Constructing Conceptual Knowledge and Promoting" Number Sense ...

1993, \Tol.5,~o.2, 124-151

Mathemntics Education Research Journal

Constructing Conceptual Knowledge and Promoting "Number Sense" from Computer-'"Managed Practice in Rounding Whole Numbers Nira Hativa Tel Aviv University This study sought to identify how high achievers learn and understand new concepts in arithmetic from computer-based practice which provides full solutions to examples but without verbal explanations. Four high-achieving second graders were observed in their natural school settings throughout all their computer-based practice sessions which involved the concept of rounding whole numbers, a concept which was totally new to them. Immediate post-session interviews inquired into students' strategies for solutions, errors, and their understanding of the underlying mathematical rules. The article describes the process through which the students construct their knowledge of the rounding concepts and the errors and misconceptions encountered in this process. The article identifies the cognitive abilities that promote student self-learning of the rounding concepts, their number concepts and "number sense." Differences in the ability to generalise, "mathematical memory," mindfulness of work and use of cognitive strategies are shown to account for the differences in patterns of, and gains in, learning and in maintaining knowledge among the students involved. Implications for the teaching of estimation concepts and of promoting students' "number sense," as well as for classroom use of computer-based practice are discussed.

Contemporary "constructi\Tist" approaches to learning, accepted almost uni\Tersally by mathematics educators, assume that mathematical knowledge is not directly absorbed but acti\Tely constructed by each learner (e.g., Bereiter, 1985; Cobb, Yackel & Wood, 1992). O\Ter the past se\Tel'al years, the role of student cognitions has been in\Testigated as an intermediary between instruction and learning. Se\Teral studies have sought to identify how students acti\Tely construct knowledge in ways which satisfy the constraints inherent in instruction (Cobb, 1988; Hiebert & Wearne,1988). Looking into the cogniti\Te processes that underlie student learning makes it possible to trace changes in students' performance to particular instructional e\Tents. It also allows researchers to identify the ways in which students' prior knowledge affects learning. Resnick (1989) suggested investigating how children construct their knowledge through examining patterns of learning in conditions which encourage students' inventive procedures. The study described here researched students' cognitive processes while they constructed knowledge from computer-based arithmetic practice that encourages student inventive procedures. Previous studies with the same computerised program (Hativa, 1988a, 1992) showed that the adaptive features of that program which enabled students to advance at their own rate in the hierarchical levels of practice, in fact accelerated the better students beyond their current class material. To solve problems in material which they had not yet learned, these students

Constmeting Conceptual Knowledge: Computer-Managed Practice 125

invented a variety of mathematical procedures and algorithms and used·a variety ()fiproblem-solving strategies. This process proved to be very challenging and e~citiI1gforthe students and promoted the.learning of new concepts. The fact that sttId~.n~s gained knowledge and understanding of mathematical concepts solely on th~l>a.sis of computer-based practice, with no further explanation, suggests that s~dE::hts .may learn new concepts by themselves through solving computerpro"icied problems. . / Res~arch has already shown that students are able to learn arithmetic concepts Clfldprpcedures by themselves by following the steps of worked examples which (:l.,te.:p.()tpr~cededby instruction (Sweller & Cooper, 1985; VanLehn, 1986; and Zhu, ~Sinl.on,1987).Because the computerised practice program in this study provides 1l§f;.1!utipns to examples when a student fails to solve them, this practice may be \T,i~~e(::la§providiriga particular type of worked example. '"Vhectlrr~ntview of the role of computers in schools is that computers should b>~J.l§~~ to promote students' cognitive skills and problem-solving abilities. From tmspempective, drill-and-practice computer programs "have a pejorative ?gl"ll10tCition" (Woodward & Mathinos, 1987, p. 2) because they are believed to be ~o.rirtgf()rthestudents and to develop algorithmic skills rather than understanding cU:H:lc;ognitive skills. However, my previous studies showed that this perception Wa,sinaccurate for the particular drill-and-practice system which I observed. Stua~nts' work with that particular system, at least for the better students, was V¢:ty . challenging and exciting and promoted conceptual knowledge and understanding, problem-solving abilities, and other cognitive skills. Because drill practice is the most frequently used computer application in school (Becker, 1991, and is commonly named CAl-Computer Assisted Instruction), it is iInportant to identify how the beneficial results of work with this computer application occur. The study presented here sought to answer this question. The study concentrated on high-achieving students because they were primarily the ones found in my previous studies to be the most successful in learning new material fr()Ill computerised drill-and-practice. Thus, the research question was: How do ~i.~h-achieving students learn new concepts and procedures in arithmetic by themselves from computer-based practice in the particular computer system? Another study (Hativa, 1992) investigated the benefits gained by lower-achieving sh.:id~nts in the same computerised practice. The findings of both studies may assist in the design of new types of computer-based practice programs which promote student problem-solving abilities and a wide range of student cognitive skills, for students of all ability levels.

N

and

Method

The Computer System The computer system investigated in this study, Integrated Instructional System (lIS), is widely used in Israel and in several other countries such as Canada, USA, Germany, Panama, South Africa, Spain, and Uruguay. Each lIS is a fully

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packaged system which presents material for a full course or for a multiple class level curriculum. The system allows many computers to be in use at one time (in a computer laboratory, for example) so that all students in a class can use them concurrently to work at his or her own personalised level and type of practice. The systems manage students' learning with minimal teacher intervention. Although this study involved a particular lIS, my observations of different systems of this type-two in Israel and two in the U.S.A.-suggest that, in spite of the' variation in detail, many aspects of the software are common to the different systems.

The Arithmetic Courseware The courseware consists of a large body of examples encompassing almost the full elementary school arithmetic curriculum, thus crossing class level boundaries. All types of practice are grouped into 15 strands, each concentrating on a single arithmetic topic (e.g., number concepts, vertical addition, decimals). The types of examples within each strand are arranged in a hierarchical sequence of normatively numbered "Levels." To illustrate the numbering system, examples of Level 45 belong to the fifth month of the fourth-grade curriculum. All examples of a certain Level within a particular strand are of the same type, and only the numerical values (generated randomly within limitations) may be different. By way of illustration, each of the three examples 27+36, 53+18, 35+49 requires the addition of two-digit whole numbers with carry-over in the units digit; these belong to the same Level within a strand of horizontal addition. Each student is presented with examples mixed from all strands that are active for him or her. Additional details can be found in Osin (1984).

The Learning-Management Software The software is designed to adapt students' work with the system to their individual needs and abilities. The management functions include (a) diagnosing each student's initial knowledge; (b) collecting continuous data on the performance of each student; (c) makingdecisions, on the basis of the student's performance of the practice examples, about which practice examples to provide; (d) making decisions about the promotion or demotion of each student through the hierarchical levels of practice; (e) providing individualised feedback-immediate (after completing the solution of each problem) and summative (of the overall performance in each computer session); and (f) reporting to the teacher the performance of each student and of the class as a whole. The following is an elaboration of some of the adaptation features. For each problem presented, the student receives three successive opportunities to provide the correct answer. After the student's response is typed in, screen-printed feedback informs the student whether the answer is correct or not. After a third incorrect trial, all steps of the correct solution are printed on the screen, without any verbal explanation, and stay there for only a short time (20-60 seconds, depending on the type of example). A student's answer is recorded by the software as correct only if the problem was correctly answered at the first trial and

Constructing Conceptual Knowledge: Computer-Managed Practice 127

within a certain time limit. An elaborate algorithm evaluates the student's mastery of each Level. The criterion for mastery of a Level takes into account a sequence of examples within this particular Level. Mastery results in promotion to the next Level within the same strand. Establishing mastery requires the correct solution of each one of the first five examples of a sequence (criterion of 100%), or of eight out of the first 10 examples (80%), or of 10 out of the first 15 (67%), etc. so that additional examples in the same sequence (within a particular Level) bring down the mastery criterion. When a student's performance is lower than the criterion for mastery,.he or she is demoted to the previous Level within the same strand. For example, failure in each one of the first three examples, or in at least six out of the first ten examples of a sequence, results in demotion to the previous Level. A weekly-produced computerised class report presents the mean Level and the problems that students face in each strand and across all strands, for each student and for the whole class.

The Concept to be Attained This concept was chosen to be "rounding whole numbers," for reasons detailed below. This topic is fully covered by six of the nine Levels 45 through 53 (material from the computer curriculum of middle fourth through early fifth grade) in the Number Concepts strand (Table 1). Levels 46, 49, and 52 which involve topics other than rounding have been excluded from the analysis and discussion below. To simplify discussion, the six Levels of rounding whole numbers have been denoted in Table 1 by the Roman numerals I through VI. Table 1

Levels of Rounding Exercises in the Computer Curriculum Level

. Range of numbers

45 (I) 47 (II) 48 (III) . 50 (IV) 51 (V) 53 (VI)

tens hundreds hundreds thousands thousands thousands

Round to

Numbers

Illustrations

tens tens hundreds thousands tens hundreds

24 624 624 3624 3624 3624

78 478 478 5478 5478 5478

Rounded respectively to 20 620 600 4000 3620 3600

80 480 500 5000 5480 5500

"Rounding whole numbers" was chosen as the topic for inquiry primarily in order to study the students' cognitive processes while they learnt new material from computer-based practice. Hiebert and Wearne (1988) suggested that the main components for studying the effects of instruction on cognitive change should be: (a) selection of a well-defined content domain; (b) identification of the cognitive processes which are the keys to successful performance in that domain; and (c) identification of an instructional sequence that is limited in scope, clearly focused,

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and promotes the use of the key cognitive processes. Rounding whole numbers satisfies all of these components: (a) the content domain is well defined-mastering this topic in this particular curriculum requires very good perception of and functioning within three whole-number domains: tens, hundreds, and thousands; (b) the cognitive processes which are the key to successful performance in this domain are also well defined-the primary processes, as illustrated and explained below, are: generalisation across different number domains, deduction of a rule from several examples, estimation, and several aspects of "number sense"; and (c) this topic is fully covered in only a few types of exercises (Table 1). Thus, the instructional sequence is limited in scope and clearly focused. The term "number sense" may be viewed as knowledge in the conceptual domain of numbers and quantities. Number sense requires cognitive expertiseknowledge which results from extensive activity in the domain of numbers. This activity involves interacting successfully with numbers, perceiving and understanding subtle patterns, solving ordinary problems routinely, and generating new insights (Greeno, 1991). The following are additional reasons for choosing the topic of rounding numbers for this study: • There is currently a renewed emphasis on teaching computational estimation (which builds on rounding) in the classroom (Sowder, 1989). Estimation is important for evaluating results of calculations with calculators and computers and for several other applications as well (Usiskin, 1986); • The promotion of students' "number sense" is a goal of growing importance (National Research Council, 1989). The development of skill areas that underlie estimation, such as rounding, were found to lead to a deeper insight into the number system, thus increasing "number sense" (Sowder, 1989, 1990); • Prior sh,ldies have shown that high-achieving students did, in fact, acquire knowledge and understanding of this particular concept from computer-based practice solely, with no teacher-directed learning (see,Jor example, Hativa; 1988a, 1992); • Solvihg examples which involve rounding does not require elaborate algorithms which may cause error (such as in subtraction and multiplication with carry-over), nor does it require the mastery of any computational algorithms as a prerequisite. These properties reduce the "noise" in investigating conceptual learning.

The Subjects The study took place in a school in an affluent suburb of Tel Aviv. The computerised class reports in that school revealed that top achievers in their respective classes started receiving practice in rounding numbers (at Level 45) towards the end of their second grade. At that time, the mean Levels in computer performance of the three second-grade classes at that school (with between 32 and 40 students each) ranged from 28 to 30. Thus, the top achievers practiced mathematical concepts which were one-and-a-half years in advance of their class mean. Interviews with the students' teachers revealed that the topic "rounding


whole numbers" was being taught in the third-grade cla$ses for use in estimating the results of arithmetical calculations. These interviews also revealed that, at the time of the study, the second-grade students learned in class to work with numbers in the tens and hundreds domain, but not yet with thousands.. Thus, rounding whole numbers was a completely new topic for the top-achieving second graders at the outset of this study, as was work with numbers in the thousands domain. Four high-achieving second graders were identified on the computer class reports as being top achievers in the computer arithmetic work in their respective classes. Two boys (Ariel and Eado) from two different classes, and two girls (Nili and Grit) respectively first and second in rank order from a third class were chosen.

The Procedure Carpenter and Moser (1983) suggested using individual interviews and observations taken in the typical setting for the task in question in order to achieve an unbiased account of students' cognitive solution processes. Prior studies done in the natural setting of the same computer-based learning system (Hativa 1988a, 1988b) showed observations and interviews as being beneficial techniques for identifying children's learning processes in computer work. The computer environment enabled every step of the students' solutions to be followed and every etror which was made identified because all steps of the solution and all types of error were displayed on the computer screen, and students' reactions to the computer-presented problems and feedback were observable. Both the student selection process and the method of observation in this study were designed to ensure that the experimental sessions pr~served the regular learning conditions of these students in the computer laboratory. The observations of each student began with Level 45 (named here Level I) in his or her regular computer work and took place during the regular class computer work, a weekly 20-minute computer session. During the session, I sat next to the student, copied (in shorthand) everything that was presented on the screen (the example presented, its strand and Level, the student's typed-in answer, the computer response, etc.), and recorded on paper a description of the student's behaviour and oral comments. Each of the observations was followed by a 30-minute interview in which the students were asked to solve on paper the examples on which they made errors during the preceding session and to explain their methods for solving the examples presented in the session. They were also asked to state the rules that they had used in solving these examples or to solve different examples of the same type. Students were also regularly asked how they gained the knowledge demonstrated in their solutions. They were asked whether they had discussed their practice material with family members, the teacher, or friends and whether they had encountered this material at a previous opportunity (in class, in the computer laboratory, or elsewhere). The class teachers were asked not to answer any question that involved the rounding of whole numbers. Figures 2 through 7 present excerpts from the protocols of the students' computer performance and of their immediate post-session interviews. Final testingsjinterviews with the students took place two weeks after the completion of computer work on the rounding examples.

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There was only one deviation from the natural conditions of the computer practice work. In order to accelerate the subjects' progress in the strand of interest to this study so that the observations could concentrate on the mathematical topics in question, two of the strands were disabled (Word Problems and Long Division); thus the students received mixed practice from six or seven strands. The reduced number of practice strands enabled the observees to cover the designated curriculum segment in approximately two months.

Development of Understanding of the Concept of Rounding of Whole Numbers The students' initial interviews checked that the students were unfamiliar with numbers in the thousands domain and had never before encountered the topic of rounding. This was true for all but Grit who had encountered a few examples of this type at the beginning of the school year, six to seven months before the study began, during the diagnosis phase for the computer work. Figure 1 describes graphically, for each observee, the performance on each of the six Levels that cover the rounding topic: the number of examples presented, the number of those solved correctly in each Level, and the promotions and demotions within Levels. CAl Level Ariel 53 5 1 50 48 47 45

53 51 50 48 47 .45

5/5

811O

Eado

5/5

Nili 53 51 50 48 47 45

5/5

1/8

o ri t 53 5 1 50 4 8 4 7 45

0/3

1 17

Rsn 9/1

1

.

50

60

6/10

° n,-_~1-,,-2/:..::2,-,,0_

7/10

20

30

40

70

80

90

Number of exercises presented

Figure 1:

Number of exercises solved correctly out of the number of exercises presented per CAl Level for each student.


The Three Steps of Identifying the Algorithm for Rounding The analysis of the students' protocols revealed that they all went through two steps in identifying the algorithm for rounding numbers to tens. These were: identifying that the targeted number has a 0 in the units digit and identifying the "direction" of the rounding-either upwards or downwards. The identification of the algorithms for rounding to hundreds and to thousands required a third st.epthe generalisation (transfer) of the rule for rounding to tens to these other domains. The following discussion provides an analysis of how the observees went through these three steps.

Step l-Identifying the Zero as the Units Digit in Rounding to Tens (Levels I and II) Grit solved all the examples she received in Level I correctly, starting with the very first, showing that she remembered the 0 as the units digit from the examples that she received during the diagnosis phase of the CAl work. For the other three students, Figures 2 through 4 present the first computer session, that is, their behaviour on their first encounter that with a rounding example. When NiH (Figure 2) received the first example of Level I, she claimed that she had never seen this type before. She immediately pressed the ADVANCE key, got the computer answer, and identified from that answerthe 0 as the units digit. After this first case, NiH did not err again in this aspect of solution. Eado (Figure 3) and Ariel (Figure 4) deduced that the target number should _have a zero as the units digit from the word "to round" which describes in Hebrew, the same as in English, the shape of a circle which resembles a O.

Step 2-Identifying the "Direction in Rounding to Tens (Levels I & II) II

In the first computer session, Grit received five numbers: 84,44, 71,21, and 84, to round to tens. By chance, all numbers required rounding"down." Grit answered _all five correctly thus satisfying the mastery requirement, and was promoted to the -next Level. In the second session, while getting examples in Level II, Grit received for the first time a number (306) which required rounding up (Figure 5). In the first trial she rounded the 306 down on the basis of her prior experience, was surprised to get an "-incorrect" response, and immediately tried rounding up. However, throughout the nine examples of Level II in this second session, Grit was not able to identify the rule of "direction." Almost every time that rounding up happened to be correct, she rounded the next example up as well, and similarly with rounding down. Because the examples that required rounding up or down were mixed, she made many errors in this session. In the post-session interview she remembered almost all of the computer-provided correct answers but without identifying the rule. Because of her failure to satisfy the mastery requirements in Level II, she was demoted to the previous Level, reached mastery at that Level a week later, and was promoted back to Level II (see Figure 1). In the third session, one week later, Grit solved correctly all five examples of Level II provided to her,

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on the first trial (Figure 5). This behaviour suggests that the elapsed tim:e (one week) allowed the cgnitive elaboration and incubation of the information about rounding to· tens o~ the basis of the solutions to many examples that she remembered. This process of "cognitive digestion" resulted in her identifying the solution to the problem. In that post-third-session interview, she correctly stated the rule for rounding a three-digit number to tens. Session 1 Level 45 (I)-Round to tens (For key to abbreviations see page 133)

Num.

Student's Response

84

ADV.

Compo Resp. & Type of error X 80

39

30 ADV.

64 71

60 80 70 30 40 20 30 60

36 27 53

76

50 70 80

Student's comments

We haven't learned anything like that! Ahah! It should end with a O.

X[i] X 40 V

X[ii] V

X[i] V

X[i] V

X[ii]

V

X[i] V

What is it? The computer is completely confused.

Post-session interview

Q. Round 84 to tens A We haven't yet studied this but I saw in the computer that it is 80 V' Q. What's the answer to 39? A40V' Q.64? A60V' Q.71? A 70 V' Q.36? A 30)( Q.27? . A 30 V' Q.53?

A50V' Q.76? A80V'

Figure 2: An excerpt from Nili's protocols. NiH, like Grit, was notable to identify the "direction" for rounding from the eight examples of the same type in Level I that she received in the first session. She also showed a remarkable memory when, in the post-session interview, she remembered correctly all but one of the answers that the computer provided during the preceding session (Figure 2). Because of her failure in Level I she was demoted to the previous Level and upon proving mastery there, was promoted back to Level I (see Figure 1). In her second session, after erroneously rounding the first example in Level I down rather than up, she demonstrated mastery of the rule for rounding to tens, and stated this rule correctly in the post-session interview. This behaviour suggests that, like Grit, it took Nili a process of cognitive

Constructing Conceptual Knowledge: Computer-Managed Practice 133'

elaboration and incubation (based on the answers that she remembered from her computer work) over the period of one week, to come up with the correct rule.

Key to Coding adopted for Figures 2-7

Num.

The number presented on the screen for the student to round.

Student's response: What the student types in as an answer to the number presented on the screen. X,V

The computer feedback to the student's response: X marks an error, recorded by the software as incorrect; V marks a correct answer.

80,40, etc:

The number which is the correct answer is printed on the screen by pressing the ADVANCE key.

ADV.

The student presses the ADVANCE key and the correct answer is printed.

[i], [ii], [iii] errors in the "direction" of rounding. an unexplained error.

[?]

[Tens], [Hundreds], [Thousands]: a "domain" error. )(

Wrong solution in the interview.

V

Correct solution in the interview.

*

A particularly difficult problem-rounding across two number domains.

Session 1 Level 45 (I)-Round to tens (For key to abbreviations see above) Student's Num. Response 84 39 64

90 80 30 40 70 60

Compo Resp.& Type of error X[ii] V X[i] V X[ii] X


Q. What is "to round"? A. That the 84 will be converted to the nearest tens, that it'll be whole tens, and therefore the answer is 90. Q. How do you know? A. When I got 84 I thought that I needed to convert it to 80 but the computer gave me an answer of 90 but sometimes I do get 80 and I don't know why. I only know that I should get a 0 and that it's either above or below.

Figure 3: An excerpt from Eado's protocol

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Session 1 Level 45 (I)-Round to tens (For key to abbreviations see page 133)

Num.

84 39

Student's Response

80 20 30 ADV.

Compo Resp. & Type of "error

Student's Comm ents

V

X[?] X[i] X

64

60

40 V

98* 87

100 90

V V

Why?


Q. Round 84 to tens. A.80V Q. What is to "round"? A. Make the number round, with a 0 Q. Round 39 to tens. A. On the computer I wrote 30 but this is 40 because 39 is doser to 40. V Q. Who told you that? A. I understood from the computer. Each number through 5 is rounded to 0 with the same tens' digit. Above 5 is rounded to the next ten. Q. Have you ever encountered rounding exercises? A. No-today is the first time.

Figure 4: An excerpt from Ariel's protocols The three examples of Level I that Eado received in the first session were apparently not enough for him to identify the rule for going up or down (Figure 3). In the second session, Eado solved correctly all the examples of Level I. The post-session interview revealed that he acquired his knowledge by asking his mother at home for explanations. Asking family members for help in problematic school work is a regular learning strategy for highly motivated students (Hativa, 1988a). To avoid further learning from his family members, I spoke with his , mother. I learned that she had taught him only rounding to tens and she promised that neither she nor any other member of the family would answer any further questions from Eado related to rounding until the study was completed. Unlike the three other high achievers, one example of each-going up and going down-was enough for Ariel to identify the complete rule, and to state it correctly (see Figure 4). However, Ariel's rounding the "5" down indicated that he was unfamiliar with the convention of rounding a "5" up, as is taught in elementary schools. To avoid this problem, the computerised algori(hm which generates the numbers for the students to round in the particular system involved excludes those numbers which are exactly in .the middle of the domain. Thus, students in their computer work are not given examples ending with a five to round to tens, with a 50 to round to hundreds, or ending with 500 to round to thousands.


Session 2 Level 47 (II)-Round to tens (For key toabbreviatfons see page 133)

Num.

374 306 462 607 913 432 586 604 421

Student's Response

Compo Resp. & Type of Error

370 300 310 470 460 610 920 910 430

V X[i] V X[ii] V V X[ii] V V

580 590 610 600 430 420

X[i] V X[ii] V X[ii] V

Student's Comments

Strange

Post-session Interview

Q. Round 306 to tens. A. 310 V Q. Why not 300? A. I don't know-I thought it was 300 and the computer said that this was an error. Q. Round 913 to tens. A. 910 V Q. Why did you write 920 in the session? A. Because I thought it was 920. Now I do this correctlybecause I remember the result from the computer and understand it better. Q. Round 604 to tens. A. 600 V Q. Round 421 to tens. A. 420 V Q. Round 676 to tens. A. 670)(

Session 3 Level 47 (II)-Round to tens ·847 289 344 236 514

850 290 340 240 510

V V V V V

Q. How do you round to tens? A. I write a 0 on the right and the next digit, for example, 43 becomes 40 and 47 becomes 50. Q. How do you round 235 to tens? A. Same as 36, it becomes 240. V

Session 3 Level 47 (lIn-Round to hundreds

437 229

600 510 501 500 420 400 200 300 400 200

V[Tens?] X[Tens] X[?] V X[Tens] V X[i] V V V

701

700

V

801 388 616

800 400 600

V V V

604 514

421 255

Q. Round 514 to hundreds. A. I didn't understand it at the beginning but then I understood that there should be two zeros. Q. Round 564 to hundreds. A.600v Q. Why 600 and not 500? A. Because the number of the tens is 6 it goes up Q. How do you round to hundreds? A. Let's take 334. It is rounded to 300 without the addition of 34. Q.From what number would you start to round to 400? A. From 350 Q. How do you know that? A. I've learned while solving exercises on the computer.

Figure 5: Excerpt #1 from Grit's protocols

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Step 3: Generalisations to Other Number Domains (Levels III-VI) It took Orit (Figure 5) two examples of Level III (514, 421) to master the "two

zeros" part of the rule for rounding to hundreds. She needed then only one additional example (255) to identify the "direction" part of that rule and solved the rest of the examples of Level III correctly on the first trial in Session 4 (Figure 1). She generalised the rounding rule correctly to thousands (Level IV), starting with the first trial. However, she erred in the first three examples in Level V and was demoted to Level IV (see Figure 1). As before, she made no errors at this Level, and was promoted again to Level V in which she failed again in the first three examples. The process of failing completely in Level V, demotion to Level IV, demonstration of mastery at this Level, and going back to Level V repeated itself three times over three sessions (Figure 1). Figure 6 presents Orit's failures in the solutions to the Level V examples in the third cycle of solving examples at this Level. Session 7 Level 51(V)-Round to tens (For key to abbreviations see page 133)

Num.

Compo Response & Type of error

Student's Response

1148

2 000 1000 1100

X[Thousands] X[Thousands] X[Hundreds] 1150

5 078

TIME 5 080 7910 7900 TIME

X

7917

V X[i] X[Hundreds] X 7920


Q. Round 8 647 to thousands, hundreds, tens. A. All correct. Q. Can you state a rule for rounding? A. If for thousands, there should be a thousand [digit] with no other number besides it and then 000. For rounding to hundred there should be a thousand, the hundred besides it, and 00. For tens there should be one O.

Note: 1150; 7 920: The correct answer is printed on the screen after 3 incorrect trials TIME: the time limit for solution was missed

Figure 6: Excerpt #2 from Orit's protocols Only in the following session did Orit identify the rule. She solved correctly eight out of the ten examples of Level V and was promoted to the next Level. This behaviour suggests that, as described before, Orit went through a process of cognitive elaboration and incubation, on the basis of the computer-provided answers to the nine examples of Level V that she received and remembered, which resulted in her identifying the rule. She then demonstrated mastery of Level VI by solving correctly 9 out of 13 examples. NiH (Figure 1) gave incorrect solutions to four of the 15 examples of Level III. In Level IV (rounding to thousands) she used the three zeros starting with the first


example but it took her three unsuccessful attempts before she identified the "direction" part of the rule. Because of these three consecutive incorrect solutions, she was demoted to the previous Level, satisfied the mastery requirements and was promoted again to Level IV. Now she solved correctly all five examples of this Level, showing again a process of incubation and cognitive elaboration which led to the identification of the rule. In Level V she generalised the rule starting with the first example and made several sporadic errors, particularly in difficult examples which involved rounding across more than one decimal place (e.g., rounding 8 597 to tens which results in 8 600). She promptly generalised the rules of rounding to Level VI, making sporadic errors in this Level. Eado quickly generalised the rules of rounding to tens to all the other number domains. After Level I, he made only three errors, two of which were in the first . examples of Levels II and III (Table 2). . Ariel, like Eado, also quickly generalised the rounding rule to all other Levels and made only very few errors, primarily in Level V (see Figure 7, and Table 2).

Session 5 Level 51 (V)-Round to tens (For key to abbreviations see page 133)

Num.

8289

8 012 6798* 9371 9364

Student's Response

Camp Resp. & Type of Error X[ThousandsI X[Hundreds] V V X[iii] V X[IIundreds] V X[ii] V

8000 8200 8290 8 010 6700 6800 9400 9370 9370 9360

Student's comments

Oops!! it was to tens

Ooff! I get terribly confused


Q. How do you round to tens? A. 1, 2, 3, 4, 5 if rounded to tens are returned to a and from 6 to 10 they increase the tens by 1 Q. Why? A. Because you go to half way Q. How do you round to hundreds? A. Exactly the same as for the tens only from 10 to 50 is the lower number and from 60 to 100 is the higher.

.Figure 7: Excerpt #2 from Ariel's protocols

A Summary of the Three Steps of the Concept Development Step 1: All students except NiH identified the zero as the units digit from the first example: two of them did this on the basis of the meaning of the word "rounding" and the third by remembering a solved example of the same type. Step 2: All students needed several examples along with their computerprovided solutions to discover the rule for the direction of rounding. Two of the students needed to go through a process of cognitive elaboration and incubation in order to make the discovery." II

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Step 3: It took two of the students four examples to generalise the rules for rounding from tens to hundreds (to Level III). The generalisation to thousands-a number domain that they had not known before-was easy for all students, and all solved the examples of Level IV with almost no error. However, all except Eado faced substantial difficulties in generalising the rules to Level V (rounding numbers in the thousands to tens). Two of the students (Nili and Orit) made some errors with each new type of example.

Verbal Statement of the Rule for Rounding Whole Numbers In order to identify how the understanding of rounding concepts evolved, students were often asked in their post-session interviews to formulate, verbally, the rules they were using. Two types of statement describing the rule for rounding emerged from the analysis of their protocols: procedural and conceptual.

The Procedural Statement of the Rule This statement consists of three components: 1. The number ofzeros in the target result. Figure 6 provides an example. By way of further illustration, the following excerpt has been taken from Orit's interview following her first computer session, before she identified the aspect of "direction." Interviewer: What does it mean "to round to tens"? Grit: It means a number in which the unit digit is a O.

2. The criterion for the "direction" of rounding. The reference to the"direction" is sometimes vague in that it does not provide a clear criterion. The illustration given below has been taken from Orit's third post-session interview: Interview,er: How do you round to tens? Grit: I write a 0 on the right and the next digit, for example, 43 becomes 40 and 47 becomes 50.

A better rule statement includes a clear criterion, using a "middle" number as in Figure 7, for example, or in Eado's fifth post-session interview: 1 Interviewer: What is the rule for rounding to hundreds? Eado: It is the same as for the tens. For the tens, from 51 to 55 you write 50 and this is the middle and if more than 55 then it is 60, the same is for hundreds-the middle is 50 so from 500 to 550 this is 500 and if more, then it is 600.

For Eado, as for Ariel above, the fact that he rounded the middle value down indicated no prior learning of the topic of rounding from sources other than the computer work.

3. The part of the given number that remains constant and the part that is to be changed. To apply .the statement of the rule to the different number domains, 1. Note that in this example, and those in the sections which follow, emphasis has been added to draw attention to specific aspects of the children's statements.


students at times differentiated between the part of the number that is not affected by the rounding, and the part that is affected. As illustrated in Figure 6, Grit stated that on rounding, some digits remain the same and others are to be replaced by a certain number of zeros. Another illustration is taken from NiH's final interview: Interviewer: What is the rule for rounding to tens? Nih: For example, 7479 to tens is 7480 because 74 is final and I find for the 79 the tens which are the closest to it.

In this illustration NiH divided the number into two parts-fixed and changeable. None of the students used all three components in a single statement of the procedural rule. Most frequently they mentioned only two (see Figures 5 and 6).

The Conceptual Statement of the Rule This statement is based on the intuitive concept of "closeness" or "nearness." Two illustrations are presented in Figures 3 and 4 ("84 will be converted to the nearest ten" and "39 is closer to 40." The following is an additional illustration from Nili's post-session interview followingher sixth computer session: Interviewer: How do you round a number to hundreds? Nili: For rounding to tens I put the number in the place which is the nearest to the given example. The same is for the hundreds.

And in her final session: Interviewer: What is the difference between rounding to tens, hundreds and thousands? Nih: I write the number which is the closest to each of these. It is difficult for me to

explain because for each of these there is another method.

The Difference in Memory Demands Between the Two Types of Rule . Statement NiH's statement of the rule reveals that the use of the notion of "closest" or "nearest" is intuitive, and easy to remember and generalise to different number domains ("The same is ...") because memorisation of only a single rule is needed for all number domains. In order to be able to apply the rule to differentnurnber domains, one needs to have a very good grasp of the structure of the number system (to have a good "number sense," as described above). In contrast, the procedural-rule statement is much harder to remember because it requires remembering a separate rule consisting of three components for each number domain.

Summary: The Statement of the Rule by the Four Children Nili, throughout all her sessions, used solely the conceptual statement of the rule whereasOrit used solely the procedural statement. Orit's first statements referred only to the number of zeros, and later on, starting with the third session, she added statements concerning also the "direction" (Figure 5). Until nearly her

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final sessions, Grit did not refer explicitly to the direction of rounding (Figure 6). However, in the last sessions, when she was asked directly, she demonstrated that she knew the criterion for the direction. For example, in Session 8: Interviewer: When do you round upward and when downward? Orit: If the number next to the tens is larger than 4 then you go upward, so, a 37 makes 40 and if the number is from 5 down, you round to 30. Interviewer: What is the rule for hundreds? Orit: Oh! I didn't think there was a rule for it. Let's see, for example, 673 goes to 700 and 645 to 600. OK, the rule is: if the number of the tens that is next to the 600is less than .2 than you round to 600 and if it is larger than 5 you round to 700. Interviewer: How about thousands? Orit: Let's say 3 575 to thousands is 4 000 and 3 450 is 3 000. OK, from 500 it starts to be rounded to 4 000, the same rule as before!

Grit deduced the rules for the different number domains from a comparison of two complementary examples per domain. It should be noted that her rule statement, although referring intuitively to the middle number, does not mention the term "middle" and is inaccurate and inconsistent. Eado and Ariel used both types of rule statement. At times they mixed both types together in the same statement, using the procedural rule as an elaboration on the conceptual rule (Figure 4). When using the procedural statement, both expressed explicitly the "middle value" as a criterion for going up or down. In the final testing/interview all students were presented with the two questions-"What is rounding?" and "How do you perform it?". They were each given 12 numbers to round. The given numbers and the target numbers were all from the tens, hundreds and thousands domains, except for one example which asked the students to round 9 680 to thousands. Data from the final interview show that the students had retained their definition of rounding as summarised above for the particular sessions. Eado and Ariel made no errors in the rounding examples whereas Nili made two errors of direction and Grit made three errors of direction and 'one domain error when' she rounded 9 680 to 1000. Nili, however, being unfamiliar with representing numbers of the order of ten thousands, wrote the rounded number as 01000.

Students' Errors Table 2 presents, for each student, the number of incorrectly solved examples out of all examples presented, at each Level, and the total of these across all students and Levels. Table 2 shows that the mean proportion of errors was higher in the initial stages of the concept learning (Levels I and II) (33%), than in the more difficult states of the concept practice which required its generalisation across different number domains (Levels III through VI) (25%). This finding suggests that the concept of "rounding" was new for the students in this study, and that once they had acquired the concept at the initial stages, they were able to generalise it relatively easily across the different number domains.


Table 2

Number and Percent of Exercises Solved Incorrectly Per Student Per Level Eado* Nih (%) Total Grit Ariel Pupil: 14/41 3+/8+ 9/18 0/5 2/10 (34) Initial I 14/44 2/10 6/15 6/14 0/5 (32) II 15/33 5+/18 28/85 6/19 2/15 Sub total 28 45 32 33 13 % errors 10/45 4/15 1/10 4/10 1/10 (22) Genera-III 0/5 5/8 8/48 3/25 0/10 (17) lisation IV 20/51 0/5 4/12 5/15 11 /19 (39) V 0/5 0/5 0/5 6/15 6/30 (20) VI 13/40 24/69 1/25 6/40 44/174 Sub total 4 25 15 35 33 % errors 28/73 30/88 6+ /43+ 72/259 8/55 Total 28 14 16 38 34 % errors *: Eado would have received more exercises in each of the different Levels if he had not been helped by his mother in the initial stage. Table 2 points also to differences in the degree of difficulty of the examples in the different Levels. The examples of Level IV seem to be th~ easiest to master (error proportion of 17%), and those of Level V the hardest (39%). A possible explanation for Level IV being easy is that it required rounding thousands to thousands and came immediately after the students had mastered rounding hundreds to hundreds (Level III). In contrast, rounding thousands to tens (Level V) came after rounding hundreds to hundreds and thousands to thousands (Levels III and IV). This seems to have had a confounding effect, and students find the adaptation of the rule more difficult. Table 2 shows that there are large differences between the students in the study. For example, Ariel and Eado appeared to be much faster learners than NiH and Grit. Ariel and Eado experienced almost no difficulties in the second stage of the generalisation of the rule (Levels III-IV). At this stage, they made 7 errors out of 65 examples (11 %) whereas Nih and Grit made 37 errors out of 109 examples (34%). Table 3 presents the frequency of error type made by each student. The errors have been broken down here into two main categories: those related to the algorithm of rounding aIld those which are not. Details of each category are given below.

""Errors" Related to the Rounding Algorithm The large majority of the student errors were related to the misuse of one of the three components of the procedural rule, primarily the first and second components. A: Incorrect number domain (incorrect number of zeros). This type of error occurs, for example, when students who are asked to round to hundreds round instead to

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tens or to thousands. These errors are marked in Figures 2-7 by [Tens] and [Thousands] respectively.

B: Incorrect "direction." This type of error occurs when the student rounds within the correct number domain but either down instead of up (marked by [iD or up instead of down ([iiD.

B1:Incorrect "direction" in difficult examples. Examples which required rounding across more than one number domain are regarded as of special difficulty and have been marked in Figures2-7 by an asterisk (*). For example, rounding 6 798 to tens (answer: 6800) requires a change in the tens' as well as in the hundreds' digit. Errors of direction for this. example occur when a student identifies the "two zetos but rounds down to 6700 rather than up to 6 800, and have been marked in Figures 2-7 by [iii]. ll

C: Incorrect part. This type of error was not observed with the students in this study.

Table 3

Number of Errors * per Error Category for each Student Total for each Grit Type of Error Ariel Eado Error Category Nili 2 7 4 13 A Domain 7 20 3 4 6 B Direction B1 Directiont 2 4 1 7 C Incorrect part 14 42 Sub total 6 18 4 ADVANCE 1 6 7 1 6 Time 1 4 Transfer 2 1 1 5 9 Unexplained 2 1 6 2 8 Sub total 6 14 2 11 33 Total errors for each 12 6 32 25 75 student * Errors within the different trials to solve the same types of problems t: Incorrect direction in rounding across more than one number domain ADVANCE: Pressing the ADVANCE key to get the final answer Transfer: The first two exercises in a sequence-when rounded to the domain of the previous sequence

"Errors" Unrelated to the Rounding Algorithm ADVANCE-key errors. When students use the ADVANCE key to receive the correct answer, (marked in Figures 2-7 by"ADV."), their solution is recorded by the computer as incorrect.

Time errors. When students exceed the time limit, their solution is recorded by the computer as incorrect (marked by "TIME").


Transfer errors. In certain cases, when the software changes the Level for the student and the student is not aware of this change, the student continues to round the first few examples in the new Level to the previous target domain. For example, when a student has ~hown mastery of Level II, rounding hundreds to tens, and is moved to rounding hundreds to hundreds, he/she continues rounding to tens, which results in an error message. The computer's error response attracts the studenes attention to the change. IIErrors" of this type are named in Table 3 as "transfer errors. II Careless errors. These are mostly" errors" of switching the order of digits when typing in the answer. For example, in her third session, NiH was assigned to round 94 to tens (answer: 90). She typed a 0 and while planning to type the 9, she got an error message (the software evaluates each digit as typed and it expects a 9 to be entered first). In the second trial, NiH typed the answer in the order expected: first a 9 and then a O. Switching the order of digits is a common type of error because in strands of vertical computations one types the digits of the answer from right-to--left and in strands of horizontal computations, from left-to-right. Since students get examples of mixed strands, it happens that they confuse the order for entering the answer (Hativa, 1988a, 1988b, 1988c; Hativa & Lesgold, 1991). This type of error occurs only in computerised practice designed so that each digit is considered separately (using an "INPUT' command). Unexplained errors. The errors which cannot be accounted for by any of the error categories listed so far are named here lIunexplained" (marked in the protocols by?). In Table 3, IIcareless errors are also counted under this category because it was not always possible to distinguish between careless and unexplained errors. ll

Summary of the Four Types of Students' Errors Ariel and Eado made only a very small number of errors in total. Nih's main errors were in identifying the domain, direction of rounding, and using the II Advance" key. Orit's main errors were in the domain and direction of rounding. She was also too slow to respond (Time errors) and appeared to concentrate on the task less than her peers (Transfer errors).

Summary and Discussion This study examined cognitive processes underlying student learning from solving computer-provided examples. As suggested in the introduction, the study attempted to identify how students actively construct knowledge in ways which satisfy constraints inherent in instruction, traced changes in student performance to particular instructional events, identified· the way in which students' prior knowledge affects learning, and examined relationships between conceptual understanding and procedural skills (Hiebert & Wearne, 1988). All this was done through examining patterns of learning in conditions that encourage students' inventive procedures. In the section which follows, the type and extent of knowledge and understanding which the students gained in rounding numbers has been

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summarised. An attempt has then been made to answer the research question concerning how high-achievers learn and understand new concepts in arithmetic from computer-based practice. Finally, the cognitive aptitudes which are shown here as promoting student learning through computerised practice have been discussed.

The Meaning of Knowledge and Understanding in Rounding Numbers The knowledge and understanding which the students in this study gained in rounding numbers may be classified into the following two categories:

1. Gaining Knowledge of Rounding Concepts Rounding is shown here as a complex concept which requires student reasoning, insight, decision-making, good understanding of the decimal number system, and good "number sense." A basic concept needed for performing the rounding is the closeness / nearness of one number to another. The task of rounding is even linked at times with the term of closeness. For example, "rounding a number to tens" may be formulated as "rounding a number to the closest ten." Had the computer program in this study stated the rounding examples in this way, the students would probably have acquired the concept of closeness from the verbal statement and would not have needed to deduce it by themselves from the examples which they solved. On the other hand, the fact that three of the four students did succeed in deducing the closeness concept from these examples indicates that, if better students are given less information, they rely on their _cognitive thinking abilities to identify concepts and rules.

2. Promotion of "Number Sense" Schoen (1986) suggested the existence of interdependency between the process of estimation in a particular domain and the understanding of mathematical concepts within that domain. He also suggested that estimation assists in number-concept development. Indeed, this study demonstrates the interdependency between the process of rounding in a particular domain and the understanding of mathematical concepts within that domain. The rounding experiences of the students were shown to promote number concepts in the following cases: • Promotion of number-size concepts: a sense of the size of numbers and of operating with numbers of different sizes; naming, reading and writing numbers in a new domain- the thousands. • Promotion of a sense of distance (closeness) between numbers~ Evaluating and estimating the "closeness" of the numbers bordering the domain to a given number in that domain. To illustrate, on rounding 5346 to hundreds, the student needs first to identify the boundaries of the nearest hundreds (5300 and 5400), and


then to evaluate which one of these two is closer to 5346. • Promotion of an understanding of the decimal number system. The students identified the upper and lower bounds, and the middle of each of the number domains involved, as well as the similarities and differences of number symbols and of operations with numbers among these domains; • Promotion of generalisation skills from one number domain to another: perception/recognition of the concept of generalisation across number domains, and the translation of a skill or property or an operation or a concept from one number domain to another.

How do High-Achieving Students Learn and Understand New Concepts in Arithmetic From Computer-Based Practice? The observations and interviews described in this study made it possible to follow the process by which the participating students have constructed new knowledge. Resnick and Ford (1981) suggested that constructivist learning consists of goal setting, identifying and solving problems of understanding, connecting old and new knowledge, constructing and testing inferences, and monitoring and evaluating learning. Indeed, the students in this study used all of these learning strategies to construct their knowledge actively. The pattern of the strategies which evolved from their protocols may be sorted under two main themes: (a) looking for meaning in the new material; and (b) looking for patterns and general rules which unify and explain the different types of examples. To elaborate:

1. Looking for meaning in the new material. This aspect was observed in the following instances: • Identifying the "a" in the units digit on the basis of the meaning of the word "to round" which describes the form of the zero; • Assigning meaning to the procedural criterion for making the decision about the direction which is the middle value of the range of numbers involved; • Assigning a "nearness/closeness" meaning to rounding.

2. Looking for patterns and general rules which unify and explain the different types of examples. This aspect was observed when students looked for a general pattern in rounding numbers of the different domains, consequently identified the "closeness" interpretation of rounding (the conceptual rule), and the role of the middle number in rounding. The strategies students used to identify the general rules were: • Trial and error to accumulate in memory an agenda of solutions to numerous problems of the same type; • Incubation or, as named here, "cognitive digestion" of the examples accumulated in memory; • Comparing and contrasting the solutions called from memory, performing analysis and synthesis on this database, considering similarities and differences, and finally identifying a unifying pattern.

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Cognitive Abilities which Promote Student Self-Learning from Computerised Practice. The aptitudes which enabled the students in this study to construct new knowledge from practice with no verbal explanations can be summarised in the following way:

High motivation to succeed in computerised practice. All students except NiH stated in their interviews and showed in their behaviour throughout the whole period of study that they enjoyed working with the CAl system and that they felt that this work was beneficial to their studies. The observations and interviews supported these claims. Eado, Ariel and Orit were very ambitious to succeed in academic learning. Nili, on the other hand, was much less motivated, perhaps because of a prior family crisis related to the academic studies of her older brother. Investing mental effort, mindful work. All except Nili seemed to invest substantial mental effort using their problem-solving abilities to overcome any impasse they encountered. On receiving a problem that they were unable to solve, they would always use all thr~e options to type in the answer, would make cognitive efforts in each of these options to provide the correct answer, and insist on discovering by themselves the method for solution. They seldom used the ADVANCE key and made very few "unexplained" errors, thus avoiding guessing on encountering an impasse. Nili was less ready than the three others to invest mental effort in the computerised practice. When she faced a new or a difficult problem, rather than using her good problem-solving abilities to find the solution by herself, she often chose the easy way out. She either made an educated guess or she pressed the ADVANCE· key which resulted in the computer displaying the answer on the screen and recording the solution to that problem as incorrect. Good ability to generalise. All four students displayed good ability to generalise algorithms and rules across the different number domains, even to the thousands' domain which was unfamiliar to them before the study began. Good memory. All students used their memory of the examples that they had encountered in the sessions and of the computer-provided solutions to these examples to identify the rounding rules. This was particularly apparent with Nili and Orit who, in their post-session interviews, remembered and retrieved, out of the approximately 60 examples that they had received in the 20-minute computer session, all those on which they had made errors, along with the correct responses provided by the computer (Figures 2 & 5). In addition, Orit remembered a rule that she had identified from a few examples that she had encountered in her computer work 6-7 months earlier. Deductive ability. All four students displayed good deductive ability in that they identified a rule on the basis of solving only a few examples. Stability of the learning. Once Eado and Ariel had identified a rule, they made almost no errors in the solution of the subsequent examples which used the same rule. This was true also for Orit for Level IV (Figure 1).


In spite of many similarities in the learning of the four students, this study also revealed large individual differences between them. Ariel and Eado, more than the other two students, looked for meaning in what they were doing, identified easily the general rule for rounding, showed greater stability and made a significantly lower proportion of errors. The process of incubation seemed to be almost instantaneous with Ariel and Eado but took substantially longer (days or weeks) with Nili and Orit. The differences in length of the incubation period and in error proportion resulted in large discrepancies in the total number of exercises needed to demonstrate mastery of the six Levels. Krutetskii (1976), investigating what comprises the ability to learn school mathematics, sorted students into four groups: "very capable," "capable," "of average capability" and "incapable." In that work, the more capable students were described as those who quickly and easily master mathematical material and who think independently and somewhat creatively while studying new material. Krutetskii assigned his subjects different tasks, several of which were of a similar nature to those which students faced in their computer work in the present study. Krutetskii found that students of different abilities differed in four dimensions, of which the two most relevant to this study, are: • The ability to generalise mathematical material: Although generalisation is usually done as a result of comparing or contrasting a series of phenomena, "capable" pupils were found to pass quickly from solving simpler problems to solving more complex ones of the same type and to generalise with only little comparing and contrasting. The most "highly capable" pupils were able to generalise, on the basis of analysing just one phenomenon, without comparing and contrasting. • Mathematical memory: The memory of pupils "capable of learning mathematics" was markedly selective: they did not retain superfluous, unnecessary data but remembered primarily generalised and abbreviated structures. This is the most convenient and economical method of retaining mathematical information. It does not load the brain with surplus information and thus permits longer and more accessible retention. Other studies carried out in other learning environments, with other subject matter and with students of different ages, have reported similar results. Silver (1979), for example, showed that good problem solvers apprehended the important structural features of the problem. Shute, Glaser and Raghavan (1988) identified, in an exploratory computer environment (Smithtown), that successful learners tended to think in terms of generalising their hypotheses and explorations beyond the specific experiment on which they were working. Successful learners were also found to show more effective memory management than the other students. The high achievers in the study presented here may be regarded as "capable" students on the basis of their top performance in arithmetic in their respective classes. The results presented in this study have shown that these students mastered new mathematical material relatively easily, through generalising from a small number of comparisons and contrasts, to rules. For Ariel, a single example was enough to deduce the general rule. The fact that three of the four high

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achievers discovered and used the conceptual statement of the rule for rounding suggests that, indeed, these students have "mathematical memory"-they remember primarily generalised and abbreviated structures, abstract mathematical relations, and patterns of reasoning.

Implications for the Classroom This study has implications for several domains.

1. Teaching of Estimation and the Underlying Skill ofRounding Because of the importance of estimation as presented above, "building a strong computational estimation strand into school mathematics programs must be a top priority for curriculum developers in the near future." (Trafton, 1986, p. 16). An instructional program designed to teach computational estimation skills (including rounding) should take into account that, like problem solving, these are higher-order thinking skills. The knowledge and understanding of these skills are developed and improved over a long ·period of time. Therefore they should be taught gradually and in chunks (Sowder, 1989, 1990). Reys (1986) advocated that "to be effectively developed, they must be nurtured and encouraged throughout the study of mathematics" (p. 31). Thus a comprehensive estimation curriculum must address the development of number sense, number concepts, and estimation strategies (including rounding) (Reys, 1986).

2. Teaching Specifically the Concept of "Rounding of Whole Numbers" I suggest that instruction of this topic be based primarily on the conceptual definition of "closeness" and on generalisations across number domains. The procedural rule may serve as an illustration of the conceptual rule and be practiced with emphasis on its three components. It is important to present a rationale for rounding, to assign each number domain a separate practice and then add practice which mixes all types of examples. Special attention should be paid to practice exercises which include the more difficult typ~s of-examples (those that require the rounding across more than one number domain).

3. Learning New Material Through Computer-Based Practice The prompt computer-provided feedback is shown here as promoting the learning of new concepts in arithmetic. The fact that the computerised practice in the system examined here was designed to provide practice of material already taught in class rather than inducing the learning of new topics, brought about the development of some difficulties in student learning. The identification of these difficulties through this study and the recognition that students do learn new material from drill and practice may contribute to improving the design of new practice systems, or to modifying those already in use. For example, this study


suggests that some of the desired features of such systems include the provision, for each new type of practice, of short verbal explanations, heIp screens, and completely worked examples so that each student arrives at deducing the general rule or algorithm. This form of system design may facilitate the conceptual and procedural learning of new material or reinforce material already taught in class. The students in this study found that this approach to learning was very challenging and exciting.

4. Change of perceptions of and attitudes towards computer.;.based practice. As discussed above, computer-based practice is usually denounced. by mathematics educators as teaching technicalities without developing understanding and cognitive abilities, and as very boring. This study shows that computer-based practice, if planned appropriately, can be beneficial for the better students, developing their understanding of mathematical concepts and their cognitive abilities, and serving as a source of learning through challenges.

Additional Issues of Interest and Suggestions for Further Studies An interesting phenomenon identified here which should be studied further is the process of "cognitive digestion" /incubation-the unconscious cognitive solving of problems over time-and the role of mathematical memory in this process. This issue which was shown here as being essential to the students' own discovery of rules and procedures, requires further study. Another area for further investigation would be the reasons for the large differences in the difficulty of generalising to the different number domains. For example, it would be of interest to identify the underlying reason why almost all students mastered the examples of Level IV rapidly and easily whereas they encountered substantial difficulties in Level V (and somewhat in Level VI as well). Understanding the cause would help mathematics educators understand better how students learn the decimal number system, and the skills related to rounding and estimation. The final area for investigation would be to improve the planning and development of the system to make it beneficial for all levels of students.

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Author Nira Hativa, School of Education, Tel Aviv University, Tel Aviv, 69978, Israel.