OP
OP
sravna
Well-known member
The concepts in Math just as in any other subject, evolve from the elementary to the advanced.
The ability we need for each level is corresponding to that level.
In elementary Math, all the student needs to do is to remember what is taught. This is basically rote ability. We are taught the basic numerals and the basic concepts of addition, multiplication etc. Innovative techniques are being used to make the child remember these but If the student just remembers what is taught, that is all that is required to pass that stage. As an example the child needs to remember 1+2 = 3 and 3*11=33 and so on. So at this level rote ability is all that is required.
Once the student goes past that rote learning stage, he needs to start understanding what the problem is because the operations do not obviously present themselves but are presented in a more abstract way than before. For example consider the problem: if a man ate 1 pizza yesterday and ate 2 pizzas today, how many pizzas did he eat in the two days? In the above problem the student should be able to understand that he has to perform an addition of the number of pizzas eaten yesterday and today, to get the result as 3 pizzas. So the ability required is the ability to understand what is stated in the problem in terms of what operations have to be performed on the numbers.
Then comes the ability to reason. This stage is usually from the high school up to the undergraduate level. The students need the ability to employ reasoning to specify the problem in an abstract way and solve the abstract representation of the problem. For example if the problem is as follows: The number of pizzas a person ate today is one more than he ate yesterday. Also if the number of pizzas he eats today is 1 more than the difference between the number of pizzas he ate today and yesterday, how many pizzas did he eat in the two days?
This can be solved by formulating equations. The equations are the follows if x and y represent the number of pizzas eaten today and yesterday respectively.
x-y = 1
(x-y) + 1= x
In the above problem, one is using the method of specifying the unknown as some alphabets and relating them with what is known, in an equation. So basically, by your reasoning you are creating a specification and a general method or procedure to solve a problem. The result is the problem need not explicitly state the value of what needs to be added, subtracted etc. but those values can be found out after following a logical procedure.
This is important because in many real world instances, we may not know the exact values but only information such as specified in the above problem. Thus we use reasoning to specify the problem in an abstract way and using a procedure to find out the exact values. Where there are no specific values for the unknown, we can from the specification of the problem find the relationship among the various unknown using a logical procedure so that with such a relationship, we can make assumptions and plug in values and see what the results are or we can find the unknown based on the values of the known. For example in both the equations given earlier, if the value 1 is not always true , we can replace them with say m and n and use appropriate values of m and n and find the corresponding values of x and y. Thus reasoning gives us the ability for an abstract specification and a method to find a solution using that specification.
The next level of ability is the ability to create the relationship as above when the information in the problem and other known information do not enable us to formulate such relationships. This is where the intuitive ability comes in. The famous Einstein equation involving energy and mass, E = m(C^2) is not something that could have been arrived at using any of the then existing relationships. One needs to think about what the relationship could be between energy and mass and have an intuitive grasp of the solution.
The intuitive grasp here is the equivalence of mass and energy. Then you arrive at that equation by using some methods as discussed in the above paragraph, based on the formulation of the problem according to the intuition. Though intuition can also happen in pure mathematics, intuition generally has to do more with one’s natural ability for it than with one’s familiarity with Math. The striking case is that of Srinivasa Ramnujan.
It may finally boil down to 1+2=3 or something similar but the effort taken to arrive at that depends on the level of abstraction of the concepts used. The advantage in identifying and separating into levels is that we can generally specify what techniques we can employ to solve the problem. For example there may be general approaches to solve a problem using logical reasoning and so such approaches can be used to solve any such problem.
We will try to focus on the general techniques for each level to help solve the problem.
The ability we need for each level is corresponding to that level.
In elementary Math, all the student needs to do is to remember what is taught. This is basically rote ability. We are taught the basic numerals and the basic concepts of addition, multiplication etc. Innovative techniques are being used to make the child remember these but If the student just remembers what is taught, that is all that is required to pass that stage. As an example the child needs to remember 1+2 = 3 and 3*11=33 and so on. So at this level rote ability is all that is required.
Once the student goes past that rote learning stage, he needs to start understanding what the problem is because the operations do not obviously present themselves but are presented in a more abstract way than before. For example consider the problem: if a man ate 1 pizza yesterday and ate 2 pizzas today, how many pizzas did he eat in the two days? In the above problem the student should be able to understand that he has to perform an addition of the number of pizzas eaten yesterday and today, to get the result as 3 pizzas. So the ability required is the ability to understand what is stated in the problem in terms of what operations have to be performed on the numbers.
Then comes the ability to reason. This stage is usually from the high school up to the undergraduate level. The students need the ability to employ reasoning to specify the problem in an abstract way and solve the abstract representation of the problem. For example if the problem is as follows: The number of pizzas a person ate today is one more than he ate yesterday. Also if the number of pizzas he eats today is 1 more than the difference between the number of pizzas he ate today and yesterday, how many pizzas did he eat in the two days?
This can be solved by formulating equations. The equations are the follows if x and y represent the number of pizzas eaten today and yesterday respectively.
x-y = 1
(x-y) + 1= x
In the above problem, one is using the method of specifying the unknown as some alphabets and relating them with what is known, in an equation. So basically, by your reasoning you are creating a specification and a general method or procedure to solve a problem. The result is the problem need not explicitly state the value of what needs to be added, subtracted etc. but those values can be found out after following a logical procedure.
This is important because in many real world instances, we may not know the exact values but only information such as specified in the above problem. Thus we use reasoning to specify the problem in an abstract way and using a procedure to find out the exact values. Where there are no specific values for the unknown, we can from the specification of the problem find the relationship among the various unknown using a logical procedure so that with such a relationship, we can make assumptions and plug in values and see what the results are or we can find the unknown based on the values of the known. For example in both the equations given earlier, if the value 1 is not always true , we can replace them with say m and n and use appropriate values of m and n and find the corresponding values of x and y. Thus reasoning gives us the ability for an abstract specification and a method to find a solution using that specification.
The next level of ability is the ability to create the relationship as above when the information in the problem and other known information do not enable us to formulate such relationships. This is where the intuitive ability comes in. The famous Einstein equation involving energy and mass, E = m(C^2) is not something that could have been arrived at using any of the then existing relationships. One needs to think about what the relationship could be between energy and mass and have an intuitive grasp of the solution.
The intuitive grasp here is the equivalence of mass and energy. Then you arrive at that equation by using some methods as discussed in the above paragraph, based on the formulation of the problem according to the intuition. Though intuition can also happen in pure mathematics, intuition generally has to do more with one’s natural ability for it than with one’s familiarity with Math. The striking case is that of Srinivasa Ramnujan.
It may finally boil down to 1+2=3 or something similar but the effort taken to arrive at that depends on the level of abstraction of the concepts used. The advantage in identifying and separating into levels is that we can generally specify what techniques we can employ to solve the problem. For example there may be general approaches to solve a problem using logical reasoning and so such approaches can be used to solve any such problem.
We will try to focus on the general techniques for each level to help solve the problem.
