Unit 2 - WordPress.com · 2016. 5. 20. · Dr. Priya Mathew St. Joseph's College of Education Mysore . 1. Pattern Recognition / Patterning the ability to see order in a chaotic/disordered

Unit 2.3

Mathematical Processes

5/19/2016 1 Dr. Priya Mathew St. Joseph's College of

Education Mysore

1. Pattern Recognition / Patterning

• the ability to see order in a chaotic/disordered

environment.

• Patterns can be found in ideas, words, symbols

and images

• key helper of our potential in logical, verbal,

numerical and spatial abilities.

• important problem-solving skill.


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Connections between Symbols

and Conceptual Understanding

Arranging dots in square patterns connects the

number 1, 4, 9 and 16 to their reference as square

numbers

▫ recognize and use connections among

mathematical ideas

▫ understand how mathematical ideas

interconnect and build on one another to

produce a coherent whole


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2. Mathematical Reasoning

• The best way for a child to learn and enjoy mathematics is to be curious, ask a lot of questions and make sense of mathematics through reasoning.

• Reasoning enables deeper conceptual understanding.

• enables to connect different ideas, gain a deeper conceptual understanding and therefore enjoy mathematics

• children are encouraged to ask a lot of questions, the teacher must be prepared to answer/facilitate these questions patiently without judging the children

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Dr. Priya Mathew St. Joseph's College of

Education Mysore

Example 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: "Sixteen plus twenty seven is equal to forty three. Forty three minus thirty is equal to thirteen. So 13 is the correct answer"

• Student 2: "Thirty is three more than twenty seven, so if I subtract 3 from 16, I will get 13. So 13 is correct to make the two sides equal"


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Components of reasoning process

• Justifying

• conjecturing

• generalizing


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Justifying

• Justifying is to make a logical argument or

reason based on an idea that has already been

understood.


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Conjecturing

• reasoning about mathematical relationships and making a statement that is thought to be true, but is not known to be true by the person making the statement

• It is usually made on the basis of 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.

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Example 2

• 6 + 4 = 10

• 10 + 22 = 32

• 4 + 8 = 12

• identify a pattern followed in this?

• student makes the following conjecture :

• " When we add two even numbers, the sum is also an even number"

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Generalising :

• A conjecture that is considered to be true is a generalization.

• Generalization involves:

reasoning about mathematical relationships

seeing patterns

and extending the reasoning beyond the scope of the original problem, where a new insight is formed.

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Example 3

• This 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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Mathematical Reasoning Leads to Mathematical

Memory Built on Relationships

A 10-by-11 rectangle built

with two staircases from 1 to

10 can help you remember

the formula for the sum of a

series of numbers


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Types/ Approaches of Reasoning

• Inductive Reasoning

• Deductive reasoning


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Inductive Reasoning:

• is bottom-up, that is, broad generalizations

are made from specific observations and

cases.

• notice a trend or a pattern and make an

informed guess (conjecture) based on that

pattern.

• inductive reasoning is a logical reasoning

process which proceeds from examples to a

general rule, known to unknown or concrete

to abstract.

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Deductive Reasoning :

• 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.

• logical reasoning process which proceeds

from a general rule to a specific instance,

unknown to known or abstract to concrete.

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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 = 2n + 2m = 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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Reasoning and Proof

Pictures of odds and

evens can help

students justify why

the sum of two odd

numbers is always

even.


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Example 2

• Theorem : Sum of the three angles of a

triangle is equal to two right angles


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3. Algebraic Thinking

• about generalizing arithmetic operations and

operating on unknown quantities.

• It is almost the same as inductive reasoning.

• It involves recognizing and analyzing patterns

and developing generalizations about these

patterns.


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Example

• Problem: Whe 3 is added to 5 times a certain number, the sum is 38; find the

u er,

• students emerging from arithmetic will subtract 3 from 38 and then divide by 5

• In contrast, they will be taught in algebra classes first to represent the relationships in the situation by using the stated operations: 5x + 3 = 38


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Alge rai Reaso i g is…

• The study of relationships among quantities.

• The study of structures.

• A set of rules and procedures.

• Generalized arithmetic.


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4. Abstraction

• Many areas of mathematics began with the study of real world problems, before the underlying rules and concepts were identified and defined as abstract structures.

• For example, geometry has its origins in the calculation of distances and areas in the real world;

• algebra started with methods of solving problems in arithmetic.

• Abstraction is an ongoing process in mathematics and the historical development of many mathematical topics exhibits a progression from the concrete to the abstract.


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historical development of geometry as

an example for abstraction

• First step : Greek – Euclidean Geometry – Plane geometry

• 17th century : Descartes- Cartesian co-ordinates - analytic geometry.

• Further steps : non-Euclidean geometries.

• 19th century : geometry in n-dimensions, projective geometry, and finite geometry.

• Finally Felix Klein's : "Erlangen program" : study of properties invariant under a given group of symmetries - this level abstraction revealed connections between geometry and abstract algebra.

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advantages of abstraction

• It reveals deep connections between different

areas of mathematics.

• Known results in one area can suggest

conjectures in a related area.

• Techniques and methods from one area can

be applied to prove results in a related area


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5. Generalization

• three meanings attached to generalization

• Abstraction : generalization is the process of finding the out the properties of a whole class of similar objects.

• extension (empirical or mathematical) of existing concept or a mathematical invention. famous example : the invention of non-euclidean geometry.

• Product : If the product of abstraction is a concept, the product of generalization is a statement relating the concepts, that is, a

theorem.


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6. Problem Solving

• students are able to achieve the expectations

in mathematics, and it is an integral part of

the mathematics curriculum.

• Students will develop, select, apply, and

compare a variety of problem-solving

strategies as they pose and solve problems

and conduct investigations, to help deepen

their mathematical understanding.


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Four Stages of Problem Solving

1. Understand and explore the problem; 2. Find a strategy; 3. Use the strategy to solve the problem; 4. Look back and reflect on the solution.


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Understand and explore the problem;

• read a problem several times, both at the start

and during working on it.

• With younger children, repeat the problem

and ask them to put the question in their own

words.

• Older children might use a highlighter pen to

mark and emphasise the most useful parts of

the problem.


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• Finding a strategy : it is a fairly simple matter to

think of an appropriate strategy. This will also help

them to understand the problem better and may

make them aware of some piece of information

that they had neglected after the first reading.

• Solve the problem: now the problem will be solved

and an answer obtained.

• During this phase it is important for the children to

keep a track of what they are doing.

• useful to show others what they have done and is

helpful in finding errors 5/19/2016 29


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looking back:

• it is good practice for them to check their working and make sure that they have not made any errors.

• it is vital to make sure that the answer they obtained is in fact the answer to the problem and not to the problem that they thought was being asked.

• children are often able to see another way of solving the problem. This new solution may be a nicer solution than the original and may give more insight into what is really going on.

• Finally, the better students especially, may be able to generalise or extend the problem.


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Advantages of Problem Solving

– build new mathematical knowledge through problem solving

– solve problems that arise in mathematics and in other contexts

– apply and adapt a variety of appropriate strategies to solve problems

– monitor and reflect on the process of mathematical problem solving


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7. Problem posing • Problem posing involves generating of new problems and

questions aimed at exploring a given situation as well as the reformulation of a problem during the process of solving it

• Advantages:

• can foster more diverse and flexible thinking, enhance stude ts’ problem solving skills, broaden their perception of mathematics and enrich and consolidate basic concepts.

• help in reducing the dependency of students on their teachers and textbooks, and give the students the feeling of becoming more engaged in their education.

• will enhance stude ts’ reasoning and reflection.

• it can foster the sense of ownership that students need to take for constructing their own knowledge.

• it results in a highly level of engagement and curiosity, as well as enthusiasm towards the process of learning mathematics.

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Algorithm • defined as a finite sequence of instructions each

of which has a clear meaning and can be performed with a finite amount of effort in a finite length of time.

• An algorithm is a procedure or formula for solving a problem.

• The most familiar algorithms are the elementary school procedures for adding, subtracting, multiplying, and dividing, but there are many other algorithms in mathematics.

• The word derives from the name of the Arabic mathematician, Mohammed ibn-Musa al-Khwarizmi.


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Algorithm and flow chart to find whether a number is

odd or even.

• Algorithm:

• Begin

• Input n

• num <- n%2

• if (num=0) then Output "even"

• else

• Output "odd"

• End

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Unit 2 - WordPress.com · 2016. 5. 20. · Dr. Priya Mathew St. Joseph's College of Education Mysore . 1. Pattern Recognition / Patterning the ability to see order in a chaotic/disordered

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