# SimCalc

## 1 Definition

SimCalc is a type of microworld

- The SimCalc "curriculum" aims to bring the
*mathematics of change and variation (MCV)*to "ordinary" children and give them the opportunities, experiences,and resources they need to develop an extraordinary understanding of and skill with MCV (Roschelle et al., 2000).

- “ The SimCalc MathWorlds Curriculum: Bring your Algebra class to life-allow students an opportunity to explore and personalize core algebraic concepts through analyzing the mathematics of motion. Activities range from Pre-Algebra through Pre-Calculus topics.” (The SimCalc MathWorlds® Curriculum, updated 11:26, 5 June 2008 (Kaput Center, University of Massachusetts Dartmouth).)

## 2 Motivation

SimCalc MathWorlds® is motivated by the idea that the central phenomenon of the twenty-first century will be change: economic, social, and technological change. Children not only need to participate in the physical, social, and life sciences of the twenty-first century, but also to make informed decisions in their personal and political lives. (Roschelle et al. 2000). To understand change, the authors claim, one needs knowledge of MCV.

Now (Roschelle et al. 2000) claim that :

- calculus sits at the end of a long series of prerequisites that filter out 90% of the population
- even the 10% who do have nominal access to MCV in calculus courses develop mostly symbol manipulation skill but little understanding (Tucker, 1990).

## 3 The SimCalc Projects

The technological mediation of mathematics and science learning community to which the SimCalc team belongs, advocates the design of visualizations and simulations collaborative expressive inquiry media, with the goals of:

- extending students' engagement with the aspects of concepts that they find problematic
- supporting shared focus of attention and part-whole analysis
- enabling gestural and physical communication to effectively supplement verbal communication
- engaging students in actively doing experiments, and providing meaningful feedback through an interface that is appropriately suggestive and constraining.

This project is based on 3 lines of innovation.

- A deep reconstruction of the calculus curriculum:
- SimCalc is seeks to build an appropriate curriculum throughout schooling (from elementary school to university).
- The most notable innovation in the SimCalc curriculum is according to Rieber (1996) the use of piecewise linear functions as the basis of student exploration. In a velocity graph, for example, a student can build a function by putting together line segments, each of the same time duration A series of joined horizontal segments denotes constant velocity and a set of rising or falling segments denotes increasing or decreasing speed.

- Rooting of mathematics principles in meaningful experiences of students (genetic seeds)
- Students bring with them a wealth of mathematical understanding that is largely untapped in traditional methods of learning calculus. The SimCalc project does not require students to understand algebra before exploring calculus principles.

- Creative use of technology in the sense of cognitive tools (or expressive digital medium:
- MathWorlds makes extensive use of concrete visual representations, coupled with graphs that students can directly manipulate and control.
- Graphs are based on data sets generated by computer-based simulations (animated clowns, ducks, and elevators), laboratory experiments, and even the students’ own body movements by capturing that are caputre with motion sensors.

- Expressive constructions

The SimCalc project has reconceptualized the teaching of mathematics at all grade levels and has put its focus on meaningful student experience based on graphs of interesting visual phenomena that students can manipulate directly. E.g. as an example the project site mentions "creating dances between characters, or choreographing a whole marching band using graphical descriptions of the marchers-either velocity or position graphs."

The SimCalc project places much value on students experiencing phenomena as the basis for their mathematical explorations. Therefor, the SimCalc curriculum is based on 4 strategies that counter traditional teaching of calculus;

- Phenomena before formalisms: phenomena are studied and understood before delving into mathematical formalisms.
- Discrete variation before continuous variation: the mathematics are based on discrete variation before turning to continuous variation.
- Accumulation and integrals before rates and derivatives: the mathematics of accumulation and integrals are taught before rates of change and derivatives.
- Graphs before algebraic symbolism: students learn to master graphs before algebraic symbolism. So, instead of requiring algebra as a prerequisite skill for studying calculus, the SimCalc project using students’ grasp of visual problem solving with graphs to enter the mathematical world of change and varying quantities.

## 4 Technology

SimCalc is the name of the overall project. Their main "product" is called MathWorlds and it is available in several variants at SimCalc MathWorlds® Software

- Java-based MathWorlds for computers (available for download)
- TI Calculator version (available for download)
- TI Calculator connected to the teachers computer (available for download)
- Some PDA version (?)

## 5 Links

## 6 References

- Rieber, L. P. (1996) Microworlds, in Jonassen, David, H. (ed.) Handbook of research on educational communications and technology. Handbook of Research for Educational Communications and Technology. Second edition. Simon and Schuster, 583-603 ISBN 0-02-864663-0

- Roschelle, J., Kaput, J., & Stroup, W. (2000). SimCalc: Accelerating student engagement with the mathematics of change. In M. J. Jacobson & R. B. Kozma (Eds.), Learning the sciences of the 21st century: Research, design, and implementing advanced technology learning environments (pp. 47–75). Mahwah, NJ: Lawrence Erlbaum Associates.

- Roschelle, J., & Kaput, J. (1996). SimCalc MathWorlds for the mathematics of change: Composable components for calculus learning. Communications of the ACM, 39 (8), 97-99.

- Tatar, D., Roschelle, J., Vahey, P., & Penuel, W. R. (2003). Handhelds go to school: lessons learned. IEEE Computer, 36(9), 30-37.

- Tucker, T. (1990). Priming the calculus pump: Innovations and resources. Washington, DC: MAA.

This article is in a large part a synthesis of Rieber 1996