A vehicle is anything through or by which something, such as thought, power, or information, is conveyed, transmitted, expressed, or achieved. The side face of the cube represents Vehicles for Understanding and Learning: mathematical structure, patterns and relations, computation and estimation, measurement and scale, and models. They can be thought of as ways of perceiving and relating ideas, and as approaches to solving problems.
Mathematical Structure, Patterns and Relations, Computation and Estimation,
Measurement and Scale, and Models
Mathematical Structure provides students with the mathematical tools necessary to design their own constructs for problem solving: What do we know about the possible answer to this question ? Can the answer be negative? Can it be a fraction? Can there be more than one answer?
Patterns and Relations give students a mechanism for seeing and developing a variety of ways of approaching problems: Do you see a pattern here? What do all of these numbers have in common? How are these two figures alike?
Computation and Estimation are procedures for checking, supposing visualizing, and setting parameters: About how many one-inch cubes would fit in this box? Is the answer on your calculator consistent with your estimate?
Measurement and Scale include measuring and using ratio and proportion: If John is four feet six inches tall, how tall should his picture be in our scale drawing?
Models can be physical or mathematical models which can help students visualize and construct mathematical knowledge and to demonstrate mathematical properties or expressions. By using physical models (manipulatives), "hands-on" activities) and computer activities, students have concrete experiences with which to solve problems: Arrange the centimeter cubes to illustrate all of the possible factors of twelve.
Mathematical models describe real-world problems and phenomena in science, social science, economics, architecture, and the like. Examples can be taken from the student's culture and experience. For example, the equation of a line can model a real-world phenomenon such as CO2 emissions from automobiles. Computers and other technological tools can be used to create a mathematical model for complex phenomena such as the prediction of food supply conditions around the world given certain conditions of supply and demand.
The Vehicles for Understanding and Doing presented in Table 2 are incorporated into investigations across many areas of mathematics and other disciplines. Like the Habits of Mind, these are ways of doing and approaching problems, and tools for problem-solving in mathematics and other disciplines.
