From exercises and tasks to problems and projects:
Unique contributions of computers to innovative mathematics education*
Douglas H. Clements
State University of New York at Buffalo
Department of Learning and Instruction
593 Baldy Hall
Buffalo, NY 14260
Running Head: From exercises and tasks to problems and projects
Those interested in using computers in mathematics education stand at a crossroads. Will computers be used to reinforce existing educational practices or can they catalyze creative innovations? Educators face these turning points with directions from classroom-based research. In this article, this research corpus is reviewed. Implications are drawn from it, centering around the need to move computer use in mathematics education from the domains of exercises and tasks to engagement with problems and projects. Unique contributions of computers to problem- and project-oriented pedagogical approaches are described and unique challenges that must be faced when implementing these approaches discussed.
From exercises and tasks to problems and projects:
Unique contributions of computers to innovative mathematics education
Those interested in using computers in mathematics education stand at a crossroads. Will computers be used to reinforce existing educational practices or can they catalyze creative innovations, following NCTM standards ? Fortunately, educators face these turning points with directions from classroom-based research. To begin, we might look at what students in mathematics classrooms are doing now with computers.
Students Using Computers: The Picture Today
Picture students using computers in their classrooms. What are they doing? Research indicates that they are using computers only occasionally, and usually only to provide "variety," "rewards," "enrichment," or "something for students to do" . In one study, 12% of teachers reported that only those students "who have finished their seat work" actually got to use the computer. Often, slower students never get to use the computer . In the elementary grades, they use mostly drill-and-practice software; their teachers state that their goal for using computers is to increase basic skills rather than solve problems. At the junior and high school levels, students spend at least one-half of their time on computers learning computer-specific skills. They spend less than one-quarter of their time working productively in all academic subjects combined. A minority of their mathematics teachers uses computers for instruction. So, students spend much of their present-day computer time with drill-and-practice exercises and tasksstructured work assigned or done as part of one's classroom duties.
When students use such computer drills or tutorials consistently, they do lead to moderate but statistically significant learning gains, especially in mathematics . This does not mean, of course, that using any software in any way guarantees such gains.
Also, research cant tell you what your goals are. Use of only drill-and-practice or tutorial software is inconsistent with NCTMs standards. While such software can develop skills in various topics, it is a weak contributor to the first four curriculum standards and other NCTM goals (Fig. 1).
1. Mathematics as Problem-solving °
2. Mathematics as Communication °
3. Mathematics as Reasoning °
4. Mathematical Connections -
5+ Achievement (content)
Beliefs about mathematics -
Affect, motivation, autonomy °
Social construction of mathematics °
Mathematical creativity -
Key: Level of contribution
- little or negative
Figure 1. Computer contributions: Drill and tutorial software
Unfortunately, from this point of view, many schools are investing heavily in highly structured "integrated learning systems." These systems automatically load one of an extensive sequence of lessons into each students computer. Evaluations of these systems also show a moderate effect on basic skills . We must, however, question other aspects of ILSs, especially diminished teacher and student control. In too many cases, ILSs represent a triumph of bureaucratic efficiency over students development.
The bottom line is this: What we as a mathematics education community are doing the most is what research and the NCTMs standards say that we should be doing the least (not none, necessarily, just the least).
Paths to Promising Problems
Although exercises and tasks predominate in the software students experience most frequently, many teachers say that computers should be used differently. In the last few years, the proportion of teachers advocating using computers from the "tool" perspective has increased, though they are still in the minority (and they more often teach language arts and English education, not mathematics) .
These teachers want to move away from drill exercises and tutorial tasks. They, along with other teachers and researchers , find that students do develop and use strategies, but that these strategies are adapted to a different goal. Students use "procedure copying" and "knowledge telling" (just say everything you know) strategies. They "do well," but do not become engaged with the subject matter. We all know students who are not lazy, but are coping with a long line of tasks. They are rewarded for completing these tasks with opportunities to do more enjoyable activitiessometimes, a computer game! They develop strategies for efficiently completing these tasks; they see that reflecting on and extending ideas do not help them.
But what do we as teachers do beyond exercises and tasks? We can move to problems and projects. Both are difficult to do well. Both can be easier to do well with computers.
Good mathematical problems
Lets look at problems first. What are the characteristics of investigations that can lead to good mathematics problems for students? Good problems :
are meaningful to the students
stimulate curiosity about a mathematical or non-mathematical domain, not just an answer
engage knowledge that students already have, about mathematics or about the world, but challenges them to think harder or differently about what they know
encourage students to devise solutions
invite students to make decisions
lead to mathematical theories about a) how the real world works or b) how mathematical relationships work
open discussion to multiple ideas and participants; there is not a single correct response or only one thing to say
are amenable to continuing investigation, and generation of new problems and questions
How computers help with a problem-centered approach
Computers, especially with their visual displays, can be a source of meaningful problems with a variety of solution strategies and solutions. Students can explore these various solution paths individually and in small groups, making decisions and receiving feedback about their ideas and strategies. Access to computers helps implement this less structured problem-centered approach, which positively affects the number and kinds of decisions students had to make and results in more active and highly motivated learners . The social nature of educational computing facilitates discussion about the problems . Most straightforward, computers allow a de-emphasis on those aspects of mathematical work that can be (and, in the world outside of school, usually is) done by machines and an increased emphasis on conceptual thinking and planning .
Building a better manipulative
One type of program is the computer manipulative. In Shapes, for example, students can choose to manipulative base ten blocks. Students concrete actions are immediately represented symbolically in two ways (see Fig. 2). Adding blocks to the block set changes the display so that it always reflects the number represented by the blocks. Students can change any block to be "one" (Fig. 3). Students can easily model operations (Fig. 4).
Figure 2. A Number Blocks screen; the "odometer" at the bottom shows the symbolic representation of the number of blocks.
Figure 3. Any block can be set to "one."
Figure 4. Addition is facilitated by separate groups of number blocks, each with their own odometer (a). If the separation is erased, one odometer shows the number in the combined group. For multiplication, students create an array and then fill it with blocks (b).
These computer blocks are not physically concrete. However, no base-ten blocks "contain" place value ideas; rather, students construct these ideas while thinking about their actions on the blocks . Actual base-ten blocks can be so clumsy that the actions become disconnected. The computer blocks can be more mentally manageable .
Research supports the notion that manipulativeson or off computerare concrete and meaningful if they make sense to the student . Sidewalk concrete is strong because it combines many separate particles in a connected physical structure (concrete means "to grow together"). A concrete idea is strong because it combines many ideas in a connected knowledge structurewe call this Integrated-Concrete knowledge . The computer environment helps links the blocks to the symbols, connecting these ideas and helping students develop Integrated-Concrete knowledge.
In addition, students can break computer base-ten blocks into ones, or glue ones together to form tens. Such actions are more in consistent with the mental actions that we want students to learn .
Graphing: Changing, rearranging, and connecting representations
Another aspect of the flexibility provided by many computer manipulatives is the ability to change and rearrange information. Most spreadsheet and data base programs will sort and reorder the data in different ways. Such flexibility can help open up new possibilities for exploring problems.
For example, Figure 5 shows a "cats" data base in the Tabletop program. Students can add, change, or sort the data in tabular form. When students move to the tabletop, they can graphically represent these data in a variety of ways. For example, in a Venn diagram form, they can show male cats. The cats move dynamically to their proper place (Fig. 6). They may wish to inquire whether more female or more male cats are heavy. Students often make another loop for female cats, then another for cats that weigh more than 11 pounds (Fig. 7). Again, the cats move dynamically into their correct position. Other representations are possible (Fig. 8).
Figure 5. The "cats" data base in the Tabletop program.
Figure 6. After moving to the tabletop, students have selected a loop and specified that it include all those cats whose sex is male. The cats move automatically into an appropriate position.
Figure 7. Adding two more loops, students can determine whether there are more heavy male or female cats.
Figure 8. Another way to view these cats is to select an "axes" representation. The cats can now be moved left and right, but not up or down, as their vertical position indicates their weight.
Chris Hancock's research shows that Tabletop offers a flexible, exploratory, approach to data analysis that encourages students to make sense of data in their own way. A graph is not a result, but one stage in a continuing process of exploration and analysis. Students can construct many representations and comparisons in a short time, making qualitative differences in their educational experience. Tabletop extends students' range; for example, they can study large composite data sets from multiple classrooms. Finally, programs such as Tabletop help students connect different types of tabular and graphic representationsagain helping students build Integrated-Concrete knowledge.
Geometric construction programs: Building dynamic connections
Another class of programs that dynamically link multiple representations includes geometric construction programs, such as The Geometer's Sketchpad . Such explorations can encourage students to make conjectures and provide insight into the reasons those conjectures might be true or false. For instance, students can construct a polygon with rays, and measure the exterior angles (Fig. 9). Then they can move parts of the polygon and watch what changes dynamically (Fig. 10). Some measures change, but the sum does not. The demonstration is convincing because the size and shape of the polygon can be changed at will. Students can repeat this experiment with other polygons, and make a conjecture. To explore why their conjecture might be true, they can use the dilate tool to "shrink" their polygon. As it approaches a point, sum of the exterior angles can be considered in a new light (Fig. 11).
Figure 9. A Sketchpad pentagon shows the measures of the exterior angles and the sum of these angles.
Figure 10. The points can be moved with the mouse, and the measures are changed automatically.
Figure 11. Using the dilation tool to shrink the pentagon begins to reveal the relationship among the exterior angles.
Feedback helps students in many ways. Betty Kantrowitz, a teacher who received the Presidential Award for Excellence in Mathematics Teaching, uses a similar geometric construction program. She states, "Kids believe the machine. I can tell them a thousand times that they cant construct a particular circle because they dont have the correct properties, and they wont believe me. Theyll insist that they can. But as soon as Geo-Explorer displays the message Cant Construct This Object, then they realize, Aha, I must be missing something" Research indicates that students using such tools score on higher-level and application questions .
Geometric construction programs also allow us to store more than static configurations. Once we finish a series of actions, its often difficult to reflect on them. But computers have the power to record and replay sequences of our actions on manipulatives. We can record our commands and later replay, change, and view them. This encourages mathematical exploration.
Computer programming: Logo
Recording and replaying students' actions is most completely realized in computer programming. Programming in languages such as Logo also focuses on the connections between the concrete and the symbolic.
An episode in one fifth-grade class serves as an example. Students had written Logo procedures to draw rectangles, then had abstracted these to write a "general" rectangle procedureone that takes two inputs for the lengths of opposite sides. Their teacher gave them a group of figures (Fig. 12) and asked them to decide whether they could draw each using their procedure and an initial turn if needed.
Figure 12. Am I a Rectangle? Students were asked to determine if they could draw each figure with their Logo rectangle procedure.
In response to the parallelogram, #7, Jonathan said, "Maybe" (he could draw it with the rectangle procedure). He estimated the initial turn, then the side lengths. After typing in his commands, he got a rectangle. He held the sheet up right next to the screen. "No."
Teacher: Could you use different inputs, or is it just impossible?
Jonathan: Maybe if you used different inputs.
(Note in the next section evidence of a transition moment.)
Jonathan: (Types in the initial turn and stares at the picture of the parallelogram; pauses.): No, you cant. (Pause.)
Jonathan: Because the lines are slanted, instead of a rectangle going like that. (traces).
Teacher: Yes, but this ones slanted (indicating #4, the oblique rectangle that Jonathan had successfully drawn).
Jonathan: Yeah, but the lines are slanted. This ones still in the size (shape) of a rectangle. This one (parallelogram) the things slanted. This thing aint slanted. It looks slanted, but if you put it back it wouldnt be slanted. Anyway you move this, it wouldnt be a rectangle.
Jonathan: (Shaking his head): "So, theres no way."
The Logo environment was important to Jonathan in figuring this out. He didnt even complete his second attempt. Rather, after making the initial turn and trying to decide on the inputs, he recognized somehow that the relationship between adjacent sides was not consonant with the implicit definition of a rectangle in the Logo procedure. The first attempt with Logo and his "running through the procedure in his head" contributed to his emerging sense of certainty .
Misunderstandings of programming. When I first came to my present university, I asked the person running the educational computer lab, "What versions of Logo do you have?"
"We dont have Logo," he replied, "I just threw it out." Retaining my proper academic composure, I ask why.
"Now that we have Dazzle Draw, what would you want to use Logo for?" But, of course, the point is not the drawing, its thinking about doing the drawing . A lot more thought has to go into deciding what should be "easy" and what should remain a struggle, in the positive sense of the word. Logo can be difficult, but worth it. As one third grade boy put it, "Logo is very hard but it had to be done. I liked doing it" .
Others believe, conversely, that Logo is for babies. In their Connected Geometry project, however, the Educational Development Center is using Logo and The Geometers SketchPad to present deep and interesting problems to high school students.
Finally, some say that we dont need programming anymore. The author of another geometric construction program demanded a direct manipulation interface only there was no need for programming! In later versions, however, the program included recorded actions then scripts then modifiable scripts. Unfortunately, the scripting language is not now as well thought out as formal computer programming languages. It is ironic that at the same time that many in computer education are downplaying programming, programming languages are showing up everywhere, in programs from word processors to the computer operating systems. The "acquisition of computer programming skills" remains important .
So, we need programming. The alternatives are less viable. The latest version of one popular program, SimLife, has 350 menu options and control options and more than 200 pages of documentation. Users want to do many things, but few want to do everything that is available . If, instead, students could extend an application with programming, there would be little need for all these extras. If students have a set of mathematical primitives as building blocks to investigate a variety of topics, they do not need the latest update. Eisenberg states that the development of complexity within the application should grow from the students' expanding ideas, realized through programming, rather than from often profit-driven additions of features.
When you program, you are constructing mathematical processes and objects. That programming uses a formal language should be welcomed, not eschewed. It offers generality and power. This is why Bulgarian educators chose to use an enhanced Logo to study secondary geometry. The tool is, in their eyes, more powerful, extensible, and creative. Students can build primitives (e.g., for the area of any quadrilateral) that are often "black boxes" in construction programs. They can also ask questions (e.g., will the same phenomenon be true of pentagons or octagons) that can not be raised within more constrained construction programs. Students build a toolbox of algorithms and geometric knowledge simultaneously.
They argue that the use of a language is central. "At the heart of this system is the philosophy that in order to do mathematics students must have LANGUAGE to express their mathematical ideas and that the notion of DEFINITION is so central to mathematics that it cannot be ignored in mathematical education" (p. 1). Also, "the most important and fundamental mathematical activity is dealing with notionsmainly composing and decomposing of notionswhich definitely needs a language" . When mathematical objects and processes are so described, they can be saved, studied, revised, generalized, utilized, and communicated. For example, if you have procedure to do greatest command divisor for integers, you can just change the MOD and the QUOTIENT procedures and it then works with polynomials.
This is not to argue against direct manipulation. In contrast, such manipulation and programming languages complement each other. Thisalong with the fact that programming environments themselves are developing and changingcan be illustrated with two new versions of Logo.
New versions of Logo. The first version has been developed by myself and my colleagues. We have tried to connect NCTMs standards and research implications to our design of this tool . For example, research supports the idea that computers can help students build mathematical knowledge out of their personal, visual knowledge, such as knowledge of walking and moving. However, it also warns that children have much difficulty with certain facets of turtle geometry work, such as measurements.
To help, Turtle Math features a turtle that turns slowly and, optionally, a ray turns with it, showing the change in heading throughout the turn. Turtle Math also provides measurement tools; for example, an on-screen protractor measures turns and angles (Fig. 13). In addition, students can use tools to label the lengths of all line segments and the measure of all turns (Fig. 14).
Figure 13. Turtle Maths on-screen protractor. One arrowhead shows the turtles heading. The other follows the cursor, which students move with the mouse. When they click the mouse, this arrowhead "freezes" and the computer displays a turn command.
Another way Turtle Math supports the growth of mathematics from the visual lies in its overall structure, which helps connect the symbolic to the graphic figure. Students enter commands in "immediate mode" in a Command window (Fig. 15). This window is long enough that students can view of step through all the commands that correspond to a figure. Any change to these commands is reflected automatically in the drawing. For example, if you change the fd 40 to fd 60, the drawing automatically shows that change (Fig. 16). This makes Turtle Math easy to use and it helps students make important connections between the mathematical commands and the turtles drawing. One student said, "You should change those three commands [gesturing at rt 90 rt 20 rt 10] to just rt 120. Because 90 + 20 + 10 is 120." A tool (the first icon on the left) copies these into the Teach window, applies a student-supplied name, and so defines the procedure.
One limitation of traditional versions of Logo has been in the lack of two-way connection between visual and symbolic representations. Students create or modify symbolic code to produce visual drawings, but not the reverse. Turtle Math provides a "Draw Commands" tool that allows the student to use the mouse to turn and move the turtle, with corresponding Logo commands created automatically (Fig. 17).
Figure 14. When you click on Turtle Maths Label Lines and Label Turns tools, the turtle shows the measures on the drawn figure.
Figure 15. Turtle Maths main windows.
Figure 16. Any change that is made to the commands, such as changing the fd 40 to fd 60, is automatically reflected in the drawing.
The cursor changes to a cross hairs, the turtle continually turns to face it, and the corresponding turn command is dynamically updated in the Command Center.
The turtle is then dragged forward or back and the corresponding fd or bk command is dynamically updated in the Command window.
Figure 17. Turtle Math provides a "Draw Commands" tool that allows students to "draw" with the mouse and automatically create corresponding Logo commands.
Another tool that allows students to build a two-way connection is the Change Shape tool. An activity sheet on parallelograms that introduces students to that tool is shown in Fig. 18. A similar tool allows similarity transformations, with the commands dynamically changing; students quickly notice that the inputs to the turn commands do not change, while the inputs to the forward commands do change, and, of course, in a certain way.
Figure 18. The parallelograms activity uses the Change Shape tool to promote students reflection.
Turtle Math provides other tools, to facilitate editing and reflection (the erasing tools and the step tool, which walks through a series of commands one-by-one), use of geometric motions, and work with coordinates and grids (Fig. 19).
Figure 19. To plan their flag (a), Ian, a child labeled as hyperactive, and his partner had to analyze the relationships between intrinsic turtle geometry (e.g., the rectangle command, with two inputs that represent two different measures) and extrinsic geometry (e.g., the coordinate jumpto commands, which have two numbers in a list that represent a single entity, a point). Their program ran perfectly the first time (b).
Our research on Turtle Math is in the initial stages. Indications are that the research-based features of the environment support and encourage students' mathematical investigations.
MicroWorlds Math Links™ (also from LCSI) is a set of electronic tools that can be manipulated both directly and through Logo programming. For instance, Fig. 20 shows the circular geoboard. You can click on the pegs to connect them, creating polygons. You can also write use Logo commands to drawn them, as in Fig. 21. What repeat command would make a regular hexagon? You can change the number of pegs; in Fig. 22, a repeat 12 [jump 5] command yields a path that touches every peg. Will repeat 12 [jump 3]? If you entered newboard 13, could you use repeat 13 [jump ___ ] and somehow not touch every peg? What jump number would you need? Consider again the 36 peg geoboard. Can you make a regular pentagon? Why not? What newboard ___ command could you use so that you then could make a regular pentagon? What one geoboard could you use to allow making a regular polygon with 3, 4, 5, and 6 sides?
Figure 20. The circular geoboard, one electronic tool from the MicroWorlds Math Links program. If you click on another peg, a "band" or line segment will be drawn.
Figure 21. A simple Logo instruction draws an equilateral triangle.
Figure 22. The command "newboard 12" creates a 12-peg geoboard. The command repeat 12 [jump 5]" touches each peg. What jump numbers will and will not do the same?
Research on Logo. Although I cant do it justice in this brief space, research on Logo has also been misunderstood. The following is a summary of the findings . Used appropriately, computer programming has been shown to help students:
develop higher levels of mathematical thinking
gain "entry" to the use of the powerful tool of algebra
develop concepts of ratio and proportion
form more generalized and abstract views of mathematical objects
develop problem-solving abilities, especially particular skills (e.g., problem decomposition, systematic trial and error) and higher-level metacognitive abilities
enhance the social interaction patterns
use the turtle metaphor to understand and integrate various topics in school mathematics
This may be why entire school systems in Australia, Brazil, Costa Rica, Greece, the UK and other countries, as well as some exemplary US schools, are emphasizing Logo and related programs (Fig. 23). For example, in one Australian school, each student uses a personal laptop computer and Logo every day. Students use such tools flexibly, across a broad range of content, to promote active, meaningful, deep, learningstrikingly consistent with NCTM guidelines. Such use of computers can make substantially greater contributions to important goals of mathematics education (see Fig. 24).
Figure 23. Holiday cards created in Logo by Costa Rica children and disseminated commercially to support education in that country.
Standard Drill Solving
1. Mathematics as Problem-solving °l
2. Mathematics as Communication °m
3. Mathematics as Reasoning °l
4. Mathematical Connections -m
5+ Achievement (content)m l
Beliefs about mathematics -l
Affect, motivation, autonomy °l
Social construction of mathematics °m
Mathematical creativity -m
Key: Level of contribution
- little or negative
Figure 24. Computer contributions: Problem-solving software
Two boys were supposed to build on what they had learned about drawing rectangles with Logo. They started drawing a man. They made a rectangle or two, argued a bit, and frittered away their time. One looked at a large rectangle. "Hey! Thats Bruce Smith!" "Yes!" answered his partner. For the next week they continued making what was to me the same drawingbut with an affective and cognitive engagement indicating that it was not so for them. It was their work, their idea, their project.
A project is an extensive undertaking requiring concerted effort. Projects can be engaging, but as teachers we can find it difficult to balance childrens interests with our educational goals. Also daunting are demands regarding materials, orchestration, and management. Computers ameliorate these difficulties in project work.
Shared knowledge building
Conscious, cooperative development of shared knowledge is the focus of the CSILE (Computer Supported Intentional Learning Environments) project. Active building of knowledge, or constructivism, is becoming increasing accepted by the educational community. Few classrooms, however, directly encourage students to view what they do in school as the construction of knowledge . CSILEs goal is to do so.
CSILE is a networked hypermedia system in which students create the entire database. They produce notestext, graphics, or bothand enter them into a communal database. These notes range from answering a specific question ("What did you find out in the heating and cooling experiment?") to making a creative drawing unrelated to any class assignment. Students write notes on their goals for learning, questions they would like to have answered, and discoveries they have made. Notes are available to all students (unless the author specifies otherwise), and readers are encouraged to engage the author in debate and suggest changes and refinements. Students store their notes with key words, and search for others notes according to these words, to complete projects or just to browse.
Students share information and strategies. One student invented a "making animals" routine that consisted of choosing geometric shapes and forming them into an animal shape, then deleting unnecessary parts of the shapes. The communal database is a particularly powerful medium for the dissemination of such inventions .
Teachers from first grade to high school are trying out CSILE. Research results indicate that such use results in increases in amounts of writing, depth of explanation, knowledge quality, question asking, solving complex mathematical word problems, and standardized achievement test scores. Also increased is student collaboration. Control groups possessed "shallow" conceptions of learning, seeing it as a matter of paying attention, doing assigned, work, and memorizing. CSILE classrooms exhibited a "deep" conception, seeing learning as dependent on thinking and understanding.
Bringing scientific processes into the classroom usually has implied a focus on individual students; CSILE focuses on the classroom functioning like a scientific community . One class was working on the inheritance of characteristics. A boy entered a telling one-line note: "Mendel worked on Karen's question." This puts the usual top-down delivery on its head. Students are part of an ongoing process of building knowledge.
What is the unique role of the computer in these classes? Perhaps that it opens up a new channel for communication, one that is not mediated through the teacher . Research comparing CSILE to classrooms with similar goals but without CSILE shows that students in the computer classrooms talk more about high-level goals. Control students get lost in the lower-level problems and lose sight of their goal. These groups, working face-to-face, varied widely in the extent to which members contributed; some hardly contributed. Competitiveness and other social factors led to rejection of good ideas. With CSILE, students participated at virtually equal levels. Finally, CSILE offers record of past goal states and solution ideas. Students review their previous work and moved on from there, where control students are more likely to flounder or start over.
Projects for knowledge design
"What is a fraction?" Fourth grader Debbie drew a circle divided into two, shaded the right half, and stated, "That's a fraction." When the investigator asked her about the other half, she replied, "No. It's not a fraction. Its nothing." Debbies fraction knowledge was limited, and disconnected from her everyday world .
Fortunately, that was pretest time. With the rest of her class, Debbie designed software to teach fractions to third graders. They had the freedom and the responsibility to create their own designs and to teach themselves about fractions and representing fractions to other children. Debbie initially was not emotionally involved in the project. But over several months, in designing her Logo software, she came to appreciate that there are many ways to represent fractions. It happened when she figured out that she could use the animated decorations for her poems to make representations of fractions more aesthetic. Other children asked her about the effects. She began to see fractions everywhere and to appreciate that people "impose" fractions on anything they wish. When asked at the end, "What is a fraction?" Debbie insightfully replied, "Fractions can be put on anything!" She used that as a theme in her software .
Figure 25. Debbie illustrates that fractions can be "put on anything!"
Harel compared three groups. The first comparison group received the same amount of exposure to Logo programming. Their work was integrated with various curriculum topics, but the problems were short and assigned by the teacher. The second comparison group received Logo once a week in a computer literacy course.
The design group showed greater mastery of both Logo and fractions than the two comparison groups. Why? The design childrens projects revealed a rich variety of ways to represent fractions. They divided a circle into four regions and flashed two on and off to show two-fourths with the written text "two-fourths"; showed an animated clock; showed a one-dollar bill with four quarters underneath, two of which moved and stopped near the written words "two-fourths of one dollar" (Fig. 26a). These led to ostensibly simple, but phenomenologically deep insights for the students. Sharifa represented fractions by using a clock, teaching users that, "Half an hour is a half of one hour!" (Fig. 26b). Sherifas ability to see the analogy between the clock and the turtle is one example of direct contributions of the computer. One might argue that this could be accomplished with other media, but the research showed that computers are critical in allowing students to create different representations of ideas, modify them over months, and share procedures and representations with each other.
Figure 26. Students created a variety of ways to represent fractions.
Repeatedly we hear similar stories. In a middle school program, immediate access to computers supported implementation of larger, more complex, and partially-specified projects. This increased students engagement, motivation, autonomy, and empowerment. Benefits varied insofar as students controlled or influenced the content, process, product, and evaluation of their own learning. Lego-Logo was especially high in offering this type of control .
The computers role was helping minimize problems managing more complex tasks. High access to computers helped "absorb" more student variation before the classroom management problems became overwhelming. Teachers found that it was easier to scaffold students project work with computers.
The physicist Richard Feynmann was asked "How do you learn problem solving?" He said that the first thing you have to do is listen to what the problem is trying to tell you, before you tell the problem all the mathematics you know. Technology allows us to listen to problems, play with them, and listen to them in different ways. And play is of the essence. Jerome Bruner has shown that students encouraged to play with materials first were far more creative in solving problems with those materials. He suggests that play loosens the coupling between ends and means and allows for exploration of different combinations. In work, we hold the end steady and vary the means until we achieve our end. But in play, we can also do the opposite. For Bruner, unprompted metacognition is nothing more or less than internalized play . Bruners research indicates that play is richest when the material provided has a clear-cut variable means-end structure, has some constraints, and yields feedback that students can interpret on their own. Interestingly, these descriptionsoriginally about puzzles and building blocksfit certain computer environments that we have discussed quite well.
Logo may make it possible to play with certain mathematical ideas creatively earlier than is currently believed . Such activity engenders both cognitive and affective involvement with mathematics. In the words of one student, "I've thought about circles in ways I've never considered before" .
In such environments, pupils experience, probably for the first time, the power to create and be in control of solving their own mathematical problems. Most of their teachers believed that this was enough reason to justify Logo programming as an activity in the mathematics classroom. Students agree: "If we didn't have the computer, what could we use to say that the electricity should flow and then it should stop? Where would we put our knowledge? We can't just leave it in our heads. We know it, we think it, but our programs would stay in our heads (Sasha, 7)" .
This stands in contrast to structured CAI activities that, due to their restrictions on solution strategies, extrinsic rewards, built-in evaluations, and lack of playfulness, may negatively affect mathematical creativity (Clements, forthcoming DC: Cite reaction paper to others work ). Figure 27 summaries the computers contribution to mathematics projects.
Standard Drill Solving Projects
1. Mathematics as Problem-solving °l m
2. Mathematics as Communication °m l
3. Mathematics as Reasoning °l l
4. Mathematical Connections -m l
5+ Achievement (content)m l m
Beliefs about mathematics -l l
Affect, motivation, autonomy °l l
Social construction of mathematics °m l
Mathematical creativity -m l
Key: Level of contribution
- little or negative
Figure 27. Computer contributions: Project software
Unique Contributions of Computers
Provide environment to test ideas and feedback
Computer programs such as those suggested here provide an environment in which students can try out their ideas and simultaneously receive feedback on those ideas. As another example, Hoyles and Noss found that on a geometric proportion task, students used additive strategies on paper-and-pencil tasks, but none adopted such strategies on the related Logo tasks. The authors trace this catalytic effect of Logo to the interaction between students formalization and computer feedback. Students formalize proportional relationships algebraically in the form of Logo programs. They receive feedback regarding their mathematical intuitions through the geometric effects. On pencil-and-paper, the first of these, while present, is less salient; the second is absent. For example, students transformed a strategy of addition with adjusted increments to a strategy of adding a fraction of a variable to that variable when they moved to the computer.
Provide mirrors to mathematical thinking
These features combine to make the computer environments mirrors of students geometric thinking. Researchers and teachers consistently report that in such contexts students cannot "hide" what they do not understand. Difficulties and misconceptions that are easily hidden by traditional approaches emerge. This leads to some frustration for both teachers and students, but also to greater development of mathematics abilities . For teachers willing to work with and listen to students, such environments provide a fruitful setting for learning to take the students perspective on analyzing mathematical situations and for discovering previously unsuspected abilities for students to construct sophisticated ideas if given the proper tools, time, and teaching.
Because students may test the ideas for themselves and receive feedback, computers can aid students in moving from naive to empirical to logical thinking and encourage students to make and test conjectures, rather than relying strictly on authority. In addition, the environments appear conducive not only to posing problems, but to wondering and to playing with ideas. Research suggests that computer environments such as considered here can enable "teaching children to be mathematicians vs. teaching about mathematics" insofar as making and testing conjectures, posing problems, and playful engagement with ideas is considered the role of the mathematician.
Link the general and the specific
Computer environments encourage the manipulation of specific screen objects in ways that assist students in viewing them as mathematical objects and as representatives of a class of objects. Such activities develop students ability to reflect on the properties of the class of objects and to think in a more general and abstract manner. Returning to Hoyles and Noss example, students using computers abandoned additive thinking because the computer provided a way to think about the general within the specific. The paper-and-pencil mode activated a fixed answer to a fixed question. The computer allowed exploration as an antidote to mental "blocks" and activated a dynamic answer. Posing the task of writing a superprocedure that would handle all cases promoted additional development. Computer environments encourage more generalized and abstract views of mathematical objects .
Numerical computation provided by spreadsheets, Logo, and other programs can help students build intuitive understanding of variables, relationships among variables, and functions. This helps students make the transition from arithmetic to algebraic reasoning .
Link symbolic to the visual
Computers can promote the connection of formal representations with dynamic visual representations, supporting the construction of mathematical strategies and ideas out of initial intuitions and visual approaches. The integration of ideas from algebra and geometry is particularly important and computer tools play a critical role in that integration .
Facilitate natural and mathematical language
Fortunately, students are not surprised that the computer does not understand natural language, so they have to formalize their ideas to communicate them. Students formalize about fives times as often using computers as they do using paper . As example from that study, students struggled to express the number pattern that they explored on spreadsheets. They used phrases such as "this cell equals the next one plus 2; and then that result plus this cell plus 3 equals this." Their use of the structure of the spreadsheets rows and columns, and their incorporation of formulas in the cells of the spreadsheet, helped them more formally express the generalized pattern they invented.
The need for complete and abstract explication in many computer environments accounts in part for students creation of richer geometric ideas . That is, in Logo students have to specify steps to a noninterpretive agent. In contrast, when "intuition is translated into a program it becomes more obtrusive and more accessible to reflection" . In one study, we attempted to help a group of students using noncomputer manipulatives become aware of these motions. However, students described these physical motions to other students who understood the task. In contrast, students using the computer specified motions to the computer, which does not "already understand." The specification had to be thorough and detailed. Students observed the results of, reflected on, and corrected commands. This led to more discussion of the motions themselves, rather than just the shapes .
Emphasize concepts and problem solving
Computers help de-emphasize routine aspects of mathematical work and emphasize conceptual thinking and problem solving. This can lead students to view mathematics as a source of models for real-world phenomena, and so to join teachers in making sense of mathematics and mathematical models collaboratively.
Scaffold problem solving
Computer environments may be unique in providing problem solving scaffolding that allows students to build on their initial intuitive visual approaches and construct more analytic approaches. In this way, early ideas and strategies may be precursors of more sophisticated mathematics. One boy wrote a procedure to draw a rectangle. He created a different variable for the length of each of the four sides. He gradually saw that he only needed two variables, as the lengths of the opposite sides are equal. In this way, he recognized that the variables could represent values rather than specific sides of the rectangle. There was no teacher intervention during this time; Logo provided the scaffolding by requiring a symbolic representation and by allowing the boy to link the symbols to the figure.
Serve as effective manipulatives
Besides the advantages already discussed, computer manipulatives can be uniquely effective in their roles of:
changing arrangement or representation,
storing and later retrieving configurations,
recording and replaying students' actions, and
changing the nature of the manipulative .
Computer environments can also facilitate larger, longer, mathematical investigations in that they help students:
communicate with one another, sometimes without teacher mediation, and with remote data bases and people;
focus on high-level goals;
participate at equal levels;
participate at equal levels; and
view, copy, modify, and use others work; and
actively explore and "play" with more activities that have scientific and mathematical content .
Encourage positive social interaction
One of the surprising research results is that the largest, most consistent benefits of the computer are in the social and emotional domain. For example:
Students cooperate more in computer environments and they cooperate on learning.
They also disagree morebut they disagree about ideas, and they are more likely to successfully resolve these disagreements, synthesizing their ideas .
The reasons for this include all the contributions already discussed, as well as the shared display and single keyboard
In conclusion, the more we move to problems and projects, the more computers make a unique contribution! And the more we see that education is revitalized
Unique Challenges of Computers:
Steering Around the Potholes
An early concern was that computers would replace teachers. Now most of us see that technology doesn't take work off our shoulders; if anything, it adds to the effort we must expend. Several evaluation projects, while showing that computers can be effective educational aids overall, also show they can impose stressful workloads, especially upon first-year teachers and principals .
Though it seems obvious, many still underestimate the importance of having adequate computer resources. Of five main factorsleadership, staff development, financial support, teachers' attitudes, and availability of computersonly the availability of computers made a difference in whether teachers believe they currently could incorporate computer technology into their classrooms . (What is not so obvious is that accepting computers previously owned by businesses may be unwise, a topic to which we shall return.)
Related is the logistics of timely installation and maintenance of the computers and software. On-site expertise is helpful in this regard .
Teachers need to create a management plan based on the number of computers available. For example,
If you only have a single computer, use it as an "electronic blackboard".
With 1-3 computers, have students rotate using the computers.
With 5-8 computers in the classroom, try having half of the students, working in pairs, use them at a time.
With a computer laboratory, involve the whole class in computer activities at the same time.
Computers should be available in the classroom rather than only in computer lab periods . I cannot emphasize too much: To use computers effectively, every mathematics classroom should have at least one computer and a large screen display available at all times, no matter what other arrangements are made.
Seriously consider notebook computers. Projects using Logo on notebooks have been very successful: " the culture of that group of children supported the formalising of the informal (the writing of Logo procedures in this case) ". The notebooks helped children work together, help each other, and share ideas. Children work beside their computers, with them shut or open, without feeling that the computer time is not being well spent.
Finally, we mathematics teachers need able machines, and not hand-me-downs. When the most computer-knowledgeable teachers are beginning to view the primary importance of computers as a tool of intellect-enhancement rather than for basic skill learning, software designers are constrained to produce software on older machines. Further, getting older machines from other sources, such as business, may be imprudent. The net value of corporate equipment donation may be negative, especially after one accounts for upgrading, upkeep, and the loss of public revenue attributable to tax deductions claimed by the donor . Equity concerns, including gender, ethnicity, and achievement differences, though not the topic of this paper, are critical here and elsewhere .
Despite the palpable importance of hardware, content and pedagogy of the software should receive more emphasis. Availability is requisite, but most attention should be given to achieving goals of educational reform through computer integration .
There are three additional general research implications for the selection and use of software. First, use an appropriate combination of off- and on-computer activities. Each has its own advantages; helping students seeing the connection between the two is perhaps the most significant. Become familiar with all software ahead of time.
Second, consider technology less as a pedagogical tool and more as a mathematical tool . 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 symbol manipulators as in the Computer-Intensive Algebra project. Our students too would see algebra as a source of mathematical models and ask "What if ?" questions: What if problem conditions change? What if the goal changes?
Third, consider technology as a thinking tool. Only a few computer-using teachers have student use any one kind of program more than five times during the year. For example, only 14% of mathematics teachers who use computer had students use tutorial CAI programs more than five times . The programs students use should become tools for thinking mathematically. We need to be wary of providing a potpourri of applications with no internal coherence. Research comparing Logo to a set of utility and problem-solving programs demonstrated that stronger feelings of control and mastery emerged with Logo . Use extensible programs for long periods of time, across topics when possible, to engage students in meaningful problems and projects.
Navigating with students
There is strong evidence that the curriculum in which computer programs are embedded, and the teacher who chooses, uses, and infuses these programs, are essential elements in realizing the full potential of technology.
Computers allow students to build their mathematical ideas. As teachers, we play a critical role in mediating students computer work, but we must remember: The student's motor is always running; our job as teachers is to build roads, place signs, direct traffic, and teach good driving, but not to drive the car. What is good mediation? We can do the following.
bring the mathematics in computer work to a conscious level of awareness and systematically extend the ideas encountered;
focus attention on critical mathematics aspects of activities, emphasizing the need for consistency and mathematical language;
facilitate disequilibrium using computer feedback as a catalyst in strengthening concepts;
construct links, or mappings, between computer and non-computer work; and
provoke reflection and prediction.
Students do not automatically transfer knowledge gained in one situation to another. Repetition is not sufficient. Questions that cause students to reflect on what they were doing are instrumental. One research recorded the following: "Ingrid rotated a rectangle [one degree at a time] 540 times and produced a circle that was filled in. When I asked her if it had gone over part of the circle again she said yes, thought for a minute and then said, I bet 360 degrees would do it. I'm going to try it" . In addition, the following teaching strategies are helpful.
Provide hands-on experience. Students need to personally manipulate computer representations and reflect on these actions to construct concepts, even after "clear" explanations and demonstrations by the teacher.
Provide adequate time. Students need time to construct, or reconstruct, ideas in computational environments. Initial strategies and conceptions may be precursors of more sophisticated mathematics, but only if we give students the time and encouragement to revise their ideas and strategies over a considerable period, using computer scaffolding to fully synthesize visual and symbolic representations.
Rethink assessment. Evaluation of learning in such environments must be re-considered, as traditional approaches do not assess the full spectrum of what students learn, especially posing and solving ill-defined problems.
Encourage collaboration. Two students working cooperatively at a computer seemed ideal in exploratory environments. But many students benefit from time working alone as well.
Conduct whole-class discussions. As mentioned, group discussions with a computer and large screen display are critical. A single location for computer work, discussion, and lecture may alleviate confusion about the overall structure of the course.
Plan for computer integration from the start. Computers will be truly useful only when they are viewed as an integral part of curriculum planning .
Plan for long term benefits. Effects are largest for students who work in technology-rich programs for more than one year . Several projects demonstrated that students find it easierover the long haulto learn more (computer environments, more than one mathematical topic)l than less.
Watch for potholes. For all types of software, be wary of students ideas about computer representations. For example, whether on computer or not, students often see graphs as a pictorial representation of a real-world situation.
Dont get discouraged. Ive noticed that many of us who believe in the potential of the computer in education get discouraged because computers are not magic. We do know, however, that it can be a valuable tool for mathematics exploration.
Computers in two classrooms
Olsons research tells a tale of several sites, only one of which is a success story . In this case, what distinguishes them was not number of computers or background of the students. All had moderate computer access; all were from working class backgrounds.
At Maple Hills, boys wishing to play games dominated the informal use of computers, precluding others from gaining expertise. The teacher in charge of computers, Jack, took a teacher-centered approach, and exhausted himself trying to give explanations in an inadequate period. Another teacher let students "experiment" but didn't know how to integrate this activity with the rest of his instruction. Teachers received little support. "Its one more thing added on, and they never take things off." So, a huge burden fell on Jack, to teach everyone. He did minimal basics and then "enrichment" for a few. Others played games and had "fun." For example, they played "Lemonade." Bankruptcy was frequent and mastery or sense of self-accomplishment rare.
In Spruce Grove, children consistently make "money" selling lemonade and the best players score high. Their teachers viewed Lemonade not as a game, but an opportunity to teach mathematical principles and the power of group organization. They would record the "best price" for a temperature and humidity, compiling these into grand lists. They complained that the game was "too easy" because it had no "randomization factor"; interesting language for sixth graders from a working class background. This was not uncommon; later they learned to program an algorithm (a term they also used) for determining whether a number was prime or not. These students learn that knowledge is valuable and that if you cooperate to acquire it, it works for you.
The computer demanded much time at all the sites. But only in Spruce Grove did teachers see it not as one more thing to do, but as an auxiliary tool to be integrated into a total program. The computer makes life more complicated, but also more interesting and rich. It is not a panacea, but it should not be merely an adjunct.
To accomplish this, teachers need computer access, support for teaching reform, technical support, validation, collegial support, and extra time. In addition, reformers should recognize the threat of unrealized potential and missed opportunities due to divergent beliefs of the various social groups involved, from teachers to students to administrators to parents to curriculum and software developers . For example, everyone in one school believed that computer access was important. However, whereas the administrators and parents believed that the lab and its schedule were adequate, the teachers believed that it was unworkable to implement the reform.
Thus, more emphasis must be given to professional development and support. Teachers need available help with technical problems, but also with pedagogic challenges of choosing software, teaching with technology, and organizing projects . This help must address the challenges of technological reform as seen through their eyes .
Thinking about computers in mathematics education should not mean thinking computers. It should mean re-thinking mathematics education. When I talked to teachers about computers, real problems, and projects, I used to be cautious even a bit defensive. I think Im going to turn that around. Now I want to ask those I talk to how they can justify a year of mathematics in school without that kind of activity.
Research offers directions at todays crossroads. An important decision is among three paths that differ in the goals and types of computer applications. Those traveling on the first path use simple computer games for "rewards" or occasionally use drill software but do not integrate it into their wider educational program. Those traveling on the second path integrate drill exercises and other structured software tasks into their programs. Those traveling on the third path use problem-solving software, geometric construction programs, function and graphing software, and Logo to engage their students in substantive problems and long-range projects.
Research suggests that the first path leads nowhere educationally useful. Teachers might better invest efforts and resources elsewhere. The second path is educationally plausible. Well-planned, integrated computer activities can increase achievement in cost-effective ways.
The third path is more challengingin time, in effort, in commitment, and in vision. This path alone, however, offers the potential for substantive educational innovation consonant with NCTM Standards. Technology actuated the world of tomorrow. Technologyused thoughtfully and creatively rather than as a teaching machinecan engender and support educational environments that will help students to flourish in this intensively mathematical world.
Finally, research shows that decisions about these paths are not final. Many teachers take the second path, return one or more years later to the crossroads, and turn to the third path. These teachers see the potential of computers, and extend their vision. I leave you, and my path metaphor, with the vision of the poet:
I shall be telling this with a sigh
Somewhere ages and ages hence;
Two roads diverged in a wood, and I
I took the one less traveled by,
And that has made all the difference.