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The resulting economy of effort is very usefulprovided that in making an abstraction, care is taken not to ignore features that play a significant role in determining the outcome of the events being studied.
After abstractions have been made and symbolic representations of them have been selected, those symbols can be combined and recombined in various ways according to precisely defined rules.
Such abstraction enables mathematicians to concentrate on some features of things and relieves them of the need to keep other features continually in mind.
As far as mathematics is concerned, it does not matter whether a triangle represents the surface area of a sail or the convergence of two lines of sight on a star; mathematicians can work with either concept in the same way.
As mathematics has progressed, more and more relationships have been found between parts of it that have been developed separatelyfor example, between the symbolic representations of algebra and the spatial representations of geometry.
These cross-connections enable insights to be developed into the various parts; together, they strengthen belief in the correctness and underlying unity of the whole structure. Many mathematicians focus their attention on solving problems that originate in the world of experience.
Or they might tackle the area/volume problem as a step in producing a model for the study of crystal behavior.
This is so for several reasons, including the following: Using mathematics to express ideas or to solve problems involves at least three phases: (1) representing some aspects of things abstractly, (2) manipulating the abstractions by rules of logic to find new relationships between them, and (3) seeing whether the new relationships say something useful about the original things.
Mathematical thinking often begins with the process of abstractionthat is, noticing a similarity between two or more objects or events.
In deriving, for instance, an expression for the change in the surface area of any regular solid as its volume approaches zero, mathematicians have no interest in any correspondence between geometric solids and physical objects in the real world.
A central line of investigation in theoretical mathematics is identifying in each field of study a small set of basic ideas and rules from which all other interesting ideas and rules in that field can be logically deduced.
To achieve this, students need to perceive mathematics as part of the scientific endeavor, comprehend the nature of mathematical thinking, and become familiar with key mathematical ideas and skills.