Douglas H. Clements
Try this problem: Write an equation using the variables S and P to represent this statement: "There are six times as many students as professors at this university." Use S for the number of students and P for the number of professors.
Most university students make this mistake: 6S = 1P. They may assume that they can directly order the symbols in the algebraic equation as they appear in the English sentence. Or they may confuse the semantics of the equation. To them, the reversed equation, 6S = P, may mean that a large group of students is associated with a small group of professors.
In contrast, the correct equation, SS = 6P, expresses an active operation being performed on one number, the number of professors, to yield another number, the number of students. It does not describe the sizes of groups literally. It describes an equivalence relation that would occur if one were to make the group of professors six times larger. In one student's words, "If you want to even out the number of students and the number of professors, you'd have to have six times as many professors" (Soloway, Lochhead, and Clement, 1982; 175). This student viewed the equation in a procedural manner -- as an instruction to act. In brief, the incorrect answer is a description of a situation, whereas the correct answer represents a prescription for action.
So would students make fewer errors in an environment that helped them take a more active view of equations? Since a program is a prescription for action, a computer-programming environment was the choice of researchers Soloway, Lochhead, and Clement (1982). They asked students to write either an equation or a computer program to represent such statements. Students got significantly more problems correct writing a program. Similarly, they were more likely to read a program than an equation correctly.
Can computers provide a similar environment for elementary school students? Might they serve to develop algebraic thinking in unique ways?
Variables in Computer Programming
Several current computer environments may help students learn algebraic thinking. Computer programming has been the most widely used and extensively studied. We focus on learning variables in Logo programming, but other tools, such as spreadsheets, may be used in similar ways (Clements 1989).
Why should programming help?
Programming is a prescription for action. Logo, in particular, was designed to help students mathematize familiar processes. Mathematizing makes something more mathematical -- more general, exact, certain, and concise. For example, students might walk and draw shapes and describe their actions. In Logo, they can translate these actions into mathematical code; for example, fd 40 commands the turtle to move forward forty steps, leaving a trail. Such activities help develop rich mathematical ideas.
In the domain of algebra, two important ideas in are variables and functions (Noss 1986). Variables are sometimes introduced in arithmetic tasks, such as __ + 5 = 12. Used in this way, they are placeholders, but they do not lend much mathematical power to students who do not use them often. In contrast, we use variables constantly in Logo. For example, we might write commands that include variables. Instead of entering FD 40, we could enter FD :length, which commands the turtle to move forward the value of the variable named "length." Similarly, repeat 4 [FD :length rt 90] (rt 90 means "turn right 90 degrees" and repeat 4 means to repeat the action inside the brackets four times) commands the turtle to draw a square with sides of that "length." We can not run these commands, however, until we give the variable length a value, as shown here.
make "length 40
repeat 4 [FD :length rt 90]
Note that in Logo, unlike other programming languages, there is a clear distinction between the name of a variable, "length, and the value assigned to it, :length. We could also write a procedure to take length as an input (see fig. 1).
Figure 1: A square defined in a version of Logo Turtle Math
Figure 1 Caption: ©Douglas H. Clements and Julie Sarama Meredith and LCSI). The commands in the command window on the left assign the variable "length" the value of 80 (Clements and Meredith 1994).
In Logo, you also can build and combine functions in a way that models mathematics. For example, the following is a simple function that divides its input, number, in half.
to half :number
If you enter square half 80, half would divide 80 by 2 and "hand off" the result to the square procedure, which would then draw a square with side lengths of 40. What would square half half 80 draw?
So logo can provide an environment in which using variables and functions is natural. It is part of the ongoing activity and authentic play.
In addition to this formal and logical side, variables -- and programming -- involve a symbolic side. Much confusion can arise because in mathematics a letter can stand for a parameter, variable, or specific unknown. Logo's use of "length for the name of the variable and :length for the value of the variable can help students keep these uses straight.
Programming does help...
Noss (1986) investigated whether learning Logo gave ten-year-olds a "conceptual framework" for learning algebra. For example, he asked them to make up rules for situations. One was, "This is a square [a figure was shown]. What could you write for the distance all around it?" Another was, "Peter has some marbles. Jane has some marbles. What could you write for the number of marbles Peter and Jane have altogether?"
Six of the eight students who had studied Logo were able to suggest names for the unknowns and to employ them in a rule that related the unknowns as variables. The two exceptions had not used variables to any extent in their Logo work. Nicola was one who had. She was solving the marble problem.
Nicola: You could use inputs again.
Interviewer: All right, show me how.
N: [Writes] : Peter + :Jane = all the marbles.
I: Can you read it out?
N: Peter plus Jane equals all the marbles. You use those two as the inputs, with as many marbles as you want to...
I: But this isn't a Logo program is it?
N: I know, but if it was...just to say that it's an input.
I: So what does the input actually mean there then?
N: That you can type in however size you want it or how many you want it... How many they want Peter to have and how many they want Jane to have.
These students also built names that stood for a range of numbers, which is counter to the natural tendency of students to interpret letters as specific numbers. Discussing the problem about the distance around a square, Julie suggested you make a "word for the length" of a side, such as LEN, and proposed a rule, LEN X 4. Julie understood that LEN could have many values, although she was probably not aware of a specific range.
In this way, Logo helped students to formalize. The metaphor of typing in a value at the keyboard can be viewed as a means of thinking about a range of numbers while only dealing with one at a time. Logo variables are assigned a specific value at the time the procedure is run, although the name of the input may stand for a large range of possible values. So as students run the procedure repeatedly and enter different numbers, they construct an initial understanding of a range of values for variables.
In normal mathematical usage, such as in y = x,the relationship between xand yis the crucial factor, not the specific examples of the relationship. This gives algebra its power and is also what students find so hard. Experience using Logo provides them a way to think about this abstract idea by linking the assignment of specific values to the variables. For example, students might write a square procedure. As they enter different values, they see that each square is one concrete example of a range of possibilities.
to square: length
repeat 4 [FD: length rt 90]
In summary, learning Logo can help students form intuitive notions about algebraic concepts (Clements and Meredith1993). However, sometimes limitations are evident. For example, students may not fully generalize the variable idea as used in Logo to other situations (Lehrer and Smith 1986). Logo experience can enhance students' understanding of algebraic ideas, but the links that they make between Logo and algebra depend on the nature and extent of their Logo experience. What types of experience should we provide?
Exposing versus laying a foundation
Some types of experience are probably inadequate. Just exposing students to variables may not help them gain a foundation. We need to help students mathematize familiar processes. Writing commands to draw a square mathematizes the actions of walking or drawing a square. For an example dealing with variables, a teacher might ask her students to produce buildings in which number of green blocks is always four greater than the number of red blocks. Then she might challenge them to create a mathematical rule for these buildings. Papert (1980) suggests that when this is fluent, natural, and enjoyable, teachers can lead students to understand the mathematical structures. Both parts are important -- sufficient experience with using variables followed by tasks and guidance to extend that experience.
Noss (1986) asked his student, Stephen, to make a rule for the green-and-red-block buildings. Stephen used Logo-influenced formulations.
IF: REDS = 10 [ MAKE "GREENS 14]
This was only a partial solution. The teacher asked about the 10, and Stephen said it was "just a random number." The need to write a number in the Logo code allowed him to start concretely, but his experience running Logo programs also convinced him that he could change that number. Then on his own he changed his formulation thus:
MAKE "GREENS FOUR MORE THAN "REDS
The teacher asked if he could formalize this rule further. He wrote:
G. = R. + 4
So he used knowledge of Logo as a catalyst in moving from a descriptive, specific-number statement to a generalized algebraic equation. Remembering the students-and-professors problem, note that the "+ 4" is on the correct side of the equation! His teacher's use of Logo experiences and guidance had laid the foundation for his algebraic thinking.
Suggestions for Teaching
Students benefit when teachers guide their experiences with variables. Introduce variables by having students use procedures with inputs. At first, recognize that students might interpret only global changes (Hillel & Samurçay 1985). For example, they might say, "It makes a bigger square." This interpretation is exacerbated by the use of such names as square :n or even square :size, which do not help students figure out exactly whatthe variable represents. It hiss better to provide and encourage students to use such names as :length.side, which are descriptive enough to support students' specific thinking about the variable's meaning. Used in this way and through such discussions, names can help students analyze the procedure and describe what is varying.
When students begin defining procedures with variables, or inputs, they should learn to do the following:
1. Identify what is varying. Students have to find all the commands within the procedure to which they need to assign a variable input, which is not always obvious.
2. Name the variable with a specific identifying name, such as :length.side.
3. Operate on the variable. This rule involves using variables appropriately as inputs to commands. It also includes passing variables to commands and subprocedures with modification (e.g., changing :length.side to :length.side + 5).
Model these steps as you introduce students to procedures with inputs.
Use tasks that illustrate the role of variables using these steps. For example, have students write a procedure to draw the letter L. Discuss with them what is varying and write a procedure that draws an L of any size. Use only one variable to determine the length of both segments.
to L: length
FD: length * .75
This procedure for example, commands the turtle to draw a vertical line segment, return to the bottom, turn right, and then draw a horizontal segment for the "foot" of the L that is 75/100 the length of the vertical segment.
Ask what happens when decimal or negative numbers are given for these inputs. Have students create the largest and smallest L's that they can. Then have them write other variable letter procedures and create designs that combine these procedures.
Sutherland (1989) used these tasks successfully to teach students about variables. She also developed the idea of function to forge links between Logo and traditional algebra. Students made a "mystery" function machines. Their partners had to guess the function and write a similar procedure themselves, which helped them see that changing a literal symbol does not change what the symbol refers to. Also, letting them choose any variable and function name appears to have been quite motivating for them.
Students can develop their intuitive understanding of pattern and structure to the point where they generalize and formalize. Giving students this type of experience in traditional algebra is difficult. We might best use Logo as a context for generalizing and formalizing, rather than attempt to contrive problems in beginning algebra. Then we can help students build links between the use of variable in the two contexts.
Working with computers can help students develop algebraic thinking. They can build on their informal methods, learning to formalize so that they can "talk to" the computer. Interaction with the computer can play a crucial role in their developing an understanding of a general method -- the heart of algebraic thinking. Computers do not work in a vacuum, however. We teachers must select tasks and guide students' experience. We can provide such guidance better if we know specifically how the computer contributes.
Clear, unambiguous syntax
Symbols are open to a variety of interpretations in mathematics. With a computer, however, only one interpretation is possible for each symbol, which is why computers require explicitness. For example, the equation's 6Scan be thought of as "six students." In contrast, the computers "6 * :S" means the operation of multiplication.
Equations on the computer are active. They represent the process of acting on an input and yielding an output. Equations on paper are rarely "run." Running equations on a computer allows students to work with the process -- testing, debugging, and exploring. For example, if students run their L procedure, and the result does not look like they intended, they can reflect on their use of variables and change the relative lengths of the two line segments.
When students try out their ideas, they receive feedback from the computer that mirrors their thinking. For example, if the L appears as , they know that this design is what their Logo program specified. They have a clear direction for changing their variables. They may have reversed the two variables (i.e., used :length * .75 instead of :length) or inverted the scale factor (i.e., used 4/3 instead of 3/4 or 0.75).
Formalizing informal ideas
Computers help students explore, express, and formalize their informal ideas. One ten-year-old stated, "I think that it helps you because you put what you think in and then you can check to see if you are right..." (Sutherland and Rojano 1993; 380). For example, students beginning to write an L procedure might not use variables first but try different numbers, such as the following.
Once students informally decide on a number relationship, such as 0.75, they can go ahead and try out this relationship by
changing the procedure to use variables.
Scaffold problem solving
Early ideas and strategies may be precursors to more sophisticated mathematics when computers provide a thinking tool. One boy wrote a procedure to draw a rectangle. He created a different variable for the length of each of four sides. He gradually saw that he only needed two variables, as the lengths of the opposite sides are equal. He recognized that the variables could represent values rather than specific sides of the rectangle. No teacher intervened during this time; Logo provided the scaffolding by requiring a symbolic representation and by allowing the boy to link the symbols to the figure. The symbols become an aid to generalization.
Linking symbols to pictures
Computers can promote the connection of formal representations to dynamic visual representations. That is, the boy's rectangle procedure was coded in the formal mathematical symbols of Logo, and it commanded the turtle to move dynamically to draw a rectangle.
In some new versions of Logo, students can change their symbols and see the picture change automatically, or actually "pull" the lines of a picture and see the symbols change (fig. 2). The integration of ideas from algebra and geometry is particularly important, and computer tools play a critical role in that integration (NCTM 1989, 125).
Figure 2: Using the Change Shape tool in Turtle Math
In Turtle Math, the Change Shape tool allows the user to change the geometric figure directly and see the effect on the commands reflected immediately. In this way, Turtle Math provides a two-way street -- the user can make or change the symbolic Logo commands and see the geometric shape change automatically and can make or change the geometric shape and see the commands change automatically. This flexibility helps build solid ideas that connect symbolic and graphic representations of geometric ideas.
For example, these commands created this
geometric shape. When the user clicks on the Change
Shape tool, the turtle disappears. To move the
vertical line segment on the right, click anywhere
on the segment and drag it to its new location.
Dragging involves holding the mouse button
down while the mouse is moved.
As one drags the line segment, the corresponding
commands in the Command Center change
dynamically as the line segment is moved.
Release the mouse button when done.
One can then change another line segment or
change a corner.
To change a corner, click on the corner. Drag
the corner to the new location. The
commands change automatically. Release the
mouse button when done.
Note: Because the turtle starts at the home position
in the center of the Drawing window, one cannot
move that corner. Similarly, the user cannot drag the
first line segment that the turtle draws. If the last line
segment it draws connects to the first one at the
home position, one cannot drag that segment either,
so that a closed shape stays closed.
Teaching and Learning Algebraic Thinking with Computers
In summary, some evidence shows that computer tools can provide an "entry" to algebraic thinking. Students perceive the use of formalizations, such as variables, as being natural and useful. Students' ability to generalize their computer-based ideas may depend on the depth of their experience and the instructional support given them in making the abstraction and generalization. Time spent learning these tools is an even better investment if we think of such activity as a medium for expressing mathematics rather than as a tool for learning it.
In this view, technology is less a pedagogical tool and more a mathematical tool (Fey 1991). Graphing tools and spreadsheets should encourage us to reconsider the algebra curriculum just as calculators and computers have made us reconsider the arithmetic curriculum. If we see algebra as primarily the study of functions and their representations, we might use function plotters, curve fitters, and symbolic manipulators. Our students, too, would see algebra as being a source of mathematical models and ask what-if questions, such as "What if problem conditions change?" "What if the goal changes?"
The potential of such a view is revealed in a story of eight-year-old Robby, who had been introduced to variables through extensive work with Logo procedures (Lawler 1985). He learned to focus on systematic changes of a single variable as a useful way of understanding the complex interactions of several variables. At a later date, Robby worked on a paper-cutting puzzle that involves joining two loops perpendicularly, taping them, and cutting around their middles (fig. 3). Surprisingly, this action produces a square. Robby did not stop there, however. He joined and cut three circles. He got two rectangles. Four circles yielded four squares. Five circles produced two identical nonplanar shapes, and so on. Robby said, "Hey! I've got a new theory: the odd-numbered circles make two and the evens all stay together." When asked how he had gotten the idea for his exploration, he explained, "It's just like what we did at Logo with the shape families. I changed one thing, a little at a time" (p. 78). Robby saw the earlier Logo activities as the embodiment of the powerful idea of systematically incrementing a variable. When asked to explain his theory, he constructed a seven circle puzzle, expecting -- and demonstrating -- that this configuration did indeed produce two figures. Hypothesis testing had emerged from a nontheoretical but orderly investigation of interesting effects. Incrementing variables had become a method of determining what is what. Robby demonstrated true algebraic thinking.
Figure 3: Robby's paper-cutting puzzle. The two loops are taped together perpendicularly, then both are cut through their middles.
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Lehrer, Richard, and Paul Smith. Logo Learning: Is More Better? San Francisco: American Educational Research Association, 1986.
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Papert, Seymour. Mindstorms: Children, Computers, and Powerful Ideas. New York: Basic Books, 1980.
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Sutherland, Rosamund. "Providing a Computer Based Framework for Algebraic Thinking. Educational Studies in Mathematics 20 (1989):317-44.
Sutherland, Rosamund, and Teresa Rojano. "A Spreadsheet Approach to Solving Algebra Problems." Journal of Mathematical Behavior 12 (1993):353-383.
Time to prepare this material was partially provided by "An Investigation of the Development of Elementary Children's Geometric Thinking in Computer and Noncomputer Environments," National Science Foundation Research grant number ESI - 8954664. Any opinions, findings, and conclusions or recommendations expressed in this publication are those of the author and do not necessarily reflect the views of the National Science Foundation.
Douglas Clements, Clements@acsu.buffalo. edu, conducts research at the State University at Buffalo, Buffalo, NY 14260, in the areas of computer applications in education, early development of mathematical ideas, and the learning and teaching of geometry. Julie Sarama, firstname.lastname@example.org, teaches at Wayne State University, Detroit, MI 48202. She is interested in children's conceptions of geometry and issues involving teachers' use of technology.
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