Section Editor: Alex Moore
In this section, we feature websites, online manipulatives, and web-based application that are appropriate for K-12 mathematics instruction. We are looking for critical reviews of technologies which focus on both the benefits and limitations of using these tools in a K-12 mathematics classroom. If you use a technological tool and wish to share with us, please respond to the Call for Manuscripts.
Kuta Software's Infinite Family
One of the well-known software packages for secondary mathematics teachers is Kuta Software’s Infinite family, which offers versions for Pre-Algebra through Calculus I. It provides a comprehensive database of problems that can be regenerated “infinitely” under selectable and specific parameters. The benefit of this software is that it makes it easy for the teacher to build custom problem sets targeting specific skill development. The software handles all the formatting and layout and produces answers to each problem, and can even print multiple version of each assignment. The creators have even built in some “Preset” options that will, with one click, produce Easy, Medium, or Hard problems of any kind. The simplest use it to click on one of the “Preset” options and identify the number of problems you want. However, Infinite, when used more thoughtfully, becomes a sophisticated tool to design problem sets. Explicitly, Infinite can be used to tailor and scaffold cognitive demand. Leinwand, Brahier, Huinker, Berry, Dillon, Larson, Leiva, Martin, and Smith (2014, p. 18) described this as switching between Lower-Level Demand problems (procedures without connections to mathematical conceptual understanding) and Higher-Level Demand problems (procedures with connections to conceptual understanding).
Consider the teacher presenting the concept of Absolute Value Equations to her Algebra I class. Creating a new problem set of absolute value equation problems is as easy as opening Infinite Algebra I, searching for “Absolute Value Equations” in the search field, clicking the question subtype of Easy, Medium, or Hard, and entering the quantity of questions desired. The assignment is done, but the teacher is missing an opportunity to develop a cognitive demand sequence in the problem set. Suppose the problem set has sixteen problems. Figure 1 illustrates the options screen with the different parameter options. First choose from the “integers,” “no operations inside,” and “no operations outside” subset of parameters. The program will produce a basic problem such as |r| = 2, which will develop a basic sense of the absolute value concept that reiterates the direct instruction the student experienced in class (Figure 2). The teacher selects two of this type of problem type for the problem set. Next, change the parameter settings to include “one operation inside,” which produces a problem such as | n / 6| = 2, which will further develop a student’s sense of absolute value in relation to the unknown quantity (Figure 3). The teacher chooses two of this type of problem to add to the problem set. Next, the complexity increases to include “one operation inside” and “one operation outside,” which produces a problem such as –1 + |6x| = 5. This builds on the previous problem types by showing the absolute value’s relationship to other terms in the equation (Figure 4). This progression continues until “two operations inside” and “two operations outside,” to produce a problem such as 4 - 2|5x - 5| = -36. This type of problem requires more process-based cognitive reasoning from the student (Figure 5). In the end, the problem set has “layers” of problems that begin with a basic form and increases in complexity (Figure 6). Building the cognitive demand, the teacher is supporting students as they build their mathematical skills from a “beginning” level (basic conceptual understanding) to an “extended” level.
|Figure 1. The options screen in Kuta Algebra I for Absolute Value Equations.||Figure 2. Example problems produced from using "no operations outside" and "no operations inside" parameters.||Figure 3. Example problems produced from using "no operations outside" and "one operation inside" parameters.|
|Figure 4. Example problems produced from using "one operation outside" and "one operation inside" parameters.||Figure 5. Example problems using multiple operations both inside and outside.||Figure 6. Outline of problem types, showing multiple groups of problems, each with one or two problems each, showing the use of intentional progression of cognitive demand.|
Leinwand, S., Brahier, D. J., Huinker, D., Berry, R. Q. III, Dillon, F. L., Larson, M. R., Leiva, M. A., Martin, W. G., & Smith, M.S. (2014). Principles to action: Ensuring mathematical success for all. Reston, VA: National Council of Teachers of Mathematics.