The Teaching and Learning of Systems of Linear Equation: A Realistic Mathematics Education (Rme) Approach
by
Menkiti Chika Cordelia Department of Mathematics, Nwafor Orizu College of Education Nsugbe +2347039880920 fvaamd@gmail.com
and
Diala Paul Ikechukwu Department of Mathematics Education Imo State University ,Owerri Imo State +2348134453365; dialapaul@yahoo.com
Abstract
Teaching systems of linear equations seems to have been a coaching problem since ancient times. Over the decades, the crude mechanistic approach has been adopted by teachers in the teaching and learning of systems of linear equation . In this approach, teachers take control over each activity while the students are passive learners. As an alternative to this traditional or mechanistic approach, there is need for an approach in which the students is actively involved in learning procedures step by step with the teacher demonstrating how to solve problems. This paper examines the teaching and learning of systems of linear equation: the Realistic Mathematics Education (RME) approach. This paper xray the meaning of RME and the core teaching principles of RME .The Realistic Mathematics Education approach shows how to formulate systems of linear equation from the construction of circuit board with its circuit diagram; resolving the resistors in each loops into a system of linear equation with current (I’s ) as the variable or unknown; the computation of the currents (I’s ) in each loop using inverse square matrix methods of solving systems of linear equation.
Keyword Mathematics, system of linear equation, realistic mathematics education approach
Introduction
Mathematics is the science that deals with the logic of shapes, quantity and arrangement. Mathematics is all around us ,in everything we do .It is the building block for everything in our daily lives ,including mobile devices, architecture(ancient and modern), art, money, engineering, and even sports.(Byline,2013) www.livescience.com what is Mathematics?
In reality, most Mathematics teachers are still teaching using the traditional or mechanistic approach which only engages the students with rote memorization and abstractness of concepts. These approach are contrary to cognitive development of students, because teachers do not provide sufficient time to actively involve the students in the learning process. Acccording to the developmental theory, knowledge cannot be transferred from a teacher to a student, but rather constructed by the students through active participation in the teaching and learning process.(Effandi and Muzakkir,2017) .To achieve high constructive ,creative and critical thinking skills and abilities in the students ,an approach to learning that is most appropriate is the realistic mathematics education approach to problem solving.
Realistic Mathematics Education (RME) approach is based on the idea of Freudenthal which says that mathematics is a human activity. Therefore, according to him ,mathematics should not be learned as a closed system but rather as an activity of mathematizing reality, which mean that students should not be confronted with readymade mathematics as an “antididactic inversion” rather they are to be actively involved in the education process. RME was developed in the Netherlands with the characteristics that rich “realistic” situations are given a prominent position in the learning process. These situation which serve as a source for initiating the development of Mathematical concepts, tools, and procedures and as a context in which students can in a later stage apply their mathematical knowledge ,which then gradually has become more formal and general and less context specific. RME involves six core principles for teaching mathematics which are inalienably connected to RME, they are
 The activity principle: here students are treated as active participants in the learning process
 The reality principle: here students are guided towards possessing the ability to apply mathematics in solving real life problems
 The level principle: here students pass various levels of understanding: from informal contextrelated solutions, through creating various levels of shortcuts and schematizations, to acquiring insights into how concepts and strategies are related.
 The intertwinement principle: here students are offered rich problems in which they can use various mathematical tools and knowledge
 The interactivity principle: here students sees learning mathematics not only as an individual activity but as asocial activity which offers them the opportunity to share their strategies and inventions with others
 The guidance principle: here teachers have proactive role in students’ learning by creating scenarios which have the potential to work as a lever to reach shifts in students’ understanding.
In Mathematics, a system of linear equations (or linear system) is a concept under the branch of mathematics called Algebra. It is a collection of two or more linear equations involving the same set of variables or unknowns. For example, a general system of m linear equations with n unknowns can be written
,

The above system of linear equation can be written or transformed into matrix form as
A = , X = , b= 
ACTIVITY ONE
The teacher guides the students in the construction of circuit board given a circuit diagram as shown below
Circuit Diagram
60 
20 
0 
Ω 
200 
Ω 
60 
Ω 
V 
240 
I 
A 
10 
0 
Ω 
I 
B 
60 
Ω 
I 
C 
10 
0 
Ω 
Materials required
 Six lamp holders
 One control switch
 One wall socket
 Two 200watts bulbs
 Two 100watts bulbs
 Two 60watts bulbs
 One junction box
 Insulated copper wire
 Small switch ( electric switch )
 Wooden board
PROCEDURES
STEP 1: Get all the necessary materials ready for the work.
STEP 2: Cut the wire (length 25cm) with a cutter and connect it to one terminal of the lamp holder labeled (H) which is the life terminal and connect wire (black) to the other terminal labeled (N) neutral.
Repeat the above step (2) for the remaining five (5) lamp holders. After the connections, mount the lamp holders on the wooden board as shown in the circuit diagram.
STEP 3: Mount the control switch on the wooden board and connect the wires to the terminal of the control switch. From the control switch, connect a switch which controls the flow of current in the circuit. After the connection, link the wires to a junction box.
From loop (L), connect the first and the second lamp holder in parallel.
At the second loop, connect another lamp holder in series to the second lamp holder.
The connection is of three loops LA, LB, and LC in the
loop A (LA),: connect 60ohms resistor and 100ohms resistor in parallel;
loop B (LB),, connect another 100ohms resistor and 60ohms resistor in parallel with 200ohms resistor in series and
loop C (LC),, connect again another 60ohms and 100ohms resistor in parallel with 200ohms resistor in series.
ACTIVITY TWO
The teacher guides the students in resolving the resistors in each loops into a system of linear equation as
LA: _{ .…Equation 1}
LB: 0 ………_{ Equation 2}
LC: 0 ………… _{Equation 3}
ACTIVITY THREE
The teacher guides the students to rearrange the systems of linear equation into matrix form thus:
160I_{1} – 100I_{2} + 0I_{3 } = 240
100I_{1} + 360I_{2} 60I_{3} = 0
0I_{1} – 60I_{2} + 360 I_{3} = 0
A=
ACTIVITY FOUR
The teacher guides the students in solving the above equations 1 ,2 and 3 using inverse square matrix method
A=
Step 1: Find The Determinants; FIRST GET THE MINORS:
DET A = 160 (129600 3600) + 100((100×360)– (60×0)) + 0
= 160(126000) + 100(360000)
DET A = 16560000
Step4 :SOLVING THE DETERMINANT OF THE MINORS OF EACH CELL THUS:
NOTE: 1,1 IS READ AS CELL OF ROW ONE ,COLUMN ONE AND SO ON
Step5: Forming a new matrix B using the determinant value of each cell thus:
B=
Step 6: Transpose B and get the Adjoints (Adj.B)
Step7: Compute the currents (I_{1}, I2 and I_{3}) thus:
I_{1}= + +
I_{2}= + +
I_{3}= + +
The individual values for each I, is the current flowing in each loop
Conclusions
The RME approach in the teaching of system of linear equation makes teaching and learning interesting and free from abstractions. It exposes the students to the interrelationship between mathematics concepts and physics concepts.
This study will equip the mathematics teachers with decisive impulse to reform the prevailing approach in mathematics education, by teaching mathematics that is relevant for students and carrying out thought experiments to investigate how students can be offered opportunities for guided inventions of mathematics.
Recommendations
 We recommend that mathematics curriculum be developed to encourage the students to inculcate research oriented abilities which will lead to inventions in mathematics.
 Teachers should adopt RME approach in the teaching of system of linear equations to promote adequate understanding and motivation to learn mathematics and apply it in real life situations.
 Mathematics teachers especially in the tertiary institutions should be trained in the use of RME approach since the new curriculum emphasized on activity centered and student or learner centered for effective teaching of mathematics.
References
Byline. (2013) what is Mathematics? Available online atwwwlivescience. comand retrieved on 10^{th} October 2018
Effandi ,Z. & Muzakkir, S. (2017)Effect of Realistic Mathematics Education Approach on Students’ Achievement and Attitudes Towards Mathematics in International Scientific Publication and Consulting Services1.3240 .Available online at www.ispacs.com and retrieved on 20^{th} June 2018
Lerman S.( 2014)(ed)Encyclopedia of Mathematics Education, DOI ,p522 Available at www.rme_ency…thed.pdf and retrieved from 7^{th} July 20