The document discusses multiple representations in mathematics education, emphasizing the importance of verbal, graphic, algebraic, and numeric modes in teaching and learning. It outlines the deductive and inductive methods of teaching, highlighting their distinct approaches and objectives for fostering student understanding of mathematical concepts. Additionally, it explains heuristic learning as a flexible method encouraging exploration and creativity in problem-solving.

1. MULTIPLE REPRESENTATION IN
MATHEMATICS
•Multiple representation of teaching and learning
mathematics
•Deductive learning
•Inductive learning
•heuristic

2. Multiple representation in learning and
teaching
• Representation is a sign or combination of signs,
characters, diagrams, objects, pictures, or graphs, which
can be utilized in teaching and learning mathematics.
• There are four modes of representation in the domain of
mathematics: verbal, graphic, algebraic, and numeric.
• Representation needs to be translated from one mode
to another mode. Translation is an important skill that
learners need to develop in order to be more proficient
in learning mathematics.

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• Different models show different aspects of same concept,
allowing students to make deeper connections. For
example, numbers can be represented as objects, set
patterns, segments on a line or scale, and points on dial.
• Multiple representations are widely used to build meaning
behind math and develop a deeper understanding of
properties or ideas connected to the same fact or
operation.
• Representations include words, symbols, graphs,
diagrams, tables, formulas, physical and virtual
manipulative etc.

4. Deductive Method
• The deductive method of teaching
mathematics is a process of reasoning from
the general to the specific. It is a method of
teaching that emphasizes the use of logic and
reasoning to solve problems. This method is
based on the principle of deduction, which
means to establish a particular truth by
showing that it follows logically from a set of
premises that are already known to be true.

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• The deductive method involves presenting a
general rule or principle, and then using
logical reasoning to apply that rule to specific
examples. This approach is often used in
geometry, where students are presented with
a set of axioms and then use logical reasoning
to prove theorems.

6. Inductive Method
• The inductive method of teaching mathematics is a process of
making generalization or formulating hypotheses based on
specific examples.
• It is process of discovering patterns, relationships, and
regularities in mathematical concept, and it often starts
concrete examples or observations and then moves to more
abstract generalizations
• It can be implemented in the classroom by providing students
with a variety of examples, encouraging them to make
observations and generalizations, guiding them in formulating
and testing hypotheses, and helping them to develop the ability
to think critically.

7. Objectives of Inductive Method
• The inductive method of teaching, where the teacher provides
students with relevant learning materials, such as examples, images,
keywords, data, etc., and guiding them to find something familiar,
identify patterns, formulate hypotheses, and test their conclusions.
• The teacher allows students to share their thought processes while
trying to answer and solve concerns. They are encouraged to
participate in open discussions and use evidence to support their
claims.
• Teacher uses open-handed questions and activities to challenge
students to explore and investigate mathematical concepts and
phenomena. The teacher creates an environment where students
can take risks to learn and apply their knowledge.

8. Heuristic
• Heuristic learning in mathematics is a way of learning and
discovering mathematical concepts and skills by using
experienced-based techniques, such as trail and error,
working backwards, drawing diagrams, making guesses,
and finding patterns.
• Heuristic learning can help students to solve complex and
non-routine mathematical problems by applying different
strategies and methods that suit the problem situation.
• Heuristic learning can also foster students’ creativity,
confidence, and interest in mathematics.

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• Heuristic proposed a four-step model for problem
solving: understanding, devise a plan, carry out the
plan, and look back. It also suggested various
method, such as using analogies, simplifying the
problem, generalizing the problem, and checking
the validity of the solution.
• Heuristic learning in mathematics is not fixed or
rigid procedure, but rather a flexible and adaptable
approach that can be used in different contexts and
level of mathematics.