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The Teaching and Learning of Systems of Linear Equation: A Realistic Mathematics Education (Rme) Approach

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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 x-ray 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 ready-made mathematics as an “anti-didactic 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 context-related 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

1. Six lamp holders
2. One control switch
3. One wall socket
4. Two 200watts bulbs
5. Two 100watts bulbs
6. Two 60watts bulbs
7. One junction box
8. Insulated copper wire
9. Small switch ( electric switch )
10. 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:

160I1 – 100I2 +    0I3                            = 240

-100I1 + 360I2    -60I3                      =    0

0I1    – 60I2 +   360 I3                   =    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(-36000-0)

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=

Step7: Compute the currents (I1, I2 and I3) thus:

I1= + +

I2= + +

I3= + +

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

1. We recommend that mathematics curriculum be developed to encourage the students to inculcate research oriented abilities which will lead to inventions in mathematics.
2. 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.
3. 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 10th 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 Services1.32-40 .Available online at www.ispacs.com  and retrieved  on 20th June 2018

Lerman S.( 2014)(ed)Encyclopedia of Mathematics Education,  DOI       ,p522 Available at www.rme_ency…thed.pdf and retrieved from  7th July 20