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.
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
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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.
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.
