Linear Counting- Addition and Subtraction
When we walk we take one step at a time. When we run we still take one step at a time! And when we jump the situation does not change. One jump at a time! Till the time we don’t experience a firewall, we shall continue with the assumption that mathematics is all about counting-measuring, weighing and finding rates being the other forms of counting.
Imagine an athlete practising in a field for a competition. He settles himself in a lane takes a deep breath and starts running. 1st step-1. 2nd step-2. 3rd step-3…..and so on. He counts the number of steps. In other words he adds 1 to the previous value to arrive at the present count. So counting is a process of addition. One added to one becomes 2. Counting and addition are not different from each other. Addition precedes counting!
Now imagine two athletes practising simultaneously. And both are counting the steps. After a fixed time, say 30 minutes they stop and compare their results. How do they do it? They approach the coach. The coach thinks and then comes up with the idea. The first lane will carry the symbol + (plus) and the second – (minus) just to differentiate one from the other. The symbol minus (-) doesn’t carry a stigma, of course. He therefore assigns the count of the first lane the symbol + and the other one -. It is something like one athlete wearing a shirt with + printed on it and the other wearing – printed on it.
Till we find a better denomination, let us designate the two lanes: the positive lane and the negative lane although the negative one is as positive as the other.
Result: + lane= +1+1+1+1= +4
Result: – lane= -1-1-1-1-1-1=-6
As is clear the symbols + & – do not tell us about the direction in which the athletes are running. Both are moving in the same forward direction much like the time which moves linearly into the future. Both athletes count their progress and when the coach looks at the result he declares the negative lane runner the winner because 6 is numerically bigger than 4. The second next thing he does is by how many numbers (or by how much) does the winner take the cake! He again counts forward from 4 to 6 to arrive at the quantified value of the win. -6+4 =-2 tells us the person in the second lane is the winner!
So, before a student develops abhorrence of the symbol minus (-) he/she should grasp that the symbol minus (-) is the identification of the number for comparative purposes only. Otherwise we are comfortable with the plus sign because we are always in motion in the real field.
It is very clear that addition has to happen in a lane of linear expansion. As is won’t in languages, the symbol 1 (one) apart from being a shape has a name or carries a sound as well. Teaching means establishing a reflex response in a student. We make a sound say nine the student writes 9. The visualisation part of learning if possible, makes it easy for the students learn faster. So the math student is learning counting. He is in lane one (+ lane) and the number 1 is moving, gaining 1 at each step. Next he visualises two lanes, +ive and –ive, where counting is in process. The idea, once again I will stress, is that before he is conscious of the difficulty of juggling with the symbol minus or subtraction, he should be familiar with it. He immediately learns that identical specie of numbers add or a linear progression is happening. Birds of the same flock fly on their designated route!
When we put the information of numbers in group formation like +8+4+9+23…, how is it to be interpreted? It means that the counting is going on in periodic intervals. Say, no sooner does the counting reach 8, it is noted. Next the pause is taken and counting started afresh. When 4 is reached (2nd pause) and the progress noted and so on. The visual information carried along with a question makes it easier for a student get connected and go about counting linearly, already grasped by him/her. It should be noted that when information is placed horizontally, the counting will start from right to left.
Again, when the information about numbers is placed in jumbled formation like: -9 +6+4-13-8 +11, the student will interpret it as score pertaining to two lanes. So he rearranges the information as follows. -9-13-8+6+4+11. Then he applies already learnt skill and comes out with the correct answer.
The most confusing application in mathematics which the students face is the operation of subtraction. This operation has become the bane of the students. So I suggest that the term subtraction should be excommunicated from the math lexicon. It should be replaced with the term, comparative linear counting. Similarly addition should be termed as linear progressive counting. The idea is that a visual image should go with each concept.
Subtraction is usually associated with examples like: if you have 5 apples and you eat 2, how many apples do you have now? This is not math. This is the applied math- a situation where a student is supposed to use the knowledge of number operations already learnt by him/her. Once a student is familiar with comparative counting, he/she will place the information mentally in the two lanes and work out the result. Since he/she is already comfortable with the symbol minus (-) the confusion associated with this term gets neutralised.
Let us take an example. Suppose a student is asked to subtract 23 from 28. This is the usual sight. The numbers don’t carry any symbols. So the mark of identification is absent. But the new age student will have no difficulty. He will straight away place 28 in + lane (because he knows that the larger number has to be placed in the 1st lane) and 23 in – lane mentally and compare the information. All he has to do is that he/she has to start counting from 24 (because counting has already been done up to 23) till 28 is reached on his fingertips. The answer in the comparative case is on the tips, which is +5. But in case the same question is repeated with the numbers reversed, say subtract 28 from 23 (+23 – 28), the student won’t be uncomfortable. He/she will put -28 in the first lane and +23 in the 2nd lane and write the correct answer -5, meaning that the larger number is large by 5 when compared with the number in the 2nd lane. The number 5 will carry the identification of minus because the student is already taught that in comparative counting the result retains the sign of the largest number or of the number in the 1st lane.
It should be noted that every time we write a number, it should carry its symbol. At present we write question like:
53 46 69
+43 +68 -31
Now this is a wrong way of placing the numbers sans their identity cards. The correct version should be:
+53 +46 +69 -83
+43 +68 -31 +31
The students are familiar with the questions in their upper avatars, which have been correctly reproduced in their 2nd version. But the 4th example is unheard of. Ask a student to solve this question and he will get confused because of the plus sign at the bottom. But the new age student will simply visualise it as a case of comparative counting in which the negative number is in lane one and give us the result -52.
While doing addition or subtraction, every student uses his/her hand with ease. Since we have only 9 numbers three fingers are used which have 3 joints on each figure. Two scenarios emerge, which are naturally associated with addition and subtraction. When we add say, 4 & 5, the counting moves forward like five,six,seven,eight,nine. Stop. Here the number to be added is in focus and the answer is in the word of the mouth- (9) the sound of nine. But in subtraction the number to be arrived at, is under observation, the counting moves forward and the answer is on the finger. This is how the difference between two different operations carried out on the same platform is made clear in the minds of the students. The differential approach has to be kept in mind at every new stage or when a new concept is introduced in the class.
Conclusion: while counting linearly (adding) the result is in the word of the mouth,
And counting comparatively (subtracting) the result is on the joint of the finger.
The process of teaching becomes easier if monotony is avoided. Learning is associated with the creation of mental images which are necessary when memory recall is sought. So we fix the rule. The addition will be done from top to bottom, if the information is arranged in rows. The student will start from the topmost number and start adding the next number. Here I may add that for practice purposes only two digit numbers should be placed in each row. But the degree of difficulty should be increased by adding rows only. This method has the advantage that the student has to concentrate and simultaneously remember the numbers which are changing at every step. This method, in my opinion improves the retention power of students and by constant practice the breadth of their imaginations improves.
It is already known that the operation of comparative counting (subtraction) starts from bottom to top. This practice also compels us to teach that the counting in case of addition should start from top to bottom. Hence the second rule. So a student should be made conversant with the rules beforehand. The practice of borrowing should not be linked to any conceptual treatment. The more operation is kept simple and less prosaic the better it is because the students in this way are liberated from the tension of making their own assessments which put them in trouble later on. Math learning is as simple as learning a skill. Repeating an operation over and over again creates reflexes which in turn incubate confidence in a practitioner.
The symbolic representation of nine numbers, 1,2,3……..quantify in numbers, some whole entities. The names one, two, three are adjectives which owe their existence in the company of a noun. An adjective can’t exist as an independent entity. One (1) is the basic corpuscle of counting. All other numbers are derived out of the progressive growth of one. So 1+1=2. Two is the symbol (diagram) which carries the denomination of two ones. It is therefore clear that counting is progress of the symbol 1 in linear direction. One step at a time till we arrive at number 9 – group of nine ones. Here we exhaust the kitty of symbols. Next we enter the field of human brilliance which by taking help from the ubiquitous geometric shape the sphere and its two dimensional projection-circle, made the counting an effortless process!
In order to count beyond 9, the human intelligence sought the help of zero to overcome the firewall of stagnation. So help for moving further was sought from the symbol 0. But zero being the natures’ economical construction, and as such being egotistical, set its own conditions of engagement. ‘ Put me on the right side of a number it will grow ten times its value. Two zeros means numberx10x10 and so on. Put me on the head of a number it won’t be able to hold me and I shall demote it to its basic value: 1. Put me on the left side of a number and I will slip away and nothing will accrue to it. Put me under the periphery of a number-well that will be too humiliating for me.’ In this way the symbol zero not only set its terms of engagement but also exhibited its distinct character as different from the numbers. Zero incidentally retains its status of a helper and nothing more. It naturally does not qualify to be designated as a number!
The concept of number line does not solve any problem of mathematics. The absurdity is that it gives zero the status of a number which actually violates the basic agreement. Second, it creates two fields, one for real numbers and the virtual field for virtual (negative) numbers. This in turn breeds the problem of transportation. How can we port numbers from one field to the other to conduct the operation of subtraction?
If we look up to a tree we find that nature does not require a virtual field to shed leaves. All the operations of mathematical nature are carried out by it in the real field. It adds and sheds leaves in the same field. We too can carry out the basic operations and as such all operations in the real field. That is my contention!