I am a teacher who teaches mathematics. And one of the most common requirements a teacher has of his or her students when working on mathematics questions is to present the workings to the solution clearly. Why is this so?
It has everything to do with reasoning. One of the main aims of education is to impart the skills of logical reasoning to students. And Mathematics is the key to understanding deductive reasoning. Everything that happened has a reason and everything that we do have a reason. The problem is not the absence of reason but the communication of reason. There are reasons which are valid and there are reasons which are not. But a teacher will never know the reasoning of a student unless the reason is communicated clearly. And that is the purpose of clear working. It is for the teacher to examine in order to validate the reasoning without which, the teacher cannot educate the student properly on the right way to exercise reasoning.
There are a few major categories of reasoning. The most fundamental of them has to be analogical reasoning, inductive reasoning and deductive reasoning.
Analogical reasoning is the act of transferring information from one context to another context. Another way of saying would be to reason based on examples. This is the most direct and straightforward way of reasoning and is commonly found especially when we see the words “for example,” in any article. This form of reasoning is useful because it is easy to digest for the learners and connections can be made instantly.
An elaborate way of applying analogical reasoning is the use of case studies. We use case studies in the educational field of humanities mainly because the subject matter simply has too many factors to be considered for the reasoning to be portable. We cannot know what will definitely happen is we were to adjust our taxes in a certain way or when we change any major governmental policies in the country. So all we can do is to case studies of similar situation and make a prediction.
We use analogical reasoning when we have no other choice due to the nature of the subject matter. This is because analogical reasoning is not always valid. In fact, analogical reasoning is a major source of errors students make in mathematics.
In algebra, we know that a(b + c) = ab + ac. This is always true. However, if we were to apply analogical reasoning, then we might say that sin (A + B) = sin A + sin B which is false. In the former, it is algebraic operation while in the latter, it is a function. Analogical reasoning failed here.
Another example is when a fellow teacher said in jest that “we learn mathematics to be mathematicians, we learn science to be scientists and we learn humanities to be human.” This is another analogical reasoning. But while is true that mathematicians and scientists tend to be good in mathematics and sciences; they are identified with their skills. But being a good human has nothing to do with skills, but rather, it has to do with our values. Hence, the reasoning is invalid here.
Inductive reasoning is stronger than analogical reasoning whereby the focus is only on a single context and that it does not use one, but many examples. Inductive reasoning is the bedrock of the sciences.
Inductive reasoning involves observing instances of certain phenomenon and then creating a general theory to explain it. This is done in all science experiments. Let’s say we hang a weight to stretch a mechanical spring. When we use a different weight, the spring stretches a different amount. By having enough sets of data from this experiment, we can induct that the amount by which a material body is stretch is linearly related to the force (weight) acting on it.
One of the most common ways to test inductive reasoning in mathematics is number patterns. Let say we have 2, 4, 6, 8, _ , _ . Getting students to fill in the blanks is a form of inductive reasoning. By using the samples and recognizing the pattern, we can induce that this is a list of even numbers.
But inductive reasoning can only provide a probable generalization because it is vulnerable to exceptions. Using algebra again, we have (p + q)/x = p/x + q/x. This is mostly true and we can test it using many different values p, q and x to test the statement. But because the statement is false when x = 0, this statement is not always valid.
It is for this reason the basis of science is that it will never claim to be the absolute truth but only offer a probable answer to any phenomenon because it allows itself to be falsified by exceptions.
Finally, deductive reasoning is the system of reasoning where we figure things out by using premises. Now, let’s say we are given that x^2 – 4x + 3 = 0, we can solve for x if we are to use the premise that when AB = 0, then either A or B must be equal to 0. Hence, we factorize the statement to become (x – 3)(x – 1) = 0. And then from the premise, we have (x – 3) = 0 or (x – 1) = 0. That gives us x = 3 or 1. We have just managed to solve this quadratic equation because of deductive reasoning.
Deductive reasoning is a form of reasoning that works all the time and is the most powerful form of reasoning. It is for this purpose that mathematics is often emphasized in schools or as a foundation for advance scholarly pursuit. Deductive reasoning is always valid.
But of course the problem with deductive reasoning is that it must begin with a strong premise. But this is not always possible, which is why the study of inductive reasoning and analogical reasoning is still necessary.
In conclusion, when we want to understand any phenomenon, we begin by trying to perform deductive reasoning like in mathematics. But when there are no suitable premises, we use inductive reasoning to make sense of the world around us like the sciences. But in cases where the factors are too numerous and complex like the humanities, we have to resort to analogical reasoning. And this is how we learn reasoning in our schools through mathematics, sciences and humanities.
It has everything to do with reasoning. One of the main aims of education is to impart the skills of logical reasoning to students. And Mathematics is the key to understanding deductive reasoning. Everything that happened has a reason and everything that we do have a reason. The problem is not the absence of reason but the communication of reason. There are reasons which are valid and there are reasons which are not. But a teacher will never know the reasoning of a student unless the reason is communicated clearly. And that is the purpose of clear working. It is for the teacher to examine in order to validate the reasoning without which, the teacher cannot educate the student properly on the right way to exercise reasoning.
There are a few major categories of reasoning. The most fundamental of them has to be analogical reasoning, inductive reasoning and deductive reasoning.
Analogical reasoning is the act of transferring information from one context to another context. Another way of saying would be to reason based on examples. This is the most direct and straightforward way of reasoning and is commonly found especially when we see the words “for example,” in any article. This form of reasoning is useful because it is easy to digest for the learners and connections can be made instantly.
An elaborate way of applying analogical reasoning is the use of case studies. We use case studies in the educational field of humanities mainly because the subject matter simply has too many factors to be considered for the reasoning to be portable. We cannot know what will definitely happen is we were to adjust our taxes in a certain way or when we change any major governmental policies in the country. So all we can do is to case studies of similar situation and make a prediction.
We use analogical reasoning when we have no other choice due to the nature of the subject matter. This is because analogical reasoning is not always valid. In fact, analogical reasoning is a major source of errors students make in mathematics.
In algebra, we know that a(b + c) = ab + ac. This is always true. However, if we were to apply analogical reasoning, then we might say that sin (A + B) = sin A + sin B which is false. In the former, it is algebraic operation while in the latter, it is a function. Analogical reasoning failed here.
Another example is when a fellow teacher said in jest that “we learn mathematics to be mathematicians, we learn science to be scientists and we learn humanities to be human.” This is another analogical reasoning. But while is true that mathematicians and scientists tend to be good in mathematics and sciences; they are identified with their skills. But being a good human has nothing to do with skills, but rather, it has to do with our values. Hence, the reasoning is invalid here.
Inductive reasoning is stronger than analogical reasoning whereby the focus is only on a single context and that it does not use one, but many examples. Inductive reasoning is the bedrock of the sciences.
Inductive reasoning involves observing instances of certain phenomenon and then creating a general theory to explain it. This is done in all science experiments. Let’s say we hang a weight to stretch a mechanical spring. When we use a different weight, the spring stretches a different amount. By having enough sets of data from this experiment, we can induct that the amount by which a material body is stretch is linearly related to the force (weight) acting on it.
One of the most common ways to test inductive reasoning in mathematics is number patterns. Let say we have 2, 4, 6, 8, _ , _ . Getting students to fill in the blanks is a form of inductive reasoning. By using the samples and recognizing the pattern, we can induce that this is a list of even numbers.
But inductive reasoning can only provide a probable generalization because it is vulnerable to exceptions. Using algebra again, we have (p + q)/x = p/x + q/x. This is mostly true and we can test it using many different values p, q and x to test the statement. But because the statement is false when x = 0, this statement is not always valid.
It is for this reason the basis of science is that it will never claim to be the absolute truth but only offer a probable answer to any phenomenon because it allows itself to be falsified by exceptions.
Finally, deductive reasoning is the system of reasoning where we figure things out by using premises. Now, let’s say we are given that x^2 – 4x + 3 = 0, we can solve for x if we are to use the premise that when AB = 0, then either A or B must be equal to 0. Hence, we factorize the statement to become (x – 3)(x – 1) = 0. And then from the premise, we have (x – 3) = 0 or (x – 1) = 0. That gives us x = 3 or 1. We have just managed to solve this quadratic equation because of deductive reasoning.
Deductive reasoning is a form of reasoning that works all the time and is the most powerful form of reasoning. It is for this purpose that mathematics is often emphasized in schools or as a foundation for advance scholarly pursuit. Deductive reasoning is always valid.
But of course the problem with deductive reasoning is that it must begin with a strong premise. But this is not always possible, which is why the study of inductive reasoning and analogical reasoning is still necessary.
In conclusion, when we want to understand any phenomenon, we begin by trying to perform deductive reasoning like in mathematics. But when there are no suitable premises, we use inductive reasoning to make sense of the world around us like the sciences. But in cases where the factors are too numerous and complex like the humanities, we have to resort to analogical reasoning. And this is how we learn reasoning in our schools through mathematics, sciences and humanities.
