Computer Environments that Engender Students’ Construction of Mathematical Ideas and Reasoning: A Constructivist Perspective
Michael T. Battista
Kent State University
D R A F T
Paper presented at the ENC Technology and NCTM Standards 2000 Conference. Arlington VA, June 56, 1998.
Partial support for this work was provided by grant ESI 9050210 from the National Science Foundation. The opinions expressed, however, are those of the author and do not necessarily reflect the views of that foundation.
Computer Environments that Engender Students’ Construction of Mathematical Ideas and Reasoning: A Constructivist Perspective
After briefly outlining the basic mechanisms in the constructivist theory of learning, I suggest several essential characteristics for instructional computer environments that can properly cultivate meaningful mathematics learning. I then illustrate these characteristics by describing two computer environments and several examples of students' work in these environments. Finally, I situate my discussion of such computer environments in an overall context of technology use in mathematics education.
Basic Constructive Mechanisms
In the constructivist theory, abstraction is the critical mechanism that enables the mind to construct the mental entities that individuals use to conceptualize and reason about their "mathematical realities." Abstraction is the process by which the mind selects, coordinates, unifies, and registers in memory a collection of mental items or acts that appear in the attentional field. Reflective abstraction takes operations performed on previously abstracted items as elements and coordinates them into new forms or structures that can be acted upon in future abstractive acts (von Glasersfeld, 1995). Meaningful mathematics learning results from the reflective abstractions students make as they accommodate their current cognitive structures to deal with perturbations–realizations that something does not work or is unexpected (Cobb, 1994; Steffe, 1988; von Glasersfeld, 1995). Understanding mathematics, however, requires more than abstraction. It requires reflection, which is the conscious process of representing experiences, actions, or mental processes and considering their results or how they are composed (von Glasersfeld, 1995).
According to the constructivist view of learning, therefore, to get students to construct increasingly powerful mathematical ideas, instruction must promote abstraction and reflection. It must (a) focus students' attention and mental acts on those aspects of phenomena that are to be mathematized; (b) encourage student reflection on the viability of their developing mathematical conceptualizations, and (c) promote appropriate perturbations to incorrect or unsophisticated student theories.
Mental Models
Mental models are integrated sets of abstractions that are activated to interpret and reason about situations that one is dealing with in action or thought. They are nonverbal experiencelike mental versions of situations whose structure is isomorphic to the perceived structure of the situations they represent (Battista, 1994; Greeno, 1991; JohnsonLaird, 1983). Individuals reason about a situation by activating mental models that enable them to simulate interactions within the situation so that they can explore possible scenarios and solutions to problems. Using a mental model to reason about a situation enables a person to mentally move around, move on or into, combine, and transform objects, as well as perform other operations like those that can be performed on objects in the physical world. Furthermore, individuals' use of mental models is constrained by their knowledge and beliefs. That is, much of what happens when we form and manipulate a mental model reflects our underlying knowledge and beliefs about what would happen if we were dealing with the objects they represent.
When we combine the constructivist theory of learning with the theory of mental models, what emerges is a picture of meaningful mathematics learning coming about as individuals recursively cycle through phases of action (physical and mental), abstraction, and reflection in a way that enables them to integrate related abstractions into ever more sophisticated mental models of phenomena. In fact, individuals' ability to understand and effectively use our culture's formal mathematical systems to make sense of their quantitative and spatial surroundings depends critically on their construction of elaborated sequences of mental models. Initial models in these sequences enable individuals to reason about physical manipulations of realworld objects. Later models permit them to reason with mental images of realworld objects. Finally, symbolic models enable them to reason by meaningfully manipulating mathematical symbols representing realworld quantities. Without this recursively developed sequence of mental models, individuals' learning about mathematical symbol systems is strictly syntactic, and their use of symbolic procedures is disconnected from realworld situations.
Characteristics of Fertile Computer Environments
By "computer environment" I include not only the computer tools made available to students, but the instructional tasks in which students are to apply these tools. Based on the above description of constructivist learning, fertile computer environments should possess four characteristics–they should be problemcentered inquiry supportive, researchbased, mental model cultivating, and reflection and abstraction inducing.
ProblemCentered Inquiry Supportive
Computer environments should engender and support genuine problem solving and inquiry. Within the overall context of solving carefully designed sequences of problems in a classroom culture of inquiry, students should be able to make and test conjectures not only about their problem solutions but about their own personally evolving mathematical conceptualizations. In fact, in an ideal environment, students would generate not only possible solutions to instructionallypresented problems, but peripheral ideas about phenomena being investigated. An ideal instructional computer environment would permit and support exploration of both.
ResearchBased
Computer environments should promote students' learning in ways that are consistent with researchbased descriptions of students' construction of particular mathematical ideas. That is, their design should be based not only on the general constructivist theory of learning, but on research dealing with how students construct knowledge for particular mathematical topics such as geometry, fractions, algebra, and so on.
Mental Model Cultivating
Computer environments should engender students' development and use of appropriate mental models for dealing with physical, conceptual, and symbolic mathematical phenomena. The instructional goal should be to encourage and support students' development of mental models and reasoning with those models that permits students to make sense of, analyze, and solve problems concerning such phenomena.
Reflection and Abstraction Inducing
Computer environments should focus students' attention on phenomena in ways that support reflection on and abstraction of mental operations necessary for properly conceptualizing and reasoning about mathematizations of the phenomenanote1. Because mathematical conceptualizations and associated mental models result from reflection on and abstraction of one's own mental actions, computer environments must make those actions and their consequences more accessible to reflection. Students' focus should be on developing and refining personally meaningful mathematical theories that guide their mathematical actions and reasoning. One especially effective device for maintaining such a focus is for students to make predictions before acting, which encourages them to consider their actions in the context of their theories. Inconsistencies between their predictions and what actually happens provides a constant source of perturbations requiring accommodations that lead to increasingly sophisticated conceptions.
The Shape Makers Computer Microworld
The primary source of mental models is our experience in dealing with the world, especially with physical objects (JohnsonLaird, 1983). To appreciate how a mental model for a parallelogram might be derived from realworld manipulation, imagine four straight rods connected at their endpoints in a way that permits freedom of movement at the connections–a movable quadrilateral is formed. Imagine now that the opposite rods are the same length. No matter how we move this physical apparatus, it always forms a parallelogram.
As we manipulate this "parallelogram maker," we not only see how its shape changes, we feel the physical constraints that we have built into it. We see and feel how one parallelogram continuously changes into others. The visual and kinesthetic experiences that we abstract from our actions with this apparatus, along with our reflections on those actions, can be integrated to form a mental model that we can use in reasoning about parallelograms.
The Shape Makers computer microworld provides students with screen manipulable shapemaking objects similar to, but more versatile than, the physical parallelogram maker (Battista, 1998). This microworld was designed to promote in students the development of mental models that they can use for reasoning about geometric shapes. In it, each class of common quadrilaterals and triangles has a "Shape Maker," a Geometer's Sketchpad construction that can be dynamically transformed in various ways, but only to produce different shapes in the class. For instance, the computer Parallelogram Maker can be used to make any desired parallelogram that fits on the computer screen, no matter what its shape, size, or orientation–but only parallelograms. It is manipulated by using the mouse to drag its control points–small circles that appear at its vertices.
Encouraging Students' Progression through the van Hiele Levels of Geometric Thought
Research has shown that students progress through several qualitatively different levels of geometric thinking (Clements & Battista, 1992). Geometric thought begins at the Gestaltlike visual level in which students identify and operate on shapes and other geometric configurations according to their appearance. It progresses to the level of description and analysis in which students recognize and can characterize shapes by their mathematical properties. At the next level, students' geometric thinking becomes abstract and relational as they see that one property can signal other properties, define classes of shapes, distinguish between necessary and sufficient conditions for classes of shapes, understand and provide "locally" logical arguments for assertions, and hierarchically classify shapes. At the fourth level, students can comprehend and create formal geometric proofs, and at the fifth, students can compare axiomatic systems.
A sequence of Shape Maker activities has been designed to encourage students to pass through the first three van Hiele levels–from the visual, to the descriptiveanalytic, and into the abstract relational (Battista 1998). In initial activities, students use Shape Makers to make their own pictures, then to duplicate given pictures. These activities encourage students to become familiar with the movement possibilities of the Shape Makers viewed as holistic entities. Students are then involved in activities that require more careful analysis of shapes–unmeasured Shape Makers are replaced by Measured Shape Makers that display instantaneously updated measures of angles and side lengths. Students are guided to find and describe properties of shapes. Finally, students are involved in classification by comparing the sets of shapes that can be made by each Shape Maker.
Examples of Student Thinking
I now present several episodes that illustrate the types of thinking exhibited by students working in the Shape Maker microworld. All examples occurred in fifthgrade classrooms with 10 and 11yearold students.
Episode 1. In his initial manipulations of several Shape Makers, MI commented:
MI: [After trying to make a nonsquare rectangle with the Square Maker and concluding that it couldn't be done] The Square [Maker] would only get bigger and twist around–so it can’t make a rectangle. [Discussing the Kite Maker] If I pull one end out, the other end goes out. 
Although MI had abstracted several movement regularities for these two Shape Makers, he did not conceptualize them as typical geometric properties. However, although his conception of the regularity for the Square Maker was holistic, he saw the regularity for the Kite Maker in terms of parts of the shape, an initial move toward the formulation of properties.
Episode 2. Three students were investigating the Square Maker.
MT: I think maybe you could have made a rectangle.
JD: No; because when you change one side, they all change.
ER: All the sides are equal.
MT, JD, and ER abstracted different things from their Shape Maker manipulations. MT noticed the visual similarity between squares and rectangles, causing him to conjecture that the Square Maker could make a rectangle. JD abstracted a movement regularity–when one side changes length, all sides change (so he couldn't get them to be different lengths, which he thought was necessary for a rectangle). ER conceptualized a traditional mathematical property.
Episode 3. Three students were considering whether the Parallelogram Maker could be used to make the trapezoidal target figure at the right. Their 
knowledge of the Parallelogram Maker was insufficient to predict that this was impossible. However, as they manipulated the Parallelogram Maker, one of the students discovered something about it that enabled her to solve the problem.
ST: [Pointing to the nonhorizontal sides in the Parallelogram Maker] No, it won’t work. See this one and this one stay the same, you know, together. If you push this one [side] out, this one [the opposite side] goes out...This side moves along with this side. 
As ST manipulated the Parallelogram Maker in her attempts to make the target figure, she detected a pattern or regularity in its movement. (Because what ST says is imprecise, we cannot know for sure exactly what she abstracted.) As she abstracted this movement pattern, incorporated it into her mental model for the Parallelogram Maker, and described it in terms of parts of the Shape Maker, she was able to infer that the target figure was impossible to make. By using the welldeveloped mental operations she had available for reasoning about physical objects, images, and motion, ST made a discovery that could, with further elaboration, form the basis for making sense of the formal mathematical property "in a parallelogram, opposites sides are parallel and congruent." She had formed a visualkinesthetic mental model of the Parallelogram Maker that she could later use as the basis for formulating its geometric properties.
Episode 4: NL is using the seven quadrilateral Shape Makers to make the design at the right. A researcher is observing and asking questions as NL tries to make shape C with the Rhombus Maker. 
NL: The Rhombus Maker on [shape] B. It doesn't work. I think I might have to change the Rhombus Maker to [shape] C.
Res: Why C?
NL: The Rhombus Maker is like leaning to the right. On B, the shape is leaning to the left. I couldn't get the Rhombus Maker to lean to the left, and C leans to the right so I'm going to try it. [After her initial attempts to get the Rhombus Maker to fit exactly on shape C] I don't think that is going to work.
Res: Why are you thinking that?
NL: When I try to fit it on the shape, and I try to make it bigger or smaller, the whole thing moves. It will never get exactly the right size.
[Manipulating the Rhombus Maker] Let's see if I can make the square with this. Here's a square. I guess it could maybe be a square. But I'm not sure if this is exactly a square. It's sort of leaning. The lines are a little diagonal. [Continuing to manipulate the Rhombus Maker] Yeah, I think this is a square maybe.
Res: When you tried to fit it on C, did you notice anything about shape C, or the Rhombus Maker?
NL: The Rhombus [Maker] could make the same shape pretty much, but if you tried to make it small enough to fit on C, it would make the whole thing smaller or it would move the shape down. And when you tried to move it up to make it smaller, it would move the whole shape up.
Res: You said the Rhombus Maker could make the same shape as shape C, what do you mean by that?
NL: It could make the same shape. It could make this shape, the one with 2 diagonal sides and 2 straight sides that are parallel. It could have been almost that shape and it got so close I thought it was that shape.
Res: So that [the Rhombus Maker] is the same shape as that [shape C]?
NL: [Continuing to manipulate the Rhombus Maker] Oh, I see why it didn't work, because the 4 sides are even and this [shape C] is more of a rectangle.
Res: How did you just come to that?
NL: All you can do is just move it from side to side and up. But you can't get it to make a rectangle. When you move it this way it is a square and you can't move it up to make a rectangle. And when you move this it just gets a bigger square.
Res: So what made you just notice that?
NL: Well I was just thinking about it. If it [the Rhombus Maker] was the same shape, then there is no reason it couldn't fit in to C. But I saw when I was playing with it to see how you could move it and things like that, that whenever I made it bigger or smaller, it was always like a square, but sometimes it would be leaning up, but the sides are always equal.
This episode clearly shows how a student's manipulation of a Shape Maker and resultant reflection on that manipulation can enable the student to move from thinking holistically to thinking about interrelationships between a shape’s parts, that is, about its mathematical properties. Indeed, NL began the episode thinking about the Rhombus Maker and shapes holistically, saying that she was trying to make the Rhombus Maker "lean to the right," and get "bigger or smaller," and that "the whole thing moves."
The fact that NL could not make the nonequilateral parallelogram with the Rhombus Maker caused her to reevaluate her mental model for the Rhombus Maker. Originally, because her model was not constrained by the property "all sides equal," her mental simulations of changing the shape of the Rhombus Maker included transforming it into nonequilateral parallelograms. Her subsequent attempts to make a nonequilateral parallelogram with the actual Rhombus Maker tested her model, showing her that it was not viable. As she continued to analyze why the Rhombus Maker would not make the parallelogram–why it would not elongate–her attention shifted to the possibility of changing its side lengths. This new focus of attention enabled her to abstract the regularity that all sides were congruent. As she incorporated this abstraction into her mental model for the Rhombus Maker, she was able to infer that the Rhombus Maker could not make shape C.
It is highly likely that NL’s conclusions about the Rhombus Maker came about partly because she had previously made a square with it. She viewed the Rhombus Maker as a transformed square, and we know from previous episodes that NL had already concluded that squares have all sides the same length. Using the Rhombus Maker took advantage of her and other students’ natural proclivity to reason about shapes by transforming them in various ways (Battista, 1994).
This example illustrates that students move toward propertybased conceptions of shapes (and therefore more sophisticated levels of geometric thinking) because of the inherent power these conceptions give to their analysis of spatial phenomena. In the current situation, NL developed a propertybased conception of the Rhombus Maker because it enabled her to understand why the Rhombus Maker could not make shape C–something that truly puzzled her.
Episode 5: Shape Makers and Classes of Shapes. One of our major hypotheses is that a Shape Maker can become for students a "concrete" way to think about classes of shapes. The Rectangle Maker, for example, can be used to think about the class of rectangles because the properties embodied by it are exactly those properties that all rectangles have. As the episode below illustrates, this Shape Maker to shapeclass correspondence enabled students to interrelate classes hierarchically, a characteristic of van Hiele level 3.
BE: A square is a rectangle, but a rectangle is not a square.
MA: I agree. The Rectangle Maker can make a square, but the Square Maker cannot make all rectangles.
SO: Every shape made by the Square Maker can be made by the Rectangle Maker because a square is a rectangle.
MA's and SO's statements show how they were using their knowledge of the Shape Makers to make sense of and justify BE's claim that squares are rectangles. Because MA and SO had properly connected Shape Makers with the classes of shapes they made, they could reflect on their mental models of the Shape Makers and draw conclusions about properties of, and interrelationships between, classes of shapes.
Concluding Remarks on Shape Makers
As students manipulate and reflect on their manipulations of a Shape Maker, they abstract its movement possibilities and relationships between its components that remain constant during movement, integrating these abstractions into a mental model of the Shape Maker. Because of the way the Shape Maker has been constructed, these abstractions reflect the geometric properties of the class of shapes made by the Shape Maker. At first, these properties are incorporated into a mental model implicitly as the properties become embodied in the model's simulated behavior. Later, the properties become explicit as they are disembedded and abstracted from the simulated behavior, and as students develop, through social interaction, the terms and concepts used to describe them in traditional mathematical language. What results are mental models of Shape Makers that enable students to make sense of traditional geometric properties of shapes and to reason about shapes and classes of shapes in increasingly sophisticated ways.
Using Computer Spreadsheets to Promote Algebraic Thinking
For students to find algebra conceptually meaningful, as well as useful in modeling and analyzing realworld problems, they must be able to reflect on, make sense of, and communicate about general numerical procedures (Kieran, 1992). Such procedures consist of set sequences of arithmetic operations performed on numbers. Examples include computing an average and performing the standard division algorithm. Thinking about numerical procedures starts in the elementary grades and continues in successive grades until students can eventually express and reflect on the procedures using algebraic symbolism.
Levels of Sophistication in Procedural Thinking
Based on the constructivist theory of abstraction, there are three levels of sophistication that students can achieve in thinking about a numerical procedure. At the first or perceptual level, students can complete the sequence of operations that constitute the procedure when the sequence's description is in full view and a specific number is operated on. At the second or internalized level, students have abstracted the sequence of actions in a procedure to a sufficient degree that they can imagine and describe the sequence while not actually performing it. They can see the procedure as being applicable to many instances rather than one particular case. At the third or interiorized level, students can reflect on, decompose, and analyze a numerical procedure. Their focus of attention shifts from performing actions to analyzing the meanings and results of actions. Because students can decompose a procedure into its component actions and consider these components as meaningful conceptual objects, they can recombine the components in ways that enable them to solve novel problems. Thinking about a numerical procedure at this level is the beginning of algebraic thought.
Number Shifters
I now describe how instructional activities with computer spreadsheets can be used to encourage students to think about numerical procedures at progressively higher levels of sophistication (Battista & Borrow, 1998). These activities can shift students’ focus of attention from merely performing operations to reflecting on, analyzing, manipulating, and communicating about sequences of operations. This shift in attention encourages students to think more generally and abstractly about the procedures. Use of computer spreadsheets also exposes students to a method of symbolically describing numerical procedures in a more algebraic manner.
The instructional activities focus on numerical functions called Number Shifters. A Number Shifter is a rule that tells how to transform one number, the input, into another number, the output, by performing a sequence of arithmetic operations. We enter Number Shifters into computer spreadsheets. For instance, Number Shifter 1 can be entered into a computer spreadsheet as shown below.
Number Shifter 
Computer Spreadsheet Code 
Once Number Shifter 1 has been entered into the computer, an output can be found by entering an input number in cell B1. The spreadsheet multiplies the input by 30, showing the product in cell B2, then adds 75 to the product, with the result appearing in cell B3. This result is also shown in cell B4 as the final output. Algebraically, we are examining the expression 30x + 75 for different values of x; that is, we are examining f(x) = 30x + 75.
The following dialogue suggests how a teacher used a Number Shifter to encourage students to move to a Level 3 analysis of a numerical procedure.
Tchr: What does Number Shifter 1 do?
DN: It first multiplies 5 by 30.
Tchr: Does the Number Shifter always multiply 5 by 30?
DN: No, but it could.
Tchr: Yes, it could. What other numbers could it use?
DN: Any other numbers.
Tchr: And where would those numbers be?
DN: In the input.
Tchr: So what does this Number Shifter do with each input number?
CR: Multiply it by 30, then add 75 to it.
Tchr: What do you mean by "it"?
CR: The number you put in.
CH: The input number.
Tchr: Do you mean that the computer will multiply the input number by 30, then add 75 to the input number?
CL: When we tried the input 7, we first multiplied it by 30 and got 210, then added 75 to the 210.
Tchr: So what do you think the computer will do with any input?
CL: First it will multiply it by 30, then add 75 to that answer. [Students agree.]
Note that the teacher used the Number Shifter to get students to be more precise when describing sequences of operations on numbers. The sequence became the object of reflection, rather than a script for acting.
Missing Input Problems
In Missing Input Problems, students determine what input will give a specified output for a Number Shifter. For instance, what input will give an output of 60 for Number Shifter 2? The analogous situation in algebra is solving the equation (6(x + 20)  30)/3 = 60.
Number Shifter 2 
Input Number 
? 
Step 1 
add the input to 20 

Step 2 
multiply the result from Step 1 by 6 

Step 3 
subtract 30 from the result of Step 2 

Step 4 
divide the result from Step 3 by 3 

Output Number 
60 
Students typically use two basic strategies to solve Missing Input Problems.
Some use guessandcheck. They choose an input number, then, based on the output,
decide what number to enter next. For instance, for Number Shifter 2, one student,
DN, reported that she tried 100, then 50, then 25, then 15 as inputs.
Tchr: How did you know to go to 15?
DN: If it [the output] was too high you would go down, and if it’s too low you would go up. I used guess and check.
It is important to note that DN was doing more than randomly entering numbers until she found the desired output. She employed a systematic procedure that took into account that larger input numbers produced larger output numbers. She reflected on the results of her inputs to decide on the next input. Other students refined this strategy, making it even more systematic. For example, with the same Number Shifter, but with a desired output of 100, MK and JD entered 10 (getting an output of 50), 20 (getting 70), 30 (getting 90), 40 (getting 110), then 35 (getting 100). These students increased their inputs by intervals of 10 until the output exceeded 100. They recognized that the corresponding increase in outputs was 20, inferring from this that 35 would be the appropriate input for an output of 100.
Guessandcheck is an important strategy that is used frequently in mathematics and should therefore not be discouraged. In fact, Kieran (1992) found that students who take a guessandcheck approach to solving equations seem to have developed a conception of equations on which more sophisticated algebraic thinking can be built.
When using the guessandcheck strategy, however, some students focus on the procedure as a whole, not on its constituent operations. To encourage students to begin the Level 3 decomposition of Number Shifters into the sequences of operations, it is important to ask students to predict the numbers that will appear for each step in a Number Shifter. Making such predictions encourages students to form a mental model of the Number Shifter that can be used to mentally run through the numeric transformations the computer performs in successive steps. Discrepancies between predicted and actual results encourage students to reflect on, analyze, and refine their mental models.
The second strategy that students use on Missing Input problems is to "undo" a procedure by performing its operations in reverse order. This strategy, too, results from a Level 3 decomposition and analysis of the procedure. For example, CR solved Number Shifter 3 as follows.
Number Shifter 3 
Input Number 
? 
Step 1 
multiply the input by 5 

Step 2 
add the result from Step 1 to 15 

Output Number 
80 
CR: You take 80 minus 15 and then divide by 5.
Tchr: Why?
CR: I just kind of know that. If you are trying to see what the first [input] number was, you take the problem that it equals, and like add turns to minus, multiply turns to divide, divide turns to multiply, and minus turns to add.
After students conjecture that the undoing strategy will work, they need to verify it by checking with a Number Shifter on the computer.
Mystery Number Shifters
In Mystery Number Shifter problems, the steps in a Number Shifter are not described, but each is performed in a different spreadsheet cell. Students must describe the operation involved for each step. When solving Mystery Number Shifter problems, students are not permitted to look into the spreadsheet cells to examine the formulas. Instead, they must experiment with a Number Shifter by entering input numbers and observing the results until they figure out the rule for each step. In essence, students are constructing algebraic models for the given inputoutput relationships. There are three types of Mystery Number Shifter problems.
In the first type, the result of each step is displayed. Students must conjecture about the operation that transforms a number in one cell to the number in the next cell. In problem 1, for example, one student might conjecture that adding 4 to an input of 1 yields 5, while another student might conjecture that multiplying 1 by 5 yields 5. 
By trying several inputs, students can test and revise their conjectures.
The second type of Mystery Number Shifter is more difficult because the result of one of the steps has been obscured. But students are shown what operation is used in that cell. In problem 4, for example, students must decide what number must be added to 6 and what operation must be performed on the resulting sum to get a result of 50.
The last type of Mystery Number Shifter is most difficult because the results of one or more steps have been hidden, with no indication of the operations used in those steps. To solve such problems, students must have interiorized a general Number Shifter sequence and they must be able to operate on that sequence by inserting conjectured operations into it.
When solving Mystery Number Shifter problems, some students develop very methodical solution strategies, as is illustrated by CR’s work.
CR: [For step 1, after observing 27 for an input of 17] It could be plus 10. I'll check. Right now I'll do 1 [she enters 1]. It’s 11, so that's correct. Then I'll do 17 [she enters 17 again]. Yeah. CR: [Predicting for the second step] It is times because it is 108. [She adds 27 and 27 to get 54, then 54 and 54 to get 108.] So it is times 4. 

CR: [Entering 1] Yeah, 1 entered is 44, so that [the operation in step 2] is times 4. [She enters 17 again.] Then it [the operation in step 3] could be minus 12. [She enters 1 as the input.] Minus 12 is correct. I'll keep it [the input] at 1. Then 16 times 2 is 32, so I'll divide it by 2 [hypothesizing the operation in step 4]. [She enters 17 to check if 96 divided by 2 is 48.] Half of 80 is 40. Half of 10 is 5. So 45. Half of 6 is 3. So yeah!
Tchr: So what is the rule?
CR: Plus 10, times 4, minus 12, divide by 2.
Tchr: Say that again, I don't understand [attempting to encourage CR to use more precise language].
CR: Add 10 to the input number, then take that and times it by 4, subtract 12 from that. Then divide what you get by 2.
CR's predictions weren't haphazard guesses, but successive conjectures about the steps in the Number Shifter. She entered an input number, examined the result for Step 1, then made a conjecture for the action taken in this step. She tested her conjecture by entering another number and examining the new result from Step 1. She repeated this conjectureandcheck strategy for the second step, the third, and so on.
Concluding Remarks on Number Shifters
The Number Shifter computer environment encourages and supports students' reflection on, analysis of, and discussion of general arithmetic procedures performed on numbers. It helps students develop those mental operations needed to decompose such procedures into their component actions and consider these components as meaningful conceptual objects that can be acted upon and recombined to solve novel problems. In so doing, it engenders students' development of precisely those mental operations that form the foundation of algebraic thought.
Putting Computer Learning Environments in a Technological Context
To fully appreciate the role that the Shape Maker and Number Shifter microworlds can play in mathematics learning, it is important to distinguish between several types of technology use in mathematics education. The first type, technological tools for doing mathematics, includes technology developed outside the field of education for the purpose of doing mathematics more easily and powerfully. In this category, I place devices such as handheld calculators and computers, and software such as spreadsheets, symbolic algebra, and graphing programs. Changes in technological tools for doing mathematics cause changes in school mathematics curricula as educators alter curricula to reflect current best practice in doing mathematics. The second type, technological tools for teaching mathematics, includes technology that has been developed with the specific intention of enhancing students' mathematics learning. In this category, I place educational software packages and instructional computing microworlds. A third type, general technological tools, includes technology whose development has little to do with mathematics or mathematics teaching. An example is webbased communication. Changes in technological tools for teaching mathematics and general technological tools change mathematics curricula as they provide educators new ways to teach particular mathematical ideas.
The Shape Maker and Number Shifter microworlds are technological tools for teaching mathematics. But an added benefit of using them is that they introduce students to technological tools for doing mathematics. That is, unlike ubiquitous computer drillandpractice programs, Shape Makers and Number Shifters actually involve students in doing mathematics with technology in ways that anticipate later use by sophisticated adults. (For instance, I used the Geometer's Sketchpad to model the planting of pine trees in my back yard. I wanted to analyze the effects of tree placement on the view from various windows in the house.) Thus, the Shape Maker and Number Shifter microworlds not only enhance students' learning of important mathematical ideas, they involve students in using powerful technological tools that they can utilize throughout their mathematical careers.
I assume here that the computer environment is easy for students to use. Clumsy interfaces or complicated useprotocols distract students' attention away from fruitful action, reflection, and abstraction.
References
Battista, M. T. (1994). On Greeno’s environmental/model view of conceptual domains: A spatial/geometric perspective. Journal for Research in Mathematics Education, 25, 8694.
Battista, M. T. (1998). SHAPE MAKERS: Developing Geometric Reasoning with The Geometer’s Sketchpad. Berkeley, CA: Key Curriculum Press.
Battista, M. T. & Borrow, C. V. A. (April 1998). Using computer spreadsheets to promote algebraic thinking in the elementary grades. Teaching Children Mathematics, 4 (8), 47078
Clements, D. H. & Battista, M. T. (1992). Geometry and spatial reasoning. In D. Grouws (Ed.), Handbook of Research on Mathematics Teaching and Learning (pp. 420464). New York: NCTM/Macmillan.
Cobb, P. (1994). Where is the mind? Constructivist and sociocultural perspectives on mathematical development. Educational Researcher, 23(7 (October)), 1320.
Greeno, J. G. (1991). Number sense as situated knowing in a conceptual domain. Journal for Research in Mathematics Education, 22, 170218.
JohnsonLaird, P. N. (1983). Mental models: Towards a cognitive science of language, inference, and consciousness. Cambridge, MA: Harvard University Press.
Kieran, C. (1992). The learning and teaching of school algebra. In Handbook of Research on Mathematics Teaching and Learning, edited by Douglas A. Grouws. New York: NCTM/Macmillan.
Steffe, L. P. (1988). Children's construction of number sequences and multiplying schemes. In J. Hiebert & M. Behr (Eds.), Number concepts and operations in the middle grades, (pp. 119140). Reston, VA: National Council of Teachers of Mathematics.
von Glasersfeld, E. (1995). Radical constructivism: A way of knowing and learning. Washington, DC: Falmer Press.