Reflections on Problem-Solving
Problem-solving is one of the major elements found in the intersection of the set of all mathematical fields. Students and professionals of mathematics are asked to problem-solve when they are instructed to find the sum of two numbers for a kindergarten teacher, to determine the height of a building for an architect, to calculate the probability of an individual's involvement in a car accident for an insurance company, or to find the most cost-effective route through a town for the postal service.
To understand how the single subject of mathematics can handle such a variety of problems, we can view problem-solving in math as decoding; every problem is translated into the mathematical language. For instance, a town becomes a graph whose intersections are vertices and whose streets are edges, and a roof bordered by a main ceiling beam becomes a right triangle. In such forms, the answer, or at least the path to the answer, often coalesces. Performing such conversions dramatically reduces the size of the problem-solving toolbox to the set of tools with which a mathematician is familiar.
However, the reality-to-mathematics translations discussed above are not necessarily intuitive. An individual needs experience to recognize that arranging non-overlapping utility pipes and wires in a town is a problem that can solved using the properties of planar graphs, for example. Without time and practice working with theorems and other tools of mathematics, one cannot hope to remember their uses or to see their applications.
The problem below is my favorite problem because it can be solved using the ideas discussed above and is an example of mathematics in biology. As an aspiring biostatistician, I will often evaluate scientific techniques and pharmaceuticals using mathematics.
To understand how the single subject of mathematics can handle such a variety of problems, we can view problem-solving in math as decoding; every problem is translated into the mathematical language. For instance, a town becomes a graph whose intersections are vertices and whose streets are edges, and a roof bordered by a main ceiling beam becomes a right triangle. In such forms, the answer, or at least the path to the answer, often coalesces. Performing such conversions dramatically reduces the size of the problem-solving toolbox to the set of tools with which a mathematician is familiar.
However, the reality-to-mathematics translations discussed above are not necessarily intuitive. An individual needs experience to recognize that arranging non-overlapping utility pipes and wires in a town is a problem that can solved using the properties of planar graphs, for example. Without time and practice working with theorems and other tools of mathematics, one cannot hope to remember their uses or to see their applications.
The problem below is my favorite problem because it can be solved using the ideas discussed above and is an example of mathematics in biology. As an aspiring biostatistician, I will often evaluate scientific techniques and pharmaceuticals using mathematics.