Tools can also provide concrete models of abstract ideas, or, when dealing with complex problems, they can enable students to manipulate and think about ideas, thereby making mathematics accessible and more deeply understood. Mathematical learning tools can be traditional, technological, or social. The most frequently employed tools are traditional, which include physical objects or manipulatives e.
Technological tools, such as calculators i.
Social tools, such as small-group discussions where students interact with one another to share and challenge ideas, can be considered a third type of learning tool. These three tools can be used independently or conjointly, depending on the type of learning that is intended. A learning tool can be as simple as an image or as complex as a computer-based environment designed to improve mathematical understanding. The key characteristic of a learning tool is that it supports learners in some manner.
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For example, a tool can aid memory, help students to review their problem-solving processes, or allow students to compare their performance with that of others, thereby supporting self-assessment. Learning tools can represent mathematical ideas in multiple ways, providing flexible alternatives for individuals who differ in terms of learner characteristics. For example, learners who have difficulty understanding the statistical ideas of arithmetic mean center and variance spread may be assisted through interactive displays that change as data points are manipulated by the learner.
A mathematical learning tool can scaffold the learner by performing computations, providing more time for students to test mathematical hypotheses that require reasoning. In the statistics example, learners can focus on why changes to certain parameters affect data—and in what ways, rather than spending all their time calculating measures.
Traditional Tools. Traditional tools are best suited for facilitating students' learning of basic knowledge and skills. Objects that can be manipulated, such as cubes, reduce the abstract nature of concepts, such as numbers, thereby making them real and tangible, particularly for younger children.
Such tools support the development of children's understanding of arithmetic by serving as a foundation for learning more complex concepts. Visualization tools, such as graphs, can support data interpretation, while paper-and-pencil tools that provide practice of computational skills can support memory for procedures and an ability to manipulate symbols.
Combining physical tools with visualization tools can substantively increase students' conceptual knowledge.
Dice and spinners, for example, can be used to support elementary school students in creating graphs of probability distributions, helping them develop an understanding of central tendency. Technological Tools. Technological tools are most effective in facilitating students' understanding of complex concepts and principles. Computations and graphs can be produced quickly, giving students more time to consider why a particular result was obtained.
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This support allows students to think more deeply about the mathematics they are learning. Electronic tools are necessary in mathematics because they support the following processes: a conjectures—which provide access to more examples and representational formats than is possible by hand; b visual reasoning—which provides access to powerful visual models that students often do not create for themselves; c conceptualization and modeling—which provide quick and efficient execution of procedures; and d flexible thinking—which support the presentation of multiple perspectives. Spreadsheets, calculators, and dynamic environments are sophisticated learning tools.
These tools support interpretation and the rapid testing of conjectures. Technology enables students to focus on the structure of the data and to think about what the data mean, thereby facilitating an overall understanding of a concept e. The graphics calculator supports procedures involving functions and students' ability to translate and understand the relationship between numeric, algebraic, and graphical representations.
Transforming graphical information in different ways focuses attention on scale changes and can help students see relationships if the appropriate viewing dimensions are used. Computers may remove the need for overlearning routine procedures since they can perform the task of computing the procedures. It is still debatable whether overlearning of facts helps or hinders deeper understanding and use of mathematics.
Technology tools can also be designed to help students link critical steps in procedures with abstract symbols to representations that give them meaning. Video is a dynamic and interactive learning tool.
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One advantage of video is that complex problems can be presented to students in a richer and more realistic way, compared to standard word problems. Students are required to solve problems encountered by characters in the Woodbury video by taking many steps to find a solution. This tool supports students' ability to solve problems, specifically their ability to identify and formulate a problem, to generate subgoals that lead to the solution, and to find the solution.
However, the information presented in a video cannot be directly manipulated in the same way that data can be changed in spreadsheets and calculators.
Learning tools that present the same information in several ways e. The ability to interpret multiple representations is critical to mathematical learning. There is evidence to suggest that multiple representations can facilitate students' ability to understand and solve word problems in functions, and to translate words into tables and graphs.
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