When the word mathematics was mentioned, I could only think about addition and subtraction. Indeed the topic gave me a lot of headache. This topic is the foundation in which mathematical principles are laid. Addition starts with a simple 1+1 and continues through mixed numbers, to complex addition of mixed units. The same applies to subtraction. In this work, I have given my biography in this topic, and expounded on how best the topic can be tackled based on research.
I developed my addition and subtraction concepts through practical experiences with countable objects. I would count my fingers when asked to add or subtract; however, I got so confused whenever I ran out of fingers i.e. in operations like 6+8. I then prepared sticks which would help me handle bigger numbers.
With bigger and bigger numbers, counting could not help. So, I was introduced to the concept of “carrying” in addition and “borrowing” in subtraction. This was the point where “place value” came into existence. I was told that, when performing addition or subtraction, digits with same place value must be operated together and never once shall I manipulate digits of different place values. After identifying the position of the decimal point, I could arrange the numbers in accordance with the place value of the individual digits, and then begin the operation. While performing an addition, I realized that some sums were more than ten (i.e. containing ones and tens). This was the point I got introduced to the concept of “carrying”, where I could write the ones digit as I carried the tens digit. The next step was to add the tens digits starting with the carried digit. The process repeated itself until all the digits were manipulated.
The subtraction was more or less similar; however, the concept of “carrying” was replaced with “borrowing”. When the manipulation was “impossible”, i.e. the second digit greater than the first digit (the subtrahend greater than the minuend), I was told to borrow from the number occupying the second place value; tens. This was my real source of headache and it took me months to understand this borrowing concept. I could not just understand how a borrowed number was equal to ten and then I add it to the minuend. It was also unclear to me how the number borrowed from reduced by one. The worst came when I was to borrow from a zero. The possibility of borrowing one from a zero was not making sense at all. However, with continued practice and constant guidance, I mastered the basic concepts and I could now handle such problems with a lot of ease.
In several occasions, I could be challenged by missing addends. In a normal operation like 5+2=7, it was not easy for me to identify the missing number (i.e. 5+_= 7). However, when it was rearranged as 7-5=_, I could easily identify the missing number.
With the natural numbers, I had developed enough knowledge and skills in handling both the addition and subtraction problems. However, this was not enough. We shifted from the natural number system to the clock system and great confusion arose. The place values were now replaced by seconds, minutes and hours. It was not making sense at all why a minute is equal to 60 seconds or why 60 minutes make one hour. “Why then can’t 60 hours make one day, or 60 days make one week?” I asked myself and my teachers, and to-date, I still ask this question. Technically, it was confusing as I could not comprehend why the sum of fifty minutes and forty minutes is one and half hours when in natural number system, the sum is 90 minutes.
Though it was ambiguous initially, we laid much emphasis on the place values of the numbers, and the equivalence of such numbers in the following higher place values. This also introduced me to the concept of units. My teacher made it clear that only similar units should be manipulated. An addition of three mangoes and five oranges neither results into eight mangoes nor oranges; rather, it remains three mangoes and five oranges. However, for generalization purposes, both mangoes and oranges are fruits, so, the operation resulted into eight fruits. The concept of common units also arose during these manipulations. Some values can be expressed in-terms of the others like in clock system, minutes, hours, and seconds can be expressed in terms of each other. The same applies to the units of distance; kilometers, meters, and centimeters can all be shifted from one unit to another. For addition and subtraction operations involving mixed units, I was required to first convert these units into one common unit. This meant that one had to have the basic knowledge of the units and their conversion. This was quite involving and difficult.
I finally wrapped my study of addition and subtraction by looking at the laws guiding these operations. Two laws were put forward: the commutative law and the associative law. With addition, reversing the order of the addends has no effect on the sum.
It is normally said that practice makes perfect. Perhaps those who said so had similar problems in mathematics as mine. It was the constant practice that enabled me to understand and master the basic concepts, which I have employed to this far. In mathematics, I owe my success in constant practice.
From my own experience and research, the concept of “counting up” and “counting down” works best. For addition, think about the biggest number first and then add up i.e. in 6+8, think 8 and count on 6 more. For subtraction, think about the smallest number first and count up i.e. in 13-5, think 5 and count up to 13. Alternatively, you can think the biggest number first and count back i.e. think 13 and count back to 5. Other than the normal counting, a proper strategy is the key to understanding and remembering the operations’ concepts. The use of number line is very important as it helps the children to identify where to start the counting, either back or forward and how to make the additions and where they land. When a student develops a mental picture, he/she retains the basic fact of the given operation. The algorithm should be taught with effective pictorial or physical models. Modeling a skill or concept helps the child to see how the skill looks like as he/she develops a clear mental imagery. Teachers should therefore use appropriate models which allow the children to visualize the procedures.
As the addition and subtraction expands into numbers with 2- and 3- digits, the need for proper alignment is necessary. Errors are minimized when the digits are kept in their proper places. In so doing, the children realize that addition or subtraction of 3-digit numbers is similar to the addition or subtraction of 2-digit numbers, or even single digit numbers.
The concept of units is equally as important as the place value. Other than natural numbers, the children must clearly identify the units in every operation so as to minimize the errors that might occur due to oversight. In case of unit conversions, the child should know the basic relationships among the individual units.
Problem solving skills in mathematics are learned and nurtured through constant practice. Unlike other disciplines, mathematical principles and concepts are understood when there are continuous applications of the principles. With clear visualization of the problem, a solution must always be found. The principles in addition and subtraction are quite abstract for children; however, if they can be made to visualize and compare the presented problems with the real life examples, the children’s understanding can be highly enhanced. Let the children do a lot of practice so that they can deeply understand and master the mathematical principles.
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National Research Council. Adding it up: Helping children learn mathematics. National Academy Press. 2001. ISBN 0-309-06995-5. http://www.nap.edu/books/0309069955/html/index.html.