Computers tn Human Behavior, Vol. 2, pp. 183-193, 1986 0747-5632/86 $3.00 + .00 Printed in the U.S.A. All rights reserved. Copyright © 1987 Pergamon Journals Inc.

The Effects of Logo and CAI Problem-Solving Environments on

Problem-Solving Abilities and Mathematics Achievement

Michael T. Battista and Douglas H. Clements

Kent State University

Abstract -- The purpose of the present study was to investigate the effects of Logo programming and CAI problem-solving software on problem solving that is dependent on specialized conceptual and procedural knowledge, problem solving that is dependent on specific executive-level cognitive skills, and mathematics achievement. No szgnificant differences were found on mathematics achievement or knowledge- dependent problem solving. However, significant differences were found on the test of executive-level pmblem solving, with the Logo group improving more than the CAI and control groups.

It has been claimed that certain compute r p r o g r a m m i n g tasks can enhance stu- dents ' unde r s t and ing of mathemat ics concepts and develop their mathemat ica l problem-solv ing skills (Hatf ie ld , 1979; Paper t , 1980). A natural choice for a pro- g ramming language to use in such activities is Logo. With Logo, children can cre- ate geometr ic f igures by typing sets of instruct ions into the compu te r that direct the m o v e m e n t of a cybernet ic " tur t le" on the compute r screen. T h e hypothesis is that in Logo p rog ramming , students learn mathemat ics by utilizing concepts that aid them in understanding and directing the turtle's movements (Feurzeig & Lukas, 1972; Papert , 1980), and that they develop problem-solving skills because they are learning "to be mathemat ic ians" ra ther than learning about mathemat ics (Papert , 1972).

Indeed, once students decide on or are assigned a figure to draw with Logo, they must devise a set of instruct ions that will make the turtle draw the figure. T h e y m a y analyze the f igure and break it into smaller parts that are more easily constructed. "The student may be called upon to build up a procedure, test it, find the errors or inadequacies, correct or improve it, test it again, and possibly refine or extend it to a more general p rocedure" (Hatf ie ld , 1979, p. 53). T h a t is, cer- tain Logo environments emphasize decomposing problems, reflecting on one's own

The authors would like to express their appreciation to the administrators and participating teach- ers and students of the Hudson and Kent City Schools.

Requests tbr reprints should be sent to Michael T. Battista, 404 White Hall, Kent State Univer- sity, Kent, OH 44242.

183

184 Batlisla and Clemenls

thinking, formalizing conceptual models, isolating and correcting conceptual errors, verbalizing goals and strategies before making overt moves toward a problem solu- tion, creating efficient problem representations, making executive decisions, and "debugging" algorithms--all essential components of problem solving (Frederik- sen, 1984). In 'fact, several studies have indicated that computer programming can increase certain problem-solving abilities (e.g., Milner, 1973; Soloway, Lockhead, & Clement, 1982; Statz, 1974).

However, other studies have failed to support the use of Logo to improve stu- dents' ability to solve standard mathematics word problems (e.g., Blume, 1984; Clements, 1985; Pea & Kurland, 1984; Seidman, 1981). It may be hypothesized that the testing instruments utilized in the latter studies were inappropriate for detecting the beneficial effects of Logo programming. First, these instruments may not have included problems whose solution depends on conceptual and procedural knowledge found in Logo programming (e.g., recursive and spatial problems). Sec- ond, the instruments may have failed to assess higher-level problem-solving skills that may be developed during Logo programming. Papert (1980), for instance, argues that Logo provides students with a vehicle and a language for "thinking about thinking," an activity that can promote the development of certain executive- level (metacognitive) problem-solving processes. There is evidence that experience with Logo can significantly increase students' executive-level skills (Clements, 1986; Clements and Gullo, 1984). Thus, there is a need to expand the range of prob- lems and processes in the study of Logo's effects on problem solving.

To see why Logo should promote the learning of mathematical concepts, con- sider what happens when students are asked to construct a sequence of commands (a procedure) to draw a rectangle. Students make their concept of rectangle more explicit by visually analyzing the component parts of the rectangle then specify- ing commands to construct these component parts. Papert claims that, " . . . the computer allows, or obliges, the child to externalize intuitive expectations. When the intuition is translated into a program it becomes more obtrusive and more ac- cessible to reflection" (1980, p. 145). This program can then be used as material "for the work of remodeling intuitive knowledge" (p. 145). That is, when students design a rectangle procedure, they are constructing a formal specification of a rec- tangle and, in doing so, they are externalizing their intuitive ideas about rectan- gles. Running the procedure allows them to test the validity of their definition, and thus to reflect on their conceptualization. Such experiences encourage students to build conceptual structures or schemas that include a more tbrmalized (abstract) notion of rectangle. These schemas can form the foundation for more sophisticated levels of mathematical knowledge (e.g., van Hiele levels; Burger & Shaughnessy, 1986). However, the findings of research studies investigating the eftects of Logo on mathematics achievement have generally not supported this hypothesis (Cle- ments, 1985; Pea & Kurland, 1984). The existence of a few positive results in this area (e.g., Findlayson, 1984) suggests that many of the nonsignificant findings may have resulted from an instructional emphasis on Logo as a programming language rather than an environment for learning mathematics.

The purpose of the present study was to investigate the effects of Logo program- ming on: (a) problem solving dependent on specialized conceptual and procedural knowledge, (b) problem solving dependent on specific executive-level skills, and (c) standard mathematics achievement. Three groups of students were studied. The experimental group received computer programming experiences in the turtle

Effects of logo 185

graphics portion of Logo. To control for the effects of development and learn- ing during the instructional period, a second group received minimal computer experience that was not related to mathematics. To control for the effects of prob- lem-solving instruction within computer environments, a third group received expe- riences with computer-assisted instructional (CAI) materials designed to promote the development of problem-solving ability.

METHOD

Subjects

Subjects were volunteers from two midwestern, middle-class school systems; 18 fourth graders from one school and 48 sixth graders from the other. (At the end of the study, 17 fourth graders and 39 sixth graders remained. Two of the origi- nal students moved to other schools; the others chose not to continue, stating that they could no longer afford to spend time in the treatment.) Within each school, students were randomly assigned to one of three groups: Logo programming, CAI problem solving, or control (computer literacy).

Procedure

Students were tested to determine pretreatment levels of problem-solving ability and mathematics achievement. At each school/grade level, the two computer treat- ments were then implemented in two 40 minute sessions per week (CAI on Mon- days and Wednesdays, Logo on Tuesdays and Thursdays) for a total of 42 sessions during the school year. The control group received a five-week introduction to com- puter literacy toward the end of the school year, immediately before the posttests. In each treatment session, students worked under the guidance of an experienced teacher familiar with the use of computers in instruction. This teacher taught all three groups of students. At the end of the 42 sessions, students were posttested over the same abilities as were assessed in the pretests.

Treatments

The goal of both experimental treatments was to increase mathematical problem- solving ability. Whole-class introduction was used by the teacher to introduce new topics. The remainder of the time, students worked in pairs at the computers while the teacher circulated about the room trying to help them discover answers to prob- lems posed. The following will describe first, the specific Logo and CAI activi- ties performed by the students, and second, the problem-solving structure and content common to both treatments.

The Logo activities were limited to turtle graphics. The focus of these activi- ties was on Logo as an environment for geometric problem solving. The concept of procedurality was introduced during the first sessions. After a brief period of exploration, students were taught how to define procedures for making shapes and how to combine procedures into superprocedures. "Accidents" or bugs (often the result of the lack of state transparency in students' procedures) led to several weeks of explorations of repeating and rotating shapes to construct different patterns. Next, students were engaged in studying the construction of regular polygons and star polygons. Specific projects were proposed as challenges and were attempted

186 Battista and Clements

by each pair of students. Finally, recursion was introduced and the "polyspiral microworld" explored briefly. After students finished work on a particular proj- ect or challenge, they were encouraged to pose and undertake problems of their own invention.

CAI problem-solving programs were selected on the basis of their potential to support students' use of the heuristics taught. Several programs from the problem- solving matrix developed by Sunburst, Inc. formed the basis of the CAI treatment, including The King's Rule, The Pond, The Factory, Gnee or Not Gnee, Puzzle Tanks, Gear.~, and Fun House Maze. Additional problem-solving programs used were Rocky's Boots and Gertrude's Puzzles from The Learning Company, Thinking With Ink and Lem- onade from the Minnesota Educational Computing Consortium, Comp-u-solve from Educational Activities, and Dzfj'~v from QED.

For the first two months, CAI students chose the program with which they wished to work. Each program's operation and goal had already been discussed by the teacher. During the last four months, specific challenges were developed for several of the programs. For example, challenges included progressing through all six levels of the King's Rule, making a product drawn on an index card with the fewest operations in The Factory, and accumulating the most money in ten days using Lemonade. Alter completing work on these challenges, students were again free to choose the program with which to work.

The teacher emphasized the same set of problem-solving heuristics in each treat- ment. The heuristics included making a table, looking for patterns, working back- wards, trying to solve part of the problem, drawing a diagram, and "keeping active" (i.e., do something--it may lead to a fruitt'ul idea). The teacher attempted to illus- trate, discuss, and provide practice with these heuristics in the context of either Logo programming or CAI problem-solving programs. Use of each heuristic was first discussed, then demonstrated, in the context of each treatment's computer environment. The teacher circulated among the pairs of students and employed guided questioning strategies to help them apply the heuristics to the given problem.

For example, the teacher used the construction of" regular polygons and regu- lar star polygons (Logo) and the programs King's Rule and Lemonade (CAI) to develop the heuristics of making a table and looking tbr patterns. The Logo grout) was provided a procedure which took two inputs as follows:

TO POLY :NUMBER.OF.SIDES :TURN

REPEAT :NUMBER.OF.SIDES [FD 50 R I G H T :TURN]

END

The goal of the students was to use this procedure to discover the relationships between :NUMBER.OF.SIDES and :TURN necessary for POLY to draw reg- ular polygons and regular star polygons. Students were guided to use a table to record the values of their exploratory inputs and resultant drawings. They looked for patterns in this table to induce the desired relationships. The CAI group used King's Rule, which presents a sequence of three numbers satisfying a rule unknown to the student (e.g., 12, 8, 46 might be presented as an example for the rule, "All numbers are even"). To discover the rule, students entered triples of numbers and received feedback from the computer regarding their consonance with the rule.

Effects of logo 187

Students were guided to use a table of values for their exploratory inputs and resul- tant feedback. They then looked for patterns in this table to induce the desired rule.

Al though no tests of knowledge of Logo p r o g r a m m i n g or of the ability to attain the goal for a C A ! p rog ram were given, observat ion indicated that all the Logo students were able to write and edit Logo procedures to draw graphic pictures, and all the C AI students were able to operate and solve at least the lowest levels of problems in the CAI programs.

Instruments

Two problem-solving tests were constructed by the authors, both consisting of five two-item subtests. Items were scored as correct (one point) or incorrect (zero points) unless otherwise noted. T h e ten items on each test were su m m ed to yield total scores.

Problem-solv ing Test 1 included problems whose solution depends on concep- tual and procedura l knowledge found in Logo p rog ramming . T h e first subtest involved combinat ions and permuta t ions . For example, "John has three blocks one red, one blue, and one green. In how m a n y different ways can he stack the three blocks one on top of the other ?" It has been claimed that the systematic think- ing requi red by such an i tem is similar to the thinking required in the design of nested Logo procedures (Papert , 1980). The second subtest assessed logical reason- ing, an essential componen t of condit ional s tatements and p rog ram flow in com- pu te r p rog ramming . For example:

P re t end you know the following two statements are true:

• If it feels warm outside, then I am dreaming . • I f I am dreaming , then ei ther I am tired or I am asleep.

Now, for the questions below, circle Y if you think the answer to the quest ion is yes, and N if you think the answer to the quest ion is no.

Y N If I am awake and I am not tired, could it feel warm outside? Y N If it feels warm outside and I am awake, must I be tired?

The third subtest consisted of multi-step arithmetic problems. For example, "An astronaut requires two pounds of oxygen per day while in space. How many pounds of oxygen are needed for a team of three astronauts for five days in space?" Such an item requires thinking similar to combining several Logo procedures to accom- plish a single task. Because turtle geomet ry is inherent ly spatial, the fourth sub- test was designed to assess students ' ability to utilize spatial representat ions. For example , "The town of Springdale lies on Freedom Lake. Th e town of Wind ing Bend is west of Springdale. M a u m e e is east of Wind ing Bend but west of Spring- dale. Bumpy Knoll is east of Valley Ridge but west of Mau m ee and Winding Bend. Which town is farthest West?" T h e fifth subtest was designed to assess the effects of experience with recursive procedures on students' ability to solve problems deal- ing with mathemat ica l recursion. For example , "A w o m an is offered a job as a bank teller at a s tart ing salary of $10,000 per year. She is told that her salary will increase $2,000 for every year she works at the bank. H o w much will her salary be dur ing her sixth year as a bank teller?"

188 Battista and Clements

Problem-solving Test 2 was designed to assess executive-level problem-solving processes. Sternberg's (1985) componential theory was used as a framework for the executive-level skills involved in solving problems. This theory posits three kinds of information-processing components: metacomponents, performance components (used in task execution), and knowledge-acquisition components. The metacom- ponents were used to design the test for the present study. They are executive pro- cesses that are utilized in planning and evaluating one's information processing, and thus control problem-solving behavior. It was hypothesized that Logo pro- gramming would positively affect each of the tbllowing metacomponents, thus, these metacomponents served as the basis for designing items tor Problem-solving Test 2. (a) Deciding on the nature of the problem--Logo's emphasis on turtle graphics allows children to pose significant problems of varying levels of complexity for themselves. They must generate ideas for projects, represent these as goals, and identify the specific problems involved in reaching these goals. Thus, they must determine the nature of each problem and subproblem. (b) Deciding on pertor- mance processes and how to combine them (i.e., choosing a strategy)--Logo's pro- cedural nature allows students to decompose a task into subtasks that can each be accomplished by individual procedures and then to plan how to combine these pro- cedures to solve the task. (c) Selecting a mental representation--Programming in turtle graphics necessitates representation of the solution process internally as an initial and a goal state, as an intended semantic solution whose organization must be verbalized tbr others, and as machine-executable code. (d) Monitoring solu- tion processes--A Logo procedure represents an external manifestation of a stu- dent's thought about a problem solution. Running the procedure allows the student to monitor the validity of his or her solution, and Logo's graphic depiction of errors and easy-to-use editor support solution debugging. This entire sequence represents a concrete example of the process of cognitive monitoring.

Two subtests of Problem-solving Test 2 assessed aspects of the first metacom- ponent, deciding on the nature of the problem (items were adapted from Schoen and Oehmke, 1979). On the first, students were asked to identify information that would be necessary to solve a problem. For example:

The distance around a rectangular garden is 176 yards. A tarmer wanted to know the length of the garden. Which of these would he need to know?

a. The garden is 36 yards wide. b. The garden is divided into 9 rows. c. Each row is 4 yards wide. d. The yard is plowed three yards deep.

On the second subtest, students were asked to identify two problems that had isomorphic problem structures. For example,

Paul is 6 inches taller than Jim. J im is 57 inches tall. To find Paul's height, the boys added 57 and 6, and got 63 inches. Which problem below could be solved using exactly the same steps?

a. Patty's and Sara's ages add to 13 years. Sara is 5 years old. How old is Patty? b. My brother is 6 years younger than I am. I am 13 years old. How old is

my brothel'?

Effects oflogo 189

c. I have 13 times as m a n y points as George. George has 6 points. H o w m a n y points do I have?

d. I have 13 dollars. I made 6 more dollars. H o w much do I have now?

T h e third subtest assessed the me tacomponen t of choosing and combining per- formance processes (i.e., selecting a solution strategy). For example, one relatively open-ended problem of this type was: "A storage room at your school holds paper. T h e teacher asks you to figure out how many sheets of paper the room would hold if it were full. Tell exactly what you would do to solve this p rob lem." A full point was given if the answer was complete , efficient, and relied on mathemat ica l pro- cedures beyond simple count ing (e.g., de te rmine the volume of the room and the vo lume of one pack of paper , divide the former by the latter, and mult iply by the n u m b e r of sheets of paper in one pack). One-ha l f point was given for a complete, but inefficient p rocedure (e.g., de te rmine how m a n y sheets of paper there are in one "row," multiply by the number of rows that would fit in the room). The tourth subtest assessed students ' ability to select an appropr ia te problem representat ion. For example , " T h e nine member s of a club decided that at the beginning of the meeting, everyone in the club should shake hands with everyone else. H o w m an y handshakes occur?" A full point was given for a correct answer; if the answer was incorrect, one-half point was given for an adequate representat ion (e.g., nine dots a r ranged in a circle with in te rconnect ing lines).

T h e fifth subtest assessed students ' cognitive moni to r ing skills. Th e items were relatively simple mathematical ly , but contained informat ion designed to lead stu- dents to an e r roneous solution strategy. For example, "When Albert was six years old, his sister was three times as old as he. Now he is 10 years old and he figures that his sister is 30 years old. H o w old do you think his sister will be when Albert is 12 years old?"

For both problem-solving tests, i somorphic problems, with only surface infor- mat ion changed, were constructed for the posttests. For instance, the example item described above for the first subtest of Problem-solving Test 1 was rewri t ten as "John has three toy c a r s - - a Mus tang , a Rabbi t , and a Corvet te . In how m a n y different ways can he line up the cars one in front of the other?" Th e example item described for the fifth subtest of Problem-solving Test 2 was rewri t ten as "W h en J o n was six years old, his sister was two times as old as he. Now he is eight years old and he figures that his sister is 16 years old. H o w old do you think his sister will be when J o n is 10 years old?"

S tandard ized tests normal ly used by the schools were utilized to measure math- ematics achievement. For the pretest, nationally-normed percentile scores on appro- priate subtests from these tests as adminis tered by the schools the previous year were collected. 1 T h e fourth graders had been given the Iowa Tests of Basic Skills ( H i e r o n y m u s , Lindquis t , & Hoove r , 1978). Th e i r pretest score consisted of the average of the percenti les on the mathemat ics concepts and the problem-solving subtests. T h e concepts subtest included items on numera t ion , n u m b e r sentences, whole numbers , fractions, decimals, geometry , and measurement . T h e problem- solving subtest assessed students' ability to solve single-step problems involving each of the four basic arithmetical operations, as well as multi-step problems. Th e sixth graders had been administered the California Achievement Test (CAT, 1977). The

I S t a n d a r d s c o r e s w e r e n o t a v a i l a b l e .

190 Battista and Clements

Mathemat ics Concepts and Applicat ions subtest percenti le score was employed. This subtest requires students to recognize concepts and solve problems in con- texts such as numera t ion , n u m b e r sentences, n u m b e r theory, n u m b er properties, graphing, ti'actions, decimals, geometry, and measurement . For the posttests, these same subtests were adminis tered by the researchers.

RESULTS

T h e means and s tandard deviat ions tbr the two problem-solving ins t ruments arc presented in Tab le 1. T h r e e - w a y repeated measure A N O V A s (Group x Grade x T ime) were per formed on both problem-solving tests. Th e re were two significant main effects on Problem-solving Test 1, Grade (F (1 , 50) = 10.27, p = .002) and T i m e (F ( 1, 50) = 9.84, p = .003) . Examina t ion of the means indicated that the

Table 1. Mean scores on the problem-solving subtests by group and grade

Logo CAI Control

Grade 4 6 4 6 4 6

Subtest Pre Post Pre Post Pre Post Pre Post Pre Post Pre Post

Problem-solving Test 1

Permutations .20 .70 .81 1.00 .50 .92 (.45) ( . 6 7 ) ( . 5 2 ) ( . 7 6 ) ( . 5 5 ) (.80)

Logic .54 .94 1.16 .92 .60 1.15 (.22) ( . 4 0 ) ( . 5 6 ) ( . 6 3 ) ( . 5 6 ) ( . 6 3 )

Arithmetic .50 .80 .85 .85 .17 .83 (.50) ( . 4 5 ) ( . 3 7 ) ( . 6 9 ) ( . 4 1 ) ( . 7 5 )

Spatial .20 .40 .62 .77 .50 .33 (.45) ( . 5 5 ) ( . 6 5 ) ( . 9 3 ) ( . 5 5 ) ( . 5 2 )

Recursion 1.00 .70 1.15 1.31 .75 1.00 (.79) (.27) ( . 6 6 ) ( . 7 2 ) (.76) (.45)

Total 2.44 3.54 4.58 4.85 2.52 4.23 (1.49) (1 .44 ) (1 .45 ) (2 .64 ) (2 .28 ) (2 .15 )

Problem-solving Test 2

Needed 1.60 1.60 1.77 1.69 1.33 .83 Information (.55) (.55) (.44) (.48) (.82) (.75)

Problem 1.00 1.20 1.23 1.08 1.33 .83 Structure (.71) (.84) (.60) (.64) (.52) (.75)

Performance .32 .38 .12 .82 .08 .18 Processes (.34) (.37) (.20) (1.34) (.20) (.33)

Representation .18 .50 .41 .73 .17 .25 (.27) ( . 5 0 ) ( . 4 9 ) ( . 6 7 ) ( . 4 1 ) ( . 2 7 )

Cognititve .40 1.00 .54 .58 .17 .17 Monitoring (.89) (1.00) (.66) (.64) (.41) (.41)

Total 3.50 4.68 4.07 4.89 3.08 2.27 (1.84) (1.08) (1.52) (2.34)(1.02) (1.65)

.71 1.07 .00 .50 .50 1.04 (.67) ( . 7 8 ) ( . 0 0 ) ( . 8 4 ) ( . 6 4 ) ( . 7 8 )

1.16 1.07 .73 1.10 .99 1.08 (.52) ( . 5 0 ) ( . 2 9 ) ( . 4 3 ) ( . 6 0 ) ( . 5 4 )

1.14 1.00 .67 .83 1.17 1.00 (.66) ( . 5 5 ) ( . 5 2 ) ( . 4 1 ) ( . 7 2 ) ( . 7 4 )

.71 .71 .17 .17 .33 .58 (.73) ( . 7 3 ) ( . 4 1 ) ( . 4 1 ) ( . 4 9 . ) ( . 9 0 )

1.00 1.25 .33 .58 1.00 .96 (.68) ( . 6 1 ) ( . 2 6 ) ( . 8 0 ) ( . 6 7 ) ( . 6 6 )

4.73 5.11 1.90 3.18 3.99 4.67 (2.26) (2.22) ( .51)(1.69) (1.94) (2.33)

1.79 1.71 1.33 1.00 1.56 1.75 (.58) ( . 4 7 ) ( . 5 2 ) (.63) (.73) (.45)

1.79 1.43 1.50 1.00 1.33 1.25 (.43) ( . 6 5 ) ( . 5 5 ) { . 8 9 ) ( . 5 0 ) ( . 7 5 )

.46 .28 .08 .13 .18 .13 (.42) ( . 5 0 ) ( . 2 0 ) (.22) (.22) (.21)

.66 .61 .08 .08 .39 .38 (.53) ( . 4 5 ) ( . 2 0 ) ( . 2 0 ) { . 4 2 ) ( . 3 8 )

.50 .36 .17 .00 .44 .58 (.76) ( . 6 3 ) ( . 4 1 ) ( . 0 0 ) ( . 7 3 ) ( . 7 9 )

5.20 4.39 3.17 2.22 3.90 4.09 (1.61) (1 .97) (1 .13) (1 .07) (1 .55) (1,26)

Note. Standard deviations are in parentheses.

Effects of logo 191

sixth graders outperformed the fourth graders and that the posttest scores were greater than the pretest scores.

The ANOVA computed on Problem-solving Test 2 revealed a significant Group x Time interaction (F(2,47) -- 5.22, p =. 009), with the Logo group's score increasing pre- to posttest and the other two groups' scores declining slightly. Newman-Keuls post hoc comparisons revealed that the Logo group's posttest score was significantly higher than the posttest scores of the CAI (p < . 05) and control (p < .01) groups. There was also a significant main effect for Grade (F(1,47) = 8.75, p = .005) with the sixth graders scoring higher than the fourth graders.

To control for variations in mathematics posttest scores due to students' incoming ability in concepts and problem-solving, analysis of covariance (Group x Grade) was performed on the mathematics achievement posttest scores, using the previ- ous year's scores as covariates. (Covariance was used because the conditions of test- ing, pre- and posttest, were dissimilar.) No significant differences among the groups were found.

DISCUSSION

No differences were found among the groups on Problem-solving Test 1. There- fore, there was no evidence that the computer treatments affected students' abil- ity to solve problems whose solution required conceptual and procedural knowledge not routinely addressed in traditional mathematics lessons. Students' abstraction of computer experience with the aforementioned conceptual and procedural knowl- edge may have been insufficient to enable transfer to related noncomputer prob- lems. Also, although it was originally intended to explore the topics of nested procedures, conditionals, and recursion at some length in the Logo treatment, stu- dents proved unable to progress very far with these more advanced topics. There- fore, it is understandable that no transfer to the solution of problems related to these topics was detected.

The Logo group evidenced greater improvement than the other two groups on the test of metacomponential problem-solving processes. This is in agreement with previous findings that Logo programming can increase certain problem-solving abilities, especially those related to executive processes (Clements, 1986; Clements & Gullo, 1984; Milner, 1973; Statz, 1974). Although statistical analysis of the sub- tests was contraindicated by the low number of items in each, it is interesting to note that most of the Logo group's increase occurred in three of the five subtests: cognitive monitoring, selecting a mental representation, and deciding on perfor- mance level processes. Cognitive monitoring may have been affected because pro- gramming in Logo encourages students to represent their thinking externally, and this may help students become accustomed to recognizing and reflecting on errors in their thinking. Selecting a mental representation may have been affected because, in Logo, students represent their thinking in multiple ways--verbally, graphically, and formally in computer code--which may lead them to become more flexible in selecting and utilizing problem representations. Finally, deciding on performance level processes may have been affected because, in Logo, students must decide how to establish and combine subtasks. In contrast, students in a typical classroom or

192 Battista and Clements

C A I problem-solving situation are encouraged to repeatedly apply a single strat- egy or heuristic.

No significant differences were found a m o n g the groups on the mathemat ics achievement test. Thus , there was no evidence that either compute r problem- solving t rea tment improved students ' knowledge of mathemat ics , even though improvemen t was hypothesized for the Logo group. It may have been that the dura t ion of the t rea tment was too short or coverage of s tandard mathemat ics too slight to lead to significant gains in mathematics achievement. Alternatively, it may be that the students did not see the connect ion between the concepts they encoun- tered in Logo and classroom mathemat ica l tasks and therefore, transfer to stan- dard mathemat ics achievement was minimal. Recent studies that have at tempted to make explicit connect ions between students ' work with Logo and their class- room mathemat ics work have found statistically significant increases on tests of mathemat ica l achievement (Findlayson, 1984; Rieber, 1983).

In conclusion, the effect of Logo p r o g r a m m i n g was investigated in three areas: problem solving dependent on specialized conceptual and procedural knowledge, problem solving dependent on specific executive-level skills, and s tandard math- ematics achievement . Evidence of positive effects was found only for executive- level problem-solving skills, the pr imary instructional emphasis of the treatments. This emphasis apparently precluded sufficient attention to the other two domains. Tha t is, time allocated for the treatments was insufficient for Logo students to mas- ter conceptual and procedural knowledge such as recursive, combinator ia l , and condit ional thinking. Likewise, coverage of mathemat ica l content or connections between Logo and classroom mathemat ics may have been inadequate.

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