Maths is an interesting subject that can get really challenging sometimes. It is a subject that interests few and repels many. However the few that it interests are those that understand the true beauty of this disciple and realize that no other subject can be studied without a basic understanding of mathematics. Moreover, almost all the processes and phenomenon that occur naturally are somehow based on maths or can be explained mathematically. For instance, when we calculate how much time remains to our lunch break or when we calculate how much change we shall receive on paying with a ten dollar bill, we use the simple concepts of mathematics. Some would argue that this is something basic and not related to pure mathematics. In that case, take the example of Fourier series that can be used to convert equations of any curve to a series of sines and cosines which represents a straight line; this is exactly what we do when we convert analog signal to digital signal or alternating current to digital current. Moving on, we can explain the movement of planets by the elliptical motion that comes under the section of conics in calculus, a branch of mathematics.
When we talk about mathematical knowledge, we commonly use the words concept, skill, theory, model etc. These are not all the same and it must be noted that specifically in the field of mathematics, these words have specific meanings and differences. The two words that we will focus on in this article are skill and concept as used in the context of mathematics. The simplest of differences between these two is that concept is merely knowing the way to do something in theory. This means that a person who knows how to perform an operation has the concept; he or she understands how a certain operation should be performed and can explain it to others. Having mathematical skill is something different. To be skilled means to be able to perform what you have the concept of. This means that a person can only be called skilled if he or she not only knows the concept but can also apply it in the proper manner. Going into further detail, a skilled person is also expected to know the various issues or problems that can arise when dealing with a mathematical operation. This is because if the skilled person knows how to perform it then he or she is expected to have performed it and realized how the operation is different from its theory.
We can also conclude from this difference that to have skill means that having the concept is a must. It is not possible to have the skill if a person does not have the concept of something. The converse of this is not true; a person need not have skill to have the concept.
Many a times in mathematics, a certain way of solving an equation or any mathematical operation is used that has certain contradictions or exceptions. This means that the formula or the way it is solved is valid at all times except when a certain condition is not met. A person who merely has the concept may not know about this as they have never actually applied it before. Even if they know about it from certain literature, they may not be able to explain the reason. On the other hand, if a person has mathematical skill, he or she can not only point out the exceptional cases but also explain the reason for the exception.
Summary of differences expressed in points
- Concept is merely knowing the way to do something in theory, a person who knows how to perform an operation has the concept, he or she understands how a certain operation should be performed and can explain it to others; skilled means to be able to perform what you have the concept of, a skilled person is also expected to know the various issues or problems that can arise when dealing with a mathematical operation, if the skilled person knows how to perform it then he or she is expected to have performed it and realized how the operation is different from its theory
- To have skill means that having the concept is a must; the converse of this however, is not true
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I am not sure that this is an accurate description of skills versus concepts. I think a person can be skilled at the solving of math problems without necessarily fully understanding the concepts involved whereas on the other hand a person who is able to conceptualize and move theories and concepts forward needs to have a firm grasp of the skills needed to arrive at specific conclusions. For example a scientist who creates a theory (concept) would need to show the path at which he arrived at said concusion (utilizing skills) but on the otherhand coud hand off his concept to a skilled scientist whom without understanding the actual concepts could still utilize a skillset to prove or disprove the concept. A math genious creates the theory of relativity and shows the math any student with the skills to follow the math could solve the problem without fully grasping the concept
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It would seem one could conceptualize without having the tools to prove or disprove as well this is done by entrepeneurs all the time. Usually leading them to those with the skills to get to the end result desired. Now this of course is probably very rare in a world as specialized as math but is nonetheless a possibility therefore neither of the two are necessities to each other.
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