Mathematical Problem Solving Latest

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PRESENTATION 3: HANDOUT 3

MATHEMATICAL PROBLEM SOLVING

PRESENTED BY:

TUN FIRDAUS

HAYATI

ATHIRAH

SYAZWANI



Mathematical Problem Solving

FIRDAUS



Mathematical Problem Solving

1. Mathematics teaching and learning primary

goal is to develop the ability to solve wide

variety of complex mathematics problem.2. To many mathematically literate people

says- mathematics is synonymous with

problem solving.



Cont.

3. When two people talk about mathematics problem solving they may not be talking about something.

4. Reitman defined a problem as when you have been given the description of something but do not yet have anything that satisfied that description.

5. Henderson and Pingry-blocking of that goal for individual.



Learning to Solve Problem is the

Principal Reason for Studying

Mathematics

NATIONAL COUNCIL OF SUPERVISORS OF MATHEMATICS



In this Chapter:

We want to review and discuss

1. Research on how students in secondary

schools can develop the ability to solve a wide variety of complex.

2. Address on how in instruction can best

develop this ability.



Instructionfundamental

goal

To developmath skills

Knowledge

Ability that transfer to tasks not explicitly

covered in

curriculum



Question?

1. Should instruction emphasize the particular

problem solving techniques on strategies

unique to each task?2. Will problem solving be enhanced by

providing instruction that demonstrated on

develops problem solving techniques or

strategies useful in many tasks?



Research on Problem Solving

1. Educational research conducted within

variety of constraints:

Isolation of variable. Availability of subjects.

Limitation of research procedures.

Availability of resources.

Balancing of priorities.



Cont.

2. Methodologies Clinical approach

Mathematical tasks situation are devised

Talk aloud while working or interviewed and asked to ref lect on their experience and especially their thinking process.

3. Basic for most mathematics problem solving research for secondary school students in past 31 years was Polya-field at cognitive psychology.



Framework

1. According to Polyas problem solving stages

2. To Polya problem solving was major theme

of doing mathematic and teaching studentto think

Understanding the problem

Making a planCarrying out

the plantLooking back





Cont.

3. This below is linear framework of U.S. mathematic textbook for problem solving

stage.

Read

Decide

Solve

Examine

Read

Know

Plan

Solve

check

OR



Cont.

4. Linear nature of that model do not promote

the spirit of Polys stage and his goal of

teaching student to think. Why?

They depicts problem solving as linear process.

They present problem solving as series of step.

They imply that solving Mathematics problem is

a procedure to be memorized , practiced and

habituated.

They lead to an emphasis on answer getting.



Cont.

5. Another aspect of problem solving is problem posing or problem formulation.

6. Not much research on that.

7. Even Polya did not talk specifically about problem posing, but much of spirit and format of problem posing is included in his illustration of looking back.

8. Framework is needed that emphasized the dynamic and cyclic nature of genuine problem solving.



Understanding the problem

Making a plan

Carrying out the plan

Looking back



PROBLEM SOLVING AS A

PROCESS

Hayati



1.) Domain Specific

Knowledge One must develop a base of mathematics knowledge in

order to become a good problem solver in mathematics.

Kantowski: those students with a good knowledge basewere most able to use the heuristics (information or

strategies that we can adopt to solve unfamiliar problems) ingeometry instruction.

Schoenfeid and Herrmann: novices attended to surface

f eatures of problems whereas experts categorizedproblems on the basis of the fundamental principles

involved.

Silver: successful problem solvers were more likely tocategorize math problems on the basis of their underlying

similarities in mathematical structure.



Example: convert all expressions of

trigonometric identities to functions

of sine and cosine and do algebraic simplification.

Question

csc x . Sin x. coc x dx



Solution

csc x . sin x . cos x dx

= 1/sin x . sin x . cos x dx= cos x dx

= sin x + C



Examples of Basic Mathematics

Knowledge

Basic Functions of Trigonometry:-

Cosec x = 1/sin x

Tan x = sin x/cos x

Basic Integration Formula:-

cos x dx = sin x +C

sin x dx= - cos x +C

sec2 x dx = tan x +C



2.) Algorithms

Is a procedure, applicable to a particular type

of exercise, which, if followed correctly, is

guaranteed to give the answer to theexercise.

Problem solving is the process of creating an

algorithm and generalizing it to a specific

set of applications.



Example: Find the minimum or

maximum point of quadratic

equations;

Question.

f(x) = 3x2 + 6x + 1



Solutionby using Completing the Square

method

f(x)= 3x2

+ 6x + 1= 3[x2 + 2x + 1/3]

= 3[x2 + 2x + (2/2)2 (2/2)2 + 1/3]

= 3[x2 + 2x + (1)2 (1)2 + 1/3]

= 3[(x + 1)2

1 + 1/3]= 3[(x + 1)2 2/3]

= 3(x + 1)2 2

x + 1 = 0 y = -2x = -1

Therefore, the value for minimum quadratic equation is (-1,-2)



3.) Heuristics

Are kinds of information, available to students inmaking decisions during problem solving, that areaids to the generation of a solution, plausible innature rather than prescriptive, seldom providing

infallible guidance, and variable in results. Synonyms terms with strategies and techniques.

Examples; simplify an algebraic expression byremoving parentheses, make a table, restate the

problem in your own words, and

draw a figure tosuggest the line of argument for a proof.



Example of the Question

Following are the numbers of passengers thosewas had transited at KTM Station on three days;

Friday until Sunday for two months. Find the

value of median by using boundary 460-464.

465 466 477 471 480 460 478 471 473

473 463 476 470 472 471 479 474 475

469 473 474 472 483 479



Solution.

o. of passenger Frequency Cumulative

frequency

Upper boundary

460 464 2 2 464.5

465 469 3 5 469.5

470 474 11 16 474.5

475 479 6 22 479.5

480 484 2 24 484.5



(472.5)

(12)

Therefore, the value of mean of the passengers is

472.5 § 473 person



Examples of Heuristics.

Guess-and-check -> Useful for solving simple problems and

together with as systematic listings, this strategy helps to

narrow down to the correct guess in the shortest time

possible.

Make a list -> Commonly used with 'Look for patterns'

strategy as it accounts for all possibilities systematically. It

organises, presents or generates information in a systematic

manner. Look for patterns -> As it involves active search from a

sequence of figures or numbers, information can be

expressed/viewed in an organised manner.



Draw a model -> Suits pupils who need to see to

understand and it translates words or information into

pictorial representations.

Work backwards -> Useful when the final result has already

been given.

Simplify the problem -> Separates the problem into sub-problems to be solved individually or in sequence.

Before-after concept -> Compare information to find the

unknown as it lists information given before and after the action.

(Extracted from 'Assessment Guide to Primary Mathematics'

by CPDD, MOE)

Examples of Heuristics.



Example of the Question

1.) Marta wants to tie a ribbon around the sides of ahexagonal box and then tie a bow. Each of side of the box is 4 inches long. She needs 14 inches for the

bow. Is one yard of ribbon enough? Explain. Thenstate exactly how long the ribbon must be.

TASK 1

»Draw a model of hexagonal box..



Example of Question

5.) A total of 28 handshakes were exchanged at

a party. Each person shook hands exactly

once with each of the others. How manypeople were present at the party?

TASK 3

»Work backwards



Example of Questions

5.) A total of 28 handshakes were exchanged at aparty. Each person shook hands exactly once witheach of the others. How many people were presentat the party?

6.) At a party recently, I noticed that every personshook hands with each other person exactly onetime. If I told you there were 66 handshakes, could

you tell me how many people were at the party?

TASK 3

» Look for patterns



4.) Managing It All

Knowledge base of domain specificinformation, algorithms, and a repertoire of

heuristics are not sufficient during problem

solving. The students must construct some decision

mechanism to select from among the available

heuristics, or to develop new ones.



Metacognition..

Long ago, Dewey emphasized self-ref lection inthe solving of problems; metacognition.

Metacognition is Ones knowledge concerning

ones own cognitive processes and products oranything related to them Flavell (1976).

Metacognition theory holds that such thoughtcan monitor, direct, and control ones own

cognition.



5.) Looking Back

Looking back is the set of activities that provides theprimary opportunity for tudents to learn from theproblem.

The phase was identified by Polya with admonitionsto examine the solution by such activities aschecking the result, checking the argument,

deriving the result diff erently, using the result orthe method for some other problem,

reinterpreting the problem, interpreting the result,

or stating a new problem to solve.



6) Problem Posing

Is a t o ol for developing and strengthening

critical thinking skills. It is an inductive

questioning process that structures dialogue in

the classroom (Sarah Nixon-Ponder).

Involves the generation of new problem about

the situation or the reformulation of a givenproblem (English, 1997; Silver and Cai, 1996)



From the definitions of problem posing, we know thatproblem posing is the strategy that encourages the

generation of new problems by changing theconditions of a current problem.

For example, given the Pythagorean Theorem where a2

+ b2 = c2. After a discussion of the attributes, the teachermay ask what if a2 + b2 < c2? Through this discussion,

the students will generate the new problems.

a

b

c



Summary of Problem Solving

as A Process

Question 1 in TASK 4:-

In isosceles triangle ABC, AB=AC and the attitude to BC is 20. if the perimeter

of the triangle is 80, find the area of the

triangle.



ProblemSolving as A Proc ess

1.) Domain specific knowledge: know the shape of isosceles triangle and the feature of the shape.

2.) Algorithms: know the procedure or concept to find the

length of both unknown sides; Pythagorean Theorem

3.) Heuristics: draw the diagram of isosceles.

4.) Managing it all: monitor and ref lect whether the way of calculation is logic or not.

5.) Looking back: check the result/answer by using other

methods.6.) Problem posing: ask self , what happen if a2 + b2 < c2 instead

of a2 + b2 = c2? Then, try to discuss with other people for

better understanding the concept.



4 Stages of Problem Solvingis

called A Processi) Understand and explore the problem (domain specific

knowledge and know the algorithm/procedure that can

be use)

ii) Find a strategy (know the heuristic/technique that can

be use)

iii) Use the strategy to solve the problem (apply the

suitable heuristic and managing it all)

iv) Look back and ref lect on solution (looking back the

solution and pose the problem/think inversely about

the result)



Address on how instruction can

best develop this ability.

Athirah





What is mathematics????



National Council of Teaching

Mathematics (NCTM) Problem solving as focus of school

mathematics ~ pose fundamental questions nature

Art of problem solving is

of mathematics

Design Mathematics instruction ~ experience mathematics as problem solving



Why problem solving????

1. Major part of mathematics~ sum and

substance of discipline

2. Mathematics have application which represent important problems.

3. Intrinsic motivation embedded~ can

stimulate interest and enthusiasm

4. Fun~ use as recreation

5. Develop art of problem solving



Teachers rationale for not include

problem solving activities

School mathematics instruction

Includes argument: Difficult

Takes too much times

Mathematics is sequential

Not in the text book



Problem solving as an instructional

method

To accomplish the instructional goal of

learning basic facts , concepts and procedures

as well as goals for problem solving is within the problem context.

To introduce concepts via lesson involving exploration & discovery



Belief about mathematics problem

solving Curriculum & Evaluation Standards ~

students need to view themselves as capable

of using their growing mathematical

knowledge

Schoenfold ~ students belief that:

1-mathematics help to think clearly and

creative

2- learned best by memorization



Secondary school mathematics students

~predisposition to view problem solving as

answer getting,

Mathematics as a set of rules, highly oriented to do well on test

Espouse belief ~ problem solving are consistent with reasonably implemented

problem solving focus in school

mathematics!!



Technology and Problem

Solving

Syazwani





This research views:

The importance of technology as a tool for mathematics problem solving.

The development of a computer program to

perform mathematical task can be challenging mathematical problem and can enhance the programmers understanding of the mathematics being used.

The technology used helped students to better understand function concepts and improved student attitudes toward problem solving.



Programming as Problem Solving

Iteration in Mathematics.

Also called successive approximation. a

problem-solving or computational method in

which a succession of approximations, each

building on the one preceding, is used to

achieve a desired degree of accuracy.



Some Programming in problem

solving

Maple software

-Maple is a general-purpose commercialcomputer algebra system.

SPSS

-Statistical Package for the Social Sciences

-Analysis of data in research



Maple software



Example:



Answer:

int(cos(x/a), x);

By using maple software:



SPSS



Evaluation on problemsolving





How to evaluate the progress in

problem solving?



The success of mathematics programs and

student learning must be evaluated by a wider

range of measures than conventional testing

NCTM



Classroom Research

Analyze the students overall progress as well

as their reactions to the notebooks in order to

asses the effectiveness of the evaluation

process.



Diagnosing students cognitive

processes

By evaluating the amount and type of help

needed by an individual during a problem

solving activity.



Dynamic Assessment

Students are given mathematics problems to solve

The assessor then begins to provide as little help as necessary to the students throughout their problem solving activity

The amount and type of help needed can provide good insight into students problem solving activities, as well as their ability to learn and apply new principles.



The Use of Hints

The power of the hint technique seems to lie in itsability to identify those particular students in

need o special kinds of help

Trismen

Problems were developed such that the

methods of solutions were not readily apparent to students.

A sequence of hints was then developed for each item.



The Use of Three Boards in

Group A learning leader (teacher or peer) guides the group

in solving the problem through the use of three

boards

The students able to discuss and ref lect on their

approaches by visually tracing their joint work

Planning

BoardDoing

Board

Representation

Board





Clinical Approach

Evaluate individuals thinking process

Evaluate progress in problem solving

Example: thinking aloud may be canonically

achieved within the classroom by placing the

students in cooperative groups.

Students may express their problem solving

strategies aloud and we may able to assess their

thinking processes and attitudes unobtrusively.



Interview & Self-Report

Students are asked to ref lect on their

problem solving experience

Example: ref lective journal

Often used in problem solving research



Scoring Students Written Work

Test on :

-personal judgment-questions about mathematical problem

solving



TESTING

VS

PROBLEM SOLVING



Testing Problem Solving

-Emphasizes answer

getting.-Need to cover a lots of materials (syllabus)

-Not wide variety of ability to solve question

-Emphasizes on how to

solve the problem.-Developing desirable and

understanding of the

knowledge

-Create variety of ability to solve question



T hus a teacher of mat hemati c s has a great

opport unity. If he fills his allotted time wit h drillinghis st udents in routine operations he kills t heir

interest, hampers t heir intellec t ual development,and misuses his opport unity. But if he challenges t he

curiosity of his st udents by setting t hem problems proportionate to t heir knowledge, and helps t hem to

solve t heir problems wit h stimulating questions, hemay give t hem a taste for, and some means of,

independent t hinking.

Polya



Conclusion

As a pre-service teacher, we should emphasize the students to involve in problem solving.

Problem solving increase students critical and

creative thinking.

We should not emphasize on exam-oriented only but

develop our students critical and creative thinking.

Mathematics teacher who must create the context for

problem solving to f lourish and for students to

become problem solvers.

Mathematical problem solving is very significant to

our life.



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