Mathematics Instruction and Technology
For students to make sense of mathematics, they must develop deep understanding that leads to “higher levels of generalizing or abstraction” according to the National Council of Teachers of Mathematics (NCTM). Technology can be used for sense making. But not all technology provides an experience that enhances mathematical understanding.
In the article, “Technology with Cognitive and Mathematical Fidelity: What it Means for the Math Classroom,” published in Computers in the Schools 26.2 (2009) the author, Beth Bos, discusses the value of various web-based formats for deepening mathematical understanding. Two criteria are used to make value judgments. The first, mathematical fidelity, is described as “an object’s conformity to mathematical accuracy.” The second, cognitive fidelity, “refers to whether a concept is better understood when the object is acted on.” Taken together, these two criteria show how well computer generated objects are able to support mathematical understanding.
The web-based formats are organized into six categories which are assessed for their purpose, strength, weakness, and cognitive fidelity. Cognitive fidelity is significant since it allows the user of the web-based format to make connections through patterns that emerge as the math object is manipulated. Equally as important, the mathematical content of the object should not be limited by the software, and interaction with the object should lead to conceptual understanding of the mathematics being explored. Based on these criteria, all technology is not created equally.
Of the six formats studied, four of them showed low cognitive fidelity. The game, informational, quiz, and static tools that generate calculations, tables, or graphs formats did not enhance student understanding of mathematical concepts. The game format enforces reactive behavior rather than in-depth exploration toward deeper mathematical understanding. The informational format is typically fact based with no interactive or intuitive components (e.g., PowerPoint presentations.) The quiz format facilitates checking for understanding which reinforces the recall level of performance. The static tools (e.g., graphing calculators) emphasize procedural steps to find the answer rather than the logic behind each step and sense-making of the logic.
The virtual manipulatives (e.g., National Library of Virtual Manipulatives) showed medium-high cognitive fidelity. These objects help to make abstract concepts more concrete. They function best as building blocks for understanding. They require direct instruction and careful monitoring to ensure that students are able to create the desired results.
The interactive math objects were the only format found to have high cognitive fidelity. The strength of this format lies in its ability to show relationships based on varying information. These relationships change with the input allowing users to make and test conjectures. Instead of focusing on whether an answer is correct, the focus is shifted to building a deeper understanding of the mathematical concepts. Since exploration relies on problem solving, less direct instruction is involved which can be considered a disadvantage. But overall, this format is advocated by NCTM.
My considerations for applying the information presented in this article are that the four formats showing low cognitive fidelity can still serve a useful purpose in the mathematics classroom. Although recall requires lower level understanding, mathematical skills are built from recalled information, as an example, addition and multiplication facts. Games can be used to build basic skills without which students are unable to function in higher level mathematics. Mathematical information in a web-based format can be easily accessed and used to survey explanations of topics as a student works on initial understanding. The quiz format is an effective means of allowing students opportunities for extra-credit and must offer more benefit than extra-credit earned by bringing in classroom supplies. The static tools are unlikely to provide cognitive fidelity, but they are no less valuable than the tables of reference data included in older textbooks printed before handheld calculators.
The virtual manipulatives, although described to be more difficult to learn how to use, would provide a means of enrichment for students who are performing at an advanced level compared with their peers. Free online resources such as these provide challenging material for students capable of being challenged. These types of resources would also provide opportunities for self-directed exploration.
The interactive math objects provide a path to learning through problem solving which is considered by some to be the ultimate mathematical instructional method. Textbooks are now being written based on the problem solving methodology, but textbook changes are expensive and slow in school districts. The interactive math objects can be used to supplement conventional mathematical texts to provide opportunities for students to develop a deeper level of mathematical understanding.
The article’s summary states that “the influence of technology on the teaching of mathematics is that not all technology results in increased understanding of mathematics.” This assessment is extremely important since technology for technology’s sake cannot be assumed to improve outcomes. Appropriate technologies used appropriately after thoughtful consideration of their added value, if any, is an important factor required for the sensible use of technology in the mathematics classroom.
