4. Strategies in Teaching Mathematics

7/31/2019 4. Strategies in Teaching Mathematics

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TUTORIAL 4

STRATEGIES IN

TEACHINGMATHEMATICS


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Definition :

helping students construct a deepunderstanding of mathematical ideasand processes by engaging them indoing mathematics: creating,

conjecturing, exploring, testing, andverifying (Lester et al., 1994, p.154). the process of reaching solutions

(Gupta, 2005). the attempt to find the solution to a

problem when the method is notknown to a problem-solver


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PROBLEM SOLVINGSTRATEGIES

Exploration

Polya Model Newman Model

Mastery Learning Direct Learning


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EXPLORATION

Students explore to solve theproblems in mathematics

Students play a very active role intheir learning - exploring problemsituations with teacher guidance andinventing their own solution

strategies.


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Obtaining knowledge for oneself. Pushing students to try out their

hyphoteses, methods, and strategies

with processes similar to those thatexperts use to solve problems.

Through exploration, learners are

encouraged to carry out expertproblem solving processes on theirown.


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Learners become independent of theteacher and begin to apply whatexperts do regarding forming and

testing hyphoteses, formulatingrules, and gathering information.

Students are force to make

discoveries on their own.


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Example Of Question

Which container contains more

marbles? Give reason.


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POLYA MODEL

George Polya was a Hungarian whoimmigrated to the United States in 1940.His major contribution is for his work inproblem solving.

Growing up he was very frustrated withthe practice of having to regularlymemorize information. He was anexcellent problem solver.

In 1945 he published the book How toSolve It which quickly became his mostprized publication. It sold over onemillion copies and has been translated

into 17 languages. In this text he


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Polyas Four Principles

First principle: Understand theproblem

This seems so obvious that it is oftennot even mentioned, yet studentsare often stymied in their efforts tosolve problems simply because they

don't understand it fully, or even inpart. Plya taught teachers to askstudents questions such as:


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Can you state the problem in yourown words?

What are you trying to find or do? What information do you obtain from

the problem

What are the unknown? What information , if any is missing

or not needed?


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Second principle: Devise a plan Plya mentions (1957) that there are

many reasonable ways to solveproblems. The skill at choosing anappropriate strategy is best learned bysolving many problems. You will findchoosing a strategy increasingly easy. A

partial list of strategies is included:Guess and checkMake an orderly listEliminate possibilitiesUse symmetryConsider special casesUse direct reasoning

Solve an equation


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Also suggested:

Look for a pattern

Draw a pictureSolve a simpler problem

Use a model

Work backward

Use a formula

Be creative

Use your head/noggen


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Third principle: Carry out the plan

This step is usually easier than

devising the plan. In general (1957),all you need is care and patience,given that you have the necessary

skills. Persist with the plan that youhave chosen. If it continues not towork discard it and choose another.Don't be misled, this is howmathematics is done, even byprofessionals.


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Use the strategy you selected andwork the problem

Check each step of the plan as youproceed Ensure that the steps arecorrect


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Fourth principle: Review/extend

Plya mentions (1957) that much can be

gained by taking the time to reflect andlook back at what you have done, whatworked and what didn't.

Doing this will enable you to predict

what strategy to use to solve futureproblems, if these relate to the originalproblem.

Reread the questionDid you answer the question asked?Is your answer correct?Does your answer seems reasonable


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NEWMAN MODEL

Anne Newman (1977) an Australianeducator

May and Newman describe problem

solving as "an internal and sequentialprocess that includes cognitive,affective, and psychomotorbehaviors."

suggested five significant prompts tohelp determine where errors mayoccur in students attempts to solve

written problems


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A student wishing to solve a writtenmathematics problem typically has

to work through five basic steps(hierarchy) :


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Newman used the word "hierarchy"because she reasoned that failure atany level of the above sequence

prevents problem solvers fromobtaining satisfactory solutions (unlessby chance they arrive at correctsolutions by faulty reasoning).

According to Newman (1977, 1983),any person confronted with a writtenmathematics task needs to go througha fixed sequence: Reading (or

Decoding), Comprehension,Transformation (or Mathematising),Process Skills, and Encoding. Errorscan also be the result of unknown

factors, and Newman (1983) assignedthese to a com osite cate or , termed


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In Newman experiment to 124 lowgrade students, she classified 3002mistakes done in a written testcontaining 40 questionsCategory Numbers of

mistakes donePercentage of

mistakes

Reading 390 13

Comprehension 665 22

Transformation 361 12

Process skills 779 26

Encoding 72 2

Careless & Motivation 735 25

SUM 3002 100


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EXAMPLE

Ali have 3 books. He buy 6 booksmore. How many books does Alihave?

Newman (1983b, p. 11) recommended that thefollowing questions be used in order to classifystudents' errors on written mathematical tasks:

1)Please read the question to me. (Reading)2)Tell me what the question is asking you to do.

(Comprehension)3)Tell me a method you can use to find and answer

to the question. (Transformation)4)Show me how you worked out the answer to the

question. Explain to me what you are doing as youdo it. (Process Skills)

5)Now write down your answer to the question.


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MASTERY LEARNING

The simplest definition of MasteryLearning is when a child achieves theunderstanding and the ability to do

certain skills in a subject area, movingahead only after showing a highcompetency level in those skills.

Student / learner evaluation =

teacher Students who have mastered the

material are given "enrichment"opportunities,

those who have not mastered it receiveJue


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Example

Stage 1 : USING PICTURE

- How many apples on the tree?

Stage 2:

- how many apples fell down?

Stage 3:- what number is missing?


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ML IS ALSO USED IN TEACHINGMATH

Involves discrimination, matching,and grouping or categorizingaccording to attributes and attribute

values. Begin working on simple

discrimination and matching with

objects that are familiar to the childand that occur naturally in his or herworld (e.g., shoes,toothbrush,

squeezetoys, blocks, etc.),


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(Example) classified according to :

* Shape (square, circle, triangle,rectangle)

* Size (large, small, big, little)

* Weight (heavy,light)

* Length (short, long)* Width (wide,narrow, thick, thin)

* Height (tall, short)


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Activities:

Give children numerous opportunities touse everyday items for matching andcategorizing: eating utensils, grooming

tools, foods, and toys for function; shoesand shoelaces for matching by size orlength.

Children can explore shapes and size

bybuilding with Legos and Unifix blocks; Ask the studnts to help in sorting

different sizes of books, different colourpaper ordifferent shapes of legos.


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Area ofInstruction

Description SuggestedInstructional

Methods

NumberSense

Basic understandings about whole numbers, decimalsand fractions, ways numbers can be representedconcretely and visually, one-to-one correspondence,part to whole relationships, etc.

Hands-on experienceswith concrete objects& Mastery Learning

Content Knowledge level - number facts, math terms,formulas, algorithms for computation, etc.

Mastery Learning

Skills Application level - rounding numbers, comparingfractions, creating graphs, interpreting function

tables, doubling a recipe, etc.

Mastery Learning

ProblemSolving

Evaluation and synthesis - solving problems in whichsolutions are not readily apparent, solvingbrainteasers, drawing on a variety of strategies totackle a complex problem, etc.

Daily Problem Solving


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Conclusion

Teaching students by stages

From lower and easy stage to higherand difficult stage

Student need to master each stageto continue to next stage.

Teacher gives remedial activities toweak students and enrichmentactivities to the fast learners.


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Direct Learning in Math WordProblems: Students With

Learning Disabilities

Anis


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Everyday acts such as deciding whetherone can afford to purchase an item

require the application of problem-solving skills. Because students with mild disabilities

will live independent and productivelives, problem-solving skills are asessential for them as for studentswithout disabilities.

However, students with disabilities areless likely to adopt a strategic approachto problem solving (Torgesen & Kail,1980); thus, they are likely toexperience difficulty in mastering theskill. It is crucial that the mathematics

program for these students include


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Two studies (Darch et al., 1984; Jones etal., 1985) evaluated the effectiveness of

direct instruction. Darch et al. compared the effectiveness

of a direct-instruction approach to thatof a basal-math approach for teachingfourth graders without disabilities tosolve word problems.

The results indicated that students who

were taught using direct instructionperformed significantly higher on theposttest than did students who weretaught by more traditional methods.


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Direct Instruction LearningVisual Concept Diagram


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Description

Based on Zig Engelmann's theory ofinstruction, DI is probably the most popularteaching strategy that is used by teachers tofacilitate learning.

It is teacher directed and follows a definite

structure with specific steps to guide pupilstoward achieving clearly defined learningoutcomes.

The teacher maintains the locus of controlover the instructional process and monitorspupils' learning throughout the process.

Benefits of direct instruction includedelivering large amounts of information in atimely manner.

Also, because this model is teacher directed,

Principles of Direct


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Principles of DirectInstruction

Introduction/Review

Topics or information to be learned is presented to thepupils or review of information sets the stage for learning.

DevelopmentThe teacher provides clear explanations, descriptions,examples, or models of what is to be learned while

checking for pupils' understanding through questioning. Guided Practice

Opportunities are provided to the pupils to practice what isexpected to be learned while the teacher monitors theactivities or tasks assigned.

ClosureTeachers conclude the lesson by wrapping up what wascovered.

Independent PracticeAssignments are given to reinforce the learning without

teacher assistance.


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Procedures

1. Introduction/ReviewThe first step in DI is for the teacher to

gain the pupils' attention. Sometimesthis step is referred to a 'focusing event'

and is meant to set the stage forlearning to take place.

At this stage, the pupils are 'informed'as to what the learning goal or outcomeis for the lesson and why it is importantor relevant.

This step can either take the form of

introducing new information or building


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2. Development

Once the goal is communicated to pupils, theteacher models the behavior (knowledge orskill) that pupils are ultimately expected todemonstrate.

This step includes clear explanations of anyinformation with as many examples as

needed to assure pupils' understanding(depending on pupils' learning needs) ofwhat is to be learned.

During this step, the teacher also "checks for

understanding" by asking key questionsrelative to what is to be learned or byeliciting questions from pupils.

At this stage, teachers can also use 'prompts'

(visual aids, multimedia presentations, etc.)


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3. Guided Practice

Once the teacher is confident that

enough appropriate examples andexplanation of the material to belearned has been modeled withsufficient positive pupil response to theinstruction, activities or tasks can beassigned for pupils to practice theexpected learning with close teachermonitoring.

It is at this stage that teachers can offerassistance to pupils who have not yetmastered the material and who may

need more 'direct instruction' from the


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4. Closure

As a final step to this model, closure

brings the whole lesson to a'conclusion' and allows the teacher torecap what was covered in the

lesson. It is meant to remind pupils about

what the goal for instruction was and

for preparing them to complete theindependent practice activities thatare then assigned by the teacher.


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5. Independent Practice

Activities or tasks related to the defined

learning outcomes are assigned in thisstep usually after pupils havedemonstrated competency orproficiency in the 3rd step.

Independent practice is meant toeliminate any prompts from the teacherand is meant to determine the degree ofmastery that pupils have achieved.

(Homework can be classified as anindependent practice because it ismeant to provide the opportunity for

pupils to practice without the assistance


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6. Evaluation

Evaluation tools are used to assesspupils' progress either as it isoccurring (worksheets, classroomassignments, etc.) or as a

culminating event (tests, projects,etc.) to any given lesson.

Evaluation of pupils' learningprovides the necessary feedback toboth the teacher and the pupil andcan be used to determine whetherexpected learning outcomes have

been met or have to be revisited in


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This study focused on the effects ofsequencing problem types and using adirect-instruction strategy for problemsolving.

This study sought to build on a study byJones et al. (1985), who compared two

variations of a direct-instruction strategy

for teaching students without disabilitiesto solve addition and subtraction wordproblems. In both variations, the "bignumber" concept (Silbert, Carnine, &

Stein, 1981) was taught. With it, students determined whether a

problem gives the big number of a factfamily. If it does, the problem requires

subtraction; if not, the problem requiresaddition.


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This approach calls for direct teaching ofarticulated strategies for translation of

word problems into equations. In the sequential variation, students

practiced solving word problemssequenced according to type; in theconcurrent variation, students practiceda balanced combination of problemtypes.

Jones et al. found that students in thesequential condition made significantlygreater gains over the 9-dayinstructional period than did the

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4. Strategies in Teaching Mathematics