Visualizing Mathematical Ideas With Technologies

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Because of the abstractness of mathematics, visualization is an important strategy in helping learners understand mathematical concepts. Such visualization tools are not always computer mediated. For example, Cotter (2000) showed that using Asian forms of visualization (e.g., abacus, tally sticks, and place cards) advanced understanding of place value, addition, and subtraction. Mathematics educators have promoted the use of manipulatives and similar visual comparative devices for many years. Snir (1995) argues that computers can make a unique contribution to the clarification and correction of commonly held misconceptions of phenomena by visualizing those ideas. For instance, the computer can be used to form a representation for the phenomenon in which all the relational and mathematical wave equations are embedded within the program code and reflected on the screen by the use of graphics and visuals. This makes the computer an efficient tool to clarify scientific understanding of waves. By using computer graphics, one can shift attention back and forth from the local to the global properties of the phenomenon and train the mind to integrate the two aspects into one coherent picture (Snir, 1995).

Visualization tools have been developed primarily for mathematics and the sciences. Mathematics is an abstract field of study. Understanding equations in algebra, trigonometry, calculus, and virtually all other fields of math is aided by seeing their plots. Understanding the dynamics of mathematics is aided by being able to manipulate formulas and equations and observe the effects of that manipulation. Programs such as Mathematica (http://www.wolfram.com/products/ mathematica/index.html), MathLab (http://www.mathworks.com/), Statistical Analysis System, and Statistical Package for the Social Sciences are often used to visually represent mathematical relationships in problems so that learners can see the effects of any problem manipulation. Being able to interrelate numeric and symbolic representations with their graphical output helps learners understand mathematics more conceptually. Those tools, because of their power and complexity, are seldom used with K–12 students. Most of the research on these tools has been conducted in universities.

Visualizing Formulas With Graphing Calculators (by Fran Arbaugh)

The National Council of Teachers of Mathematics (NCTM) recommends that mathematics instruction at all grades enable students to create and use representations to organize, record, and communicate mathematical ideas; select, apply, and translate among representations to solve problems; and use representations to model and interpret physical, social, and mathematical phenomena (NCTM, 2000, p. 360). Handheld graphing calculators (such as those made by Casio, Hewlett-Packard, and Texas Instruments) are portable tools that students can use in the classroom or at home to support their mathematical sense making.

Students often have difficulty distinguishing important features of functional relationships. For instance, to build understanding of linear relationships, students can use different representations, generated by the graphing calculator, to make connections between what is happening contextually, numerically, graphically, and symbolically for a particular mathematical relationship.