7/31/2019 4. Strategies in Teaching Mathematics
1/43
TUTORIAL 4
STRATEGIES IN
TEACHINGMATHEMATICS
7/31/2019 4. Strategies in Teaching Mathematics
2/43
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
7/31/2019 4. Strategies in Teaching Mathematics
3/43
PROBLEM SOLVINGSTRATEGIES
Exploration
Polya Model Newman Model
Mastery Learning Direct Learning
7/31/2019 4. Strategies in Teaching Mathematics
4/43
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.
7/31/2019 4. Strategies in Teaching Mathematics
5/43
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.
7/31/2019 4. Strategies in Teaching Mathematics
6/43
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.
7/31/2019 4. Strategies in Teaching Mathematics
7/43
Example Of Question
Which container contains more
marbles? Give reason.
7/31/2019 4. Strategies in Teaching Mathematics
8/43
7/31/2019 4. Strategies in Teaching Mathematics
9/43
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
7/31/2019 4. Strategies in Teaching Mathematics
10/43
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:
7/31/2019 4. Strategies in Teaching Mathematics
11/43
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?
7/31/2019 4. Strategies in Teaching Mathematics
12/43
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
7/31/2019 4. Strategies in Teaching Mathematics
13/43
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
7/31/2019 4. Strategies in Teaching Mathematics
14/43
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.
7/31/2019 4. Strategies in Teaching Mathematics
15/43
Use the strategy you selected andwork the problem
Check each step of the plan as youproceed Ensure that the steps arecorrect
7/31/2019 4. Strategies in Teaching Mathematics
16/43
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
7/31/2019 4. Strategies in Teaching Mathematics
17/43
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
7/31/2019 4. Strategies in Teaching Mathematics
18/43
A student wishing to solve a writtenmathematics problem typically has
to work through five basic steps(hierarchy) :
7/31/2019 4. Strategies in Teaching Mathematics
19/43
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
7/31/2019 4. Strategies in Teaching Mathematics
20/43
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
7/31/2019 4. Strategies in Teaching Mathematics
21/43
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.
7/31/2019 4. Strategies in Teaching Mathematics
22/43
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
7/31/2019 4. Strategies in Teaching Mathematics
23/43
7/31/2019 4. Strategies in Teaching Mathematics
24/43
Example
Stage 1 : USING PICTURE
- How many apples on the tree?
Stage 2:
- how many apples fell down?
Stage 3:- what number is missing?
7/31/2019 4. Strategies in Teaching Mathematics
25/43
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.),
7/31/2019 4. Strategies in Teaching Mathematics
26/43
(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)
7/31/2019 4. Strategies in Teaching Mathematics
27/43
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.
7/31/2019 4. Strategies in Teaching Mathematics
28/43
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
7/31/2019 4. Strategies in Teaching Mathematics
29/43
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.
7/31/2019 4. Strategies in Teaching Mathematics
30/43
Direct Learning in Math WordProblems: Students With
Learning Disabilities
Anis
7/31/2019 4. Strategies in Teaching Mathematics
31/43
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
7/31/2019 4. Strategies in Teaching Mathematics
32/43
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.
7/31/2019 4. Strategies in Teaching Mathematics
33/43
Direct Instruction LearningVisual Concept Diagram
7/31/2019 4. Strategies in Teaching Mathematics
34/43
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
7/31/2019 4. Strategies in Teaching Mathematics
35/43
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.
7/31/2019 4. Strategies in Teaching Mathematics
36/43
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
7/31/2019 4. Strategies in Teaching Mathematics
37/43
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.)
7/31/2019 4. Strategies in Teaching Mathematics
38/43
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
7/31/2019 4. Strategies in Teaching Mathematics
39/43
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.
7/31/2019 4. Strategies in Teaching Mathematics
40/43
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
7/31/2019 4. Strategies in Teaching Mathematics
41/43
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
7/31/2019 4. Strategies in Teaching Mathematics
42/43
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.
7/31/2019 4. Strategies in Teaching Mathematics
43/43
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