This is the second Part of CTET Mathematics Pedagogy. In this part we will discuss Problem of Teaching Mathematics, Error Analysis, Diagnostic and Remedial Teaching.

1. CTET MATHEMATICS
Mathematics Pedagogy
Part - 2

2. • This is the second part of Mathematics Pedagogy.
In previous part we discussed Nature of
Mathematics, Need of Mathematics in
Curriculum, Language of Mathematics,
community of Mathematics and communication.
• In this we will discuss Problems in Teaching
Mathematics in context of teacher and students,
Evaluation, Error Analysis, Diagnostic and
Remedial Teaching.

3. Evaluation
“Evaluation is the systematic assessment of the worth or merit of
some object.”
“Evaluation is the systematic acquisition and assessment of
information to provide useful feedback about some object.”
Both definitions agree that evaluation is a systematic endeavor and
both use the deliberately ambiguous term 'object' which could
refer to a program, policy, technology, person, need, activity, and
so on. The latter definition emphasizes acquiring and assessing
information rather than assessing worth or merit because all
evaluation work involves collecting and sifting through data,
making judgments about the validity of the information and of
inferences we derive from it, whether or not an assessment of
worth or merit results.

4. Education Evaluation
• Educational evaluation is the evaluation process of characterizing
and appraising some aspect/s of an educational process.
• There are two common purposes in educational evaluation which
are, at times, in conflict with one another. Educational institutions
usually require evaluation data to demonstrate effectiveness and
to provide a measure of performance for marketing purposes.
Educational evaluation is also a professional activity that individual
educators need to undertake if they intend to continuously review
and enhance the learning they are endeavoring to facilitate.
• The Joint Committee on Standards for Educational Evaluation
published three sets of standards for educational evaluations. The
Personnel Evaluation Standards was published in 1988, The
Program Evaluation Standards (2nd edition) was published in
1994, and The Student Evaluations Standards was published in
2003.

5. Purpose and Function of Evaluation
• To determine the present status of the children in the teaching-
learning process.
• To motivate children for better learning.
• To provide basis for guidance and counseling to children.
• To determine the rate of progress of the children.
• To collect evidence for improvement in curriculum.
• To diagnose children’s weaknesses and strength for further instruction.
• To measure the effectiveness of the examination system.
• To improve the efficiency and success of the teacher.
• To discover innovative and effective methods of teaching.
• Classify children in different categories.
• To improve the teaching - techniques and strategies.
• To provide basis for remedial teaching on the basis of strength and
weakness of the children.
• To give reinforcement and feedback to both the teachers and pupils.

6. Good Evaluation
• A good evaluation programme should have following characteristic:
• It measures the direction and extent behavioural changes.
• Evaluation is quantitative and qualitative estimation of specific
changes in children’s behaviour.
• Evaluation is diagnostic so that it may provided basis for remedial
teaching.
• Evaluation is a continuous and comprehensive process.
• Evaluation offers feedback to the entire educational system.
• Evaluation is pupil oriented
• Evaluation is activity based.
• Evaluation used to improve instruction, curriculum, methods and
examination etc.

7. Process of Evaluation
• Selection and Formulation of objective.
• Defining objectives in terms of behavioural
changes.
• Selection of tools and techniques of evaluation.
• Use of tools and techniques and making results.
• Interpretation and generalization of results.
• Feedback of the results for improvement of
teaching-learning process.

8. Formative Evaluation
• Formative evaluation is generally any evaluation that
takes place before or during a project’s
implementation with the aim of improving the
project’s design and performance.
• Formative evaluation complements summative
evaluation and is essential for trying to understand
why a program works or doesn’t, and what other
factors (internal and external) are at work during a
project’s life.
• Formative evaluation does require time and money
and this may be a barrier to undertaking it, but it
should be viewed as a valuable investment that
improves the likelihood of achieving a successful
outcome through better program design

9. Need and Importance of Formative
Evaluation
• Provides information to teachers for modifying instruction and
teaching
• Helps to prescribe group and individual remedial programmes.
• Helpful in monitoring pupil’s learning progress.
• Provides feedback to both the teachers and pupils.
• It facilities retention and transfer of learning.
• Enables teacher to change the mode of instruction according to
need of the students.
• It also reinforce the learning of high achievers.
• Helps in better performance in the simulative evaluation.
• It can also be used as a self evaluation device by the children.
• Provides immediate feedback to the children.
• Helpful in diagnosing child’s strength and weaknesses.

10. Summative Evaluation
• Summative evaluation looks at the impact of an intervention on the target
group. This type of evaluation is arguably what is considered most often as
'evaluation' by project staff and funding bodies- that is, finding out what the
project achieved.
• Summative evaluation can take place during the project implementation, but is
most often undertaken at the end of a project. As such, summative evaluation
can also be referred to as ex-post evaluation (meaning after the event).
• Summative evaluation is often associated with more objective, quantitative
methods of data collection. Summative evaluation is linked to the evaluation
drivers of accountability. It is recommended to use a balance of both
quantitative and qualitative methods in order to get a better understanding of
what your project has achieved, and how or why this has occurred. Using
qualitative methods of data collection can also provide a good insight into
unintended consequences and lessons for improvement.
• Summative evaluation is outcome-focused more than process focussed. It is
important to distinguish outcome from output. Summative evaluation is not
about stating that three workshops were held, with a total of fifty people
attending (outputs), but rather the result of these workshops, such as increased
knowledge or increased uptake of rainwater tanks (outcomes).

11. Formative VS Summative Evaluation
• Formative evaluation is qualitative while summative
evaluation is quantitative.
• Formative evaluation is a continuous process while
summative evaluation is an event that takes place at the
end of an instructional unit.
• Summative evaluation is formal and takes the shape of
quizzes and written tests whereas formative evaluation
is informal such as homework and projects.
• The aim of formative evaluation is to improve upon what
has been learnt whereas the aim of summative
evaluation is to prove the amount of learning that has
taken place.

12. Problems of Teaching Mathematics
• A sense of fear and failure among a majority of
children.
• A curriculum that frustrates both a talented
minority as well as the non-participating majority
at the same time.
• Crude ways of assessment making perception of
Mathematics as mechanical computation.
• Lack of teacher preparation and support in the
teaching process Mathematics.

13. Fear and Failure
• Mathematics anxiety and math phobia are the
consequences of that emotional comment. Mathematics
is taught in the school in such a way that it alienates the
children and causes them to stop attending schools.
Such fear of Mathematics is closely linked to a sense of
failure. The cumulative nature of Mathematics mainly
causes such Fear and Failure.
• The other main reason is the predominance of symbolic
language. When symbols are manipulated without the
understanding, then after a certain point children start
dissociating from the subject.
• The language of Mathematics learnt in school is
completely different from their everyday speech and
becomes a major force of alienation in its own right.

14. Disappointment Curriculum
• Any mathematics curriculum that prioritizes procedure
and memorization of formulas over understanding of
concepts, enhances the anxiety and phobia.
Mathematics is the subject that sees great motivation
and talent even at an early age in a small number of
children. These children take to quantisation and
algebra easily and carry on with great facility. The
curriculum does not offer conceptual depth and thus
settles for the minimal use of children’s motivation. The
learning procedures may be easy for them, but their
understanding and capacity for reasoning remain un-
utilised.

15. Crude Assessment
• The crude method of assessment are the barriers for
the effective teaching and learning of mathematics.
Those method of assessment encourage perception
of mathematics as mechanical computation.
• In all the subjects except the mathematics, partial
knowledge may be tried and demonstrated but in
case of mathematics such as perception is easily
coupled to anxiety and panic.
• The evaluation procedure in mathematics has seen a
little change in our school over a hundred year or
more.

16. Inadequate Teacher Preparation
Mathematics education depends very heavily on
teacher’s understanding of mathematics and his
preparation of teaching mathematics. It is the
teacher’s knowing of pedagogic techniques that
makes the teaching and learning of mathematics
so fascinating.

17. Other Systemic Problems
• Compartmentalisation
This includes very little systematic communication between the high
school and college teachers of mathematics. Most of the school
teachers have never even seen or interacted with the research
mathematics
• Curricular Acceleration
It involves the rapid and frequent change of the course curriculum
making the shape of mathematics education as taller and more
spindly, rather than broad and rounded.
• Gender Issue
Mathematics is the subject which tends to be regarded as a masculine
domain and this perception is further aided by the complete lack
of references in text book to women mathematicians. It has been
found that the teachers tend to address boys more than girls.

18. Error Analysis
The purposes of error analysis are to
(1) identify the patterns of errors or mistakes that
students make in their work,
(2) understand why students make the errors, and
(3) provide targeted instruction to correct the
errors.
When conducting an error analysis, the teacher
checks the student’s mathematics problems and
categorizes the errors.

19. Addition and Subtraction
• Lack of understanding of regrouping
• Confusion of 1s and 10s in carrying and writing
• Forgetting to carry 10s and 100s.
• Forgetting to regroup when subtracting 10s and 100s.
• Regrouping when it is not required.
• Incorrect operation (the student subtracts instead of adding or vice
versa).
• Lack of knowledge of basic number facts.
Multiplication and Division
• Forgetting to carry in multiplication.
• Carrying before multiplying.
• Ignoring place value in division.
• Recording the answer from left to right in multiplication.
• Lack of alignment of work in columns.
• Lack of knowledge of basic number facts.

20. Fractions
• Incorrect cancellation.
• Failure to reduce to lowest common denominator.
• Ignoring the remainder.
• Incorrect conversion of mixed numbers to fractions.
Word Problems
• Difficulty in reading.
• Inability to relate to context of problem.
• Inability to understand the language and vocabulary of the
problem.
• Difficulty in identifying the relevant and the irrelevant
information.
• Difficulty in identifying the number of steps required to solve the
problem.
• Trouble in doing mathematical operations (addition, subtraction,
multiplication, division).

21. Diagnostic and Remedial Teaching
Diagnostic Evaluation
Diagnostic assessment is a form of pre-assessment that allows a
teacher to determine students' individual strengths, weaknesses,
knowledge, and skills prior to instruction. It is primarily used to
diagnose student difficulties and to guide lesson and curriculum
planning.
Imagine being a teacher in a new classroom. You begin teaching a
lesson only to be met with stares of confusion from your students.
When you ask the students if they understand what you are
teaching, they reply that they have no idea what you're talking
about. Now imagine teaching that same class after conducting a
pretest to determine what the students already know about the
topic. Which scenario sounds preferable? Which would result in a
better experience for both the teacher and the students?

22. Benefits of Diagnostic Assessment
Diagnostic assessment benefits both the instructor and the
students.
• First, it allows teachers to plan meaningful and efficient
instruction. When a teacher knows exactly what students
know or don't know about a topic, she can focus lessons
on the topics students still need to learn about rather
than what they already know. This cuts down on student
frustration and boredom.
• Second, it provides information to individualize
instruction. It may show a teacher that a small group of
students needs additional instruction on a particular
portion of a unit or course of study. He can then provide
remediation for those students so that they can fully
engage with new content.

23. Similarly, if a teacher discovers that a group of students
has already mastered a large portion of a unit of study,
he can design activities that allow that group to go
beyond the standard curriculum for that topic through
independent or small group study.
• Finally, it creates a baseline for assessing future
learning. It shows both the teacher and the students
what is known before instruction has occurred. Thus, it
sets a baseline on a topic. As the students move through
instruction, they can see what they are or aren't
learning, and the teacher can provide remediation or
enrichment as needed.

24. Remedial Teaching
Remedial education (also known as developmental
education, basic skills education, compensatory education,
preparatory education, and academic upgrading) is
assigned to assist students in order to achieve expected
competencies in core academic skills such as literacy and
numeracy.
Whereas special education is designed specifically for
students with special needs, remedial education can be
designed for any students, with or without special needs;
the defining trait is simply that they have reached a point
of under preparedness, regardless of why. For example,
even people of high intelligence can be underprepared if
their education was disrupted.