literature relating critical skills for problem solving in mathematics and in computer programming

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Literature Relating Critical Skills forProblem Solving in Mathematics andin Computer ProgrammingLeah P. McCoyDivision of EducationIndiana University at South BendSouth Bend, Indiana 46634

Mathematical problem solving is important because a major goal ofmathematics education is to prepare students for the future. Since the future isdifficult to foresee, students must be taught to apply mathematical skills andconcepts to novel situations. The National Council of Teachers ofMathematics (1980) has targeted problem solving as the focus of schoolmathematics of the 1980s. Similarly, the National Council of Supervisors ofMathematics (1977) stated that "learning to solve problems is the principalreason for studying mathematics" (p. 2).

Nearly every mathematics curriculum K-12 includes some goals aimed atdeveloping mathematical problem solving skills. Even with this attention,students* mathematical problem solving skills are often poor. In a review ofthe results of the second National Assessment of Educational Progress,Carpenter, Corbitt, Kepner, Lindquist, and Reys (1980) noted the widespreadfailure of students on items dealing with solving mathematical problems.There is much current speculation about the value of computer

programming as a means to develop problem solving skills. Several authorshave argued that computer programming experience improves problem solvingability (Lepper, 1985; Nickerson, 1982; Papert, 1980; Shneiderman, 1985).Much of the logical justification for the theory that computer programmingskills may transfer to other problem solving activities is based on the similarityof the two exercises.

Five skills are critical ingredients for success in problem solving inmathematics and in computer programming. These are general strategy,planning, logical thinking, variables, and debugging. The literature providesevidence of the relationship between mathematical problem solving andcomputer programming in each of these five areas.

General Strategy

A classic model of problem solving presented by Polya (1957) includes thefollowing steps: (a) understanding the problem, (b) devising a plan, (c)carrying out the plan, and (d) looking back. Even though the model seemslinear, in practice it is recursive. Current and previous steps are often repeated

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Critical Skills for Problem Solving 49

as evaluation occurs throughout the process. This model can be seen as anoverall strategy that drives the human information processing system duringproblem solving. It functions as a general framework within which problemsolving is carried out. As such, it is a general strategy for solving a problem�whether mathematics or computer programming or some other content area.A few studies have found evidence that experience in computer

programming has a positive effect on problem solving in other contexts. Theseresults suggest that general problem solving strategy transfers from oneactivity to the other. Several researchers have hypothesized that transfer ofproblem solving skills is the underlying cause for relationships that they haveobserved.

In studying the overall relationship of problem solving ability and computerprogramming performance, Nowaczyk (1984) constructed a 7-item test ofgeneral problem solving skill. His test included items dealing with logicaloperations, algebraic solutions, transformations, and identification ofmathematical relationships. He compared the score on this problem solvingmeasure with final grades in computer courses for 301 college undergraduatesubjects. He found that the problem solving score was related to students’programming performances.

Despite the fact that his problem solving test was questionable because itwas short and he presented no reliability or validity statistics, the results ofthis study offer important tentative evidence. Nowaczyk’s conclusion was thatthere is a positive relationship between problem solving ability and computerprogramming.

Using an adaptation of the Nowaczyk instrument as a measure of generalproblem solving, McCoy and Orey (1988) studied 120 middle and high schoolstudents enrolled in BASIC courses. Their results revealed that after onesemester of BASIC instruction, general problem solving scores hadsignificantly increased. This study did not include a control group forcomparison, and thus the cause of the increased problem solving scores issuspect; however, the results indicate a possible effect.Reed and Palumbo (1988) studied 23 college students enrolled in a 7-week

BASIC course. Following the BASIC instruction, they found that thesestudents had statistically significant gains in problem solving, as measured byan instrument consisting of sections from the Ross Test of Higher CognitiveProcesses and the Watson-Glaser Critical Thinking Appraisal. These resultswere replicated in a similar study using 21 students in BASIC and Logocourses (Reed, Palumbo, & Stolar, 1988). Both of these studies used the sameproblem solving measure, for which no reliability or validity data arepresented. Neither study included a control group, and both used intact classeswhich were quite small; however, this is preliminary evidence of a relationshipbetween computer programming and problem solving.Using longitudinal data of mathematically talented college students,

Johnson and Harding (1979) found that those students who completed elective

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50 Critical Skills for Problem Solving

computer courses scored higher on a departmental mathematics examinationthan those who did not elect the courses. This study involved mathematicsmajors at Cambridge University who were required to pass a yearly Triposexamination at the end of each year of their 3-year course of study. Thisexamination is an extensive and rigorous measure of mathematical problemsolving ability. Students were allowed to choose the computer course as one oftheir electives. To control for this self-selection, the researchers used thefirst-year Tripos score (before computer courses were available) to adjust theposttest score. Regression analysis was then used. The researchers concludedthat the computer course had a significant positive effect on the mathematicalproblem solving ability of these mathematically talented students.

In a study of the processes of problem solving, Foster (1972) examined theeffect of instruction in use of the computer and flow charts for solvingproblems on problem solving performance. His subjects were 68 eighth gradealgebra students who were assigned randomly to one of four instructionalgroups: flow charts, programming in BASIC, both flow charts andprogramming, or neither flow charts nor programming. At the end of the12-week treatment, all subjects were tested on the Problem Solving AbilitiesTest, a 24-item pencil and paper test that measures selected problem solvingprocesses.

Foster (1972) reported a difference between the computer group and theneither computer nor flow chart group. He found that the computer grouphad significantly higher total scores on the posttest (Problem Solving AbilitiesTest). He found no significant differences in performance involving the othertwo groups, flow charts or both computer and flow charts. His conclusionw^as that programming the computer positively influences development of theselected problem solving behaviors measured by his test. Again, the measurewas researcher-constructed and no reliability or validity data were presented.

Wells (1981) also studied the processes involved in the activities of computerprogramming and mathematical problem solving. Her subjects were 15 highschool students. She had them talk aloud as they solved 12 mathematicsproblems and 5 programming problems. She then analyzed the protocols andsought to identify and compare the processes. She found that the sameprocesses were used for both activities. She also found a significant correlation(r = 0.77) between mathematical problem solving success and computerprogramming problem solving success.

Clements and Battista (1987) studied the problem solving processes of 12fourth grade students. The students were matched on the basis of mathematicsachievement and problem solving ability. They were then assigned to 20-weektreatments of either Logo or word processing. The Logo group*s problemsolving behavior improved more. The problem solving measure consisted offive items designed by the researchers to assess executive-level problem solvingprocesses.

Thus, four studies generally relate problem solving in computer



programming and in mathematics. All of these studies found evidence thatthere is a relationship, and their conclusions were that the general strategyand/or processes involved in the two activities are similar. Additionally, fourother studies reported results indicating an increase in problem solving successas a result of computer programming experience, indicating possible positivetransfer. Therefore, it appears that the general strategies for problem solvingin mathematics and in computer programming are related.

Planning

The next critical area for problem solving in computer programming and inmathematics is planning. The planning process has been studied in a generalsense and also from the view of using particular programming skills in othertypes of problem solving.White and Collins (1983) studied the effect of programming experience on

planning. Their subjects were students age 10 to 14 at a summer computercamp. Each subject was asked to write a plan for a familiar activity (making apeanut butter and jelly sandwich or brushing one’s teeth) both before andafter programming instruction. The plans were evaluated for completeness.The treatment for the subjects was eight hours of computer instruction whichemphasized planning. Postmeasure scores were significantly greater thanpremeasure scores. The researchers concluded that planning, as a componentskill of problem solving, can be taught in a computer programming context,and that this skill transfers to general problem solving.The White and Collins (1983) study had several weaknesses. The treatment

of eight hours was hardly long enough to expect effects. The two planningtasks were so general that they may have little application within aneducational setting, and the results of the pretest were discussed with thestudents prior to the postmeasure. It is likely that this direction had a largeinfluence on the posttest scores. Even with these problems, the results of thisstudy suggest that computer programming experiences may improve planningskills. Since no replication is available, these results are considered weak, butpositive, evidence of the effect of computer programming on planning.Another study of the effect of computer programming experience on

planning skill was conducted by Pea and Kurland (1984). Their subjects werechildren, age 8 to 12, some of whom had completed one year of Logoprogramming and some with no programming experience. The two groups ofchildren were compared on a planning task, which required them to design aplan for cleaning a classroom. The results showed that there was no differencein planning skill between the programming and nonprogramming groups.Pea and Kurland (1984) suggested two reasons for their nonsignificant

results. First, they state that their task may not have been an appropriatemeasure of planning skill. The second area of concern, which they consideredthe probable cause of the lack of planning skill transfer, was the discoverylearning pedagogy associated with the Logo instruction given to their



treatment group. They seriously questioned this teaching method andsuggested that students need "instructional guidance" to develop "advancedthinking strategies" (p. 44).A year-long study compared high school students who had no programming

experience, some programming experience, and extensive programmingexperience (Kurland, Pea, Clement, & Mawby, 1986). The programmingcourses consisted of one year of BASIC and of short units of variousprogramming languages: BASIC, COBOL, Logo, FORTRAN, MACRO, andPascal. The planning measure consisted of a scheduling task similar to thatused by Pea and Kurland (1984). The task was modified to feature a robot ina computerized environment. Results indicated that there was no difference inplanning ability of the different levels of programming experience.

Project ACCCEL (Dalbey & Linn, 1985; Linn, 1985; Linn & Dalbey, 1985)was a series of studies of the programming behavior of precollege students.This research was based on a theory of a "Chain of CognitiveAccomplishments" to describe the achievement of programming students. Thischain consists of three levels: syntactic knowledge of language features, designskills, and problem solving skills. According to the theory, in order forstudents to be able to transfer the design or problem solving skills, they mustmove to that point along the chain.One of the ACCCEL studies involved using Spider World, a sublanguage

that emphasizes design. Three 12-week middle school programming classeswere instructed as follows: (a) using Spider World for three weeks and BASICfor nine weeks, (b) using Type Attack (a typing instruction software package)for three weeks and BASIC for nine weeks, and (c) using BASIC for theentire twelve weeks. After the treatment, the Spider World group had betterdesign skills (Dalbey & Linn, 1986).One problem with this study is the appropriateness of the instrument used

to measure design skills. It contained items which were similar to the SpiderWorld activities. Thus, the higher scores of the Spider World group mayreflect their familiarity with the format and the material rather than betterdesign skills.Widespread use of the sublanguage may or may not be feasible; however, it

should be noted that there was instruction particularly directed toward design,and it was shown to be effective. It has not been established whether mostsecondary programming courses emphasize design and problem solving, orwhether they concentrate on language features. The ACCCEL studies foundwide variation in the nature of the courses in the schools they observed(Dalbey & Linn, 1985).Another of the ACCCEL studies was also concerned with planning.

Instruction was given to students in the use of a structure diagram to facilitateplanning. The researchers observed that those students who had been taughtto use a structure diagram were better at planning (Dalbey, Tourniaire, &Linn, 1986).



Blume and Shoen (1988) studied 54 eighth grade students, half of whomhad completed one semester of BASIC instruction. They had the studentsthink aloud as they solved five problems. Their results indicated no differencein the planning processes of the programmers and nonprogrammers.

Thus, there is conflicting evidence of the possible effect of computerprogramming experience on planning and design skills. One study, eventhough it had methodological flaws, provided at least weak support for thetheory that planning activities associated with programming may improvegeneral planning (White & Collins, 1983). Nonstructured discovery lessons inLogo (Pea & Kurland, 1984), a 1-semester BASIC course (Blume & Shoen,1988), and varied experience with programming (Kurland et al., 1986) wereshown not to improve planning skills. However, other studies have found thatinstruction in planning and design has a positive effect on planning skills, witha planning language (Dalbey & Linn, 1986) or with the use of structure charts(Dalbey et al., 1986). The key to transfer of planning skills seems to be theinstructional approach. When planning skills were explicitly taught in aprogramming context, there was positive transfer to mathematics and generalproblem solving. This transfer did not occur when planning was not givenparticular attention.

Logical Thinking

The third critical area for problem solving is logical thinking. Even thoughlogical thinking is important and desirable, it is one area which is difficult todefine and measure. Five studies attempted to assess the influence of computerprogramming on logical thinking.One study examined changes in logical thinking as a result of Logo

programming activities (German & Bourne, 1983). For one school year, thirdgrade students were given Logo instruction and then allowed to experiment atthe computer either one-half or one hour per week. On a postmeasureconsisting of slides of combinations of color, shape, size, and number,students were to determine the rule for inclusion based on a conditional (If. . . Then . . . ) relationship. Results showed that the one hour group madesignificantly fewer errors than the one-half hour group. This is presented asevidence of improvement in logical thinking.

Similarly, one of the items on the general problem solving test developed byNowaczyk (1984) also dealt with the conditional rule. In this case the rule was"If a letter is sealed, Then it must have a 20 cent stamp on it" (p. 154). Theresults revealed that the students who got this item correct had significantlyhigher programming grades than those who missed it.Kurland et al. (1986) also examined logical thinking ability at their three

levels of programming experience in high school students. They constructedboth verbal and nonverbal tasks to measure reasoning associated with the "If

. . Then ..." structure commonly used in programming. They found nosignificant differences among the no programming, some programming, and



extensive programming groups.Shaw (1986) found no differences in reasoning skills between programmers

and nonprogrammers. Her subjects were 132 fifth grade students, who weregiven the New Jersey Test of Reasoning Skills. The content of this test seemsvalid and reliability ranges are acceptable (.84 to .91). Using ANCOVA toadjust for pretest scores, she found no significant differences following a7-week programming treatment.Another study, also using the New Jersey Test of Reasoning Skills, found

that Logo instruction had a positive effect on reasoning skills (Many,Lockard, Abrams, & Friker, 1988). Seventh and eighth grade students (n =

171) were randomly assigned to either a 9-week Logo treatment or to a controlgroup who did not have Logo. The posttest revealed that the Logo groupscored significantly higher on the reasoning measure.The results of these five studies of the relationship between computer

programming experience and logical reasoning present conflicting evidence.Some studies found a positive effect of programming experience on logicalreasoning (German & Bourne, 1983; Many et al., 1988; Nowaczyk, 1984;),while others found no evidence of a relationship (Kurland et al., 1986; Shaw,1986). It appears that the major problem in this area is the definition andmeasurement of logical thinking skills. The studies cited used various formaland informal measures to quantify logical thinking. This variation mayaccount for the difference in results.

Variables

The critical skill area with the most empirical evidence of a transfer of skillsfrom computer programming to mathematical problem solving is variables.This is due, at least in part, to the fact that programming languages usealgebraic variables in a manner similar to their mathematical use, but givethem an operational context. Given the fact that many mathematics studentshave difficulty with algebraic variables, it may be that this area is the mostimportant short-term effect of computer programming instruction.

In an informal study. Hart (1982) observed that children who hadcompleted computer programming courses performed better on algebra tests.His explanation was that programming helps students to see "letters as labelsof stores whose contents can vary" (p. 52).

Soloway, Lochhead, and Clement (1982) studied the relationship ofcomputer programming and the use of variables. Previous studies had foundthat college engineering students had difficulty in correctly representingrelationships in algebraic equations (Clement, 1982; Clement, Lochhead, &Monk, 1981). Soloway et al. (1982) designed two experiments which sought tofind out whether students would have similar difficulties with computerprograms and with algebra equations in this area. In the first experiment, twogroups of college computer students were given the same problem but wereasked to express it as either an equation or a program. Significantly more

School Science and MathematicsVolume 90 (I) January 1990


subjects in the programming condition were able to express it correctly. In thesecond experiment, two groups of college subjects were asked to "write asentence in English" to express the relationship given in both an equation anda program. The number of students who got the equation correct and theprogram incorrect was significantly less than the number of students who gotthe program correct and the equation incorrect. The conclusion was thatcomputer programs facilitate the understanding of algebraic variables.Oprea (1985) studied the effects of programming instruction on

generalization and understanding of variables. Her sample w^as three intactgroups of sixth graders who received six weeks of programming instruction.The groups were (a) wholistic (taught at the whole program level formathematically relevant problems), (b) elemental (taught the individualcommands necessary to write a complete program), and (c) control (noprogramming instruction). She developed three instruments for measuringprogramming ability, generalization, and understanding of variables. Usinganalysis of covariance to control for pretest scores, she found that both of theprogramming groups scored significantly higher on all three measures than thecontrol group. She concluded that programming enhances generalization andunderstanding of variables. There was no effect for the different instructionalmethods.McCoy and Burton (1988) studied 21 students, ages 10 to 17, at a summer

computer camp. At the end of a 2-week intensive program of BASICinstruction, they found that their subjects showed a significant improvementin ability to use mathematical variables and mathematical problem solving.These gains were measured pre and post by subtests of the Algebra ReadinessTest. Again there was no control group, but the results are an indication of apossible effect.Mayer (1975, 1976, 1979, 1985) explained computer operation in terms of

transactions. A transaction is an event in the computer that involves someoperation on some object at some location. He proposed that this concreterepresentation would make programmers better able to understand and usealgebraic variables. He verified this idea in a subsequent study (Mayer, Dyck,& Vilberg, 1986), in which experienced programmers were found to have abetter understanding of and facility with mathematical variables. He studiedcollege students in a beginning BASIC course. When he compared them with acontrol group, he found that the BASIC students gained significantly more inword problem translation, word problem solution, and procedure compre-hension. He concluded that all of these specific skills are components ofgeneral problem solving, and therefore, BASIC instruction improves problemsolving.Kurland et al. (1986), in their study of high school students with varying

levels of programming experience, found no difference in ability to usevariables. That is, the students who had extensive programming experiencewere no different in ability to use variables from students who had some or no



programming experience. Blume and Shoen (1988) also report no differencebetween junior high school programmers and nonprogrammers in variablesand equations.McCoy (1987) studied 46 junior high and high school students at a

computer camp. She found a significant relationship among ability to usevariables, amount of computer programming experience, and mathematicsexperience. There was a stronger relationship between variables and computerprogramming experience than between variables and mathematics experience.The above studies have investigated whether computer programming

improves understanding and skill in algebraic variables. Even though twostudies found no improvement in understanding of variables (Blume & Shoen,1988; Kurland, et al., 1986), several other studies have found a positive effect.In studies of young children (Hart, 1982), secondary school students (McCoy,1987, McCoy & Burton. 1988; Oprea, 1985), and college students (Mayer et

al., 1986; Soloway et al., 1982), experience in computer programming wasfound to have a positive effect on the knowledge of algebraic variables.

Debugging

The term debugging is peculiar to computer programming, but the conceptis not. In any kind of problem solving, the solution is checked and, if foundnot to be acceptable, is reevaluated. This is the fourth step in Polya’s (1957)Problem Solving Model, looking back.Even though debugging is an essential component in problem solving, it is

not always considered as a separate skill area; it is more usually thought of asa component of other skills. In a study mentioned earlier (Soloway et al.,1982), one conclusion was that the better understanding of variables forprogramming students was partially due to experience in debugging. Whilestudents do not run equations in mathematics, they can and should performactual number testing. This debugging activity would be useful inmathematics, just as it is in programming. A few studies have considereddebugging as a distinct skill that may be developed and improved.

Clements and Gullo (1984) concluded that programming enhances thinkingskills. They studied 18 six-year-olds. Half the children had Logo programmingfor 12 weeks, while the other half had 12 weeks of Computer AssistedInstruction (CAI). The results showed that the Logo group scored higher onmeasures of reflectivity, divergent thinking, and metacognitive ability. In thisstudy, reflectivity is a more general label used for debugging. The researchers’explanation of these results was that programming activity makes the thinkingprocess conscious, and so the programming children were better thinkers.

In a part of the ACCCEL Project, programming students were observed tobe superior to nonprogrammers in debugging (Dalbey & Linn, 1985). Theresearchers in this study concluded that debugging is ^a generalized skillwhich can be applied outside the domain of programming" (p. 254). Theyfurther stated that the computer programming environment is an ideal place to


Critical Skills/or Problem Solving 57

learn debugging because of the immediate feedback.Kurland et al. (1986) examined the debugging behavior of high school

students with varying levels of programming expertise. They studied talk aloudprotocols and concluded there was no difference in students with extensive,some, or no programming experience in their debugging behavior.

Conversely, Blume and Shoen (1988) found that their programming group(one semester of BASIC versus no programming) used significantly morelooking back (or debugging) processes. They cited this difference as evidencethat programming develops skills that transfer to other problem solving tasks.Thus, the evidence seems to verify the positive effect of computer

programming experience on debugging skills. Although one study reported noeffect (Kurland et al., 1986), several others have found positive effects (Blume& Shoen, 1988; Clements & Gullo, 1984; Dalbey & Linn, 1985; Soloway et al.,1982).

Summary

The results of studies relating problem solving in computer programmingand in mathematics are generally consistent. The five areas identified ascritical for problem solving (general strategy, planning, logical thinking,variables, and debugging) seem similar in both contexts. The research in thisarea has considerable variation in research methods, in subjects, in treatment,and in the particular aspect of problem solving studied. In each of the fiveareas, studies have found indications that computer programming may have apositive effect on mathematical problem solving.

In many cases, however, these results have not been overwhelming. Becauseof various methodological shortcomings and lack of replication, the evidencemust be considered tentative, and there have been negative results; somestudies have found a lack of positive effects. Therefore, the conclusion of thisreview is that much more research is needed in this area.

Further research should concentrate on a number of pertinent questions:Exactly what is a problem solving skill and how can we effectively measure it?What is the effect of programming instruction when participants are comparedto relevant control groups? Do experience and expertise in programmingproduce the same effects? What are the long-term effects of programminginstruction? What is the interaction of programming treatment with differentdevelopmental levels? Is it possible to design programming instruction tofacilitate transfer of problem solving skills?

References

Blume, G. W., & Shoen, H. L. (1988). Mathematical problem-solving perfor-mance of eighth-grade programmers and nonprogrammers. Journal forResearch in Mathematics Education, 7P(2), 142-156.

Carpenter, T. P., Corbitt, M. K., Kepner, H. S., Lindquist, M. M., & Reys,R. E. (1980). NAEP note: Problem solving. Mathematics Teacher, 73(9),427-433.



Clement, J. (1982). Algebra word problem solutions: Thought processesunderlying a common misconception. Journal for Research in MathematicsEducation, 13(\), 16-30.

Clement, J., Lochhead, J., & Monk, G. S. (1981). Translation difficulties inlearning mathematics. American Mathematical Monthly, 88, 286-290.

Clements, D. H., & Battista, M. T. (1987, April). The effects of Logo onmathematical conceptualizations and problem-solving abilities. Paperpresented at the annual meeting of the American Educational ResearchAssociation, Washington, DC.

Clements, D. H., & Gullo, D. F. (1984). Effects of computer programming onyoung children’s cognition. Journal of Educational Psychology 76(6),1051-1058.

Dalbey, J., & Linn, M. C. (1985). The demands and requirements ofcomputer programming: A literature review. Journal of EducationalComputing Research, 7(3), 253-274.

Dalbey, J., & Linn, M. C. (1986). Cognitive consequences of programming:Augmentations to BASIC instruction. Journal of Educational ComputingResearch, 2(1), 75-93.

Dalbey, J., Tourniaire, F., & Linn, M. C. (1986). Making programminginstruction cognitively demanding: An intervention study. Journal ofResearch in Science Teaching, 23(5), 427-436.

Foster, T. E. (1972). The effects of computer program experiences on studentproblem behaviors in eighth grade mathematics. Doctoral dissertation,University of Wisconsin, Madison.

German, H., & Bourne, L. E. (1983). Learning to think by learning LOGO:Rule learning in third-grade computer programmers. Bulletin of thePsychonomic Society, 21(3), 165-167.

Hart, M. (1982). Using computers to understand mathematics. MathematicsTeaching, 52-54.

Johnson, D. C., & Harding, R. D. (1979). University level computing andmathematical problem-solving ability. Journal for Research in MathematicsEducation, 10, 37-55.

Kurland, D. M., Pea, R. D., Clement. C., & Mawby, R. (1986). A study ofthe development of programming ability and thinking skills in high schoolstudents. Journal of Educational Computing Research, 2(4), 429-458.

Lepper, M. R. (1985). Microcomputers in education: Motivational and socialissues. American Psychologist, 40(1), 1-18.

Linn, M. C. (1985). The cognitive consequences of programming instructionin classrooms. Educational Researcher, 14(5), 14-29.

Linn, M. C., & Dalbey, J. (1985). Cognitive consequences of programminginstruction: Instruction, access, and ability. Educational Psychologist, 20(4),191-206.



Many, W. A., Lockard J., Abrams, P. D., & Friker, W. (1988). The effect oflearning to program in Logo on reasoning skills of junior high students.Journal of Educational Co/reputing Research, 4(2), 203-213.

Mayer, R. E. (1975). Different problem-solving competencies established withand without meaningful models. Journal of Educational Psychology, 67(6),725-734.

Mayer, R. E. (1976). Some conditions of meaningful learning for computerprogramming: Advance organizers and subject control of frame order.Journal of Educational Psychology, 68(2), 143-150.

Mayer, R. E. (1979). A psychology of learning BASIC. Communications oftheACM.22, 589-593.

Mayer, R. E. (1985). Learning in complex domains: A cognitive analysis ofcomputer programming. The Psychology of Learning and Motivation, 1989-130.

Mayer, R. E., Dyck, J. L., & Vilberg, W. (1986). Learning to program andlearning to think: What’s the connection? Communications of the ACM,29(7), 605-610.

McCoy, L. P. (1988, April). General Variable Skill, Computer Programmingand Mathematics. Paper presented at the annual meeting of theInternational Association for Computing in Education, New Orleans, LA.

McCoy, L. P., & Burton, J. K. (1988). Correlates of computer programmingin children. Computers in the Schools, 4 (3/4), 159-166.

McCoy, L. P., & Orey, M. A. (1988). Computer programming and generalproblem solving by secondary students. Computers in the Schools, 4 (3/4),151-157.

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National Council of Teachers of Mathematics. (1980). An agenda for action.

Reston. VA: Author.Nickerson, R. S. (1982). Computer programming as a vehicle for teaching

thinking skills. Thinking: The Journal of Philosophy for Children, 4, 42-48.Nowaczyk, R. H. (1984). The relationship of problem-solving and course

performance among novice programmers. International Journal of Man-Machine Studies, 21, 149-160.

Oprea, J. M. (1985). Computer programming and mathematical thinking.Proceedings of the seventh annual meeting psychology of mathematicseducation�North American chapter, 212-217.

Papert, S. (1980). Mindstorms: Children, computers, and powerful ideas. NewYork: Basic Books.

Pea, R. D., & Kurland, D. M. (1984). Logo programming and thedevelopment of planning skills (Tech. Rep. No. 16). New York: Bank StreetCollege of Education, Center for Children and Technology.

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60 Critical Skills/or Problem Solving

Reed, W. M., & Palumbo, D. B. (1988). The effect of the BASICprogramming language on problem solving skills and computer anxiety.Computers in the Schools, 4 (3/4), 91-104.

Reed, W. M., Palumbo, D. B., & Stolar, A. (1988). A comparison of theeffects of BASIC and Logo instruction on problem solving skills.Computers in the Schools, 4 (3/4), 105-118.

Shaw, D. G. (1986). Effects of learning to program a computer in BASIC orLogo on problem-solving abilities. AEDS Journal, 19, 176-198.

Shneiderman, B. (1985). When children learn programming: Antecedents,concepts and outcomes. The Computing Teacher, 12(5), 14-17.

Soloway, E., Lochhead, J., & Clement, J. (1982). Does computerprogramming enhance problem solving ability? Some positive evidence onalgebra word problems. In R. J. Seidel, R. E. Anderson, and B. Hunter(Eds.), Computer literacy: Issues and directions for 1985. New York:Academic Press.

Wells, G. W. (1981). The relationship between the processes involved inproblem solving and the processes involved in computer programming.Doctoral dissertation, University of Cincinnati, Ohio.

White, K. B., & Collins, R. W., (1983). An experimental investigationutilizing the computer as a tool for stimulating reasoning skills. AEDSJournal, 16, 234-243.


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