Interaction and Interactivity

By George Collison

In 1999 Dr. Judah Schwartz challenged mathematics educators: “Can technology help us make the mathematics curriculum intellectually stimulating and socially responsible?” Schwartz envisioned a major paradigm shift in the teaching and learning of mathematics in which web-based technology would add representational power and thus new dimensions to visualizing, as well as capacity to communicate mathematically with others. The Seeing Math project, in support of that vision of stimulating and socially responsible teaching, has developed eight web-based interactive software tools for middle and high school mathematics. The Seeing Math software design represents functions, data sets, and proportional relationships in a way that is not possible on current handheld devices, like the graphing calculator. Through web communication, Seeing Math interactives enable users to share with others images of their mathematical products as part of an online discussion board.

The Piecewise Linear Grapher displaying piecewise, discontinuous, and step functions.

Researchers have identified two qualities of applets (small interactive software that runs on the Web) that make them effective cognitive tools to support growth of understanding: interaction and interactivity. Interaction characterizes the “conversation” between the tool and the learner. Seeing Math interactives support the learner/applet conversation in many unique and highly effective ways. In some, users move a slide bar below the x-axis to draw the graph. The design emphasizes the function as the link between the independent and dependent variables. Users can shift readily from one algebraic form of a function to another equivalent form by clicking a tab. The graph of a function and its symbolic representation are dynamically linked, so changes in one affect changes in the other in real time. For instance, vertical or horizontal shifts (transformations) change symbolic representations. The Quadratic Transformer enables users to experiment directly with f(x) notation. Questions like “How is a(f(x)) different from f(ax) or from f(x-a)?” become the focus of individual and group inquiry. Immediate feedback through linked representations draws users deeper into exploration of mathematical content.

Interactivity refers to the feel, form, properties, and quality of the interaction with the tool. Seeing Math applets permit online sharing of screen shots, which supports online dialogue. Because they explicitly and dynamically link graphical representation and symbolic and numerical forms, the tools abound with “what if” possibilities to explore mathematics. For example, the Piecewise Linear Grapher permits students to define linear relationships that are not functions. If a user has created a graph with multiple y values for some values of x, a warning box indicates the double y-value assignment for that region, but does not suggest how to fix the problem. The animation of the function simply disappears for the region of the domain for which the function is improperly defined. The design turns an annoyance (the disappearing animation) into a learning opportunity. The student asks herself, “What changes in the domain will fix the problem?”

The eight Seeing Math algebra interactives support an approach to algebra using the function concept as a central theme. Traditional approaches offer sets of exercises detailing proper manipulation of symbols and equation solving. Teachers and students miss many opportunities to make connections to real-world, understandable mathematics. The function concept unifies later study in algebra and the study of change in calculus; introducing functions earlier aids students’ understanding of mathematics significantly.

Seeing Math interactives

The interactives, plus warm-up exercises, User Guides, FAQ’s, and activities are available on the CD included with this newsletter, as well as at the Seeing Math website. Try them out yourself and with your students. (Note: you need Java 1.3.1 or higher to run them.)

Piecewise Linear Grapher

A huge variety of problems, from discrete rate problems like taxi fares, cell phone plans, or purchase of building materials are fundamentally linear, but employ variable rates depending on quantity. The study of piecewise functions can forge important real-world links for beginning algebra students. Piecewise functions also provide a concrete, readily understandable entrance to key ideas like range and domain. Open the Piecewise Linear Grapher, and do the warm-up exercise to get familiar with the layout of the interactive. Next, try the sample activity, which explores a cell phone problem. After completing these, you may wish to try something more challenging. Not all real-world problems have unique solutions. Use your knowledge of slopes of line segments to solve the following challenges.

A hotel with four elevators

A shopping mall has four elevators. They all move at different speeds, but each moves at a constant speed, whether up or down. Elevator One starts on Floor 1, and Elevator Two starts on Floor 10.

All four elevators start moving at the same time. After 4 minutes, two of the elevators are on Floor 6. After 6 minutes, three of the elevators are on Floor 4. (None of the elevators change direction.)

At what floors did Elevators Three and Four start? Which elevators are on the same floor after 4 minutes? After 6 minutes?

Consider the approaches you used to solve the problems (graphic, symbolic, or something else altogether). Did you feel the need to use a symbolic approach? Did you use a combination of approaches? Which felt more natural, and which was more helpful?

Design and solve your own 2 or 3 elevator problem with the elevator(s) stopping at some floors for specific lengths of time.

System Solver

The System Solver permits representation and manipulation of systems of linear equations in rational form. It is not an artificial intelligence support for solving that suggests the next steps in the process. The System Solver directly links the graphs of intermediate steps and associated numerical tabulations with the solution process and with the solution set. The software clarifies how legitimate manipulations can result in perplexing equations like 7=2 or –6 = –6. Try the warm-up and sample activities, then create your own systems of equations and solve them.

Conclusion

The Piecewise Linear Grapher and the System Solver demonstrate how excellent software design can support the conversation of the learner with the content through fluidly linked representations, and with other learners through sharing on the Web. The interactives also show how web-based devices can be effective cognitive tools that open up opportunities to develop understanding in mathematics for all students, not only those adept with symbols. With such tools, the teaching and learning of mathematics can be an intellectually stimulating and socially responsible activity for all.

George Collison (george@concord.org) is an Associate of the Concord Consortium and Senior Curriculum Author for the Seeing Math project.