PROCESSES IN
MATHEMATICS:
- Mathematical reasoning.
- Pattern recognition.
- Algebraic thinking.
- Geometric thinking (Van Hiele model
of geometric thought).
- Problem solving in mathematics.
- Creative thinking in mathematics.
Mathematical
Reasoning
The best way for child to learn and enjoy mathematics is to be curious,
ask a lot of questions and make sense of mathematics through reasoning. It is a
teacher’s responsibility to encourage this curiosity in the mathematics class
as well. The best way for children to make sense of mathematics is through
reasoning. Children learn to use mathematics in meaningful ways by asking a lot
of questions and thinking. This enables them to connect different ideas, gain a
deeper conceptual understanding and therefore enjoy mathematics. For this, they
classroom environment must be such that children are encouraged to ask a lot of
questions, the teacher must be prepared to answer/facilitate these questions
patiently without judging the children; essentially the classroom environment
must be free from fear.
What
is reasoning?
See the example in Table-1 below.
Both Student-1 and Student-2 have reasoned their answers. Notice that the
teacher was not satisfied with the correct answer, but asked the students to
explain how they solved it. By explain the solution, students are able to make
sense of the mathematics. The teacher clearly knows from the students’ response
that they not only know how to add, but they have also conceptually understood
what the”=” sign represents in the problem. Reasoning enables deeper conceptual
understanding.
Table-1
Teacher
(to the class): Can you solve this problem?
16+27= ( ) + 30
Student-1:
13
Student-2:
13
Teacher:
How did you solve it?
Student-1:
Thirty is three more than twenty seven, so if I subtract 3 from 16, I will
get 13.So 13 is the correct to make the two sides equal”.
Student-2:
“Sixteen plus twenty seven is equal to forty three. Forty three minus thirty
is equal to thirteen. So 13 is the correct answer.
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In mathematics theory we say that
there are three main components to the reasoning process. They are justifying,conjecturing and generalizing. These terms may sound
complicated, but they are very simple to understand.
1.
Justifying:
Justifying
is to make a logical argument or reason based on an idea that has already been
understood. The explanation that Student-1 and Student-2 gave in table -1 above
are examples of justification. These students have already understood addition,
the meaning of the equal to sign (=) in the equation the teacher wrote and
justified their answers to the given problem.
2.
Conjecturing:
A
conjecture involves reasoning about mathematical relationships and making a
statement that is thought to be true, but is not known (by the person making
the statement) to be true. It is usually made on the basis of incomplete
information. In the example below (Table-2), the student sees a pattern; (every
time two even numbers are added the sum is always an even number) and makes
this conjecture. The student did not know this previously and makes this
statement based on just the sums she/he sees on the black board (incomplete
information). It is thought to be true by the student but he/she has not proved
it to be true by any formal methods.
Table-2
Teacher presents each
of the following problems. He guides students to identify a pattern (that
when two even numbers are added the sum is also an even number).
2 + 4 = 6
12+14 =26
4 + 8 =12
One Student makes the
following conjecture:
“When we add two even numbers, the sum is
also an even number”
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3.
4. Generalizing: A conjecture that is considered to be true is a generalization. Generalization is also involve reasoning about mathematical relationships, seeing patterns and extending the reasoning beyond the scope of the original problem, where a new insight is formed.
4. Generalizing: A conjecture that is considered to be true is a generalization. Generalization is also involve reasoning about mathematical relationships, seeing patterns and extending the reasoning beyond the scope of the original problem, where a new insight is formed.
For
example, in table-3 student and teacher make a generalization.
Table-3
The teacher is
provoking students to see a pattern and make conjectures.
2 + 0 = 2
3+ 0 = 3
4+ 0 = 4 and
so on
One student makes a
conjecture by saying
“When we add zero to
any number, the sum is always the same number”.
The teacher explains
that this is true for all positive integers and writes it more formally as,
X + 0 = X; where x is any positive integer.
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It is important to remember that when
every child has an opportunity to reason and make sense of the mathematics that
is being taught in school, that they develop deeper conceptual understanding of
topics in mathematics. When children develop conceptual understanding of their
mathematics then they begin to enjoy learning and doing mathematics. It is
interesting to observe that there are different types of reasoning done in
mathematics. It is interesting to observe that there are different types of
reasoning done in mathematics.
Mathematics is also called the science of
reasoning. According to Locke, “Mathematics is a way to settle in mind a habit
of reasoning. “ Reasoning in mathematics is of two types: 1) Inductive
reasoning and 2) Deductive reason.
Inductive
reasoning
When statements containing mathematical
truths are based on general observations and experience, reasoning is called
inductive reasoning.
Inductive reasoning is
bottom-up, that is broad generalizations (or general rules) are made from a
specific observations and cases.
Inductive reasoning is a logical process in which multiple premises, all
believed to be true or found true most of the time, are combined to obtain a
specific conclusion. In Inductive reasoning, we notice a trend or a pattern and
make an informed guess (conjecture) based on that pattern. The resulting
generalization may not necessarily be true.
All the above examples
shown in Table 1, 2 and 3 are examples of inductive reasoning.
Inductive reasoning (as opposed to deductive reasoning or adductive) is reasoning in which
the premises are viewed as supplying strong evidence for
the truth of the conclusion. While the conclusion of a deductive argument is
certain, the truth of the conclusion of an inductive argument is probable,
based upon the evidence given.[1]
Many dictionaries define inductive reasoning as reasoning that derives
general principles from specific observations, though some sources disagree
with this usage.[2]
The
philosophical definition of inductive reasoning is more nuanced than simple
progression from particular/individual instances to broader generalizations.
Rather, the premises of an inductive logical argument indicate some degree of support (inductive
probability) for the conclusion but do not entail it; that is, they suggest truth but do not
ensure it. In this manner, there is the possibility of moving from general
statements to individual instances (for example, statistical syllogisms,
discussed below)
DEDUCTIVE REASONING
Deductive reasoning is a
type of reasoning that is top-down. It is a process by which we begin with one
set of facts and deduce some other sets of facts. Deductive reasoning is the
process of resoning from one or more statements to reach a logically certain
conclusion. It is a logical reasoning process which proceeds from a general
rule to a specific instance, unknown to known or abstract to concrete.
The example in table-5 below is a process
of deductive reasoning. In this example
we started with some facts (definition of even integers) and logically deduced
that the sum of two even integers is even. This conclusion is logically
certain.
Table-5
Example-1:
Hypothesis:
The sum of any two even integers is even.
Proof:
Let a and b be even integers.
We have that a=2n and
b=2m. (by definition of even numbers)
Consider the sum a +
b= 2m + 2n= 2 (n +m)=2k where k= n + m is an integer.
Therefore the sum of
a + b= 2k (by definition of even numbers we have shown that the hypothesis is
true)
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 Example-2
D B E
A C
Theorem:
“Sum of the three angles of a triangle is equal to two right angles”
Data: ABC
is a triangle.
To
Prove:
∟BAC
+ ∟ABC +∟ACB
Construction:
Draw a line DE parallel to side AC and passing
through vertex B.
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Deductive reasoning, also deductive
logic, logical deduction is the process of reasoning from one or more statements (premises) to reach a logically certain
conclusion.[1] It differs from inductive
reasoning and deductivereasoning.
Deductive
reasoning links premises with conclusions. If all premises are true, the terms are clear, and the rules of deductive logic are
followed, then the conclusion reached is necessarily.
Deductive reasoning (top-down logic) contrasts with inductive
reasoning (bottom-up
logic) in the following way: In deductive reasoning, a conclusion is
reached reductively by applying general rules that hold over the
entirety of a closed
domain of discourse, narrowing the
range under consideration until only the conclusion(s) is left. In inductive
reasoning, the conclusion is reached by generalizing or extrapolating from
specific cases to general rules, i.e., there is epistemic uncertainty. However, the inductive reasoning mentioned here
is not the same as induction used in mathematical proofs – mathematical
induction is actually
a form of deductive reasoning.
Deductive
reasoning, this type of reasoning is based on certain postulates or
axioms and in this statements are products of mind. We try to compare and
contrast various statements and then draw out conclusions from such a
comparison. Essentials of a deductive reasoning are as follows:
* Undefined terms-----point, surface etc.
*
Definitions
* Postulates and Axioms.
Reasoning:
- Justify
why an answer or approach to a problem is reasonable;
- Make
and test generalization or observation;
- Make
prediction or draw conclusions from available information;
- Analyze
statements and provide examples which support or refute them;
- Judge
the validity of arguments by applying inductive and deductive thinking;
Problem
Solving:
- Use
information to identify and define the question/s within a problem;
- Make
a plan and decide with information and steps are needed to solve the
problem;
- Choose
the appropriate operation/s for a given problem situation;
- Select
and apply appropriate problem-solving strategies to solve a problem from visual(draw
picture, create a graph), numerical (guess and check, look for a pattern),
and symbolic(write an equation) perspectives;
- Organize,
interpret, and use relevant interpretation;
- Select and use appropriate tools and
technology;
- Show
that no solution or multiple solutions may exist;
- Identify
alternate ways to find a solution;
- Apply
what was learned to a new problem.
Creating
classrooms that Encourage Reasoning
There are many ways to create a mathematics classroom environment that
encourages mathematical reasoning. Some useful pointers are given below:
1. The
teacher must encourage children to ask plenty of questions and also ask many
questions.
2. A
significant amount of time must be spent discussing and talking mathematics in
the class. Teacher can make statement like-“when we add an odd number and an
even number the sum is an odd number” and ask the children if this statement is
(Always, sometimes or never) true. The teacher must also ask the students to
justify their answers.
3. Teachers
should not dismiss wrong answers, but encourage the children to justify their
answers and in this process sometimes children self-correct their mistakes.
Otherwise the teacher can intervene at the appropriate time and help the child
find his/her error.
4. Mathematics
classes are not only about drill and solving sums in notebooks. The classroom
must include talking mathematics. Only when children talk mathematics they are
able to available on the internet, the teacher can choose suitable age and
topic
5. Encourage
children to create their own word problems, this also helps children to reason.
6. Teachers
must frame rich questions, in such a way that children learn how to make
conjectures, generalizations and justify their answer. (Give an example).
7. connect
the language they speak with the symbols and language of mathematics and make
connections with the real world and mathematics.
8.
There are appropriate puzzles for
students to solve in groups, alone or as a whole class.many mathematics puzzles
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ReplyDeletethankyou sir
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