Revisiting addition and subtraction: Mathematics made simple
Morvyn Ngoni Nyakudya - Abidile Lebotsamang | Monday July 11, 2022 06:00
Botswana, like any other country, is facing its fair share of challenges around mathematics education based on poor results for many years, at all the crucial exit levels such as Standard 7, Form 3, Form 5, and Form 6 (Botswana Examinations Council) and even at University.
I have taught math in Botswana at the university level (Bachelor's and Master's degrees) since 2003 and I have concluded that there is a dire need to re-focus when it comes to math learning and teaching.
It is not uncommon to teach a university student who cannot perform simple addition and subtraction, or at worst, one who engages the use of a scientific calculator for an operation like 15 minus 7.
I publish extensively on mathematics anxiety, a phenomenon that manifests itself in many, if not almost all, students. Math anxiety has been defined as feelings of apprehension and increased physiological reactivity when individuals deal with math, such as when they have to manipulate numbers, solve mathematical problems, or when they are exposed to an evaluative situation connected to math. If all teachers of math, individually and collectively, can tackle this problem called math anxiety, Botswana will be a better country, with men and women capable of handling mathematics, a subject required for, among others, constructing bridges, airports, and smart cities.
Several students across the country are re-taking some subjects they failed in Form 3 and/or Form 5. These subjects include mathematics. Institutions like CRACKiT Botswana, with 54 branches across the country, and still growing, must be applauded for making great efforts to re-teach the students. They have produced excellent results so far.
Perhaps there is a need to re-write and re-teach most or all mathematics topics in simpler terms as compared to packing the subject with the usually complex and sometimes confusing jargon.
This article digresses from the traditional textbook concept that is usually inundated with symbols that students do not understand. A case in point that I have also demonstrated to students in a class situation is:
“∀ a∈A,∃ b∈B s.t.ab=x"
The above statement simply means:
“For all elements ‘a’ in A, there exists an element ‘b’ in B such that a×b=x”.
It is evident, from these two supposedly synonymous statements that it becomes extremely difficult for students to fully appreciate and understand mathematics as presented in the former expression. They will end up fearing Mathematics altogether leading to delayed uptake of the course or total avoidance.
The first case in point is to deal with the number systems. There are just but 10 numbers that we have to grapple with in all our mathematics calculations daily in cases as simple as a kindergarten child learning to count using their little fingers, through to cases where a student is working on secondary school mathematics rigour, to very complex cases such as applying advanced mathematics to, among many, construction of bridges, under-sea airports, flying air buses and aircraft control. The 10 numbers are simply 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9. These numbers are significantly incredible in our daily lives as they create an infinite array of other numbers that tell the stories of humankind. Some of these stories include our ages, what we buy and keep, our insurance premiums, the population of a country, bank interest rates, and a lot more.
From my own experience, this article is intended to teach learners, at all levels of education, how to add and subtract numbers without making mistakes (I have seen mistakes at the highest level).
Number systems
The major classifications of numbers are unpacked and explained in simpler terms. In this article, the number line and the mathematical operations of subtraction and addition will be presented using simple techniques. The number system is one complex piece of mathematics that needs to be interrogated further to gaining early math competency in learners.
Various number systems define an array of numbers. They can be divided into groups of Natural (1, 2, 3, 4 ...), Integer Numbers (...3, −2, −1, 0, 1, 2...), Rational Numbers, Real Numbers, and Complex Numbers. Understanding these number systems is crucial in ensuring that learners eliminate difficulties in the areas of mental computation, estimation, and quantitative judgment, all of which are important components of the understanding of numbers.
Given the poor performance reported by the Botswana Examinations Council (BEC) over the years, there is a serious need to consider re-writing math books, retooling the teachers of mathematics and the introduction of an interactive teaching platform where the written math language is easy to understand. This article explains addition and subtraction in very simple terms.
Addition and subtraction
The operations of addition and subtraction seem easy as they appear. Yet many authors have opined multi-paged comprehensive papers to explain these two seemingly easy concepts. The authors use a simple, empty and vertical number line to try and raise the level of math awareness of students in every grade. They further note the absurdity of negative numbers. It was and still is hard to think of numbers below zero and how they can be included in mathematical operations of addition and subtraction. This is why, up to this day, negative numbers are difficult to study especially when dealing with complex number systems.
The biggest problem with negativity (negative numbers) turns out to be “how do you model negative numbers?” It does not feel strange to carry out the operation 10-7 (ten minus seven) because the mind is premised to think of reducing 10 by 7 (like eating 7 sweets from a packet containing 10 sweets, or spending P7 from a P10 note one has).
It now becomes difficult to perform the operation 7-10. This operation raises answers like ‘it can’t’ a clear statement that means ‘it is not possible’. It is more natural to say ‘I ate 7 sweets from 10 sweets in the bag’. This is why the number systems and the operations of subtraction and addition should be mastered early in school life.
Jordan, Kaplan, Nabors Oláh, and Locuniak (2006) stress the critical need for children entering kindergarten and first grade to be prepared to embark on formal math learning by first acquiring adequate math skills as children at the initial growth of their cognitive development. This has been seen as a challenge as Jordan et al. (2006) conclude that there was wide variation in the level of math skills children acquire during the preschool years. Many more recent authors have expounded further on this concept. Addition and subtraction are important skills for children when they register for formal school (Nunes, Bryant, Evans, Bell, Gardner, Gardner, et al., 2007). This was also observed by Ching and Nunes (2017) who noted that conceptual knowledge of arithmetic is an important element in readying preschool children to handle math easily.
Several authors have argued that to have a good understanding of numbers, the number line should be used to explain some of the operations that learners go through.
The Number Line
The number line has a zero (0) in the middle with negative numbers on the left of the zero mark while positive numbers are on the right of the zero mark. Diagrammatically, the number line is as follows:
-5 -4 -3 -2 -1 0 +1 +2 +3 +4 +5
The negative numbers go as far as minus infinity (-∞) while the positives also reach positive infinity (+∞).
The number line can assist us greatly especially when adding and subtracting. You can position yourself on the number line to perform calculations such as:
-4 – 10 OR -21+6 OR -56+78, just to use a few examples.
The number line aids us in getting the right direction to follow. The negative sign tells us to go towards the left () while the positive sign indicates that we have to go towards the right (). So the positive tells you to move to the right while the negative says move to the left.
Let us do an example, in -4-10,
Position your finger at -4: -14 -13 -12 -11 -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0
Because -4 is followed by a minus (as in -10), move 10 steps to the left of -4.
-14 -13 -12 -11 -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 You stop at -14
-14 -13 -12 -11 -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0
So, -4-10=-14.
Another example is -21+6,
Position your finger at -21: -21 -20 -19 -18 -17 -16 -15 -14 -13 -12 -11 -10 -9
Because -21 is followed by a plus (as in +6), move 6 steps to the right of -21.
-21 -20 -19 -18 -17 -16 -15 -14 -13 -12 -11 -10 -9
You stop at -15
-21 -20 -19 -18 -17 -16 -15 -14 -13 -12 -11 -10 -9
So, -21+6=-15
The example -58+78 can be written as +78-58 by interchanging the numbers. The result of this is 20. Using a number line for this example can be cumbersome.
Note that each number carries its sign on the left. For 50-100, 50 carries a “+” while the 100 carries a “-”. Also note that the ‘+’ on the 50 is silent (think of you saying ‘I have +10 sweets in my pocket’: this statement is mathematically correct but rather absurd or awkward!).
50-100 can be written as -100+50 through the interchange.
It is therefore important to be cautious about the signs whenever you are adding or subtracting numbers.
Let us join hands and follow all articles that will come in future, perhaps that child who is struggling with mathematics can start seeing light. The next series will discuss multiplication and division.
*Morvyn Nyakudya is the Dean of the School of Graduate Studies and Research at BA ISAGO University. He has a passion to improve math performance in Botswana and he believes that we should all start at the lowest levels of learning. morvyn.nyakudya@baisago.ac.bw
Lebotsamang Abidile is the Research Assistant in the School of Graduate Studies and Research at BA ISAGO University. lebotsamang.abidile@baisago.ac.bw