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Learning To Think Spatially
of working and final results (the functional equivalent of short-term memory)
Assistance: provides prompts, feedback, hints, and suggestions to guide the choice of data analysis steps and to manage the flow of work
Display: provides a flexible display system for the representation of working and final results to oneself and to others—in physical form (e.g., a graph on paper, a three-dimensional model of molecular structure) or in virtual form (e.g., on-screen, for hard-copy printing, for export to other software packages)
For example, throughout history, we have developed, taught, and used a suite of tools—abacuses, compasses, Cuisenaire rods, protractors, graph paper, measuring and slide rules, and mechanical and electronic calculators—to facilitate calculations in the process of mathematical problem solving. Today, with the advent of sophisticated computer technologies, we are beginning to teach students to use software such as spreadsheets, database management programs, computer programming languages, and statistical analysis programs to perform calculations and to solve mathematical problems.
By routinizing basic mathematical operations (simple—such as addition, subtraction, multiplication, and division, complex—such as percentages, square roots, exponentiation, or generating trigonometric functions), tools and technologies provide ways of performing calculations and tracking the flow of sequences of chained operations. They can speed up the process of problem solving and increase the chances of arriving at a correct answer. They also provide ways of representing the working and final results to oneself and to others. Similar suites of tools and technologies can support spatial thinking in other knowledge domains (e.g., in architecture: pencil sketches, colored perspective drawings, sections [plans, elevations, etc.], balsa wood and cardboard models, CAD systems, virtual reality displays; in sea navigation; portolan charts, astrolabes, compasses, sextants, modified Mercator projection maps, chronometers, celestial tables, Loran, GPS).
In any knowledge domain, the components of a suite of support systems serve different functions in different contexts, for example, trading off speed and simplicity (in terms of data needs and the execution of operations) for depth and complexity. The elements of a suite of tools are not necessarily built in a coordinated fashion, either in terms of a division of functions or in terms of common design principles: they are assembled over time, with new tools adding to or replacing existing ones. However, their alignment along a low- to high-technology continuum is not necessarily synonymous with worse to better. Because of their simplicity, transparency, and intuitive nature, low-technology tools are often taught and used as precursors for understanding the complex, nontransparent, and non-intuitive operations of high-technology tools. Indeed, in many instances, the “back-of-the-envelope” answers generated by low-technology tools are perfectly adequate to the task at hand. However, with the increasing link between workforce demands and digital information technology, familiarity with and indeed mastery over high-technology tools is increasingly important.
6.4TOOLS FOR THOUGHT: THE LIMITS TO POWER
Support systems, especially those that are computer based, are the cognitive equivalent of power tools. With the promise of access to such power comes costs and challenges. These include the time and committed effort it takes to learn to use a support system (and continuously upgrade to new versions), the need to understand the system’s range of appropriate and inappropriate uses, and the need to appreciate the system’s characteristic limits and idiosyncrasies.
There is a wide range of support systems available in science, mathematics, and design (see Box 7.1 for a description of hi-tech support systems for spatial thinking). These support systems can be a boon or a bane to the learner. Experts in a knowledge domain start with an understanding