"Domain knowledge" refers to knowledge and skills specific to a particular discipline or cognitive aptitude. Everyone brings a unique combinations of domain knowledge to problem-solving tasks. To a greater or lesser extent, these differences can reasonably be assumed to affect the ways in which those tasks are negotiated. In the current study, three sorts of domain knowledge were deemed relevant to the tasks at hand: students' knowledge of Logo programming, their mathematical understanding, and their spatial intelligence (Gardner,1983). Each of these is outlined in Table 1 and discussed in more detail in the sections that follow.

Figure1. Logo Programming Problems

Logo Programming. Just as reading entails more than simply decoding written language, so the knowledge of Logo progaramming entails more than simply knowing the meaning of its commands. Condering that they were all novice programmers, the students involved in the study exhibited a surprising range of understanding of basic Logo commands. While the majority of students demonstrated that they knew basic graphics commands (i.e., FD, BK, RT, LT, PU, PD), three students were very shaky with even these. At the other extreme, one student easily used the more advanced SETH ("set heading") command. Some students showed they had a good sense of units of distance and degrees ofturns; others showed they didn't. Some students understoodthe reversibility of FD and BK, RT and LT; others didn't. Finally, although several students seemed to have a relatively automated idea of how to make a square, the majority had to work the whole figure out.

For the purposes of this study, three levels of Logo programming knowledge were distinguished (see Table 1). Students who exhibited a ready grasp of Logo graphics commands including a good sense of distances and degrees of turns, who used more advanced Logo commands, and/or who showed that they knew the formula for a square were categorized as "very good" (VG) programmers. Students who exhibited a working knowledge of basic Logo graphics commands and some sense of distances and turns where categorized as "average" (OK) programmers. Students who had difficulty with basic Logo graphics commands and/or distances and turns were categorized as "not good" (NG) programmers.

Mathematics. The students in the study also exhibited a range of mathematical knowledge and skills. Many students were obviously able to mentally combine and/or split apart quantities, but some could only combine certain quantities, and at least one had trouble combining even simple quantities.

For the purposes of this study, three levels of mathematical knowledge were distinguished (see Table 1). Students who exhibited the ability to mentally manipulate quantities, including quantities over 100, and who exhibited mathematical thinking were categorized as having "very good" (VG) mathematical abilities. Students who minimally manipulated quantities mentally, but never numbers greater than 100, and who exhibited no mathematical thinking were categorized as having "average" (OK) mathematical abilities. Students who exhibited difficulty in manipulating simple quantities mentally and gave no evidence of mathematical thinking were categorized as "not good" (NG) in mathematical abilities.

Spatial Discrimination/Directionality. Two students in the study seemed yet to have mastered directionality or they had a difficult time distinguishing between left and right. In one case, this made figuring turns especially laborious. The other student, Kathy, is the student who had mastered the SETH command. Indeed, Kathy had probably the most complete domain knowledge of Logo programming of any of the students involved in the study. She stated that she used SETH for all her turns (except the ones embedded in the square, which she had automated) precisely because she had a hard time distinguishing left from right. Kathy's use of SETH is a good example of the use of ability in one domain to compensate for a lack of knowledge or skill in another.

For the purposes of this study, students' knowledge of directionality was simply categorized according to whether they could (OK) or could not (NG) readily distinguishing left and right (see Table 1).

In their recent article on styles and voices within the computer culture, Turkle and Papert (1990) used the term "styles" to refer to different approaches to knowledge in the Piagetian model. They argued, however, that suchcognitive styles do not represent stages in the evolution of formal reason (Piaget, 1971) but rather that each is equally valid on its own terms. Whether or not cognitive styles represent developmental stages or mature diversity, they could affect the ways in which problems are solved. This article employs the term "cognitive styles" in this manner, and it is in this sense that several distinct cognitive styles (as compared with the single dichotomy described by Turkle and Papert emerged from analyzing the integrated protocols. These can be categorized in terms of the following dichotomies: global versus local problem representation, abstract versus con thinking, and top-down versus bottom-up planning. Each of these is outlined in Table 2 and described in more detail in the following sections.

Global Versus Local Problem Representation. The distinction between global and local problem representation is found in whether or not students consider specific parts of a problem in relation to others (see Table 2). The problem representations of students, for example, who considered the parts of the triangle and the square in Problem 1 (see Figure I ) in relationship to each other, or who used a formula to describe the various squares, were categorized as "global" (GL). April's solution to the third problem, for example, contains many good examples of global thinking:

(In this and in all examples given in this article, Logo commands are shown in plain type, student comments are shown in italics, the researcher's (real time) comments are in italics inside angled brackets (>< ><), and researcher's laterexplanations are given in plain text inside parentheses. Students' names have been changed to ensure confidentiality.)

... LT
90
FD 130 - (this sends the turtle right to the lower left corner)
(Oh wow, how'd you know 130)
I don't know, I guessed.
RT 90 - (she takes the turns for granted)
FD 130
Cause that's about the 1ength of that - (she compares the line she has already drawn half way across the bottom of the screen with the distance she needs to cover up the left side of the screen) - Oops, not quite enough
FD 20
All right.
RT 90
Now I'm going to go forward um... OK so now I'm going to go... Let's see, it was 150 to there - (she examines the line she has drawn on the left side of the screen, then notices the line she has drawn half way across the bottom and chooses that as her measure.) - Let's see, it was...we11, it was 130 to there - (points half way across the top of the screen comparing it to the bottom) - so that's probably ... 260, I guess... All right
FD 260
... (APRIL, PROBLEM 3)

On the other hand, the problem representations of students who considered each command sequence in itself or in relation to the commands immediately preceding it were categorized as "local" (LOC). An extreme example of a local problem representation accounts for Laura's divergence from the given drawing to relate what she was doing to what she had already drawn:

... So I have to make a square ...
- (Laura doesn't seem to see that the square is in the upper right corner: she just makes a square. In the previous problem, she similarly didn't include the line splitting the triangle and the square.)
FD, hmmm...
FD 40
... (LAURA, PROBLEM 2

Abstract Versus Concrete Thought. Piaget (1971) made the classic distinction between formal mental manipulation and thinking that is grounded in concrete objects (see Table 2). In this study, planning ahead, systematic testing, use of formulas, and mentally combining or splitting apart quantities to calculate distances or turns were all considered evidence of "abstract" (ABS) thought. The lack of these, together with the use of idiosyncratic numbers, was considered evidence of "concrete" (CON) thought. In addition, a distinction was made between abstract and concrete uses of reversibility.

Reversibility has to do with the fact that in Logo the left (LT) command is the opposite of the right (RT) command and the back (BK) command is the opposite of the forward (FD) command. When students turned the turtle in the wrong direction or moved it too far, they could correct their errors by making use of this concept. Some students simply turned or moved the turtle the distance they had gone in the wrong direction plus the distance they wanted to go in the first place. These students were categorized as abstract (ABS) thinkers. Other students seemed to need to get the turtle back to where they went wrong, to anchor their thinking in a concrete experience, before they turned or moved in the correct direction. These students' thought was categorized as concrete (CON).

For example, in working on the second problem (see Figure I ), April exhibited abstract thinking when she kept going with the square she was drawing, even though she forpot to put the pen down:

... FD 40
Oops, I forgot to put the pen down.
PD
RT 90
FD 40
RT 90
FD 40
RT 90
FD 40
RT 90
FD 40
I did it.
HT (APRIL. PROBLEM 2)

In contrast, when Lilly made a similar mistake in the second problem, she went back and put the pen down before drawing the square. When she turned the wrong way, she first turned back to where she started and then made the correct turn. Lilly's thinking was therefore categorized as concrete.

... FD 30
Oh, I keep on forgets i ng to put the pen down.
BK 30
PD
FD 30
There!
-(plays turtle)
RT 90
- ( Lilly shakes her head because theturtle has again turned the wrong way.)
LT 90
LT 90
FD 30
( LI LLY, PROBLEM 2 )

Top-Down Versus Bottom-Up Planning. The distinction between behaviors categorized as top-down (TD) or bottom-up (BU) planning involved whether students apoached a problem as if it consisted of distinct smaller problems, or whether they approached that problem as a single command sequence to be discovered one step at a ne (see Table 2). Students such as Tom, who used a top-down approach, planned what they were going to do before doing it:

...Yeah. See I'm going to be going FD30 all around this thing and then I'm going to go RT 90 to make the turns, so that will be easy, but then when I get to the triangle, I'm going to have to figure out a bunch of different stuff.
... (TOM, PROBLEM 1)

On the other hand, students who used a bottom-up approach "discovered" their plan from the problem-solving experience itself. Lilly, for example, considered each piece of her solution as it unfolded in the programming sequence rather than in relationship to a larger, preplanned design:

...(What are you thinking about?)
How far up the line should be.
(Oh, OK, the first line that you're drawing?)
Yeah.
...(Now what are you thinking about?)
Which way I should turn the turtle to make him go down.
...(OK, now what areyou going to do?)
I 'm going to put the pen down, go up to the top, actually go up on this side.
...( LILLY, PROBLEM 1 )

A number of distinct problem-solving strategies can be distinguished within general problem-solving behaviors (Newell & Simon, 1972; Wicklegren, 1974). Certain of these seem more applicable to programming problems in general and to children's programming in particular (Clements & Gullo, 1984; Lawler, 1985). In previous work (Swan & Black, 1989), five such problem-solving strategies that seemed well-represented in Logo programming environments-forward chaining, analogy, alternative representation, subgoals formation, and systematic trial and error-were isolated. These strategy distinctions, as explained in Table 3 and in the following sections, were carried over to the current study.

Forward Chaining. Forward chaining is a species of means-ends analysis (Newell & Simon, 1972) that involves working from what is given in a problem toward the problem goal in step-by-step, transformational increments.

Table 3.
Problem-Solving Strategies

forward chaining (F. CHAIN)	quantities to get progressively closer to the drawing of a single line or angle
analogy (ANALOGY)	evidence that student has used solution to one problem to help solve another
alternative representation (ALT. REP.)	evidence that student has reconceptualized a problem to solve it
subgoals formation (SUBGOALS)	student breaks the problem inot distinct, chunked parts
systematic trial & error (SYST. T&E)	student systmatically tests quantities needed for distances or turns by picking the pen up and trying them

Of course, most programming activity involves some forward chaining. In the current study, programming behaviors were categorized as utilizing forward chaining (F.CHAIN) strategies only when students used forward chaining in the extreme; that is, only when they used repetitions of known quantities to move the turtle a single distance or turn, i.e.:

... PU
FD 100 - X 2 - (using the arrow and return keys)
(So you're moving it forward first... )
Whoooh, it wrapped around the screen.
FD 10 - X 5 - (using the arrow and return keys)
(Why did you go back over your old stuff?)
Well, I was trying to figure out, I was trying to... I messed myself up, I shouldn't have I was trying to change the LT to something else.
(Oh.)
RT 90
FD 20 - X 2 - (using the arrow and return keys)
LT 90
(Now why are you turning 1eft?)
So I can go up a little bit.
(Go up for what?)
To get it a little bit up more on the screen.
FD 5 - X 2 (using the arrow and return keys)
RT 90
(Now why are you turning it right?)
So I can move it over here.
FD 5 - X 5 - (using the arrow and return keys )
RT 90
(Now you turned it right again. What did you do that for? So I can start making my square. . .. (TOM. PROBLEM 2)

Analogy. Analogy involves the discovery of a particular similarity between two problems and "a mapping of knowledge from one (the base) onto another (the target), predicated on a system of relations that holds among the objects of both domains" Gentner, 1987). In the current study, students were categorized as using analogy (ANALOGY) strategies when they used techniques from one problem to solve a part of ther problem. Kathy, for example, used an enlarged square procedure to draw the border for Problem 3 (see Figure 1). In the following protocol, notice also her use of systematic trial and error and abstract reasoning:

...I'm going to go to one of the corners and make a square with, instead of40, 1ike lOO or something.
SETH 135
PU
FD 85 FD 12
SETH O REPEAT
1[FD 100]
(What's the matter?)
I was just seeing if I might just want to do it once to see if it will work.
(Without the pen down?)
Yeah.
(Oh, I see what you're doing. That was a good idea.) - (started with a regular square then amended to this as she realized that 100 steps for each side might not be enough) FD 60
To get the turtle to the edge of the screen.
BK 10
(Now what are you doing?)
I'm going to go back 150 and...
BK 150 - (did figuring in her head here: 100 + 60 - 10)
(How'd you know 150?)
Because I did FD lOO and it was about here, so I did FD 60 and it went too far. so I did back lO.
So you added and subtracted.)
Yes.
REPEAT 4[FD 150 LT 90] - (changes RT to LT in square formula)
Whoops.
PD
REPEAT 4[FD 150 LT 90]
HT
(KATHY. PROBLEM 3)

Alternative Representation. Alternative representation involves reconceptualizing problem from differing perspectives. In the current study, students were judged as using alternative representation (ALT. REP.) strategies when there was clear evidence of reconceptualization. Interestingly, in all such cases reconceptualization took place as a result of the lack of the specific domain knowledge necessary to solve a particular problem. Robert, for example, made use of the square procedure he had saved to disk when he encountered difficulty making a square from scratch:

... Can I write "square"?
(What happens if you write "square"?)
Then I'll have a square.
(OK, do that.)
No.
(What's the problem?)
I didn't have a square in there.
(You didn 't make a program that draws a square, right?)
No.
(Do you know how to write one? Do you know how to make a program that draws a square?)
Uh uh. - (no)
(Well, then, I guess you can't do that. Let's do something else. What do you know how to do?)
Well, is this my program?
(No, do you want your program?)
Yes, 'cause that wi 11 be easier.
- (gets his disk, 1oads it, goes to the page with shapes on it, checks flipside for name of procedure)
Oh, SQ is it.
SQ
(ROBERT PROBLEM 1)

Subgoals Formation. Subgoals formation refers to breaking a single difficult problem into two or more simpler problems. In the current study, students showed subgoals formation (SUBGOALS) when they explicitly stated that they were breaking problems into distinct parts, i.e.:

... (So what are you going to do?)
Well, I already know the direction so I can start over again.
RG
OK, now I'm going to make the square and then make the triangle.
... (DOUG, PROBLEM 1)

Systematic Trial and Error. Systematic trial and error involves the testing of possible solutions in a systematic, guided fashion. This differs from forward chaining in that it includes negative inference (problem-reduction through the elimination of possibilities) as well as positive inference, and in that its power lies in the systematic testing of such possibilities rather than in the immediate choice of the best one. As with forward chaining, almost all programming involves some use of systematic trial-and-error strategies. In the current study, therefore, only when students picked the pen up and tested a move before implementing it was their behavior categorized as involving systematic trial-and-error (SYST. T&E) strategies, i.e.:

(Now what are you thinking?)
I'm trying to figure out how much to turn- (plays turtle) -OK.
RT 70 - (draws on screen with his finger)
PU
FD 10
BK 10 - (tests it)
LT 40
(Why are you using LT now?
To turn it back.
(Oh, good.)
FD 10
See if that's right, I'm not sure.
BK 10
Yup.
PD
...(SAM, PROBLEM 1)

Because explicit operational definitions were developed for each of the preceding problem-solving strategies, one or more students could possibly not have demonstrated the use of any of these strategies. This, however, was not the case. All the students who participated in the study used at least one of the identified strategies. This finding seems to indicate that people naturally employ some strategy when faced with problems to solve. The question then becomes: When and why do they use differing problem-solving strategies? The following section explores this question by exploring patterns of strategy choice among the students.

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