Logo Programming, Problem Solving, and Knowledge-Based ...

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IR 015 847 Swan, Karen; Black, John B. Logo Programming, Problem Solving, and Knowlc,Ige-Based Instruction. Apr 90 39p.; Paper presented at the Annual Meeting of the American Educational Research Association (Boston, MA, April 16-20, 1990). Reports Research/Technical (143) Speeches /Conference Papers (150)

MF01/PCO2 Plus Postage. *Abstract Reasoning; Computer Assisted Instruction; Educational Strategies; Expert Systems; Intermediate Grades; Junior High Schools; Microcomputers; *Problem Solving; Programing Languages; *Thinking Skills *LOGO Programing Language

ABSTRACT

The research reported in this paper was designed to investigate the hypothesis that computer programming may support the teaching and learning of problem solving, but that to do so, problem solving must be explicitly taught. Three studies involved students in several grades: 4th, 6th, 8th, 11th, and 12th. Findings collectively show that five particular problem solving strategies can be developed in students explicitly taught those strategies and given practice applying them to solve Logo programming problems. The research further demonstrates the superiority of such intervention over Logo programming practice alone, explicit strategy training with concrete manipulative practice, and the instruction in content areas that is traditionally prescribed for the teaching and learning of problem solving. Knowledge-based instruction linking declarative to procedural knowledge of problem solving strategies is recommended as a means to this end. The results also suggest, however, that computing environments may be uniquely conducive to the development of problem solving skills as they help learners bridge the gap between concrete and formal understanding. (Contains 45 references.) (DB)

*********************************************************************** IeprouucLIons suppli ea oy LbKb are cne DesE that can be made from the original document. ***********************************************************************

U.S. DEPARTMENT OF EDUCATION 011.ce it Educabonal Research and imeanvement

EDUCATIONAL RESOURCES INFORMATICN CENTER tERICI

II: This document has been reproduced as tecenved bon, the cretson or organizahon br.g.nahng .1

same, changes ha.e been made tc mptowe reproduction duaith, Points 01 mew or opinions slated .n ihts dot..

ment do not necessenty represent °Moe. OE R. 005.1ton of poitcy

LOGO PROGRAMMING, PROBLEM SOLVING, AND KNOWLEDGE-BASED INSTRUCTION

Karen Swan, University at Albany John B. Black, Teachers College, Columbia University

ABSTRACT

Not very long ago, computer programming was touted as the solution to the problem solving crisis in American education, a discipline through which students would automatically acquire logical thinking and problem solving skills. More recently,

however, such notions have gone the way of similar ideas concerning Latin and geometry.

Research has indicated that problem solving abilities are not automatically acquired through computer programming, and programming is accordingly being deemphasized in computer education. Some researchers, however, maintain that computer programming might well support the teaching and learning of problem solving, but that to do so, problem solving must be explicitly taught. The research reported in this paper was designed to investigate such hypothesis. Three studies are described which collectively show that five particular problem solving strategies can be developed in students explicitly taught those strategies and given practice applying them to solve Logo programming problems. The research further demonstrates the superiority of such intervention over Logo programming practice along, explicit strategy training with concrete manipulatives practice, and instruction in content areas traditionally prescribed for the teaching and learning of problem solving. The results indicate that problem solving strategies will not be developed through Logo programming alone, rather must be explicitly taught and practiced. Knowledge-based instruction linking declarative to procedural knowledge of problem solving strategies is recommended as a means to this end. The results also suggest, however, that computing environments may be uniquely conducive to the development of problem solving skills, in that they support quasiconcrete, malleable representations of abstract concepts that can help learners bridge the IN4`b

gap between concrete and formal understanding.

NI"b 43d_ "PERMISSION TO REPRODUCE THIS MATERIAL HAS BEEN GRANTED BY

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Karen Swan

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communication was therefore more influential in shaping our thought than the content of the message it transmitted. Whether or not one accepts McLuhan's hypothesis en toto, it seems dear that

each medium of communication entails unique formal attributes that matter, or that can be made to matter, in learning. Salomon (1981), for example, has shown how differing filmic presentations can "activate," short-drcuit," or 'model' particular cognitive processes. In this vein, the computing medium has been singled out in recent years as particularly supportive of the development of problem solving abilities (Feurzig, Horowitz & Nickerson, 1981; Harvey, 1982; Mayer, Dyck & Vilberg, 1986; Linn, 1988; So loway, 1986). The Logo programming language, in particular, has been described as an environment designed to 'help children to develop problem solving skills, to think more dearly, to develop an awareness of themselves as thinkers and learners' (Watt, 1983, p. 48).

Indeed, Seymour Papert (1980), Logo's creator, maintains that computers are truly revolutionary educational tools because they support 'transitional objects to think with.' His idea seems to be that abstract ideas can be represented, manipulated, and dynamically tested in computing environments, thereby relieving burdens to working memory and providing students with quasi-concrete models of cognitive processes that can be easily internalized. Subgoals formation, for example, is a problem solving strategy that is given

quasi-concrete representation in Logo programming in that Logo programs are composed of small subprocedures each of which define the means to satisfying particular parts of a larger programming problem. Each subprocedure, moreover, can be written, tested, and refined by itself before the parts are assembled to create the larger program. Programming in this manner, it can be argued, provides a

model of the ahstract process of subgoals formation which can be easily internalized and generalized. Papert writes (1980, p. 23), 'I began to see how children who had learned to program computers could use very concrete computer models to think about thinking and to learn about learning . ."

Papert moreover contends (1980, p. 9), . . that the computer presence will enat..e us to so modify the learning environment outside the classroom that much if not all the knowledge schols presently try to teach with such pain and expense and such limited success will be learned,

as the child learns to talk, painlessly, successfully, and without organized instruction.' Unfortunately, researchers investigating the effect of computer programming ce children's problem solving skills found no such link between programming practice and the automatic development of problem solving abilities (Papert, Watt et al, 1979; Ehrlich, Abbott, Salter, and Soloway, 1984; Pea & Kurland, 1984; Leron, 1985; Patterson & Smith, 1986; Shaw, 1986; Mandinach & Linn, 1987). Their findings have led many educators to discount all notions of the unique suitability of programming environments for the development of thinking and problem

solving skills, and programming is being accordingly de-emphasized in educational computing

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4. Recursion. Otherwise, return to step 1 and specify a new base domain. Apply steps 2 and 3 to it. Continue in this manner until an adequate representation is discovered. Programming environments inherently support the development of analogy in that one is always mapping between computer code (a formal representation) and program output (a concrete representation), and recursion, an important Logo programming technique, is a form of analogical reasoning. Recursion, however, is confusing to most adults. None-the-less, Doug Clements (1987) found significantly better onnlogical reasoning among students with prior Logo experience. Analogy, moreover, is the basis for all transfer, thus a critical clement in the research we designed.

***** INSERT FIGURE 1 ABOUT HERE *****

These six problem solving strategies -- subgoals formation, forward chaining, backward chaining systematic trial and en or, alternative representation, and analogy can be more or leas concretely represented, then, within a Logo programming context. We accordingly designed our instruction and our testing procedures around them.. The instruction was split into units, one for each strategy. Each unit began with explicit instruction on a particular strategy (declarative knowledge) in which wall posters which enumerated the steps involved in each strategy (FIGURE 1) were presented and discussed. This was followest by mediated practice (Feuerstein, 1980;

Delclos, Uttlefleld & Branford, 1985; Como, 1986) solving problems designed to be particularly amenable to solutions employing that strategy (procedural knowledge). An example of a problem

set is given in FIGURE 2. We likewise created six separate tests, each designed to measure students' facility in applying specific strategies to non-computing problems. (See APPENDIX.) Our goal was for students to transfer the strategies learned in the intervention to the paper and pencil tasks of the problem solving tests.

" ** * INSERT FIGURE 2 ABOUT HERE *"**

STUDY ONE Study One was a pilot study concerned with testing the efficacy of the intervention we designed for supporting the development of six particular problem solving skills. Within that context, two factors we believed might effect such development were also manipulated and examined. The first of these was thespecffic Logo practice environments utilized. In particular, we thought that students might be more likely to develop problem solving strategies practiced within both the graphics and the list-processing programming environments, butwe also thought it was possible that one or the other of these environments alone would be more supportive of the development of particular strategies. The second factor we thought might influence the effectiveness of the instruction was the grade levels of the subjects involved. Specifically, we thought there might be developmental differences in students' propensity to develop particular problem solving strategies. Study One thus examined three Interrelated questions:

to

1. Can the Logo procananing environment support the development of subgoals formation, forward chaining, backward chaining, systematic trial and error, alternative representation, and/or analogy strategies when those strategies are explicitly taught and practiced? 2. Do differing practice environments within Logo programming differentially effect the likelihood that such strategies will be developed within that instructional context? 3. Do developmental differences differentially effect the the development of such strategies will be developed within that instructional context?

Subjects. Subjects were 133 fourth through eighth grade students in a private elementary school. All subjects had at least 30 hours previous Logo programming experience.

Methodology. All subjects were given paper-and-pencil exercises testing their ability to apply six problem solving strategies subgoals formation, forward chaining, backward chaining, systematic trial and error, alternative representation, and analogy -- and randomly assigned by grade to one of three treatment groups receiving respectively graphics, list processing, or graphics and list processing practice problems. All subjects received training in each strategy, then were asked to solve four practice problems particularly amenable to solutions involving

that strategy. On completion of all six strategy units, subjects were post-tested using different but analogous exercises. Mean pre-test scores were compared between groups using one-way analysis of variance and found to be statistically equivalent (F2, 130 °" 1.50, P > .10), hence,

treatment groups were assumed to be generally equal in problem solving ability before treatment. Raw scores on all tests except those for alternative representation were converted to

percent correct scores and compared using a four-way analysis of variance. Independent variables were test, strategy, class (grade level), and group. The 'dependent variables were scores on the strategies tests. Because they had ne maximum possible correct, alternative representations measures were evaluated separately using a three-way analysis of variance.

Results. Significant differences were found between subjects' mean pre- and post -test scores for all problem solving strategies except backward chaining for subjects in ail treatment groups.

These results demonstrate the effectiveness of the instruction we designed for supporting the

development of subgoals formation, forward chaining, systematic trial and error, alternative representation, and analogy strategies. The intervention was not shown to be effective for the teaching and learning of backward chaining strategies, nor were differing practice environments found to differentially effect strategy development. The results also revealed developmental differences in students' facilities for both using and developing particular problem solving strategies. Not surprisingly, older students were better than

119

younger ones at applying all strategies. They were also more likely to benefit from instruction

and practice in alternative representation and analogy strategies, while younger students benefited more than older ones from instruction and practice in subgoals formation strategies.

TABLE 1

Study One; ANOVA Table for Subgoals Formation, Forward Chaining, Backward Chaining, Systematic Trial & Error, and Analogy SS

OF

MS

F

PROB

MEAN CLASS GROUP CO ERROR

4952023.5 67283.5 4388.2 18297.7 154400.6

1

4952023.5 16820.9 2191.1 2287.2 1308.2

3784.56 12.86

0 0000 0.0000 0.1914 0.0944

TEST TC TO TCG ERROR

40134.8 947.3 490.9 1286.8 27717.9

40134.8 236.8 245.5

170.86

0.68

STRATEGY SC SG SCO ERROR

TS TSC

TS0 TSCO ERROR

229966.2 41420.0 2205.1 15784.3 158028.5 14457.3 10124.9 2064.6 7842.1 115325.9

4 2 8

118 1

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1.68 1.75

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160.8

118

234.9

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3614.3 632.8 258.1 245.1 244.3

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32 472

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472

7.73 0.82 1.47

1.01

0.0000 0.4061 0.3549 0.7042

0.0000 0.0000 0.5823 0.0495

0.0000 0.0007 0.3928 0.4654

TABLE 1 shows the results of the four-way analysis of variance comparing students' strategy measure scores in Study One. Significant main effects for class (F2,118 as 12.86, p < .01), test (F1,118

170.86, p < .01), and strategy (F4A72

1.71.72, p < .01) were found. No

significant main group effect was discovered (F2,118 - 1.68, p > .10). TABLE 2 shows the

results of the three-way analysis of variance comparing students' alternative representation scores.. Again significant main effects were found fir class (F41118

6.02, p < .01) and test

(1,118 - 1.37, p < .01), but no main group effect was found (142,118

1.05, p > .10).

The main class effect indicate.) developmental differences in students' problem solving abilities. Students' mean scores for all problem solving strategies rose with increasing grade levels, indicating that older students were more adept at utilizing the strategies than younger ones. This effect, although expected, supports views linking problem solving with formal operational cwt.; The main test effect indicates sigrtificant pre- to post -test differences across strategies for all grade levels and treatment groups. Such increases argue for the success of the instruction tested. The main strategy effect indicates significant differences between mean scores on the various 10

12

stl ategy measures. Because these measures were not designed to be equivilant, however, the effect is not meaningful. The lack of a group effect indicates no significant differences between treatment groups, thus that differing practice environments within Logo had no effect on the efficacy of the instruction we designed.

TABLE 2 Study One; ANOVA Table for Alternative Representation SS

DP

MS

F

PROB

MEAN CLASS GROUP CO ERROR

3389956.9 36310.0 3170.5 19551.9 177911.7

1

3389956.9 9077.5 21589.2 2444.0 1507.7

2248.39 6.02

0.0000 0.0002 0.3518 0.1259

TEST TC TO TCG ERROR

115234.5 14196.0 1599.8 7734.4 98694.5

115234.5 3549.0 799.9 966.8 836.4

137.78

4 2 8 118 1

4 2 8 118

1.05 1.62

4.24 0.96 1.16

0.0000 0.0030 0.3873 0.3316

The four -way analysis of variance also examined eleven interaction effects. Significant strategy by class (F16,472 - 7-73, p < .01), test by strategy (F14,472 " 14.79, p < .01), and test

by strategy by class (F16,472

2.59, p < .01) effects were discovered. The test by strategy effect

indicates differences in pre- to post -test changes between the five problem solving strategies tested.

An examination of the simple test effect at each level of strategy reveals significant test effects for

all of these except backward chaining, indicating that students developed subgoals formation, forward chaining, systematic trial and error, and analogy strategies as a result of the intervention (FIGURE 3). A similar result was found for measures of alternative representation. Such results argue strongly for the success of the intervention, and support the finding of a main test effect.

***** INSERT FIGURE 3 ABOUT HERE ***** Likewise, a closer examination of the strategy by class interaction supports the finding of a main effect for class Simple class effects were found at all strategy levels except backward

chaining, indicating developmental differences in students' problems solving abilities, with older students generally scoring higher than younger ones on the various strategy measures (FIGURE 4). In addition, a finer grained analysis of the test by strategy by class interaction indicates

developmental differences in students' propensity to develop particular strategies. Progressively

weaker test effects were found at increasingly higher grade levels for subgoals formation measures, and progressively stronger test effects were found at increasingly higher grade levels for

measures of analogy. A seperate analysis also revealed progressively stronger test effects at increasingly higher grade levels for alternative representation measures. Such results hint at developmental differences in students' readiness to develop certain problem solving strategies, in particular, that younger studentswere more ready to develop subgoals formation strategies, and that

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strategy measures were not designed to be equivilant. The test effect indicates significant pre- to post-tint changes, but these could have resulted from practice or maturity and not from the various interventions.

TABLE 3

Study Two; ANOVA Table for Subgoals Formation, Forward Chaining, Backward Chaining, Systematic Trial & Error, and Analogy SS

1W

MEAN GROUP ERROR

2074999.0 33907.5 128357.9

1

TEST TG ERROR

1576.6 6871.7 25735.0

STRATEGY SG ERROR

283190.9 13478.6 132630.9

TS TSG ERROR

2274.2 743.4 60540.8

MS

P

PROB

1568.08 12.81

0.0000 0.0000

5.94 12.96

0.0166 0.0000

94397.0 2246.4 455.8

207.11 4.93

0.0000 0.0001

758.1 123.9 208.0

3.64 0.60

0.0132 0.7338

2074999.0.

2 97

16953.7 1323.3

1

1576.6 3437.3 265.2

2

97 3

6 291 3

6 291

The group effect, however, clearly indicates differences between groups resulting from the various interventions. Because the groups were statistically equivalent before receiving treatment, but significantly different overall, the group effect indicates differences in problem solving strategy skills resulting from differences in the interventions. This result is corroborated by the finding of a

test by group interaction (F2,97 - 12.96, p < .01), indicating differences in pre- to post-test. changes resulting from the differing treatments. This interaction was examined in greater detail by assessing the simple test effect at each level of group. A strong test effect was found for the group recieving problem solving instruction and Logo programming practice, and for that group alone. Students in receiving problem solving instruction and Logo programming practice improved an

average of 11.1 percentage points on the four strategy measures, while the scores of students receiving similar instruction but cut-paper manipulatives practice remained essentially the same, and the scores of students receiving Logo programming practice alone actually dedined although not !dgnificantly. FIGURE 5 illustrates twee results. The findings demonstrate that the intcrvention we designed, and that intervention alone, resulted in significant increases in students' problem solving abilities in four areas -- subgoals formation, forward chaining, systematic trial and

error, and analogy. They argue for the superior* of explicit problem solving instruction and Logo programming practice over both similar instruction with msmipulatiws practice tnd Logo programmingalone. * * * * * INSERT FIGURE 5 ABOUT HERE *****

17

The results of the analysis of variance for alternative representation measures (TABLE 4)

is problematic. Significant main effects were found for group (F2,97 - 4.13, p < .05) and test (

F2,97 - 10.55, p < .01). Because he groups were not equivalent to begin with, the group effect is not meaningful. Indeed, an examination of group means shows that all groups improved on alternative representation measures, thus, the test effect is not meaningful either. What would be meaningful would be a solid test by group interaction. Unfortunately, the analysis of variance reveals only weak significance for the interaction (F2,97 - 2.57, .05 < p < .10).

TABLE 4 Study Two; ANOVA Table for Alternative Representation SS

DF

MS

F

PROF

MEAN GROUP ERROR

19113259.6 28058.5 329431.8

1

19113259.6 14029.3 3396.2

563.35 4.13

0.0000 0.0000

TEST TO ERROR

24819.1 6631.9 125363.8

24819.1 3315.9 12922.4

10.55 2.57

0.0016 0.0821

2

97 1

2

97

The simple test effect was none-the-less assessed at each level of group to examine the test

by group interaction in greater detail. A strong test effect was found for the group receiving problem solving instruction and Logo programming practice (F1,97 = 18.91, p < .01), whereas

only a weak test effect was found for the group receiving problem solving instruction and cut-papermanipulativespractice (171,97 - 3.61, .05 < p < .10), and no test effect at all was found for the group receiving Logo programming practice alone (F1,97 - 1.81, p > .10).

FIGURE 5

illustrates the differences in pre- to post-test changes between groups. It can be seen that the group

receiving problem solving instruction and Logo programming practice improved more than twice as much as either the group receiving instruction with cut-paper manipulatives practice or the group

receiving Logo programming practice alone. Because students in the explicit instruction - Logo programming group had lower scores on alternative representation measures to begin with, however, the greater gains they made might be attributed to differential ability levels rather than the intervention. Thus, the most we can conclude is that it is possible that students in the Logo graphics group showed greater increases on measures of alternative representation as a result of the intervention we designed.

Discussion The results of Study Two argue strongly for the superiority of explicit problem solving instruction and Logo programming practice over both similar instruction with manipulatives practice and discovery learning in Logo programming environments for the development of four problem solving strategies subgoals formation, forward chaining, systematic trial and error, and analogy. Indications are that such pedagogy may be most effective for the teaching and learning of

18

alternative representation strategies as well. In terms of the research questions, then, we can conclude that explicit problem solving instruction with Logo programming practice is more supportive than Logo discovery learning environments of the development of those strategies, and that the Logo programming environment itself is more supportive of such development than concrete manipulatives. The findings thus support Papert's (1980) contention that abstract ideas can be concretely represented on computers in ways that help students bridge the gap between concrete and formal thought, but argue against his claim that such transition will take place automatically when students workwithin Logo programming environments.

STUDY THREE Because Studies One and Two utilized the same teachers and similar student populations, we had some question about the general application of the instruction we designed. Study Three was thus concerned with validating the results of the first two studies with a different teacher and a

differing student population.. Because the controls used in this study involved regular classes in formal mathematics and programming, it also investigated the differing efficacies of that instruction vs. regular instruction in domains typically prescribed for the teaching and learning of problem solving. Study Three, then, explored the following questions: 1. Does explicit instruction and mediated Logo programming practice support the teaching and learning of five particular problem solving strategies subgoals formation, forward chaining, systematic trial and error, alternative representation, and analogy among high school students?

2. Is explicit problem solving instruction with Logo programming practice superior to regular instruction in mathematics and programming for supporting the development of such strategies?

Subjects. Subjects were 40 eleventh and twelfth grade students at an American school in Switzerland enrolled in one of three classes a Logo class, an Advanced Placement (AP) Pascal dams, or a

Pre-Calculus class. No subject had any previous Logo programming experience.

Methodology. An subjects were given paper-and-pencil tests of their ability to apply the five problem solving strategies on which subjects improved in the first study. Subjects in the Logo class received explicit problem solving instruction and Logo programming practice in each strategy. Subjects in the AP Pascal and Pre-Calculus dames received regular content area instruction. Mean pre-test scores were compared between groups using one-way analysis of variance and found not to be statistically equivalent (F2,37 12.76, p < .01), hence groups could not be assumed

generally equal in problem solving ability before treatment. An examinatke of group means revealed that the Logo group scored tigulficantly lower than students in both the AP Pascal and

19

Pre-Calculus groups on pre-test measures. On completion of all treatments, subjects were post-tested using different but analogous tests. Raw scores on all tests except those for alternative representation were converted to percent correct scores and compared using a three-way analysis of

variance. Independent variables were test, strategy, and group. The dependent variables were scores on the strategies tests. Because they had no maximum possible correct, alternative representations measures were evaluated seperately using a two-way analysis of variance.

Results. Significant differences in pre- to post-test increases were found between groups. Further analysis of this finding revealed that subjects in the Logo class showed significantly improved subgoals formation, forward chaining, and systematic trial and error strategies. Increased ability in applying alternative representation strategies was also indicated but not conclusively demonstrated

for this group. The results argue for the superiority of explicit strategy training and Logo programming practice over regular instruction in subjects traditionally prescribed for the teaching and learning of problem solving and demonstrate the efficacy of the instruction we developed with a very differentstudentpopulation.

TABLE 5 Study Three; ANOVA Table for Subgoals Formation, Forward Chaining, Backward Chaining, Systematic Trial & Error, and Analogy WEAN

GROUP

FOR TEST TO ERROR

SS

AF

MS

F

PROB

1220349.2 1509.4 44494.6

1

1220349.2 754.7 1202.6

1014.79

0.63

0.0000 0.5395

1540.2 1768.2 152.9

10.07 11.56

0.0030 0.0001

29.56 4.13

0.0000 0.0009

111

9704.6 1357.4 328.3

3

494.4

154

6

180.5 139.6

1.29

0.0170 0.2663

1540.2 3536.4

5657.4

STRATEGY SG OR

29113.7 8144.3 36444.4

TS

1483.3 1083.8 15493.5

?SG ERROR

2 37 1

2 37 3

6

111

TABLE 5 shows the results of the three-way analysis of variance for subgoals formation, forward chaining, systematic trial and error, and analogy in Study Three. Significant main effects

were found for test (F2,37 is 10.07, p < .01) and strategy (F3,111

29.56, p < .01), but not for

grouP (F2,37 - 0.63, p > .10). Significant test by group (F2,37 - 11.56, p < .01), test by 3.54, P < .05), and grater/ by grouP (F6,111 - 4.13, p < .01) interactions were also found. The results of the analysis of variance of scores on alternative representation strategY (F3,111

measures is shown in TABLE 6. It reveals significant main effects for group (F1,37 - 7.69, p
.10). A possible explanation for tAe lack of

improvement on tests of analogy (especially considering that older students in the first study improved the most on these measures) is that the analogy test itself was too easy for the high school students tested in Study Three. Indeed, students in this study scored so high, on the analogy pre-test that there was little room for improvement on the post-test.

A significant simple test effect was also found for students in the Fre-Calculus group on the

systematic trial and error measure (p < .05), indicatin,g that students in this group developed systematic trial and error strategies to some extent, but significant differences were still found between this group and the Logo group on the systematic trial and error measure. No simple test 21 18

effects were found on any other strategy measures for members

of the Pre-Calculus group, or on any strategy measures for members of the AP Pascal group. We can conclude, then, that the intervention we designed was effective in increasing participating students' ,abgoals formation, forward chaining, and systematic trial and error problem solving skills, and that it was more effective in this respect than regular instruction in subject areas traditionally prescribed for the teaching and learning of problem solving. An examination of the simple test effect at each level of group for alternative representation measures also reveals a strong test effect for the Logo group (p < .01), but not for the other two groups (p > .10). Because no test by group interaction was found in the analysis of variance for this measure, however, the most we can conclude is that it is possible that students in the Logo group showed greater increases on alternative representation measures as a result of the interventio we designed.

Discussion The results of Study Three support the efficacy of the instruction we designed for the teaching and learning of subgoals formation, forward chaining, and systematic trial and error strategies among high school students, and argue for its superiority over regular instruction in subjects typically prescribed for such teaching and learning. Indications are that explicit instruction and Logo progrannting practice is also effective fat the development of alternative representation strategies among such population. The results thus strongly support Thomdike's (1907) contention that the transfer of problem solving skills from formal disciplines does not occur automatically, but rather occurs to the extent that such skills are explicitly taught and practiced .

within such formal study.

CONCLUSIONS The results of our research demonstrate the effectiveness of, a knowledgebased approach for developing four problem solving strategies subgoals formation, forward chaining, systematic trial and error, and analogy through Logo programming practice. Indications are that alternative representation strategies may also be similarly developed. They demonstrate that explicit instruction and Logo programming practice is more effective than explidt instruction with concrete manipulatives practice, Logo discovery learning, and instruction in subjects traditionally prescribed for the teaching and learning of problem solving. They thus suggest that Papert (1980) was correct in claiming that computing environments can uniquely support the development of problem solving and critical thinking skWs, but that he was incorrect in assuming that such development would take place without for al instruction. That explicit instruction is integral to the development of problem solving skills through Logo programming is shown by Studies INvo and Three. These findings are corroborated by other research (Carver, 1987; Lehrer & Randle, 1987; Thou/peon & Wang, 1988; de Corte et al, 1989;

22

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0__6

Gorman, H., Jr. & Bourne, L. a (1983) Learning to think by learning Logo: rule learning

in third-grade computer programmers. bulletin of the__Euchollook3adety. 165-167.

Greeno, J. G. and Simon, H. A. (1984) Emblem Solving and_Ring (Technical Report No. UPITT/LRDC/ONIVAPS-14). Washington, DC: Learning Research and Development Center, Office of Naval Research.

Harvey, B. (1982) Why Logo? byte. 7 (8), 92-95.

Holyoak, K. J. & Koh, K. (1987) Surface and structural similarity in analogical transfer.

Msmoci_gisiCoznitimal 332-340. Lawler, R. W. (1985) campi=ExtLeidg-sr,__Euld___Cognitive Devglzmglit A Child's

Isatninginafonzitm. New York: Halsted Press.

Lehrer, R. and Randle, L. (1987) Problem solving, metacognition and composition: the effects of interactive software for first-grade children. Journal of niFutrna Research. (4), 409-428. Lehrer, R., Sancilio, L. and Randle, L. (1989) Learning pre-proof geometry with Logo. Cognition and Instruction. 6 (2), 159-184. Loon, U. (1985) Logo today: vision and reality. The Computink Teacher. 12 (6), 26-32.

Mandinach, E. B. & Linn, M. C. (1987) Cognitive consequences of progyamming: achievements of experienced and talented plogannners. Journal of Educational CsanautizigRocarla, (1), 53-72. Mayer, R. E, Dyck, J. L. & Vilberg, W. (1986) Learning to program and learning to think. what's the camection? Communications of the ACM. 29, (7), 605-610. McLuhan, M. (1964) Understandingiledia. NY: New American Library. Miller, G, E. & Emihovich, C. (1986) The effects of mediated programming instruction on preschool children's self-monitoring. Journal of Educational Comnutinit Research. 2 (3), 283-297.

Newell, A. & Simon, H. A. (1972) liunualationASshlibig. Englewood Cliffs, NJ: Prentice-Hall.

Papert, S. (1980) Minskirrnvi. New York: Basic Books.

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Pea, R. D. & Kurland, D. M. (1984) On the cognitive effects of learning computer programming. New Ideas in Psychology. 2 (2), 137-167. Polya, G. (1973) flow To Solve It. Princeton, NJ: Princeton University Press.

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25

Soloway, E. (1986) Why Kids Should Leant To ,gram. New Haven, CT: Cognition and Programming Project, Department of Computer Science, Yale University.

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26

SUBGOALS FORMATION ®What is the problem? ()What little problems are a part of the problem? Make a tee.

OCan you solve them? If yes, solve them and use yo ir tree to reassemble the parts.

()If no, go back to Ctraki red o

and ((for

each of your smaller problems .

FIGURE 1

Wall Chart for Subgoala Formation

analogy 1.

Horizontal symmetry

Put together at least 3 shapes to create a design in the upper right quadrant of the screen, then draw the mirror image of your design in the lower right quadrant.

2.

Vertical symmetry

Put together at least 3 shapes to create a design in the upper right qui. drant of the screen, then draw the mirror image of your design in the lower left quadrant.

3. Diagonal symmetry

Put together at least 3 shapes to create a design in the upper right quadrant of the screen, then draw the mirror image of your design in the lower right quadrant.

4.

Mirrors

Put together at least 3 shapes to create a design in the upper right quadrant of the screen, then draw the mirror images of your design in the other three quadrants.

FIGURE 2

Problem Set for Analogy

28

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forward chaining

backward systematic chaining trial & error analog

alternative representation (raw score totals)

STUDY ONE: PRE- TO POST-TEST DIFFERENCES BY STRATEGY

FIGURE 3

STUDY ONE: PRE- TO POST-TEST DIFFERENCES BY GRADE LEVEL

FIGURE 4

20

STUDY TWO:

19.1

MEAN PRE- TO POST-TEST DIFFERENCES BY STRATEGY AND GROUP

18 16

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31

alternative representation (raw score differences)

STUDY THREE: PRE- TO POST-TEST DIFFERENCES BY STRATEGY AND GROUP

alternative representation (raw score differences)

FIGURE 6

3

32

APPENDIX Problem Solving Strategy Measures

33

The Sport Shop sold sneakers for one week at $30 a pair and-made $1800.00.- -The next week they reduced the price to How much money did $20.00 a pair and sold twice as many. they make altogether? 3.

that cost $1.00. Jean and Marie split a milk shake quite enough money to pay for her half. Marie didn't have How much money did Jean put in She still owes Jean $.10. for the milk shake? 4.

Subgoals formation.

Our measure of students' ability to decompose complex problems into smaller subgoals units consisted of five mathematical word problems that required decomposition for correct solution. dents were asked not only to solve the problems but to show S They were given credit for how they broke them into parts. correctly identified subgoals as well as for the correct answer, with a possible total of five points per question.

34

0 GRAY THINGS THAT ARE NEITHER. TRIAN3LES NOR CIRCLES

a

EVERYTHING THAT ISN'T BLACK

al :II

CROSSES OR STRIPED TRIANGLES

azi

...

II

EVERYTHINO THAT IS EITHER A CIRCLE OR BLACK OR STRIPED

12 ,

ii E

THINGS THAT ARE GRAY BUT NOT CROSSES

. .

Forward Chaining.

skills The test designed to measure subjects' forward chaining computer game, Rocky's was a paper-and-pencil version of the In Rocky's Boots, symbolic Books (The Learning Company, 1982). combined to produce machines "AND", "OR", and "NOT" gates are attributes and sets of attributes that respond to targeted striped circles, everything (e.g., gray triangles, corsses or Combinations of gates must be built that is not black, etc.). correct solutions. chaining manner to achieve up in a forward the required Our paper-and-pencil version had subjects draw questions which were There was a total of fifteen connections. given one point each for correct solution.

35

Backward chaining.

The test designed to measure subjects' backward chaining skills was a paper-and-pencil adaptation of the computer game, The Factory (Sunburst, In The Factory, players are shown a finished product and asked 1984). to combine various machines to produce a similar product. Thus, players must work backwards from the product to deduce a correct sequence of machines that will produce it. Our paper-and-pencil version had subjects list the required machine sequences. The test consisted of fifteen questions which were given one point each for correct solution.

SHIFTED ALPHISI3ET: A SAYING: 0 fcznitu gbcbs uohints be acgg.

-V

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-H

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-M

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+ DREAMS E NTREP

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Systematic trial and error.

Cryptography involved systematically trying and testing different symbol We chose two decoding combinations to attain coherent decoding systems. exercises to test subjects' abilities to systematically utilize trial and error strategies. The first of these was a shifted alphabetical The second involved variations on a number code problem devised code. Students were given ten points for by Newell And Simon (1971). For partial solutions on the shifted _correctly decoded problem. alphabet problem, one half point was given for each correctly identified On the number code problem, a full point was given for each letter. correctly identified letter.

37

Use the squares to create as many interesting and unusual drawings as you can.

Alternative representation The measure of subjects' ability to create alternative representation used was derived from the Torrance Test of Creative Thinking (Torrance, 1972). Students were given sets of geometric fitures (squares or circles) and asked to use these to produce as many interesting and unusual drawings as they could. The resultant drawings were scored for quantitiy, diversity, originality, and elaboration.

38

3.

4.

5.

XYZ : ZAX :: ABC : XYZ : Z.iiX XYZ :

01

6. B 7.

E:

p:

8

10.

Analogy

Subjects' skill at analogical reasoning was measured with completion exercises consisting of ten verbal and ten visual analogies. Students were given one point for each correct answer.