Download pdf - Teaching Strategies in Mathematics

7/31/2019 Teaching Strategies in Mathematics

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INNOVATIVE TEACHING

STRATEGIESIN MATHEMATICS

By:

JOCELYN G. TAMARES

Maruhat national High School

Gapan City Division

MERCY LEGASPI

Mayapyap National High School

Cabanatuan City Division


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A teachers primary job is touncover the important ideas

in subjects, not cover atextbook.

-Anonymous


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Challenges in Teaching

Engaging the learners;

Empowering the teachers;

and

Enabling research.


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Promoting Inquiry

Inquiry Question Openers


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What is an inquiry-based classroom?

It is a classroom which allows a student to play the

role of an experienced co-researcher rather than

of someone with all the answers.

Here, the teacher:

gives enough hints and poses probing questions;

offers encouragement for good thinking, not

just for right answers. treat answers, right and wrong, as discussion

topics until the class the research team - reaches

a consensus.


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An inquiry-based classroom demands flexibility

in responding to students ideas.

Here, the goal is for students to experience mathematics as a

process of finding and connecting ideas so that

justifying ideas and problem solving become moreimportant than the actual solutions;

the teacher spends time in planning and thinkingof how students might address the problem under

investigation; and

the teacher leads students to know that thethinking and problem-solving skills they develop

can serve them in all aspects of their lives.


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Some Effective Mathematics

Teaching Methods


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Indicators of Effectiveness of a

Teaching Strategy/Method

Students ability to

perform the desired competencies;

transfer learning and use whateverconcepts and skills learned to solve a similaror related problem;

accomplish tasks that require higher orderthinking skills; and

appreciate the application of the concept

or principle to real-life situations.


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CONCEPT MAPPING

A concept map

is a diagram or network indicating

interrelationships among concepts and

representing conceptual frameworks within aspecific domain of knowledge (Novak, 1990).

Here, the nodes represent concepts, the lines

linking the nodes represent relationships, andthe labels on the lines represent the nature of

the relationships.


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generates and communicates ideas which can

aid collaborative projects (brainstorming, etc.);

aids learning by explicitly integrating new and

old knowledge;

helps in assessing understanding or diagnosing

misunderstanding;

enhances the problem-solving phases of

generating alternative solutions and options; encourages positive self-concept.

(Seaman, 1990; Gaines and Shaw, 1995;

Plotnick, 1997; Williams,1997


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Sample Concept Map


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FLOW CHARTING

an outline of the sequence of processes indiagram form


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MINDS-ON HANDS-ON

, focusing on the core conceptsand critical thinking processes needed for

students to create and re-create

mathematical concepts and relationships intheir own minds.

, experimenting first-hand with

physical objects in the environment and

having concrete experience before learning

abstract mathematical concepts.


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PROBLEM-BASED LEARNING

This is a curriculum delivery system that

recognizes the need to develop problem

solving skills.

The teachers act as facilitators of learning.

The students are given guidelines on how to

approach the problem.

(Here, students act as professionals andconfront problems as they occur with fuzzy

edges, insufficient information - to determine

the best solution.)


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Problem-based Learning

EVALUATION

EXPLANATION

ELABORATION

ENGAGEMENT

EXPLORATION


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Sample Problem

(Systems of Linear Equation)

You have decided to purchase a cell phone and areconsidering which company offers the mostaffordable plan for you.

Talk-a-lot charges P199.90 per month and

P15 per minute

Motor mouth charges P399.90 per month

and P5 per minute

1. Write an equation that represents each phonecompany.

2. Is there ever a time when both would charge the

same amount? When? How much?


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COOPERATIVE LEARNING Think-Pair-Share- a strategy designed to

provide students with food for thought ona given topic enabling them to formulateindividual ideas and share these ideas with

another student. Uses: Note check, vocabulary review, quiz

review, concept review, lecture check,

Management Ideas: Assign partners, changepartners, give think time, monitor discussion,timed-pair-share, randomly select students


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CONSTRUCTIVIST TEACHING METHOD

Orientation Students are given the opportunity

to develop a sense of purpose and motivation forlearning the topic.

Elicitation Students are provided with questions

or problems for them to think about. Their initialideas about the concepts covered in the lessonare solicited as well as their answers to the givenquestions.

Discussion This includes (a) clarification andexchange of ideas; (b) construction of new ideas;and (c) evaluation of new ideas.


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Constructivist

Application Students are given the opportunity

to use their developed ideas in a variety ofsituations. In cooperative groups, they solve,

explain and justify the solution of practical and

textbook exercises. Summary and Reflection Students are asked to

summarize the principles learned and write their

reflections on how they learned the concepts andprinciples and how the method helped them in

dealing with mathematical activities.


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MATHEMATICAL INVESTIGATIONS


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Profitable Activities in Teaching Mathematics

1. Use of Music2. Visualization

3. Games and Puzzles

4. Math Lab5. Arts and Aesthetics

6. Journal Writing

7. Mathematics in

Context: Story

Mathematics

8. MathematicalConnections

9. Using News Items and

Clippings10. Research Reports

and Projects

11. Integratingtechnology

12. Use of manipulatives


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To make the students relaxed, entertained,and eager to learn, especially when they aretired of routine work in mathematics.

This technique works very well as long as thevolume remains low enough for the studentsnot to compete against but loud enough to be

heard and to eliminate squeaks and other littledistractions.


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2. Visualization

Provide projects and hands-on materials

and draw three-dimensional figures from

real models.


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Some manipulatives in the form of

mathematical games and puzzles aretangrams, Rubiks cube and Soma cubes,

and problems which can be solved

through logical reasoning.


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3. Games and Puzzles

useful in improving problem solving andvisualization skills;

promote high-level skills such as analysis

and synthesis; reinforce concepts and skills;

improve students attitude toward learning

math;

add variety to the instructional program.


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4. Math Lab

offers opportunity for math to come aliveand for students to get involved and see the

positive relationship between mathematics

and everyday life; provides opportunity for encouraging

creativity and social skills development.


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Sample Math Lab Activities

A. Assume that you have Php10,000 whichyou can spend for a party.

1. Using the ads of a certain restaurant,

plan your expenses.2. Draw up a budget that will include all

party expenses.


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B. Clip a recipe for cooking your favorite

dish and make a new recipe for 20 servings.

1 lb spaghetti

cup heavy cream

4 eggs

lb bacon, cut into-inch dice

1 tsp extra virgin

olive oil

2 Tbsp chopped

Italian parsley

Kosher salt and

freshly ground black

pepper, to taste

cup freshly grated

pecorino-romano

cheese (parmesan

may be substituted)

Serving: 4


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Fill a large soup pot with cold water and add a

handful or so of Kosher salt. Stir and taste; it should

taste like seawater. Cover the pot and heat the

water until it boils.

Add the diced bacon to a cold saut pan and cook

slowly over a low heat for 10 to 15 minutes or until

crisp. Remove bacon from pan and drain on papertowels.


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Drop the spaghetti into the boiling salted

water and cook according to package

instructions, about 6 to 9 minutes or until aldente, or tender but still firm to the bite.

While the pasta cooks, combine the eggs,

cheese, cream and olive oil in a bowl and beatwith a whisk until completely mixed.

Drain pasta, toss with the egg and cream

mixture, then add the cooked bacon andchopped parsley. Serve right away, with

additional grated cheese if desired.
http://culinaryarts.about.com/od/glossary/g/Al-Dente.htmhttp://culinaryarts.about.com/od/glossary/g/Al-Dente.htmhttp://culinaryarts.about.com/od/glossary/g/Al-Dente.htmhttp://culinaryarts.about.com/od/glossary/g/Al-Dente.htm


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C. Turn to the Computers for Sale section

of the classified ads.

1. Figure out the average cost per

computer in four different outlets.

2. Draw a bar graph which shows theaverage cost per computer in the four

outlets.


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D. Have everyone in the class select a stock

and follow it up for a week.1. Each student follows the progress of his

chosen stock and draws a line graph

depicting its behavior.

2. Ask students to describe the behavior of

the said stock.


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E. Turn to the sports page of any newspaper

and find the standings of any professionalsport.

1. Add up the total number of wins and losses

for each division.2. Which division appears to be the

strongest? the weakest?

3. By how much do their average winnings

differ?

F Mathematical Magic


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F. Mathematical Magic1. Ask a student volunteer to write down any

three- digit number whose first and last digitsdiffer by more than one.

2. Then form another three-digit number by

reversing the digits of the first number(i.e., 235 becomes 532).

3. Subtract the smaller number from the larger

number, reverse the digits of the answer, and

add this to the result of subtraction.

4. What is the sum? Justify your answer.


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5. Arts and Aesthetics

The use of art as teaching aid can lead studentstowards better understanding of, andappreciation for mathematics.

Tessellations, modular art, and

curve stitching may be done in

the classroom.


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6. Writing Math/Journal Writing

Give students the opportunity to create andcommunicate mathematics problems of their

own.

Write

- an algorithm, a definition, a problem;

- generalizing a concept/relationship;

- reporting a project;

- accomplishments, reactions


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7. Mathematics in Context: Story

MathematicsPlace a mathematical topic in the context

of a brief story to further explain it and

motivate students to carry out problem-solving activities.


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Sample of Mathematics in Context

Suppose that a classical concert ticket

costs twice as much as a rock concert

ticket.

Is it cheaper, more expensive, or equally

expensive for you to take 100 friends to a

rock concert or two friends, 25 timeseach, to a classical music concert?


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-Which of the regions in the Philippines is thelargest in land area?

-Which is the smallest?

-How many times greater is the largest?

Which region has the largest population?

-What is the population density (number of personsper square mile)of a particular region?

-What is the population density in persons per

hectare of the region in which you live?


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-What change in the area occurs in a leaf of a fast-

growing plant as the growing season advances?-What is the rate of the change based on a simple

graph of area versus time?

-How does the rate of change in the area comparewith the rate of change in the length of the leaf?

Ifthe distributed electric current through a metalstrip of uniform but irregular cross section is one

ampere, what is the current density per square

centimeter across the cross section?

9 News Items that Provide Real


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9. News Items that Provide Real

Mathematical Experiences

Decimals can be computed from batting orshooting averages and percentage of passcompletions for a favorite sports activity.

Common fractions are found in recipes. Find arecipe serving six persons and have studentsconvert the quantities so the recipe will serve15 persons.

Metric system can be introduced or reviewedby reading and interpreting articles,especially from foreign news services.


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Graphs are common in most newspapers,

and students can draw their own from data

they find in articles.

Bar graphs may be constructed from sports

statistics such as baskets scored by each of

the baseball or basketball teams.

Election results can be depicted through a

circle graph in which sectors indicate thepercentages of vote received by each

candidate.


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Line or bar graphs can record daily

temperatures of hours of daylight (from a

certain date to another), height of annualrainfall or the rise and fall of tide in the

locality, or the households monthly water

and energy consumption.

10 Use of Research Reports


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10. Use of Research, Reports,

and Projects

Encourage students to pursue individual areas of

interest and open up new dimensions of study for

them.

This provides teachers excellent opportunities to

know their students better and work with themon an individual basis.

Sample of Use of Research Reports


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Sample of Use of Research, Reports

and Projects

Finding the common fraction or decimal

approximation of a given number to correct to

the given number of decimal places

The Pythagorean Theorem has over 300 proofs

(see, for example, Loomis 1968).

The intriguing secret order of the Pythagoreans

may be of interest to some students.


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The mathematics of the honeycomb was studiedby Pappus, a fourth-century Alexandrian

mathematician. No other shape of cell is aseconomical of space and materials as is thehexagonal structure of the honeycomb.

Mathematics and postage stamps will interestsome students as a project. Mathematics inmeasurement, computation, cartography

(mapmaking), and design is an interestingresearch topic.

Other Mathematical Activities


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Other Mathematical Activities Gas Station Map mathematics which uses road

maps can be a nice source of mathematical

activities.Some of these are:

1. Tracing a highway across a region an countingthe towns it passes through.

2. Measuring the distance between two citiesshown on the map, then computing how farapart these cities are in kilometers.

3. Using the mileage chart on the map to finddistances between cities and using yarn orstring to find distances through the mapsconversion scale. Why are these distances not

the same?

4 I f M l Il l h h l


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4. In a trip from Manila to Iloilo, how much lesstime does the airplane have to cover than aferry boat?

5. Find the areas of some cities and compare resultswith figures found in official records.

Kite building can lead to the study of geometricshapes (such as Graham Bells famoustetrahedral kites), to methods of heights (such

as using the hypsometer), and to studies of thephysics of flight.


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Graphing pictures through point plotting is

instructive and entertaining. Students find

solution sets for equations and plot them bygiving pictures of a variety of objects or pictures.

On top of the lesson on trigonometry of triangles,

students may be asked to investigate thetrigonometry of squares.

Probability and statistics provide a rich source of

interesting, real-life problems. Initially, chanceoutcomes can be shown using coins, dice, a

table of random digits or a deck of cards


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11. Use technology

Using internet sources

Blended e-learning

Use powerpoint presentations

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12.Use of Manipulatives

to represent mathematical concepts,numbers and operations

Algebra tiles

Geometric figures

Other mathematical gadgets

T hi P bl S l i


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Teaching Problem Solving

Provide opportunities for students toactively contribute to the formulation of the

problem, to write the problems themselves,

and to change conditions in the problemsand to critique a given solution.

Problem Solving


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Problem Solving

(Mathematics and Science

Education Center)

EliminatePossibilities

Break itdown intosubordinateproblems

Draw apicture ordiagram,or make amodel

Solve a simpler

related problem

Look for Patterns

Workbackwardsfrom whereyou want toend up

Make atable,chart ororganizedlist

Guess,estimate,checkand revise

Cl f f P bl


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Classification of Problems

1. Problems without numbers

2. Problem situation without questions

3. Problems with insufficient or

extraneous data4. Problems that can be solved by drawing

a diagram

5. Problems whose are also

problems

6. Open-search Problems


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2 Problem situation without


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2. Problem situation without

questions

Joy and Sally go to the store. Joy hasPhp580 and Sally has Php620. They want

to buy a gift that costs Php920.


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3. Problem with Insufficient or

Extraneous Data

Two sides of a triangle have lengths 4cm

and 6cm. Find the perimeter of thetriangle.

4 P bl h


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4. Problems whose are

also problems

Examples:

1. a) 337 + 456 = ?

b) 3_7 + _ _6 = 533

2. a) What are the prime factors of 90?

b) List the 12 smallest numbers the prime

factors of which are 2, 3, and 5.


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3. a) Round off 5.77 to the nearest tenth.

b) What numbers when rounded off to

the nearest tenth give 5.8?

4. a) What is the surface area of a box withdimensions 2 cm x 6 cm x 10 cm?

b) Design a box with a surface area of

128 sq cm.

5 Problems that can be solved by


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5. Problems that can be solved by

drawing a diagram

Activity: Ask the students to prepare a sketch

that illustrates a given problem and solve the

problem based on the figure.

1. John, Alex, and May live on the same road.

John lives 10 km from Alex. May lives 2km

from Alex. How far does John live from May?

2. Ninety-six students are to be placed in rows,

with eight students in each row. How many

rows will there be?

6 O h P bl


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6. Open-search Problems1. Find all primes that are one less than a

fourth power.

2. Draw plane figures with equal areas.

3. Design a basketball court, given the size ofthe lot.


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Conduct problem-solving activities

Activity: The students may be asked tocollect problems: textbook problems, media

problems, non-routine problems, puzzles,

and strategy games.

Some of these strategy games are Tic- Tac-Toe and

the Tower of Hanoi.

T lk b t th ti l li ti f


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Talk about the practical applications of

mathematics.

Activity: Ask students to interpret graphsand charts, cause and effects of certain

day to day happenings, estimate and

evaluate responses.

Elements of a Thoughtful Learning


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Elements of a Thoughtful Learning

Environment

A classroom layout that invites thinking, i.e. a

seating arrangement where students face

each other in groups or clusters, or around a

hollow square, which makes possible

student-to-student interchanges, provides

opportunities for active student learning and

thinking

Classroom interactions that involve


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Classroom interactions that involve

information processing rather than information

receiving and repeating students in posing problems, comparing,

analyzing, formulating hypotheses, judging the

worth of arguments, and in asking and answeringquestions.

In a thinking classroom, students dissect, reflect on,

and add to what they read, hear, see, or feel togive new meaning to it, rather than simply

remember the meanings provided by authorities in

the field.

The Emerging Role


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The Emerging Role

of the Mathematics

Teacher

Creator of a thoughtful learning environment

that promotes and support higher-order thinking

and a classroom climate that nurtures thinking

beyond the levels of recall and translation, as well

as an environment in which more complex thinkingcan occur

Originator of precise thoughtful


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Originator of precise, thoughtful

language rather than vague terminology

or generalizationsPromoter of higher order thinking skills

through questioning, structuring,

responding and modeling

Organizer of classroom study and courses

around thoughtful questions

Makes innovative changes and focuses on

something other than memorization of

contents;


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Realistic and relevant applications of

the topic under considerationEncourages independent thinking

through the use of divergent strategies

in solving problems;Makes the teaching-learning process

more credible, convincing, meaningful,

and interesting.


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Use of manipulatives to provide

experiences in actual problem-solvingsituations and analyzing sensory data

Have field trips since educational

excursions can be used fruitfully to enrichinstruction in mathematics


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Life is good for only two things,discovering mathematics and

teaching mathematics."

Simon Poisson


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THANK YOU!