Michael T. Battista
Kent State University, Kent, OH 44242
Douglas H. Clements
University of New York at Buffalo, Buffalo, NY 14260
Why is it that students often do not learn what they are taught? On what do they base their thinking? What can we as teachers do to help them construct accurate and robust understandings?
Our research has convinced us that furnishing Logo tools for manipulating embodiments of geometric objects helps students construct more abstract and coherent concepts. It also consistently supports several related constructivist themes. First, contrary to institution, students frequently do not use verbal definitions and rules when they think (Cements and Battista 1990). They use conceptual structures that they have constructed out of pieces of in- and out-of-school experiences. Even when students' verbal definitions or descriptions are correct, these conceptual structures-when faulty-actually rule their thinking. Second, students need personally to manipulate ideas and reflect on them to construct concepts, even after "clear" explanations and demonstrations have been given by the teacher. We shall illustrate our findings with several examples of students'learning.
Constructing Concepts versus Learning Verbal Definitions
In one study we conducted, one group of students had worked with Logo during a free period of the day; the other had not (Clements and Battista 1989). The two groups received identical classroom instruction in geometry. Even after being taught the standard textbook definition, most of the non-Logo students'definitions of angles reflected mainly the influence of common language uses. Typical responses included the following: "It's like a line"; "Like a line that goes on a slant or a corner"; and "Like if you were in the classroom here (to the side) and the chalkboard were here, then that would be a bad angle to see." Students who had worked with Logo displayed different conceptual structures. Their responses emphasized more dynamic and abstract ideas, for example. "An angle is a turn"; "Sort of a line from a turn"; and "It's when two lines meet each other and they come from two different ways."
Students exhibited other clear signs that they were not tied to verbal definitions-even their own. For instance, we asked students to identify figures that were angles. Several Logo students had defined an angle only as a turn, but they nevertheless correctly identified all the angles. Other Logo students correctly defined angles, but also included diagonal line segments. Note that drawing such segments in Logo would require a turn if the turtle started from its "home" heading.
More than twice as many Logo students responded correctly to the question "How many angles in this figure (an 'X')?" Again, however, alternative conceptions emerged in both groups. One Logo student responded "zero" She explained, "When you turn after drawing a straight line, that's an angle." For her, the figure had no angles but would have had two "if you had turned." A non-Logo student had a different notion: "If it were two halves of two triangles, it would be two angles. If it was an 'X', it wouldn't be any angles." Thus, the way a figure was made or viewed was important to these students in deciding whether or not it was an "angle." Neither their textbook nor their teachers had ever mentioned such an idea.
Tasks for Integrating Turns and Angles
In our present work with Logo, we encourage students to reflect on their physical and mental actions. Logo then serves as a transitional device between physical experiences, such as moving and turning, and abstract mathematics. It allows students actively to explore and invent mathematical models and ideas.
For example, you might ask your students the following questions: "What is a turn? Can you turn a whole turn-all the way around? Can you turn a half turn? Turn to the right or do a 'right face'; how much of a whole turn is that? Which way would you be facing if you turned four (or three or two) of these turns? So what portion of a whole turn is it?"
For some students it is beneficial to introduce turns that are smaller than quarter turns with a "floor clock." Focus on the amount of turn from one "clock number" to another. Say, "Point at two o'clock. Turn right two units. Where are you pointing now? How many turns of this type must you make before you are facing in the same direction you started?"
Explain that the Logo turtle uses much smaller units. Show a transparency with 360 radii superimposed on the clock transparency. Explain that 360 turns of this size (degrees) are required to turn all the way around. Ask students how many degrees are needed to turn halfway around; a quarter way around; and on e to six clock units.
Have students play simple Logo games, such as "Hit the Spot," that emphasize estimation of the amount of turn. One partner places a finger or small sticker anywhere on the screen. The other types in as few Logo commands as possible to place the turtle directly beneath the other's marker. A related activity has one partner create a list of commands off the computer. The other partner puts a sticker on the screen a the place her or she believes the turtle will stop. The prediction is checked by typing in the commands. In these and all following work with Logo, encourage students to reflect on their use of reference measures (e.g., "That looks like ninety degrees and about half of ninety degrees more. I'll try one hundred thirty-fine degrees"), thereby laying a foundation for the notion of angle classification.
Furnish activities to help students integrate their understanding of turn with the concept of angle. For example, give students sets of commands (in the pattern FD 50 RT __FD 50) and asked them to predict the path the turtle will draw for various entries of RT. They check their predictions on the computer. Explain that these special paths are called angles. Ask the students to describe angles in path language. Older students can use protractors to measure the angles formed. By examining a table with the headings "amount of turn along path" and "measure of angle formed," they can discover the relationship between these two quantities.
Group Discussions and Personal Constructions
A fifth-grade teacher is sitting at a computer that is connected to a screen that al students can see. She is typing commands given by the students. They are attempting to write a procedure to cause the turtle to maneuver through a maze to each of several points on the screen, returning home after each. The path home is to be the same as the path to the destination. The students have directed the turtle to the first destination, a restaurant.
|Teacher:||How are we going to get the turtle back?|
|Jonathan:||Everything that was a forward is a backward, and everything that was a right is a left, and everything that was a left is a right, probably.|
|Teacher:||So you're telling me what?|
|Sally:||Reverse the stuff.|
|Jonathan:||You reverse, go from the bottom (of the procedure) to the top.|
|Teacher:||What comes first?|
The students now correctly start giving the commands to "undo" the original path.
|Jonathan:||We've got a problem here because here's the restaurant; it goes out of the restaurant BK 10; then right would be up here, so it's going to get screwed up.|
|Robin:||Now its going backward.|
As the students discuss Jonathan's misgiving about the proposed solution, Andy figures out his mistake by standing up and acting out the commands. A consensus is reached about the commands to enter. As the commands are then entered into the computer, the students agree that their proposed solutions is correct.
In this episode, the students are intensely involved with solving a problem. Consistent with the NCTM's curriculum standards (1989), the students are discussing mathematical ideas and making and evaluating conjectures and arguments. Although the teacher is entering the students 'commands into the computer, she does not judge the correctness of the students' ideas. The students judge their ideas by examining the results on the computer screen.
At this point, students are sent to the computers in pairs to continue the activity. Emily and Ryan excitedly attempt to get the turtle to the second destination. As they try the commands for the path they have planned, they express no surprise or disappointment when it does not work. They anticipated that it would not be perfect and that they would have to alter it. Because the computer permits Emily and Ryan to evaluate their conjectures, it enables them to strike out on their own in solving the problem promoting the development of autonomous thinking in mathematics.
Immediately on seeing that their solution is correct, Emily says excitedly, "Let's get him back." Initially, she thinks that the first command in getting the turtle back to its starting point is the reverse of the first command in the procedure for getting the turtle to the destination. But Ryan says that it must be the reverse of the first command in the procedure for getting the turtle to the destination. But Ryan says that it must be the reverse of the last command. They examine the instructions for returning the turtle from its first destination ( written as a class). When asked, "Why does it have to be this [the last] command?" Emily and Ryan simultaneously offer several explanations, al of which have correct elements but do not answer the question. Finally, Emily explains the decision: "We started from her [turtle's starting point] and ended here [draws in the air a curved path that is not a copy of the screen path], so we want to start here this time [indicates destination], so we have to start with the last one [points to the last command in the original procedure]."
Even though the idea of undoing a path had been discussed the previous day in class, Emily and Ryan needed to apply and reflect on it themselves to make the idea personally meaningful. Thus, this episode illustrates the power of individual work on computers and the relationship between this work and class discussions. The individual work seems essential to students' constructions. The actual construction does not take place until the students themselves manipulate the ideas. Logo allows them to do so -- personally to manipulate the ideas and decide on their validity and exact form.
Maze tasks like those we have used in our project can be created in various ways. The easiest might be to draw mazes on overhead transparencies and tape them to the computer screens. We used Logo and "paint" programs to create and save screen mazes.
Be sure to direct students not only to get the turtle to various locations in the screen mazes but to return it along the same path. This task lets them encounter the fundamental idea of "undoing" a sequence of operations. It is the same idea encountered in such problems as "I'm thinking of a number. When I multiply it by three, then subtract five, I get sixteen. What is my number?"
An extension of the original maze problem is possible in versions of Logo that have a "window" commend that allows the turtle to go off the screen. Give students the directions for some destination that is off the screen. Give students the directions for some destination that is off the screen. Ask them to return the turtle to its starting point. You might also give students directions from one off-screen destination to another. Have the students get the turtle to one destination after the other, then return the turtle to its starting point.
Clements, Douglas H., and Michael T. Battista. "Learning of Geometric Concepts in a Logo Environment." Journal for Research in Mathematics Education 20 (1989): 450-67.
"Research into Practice: Constructivist Learning and Teaching." Arithmetic Teacher 38 (September 1990): 34-35.
National Council of Teachers of Mathematics Commission on Standards for School Mathematics. Curriculum and Evaluation Standards for School Mathematics. Reston. Va.: The Council. 1989.
For more information on Logo Geometry, contact: Rachel Auslander, Instructional Technology, Silver Burdett & Ginn, 250 James St., Morristown, NJ 07960. (201) 285-8144.
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