Multiplication is one of the basic school mathematics topics which is seen as very easy to work with but alas, in-service and pre-service or prospective mathematics teachers are not able to give conceptual meanings to its mechanical and algorithmic solutions. This is made evident from many research findings in other countries order than Sub-Saharan African (SSA), where in-service mathematics teachers are not able to make sense of the multiplication of two-digits or more whole numbers using algorithms or approaches which make no meaning to them: not to talk of those being trained in their respective teacher training institutions and colleges of education. They are not able to give explanation to the algorithms or the approaches with understanding or meaning.

Teaching is an act and every mathematics instructor or facilitator adopts the strategies that is best known to him or her. In other words, in-service or practicing mathematics teachers use the approach which they have much knowledge and confident. This same rule is seen in pre-service or prospective mathematics teachers. The topic, “Multiplication of Whole Numbers” at the basic school level happens to be a challenging and confusing task for both mathematics teachers and students. More especially, when teachers use different multiplication algorithms without understanding. 

Many scholars and researchers have investigated how multiplication is taught to students and sought to find the sense that both teachers and students make from the approaches used. This paper reports on an effort to understand how in-service mathematics teachers only make meaning of multiplication algorithm. Permanent in-service or practicing mathematics teachers in public and private basic schools were engaged in an unstructured interview and discussion in the comfort of their places of work. A qualitative exploratory research design was used for this study.

Literature Review 

At this section, the theoretical framework for the study would be discussed, the conceptual framework for the study and thematic areas as education in the central region of Ghana, Area Model, Realistic Mathematics Education, Long Multiplication algorithm, approaches in teaching multiplication of whole numbers. 

Theoretical Framework for the Research 

The theoretical framework clarifies the important concepts in this study. A theoretical framework suggests my assumptions and beliefs about possible variables and paths to conduct the research. Radhakrishna et al. [1] mentioned that a theoretical framework is a conceptual model of how one theorizes or makes logical sense of the relationships among several factors that have been identified as important to the problem. It determines which questions and statements are to be answered by the research and how empirical procedures are to be used as tools to answer these questions or verify the statements [2]. Radhakrishna, Yoder and Ewing further indicated that, a theoretical framework integrates key pieces of information, especially variables, in a logical manner and to conceptualize a problem that can be tested.

One of the latest studies conducted by Flowers, Kline and Rubenstein in Apsari et al. [3] acknowledges that, operations on multiplication are quite challenging to both the in-service and pre-service mathematics teachers, as well as, students [4-7]. Yet, Larsson [8] in his paper on students’ understanding of multiplication, upholds that multiplication happens to be one of the basic operations of mathematics which are needed at early level of every student before entering higher grades. This study seeks to find out what Ghanaian in-service mathematics teachers make out from the use of the algorithm they use in their classroom. In so doing, introduced the Area Model in multiplication of whole numbers and observed what they made out of it. Varied questions and discussions were engaged between the researcher and the participants. 

Similarly, Apsari et al. [3] recent study on prospective or pre-service mathematics teachers making sense of multiplication algorithm in Indonesia agree to my position observed that, about 82% of the teachers who participated in the study were not able to justify the brain behind their process. According to Kwon et al. [9], the use of the Area Model, an aspect of the Realistic Mathematics Education (RME) Approach, will work well and to explain the logic behind multiplications of whole numbers by pre-service teachers in the United States. This model can be used in teaching operations on fractions as well. 

In Sub-Saharan African countries, including Ghana, the Four-Year Bachelor of Education Degree, Eight Semester, Initial Teacher Education Curriculum, designed by the University of Education, Winneba, spelt out clearly at page 280, unit 6, the topic “Multiplication of Whole Numbers”, subtopic, mental strategies for computing and estimating 1-digit products, but did not recognize the use of Area Model as a teaching approach. In the column for teaching and learning activities to achieve the stated topic and subtopics, the authors were not silent on the use of other approaches in teaching multiplication operations, which has low conceptual fluency, but were silent on the use of Area Model to help pre-service teachers grasping multiplication concepts. Yet, they need to adapt strategies which would help in transferring much knowledge to the pupils in developing their 21st century skills [10]. Without that, prospective mathematics teachers would not be able to perform their roles as “coaches” or “facilitators” in the classroom.

In similar vein, NaCCA and the Ministry of Education has positioned in the new Ghanaian standard curriculum, for Early Grade, Basic 3 that, pupils demonstrate an understanding of multiplication of 5 by 5 as content standards. In the indicators and exemplars, five examples were elaborated but never showed the approach of using an aspect of the Realistic Mathematics Education (RME) Approach which has conceptual understanding for related topics in mathematics [11]. All the approaches given leads to procedural learning. In RME approach, the activities to be learned starts from the symbol of real world which is horizontal mathematization. In this case, students can get along and connected in the teaching and learning process with understanding. In-service and preservice mathematics teachers’ role primarily here is to mentor and facilitate for students in the process of reconstruction of mathematical ideas and concepts with greater meaning [12]. 

Pre-service Mathematics Teachers apparently, in many West African countries, are moving on a rapid pace to catch-up with technology usage in their classroom. Some are even transforming their usual intended or planned curriculum to meet international standards. In Ghana, there is a transformation of curriculum from Objective-Based Curriculum to Standard-Based Curriculum for the first cycle institution which starts with early childhood or early grade. This calls for all restructuring of resources which must be met in order to meet the international standards, such as, the practices at PISA, TIMSS, etc. [13]. For this to be realized, young children and prospective teachers whom much depends in order to merit this must be exposed to a variety of modernized, digitalized and technological approaches in doing things. Some are using manipulatives in teaching and learning mathematics with conceptual understanding instead of procedural understanding.

According to Hendriana et al. [14], many students in the rural areas in Indonesia have face greater challenges in learning concepts which involves number operations in multiplication. This became evident when their study on the innovation of learning trajectory on multiplication operations revealed that in-service mathematics teachers in Asia pacific use formulas when introducing multiplication concepts. Lee [5] explored the Egyptian Multiplication and the Russian Peasant Multiplication to address the difficulties prospective teachers face in deciphering multiplication algorithms with the Area Model. In his study, it was observed that this approach is the most powerful and useful which allows students to record products without worrying about regrouping. 

Realistic Mathematics Education (RME) Approach 

Since 1970s, Utrecht University owned a research institution which always attempted to renew mathematics learning. The Freudenthal Institute was pioneered by Hans Freudenthal. It was based on the Netherlands and had been active from 1905 to 1990. Hans’ work was called Realistic Mathematics Education (RME). RME was developed on daily life concepts. It was then used in many countries such as the United States and some African countries. Research conducted in some countries (including developing countries such as Indonesia) has proved that RME is a promising approach to fix and improve students’ understanding of mathematics concepts. 

RME has a purpose to change mathematics learning into more fun and meaningful for students by introducing them into problems within contexts. RME starts with picking up problems relevant to students’ experiences and knowledge. The teacher then acts as a facilitator to help students solve the contextual issues. This problem-solving activity which is contextual is believed to bring positive impacts to students’ cognitive achievement especially related to their ability in understanding mathematics. 

The best way to teach mathematics is to provide students with meaningful experiences by solving issues that they face every day or in other words by dealing with contextual problems. Mathematics learning would be more effective if students are able to work to process and change information actively. RME has emphasized the use of learning aids in learning which is related to students’ ability. Realistic refers to asking students questions that they can think of. It was then followed by students solving mathematics problems. Instructions in RME are mainly focused on students and the development of their ability in learning mathematics. Students’ activities are mostly interactive and they are designed to build students’ interest in studying mathematics. 

RME can increase students’ logical, critical and creative thinking. It helps construct learners’ cognition at every stage of creative thinking. Based on some literature and research, creative thinking process is actually more oriented and concentrated on individuals’ cognitive and intellectual functions, particularly in creative problem solving. 

The structure of the intellectual ability is systematically perceived as a boost for students’ creative thinking and achievement. RME is oriented on empowering mathematization as a key process to mathematics learning. Mathematics is not only for mathematician, but it is involved in someone’s daily life. Mathematization helps students connect ideas to rediscover, which means that it constitutes a process in which students formalize their informal understanding and intuition. Freudenthal uses the mathematization concept in developing RME. This process includes two aspects that are horizontal and vertical mathematization. Horizontal mathematization is related to transforming problems found every day to symbols meanwhile vertical mathematization is a process that occurs within the scope of the symbols. RME draws these two approaches closer so that they can be related to each other and sustainable. In other words, learning starts from an informal step which later directs students to do mathematization of real-world problems represented by symbols. After that, the students can continue with vertical mathematization by using models to draw more general conclusion. 

RME Learning Characteristics 

Since RME was introduced, it has established mathematics curriculum and pedagogy. Clements and Sarama, state that main characteristics of RME include the application of meaningful contexts, the development of model which lets the transformation happen from contextual to formal mathematics, the recreation of mathematics concepts by the students, the interaction between students and teacher and the perception of mathematics as an integrated subject. These characteristics lead to the progressive mathematical process which lets learners associate problems with contexts, identify relevant mathematical concepts, solve problems and interpret the solution based on their contexts. Yuwono, simplifies RME characteristics into understanding contextual problems, discussing the problems and providing solutions to the problems. 

Conceptual Framework 

This study will be making use of the RME model and be conceptualizing the Area Model as the conceptual framework (Figure 1).

Figure 1: A Prototype Conceptual Framework Designed for the Study

This study will dwell on the framework as Figure 2 using.

Figure 2: Conceptual Framework for the Study

Research Design: The purpose of a research and its objectives determines the type of research design to be used by a researcher. A research design refers to the overall strategies and procedures that a researcher chooses to integrate in his or her study in a sequential and systematic way [15]. Koul [16] also defined research design as a number of procedures needed in order to collect data. The design therefore, constitutes a frame for the collection, measurement and analysis of data. Fink [17] describes research design as all the stages and the processes involved in reaching the respondents. A research design will typically include how data is to be collected, what instrument will be used and the intended means for analyzing the data collected. In addition to these, Creswell [18] refers to a research design as the plan of action that links the philosophical assumptions to specific methods. Different researchers use different designs based on their philosophical background.                 

This study is a qualitative exploratory research method where descriptive research design was adopted to collect data and were analyzed. For this exploratory study, four expert in-service mathematics teachers participated. It was conducted in the Komenda Edina Eguafo Abirem (KEEA) municipality in the Central Region of Ghana. Exploratory research is done to investigate a problem that has not been studied or profoundly investigated in the past. In exploratory research, researchers try to gain familiarity with an existing phenomenon and obtain new insight into the details of the problem. In exploratory research, the research process varies according to the finding of new data or insight. The problem of in-service mathematics teachers not able to make sense or meaning of the multiplication algorithm. The researcher has observed that using other forms of approaches by the in-service mathematics teachers in teaching multiplication would make sense to both teachers and students and it will assist students to apply learned skills in other topics in mathematics. 

Sample and Data Collection

Sampling: Non-probabilistic sampling was used to select the participants for the study. Nonprobability samples might be used when researchers are conducting qualitative (or idiographic) research, exploratory research, student projects, or pilot studies. The researcher visited the various schools randomly and got introduced as a research student from the Department of Mathematics and ICT Education of the University of Cape Coast. Three different basic schools were visited: two public and one private because public schools are more than private schools in the municipality. Table 1 shows the sampling procedure that was used for the selection of participants for this study.

Table 1: Sampling and Sampling Procedure Table 

Two male and one female teacher were attended in the public schools and one male teacher was attended to from the private school. This was so because, on the average, there seems to be more male teachers in the public basic schools than females. Also, there are more male teachers in the private schools in the basic schools than females. In the public school, one male teacher was in the Junior High School while the other two were in the Upper Primary levels. In the private school, the participant was in the Upper Primary level as well.

To draw a convenient sample, a researcher simply collects data from people or other relevant elements that they can access conveniently. Also known as availability sampling, convenient sampling is the most useful in exploratory research or student projects where probability sampling is too costly or difficult.

Data Collection

Unguided interview also called unstructured interview was used to collect the data. On Commerce Papers web page, unguided interview is a design used to let participants to speak his or her mind freely. Also, directed interview was used where questions were asked to measure participants’ job knowledge on multiplication operations. Interview helps to know the opinion of participants from different angles [19].

Table 2: Interview or Research Questions which Contains some Demographics of Participants

Source: Field work

Table 2 shows the research questions for the study and some demographics of the participants.

Interview Guide 

Validity of Data: Validity with the would be constructed instrument will be a concern to me, the researcher because no existing instruments are available to use for validity estimates and construction of the instrument since I started searching on this area of study. There might be difficulty in obtaining adequate and precise construct validity even though the survey would be grounded in literature related to measuring constructs professed to be related to the instrument. However, some types of validity measures can be used with this study such as face and content validity. Face and content validity were obtained through the piloting of the interview. Experts in the use of Realistic Mathematics Education (RME) model such as the Area Model would be contacted enhance the qualitative instrument.

Question 1

What Multiplication Algorithm does In-Service or Practicing Mathematics Teachers use in teaching Multiplication of Whole Numbers? From the interview, it was observed in Figure 3 that almost all mathematics teacher in both the private or the public teachers in the basic schools in the KEEA municipality use the Long Multiplication algorithm (Standard algorithm) in the classroom.

Figure 3: A 100% Stacked Bar Chart Showing the Approaches that In-Service Mathematics Teachers Use

This is due to the fact that, that is what they were taught and it was used when they were students in their various Colleges of Education. Also, this is the only straightforward approach readily available in the teaching syllabus and text books this attest to the fact that Ghanaian students who attend international test and assessment will perform abysmal since in-service mathematics teachers use old and approaches which have no bearing on other topics in their teaching. 

It was also observed that, even if students do not understand the use of the algorithm or make meaning of it, in-service mathematics teachers keep pushing it through and refer students to their mates for remedial teaching. Remedial teaching is an auxiliary for teaching and providing support for students who do not comprehend a teaching. In this way, other teaching methods could be employed by their classroom teachers but here is the case that in-service teachers rather make emphasis on an approach which does not go well learners. When participants were asked to give explanation to the long multiplication algorithm used, they could not explain.

A 100% Stacked Bar chart is used to compare the percentage that each value contributes to a total.

What other Approach do you use to Teach Multiplication?

From the unguided or the unstructured interview with the in-service mathematics teachers, it was observed that almost 100% (Figure 4) of the teachers interviewed to use the Long Multiplication approach which is considered to be the Standard method. And this style of teaching makes the them make room from equity and inclusion. They use differentiation in their classroom but all boils down to rote learning; teaching which do not allow what students have learnt to be transferred into other areas and topics in mathematics. There is much time and period allocation to mathematics. Same is seen even in assessment of students. In-service or practicing mathematics teachers only rely on procedural teaching and learning instead of conceptual teaching and learning for better understanding.

Figure 4: A Pie Chart Showing Responses on the Awareness of Area Model Multiplication

It was also observed that, in-service or practicing mathematics teachers tend to use grouping of objects as a form to let students understand simple multiplication of whole numbers as well as repeated addition. Some too try to reduce the pace in delivery with the assumption that students should be able to make meaning of multiplication algorithm. This shows that mathematics teacher’s research domain is limited and this in line with Akayuure [21] whose study was on classroom assessment literacy levels on mathematics teachers in Ghanaian Senior High Schools. 

Another observation was that, in-service mathematics teachers are static: they tend to rely only on approaches and procedures which are handy to them (Figure 5).

Figure 5: A bar Chart Showing Gender in Public and Private Schools

As reviewed literatures have shown the importance of multiplication operations on students’ learning, it was observed in this study that approximately 100% of in-service or practicing mathematics teachers in the municipality do not make sense of the algorithm they use in their classroom. This was made more evident when the researcher probed further to find out the actual meaning that these permanent in-service mathematics teachers made out themselves. From the findings in this exploratory study, many in-service mathematics teachers at the basic school level are mainly, male dominated. One interesting finding was that, all teachers use the algorithm to teach the students. There have never been any teacher professional development courses in their specific areas for these teachers which make them lag behind in new and conceptual approaches in teaching multiplication of whole numbers. 

In Figure, it was observed that, 0%:100% of the in service mathematics teachers says Yes: No to use of other approaches in their classroom discourse. From it again, it is made evident that, almost all in-service mathematics teachers in the municipality use the Long Multiplication algorithm with 100% who said, Yes, as against 100%, who said, No, for Rectangular Array, Lattice Method and “Other” which includes the use of the Area Model, which happens to be an aspect of RME approach. 

The findings of teaching of multiplication operation by in-service mathematics teachers have a significant role in students’ learning. The findings suggest the similarities across teachers’ approaches in terms of teaching multiplication of whole numbers [22]. Using Area Model which is an aspect of Realistic Mathematics Education (RME) approach has an extreme knowledge gain when in-service mathematics teachers and students use it. The model helps them to make real meaning of multiplication and make apply many other topics in mathematics. Multiplication is not only done using algorithms, repeated addition, lattice and multiplication facts which do not allow both in-service mathematics teachers and their students to transfer, learning.This study provides evidence of some commonalities with teachers’ approach. This study will offer the in-service and pre-service mathematics teachers an opportunity to apply and extend their prior knowledge and teaching strategies to unfamiliar contexts. This is in line with the study conducted by Kwakye [23] who observed in his study that, using of varied and multiple approaches in teaching and learning will boost the confident level of both teachers and students. 

Recommendations

It is recommended that; in-service mathematics teachers be allowed to go for further studies and attend professional development sessions to let them be abreast with new approaches in teaching mathematics. It is as well recommended that, in-service teachers try to use the realistic mathematics education approach in their classroom discourse. 

Limitations

This study was limited to the Komenda Edina Eguafo Abirem (KEEA) Municipality in the Central Region of Ghana. Also, it was limited to only in-service mathematics teachers at the basic school.

Acknowledgements

I show appreciation to the four in-service mathematics teachers who availed themselves for the interaction for this study and which has made it successful. 

Funding

Personally funded.

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