Composable Components for Calculus Learning1 - Semantic Scholar

SimCalc MathWorlds: Composable Components for Calculus Learning1 Jeremy Roschelle and James J. Kaput University of Massachusetts, Dartmouth Introduction: Reforming the mathematics curriculum SimCalc's mission is to enable all children to learn the mathematics of change beginning in the early grades. This mathematics is conventionally sequestered in elite calculus courses after many prerequisites, which denies access to critically important ideas of rate, accumulation, approximation, limit, mean value, etc. We aim to teach these essential ideas to mainstream children, as part of a restructured K-12 mathematics curriculum. By supporting conceptual growth of powerful ideas, we will prepare all citizens to describe, discuss, design, and analyze processes of change (Kaput, 1995). Simulations can enable children to learn difficult concepts at an early age. For example, White (1993) successfully taught physics to middle school students. 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. However, successful educational software requires more than simulations. Software should be integrated with e-mail, presentations, notebooks, and real-time data collection. Furthermore, software should enable teachers, curriculum designers, and assessment providers to adapt activities for particular children, settings, and goals. The SimCalc team is designing its software as “composable components” to allow (a) re-use of basic elements, (b) integration with complementary software, and (c) easy activity authoring. MathWorlds: The Mean Value Theorem Our sequence for teaching the Mean Value Theorem (MVT) illustrates the SimCalc approach. In each activity, we cycle through three stages: warm-up, construction and application (Lesh, Amit, & Schorr, in press). In the MVT warm-up, students build variable velocity motions with 1

This material is based upon work supported by the National Science Foundation under Grant No.

9353507. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the author(s) and do not necessarily reflect the views of the National Science Foundation.

constant velocity rectangles. Then students construct the constant function asserted by the MVT using these rectangles. Finally, the students apply rectangular approximation to integrate the area under a curve.

Figure 1: Dropping rocks in the crusher using a piecewise constant velocity graph.

The warm up activity involves flying a UFO to pick up space rocks and drop them in a crusher (Figure 1). The UFO moves according to a sequence of two different constant velocities, which the students can adjust using the mouse. By playing this game, students experience the relationships among rate of motion, distance, and time in constant velocity graphs.

Figure 2: Enacting the mean value theorem with animated characters.

In the construction activity, students set the constant velocity of a clown so that he arrives at the same final location as a flying woman (Figure 2). She flies with constant acceleration. Solving this challenge is tantamount to finding the mean value of the changing graph. In the application activity (not shown), a mother and baby duck swim in a pond. The mother varies her velocity, while the baby tries to follow with discrete, constant velocities. The mother quacks angrily if the baby drifts away. The student doubles the number of graph segments and shrinks their duration, trying to help the baby achieve a good approximation to the mother’s motion. This illustrates the process of integration by approximating a curve with rectangular areas, shrinking the time delta to the limit. It is one approach to the Fundamental Theorem of Calculus. Composing activities by dragging, scripting, and guiding Software re-use, integration, and activity authoring are critical for educational software success (Roschelle & Kaput, 1995). 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 the MVT activities, we dropped graphs into place and dragged actors to the graphs to link their motion to the graph plot. The make-new-challenge (bull’s eye) button in the toolbar is actually a script that was dragged from the desktop 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 (Thorton, 1992) software to digitize real motions. Scripts also generate varied challenges and can carry out complicated actions in a single click. Finally, we are 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. Towards curricular reform supported by component software We believe the promise of educational software is not teaching efficiency, but rather fundamentally altering the curriculum. Students who learn only 19th century mathematics will not thrive as 21st century citizens. Educational technology must enable more students to engage with sophisticated subject matter, at a younger age. In on-going field tests in elementary, middle and high schools, and remedial college classes, we are finding that SimCalc can help students who would otherwise never reach, much less pass, a conventional calculus course. These students can develop conceptual understanding of key concepts in calculus. Further work will involve detailed analyses of young students' conceptual growth as well as using SimCalc with university courses. Using educational software to achieve systemic impact will require innovative software engineering. Stand-alone, closed applications will ultimately frustrate teachers and schools with fragmentary solutions. Although we have found that drag and drop, scripting, and guiding can open up applications in useful ways, education needs a full component software approach. Educators need to compose rich activities using diverse modules including graphs, tables, algebras, calculators, simulations, data collectors, multimedia notes, red ink markup, instructional and assessment systems, etc. — regardless of which vendor developed the component. With other interested research and commercial projects, we are beginning to explore the extent to which OpenDoc or other component software frameworks could enable a collective effort to engineer a suite of tools for 21st century schools.

URL http://www.simcalc.umassd.edu/