TECHNIQUES OF LEARNING MATHEMATICS:

TECHNIQUES OF LEARNING MATHEMATICS:

Problem solving technique:

     The problem-solving technique is one involves the use of the process of problem-solving or reflective thinking or reasoning. Problem-solving method/technique, as the name indicates, begins with the statement of a problem that challenges the students to find a solution. In this process of solving the problem the students may be required to gather data, analyse and interpret the information, to arrive at a solution to the problems.

Definitions of Problem-solving

1).”A problem occurs in a situation in which a felt difficulty to act is realized. It is a difficulty that is clearly present and recognized by the thinker. It may be a purely mental difficulty or it may be physical and involve the manipulation of data. The distinguishing thing about a problem, however, is that it impresses the individual who meets it as needing a solution. He recognizes it as a challenge”. -----Yokam and Simpson

2). “Problem solving is a planned attack upon a difficulty or perplexity for the purpose of finding a satisfactory solution”. ---- Risk. T.M

3).”Problem-solving is an Educational device whereby the teacher and the pupils attempt in a conscious, planned, purposeful manner to arrive at an explanation or solution to some educationally significant difficulty”. --- James Ross.

From the above definitions, problem solving involves the following,

·         A goal to be reached

·         A felt difficulty to reach the goal

·         Challenging the felt difficulty through conscious, planned and purposeful attack

·         Reaching the goal or arriving at a satisfactory solution to the problem at  hand

Main objectives of Problem-solving technique

·         to stimulate reflective and creative thinking of the students.

·         It involves the thought process the result from a doubt, a perplexity or a problem.

·         The approach leads to the formulation of generalisations that are useful in future situations involving the solution of similar problems.

·         The solution of a problem, whatever be its nature, practical or informational involves the process of reflective thinking.

Steps in problem-solving

1. Identifying and defining the problem

The problem arises out of a felt need and out of existing student activities and environment activities. The students should be able to identify and clearly define the problem. The problem that has been identified, should be interesting, challenging and motivating for the students to participate in exploring.

2. Analyzing the problem

      The problem should be carefully analysed as to what is given and what is to be found out. Given facts must be identified and expressed, if necessary in symbolic form. The relationships are to be clearly stated. Relations that are not explicitly stated may be supplied by the students.

3. Formulating tentative hypothesis

  The focus at this stage is on hypothesising-searching for a tentative solution to the problem. Analysis of the given data, and analysis of interrelationships among the given facts help the students in formulating hypothesis or educated guesses as the solution to the problem at hand.

4. Testing the hypothesis

    Appropriate method should be selected to test the validity of the tentative hypothesis as a solution to the problem. If it is not proved to be the solution, the students are asked to formulate alternate hypothesis and proceed.

5. Checking the results or verification of the result

         At this step the students are asked to determine their results and substantiate the expected solution. The students should be able to make generalisations and apply it to their daily life.

Approaches and Techniques to Problem-solving  

 Problem solving advocates the following approaches

·         Analytic and synthetic methods.

·         Inductive and deductive methods.

·         Method of analogies:

     In analogy, problems are solved by comparing them with similar problems that have been solved before. Thus the method of solution becomes explicit and clear.

·         Restatement method:

    Problem solving becomes easier if the student is able to redefine the given problem using his own language and symbols. This approach is known as restatement method.

·         Method of Dependencies:

     In this method, the problem is solved by focusing on mutually dependent components in the problem. The analysis of the problem into its constituent elements throws light on the mutually dependent elements in the problems. The interrelationships among the elements can be made use of for reaching the correct solution of the problem.

·         Graphic Method:

    In this method, the problem is represented using diagrams and figures. The graphic representation aids the students in determining fundamental relationships that exist among the given data and to look for further details and relationships necessary for solving the given problem. This method is very helpful in proving theorems, solving riders, problems relating to mensuration etc.

Teacher’s role in Problem-solving technique 

     The teacher plays a significant role in problem solving method. The teacher’s role is to:

1.       Ensure an atmosphere of freedom in the class.

2.       Create the problem situation.

3.       Assist the students in accepting, defining and stating the problem.

4.       Helps the students in analyzing the problem and in breaking up the problem into simple units.

5.       Help the students keep their attention focused on the main problem all the time.

6.       Guide the students in locating relevant source materials.

7.       Encourage the students in seeking important relationships in the data.

8.       Helps the students develop an attitude of open mindedness and critical enquiry.

9.       Exhibit spirit of enquiry and discovery.

Characteristics of Good Problem

1.       The problem should be real rather than an artificial one.

2.       It should facilitate the integration of old and new process.

3.       The solution of the problem should result in learning new higher order rules.

4.       The solution of the problem should help in transfer of knowledge.

5.       The problem should be educationally significant, productive of important and worthwhile learning.

6.       It should be possible of solution. The students should be equipped with background information and skills which are prerequisite for solving the given problem.

7.       It should be related the sub-unit, the unit and the course.

8.       It should form the basis for further learning.

9.       It should be clear and free from ambiguities.

10.   It should be interesting and challenging.

11.   It should arouse the curiosity of the students.

12.   It should occur frequently in life situations.

13.   It should provide best mental discipline to the students.

14.   It should facilitate realization of the objectives of teaching mathematics.

Reasons for Difficulties in Solving Problem

1.       Lack of interest and motivation.

2.       Lack of language clarity in understanding the problem.

3.       Inability to analyse the problem thoroughly.

4.       Lack of focus on the key relationships.

5.       In ability to identify the interrelationship among the given data.

6.       Lack of fluency in the mental visualization or diagrammatic representation of the problem.

7.       Inability to recall and apply appropriate rules and formulae.

8.       Lack of skill and practice in solving problems.

9.       Lack of proficiency in the fundamental arithmetic operations.

10.   In adequate knowledge of fundamental mathematical concepts, rules and formulae.

11.   Difficulty in reading, identifying and using mathematical symbols.

Types of Mathematical Problems

1.       Puzzle Problem: These are problems designed for the exercise of ingenuity and patience, as these problems create some bewilderment or perplexity in the individual who faces it, sometimes people solve them as a leisure time activity merely for the sake of joy and pleasure that they derive. However, such problems preserve the curiosity of the student and he feels joy in solving them.

2.       Catch problem: These problems display a jugglery of words. Such problems check the mental alertness of the students, but have little bearing on training mental faculties.

3.       Real Problems: These problems are directly related to the real life experiences of the students. They emerge from the real life situations. The solution of such problems help the students in facing future life problems with ease and confidence. The solution of such problems stimulates the curiosity and help in training the mental faculties of the children.

4.       Unreal problem: Problems which are beyond the purview of real life situations are called unreal problems. Such problems give false information to the students.

Merits of Problem-solving method

1.       Problem-solving provides a real life like experience to the children.

2.       It develops in pupils good habits of planning, thinking, reasoning and independent work.

3.       It develops initiative and self-responsibility among the students.

4.       It takes into account individual differences.

5.       It helps the students to develop reflective thinking.

6.       It helps the students to approach future problems with confidence.

7.       It builds a mental attitude for effective learning based on critical thinking.

8.       It helps the children develop mental traits of open-mindedness and tolerance as the children see many sides to a problem and listen to many points of view.

Demerits of Problem-solving method

1.       Not all students are problem solvers.

2.       The problem solving method becomes monotonous if used too frequently.

3.       It is time consuming and consequently it is not possible to cover the syllabus on time.

4.       The success of this method depends upon mathematics teachers who are well versed in critical thinking and reflective thinking. Not all mathematics teachers are well versed in these type of thinking.

5.       Reference and resource materials may be difficult to come by.

6.       Only a skilled and resourceful teacher will be able to make an effective use of this method.

7.       All topics in mathematics cannot be taught through this method.

8.       Lack of interest and motivation on the part of the students can spoil the effectiveness of this method.

ORAL WORK

  Oral work is a part of the written work. Oral questions can be solved mentally or orally without the use of any writing material. In carrying out oral work, the pupils have to solve mental problems without the use of paper and pencil.

Importance of Oral Work

1.       Most mathematics work we use in our practical life is oral. The day-to-day sale and purchase of commodities, a small transaction of money etc., are all made through oral work.

2.       It gives enough practice for independent thinking and analysis of the problem. The use of oral work provides good mental exercise. It increases the speed and sharpens the intelligence of pupils.

3.       Oral work helps in the process of teaching, work as

a)      In testing the previous knowledge of the pupils.

b)      Oral work in the form of question and idea helps in the presentation of subject matter.

c)       It proves as one of the best attention catching devices.

d)      It helps the teacher, may easily pay individual attention.

e)      Much in carrying out revision work.

4.       In solving problems much of the time and energy of both the teacher and pupil can be saved if the oral work is made use of.

5.       It helps in creating interest as well as maintaining interest of the student in the study.

6.       Oral work develops healthy competitions among pupils.

Merits of Oral Work

1.       Once a pupil has become well as oral mathematics be enters written work with full confidence.

2.       It develops accuracy, precision and motivation in the learner.

3.       Oral problems can be employed to break the monotony of the class.

4.       As it is backbone of written work it helps the pupil to have better performance in written work.

5.       It removes shyness of the pupil.

6.       It is an effective means of maintaining discipline.

WRITTEN WORK

       Problem solving in mathematics requires the written work. It facilitates and pushes ahead the work that has been carried out orally. In written work the help of writing material is essentially taken care off. Oral work gives the start and the written work follows it.

Purpose of written work in mathematics

1.       The teacher can able to know the amount of work done by the pupils.

2.       It can be also possible to make the pupils in solving problems according to certain rules and processes.

  Merits of Written work

1.       Through written work it is possible to have clarity of the thought and proper reasoning.

2.       It is possible to have lengthy problems.

3.       Written work is more permanent in nature and so it is possible to judge achievement of the pupils.

Precautions, which are taken in written work:

1.       The teacher should give proper instruction to the pupils with regard to the work, time limits and other facts.

2.       It should not be beyond the psychological limit of the pupil.

3.       It should be such as to keep the entire class busy.

4.       The problems that are given are clear and definite.

5.       As far as possible the written work should be verified to ensure the correctness of the result.

6.       There should be use of black board for written work.

7.       Students should be made habituated of doing the work correctly.

DRILL WORK

      Drill work usually to drill the minds of the pupils on the lesson taught. Drill is an essential part of all mathematics work. It provides opportunity for self-improvement. The basic facts and operations of mathematics have to be memorized through sufficient drill work. Drill is nothing but the ways to revise the lesson that is already taught. Drill must be recognized as an essential means of attaining some of the desired controls. The acquisition of facility in operations can be secured only though systematic and repeated practice.

Principles (Precautions) of Drill work

1.       Drill period should be very short say 10-15 minutes.

2.       The learner should understand what he is practicing and appreciate its significance.

3.       The learner should be an active participant.

4.       The drill should follow developmental and discovery stages of learning.

5.       Drill sums should be well graded.

6.       Drill should be based on thinking and insight so that it never become a mere mechanical repetition.

7.       The achievement of the learner during drill sum stage must be frequently checked.

8.       Pupils of lower mental abilities require more drill sums.

9.       Drill sums should not be used as a punishment.

      When drill is provided to develop meaning, it should increase understanding. Thus as effective drill not only develops knowledge and skills but also a means of maintaining good habits. In this direction the pupils learn better in mathematics through drill work because it helps them to practice in solving more problems. The practice leads them to attain mental satisfaction and ultimately the stage of perfection, because the practice makes the man perfect.

CONCEPT MAPPING

     The term mapping denotes a procedure which requires children to “map” out what they have learnt and how it appears to fit” Children might be helped to draw a web or flow chart to show what they have been learning about. Such a chart represents the ideas, concepts and knowledge that the children have been working with during a particular unit, as perceived by the child. The procedure might begin by listing aspects of the topic which were covered. The children can then map the relationships between the different items which explain how they see the links. This provides a way of seeing what they have understood. It could then provide a basis for teacher and the student to talk over understandings and misunderstandings.

     A Concept map is a diagram that depicts relationships between concepts. It is graphical tool that we can use to organize and, sometimes more important, to visualize content of lesson or theme. The terms (concepts) are commonly written in the “balloons” and they are linked to each other with lines and, if needed, words that describe the relationship between them. There are few graphical presentations similar on the first sight but different in their approach and functionality so as in use. The most similar among them are mind maps but mind maps serve a different purpose. They help with memory and can be used in brainstorming as a very effective tool.

         Mind maps are collections of words structured by the mental context of the author itself with visual mnemonics, and, through the use of colour, icons and visual links. Also, algorithm may look like concept map but just on the first sight. Algorithm is a step-by-step procedure for calculations. For making scheme of algorithm we use special notation and symbols.

       Concept maps have hierarchical structure. Mapping is the creative process of organizing content and can be used in planning lessons, learning, individual and group work, developing mathematical literacy and fostering mathematical thinking. Conceptual mapping can be easily applied to other school subjects and to everyday life. Once accepted, making concept maps becomes the way of successful learning. Conceptual mapping technique was introduced in the education by Joseph Donald Novak.

Simple example of meta map is given below:

Benefits of using a conceptual map/Advantages of conceptual map

1.       Perceive the concepts and relationships among them.

2.       Visualize, organize and distinguish concepts by their importance.

3.       Develop mathematical literacy.

4.       Connect a new knowledge with the old one.

5.       Evaluate learning process.

6.       Expand their knowledge.

7.       Apply mapping method to other contents.

8.       Be more active.

9.       Get better results by working in groups or pairs.

10.   Develop their communication skills through the presentation of conceptual maps and discussions.

Maps allow teacher to:

1.       Teach students how to learn without understanding.

2.       Provide comprehensive view of the lesson.

3.       Organize teaching material.

4.       Visualize the teaching process.

5.       Introduce new concepts and link them with the known.

6.       Decompose complex ideas.

7.       Check the level of understanding.

8.       Identify weak points.

9.       Explore the reason for misunderstanding among students.

10.    Encourage student activities.

11.   Connect interdisciplinary.

Disadvantages of the concept mapping

Technical:

1.       Paper (if we restrict ourselves to A4).

2.       Duration of a lesson (45 min).

3.       Related to the content.

4.       Lessons with a lot of new or similar concepts.

5.       Lessons that have linear structure.

6.       Mapping cannot be used at any time (for different reasons), but we can use already made maps.

Concept maps are covering higherlevels in learning process which can be shown schematically

 

<--------               <--------

In the Bloom’s taxonomy, learning at the higher levels in dependent on knowledge and skills at lower levels. Visualization through concept maps can help to link those parts and estimates answers on cognitive verbs.

     Concept map are facilitative tools which help to improve learning, creating and using knowledge based on reasoning and sense making. They help to develop a way of organized thinking that can be applied as well in every day line.

3.1.4: Collaborative learning and Cooperative learning strategies

COLLABORATIVE SKILLS

Definition

        Collaborative skills are the behaviors that help two or more people to work together and function well in the process. Teachers can train their students in the skills of collaboration so they will be able to accomplish group tasks.

Examples

Basic skills of collaboration are similar to skills of communication, which can be taught to younger children. The University of Vermont's Department of Education has identified a list of skills of collaboration for the classroom. They require students to learn how to:

• Begin a conversation
• End a conversation
• Ask for help
• Ask a favor
• Give a compliment
• Join in
• Accept criticism
• Follow directions
• Ask questions
• Say 'thank you'
• Say 'no'
• Accept 'no'
• Encourage others
• State feelings
• Negotiate
• Express concern for others
• Listen
• Take turns
• Take responsibility

          Collaboration is the act of working together for a common goal. The Partnership for 21st Century Skills says that mastering collaboration skills requires the ability to work effectively with diverse teams. It also requires the ability to "be helpful and make necessary compromises to accomplish a common goal."

Time for productive collaboration is a must in today's classrooms.

• Phillip Schletchy identifies qualities of the work teachers give students that affect engagement. Affiliation, that is, opportunities to work with others, can be a positive influence on student engagement.
• A study on cooperative learning found that "subjects who worked cooperatively spent more time working on practice exercises and reported greater satisfaction than those who worked individually."
• "Studies have shown that groups outperform individuals on learning tasks, and further that individuals who work in groups do better on later individuals’ assignments as well (Barron, 2000b, 2003; O'Donnell &Danserau, 1992)."Powerful Learning by Linda Darling-Hammond, page 19.
• Having the capacity to collaborate is an important component in project-based learning and an essential personal and professional skill.
• The Partnership for 21st Century Skills, a national organization formed by government, corporations, associations, and individuals, has developed a framework that fuses the 3 Rs with the 4Cs. The 4Cs are:
• critical thinking and problem solving
• communication
• creativity and innovation 
• collaboration

         Working effectively with others is an extremely complex endeavor. Collaboration skills are complicated to learn because they are actually people skills. Learning these skills takes guided practice and quality feedback. Teacher's shouldn't expect their students to work together effectively without explicitly teaching and modeling collaboration skills. These skills include:

• Active listening
• Respect
• Manners
• Positive Attitude
• Focused
• Social Awareness

Simply telling students to work together won't lead to productive collaboration. Teachers need to develop activities and projects where students have reasons to collaborate. We must teach students how to be good group members through modeling, role playing, discussion, and facilitating. Collaboration can be taught and learned by

• Assigning clear responsibilities
• Showing students examples
• Assigning a leader
• Encouraging self-direction
• Charting progress
• Conducting group and self-evaluations
• Designing rubric to measure the process and product

Cooperative learning techniques:

     Cooperative learning is a successful teaching strategy in which small teams (each with students of different levels of ability); use a variety of learning activities to improve their understanding of a subject. It is an instructional arrangement for teaching academic and collaborative skills to small heterogeneous groups of students. Cooperation means working together to accomplish shared goals. Hence, students work in mixed ability groups and rewarded on the basis of the success of the group. Students work together to maximize their own and each other’s learning. It is a teaching strategy involving children’s participation in small group learning activities that promote positive interaction. Each member of a team is responsible not only for learning what is taught but also for helping teammates learn, thus creating an atmosphere of high achievement.

The main purpose of co-operative learning is actively involving students in the learning process.

Steps for co-operative learning technique

1.       Content to be taught is identified, and criteria for mastery are determined by the teacher.

2.       The most useful cooperative learning technique is identified, and the teacher determines the group size.

3.       Students are assigned to groups.

4.       The classroom is arranged to facilitate group interaction.

5.       Group processes are taught or reviewed as needed to assure that the groups work smoothly.

6.       Teacher develops expectations for group learning and makes sure students understand the purpose of the learning that will take place. A time line for activities is made clear to students.

7.       Teacher presents initial material.

8.       Teacher monitors student interaction in the groups, and provides assistance and clarifications as needed. Teacher reviews group skills and facilitates problem solving when necessary.

9.       Student outcomes are evaluated. Student musts individually demonstrate mastery of important skills or concepts of the learning. Evaluation is based on observations of student performances.

Steps of Most often used techniques

Learning together technique:

Steps:

1.       Determining the instructional objectives and content.

2.       Deciding the group size.

3.       Dividing the students into groups.

4.       Arranging of the class.

5.       Planning of educational materials.

6.       Giving the roles to the group members in order to provide dependence.

7.       Explaining the academic work.

8.       Creating the positive objective dependence and cooperation among the groups.

9.       Explain the criterions and behaviours necessary for achievement.

10.   Guiding the student behaviours and helping the group work.

11.   Finishing the lesson.

12.   Evaluation of individual student’s qualitative and quantitative learning.

13.   Evaluating the performance of the group.

Jigsaw technique

       The jigsaw strategy is a cooperative learning technique and efficient teaching method that also encourages listening, engagement, interaction, peer teaching, and cooperation by giving each member of the group an essential part to play in the academic activity. Just as in jigsaw puzzle, each piece, each student’s part is essential for the completion and full understanding of the final product.

Steps:

1.       Students are divided into home groups of three to six students.

2.       Individual members of each group then break off to work with the “experts’ from other groups.

3.       “Experts” research a subcategory of the material being studied.

4.       “Experts” return to their home group in the role of instructor for their subcategory.