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Mathematics Education Research Journal, VolA, No.1, 1992.

GOOD STUDENTS BEAT THE COMPUTER SYSTEM: STRATEGIES FOR SELF LEARNING FROM COMPUTERIZED PRACTICE IN ARITHMETIC Nira Hativa, Tel Aviv University

Naturalistic methods of inquiry were used to investigate learning processes of above-average second, third, and fourth graders while practicing arithmetic with a computer. Because the sofn-vare enabled the better students to accelerate through the practice material, they received practice in topics that had not yet been covered in class, and thus were attempting material which was new to them. It also happened that they encountered exercises that they had learned in class but hadforgotten how to solve. This study reveals that when confronted by exercises they do not know how to solve, above average students use a variety of strategies that lead to their identification of solution algorithms, while not always understanding the underlying concepts. The article identifies the different problem solving strategies that students used, sorts them into categories, and illustrates them with examples from students' protocols. On the basis of the findings, suggestions are made for designing computer software for arithmetic practice that promotes student problem solving strategies along with mathematical understanding.

One of the major advantages expected of the computer as a curriculum learning tool is that it enables each child to interact individually with the software and to learn at his or her own pace, style, and ability level. In learning or practicing mathematics with computers, the material is arranged sequentially according to either increasing difficulty or to the curriculum topics. Each student is able to advance through this sequence at hislher own rate. However, if students in the same classroom, but with heterogeneous aptitudes are allowed to advance each at their own rate through the curriculum material, we may expect that after a while they will develop large discrepancies among their placement in the curriculum units. Indeed, several studies (e.g., Osin & Nesher, 1979; Hativa, 1986, 1988) reveal large differences in the mean level of arithmetic practice between high and low achievers of the same classroom. Such differences grow with the length of use of the computerized work. The better students advance in the arithmetic computer curriculum levels at a much faster rate than the low achievers. In addition, there can be large initial discrepancies in the starting level of practice among students as a result of an initial diagnosis administered at the beginning

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of the school year, before starting the computer practice work. Thus, if the better students are not diverted from the regular computer curriculum to work on some enrichment material, they are accelerated by the computer software so that they get exercises that are part of a curriculum for higher grades than the one they currently attend. Their advancement in the practice levels is based on their success in proving mastery of each level of practice. This raises an interesting question: how do the better students succeed in proving mastery of material that they have not yet been taught inclass, and for which they do not get any adequate i n s t r u c t i o n ? ' The present study has been designed to answer this question. It aims to reveal what are the problem solving strategies and the learning techniques used by above average students when they solve arithmetic exercises that belong to material with which they are not familiar. Method The study used qualitative methods of inquiry to identify students' cognitive processes during the computer practice. Carpenter and Moser (1983) suggest that for researchers to achieve an unbiased account of students' cognitive processes; individual interviews and observations should be conducted within a setting typical for the task in question. However, Ginsburg, Kossan, Schwartz and Swanson (1983) argue that observations in natural settings might be "exceedingly difficult to make, partly because much thought is private and partly because the occasions on which thought is public are few and far between" (p. 17). Be this as it may, my substantial experience in observing students working with different computer systems suggests that the naturalistic method of observation coupled with individual interviews is particularly appropriate for investigating students' solution processes in a computerized learning environment. The computer context makes it possible to observe every step in the students' solutions and every error made, because these are displayed on the screen. As well the students' reactions to the computer presented problems and feedback are easily observable.

The Computer Learning Environment The computer system observed in this study was adapted to Israeli schools in 1975 from an American system marketed by Computer Curriculum Corporation (CCC) of Palo Alto, California. Both systems, the Israeli and the American, are widely used in their respective countries. The Israeli system was used in Israel in 1990 by more than 120,000 elementary school students. With a translated curriculum, it is being exported successfully, under the name of DEGEM, to Can~da, USA, Germany, Panama, South Africa, Spain, and Uruguay. In these countries there are altogether thousands of systems (each with 30-40 terminals). The CCC arithmetic curriculum was extensively field tested and successfully validated as an effective supplement to instruction (e.g., Gourgey, 1985; Ragosta, Holland & Jamison, 1982).

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Each system in Israel is rented, rather than bought, from the Center for Educational Technology (CET), with its operation strictly monitored by the CET. Each school is visited often by a CET representative who supervises the administration of the system's operation. This centralized method reduces differences in operations between the various schools so that the system is operated in almost the same manner in all regular schools. The configuration of a school system in Israel is a minicomputer with 40 terminals located in a computer lab. Since 1991, the hardware is being replaced by locally networked microcomputers. The system provides practice in arithmetic, reading comprehension, and a few other topics. For the arithmetic topics in a typical school, each second through sixth grade class, together with the teacher, visits the computer lab twice a week, each session lasting for twenty minutes. This practice is designed to complement three regular 45 minute sessions of in-class arithmetic instruction by the teacher each week. The software of the computer system in this study consists of a large number of timed exercises grouped into 15 strands, encompassing the elementary school arithmetic curriculum. Each strand concentrates on a single arithmetic topic (e.g., vertical.multiplication, decimals). The exercises in each strand are hierarchically arranged in a sequence. A student. working individually with the system receives exercises in a random order from all strands that are operative for him or her. If a student does not succeed in solving a particular exercise in three trials, the full solution (including all intermediate computational steps) is printed on the screen. The computer work is timed and therefore excludes the option of making computations on paper. Advancement through the levels of a strand is based on principles of mastery learning. Within each strand, the student receives a group of exercises of the same type that differ only in the digits. For example, the following are three exercises of the same type, vertical addition of two digit whole numbers with carry-over in the units' digit: 22 48 37

+39

+34

+13

When a student satisfies a certain criterion for mastery of a particular type of exercise, he or she is advanced by the software to the next type of exercise. The whole process of managing the studenfs work with the computer (e.g., initial diagnosis, testing for mastery, going up or down the hierarchy of exercises) is carried out by the computer, usually with no teacher intervention.

The Procedure There were 42 students in this study drawn from second, third and fourth year classes in a Tel-Aviv suburban school for middle to upper SES families .. Subjects were above the class average in computer performance. Above average participants were chosen because a previous study (Hativa, 1986) had indicated that they used problem solving strategies more frequently

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than less good students. During the four months of this study, each subject was observed over a ten minute computer session at least three times, once in every 10 consecutive sessions. The time interval between two consecutive observations of a particular student was such that in each of the observations, the student received a completely new set of exercises to solve in the computer work. Two observers shared the load of observations. In order to develop a common procedure of observation and interviewing, three students were observed by both observers, each one in a single computer session. During the computer session, the observer recorded on paper every computer exercise the student received, all the steps of the student's solution, as well as the computer response. Immediately on completion of the computer session, the observer interviewed the student and asked him or her to describe or explain the method for solution for each type of exercise and the source of the student's knowledge for this solution. The observer also used probing questions to identify the level of the student's understanding of the mathematical concepts underlying the practice.

Above Average Students' Strategies for Advancement in the Computer Practice The main goal as designed for the computer system in this study is to provide drill and practice 0l! material students have already learned in class, arid to reinforce and improve mastery and understanding of already familiar concepts and algorithms. However, our observations show that in the actual use of the computer program, the name of the game particularly for high achieving students, is the fastest possible advancement through the hierarchical levels of the computer practice. This means the material is mostly unfamiliar to them, and is completed without full understanding. Even when the less able students have practice in materiai already taughtin class, if the teaching of this material does not occur very close to the common practice (which is regularly the case), they appear to forget the solution algorithm (Hativa, 1986). Thus, in many cases. students work on computer exercises they either do not fully understand, or do not remembcr how to solve. Our observations reveal that very frequently students do manage to provide correct answers to such exercises by resourcefully applying problem solving strategies. The rest of this article demonstrates the strategies students use to meet this goal and examines the understanding underlying the students' solutions to such exercises. Pressley (1987) defines learner strategies as goal directed processes which are potentially conscious and controllable, and when matchcd to task requirement. facilitate perfonnance. Mayer (1983) defines them as techniques that serve as a guide in the problem solving process. although they do not guarantee a solution. Students' problem solving strategies have been widely studied within the context of general domain knowledge and also of domain specific knowledge, as related to a particular subject matter. Looking at the general

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problem solving strategies identified in research literature (Clement, 1984; Gick, 1986; Greeno, 1978; Polya, 1957), the following lists ones relevant to this study: 1. analogy, 2. generalization, 3. means-end analysis, 4. induction, 5. embedding the problem into a larger one, 6. using heuristics to reduce the number of paths for solution, 8. comparing solutions with worked examples, 9. trial-and-error, 11. connecting the problem to existing knowledge, 12. searching through the problem space (the problem solver's view of the problem), 13. identification of patterns among the relations, 14. problem decomposition, and 15. analyzing relations among problem elements. Our observees used all these strategies as well as additional ones in order to prove mastery of material they had not yet learned. Additional strategies identified in this study are; 7. 10. 16. 17.

probabilistic considerations, getting human assistance, elimination, and going backward.

The next section of this article discusses and illustrates the ways students apply these domain general strategies to the specific context of computerized practice in arithmetic. Strategies numbered I through 10 are presented under respectively numbered headings. The rest of the listed strategies are mentioned when shown to be used. The illustrations are taken from protocols of students' computer sessions. The questions by the observer (Q:) and answers by students (5:) are taken from the protocols of their postsession interviews. To indicate computer responses, 'C:' is used.

1. Using analogy by keeping to a pattern. Analogy seeking is a key learning mechanism in solving problems. "The efficient learners seek relationships between what they know and what there is to be known" (Brown, Pallinscar & Purcell, in 1985, p.126). "Learning by analogy is the mapping of knowledge from one domain over to the target domain, where it is applied to solve problems" (VanLehn, 1986, p.152). The students in our study frequently used solving-by-analogy stra,tegies, particularly in their mapping from the familiar domain of whole numbers to the yet unfamiliar domain of decimals and negative numbers. The strategies are sorted here into categ9ries representing the types of analogies

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induced by the different types of exercises. Each category is illustrated by one or two examples. 1.1 Imitations of worked examples Several types of exercises in the computer curriculum come with builtin imitation factors which encourage students to follow a given pattern without the need to understand what they are doing. This type of analogy is the simplest form of "example-exercise mapping" (VanLehn, 1986).

Example 1: If 2 = 0.2 then 1 =? [fifth-grade curriculum, answer: "0.7"1 10 10 S [third grader]: Very easy -- the answer is zero, then a period, then a seven. Q: Why? S: It is like the given 0.2, I've just replaced the 2 with a 7. Q: What does your answer [0.7] mean? S: This line [the fraction line] is being converted into a period and the zero [in the 10] goes in front of the point. The answer reveals imitation of the given example, with no understanding of the notation at all. The boy concluded from the computer examples that 0.7 is just another notation for writing the fraction 7/10. This type of imitation was observed with many students, not only the good ones.

1.2 Transferring the pattern of arithmetic operations from familiar to new notation

Example 2: (-1)+(-4)=? [fifth-grade curriculum, answer: "-5"] S [second grader]: 1 plus 4 is 5 and I put the subtraction sign before the 5. Q: Why? S: I don't know but I think it should be there because it comes also before the 4 and the 1 [in the given example] ..

EX£lmple 3:

0.938 - 0.552 ????

[fifth-grade curriculum,answer: "0.386"]

S [third grader]: I subtract as usual: 938

-ill Q: S: Q:

386 and then I copy the period and zero. Why do you copy the period and the zero? Because they put them there in the [original] numbers. What do the "period and zero" mean?

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S: Q: S:

I'm not sure exactly but they put them in many numbers and this makes the numbers small. How do you know that? I asked the teacher and this is what she said.

The dialogues in examples 2 and 3 show how students solve the problems as if they were in whole numbers, then adjusting the answer to the external features of the numbers (the period or the minus sign), without understanding the meaning of these features. The latter dialogue also demonstrates that the student approached his teacher to get an explanation for the type of numbers that he did not understand [strategy #10] and that the teacher looked for a way to get out of the question without really providing the appropriate explanation of decimals. 1.3 Transferring the pattern of a sequence from familiar to new notation

Example 4: What is the missing number in "_45, -40,? ,-30, _25"? [fifth-grade curriculum, answer: "_35"] S [third grader--types in promptly]: "_35" Q: How? S: This is very simple. The numbers become smaller by fives so that the missing number is 35 and I add the minus to make it look like the other numbers. Q: What is the meaning of the "minus"? S: It is below zero. Q: What is "below zero"? S: I'm not sure. My father told me that and he promised to explain it to me. We see here again a solution by imitation of whole numbers without understanding. We see also that the student is not comfortable with solving without understanding and that he approaches his father for getting explanations [Strategy #10]. To summarize students' use of analogy: Students accomplish problem solving in the new domains by (a) searching in memory for similar notations, operations, algorithms, patterns, or rules with whole numbers [Strategies #11,#12,#131; (b) performing the required operations with whole numbers; and then (c) adjusting the answers to the domain of decimals and negative numbers by adding the decimal point or the negative sign, respectively. This simple problem solving process of maintaining the pattern of the given exercise enables students to solve correctly exercises in number domains that are unfamiliar to them. However, the students do not seem to get to understand the meaning of the new types of numbers unless somebody more knowledgeable (the teacher, a parent, peers, etc.) explains it to them. This finding may .guide designers of computerized drill-and-practice programs to give special attention to providing meaning to those examples that

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can be solved using analogy to whole numbers. This type of example is frequently found in computerized practice programs.

2. Generalization. Generalization is an advanced type of analogy that requires, more than simple imitation, a certain level of knowledge transfer to new situations. Our observations identified generalizations to new situations of arithmetic notations, operations, principles and rules. 2.1 Generalizing notations to new situations

Example 5: A fourth-grade student who had not been taught decimals in class, learned from final computer answers (like in Example 1) to solve correctly exercises of the type: 53 =:: ? [fifth-grade curriculum, answer: "5.3"] 10 A week later he received a new type of exercise consisting of a three digit number to be divided by 100: 537 -_?. 100 S [types in]: 5.3.7 Q: How did you come up with your answer? S: I know that when I have only one zero below [the denominator of the first example] and two numbers above [i.e., two digits, as in 53 in the numerator], I put one period between the 5 and the 3 [as in 5.3] and if I have two zeros and three numbers, I thought I should put a second period. This explanation reveals the strategy of generalization of the decimal notation to the new situation. The student deduces from the first type of examples that the zero of the 10 is being converted to a period and thus he converts each of the two zeros in the 100 to a period. This resourceful solution also reveals the student's lack of understanding of the concept of the decimal point. 2.2 Generalizing operations to new situations

Example 6: 63 x 64 = 6? [fifth-grade curriculum, answer: "7"] S [second grader--types in]: "12". C: [Computer response]: Try again! S: I don't understand why ... it is written here to multiply so I multiply 3x4. The computer is wrong [types again "12"]. C: The answer is 63 x 64 =:: 67 S: Oh! I see! When the numbers are above [in the exponents], one should add rather than multiply.

Strategies for Self Learning ofArithmetic 69

Example 7:

1

First try for many students was: 9 that is, they added separately the numerators (2 + 5 = 7) and denominators (3 + 6 = 9). These two examples illustrate how students initially apply rules and arithmetic operations from the familiar domain of whole numbers to the new domains [Strategy #11]. Then, on failure, they learn from the computer response [trial-and-error: #8] the correct, new algorithm for solution without getting any explanation of it. 2.3 Generalizing principles and rules to new situations

Example 8:

Q: S:

A second grader proved mastery of computer exercises of the type: "Given 24, 87, 50, 62 -- which one is divisible by 10?" [answer: "50"]. A few weeks later, she received exercises of the type: "Given: 2470,4800,5004,6203 -- which one is divisible by 100?" [answer: "4800"]. She promptly typed in "4800".

How did you know that? I knew that if I had one zero it was divisible by 10 so I understood that two zeros were djvisible by 100.

Her answer reveals that she generalizes the rule of divisibility by 10 to divisibility by 100, using without understanding a completely technical algorithm.

Example 9:

Round 3.6 to a whole number [seventh-grade curriculum, answer: "4"]

S [fourth grader--types in]: "4" Q: How did you know that? S: Last year I received from the computer exercises such as to round 36 or 360 and I remembered that I had to add one number to the answer [Le., to round upward to 40 and 400 respectively]. This explanation demonstrates that she applied a familiar rule (for whole numbers) to a new number domain (decimais). It also demonstrates her excellent memorization ability; a feature shown frequently with the high achievers in my observations. Although she had not yet studied the concept of rounding of whole numbers in class and the practice with the previous

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rounding exercises had occurred more than a year previously, she was able promptly to generalize the rule to the new situation. To summarize, generalization requires a greater thinking effort than analogy but the processes are similar. For generalization, students solve the problems by (a) searching in memory for similar notations, operations, algorithms, patterns, or rules with whole numbers [Strategies #11,12,13]; and (b) applying the operations, algorithms, etc., to the new situation, but not by way of imitation, as in analogy. Although students here also correctly solve exercises by generalizing without full understanding of the new number domain, I suggest that using this strategy still develops students' abilities to generalize from one number domain to another.

3. Using means-end analysis. Means-end analysis is the strategy of reducing the difference between the current state and the goal of the problem by applying appropriate problem solving operators. It is useful whenever the initial state and goal of a problem are fairly well specified (Gick, 1986). In the computer curriculum examined here, only a small proportion of exercises come with a final answer or with an explicit goal. The use of means-end analysis in such exercises is illustrated in example 10. However, students also appeared to use this strategy when the goal was not specified, as illustrated in example II.

Example 10:

If 40 + a =63 then (choose the right option for an answer): 1. a =63 + 40 2. a = 63 - 40 3. a =40 - 63 [fifth-grade curriculum]

This exercise is designed to provide practice in the rules of equality. For its intended solution, one should subtract the same number (40) from both sides of the given equation and get an equivalent expression (40 + a- 40 =63 40). Thus the COITect answer is option # 2 (a = 63 - 40). However, the type of a multiple choice exercise which provides the "given" [here 40 + a = 631 and the goal [identifying the correct answer from the given options], calls for using the strategy of means-end analysis. The following are the solution methods of two children who had not studied these JUles in class before getting them in the computerized practice. The methods for solution that are manifest through their explanations were also observed with students who had studied the related rules but who still preferred other solution methods than using the rules. Student I [second grader]: The computer asks me what number I should add to 40 in order to get 63. This is a very easy computation -- it is a "23". Then I compute the result of each of the three given answers and see which of these gives me a 23. The first roption I

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with 63 + 40 is too large but the second with 63 - 40 gives me exactly 23 which is the correct answer. Stude!1t 2 [third grader]: I need to add a number to 40 in order to get 63. The solution of the first exercise (i.e., the first option a = 63 + 40) is over 100. it is too large rto be added to 40 to get 63]. Trying the second exercise yields that its solution, 23, exactly fits when added to 40 [i.e. 40 + 23 = 63]. Both students compared the given with the goal (i.e., the three optional results) and worked towards reducing the differences between them, using elimination [#16] and going backward [#17]. The students' explanations demonstrate that although they solved the problem fully understanding what they were doing, they did not satisfy the objectives set by the curriculum designers for this type of exercise.

Example 11:

40 =5 X 2? [sixth-grade curriculum, answer: "3"] This type of exercise comes after the students practice extensively the factorizing of whole numbers into prime numbers, exponents, and the solving of exercises of the type "40 = 5 x?". The expected method for solving the given exercise was to start by 40 + 5 = 8 and then to factorize 8 to give 23.

S [fourth grader]: First, I divided 40 by 2, this is 20. Then I divided again 20 by 2, this is 10. Then 10 divided by 2 is 5 and 5 is the final answer. Now I remembered how many "two's" I had used and this was 3 so this was the answer. Thus, this student first set the component of the multiplication that is known (5) as a target, the "end"; divided 40 (the "given") successively by 2 (thus reducing the difference between the "given" and the "end"), until she eventually arrived at the target of 5. Examples 10 and 11 demonstrate, in addition to the means-end strategy, how students use invented solution processes of their own which are very different from those planned by the curriculum designers.

4. Induction from a single case. This strategy is manifested in exercises designed to examine knowledge of rules of arithmetic (e.g., order of operations, the commutative, associative, and distributive laws).

Example 12:

Is the equation: a x blc = a I b x c true for every a,b,c? [Fifth-grade curriculum, answer: No.]

Many of the observed students developed the following strategy: they substituted the letters in the given equation with numbers (e.g., a with 1, b

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with 2 and c with 3) thus connecting the problem to existing knowledge [Strategy #11]; computed mentally (1 x 2/3 = 2/3 and 1/2 x 3 = 3/2); and drew conclusions on the basis of the results of such computations (that the equality is not true for every a,b, and c). In this way, although often their answer was correct, the mathematical principle underlying it was wrong; they used a single special case to infer a general rule. Indeed, some substitutions caused erroneous solutions. For example, several of the observed students substituted all the different letters with one single number, producing in the given example the following kind of equality: , 5 x 5/5 = 5/5 x 5 = 5 On the basis of this equality, they concluded that the rule specified was true (for all numbers). Similarly, students who substituted 1 for c in the equation "a+b/c=(a+b)/c" deduced erroneously that the rule held for every a, band c. Again, then, we observe students' solution processes that are different from the' objectives of the curriculum designers. This time, the students' processes may lead to incorrect solutions.

5. Embedding a problem into a larger one. Example 13:

Which of the four numbers: "59, 70, 78, 96" is divisible by 4? [fifth-grade curriculum, answer: "96"]

S [third grader--types in] "96" Q: How? S: 59 is odd. 72 and 80 are divisible by 8 and therefore also by 4 so that 70 and 78 are not [divisible by 4]. I know that 96 is 80+8+8 therefore it is divisible by 8 and also by 4. This student knew by heart the multiplication table of 10xl0. However, this knowledge was not enough to find out whether the numbers 70, 78, 96 are divisible by 4 because they were outside of the table of multiplication by 4. Thus, he embedded the problem of division by 4 into the problem of division by 8, a multiplication table that he knew by heart up to 8 x '10 = 80. He used here two strategies; (a) If a number (72, 80) is divisible by 8, then it is divisible by 4, and (b) adding or subtracting 2 to a number divisible by 4 gives an even number (70, 80) that is not divisible by 4. This student shows very good understanding of multiplication and divisibility by components of a factor. In fact, he could end here, deducing that 96 is the answer, by eliminating all other options. However, he decided to further examine this answer, to see whether 96 is indeed divisible by 8. For this examination he applied another strategy; he added 8 repeatedly to 80 (both numbers are divisible by 8 and therefore by 4) until he reached the 96. This student used also strat~gies "connecting the problem to existing knowledge", "problem decomposition", and "analyzing relations among problem elements" (#11, #14 and #15).

Strategies for Self Learning ofArithmetic 73

6. Using heuristics to shortcut the search for solution. Several of the observed students, when faced with an impasse, used heuristics to eliminate several of the possible computational steps in order to shortcut the search for the correct solution. Example 14:

A fourth grader, after practicing a sequence of exercises with multiplication of exponents ("53 x 54 = 57"), received for the first time vertical division of exponents: "57 / 54 = 5?". [seventh-grade curriculum] She typed correctly a "3" (= 7 - 4) on the first trial. She explained:

S: I assumed that the answer could not be an addition [of the exponents: 7+4] because addition had already worked well in the multiplication exercises [e.g., 57 x 54 = 5 11 ] and therefore it could not be good at the same time for the division exercises. The division of 7 by 4 is not good [i.e., it does not yield the whole number that she believed was expected by the computer as an answer], so I tried subtraction and this time it worked.

Example 15:

A third grader, after completing a sequence of exercises in. factoring whole numbers into prime numbers, received a new type of exercise that consisted of three steps [sixthgrade curriculum]: Step 1: Factorize 42 to prime numbers; she typed in correctly (2x3x7) Step 2: Factorize 45 to prime numbers; she again typed in correctly (3x3x5) Both pairs of questions and answers remained on the screen and then the next step was: Step 3: "What is the largest common divisor of 42 and· 45?" [TIie objective is that the student identifies that there is one factor, a "3", which is common to the two give numbers, 42 and 45, and thus it is the answer.]

What is the meaning of these long words? [In Hebrew the term consists of four words] Researcher: You know that I'm not allowed to answer your questions. Try to answer this by yourself. S: [after a short time for thinking, typed in] "3". S:

I was very surprised to observe this, because she had almost no chance of understanding the question without any explanation. After several exercises in other strands, she received a second exercise of the same type. This time

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the two given numbers were: Step 1: 45 (= 3x3x5) and step 2: 44 (= 2x2x1l). [Since there is no common factor to these two numbers, their largest common divisor is "1 "]. To my growing astonishment she immediately typed in the correct answer, a "1". After the session was over she explained: I didn't know what these long words [the largest common divisor] meant. Thus, I thought, what can the computer ask me to do? It could not be an addition because the numbers [42,45] were too large to add by heart and so I assumed that it was subtraction and indeed it worked" [45 - 42 = 3; 45 - 44 = 1] .... Examples 14 and 15 show that when students face the need to use an unfamiliar operation, they try to overcome the impasse by using another operation which is familiar to them. To shortcut the procedure of choosing the most appropriate of the four arithmetical operations, they first eliminate the operations that they believe "the computer could not ask them to do". These heuristic considerations were based on knowledge of arithmetic operations or on familiarity with the way the particular computer curriculum worked. These students knew what level of difficulty of mental computations they could expect from the computer curriculum, considering the fact that making computations on paper while on-line was forbidden. A typical rationale provided by students using heuristics was "It looks logical to try this operation", or "The computer could not ask me to do such a difficult computation".

7. Using probabilistic considerations. When students encountered multiple choice exercises they could not solve, they often used educated guesses, based at times on probabilistic considerations. EX£lmple 16:

Which of the four numbers "423, 610, 498, 536" is divisible by 9? [Fifth-grade curriculum, answer: "423"]

The observed third grader did not know the rule of divisibility by 9. He was not even aware that rules of divisibility existed. S: I try first the 610 but 61 is not divisible by 9 because only 63 is divisible [by 9]. Then I see that there are two numbers that start with a 4, so I type in the 4 [as the first digit]. If the computer tells me that I'm wrong, I know that the answer is 536. If I'm right, I quickly divide [mentally] 423 by 9. If the computer would have told me that I was wrong, I woulq know that the correct answer is 498. To summarize this student's strategy: he uses elimination (strategy #16) to get rid of the wrong solutions. For this aim, he first looks for a number that ends with a "0" because, according to his explanation, "these numbers are easy to divide". Then he applies a probabilistic strategy: If two of the remaining three numbers start with the same digit, he types in this particular

Strategiesfor Self Learning of Arithmetic 75

digit I with probability for success of 2/3]. If the computer responds that this digit is incorrect, these two numbers are eliminated at once and the remaining number is the answer. If this digit is accepted by the computer, he divides mentally by 9 only one of the two numbers that start with that digit. Again, we see here a method for solving an exercise correctly but that does not at all achieve the objective underlying the exercise.

8. Inducing from worked examples. When all the previously listed strategies fail, students resort to getting help from the computer. In such a case they either press any key three times (for three trials) or they press a particular "Advance" key. Both actions result in a display of the correct answer with all the interinediate steps. This answer remains on the screen for only a short time (from 6 to 9 seconds). The students concentrate on the displayed answer, trying to identify the algorithm used. In this way, the solution displayed on the screen serves as a worked example from which students frequently induce the full algorithm for solution. VanLehn (1986) suggests that learning inductively from worked examples is a very common method for gaining knowledge". There is evidence that students flip through the textbook to locate a worked problem that is similar to the one they are currently trying to solve. The situation with the computer system used here though, differs from worked examples presented on the chalkboard or in the textbook in two aspects: (a) the final solution appears for student's consideration very briefly, although not long enough in many cases, to identify the underlying procedure; and (b) the full solution to exercises are presented to the learner without any verbal explanations (unlike in textbooks with their printed explanations). Verbal explanations are crucial for indicating the particular kind of induction to be performed on the examples; what aspects of the examples should be generalized, and how to integrate them. That is, verbal explanations have an indirect effect on students' learning from examples (VanLehn, 1986).

9. Using trial-and-error. When students in our study induced the algorithms for solution from final answers to exercises, they often used the trial-and-error strategy. This strategy was observed to be used frequently by all students, not only the better ones. The better students, however, succeeded in identifying the correct algorithms from final answers more frequently and after less trials than their peers. The next four examples illustrate the use of the trial-and-error strategy: Example 17:

On receiving exercises of the type "100000= 1O?" ,[fifthgrade curriculum, answer: "5"], almost all students observed, irrespective of their achievement level, discovered very easily from the final answers that the algorithm for a correct solution was to count the number of ze~os. M~my of these students, after typing in the correct solution, were not able to explain their solution.

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Example 18:

Replace the "?" by>,