"Green Globs and Graphing Equations"
An Analysis
A1. Specific Virginia SOLs
Algebra II SOL Introduction
"
Students will vary the coefficients and constants of an equation, observe
the changes in the graph of the equation, and make generalizations that
can be applied to many graphs."
AII.15
The student will recognize the general shape of polynomial functions,
locate the zeros, sketch the graphs and verify graphical solutions
algebraically.
AII.18
The student will identify conic sections (circle, ellipse, parabola,
and hyperbola) from his/her equations. Given the equations in (h,k)
form, students will sketch graphs of conic sections using transformations.
It is also important to note that Algebra II students cannot be successful with the course SOLs without a solid foundation in Algebra I where the following objective, directly addressed by this software, appears.
A2 Rewritten Objectives:
Given the equations of lines, parabolas, circles, ellipses and hyperbolas,
the student will:
- identify basic shapes of the graphs of the equations
- analyze relationships between equations and their graphs
- analyze characteristics of various types of graphs
- manipulate graphs by altering the corresponding equation
the student will:
- Construct equations to fit specified criteria
- Transform basic graphs by modifying the constants and coefficients of equations
- Apply algebraic concepts and techniques graphically
- Translate graphs horizontally and vertically
- Apply the geometric property of symmetry
B1. Title: Green Globs and Graphing Equations
B2. Designed by: Sharon Dugdale and David Kibbey, University of Illinois
Published by Sunburst
B3. This program provides "two interactive learning environments" and two strategy games. The documentation includes suggestions that the first module (equation plotter) is effective as a teacher demonstration tool or as a small group activity while the remaining three modules are directly designed to be used as an individual or small group activity. Printed worksheets are included to guide and direct students in developing strategies.
B4. The attitudes of teacher and students are pivotal to the success of this program. The teacher’s role is described as that of a facilitator who injects carefully chosen comments and questions while encouraging students to explore their own ideas. The student must adapt to the role of investigator, developing basic strategies that can be applied in many situations. This is not an exercise in memorizing formulas. Both teacher and student must experiment, use trial and error, learn from mistakes and be creative.
B5. The computer based material is organized into four modules. Two are described as "interactive learning environments" which is the vendor’s term for situation exploration. Two modules are strategy games that use the math and graphing concepts presented, discovered and practiced in the exploration modules in the pursuit of game related goals.
B6. This program is designed for grades 6 through adult. This wide range can be accommodated by appropriate selection of activities and level of difficulty. Students in Algebra I and Algebra II are the primary target audience.
Instructional Context
Context Description:
C1. Green Globs and Graphing Equations is an example of situation exploration software. Learners are provided a shared learning space where as contributors in a whole class activity or as members of a cooperative learning team they can investigate the relationship between algebraic and graphing concepts. The knowledge and skills acquired during exploration are directly applicable to the games included in the software package where the learners use either competitive or cooperative methods to devise problem-solving strategies. The introduction to the Algebra II SOLs expresses the philosophy that permeates the individual standards of the course - "build a strong connection between algebraic and graphic representations of functions. This program is a deliberate attempt to fulfill that goal.
C2. The context is primarily instructional and to a lesser degree transfer. Compared to the popular multimedia programs currently on the market, the format of Green Globs and Graphing Equations is rather austere. The program relies on the teacher to have provided the orientation function. The student is presented with pull down menus listing the four modules available. Instructions are included concerning the proper format for entering equations, definitions of function names and directions for the copying, erasing, saving and using the zoom feature. A practice window is provided to help the student determine if he has understood the directions before progressing.
The Equation Plotter module is the basis for the program. The instructional activities provide students with the opportunity to explore equations and graphs of the functions and relations which constitute a major part of the curriculum of the algebra course sequence (lines, parabolas, circles, ellipses, hyperbolas). The analysis of exponential and trigonometric functions is also an option for advanced students. Worksheets are included to guide students’ exploration and they are directed to record and describe their results and search for patterns.
The practice opportunities and feedback received by the student during Equation Plotter prepares him for the Green Globs game. Here the student is presented with a random distribution of "green globs" on a coordinate grid. He must analyze the placement of the targets and construct equations that when entered at the keyboard intersect as many globs as possible. The exponential scoring system rewards careful analysis. The student earns one point for the 1st glob, 2 points for the second, 4 points for the 3rd, 8 points for the 4th, etc.
The two remaining modules provide additional opportunities for practice, feedback and review as well as an extension of knowledge and skills. In the Linear and Quadratic Graphs module a target graph is drawn by the program. The student is prompted to type the equation that the graph represents. Different colors are used for differentiation. A correct answer takes the user to a problem of greater difficulty. If incorrect, the student observes how their response differed from the target and tries again. An option of seeing the correct equation is provided but the student then remains at the same level.
The last module is a game called Tracker where the student tries to identify hidden graphs by "firing" equations called probes. If a probe crosses the path of a hidden graph a symbol is recorded on the coordinate plane. When the student has enough information, he types his guess as to the equation of the hidden graph. Once again the scoring system rewards careful analysis. The student’s score is charged for each probe fired and each incorrect guess. Points are earned for each type of graph discovered according to the level of difficulty - 30 for a line, 35 for a parabola, 40 for a circle, 45 for an ellipse and 50 for a hyperbola. In both games novice and expert options can be chosen and a record of high scores is kept.
C3. This program serves its instruction and transfer functions well. I believe that students who can score well in Green Globs and Tracker will also score well on the relevant questions on the SOL exam. Such a result would be direct evidence that the student has mastered the SOLs to be learned.
C4. The motivational appeal of this program is high especially when
compared with the mathematical alternatives of practicing a series of
textbook problems with pencil and paper even with the assistance of a
graphing calculator. Using Keller’s ARCS model specific strategies in
all areas can be identified. Many of these assume teacher supervision,
support and intervention as described in the documentation.
| Attention | Using creativity techniques to create associations Problem solving activities are the main focus of the program Games require learner participation |
| Relevance | The order of difficulty specified by the type of
graph builds on the learner’s existing skills. The worksheets provide opportunities and options for organizing one’s work |
| Confidence | Each module format provides self-evaluation tools
clearly based on the stated rules. The scoring criteria is clearly explained Materials are organized on an increasing level of difficulty Students can set realistic goals based on current knowledge and skills Students are encouraged to verbalize and record results and hypothesis for both successful and unsuccessful attempts in cooperative learning settings. Students become more independent as they progress from the teacher led demonstration to cooperative learning teams and individual practice. |
| Satisfaction | Students in cooperative learning teams share their
insights with others. Informative, helpful feedback is the essence of the program structure. |
Context Evaluation Summary
C5. This program is a comprehensive attempt to address the general philosophy of the NCTM standards and specifically stated SOL standards for the Algebra sequence. It has been designed to be used with teacher supervision, support and intervention in a classroom climate where the attitudes of teacher and students are pivotal to the success of the program. Improvements to these external conditions would increase the effectiveness of the program more than any specific changes to the software itself.
Instructional Components and Conditions
Descriptions and Evaluation
Instruction Components/Conditions Descriptions
D1. Green Globs and Graphing Equations primarily addresses the Information/Activities and Practice & Feedback components of Instructional Design. Throughout this program the activities are structured in accordance with situated learning guidelines and strategies. Specifically:
- Support collaborative construction of knowledge
- Promote reflection to enable abstractions to be formed
- Promote articulation to enable tacit knowledge to be made explicit
- Provide coaching at critical times
D2. These conditions (from Appendix B) are addressed in this program:
Information/Activities:
- Activities are structured to conform to "Situated Learning" guidelines and strategies
- Present a variety of clear, concrete examples and nonexamples
- Provide opportunities for learners to explore the learning environment with minimal instructor guidance and intervention.
- All should get practice
- Feedback as immediate as possible.
- Present cues within the new context which aid learners in selecting and applying the appropriate previously-learned SKA
- Clearly identify the incentives/rewards (of) succeeding within the new context.
These conditions (from Appendix D) are addressed in this program:
Intellectual Skills:
- Present varied examples or instances of concepts and rule applications, calling attention to the distinctive features of examples, definitions, and procedures.
- Encourage learners to recall previously learned information or examples that illustrate concepts or rules being presented
- Provide learners with opportunities to "play" with concepts.
- Present a variety of contexts or experiences that allow the students to practice applying the rules or identifying/describing concepts, providing guidance throughout early stages of practice.
Instructional Components/Conditions Evaluation Summary
D3. Practice with Immediate Feedback is the cornerstone of this software. It occurs in every module. Some of the activities in this software represent common forms of feedback. In the Linear and Quadratic Graphs module students are told they are "right" when they have correctly typed in the equation of the graph presented on the screen. If incorrect, the graph of their guess is plotted in a contrasting color. These visual clues allow the students to adjust their equation appropriately before trying again.
In the Tracker game, once the student has gathered enough information from his probes to input his guess, i.e. the equation of one of the hidden graphs, the program responds with either "right" or "tracker missed". The student is given no additional visual clues, but must fire more probes to gather more information and try again.
More common however is the open-ended feedback offered by the program as the student uses the software to investigate relationships between equations and their graphs. The immediate graphical feedback adds to the student’s store of knowledge and guides him to further investigation. This occurs in Equation Plotter and Green Globs where there is no "right" answer to discover. Equation Plotter provides an endless opportunity to experiment, receive visual feedback and add to one’s knowledge. Green Globs offers a dynamic form of feedback as each glob "explodes" when hit, but there are numerous possibilities as to the equations that might result in success.
D4-D6. Except for the task of introduction which is left to the teacher, this program does a thorough job of incorporating the components of instructional design
The primary outcome being facilitated is the intellectual skill evidenced when students must analyze a situation and decide which principles are needed to perform the task at hand. This program subtly strives for a change in attitude as well. Through exploration and extensive feedback, students construct the relationships between equations and their graphs. This is a departure for many students accustomed to experiencing mathematics as an endless array of formulas to be memorized.
Instructional Message Display Description and Evaluation
General Message Display Description
E1. The screen display follows traditional horizontal-vertical and left-right organization with relevant information easily-identifiable. Every module mirrors the same basic layout. The majority of the screen is reserved for the Cartesian coordinate grid. Pull-down menus occupy the top left corner of the screen in the Window95/98 tradition. Prompts and instructions are located in the left and bottom margins of the grid area and are printed in contrasting colors. Special function buttons appear in the right hand margin of each module. The program’s creators have observed the "keep it simple" rule of design.
General Message Display Evaluation Summary
E2. Although reference to navigation assistance and equation entry format is only a click away in a pull down menu, the instructor may want to include a printed version of this information with the worksheets provided to the students. It can then be consulted as needed without distracting the student from the visual activity with which he is engaged.
Text Use and Display Description
E3. The program observes the basic rules of effective text use and display regarding line breaks, annotations in the margins, mixed upper and lower case letters and text/background contrast. Note that the color scheme of the Windows version (black text on a white background with gray margins), replaces the blue background used as the standard by the old DOS and Apple versions. The text of the equations is recorded in the gray margins in either red or blue.
Text Use and Display Evaluation Summary
E4. Given the nature of this software, there is not much need for extensive text display. The graphs of the functions and relations visually say it all. Menus are extremely straightforward and easily understood by even the youngest students. Definite care has been taken to create an uncluttered workspace with color used for contrast to direct one’s attention and emphasize major elements.
Pictures and Illustrations Use and Display Description/Evaluation Summary
E5-E6. Pictures and illustrations are not applicable to this program. The Cartesian coordinate grid and the graphs displayed are clearly done.
Diagrams Charts and Graphs Use and Display Description
E7. The majority of the screen is occupied by the Cartesian coordinate grid, the shared workspace the students will use. Students are given control to change the boundaries of the grid, zoom in and out and change the center point of the zoom feature.
Diagrams Charts and Graphs Use and Display Evaluation Summary
E8. The visual arrangement of the screen helps focus the student on the task at hand Equations are plotted in red or blue, "globs" remain their traditional green while the text of the equations are color correlated to that of the related graphs. The plotting of the graph has been purposely slowed down to direct ones attention to the activity. In each module the student has the option of saving or printing the screen for further study. Great attention has been paid to visual clarity.
Audio Use Description:
E9. The only audio is the subdued "swish" as a targeted glob explodes. Even this modest noise can be turned off.
Audio Use Evaluation Summary:
E10. Audio is not a major need or feature of this program. Additional sound would only be distracting, especially an entire lab full of such sounds. Headphones have an isolating effect and reduce the collaborative nature of this activity.
Navigation Description and Evaluation Summary
E11-E12. Navigation is accomplished through pull down menus and buttons for special module options such as "new game", "see answer" and "zoom". The screens of instructions contain buttons for moving to the previous or next screen. The navigation format is straightforward and easy to use.
Summary: Final Thoughts
Green Globs and Graphing Equations has been on the market for over 15 years. It has outlasted similar more expensive programs, which have been marketed with far more hype. I believe that this is because it is a well designed, motivating and visually appealing program that is a useful tool for teaching the specific SOLs as well as fostering an exploratory mindset in the students who use it. It should be part of the software inventory of every math department and an integral part of the Algebra curriculum.
This program is designed to be used with the active participation of the teacher in creating a classroom atmosphere where open inquiry is practiced and valued. Some observers may interpret this need for teacher support as a weakness. I consider it a strength when the value of the teacher in the instructional experience is recognized as important. There are teacher independent programs on the market where the software attempts to provide the majority of instructional components. This is not one of those programs.
While the specific SOLs regarding the relationship between equations and their graphs are mastered, the student is learning a more general lesson. Mathematics is not a series of formulas to be memorized but a structured universe to be explored.