Visualizing Mathematical Ideas With Technologies

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.

Students using a graphing calculator can easily move between the symbolic, graphical, and numeric representations of the two functions. They can trace along both functions to find (x, y) values graphically. They can then compare those values to the (x, y) pairs in the table. Students can find x and y intercepts on the graph and table and discuss how to manipulate the symbolic representation to find the same information.

Research indicates that the use of graphing calculators has a positive influence on students’ understanding of mathematics (Ruthven, 1990). In addition, Dunham and Dick (1994) report that students who use graphing calculators are more flexible problem solvers, are more persistent when faced with a new problem situation, and are highly engaged in the act of problem solving. As more and more mathematics textbooks incorporate the use of a graphing calculator in learning and teaching high school mathematics, more research needs to be conducted on the impact of this technology on student understanding.

Tinkering With Data Sets

Data analysis and interpretation of statistics are key skills, according to standards published by the NCTM. The Technical Education Research Center in Cambridge, Massachusetts, created a simple-to-use database program called TableTop to support database construction and analysis by school-age children (Hancock, Kaput, & Goldsmith, 1992). Tabletop works with existing databases or with databases students create themselves. Data are visually represented by mobile icons that can be arranged into box plots, cross tabulations, histograms, scatter plots, and Venn diagrams. Students develop mathematical understanding of attributes, logical relationships, place value, and plotting and learn to perceive the stories and patterns that lie within the data they collect.

TableTop has been replaced by new data visualization software called TinkerPlots (http://www.umass.edu/srri/serg/projects/tp/tpmain.html). Developed with a grant from the National Science Foundation at the University of Massachusetts, TinkerPlots is data visualization software for grades 4 to 8 that enables students to see different patterns and clusters in statistical data. Students begin by asking a question that requires a prediction or inference (see chapter 3). They collect data (e.g., shoe size and height), assign units to the data (e.g., size and inches), and then represent the data graphically in many ways. With all the data points on a graph, students can group them in clusters, sort them by amount or other sequence, and display them in a seemingly infinite variety of formats. Students are able to use rich data sets or generate their own data sets based on problems they invent and construct their own graphical displays to help them solve the problem. Students learn to reason with data.

Cliff Konold (2006), the designer of TinkerPlots, introduces the use of the software by asking the class whether they think students in higher grades carry heavier backpacks than do students in lower grades. He has them explore a data set to see whether the data support their expectations. To help them answer the question, students can separate the cases into four bins according to the weight of the backpacks. To view the data in different representations, the icons representing each case can be stacked, then separated completely until the case icons appear over their actual values on a number line. By selecting the attribute Grade, the fifth-grade students were separated vertically from the other grades. By pulling out each of the three other grades one by one, students could then see the distributions of PackWeight for each of the four grades in this data set (grades 1, 3, 5, and 7). These different views enable students with different cognitive styles to find a mathematical representation that makes sense to them. TinkerPlots can also import Microsoft Excel spreadsheet files to enable students to visualize data in more ways than those afforded by Excel. Students can assign different icons to the data points and generate numerous comparative plots that Excel cannot.

Fathom Dynamic Statistics Software (by Fran Arbaugh)

Like TinkerPlots for elementary and middle grade students, Fathom Dynamic Statistics Software (Finzer, Erickson, & Binker, 2001) allows high school students access to powerful tools for making sense of large data sets.

Visual Geometry With Geometric Supposer

One of the best-known visualization tools is Geometric Supposer (http://cet.ac.il/ math-international/software5.htm), a tool for making and testing conjectures in geometry through the process of constructing and manipulating geometric objects and exploring the relationships within and between these objects (Schwartz & Yerushalmy, 1987). Geometric Supposer allows students to choose a primitive shape, such as a triangle, and construct it by defining points, segments, parallels or perpendiculars, bisectors, or angles (Yerushalmy & Houde, 1986). The program plots and remembers each manipulation and can apply it to similar figures. For example, if the students conjecture that “a median drawn from the vertex of any triangle to the opposite side bisects the angle,” they can test it easily by asking Geometric Supposer to measure the angles or by applying the relationship to several other triangles. The students will learn immediately that the conjecture is not true. Constructing these test examples manually would require more effort than students are likely to generate, but the computational power of the computer makes this testing very easy.

Geometry instruction is traditionally based on the application of theorems to prove that certain relationships exist among objects. This top-down approach requires analytic reasoning, which a majority of students find difficult. Geometric Supposer supports the learning of geometry by enabling the student to inductively prove these relationships by manipulating the components of geometric objects and observing the results. Rather than having the student apply someone else’s logic, Geometric Supposer makes explicit the relationships between visual properties and the numerical properties of the objects (Yerushalmy, 1990). Rather than using the computer to provide conclusive results, the computer calculates the results of students’ experiments. The research results with Geometric Supposer have been consistently positive.