Advanced Technology for Core Curriculum Reform
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The mission of the SimCalc Project is to democratize access to big ideas in mathematics. We believe that interactive tools for visualizing, transforming and simulating mathematical objects will make it possible for every student to achieve solid conceptual understanding of the mathematics of change.
An Interplay of Simulations and Physical Experience
Simulations can enable children to learn difficult concepts at an early age. SimCalc MathWorlds provides animated worlds in which actors move according to graphs. These graphs can be directly edited with the mouse, and during edits they exhibit dynamic links that reveal mathematical relationships. Games and challenges engage students in learning central ideas in calculus and set the stage for algebra, dynamical systems, and other topics.
| For example, in the activity pictured to the right, students try to adjust a velocity graph to enable the baby ducky to catch up to her mother. The baby duck starts out several meters behind the mother, and must go fast to catch up, but then slow down at the right time to match the mother's motion. The mother quacks worriedly if the baby gets too far behind. |  |
Educational software, however, requires more than simulations. Using MathWorld's link to Microcomputer Based Lab (MBL) software, students can enter their own bodily motion into MathWorlds. For example, a student can walk across the front of the classroom. Their walk will then become the motion of an object in MathWorlds, perhaps represented by a clown. Then the students can explore the mathematical properties of their own motion, perhaps explore its mean value.
To complete the cycle, TERC's Line Becomes Motion (LBM) apparatus allows a graph to drive a physical device, such as two cars moving on a track. Using LBM, students can experience graphs that they create as physical motions in the real world.
In addition, SimCalc is creating new devices to enable students to explore unfamilar mathematical systems in physical form. For example, TERC's Bouncing Cart enables students to explore a chaotic system and control the amplitude and frequency of the piston that drives the behavior.
Newly Expressive Mathematical Representations
A commonplace image of the calculus classroom features a blackboard covered with inscrutable formulas and exotic symbols. For most people, this image generates fear and anxiety. Educational researchers have found this fear is warranted, for very few people come to understanding the big ideas behind calculus from these traditional representations.
Modern technology supports a more friendly alternative: using dynamic graphics to represent mathematical ideas. Graphs draw upon students' vast spatial reasoning skills, and support learning the mathematics of change even before students reach an algebra course. Graphs provide a powerful means for students to interpret the real world, and to explore simulations.
SimCalc has developed a special type of graphs for teaching the mathematics of change to young children. In their simplest form, these graphs depict rectangular chunks of velocity. The height of the rectangle means "how fast" and the width means "for how much time." Discrete rectangular shapes draw upon students understanding of related skills like signed numbers and the area multiplication model. These understandings can then be readily extended to the concept of integration.
 | SimCalc's position, velocity, and acceleration graphs are dynamically linked. Thus, if students make a change to a velocity graph, they instantly see the corresponding change in the position graph. And vica versa, a change to position is automatically reflected in the velocity graph. |
As students' understanding of velocity rectangles grows, our activities apply this representation to core concepts in the mathematics of change such as the Mean Value Theorem, Limits, and Approximation. Moreover, from the simple velocity rectangles, students move on to consider piecewise linear polynomial, periodic, and exponential functions. At TERC, we have also been exploring student's understanding of novel representations such as phase space plots.
Supporting Educational Authoring
Software re-use, integration, and activity authoring are critical for educational software success. Education cannot afford to rewrite components for each application. To this end, SimCalc has been exploring three MacOS System 7.5 innovations: Drag and Drop, AppleScript and AppleGuide.
MathWorlds uses drag and drop to enable easy layout of an activity. To make MathWorlds activities, we drop graphs into place and dragg actors to the graphs to link their motion to the graph plot. Scripts and tools can be dragged from the Macintosh desktop directly into the toolbar.
Scripting is simple, lightweight programming. The activity author or teacher can put scripts in the menu bar or the tool bar, and these scripts can control of the MathWorlds interface. In assembling activities, we use scripts to connect MathWorlds to Eudora for e-mail, and to MacMotion software to digitize real motions. Scripts also generate varied challenges and can carry out complicated actions in a single click.
| We are also exploring AppleGuide for presenting instruction. AppleGuide can present both general and step-by-step information. AppleGuide draws students' attention to elements of the screen with red "coach marks." Because guides are modular, any teacher can author or modify a guide to suit their students. |  |
Finally, SimCalc is on the forefront on the component software revolution. Emerging new standards such as OpenDoc and Java are allowing us to create educational modules (such as graphs, tables, and simulations) that plug and play with other educational software. Component software will provide the ultimate educational authoring environment, by letting teachers and students freely combine content from a variety of research projects and vendors.
Beyond The Desktop Computer
Educational technology research has often centered on desktop computers. Although desktop computers offer great flexibilty and power, costs have remained high and the hardware lacks mobility. Hand-held devices, such as graphing calculators, offer strong mathematical power at a low price, and can travel with the student. We believe desktop computers and calculators should evolve into complementary, rather than competitive roles.
We are developing a model classroom for the implementation of a scalable, dynamic and technology rich math and science curriculum. The classroom itself is a node in a district-wide intranet, and is equipped with a local network of one or more computers and an overhead projector, CBL devices and hand-held devices such as graphing calculators. We are also designing curriculum exercises with different components that are scaled to the many technologies in the model classroom network, from the Internet to the graphing calculator. We believe that such a model will prove immensely valuable for many reasons:
| Cost effectiveness | This is an inexpensive way to integrate technology into the curriculum. It leverages the capabilities of inexpensive, mass produced devices like CBLs and graphing calculators by coupling them to more powerful computers via the network. |
| Appropriate technology | Well designed, special-purpose devices like graphing calculators supply an activity context that is complementary to the personal computer. They give individual students a powerful and readily available set of tools for the investigation of complex mathematical and scientific phenomena that can be modelled or simulated in more complex PC software. By distributing curriculum activities across a whole network of integrated devices, we can also ensure that students do not think that mathematical ideas 'live' in the computer, software, calculator or any other device — they 'live' in our minds! |
| Reduced complexity | More computers means more complexity and maintenance — hence more expense and more down time. Often, the burden of maintenance falls to the classroom teacher, who is (understandably) ill-suited to the job of computer system administrator. The result is an implementation that is both expensive and under utilized. Embodying more of the classroom's technological complexity in hand-held devices and network resources shifts the burden of maintenance from the classroom teacher to the device and system designers, where it belongs! |
Our pilot implementation is based on technology currently available: the Macintosh computer, TI-83 Plus and CBL devices, the TI GraphLink software and our own MathWorlds simulation and authoring environment. We are currently working to develop a set of specifications for useful GraphLink functionality. Our envisioned design will enable us to develop graphing calculator programs that embody portions of curriculum activities, bundle them with our own software documents and configue the whole classroom network with a single activity configuration script.
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