INTRODUCTION
Importance of using Teaching and Learning Materials (TLM) in teaching and learning. This article covers the literature review of how basic school pupils were helped to under addition and subtraction. Continue reading.
Literature review is basically concern with drawing information or ideas from other people work or theories about the topic under study and how the present study intends to solve or improve upon the existing theories and information.
For that matter, Steven Shade said that, the functions of the literature review is to discuss relevant research carried out on the topic and also to prevent duplication of ideas and theories.
HISTORICAL PERSPECTIVE
The goal of understanding students‘ learning of mathematical topics is not new to the field of education as efforts have been made since the days of Pestalozzi and Montessori during the last few centuries. Specifically, John Dewey advocated that the most effective learning tends to be self-directed, guided by theory, and ideally attached to experiences.
He emphasized that learning does not represent a set of disconnected events which take place in isolation, but rather as an integrated lifelong process. Dewey provided a multi-step approach to the learning process by identifying an initial stage of experience, followed by a reflection phase.
During this phase of reflection, the emphasis is on the synthesis of experience with theory as one revisits and generalizes based on the reflection. The postulation of the learning process concludes with a new generalization which may be verified and tested in the realm of practice guiding new learning cycles.
Learning, according to Piaget, is understood as a process of conceptual growth which requires the formation and reorganization of concepts in the mind of the learner. This knowledge may not be communicated but rather is constructed and continuously reconstructed by individuals through an active process of doing mathematics.
Empirical evidence has repeatedly supported this argument which favors learning and teaching mathematics for understanding Polya. In Brown well view the specific domain of arithmetic and understanding whole number concepts and operations, Brownell contended, If one is to be successful in quantitative thinking one needs a fundamental of meanings, not a myriad of automatic responses.
If this notion is to be meaningful, the instruction needs to emphasize the teaching of arithmetical meanings and making these notions sensible to children through the development of mathematical relationship.
In the 10th year book published by National Council of Teachers of Mathematics Brownell addressed whole number concepts and operations by recommending an intelligent grasp of these topics for children in order to deal with proper comprehension of the mathematics as well as their practical significance.
Upon the discussion of the critical nature of learning whole number concepts and operations, the next natural question should address who will be responsible for teaching these 12 significant notions and how should they proceed? Clearly, since elementary school children spend a majority of their time dealing with such topics, elementary school teachers carry the vast majority of the responsibility for establishing and developing these topics in children.
In the words of Bruner, When we try to get a child to understand a concept the first and most important condition, obviously is that the expositors themselves understand it . Therefore, current and future elementary school teachers should have the foundational understanding and appropriate education to accomplish this vital task.
Furthermore, the teaching and learning of mathematics for understanding has and continues to be consistent with the recommendations of National Council of Teachers of Mathematics and the National Research Council.
Historically, the reform programs and movements that were implemented through careful deliberation and by people who understood the intricacies of learning in children and prospective teachers have had-often dramatic results National Council of Teachers of Mathematics.
As the primary focus of this research, this study intentionally examined the conceptual understanding of prospective teachers within the domain of whole number concepts and operations.
As previous research in this topic has indicated Andreasen, prospective teachers tend to progress through levels of development very similar to the stages that children experience in learning whole number notions.
Hence, this review of literature shall next describe the ways that children typically develop their thinking in order to inform the understanding of prospective teachers. Children’s Development of Whole Number Concepts From an analysis perspective, it should not be difficult to fathom the reasons behind children‘s struggles with the notion of number.
After all, in the base-ten system, the value of a 13 digit is comprised of a dual meaning: the value of the digit and its position within the numeral. That is to say a single numerical symbol can simultaneously represent different notions depending on its placement within a number.
The compactness and sophistication of this number system are intertwined with its inherent initial complicated nature. When children begin to learn to count, they do not attend to a difference between a single-digit and a multi-digit number.
From a child‘s perspective, the number 10 follows the number 9 very much in the same way that the number 9 followed the number 8. The numbers aforementioned simply represent a number of items.
In fact, research has illustrated that children observe that 10 only differs from its precedent number 9 in that 10 represents one additional item. Some children tend to count by ones, whereas others with a more developed understanding of whole number concepts may directly model the problem to eventually arrive at a recognized operation.
Many young children come to elementary schools with the ability to solve single digit addition problems. As Fuson and Steffe have indicated, children who did not possess an understanding of the place value notion tended to struggle at this stage and beyond.
In the next few sections, some of children‘s strategies and invented algorithms will be presented to provide insight into the mathematical understanding that is needed to use each of these approaches.
Andreasen and Roy have shown that placed in a new context, prospective teachers followed a similar progression to children in developing understanding of whole number concepts and operations.
Hence, this research project benefited from having an understanding of the path children follow to gain proficiency with whole number operations in order to inform the path that prospective teachers ultimately followed.
THE CONCEPT OF ADDITION AND SUBTRACTION
Addition in math is a process of combining two or more numbers. Addends are the numbers being added, and the result or the final answer we get after the process is called the sum. It is one of the essential mathematical functions we use in our everyday activities Andreasen.
Addition usually signified by the plus symbol (＋) is one of the four basic operations of arithmetic, the other three being subtraction, multiplication and division.
The addition of two whole numbers results in the total amount or sum of those values combined. The example in the adjacent image shows two columns of three apples and two apples each, totaling at five apples.
This observation is equivalent to the mathematical expression “3 + 2 = 5” (that is, “3 plus 2 is equal to 5”). Besides counting items, addition can also be defined and executed without referring to concrete objects, using abstractions called numbers instead, such as integers, real numbers and complex numbers.
Addition belongs to arithmetic, a branch of mathematics. In algebra, another area of mathematics, addition can also be performed on abstract objects such as vectors, matrices, subspaces and subgroups.
Addition has several important properties. It is commutative, meaning that the order of the operands does not matter, and it is associative, meaning that when one adds more than two numbers, the order in which addition is performed does not matter see Summation.
Addition also obeys predictable rules concerning related operations such as subtraction and multiplication.
Subtraction is the operation or process of finding the difference between two numbers or quantities is known as subtraction. To subtract a number from another number is also referred to as ‘taking away one number from another’ Asafo-Adjei .
Subtraction, denoted by the symbol (－) , is the inverse operation to addition. Subtraction finds the difference between two numbers, the minuend minus the subtrahend: D = M − S. Resorting to the previously established addition, this is to say that the difference is the number that, when added to the subtrahend, results in the minuend: D + S = M.
For positive arguments M and S holds:
If the minuend is larger than the subtrahend, the difference D is positive.
If the minuend is smaller than the subtrahend, the difference D is negative.
In any case, if minuend and subtrahend are equal, the difference D = 0.
Subtraction is neither commutative nor associative. For that reason, the construction of this inverse operation in modern algebra is often discarded in favor of introducing the concept of inverse elements where subtraction is regarded as adding the additive inverse of the subtrahend to the minuend, that is, a − b = a + (−b).
The immediate price of discarding the binary operation of subtraction is the introduction of the (trivial) unary operation, delivering the additive inverse for any given number, and losing the immediate access to the notion of difference, which is potentially misleading when negative arguments are involved.
Some Children’s Addition and Subtraction Strategies includes;
Counting-on: Elementary school children often times use the counting on strategy when one addend is small. For instance, in solving 7 + 4, children begin with 7 then counts on four more: 8, 9, 10, 11. Children typically begin with the larger addend and then count on as many numbers as needed according to the other addend.
In a similar scenario given 3 + 14, children may start by counting on 14 from 3, or in some cases start with the larger addend 14 and count on 15, 16, 17 even though the larger addend is the second addend in the expression.
Near doubling; Another strategy used by children in solving problems of the type 7 + 8 is called near doubling. Some children will use the addition fact of 7 + 7 = 14, and then simply add 1 to finalize their solution.
This method suggests a double plus 1 approach. Other children will take a similar problem, 8 + 7 and use the addition fact of 8 + 8 = 16, and then subtract 1 to finalize their solution. This method utilizes a double minus 1 approach in solving whole number addition problems.
Adding to ten; When adding pairs of numbers whose total is greater than ten, many children use adding to ten strategies. Taking the previous example of 8 + 7, children will think through the transitory stage of what do I need to add to 8 to make a 10?
Next, they regroup 2 of the 7 objects and group them with the 8. The next question to be answered involves ten and 5 more makes how much?
Adding one-less-than-ten;This particular strategy is used by children in the particular instances when adding nine as one of the addends. The one-less-than-ten strategy becomes very efficient when children are asked to solve problems such as 6 + 9.
More advanced children solve this problem by performing 6 + 10 = 16. Next, knowing that 9 is one-less-than-ten, then the result should be one less than 16. In other words, through familiarity with adding by place value, children can quickly arrive at the result of 6 + 9 = 16 – 1 = 15. This particular strategy can be extended to multi-digit addition problems in scenarios when one of the addends has a 9 such as an addend of 39 or 59.
Similar to addition strategies discussed above, many subtraction strategies also require a solid foundation of place value understanding. Children need to realize that a number such as 23 is not simply a 2 and 3.
According to Cobb and Wheatley, children arrive at the understanding of conservation of number as well as seeing ten as an abstract notion in addition to an iterable unit. A few of the subtraction strategies consistently used by children have been illustrated in the following
Examples of Subtraction Strategies
Front-end subtraction; In solving a problem such as 62 – 38, many children who have not yet been taught standard algorithms for multi-digit subtraction begin by taking away the ten first. In other words, their thought process begins at the ―front of the number and can be modeled by 62 – 30 = 32.
Next, children using this subtraction strategy will subtract 8 from the resultant number. By performing 32 – 8 = 24, these children have not followed the traditional subtraction algorithm which begins with subtracting the ones unit first.
Using this technique, children need to have an understanding of decomposing numbers according to place value to arrive at 62 – 38 equals 24.
Compensation Continuing; With the subtraction problem 62 – 38, some children will compensate by adding what it takes to make the bottom number a multiple of ten.
In this case, children would add 2 to both numbers and instead solve the problem 64 – 40 (compensating by 2). The new problem of 64 – 40 appears to be much easier for children and can be solved using a variety of methods to arrive at the result of 24.
Through the use of compensation strategies, children have once again solved the problem 62 – 38 to get 24.
Taking extra and adding back; In a somewhat different approach from the compensation strategy, some children only change one of the numbers and then adjust to make it model the original problem.
In the case of the same example 62 – 38, some children might take an extra two away so that the problem could be rewritten as 62 – 40 = 22. Then, knowing that they have taken an extra two away, they will add the amount 2 back to the result. Therefore, 2 must be added to 22 to get 24 as the final answer.
Note that this strategy is different from compensation as children only adjusted one of 23 the two original numbers to perform this subtraction problem in a manner that made sense to them individually.
Using place value understanding to subtract by equal additions; A related – yet distinct approach from the compensation strategy discussed earlier involves using place value understanding. Children who possess an understanding of 1 group of tens simultaneously equaling 10 ones use the ideas of decomposing numbers and an algorithmic approach to subtract by equal addition.
In this approach, children will add the same amount but add it in different place values – such as 10 ones in the ones column and 1 ten in the tens column – to solve a desired subtraction problem. Figure 4 revisits the same subtraction problem 62 – 38 in order to demonstrate subtracting by equal addition.
Note the way children will change the 2 in 62 and the 3 in 38 by adding the same amount to both numbers.
IMPORTANCE OF TEACHING AND LEARNING MATERIALS
Teaching and Learning Materials are materials used during the teaching and learning process to enhance pupils’ understanding.
The major significance of teaching-learning materials is recognized within the classroom environment by providing support and assistance to the educators with the presentation and transmission of educational content and the achievement of educational objectives Ibe-Bassey.
The teaching-learning materials are put into practice by the educators with the primary objectives of imparting learning among students regarding the academic concepts and enabling them to achieve their goals and objectives.
The significance of teaching-learning materials is usually recognized in terms of five aspects Kimago. These are student motivation, developing creativity, evoking prior knowledge, encouraging the processes of interpreting, understanding, organizing, and amalgamating the educational content, logical thinking, reasoning, and communication, and contributing to the development of different skills, values, and attitudes among students, and enabling them to acquire an efficient understanding of the academic concepts.
The teaching-learning materials are defined as the instruments of presentations and transmission of the prescribed educational material Busljeta.
The teaching-learning materials can be differentiated in terms of various characteristics that are apparent at first glance.
When these are used in different classroom environments, then the students are required to possess various skills. For instance, when they are making use of technology to prepare their assignments, reports, or projects, then they need to acquire efficient knowledge Makombe et al.
In the field of didactic theory, as well as in teaching practice, the classification of teaching-learning materials into visual, auditory, and audio-visual is universally acknowledged Busljeta.
Furthermore, the educators, as well as the students, need to possess effective communication skills, especially when they are making use of any teaching-learning materials.
When the educators will be able to communicate effectively, then they would facilitate understanding among students. Whereas, when students augment their communication skills, then they will be able to acquire an efficient understanding of the academic concepts.
The educators and the students need to collaborate in the development of teaching-learning materials.
Finally, M.O.E , pages 42-43 states that when teaching and learning materials are used, they help focus pupils’ attention and interest in the particular concept being taught.
Functions of Teaching-Learning Materials
The primary functions of teaching-learning materials are to motivate students’ acquisition of education. Teachers to provide assistance and support to the learners to achieve academic outcomes primarily use these. The major objectives have been stated as follows:
1. Motivate Learners
The teachers make use of not only one, but various forms of teaching-learning materials within the classroom setting. When they are making use of them, they ensure that students are able to feel pleasurable and get motivated towards learning. Therefore, students develop interest and enthusiasm, and develop motivation towards learning Onche.
2. Development of Knowledge and Skills among Teachers
Through the implementation of teaching-learning methods in an effectual manner, the teachers are able to develop their knowledge and skills. They can generate awareness, regarding how to make use of this knowledge in performing their job duties well.
They need to make use of these skills and knowledge in the achievement of educational objectives Owoeye.
3. Help in Longer Retention of Information
The TLM, when implemented should ensure that they help in the longer retention of information. When learners pay appropriate attention to TLM, then they are not only able to acquire an effective understanding of the concepts, but also are able to promote longer retention of information.
4. Facilitate Holistic Learning
Through TLM, the learners are not only able to acquire an efficient understanding of the academic concepts, but the teachers also assist and support them in the augmentation of psycho-motor, cognitive and intellectual development.
As a development of these aspects is regarded as essential for promoting effective decision-making processes and rational thinking Udosen. From the above thoughts, I strongly ascertained that Teaching and Leaning Materials help in relating abstraction to concreteness especially in subject like mathematics.
They also encourage verbal and written expressions i.e. they form a foundation upon which the pupils can take a sure position to apply his/her learning to other areas of learning. The researcher, therefore, decided to use the bottle tops coupled with a detailed explanation so as to address the gap.
The teaching and learning materials were also developed from the local materials which becomes easy for every student to get on their own and hence making practicing on their own in their homes during their leisure thereby improving upon their learning capabilities in addition and subtraction and mathematics as a whole.
The teaching method which is pupil-centered with enough activities would also help the students to have in-depth meaning of the whole concept.
SUMMARY
In conclusion, the review has shown that all the previous study on the concept of addition and subtraction has individual perceptions though author talks about the subject, importance to the society and also its relation with life.
Again, authors express and stress on the importance of introducing or teaching a maths concept with appropriate, concrete, manipulative or teaching and learning materials. Adopting the outlined researchers teaching could make children to develop much interest in the subject.
However due to the importance of the topic to related concept there is the need for more research on the topic to find other possible ways to make people up and doing in mathematics. These research will therefore go a long way to apply some of the suggestion in different content.
SOURCE: bbcpulse.com