacher, without whose cooperation and teachingskills this paper would not have resulted.

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Graduate Institute of Mathematics Education,National Chiayi University,85, Wen Lung, Ming-Hsiung, 621,Chiayi, TaiwanE-mail: [email protected]

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