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Introduction

Discussions are still going on the different methods of teaching that best suit to students ability and potentiality. The cliché that a student should listen to the teacher passively has been changed and many theoreticians have brought new innovations to this field. Among these theories, personalized learning plays a significant role; especially in promoting new methods of teaching that really make studies enjoyable to both the teachers and the students. Personalised Learning means, Assessment, teaching, curriculum and school strategies designed to ensure that the talents of every individual can be fully realised. It aims to raise standards for all by setting high expectations based on a sound knowledge and understanding of every child. (What does personalised learning mean to you 2008). Personalized learning is a strategy to accomplish individual students need. The Personalized Learning approach emerged in the publics charter school system in California, in1990s. (Personalised learning 2009).

It can be class room based or non-class room based. Just different from the traditional method of teaching, personalized learning identifies the skills, passion and trend of individual students. As it is more concerned with the practical side of learning than that of the strict theoretical side in the class rooms, there is possibility for it to be misinterpreted, that is, a system that wastes the valuable hours of a student. Many researches have propounded on Personalised Learning and one of the major researchers of this field was Charles Desforges, an ex-primary teacher and researcher. When referring to the application of the Personalised Learning in school children, he observed that though it is some what easy to learn, its application is difficult. So he suggested that for the application of Personalised Learning the trick we need to perform is to bring the practical knowledge of teachers and the theoretical knowledge of the researchers together in order to promote advanced teaching practices. But to a more practical level, one can see that personalized learning is much higher than the traditional method of teaching. Students can depend on any kind of learning styles; this doesnt mean that their intelligence level is poor.

The aim of personalized learning is making learning most productive and meaningful way to rejuvenate student in the learning process. According to the social constructivists each and every child has an innate ability to acquire knowledge. It is very clear that every student has his own individual strengths, weaknesses, and his own particular way or approach of learning. Putting this common notion into practice one can easily find that our current education system has not been successful in individualizing learning experiences. Majorities of the countries (UK, US, India, China etc.) are following class room learning process rather than individual learning approach. We can see the consequences in general, only 50% or below secure good score. Students intelligence level and interests varies, so that some of them can move ahead with class room learning whereas others cant. The following statement from the online article entitled Personalized Learning rightly comments about the drawbacks of current education system. It says: The Personalized Learning model is the solution to turn the fate of our public education system from continuing and increasing failure rates to one of success for all students, one that will truly ensure that no child is left behind. (Personalised learning 2009).

Personalized learning process props up to meet the demands and aspirations of a student. Traditional classrooms provide typed learning pleura for many years and learning becomes passive. In present decade, with the emergence of learning theories one can notice a gradual change from teacher-centred to student-entered. Individual difference of a student demands personalized learning experience in classroom. Personalized mode of education promotes motivation to the child and it helps him to acquire self-direction and self-reliance. Personalized way of learning follows the process such as planning, learning, reflecting and feedback. Here the learner will have a choice about what is learned, when and how it is learned. Personalized learning is possible for traditional learning establishments like schools and colleges. Personalized learning claims learning that happens anywhere. Based on a learning plan, personalized learning becomes more effective; assignments, group works, projects etc improve students cognitive ability.

Personalized learning is multi-faceted, which includes parents involvement, peers involvement, smaller class sizes, technology, articles, guides, individual tuitions, more one-on-one teacher and students involvement, teacher and parent-teacher association, independent study, home schooling, media etc. (Rice 2004). Personalized learning leads the teachers and the students for innovative teaching techniques. A study reports the subsidiary benefits of personalized learning. It says, Schools that reported more extensive implementation of personalize learning strategies often mentioned experimentation: in teaching and learning techniques. (Sebba, et al 2007, p.26) Parents have a very notable role in promoting personalized learning. As parents are well aware of the potentiality of their children, they can help their children to select the type of education that best suits to their mental ability. Based on that, they can arrange the kind of learning they want. Peers also contribute their role in learning styles. A well understood student can help his peers. They can discuss among themselves; each will be having strong areas and week areas, but when they combine together it results in better understanding. Some students feel difficulty in big class room; they prefer schools with smaller class size. Media and articles also play a big role in personalized learning. If a student needs much more help he can depend on individual tuition. Generally the class room based teaching means 1:40 ratio, that is, one teacher for teaching 40 students in the class room. (Personalized Learning 2004)

The teacher gives lecturing, notes and goes with other regular routines. Some students may be able to follow, some may not. This is the reason for massive failures happening in public education system. Especially in the case of mathematics students need more guidelines. Once they catch the idea they can go ahead. Let us look into benefit of personalized learning in linear equations. The teacher can find out the individual tastes of students through a set of questionnaires. Questionnaires are attached at the end. Questionnaires are done as a test and will help the teacher to identify individual passion. The test will help the teacher to acquire enough knowledge about the Intellectual Qualities of Various students in a class room. The application of personalized learning will be based on the test result. After analyzing the questionnaire teacher can decide the individual passion. Certain students prefer demonstration while others depend on explanation and practical examples. When a student is given problem he/she may depend on text books to solve the problem. Some may depend on their peers who are better on it. And finally when it turns to a bright student, he/she tries to solve himself/herself.

Students are asked to solve the problems after giving the basic ideas. They can depend on multiple resources like technology access (computer and internet), students group, text books or peers and guides etc. It s highly recommended that one should have a clear concept before moving ahead with problems.

Here conduct a case study for finding the effectiveness of personalized learning in modern classroom. The study is based on a group of students consisting of fifteen ten year old students. Each student comes from various socio economic backgrounds. Based on the questionnaire one can easily find that five students are going above average, and five of them have an average cognitive level, rest of them are below average. Average students need individualized attention. With the help of displays, charts, flash cards, flip charts and demonstrations students understand the problem in their own way. Classroom assessment and tests are the effective ways for finding these problems. Students have failed to understand the question sand they forget the steps of mathematical problems.

Introducing new methods like drills, pictorial representations, and numerical games students can develop a positive attitude to the classroom. After the analysis of the results it is clear that other students faced some problems to understand classroom activities. The learning material considers students cognitive level, difficulty level, social and cultural backgrounds, infrastructure facilities. Most of the students need individual attention in their learning. Consider the scenario where a students misinterpret n*n as 2n instead of n^2, and also n+n as n^2 instead of 2n.These are detected after small assessments. Students are made to realize the difference between the terms n^2 and 2n via substituting the values for n. They can understand the consequence hence after, the total calculation goes wrong. Therefore a student should possess the basic ideas. Here is a planned learning material with certain learning objectives. It includes a lesson plan about linear equation. Number of personalized learning activities and exercises are included in this lesson plan.

Different learning styles of solving linear equation

Different learning styles introduced for these topics are practical examples, basic ideas with demonstration, steps with explanation (depend on text books, articles, internet etc) and online games. Most of these learning styles introduced are referenced with national strategies. These are designed for class 10 pupils for increasing their standards, implemented in UK. These are introduced to reduce the errors, misconceptions and also especially when they are not able to follow with basics of linear equations. These personalized learnings are experimented soon after few assessments, evaluating students strengthens and weakness.

Illustrate the practical examples

Algebra is a division of mathematics designed to help solve certain types of problems quicker and easier using linear equations. The general idea of algebra can explain through language. Before stepping into problems they should develop a meaningful idea about it. This is possible via explaining them through language. In terms of language equation can be explained as follows-It is based on the concept of unknown values called variables, unlike arithmetic which is based entirely on known number values. Let us discuss an important concept in algebra the equation. The idea of an equation can be compared with equipment which is used for balancing weights, with weights bar on one side and equal substance on other side. We have to add or subtract weights on one side or on either side to make it balanced. Consider this scenario of balancing machine with 20 units on one side and 30 units on other side. Here, for balancing, either we have to add 10 units on one side or subtract 10 units from other side. In linear equation instead of weights, numbers or constants are used. The equation can also be balanced by a substance called variable. A variable is an unknown number represented by any letter in the alphabet. Several symbols are used to relate all of variables and constants together. These symbols are:

• * Multiply
• / Divide
• + Add
•  Subtract
• () Calculate what is inside of the parentheses first

Let us simulate a model to motivate students to solve a practical problem through algebra. Consider a typical problem- investment of four people, they invest ¬40000. They have equal share in this investment. What is the individual share? This is explained in this material later. (Basic of algebra 2005).

Conveying the basic ideas through demonstration

Know that letters are used to stand for numbers in algebra.

For example

In the equation 4z+3=22, z is particularly unknown number, but in the equation x+y=24, x and y can take many different values

Recognize algebraic conventions

4*n or n*4 can be thought of as 4 lots of n or n+n+n+n, and can be shortened to 4n.

In the expression of 4n, n can take any value, but when the value of expression is known, an equation is formed, i.e. if 4n is 18, the equation is written as 4n=18.

Use the equals sign appropriately and correctly.

• Recognize that if x=y then y=x and that x+y=z can also be written as z=x+y
• Use letter symbols to write expressions in meaningful contexts.

For example

Add 8 to a number x+8

Subtract 5 from a number m-5

A number multiplied by 3 then added by 4 m*3+4

Understand the difference between expressions such as

2x and x+2 4(a+6) and 4a+6

a^2 and 2a 2a^2 and (2a)^2

Explain the distinction between equations, formulae and functions. For example

• In the equation 5z+4=2z+40, z is particularly unknown number.
• In the formula v=u+at, v, u, a and t are variable quantities, related by the formula.
• In the function y=8x+23, for any chosen value of x, the related value of y can be calculated.

Understand inverse operations

Analyse that any one of the equations:

• x+y=z, y+x=z, z-x=y, and z-y=x
• ab=c, ba=c, b=c/a and a=c/b

Implies each of the other three in the same set, as can be verified by substituting suitable sets of numbers into the equations

Apply inverse operations when two successive operations are involved. For example, the inverse of dividing by 5 and subtracting 6 is adding 6 and multiplying by 4, i.e.

If x/5-6=y, then x=5(y+6).

Use a spreadsheet to verify this, entering different numbers in column A, including negative numbers and decimals. If column C always equals column A, the inverse is probably correctly expressed.

(Algebra 2008).

Problems solving using linear equations

The equation is similar to problems in mathematics. For example consider this expression +16=20.We could easily guess equals 4. Here in this problem stood for an unknown number. We can convert this into algebra as

x+16=20.

i.e. x=20-16=4. And answer is x=4.

Understanding Coefficients

The coefficient of a variable is the number which the variable is being multiplied. Consider following equation

x+10=4x+20

i.e. 4x-x=20-10

i.e. 3x=10

Here in this equation 3 is the coefficient of x. Consider following terms to understand coefficients better.

Term Coefficient of x

4x 4

0.34x 0.34

x 1

-x -1

In order to solve 3x=10. i.e. to get the value of x we need to divide both sides by 3.We can compare this with a classic problem. Consider the investment of three people, they invest ¬30000. They have equal share in this investment. What is the individual share? i.e. 30000 divided by 3. Answer is ¬10000.We can convert this to algebra as x+x+x=30000, where x represents individual share i.e. the unknown variable.

To solve this equation x+x+x=30000

i.e. 3x=30000.

To get the value of x, divide with coefficient of x on both sides i.e. x=30000/3 =10000

(Basic of algebra 2005)

Steps with explanation

Here the students are given wide range of resources such as text books, articles, internet etc to solve the problems.

General concept: Move variable on one side and everything else on other using inverse operations

Step 1: Simplify both sides, when necessary.

This would involve things like removing (), removing fractions, removing decimals and adding similar terms

Step 2: Use addition or subtraction to move variables to one side and all other terms to other side

Step 3: Use multiplication or division to eliminate any values that are in front of the variable.

Step 4: Check the answer. Answer can be verified by substitution manually or using excels.

Example 1

Solve the equation 10-4x=6

10-4x=6

10-4x-10=6-10 Inverse of addition by 10.ie sub by 10

-4x=-4 Bring all x terms one side

-4x/-4 = -4/-4 Inverse of multiplication by -4 i.e. div by -4

x=1

Understand the term -4x i.e. the operation involved is multiplication i.e. -4 multiplied with x.

We can verify the answer by substituting the value obtained in equation

.i.e. 10-4*1=10-4=6

Flash exercise by Jay Austin and Jordan Noll, College of Design; Brad Hokanson, faculty, College of Design, University of Minnesota, St. Paul, MN; with refinements by: Steve Kladstrup, Independent Flash Developer, Minneapolis, MN. (Solving linear equation).

Verify with excel as follows

Value is the answer we obtained. In LHS columns A2 takes the value of Values (A) columns. If LHS columns and RHS columns evaluate the same we can assume that the answer is correct!

Example 2

Solve the equation 2(x+3)-2=4(x-2)

2(x+3)-2=4(x-2)

2x+6-2=4x-8 remove the parenthesis

2x+4=4x-8

2x-4x=-8-4 Get all x terms one side

-2x=-12

-2x/2=-12/2

-x=-6

-x/-1=-6/-1 Inverse of multiplication by -1 is div. by -1

x=6

If you place 6 back in for x i.e. 2(6+3)-2=4(6-2)

2*9-2=4*4

18-2=16

16=16

Verify with excel. (Nancy 2009).

Example 3

Solve the equation 1/4x+1/2=4x-1/2

(1/4) x+1/2=4x-1/2

4*(1/4) x+4*1/2=4*4x-4/2 to get rid of the fractions, multi.both sides

by the LCD of 4

x+2=16x-2

x-16x=-2-2 bring all x terms one side

-15x=-4

-15x/-15=-4/-15 Inverse of multiplication by -15 is div. by  15

x=4/15

If we put back x=4/15 we can see the equation is balanced.

(Solving One-Step Linear Equations 2009).

Example 5

Solve the equation. 25z-.2=.45z-0.1

100(0.25z-0.2) =100(0.45z-0.1) to get rid of the decimals mul. Bothe sides by 100.

25z-20=45z-10

25z-45z=-10+20 Get all z terms one side

-20z=10

-20z/20=10/20 Inverse of mult. By 20 is div 20

-z=1/2

z=-1/2

Verify with excel.

We can see that LHS and RHS coincide. Hence the value -1/2 substituted is correct. (Beginning algebra Tutorial 14: Solving linear equation 2008).

Learning through online games

Online games inspire the students more. They feel more enjoyment while doing the problems rather than learning something in strict environment. Students can strengthen their skills through online games. Some of the games which help students in linear equations are

Postman Phil

Help Postman Phil deliver letters by solving algebraic expressions by substitution.

Planet Hop

Travel deep space and find the coordinates of the planets and the equation of the line.

Planet Blaster

Solve linear equations and blast the planets

Equation Match

Help ancient mathematicians find matching pairs by working out the value of x in each equation.(Algebra math game 2008).

The following games also used for stimulating the students:

1. Think of a Number  interactive discussion starter.
2. Substitution  Evaluate the expressions by substituting numbers for letters.
3. Sequences  Work out the 4th, 5th, 10th, 20th and nth terms in a number sequence.
4. Algebra Cross number  To solve this cross number puzzle, you also need to work out what numbers x and y represent.
5. Functions Machine  To solve this problem, you must work out how to continue the sequence of numbers  drag the operators into the boxes on the function machine to help. Now available as 2 step and 1 step puzzles, for those of you who prefer to solve simpler math problems

(Mathematical Interactivities  puzzles, games and other online educational resources).

Assessment programs

Assessment programs should be conducted in order to find whether the learning styles introduced are effective. Personalized Learning schools combine multiple assessment levels of student academic achievement. Certificated teachers oversee student progress toward satisfying state academic standards through the regular recording of detailed learning records, compiling extensive student work portfolios, and conducting annual state mandated testing programs. (Personalized Learning 2004)..

Assessment program can also be done via day-to-day observations, periodic observations and interactive section in class room. When a weak student gradually shows improvement, we can give positive attitude to the learning styles introduced. Some students may feel difficulty in assessments program. Even if they are talented, they will be blank during assessments. Self assessments and peer assessments can be used for these folks. Gradually they will be out ridden of fear. Assessments can help to improve the motivation and self-esteem of individual students.

Next let us think what should be done for a student who doesnt show any kind of development. The teacher should give special kind of attention to those students. Try to find out what exactly their problems are. If they are haunted by house problems or any other kinds of threat by which they are lagging behind, consult their parents. If they are not facing any kind of problems, check their intelligence level and interest for learning and reorganize the learning styles. Test them through different learning process and tailor them to achieve success. Frequent assessment feedback helps to enhance their learning. Any kind of improvement shown by them is important. There is a tendency for low attaining students to cease their studies, later they become victims of serious social problems (Black & William 2001).

The following questions can be used for assessments (a) 6-7(x-3) =-1 (b) 1/2x-1/8=3/8x-5/8 (c). 25a-0.5=.2a+0.15 (d) 7(x-3) ==7x+2

(Beginning algebra tutorial 14: Solving linear equation 2008).

Evaluation

Evaluation is done to find out what went well and what needs improvement. Self evaluation is a vital part of personalized learning. Here the student is responsible for all activities. The following statements rightly comment about the role of a student in a student-centred classroom. The words of Dr Valentina make it clear when he says: In this study this was evident in learner-centred contexts where students were encouraged to take increased responsibility for their learning. This was achieved by student planning, organizing, evaluating and implementing strategies to improve their learning. In a teacher-centred approach these elements of the learning process are controlled by the teacher. (Klenowski 1996) In traditional classrooms teachers play a significant role because the teacher is fully responsible for improving a controlled learning environment and he or she is responsible for giving support and guidance for students. Students self evaluation is an effective method of assessment in personalized learning. Here each student will get an opportunity to evaluate and monitor their performances.

Evaluation is done generally after the assessment program. A highly scored student can tell what exactly motivated him/her. Whether it is computer games or articles or individual guides or whatever it is. A low scored student can be considered for what needs improvement. Some students may need a mixture of all types of learning. In the evaluation process of year 10 group as a case study, it was found that 6 students have scored highly. They followed all kinds of learning styles. Students who only depended on explanation could score average marks. Students group also could promote the weaker students. Demonstration through computers and internet games also could influence the students to a great extent. Yet few of the students are not able to follow the full concepts of algebra. The learning styles have to be reorganized after evaluating their intelligence level and interest for learning.

Theories of learning and assessment

Increasing the standards of learning that are achieved through school education is very important. Government is carrying out tremendous changes in curriculum and assessment programs to achieve the standards. Learning is driven by what teachers and students carry out in class rooms. Here the teachers play the major role. They have to tailor the students to meet the demands. Technically we can treat class room as a black box. When we treat class room as a black box there are components involved in it  a) the input b) the out put c) the process inside the black box. Inputs from the outside are fed in to the system i.e. the pupils, teachers, other resources, school rules, parents demands, assessments with high expectation, etc. Output follows  pupils are more knowledgeable and talented, better test results, teachers who are more or less satisfied etc. Let us consider the process inside. Ultimately the teacher makes the process inside. They have to make the inside work efficiently. They have to build the students to meet the challenges through regular practices and assessments. (Black & Wiliam 2001)

Some times they will have to introduce new learning styles to achieve the target. Number of learning theories and teaching strategies are promoting new learning experiences for students. The Behaviourist learning theories are very important in the process of personalized learning. Here learning is brought about through the process of giving stimulus, response and rewards. This helps the students to self search and assessment. Positive reinforcement and positive feedback play an important role in personalized learning. The journal article titled Theories of Learning and Their Implications for On-Line Assessment gives some comments about behaviourist approaches of learning. The article says: The behaviourists take the view that learning is brought about through stimulus, response and reward, a form of conditioning process or associative learning. (Francis Underhill 2006) Through gradual and sophisticated chains of behaviour student will attain the reward of positive reinforcement. Another major learning theory is cognitive theory. The advocates of cognitive approach focus on the qualitative development of thinking.

The exponents of cognitive theory suggest an experimental learning having four different stages. These are conceptualization, planning, experience, and reflection. In a personalized classroom students will get a lot of chances for thinking and practicing. Cognition refers mental ability and a cognitive classroom includes the process such as thinking, remembering and continuous learning. In a personalized classroom the teacher will get the chance to monitor students mental actions and control learning process. As a result of continuous thinking and remembering of concept or problem the student will acquire the steps or equations. Social constructive theory plays an important role in present education system. A social constructivist develops his theory on the basis of the process, scaffolding. According to the social constructivists man has an innate ability to learn something. The duty of a teacher is to modify this ability. Teacher plays the role of a facilitator and a good guide. In a personalized class room teacher gives a planned learning material which includes number of learning objectives. Students read the materials and analyze the problem in a detailed manner. In a constructivist classroom student constructs knowledge himself. Individual difference is unique in every person. Knowledge acquisition in a classroom is based on the cognitive level of the students. Especially in a mathematics classroom some problems like fraction are not very difficult for average students. But a student of below average needs personalized instructions. All these learning theories suggest motivation as an important factor in a classroom.

Humanist learning theories are also helpful for a personalized classroom. The advocate of Humanist learning theories concentrates the development of the students self concept. Self esteem is an important motivating factor for a student. Humanists say that

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