teaching algebra revised

MODELS OF TEACHING ALGEBRA

Carlo Magno, PhD

De La Salle University, Manila

Teaching Principles of School Mathematics

Learning Principles in Mathematics

3 part structure in algebra Representing the elements in an

algebraic form Transforming the symbolic expressions

in some ways Interpreting the new forms that has been

produced

Perspectives

Algebra is useful to those who work in the fields like science, engineering, computing, and teaching mathematics.

Algebra is interesting because it affords admirable example of ingenuity.

Principles and Standards for school mathematics Algebra should enable students to:

Understand patterns, relations, and functions

Represent and analyze mathematical situations and structures using algebraic symbols

Use mathematical models to represent and understand quantitative relationships

Analyze change in various contexts

Learning algebra

Demonstrating worked examples practice exercises (working for fluency)

Demonstration and practice application Problem link to topics (working for

meaning) Construct and reflect on the meanings

for expressions and equations (working for meaning)

Models of teaching

Concept attainment model Inductive thinking model Advance organizer Inquiry training model

Concept Attainment Model

Provide students with examples, some that represent the concept and some that do not.

Urge students to hypothesize about the attributes of the concepts and to record reasons speculations. The teacher my ask additional questions to help focus student thinking and to get them to compare attributes of the examples of nonexamples.

Concept Attainment Model When students appear to know the concept, they

name (label) the concept and describe the process they used for identifying it. Student may guess the concept early in the lesson, but the teacher needs to continue to present examples and non examples until the students attain the critical attributes of the concept as well as the name of the concept.

The teacher checks to see if the students have attained the concept by having then identify additional examples, say yes or no, tell why or why not their examples, and generate examples and nonexamples of their own.

Concept Attainment ModelPhase 1Presentation of data and identification of concept

Teacher presents labelled examplesStudents compare attributes in positive and negative examplesStudents generate and test hypothesesStudents state a definition according to the essential attributes

GCF Grouping 5x2 - 25 16x + 8 27y2 - 9

3ax + 6ay + 4x + 4y 2ac + 4ax – 5c + 10x

1. Characterize each set. What do you observe about factoring the equation in the first set? How about in the second set? (characterizing examples)

2. What is the difference between the two sets when they are factored? (comparing attributes)

3. Given the characteristics mentioned, what is factoring using GCF? What is factoring using grouping? (defining)


Phase 2Testing attainment of the concept

Students identify unlabeled examples as yes or noTeacher confirms hypothesis, names concept, and restate definitions according to essential attributesStudents generate examples

Concept Attainment Model Which of the following requires factoring by

GCF? Grouping? (confirming hypothesis)36x2 – 544ax + 16ay + 4x + 4y20ac + 14ax – 5c + 20x81 + 18y2

Factor the terms. (confirming the hypothesis) Give your own examples of terms that

requires factoring by GCF? Factoring by grouping? (generating own examples)


Phase 3Analysis of thinking strategies

Students describe thoughtsStudents discuss role of hypotheses and attributesStudents discuss type and number of hypotheses

Concept Attainment Model Why will grouping not work for: (describe

thoughts)36x2 – 54 81 + 18y2

Why will GCF not work for: (describe thoughts)4ax + 16ay + 4x + 4y20ac + 14ax – 5c + 20x

Inductive Thinking Model

Three postulates:Thinking can be taughtThinking is an active transaction between

the individual and the dataProcesses of thought evolve by a sequence

that is lawful

Inductive Thinking Model Strategy 1: Concept formation

Enumeration and listingGroupingLabelling and categorizing

Set A Set B y=4x + 3 C=10x + 5 D=3x + 2

D = 3x2 + 4x + 5 A = 6c2 + 10c + 3 Y = 3x2 + 4x +6

1.Look at the two sets of data, why are they separated? (listing)

2.What is the difference between them? (categorizing)

3.What do you call the equation in set A? How about for set B? (labeling)

Inductive Thinking Model Strategy 2: Interpretation of data

identifying critical relationshipsexploring relationshipsmaking inferences

Set A Set B y=4x + 3 C=10x + 5 D=3x + 2

D = 3x2 + 4x + 5 A = 6c2 + 10c + 3 Y = 3x2 + 4x +6

1.Can the equations be plotted? (critical relationship)

2.What can be produced for each set of data? (exploring relationships)

3.Will there be difference in the graphs for seta A equation and set B equation? (inferences)

Inductive Thinking Model Strategy 3: Application of principles

Predicting consequences, explaining unfamiliar phenomenon, hypothesizing

explaining and/or supporting the predictions and hypothesis

Identifying the prediction

Set A Set B y=4x + 3 C=10x + 5 D=3x + 2

D = 3x2 + 4x + 5 A = 6c2 + 10c + 3 Y = 3x2 + 4x +6

1. If we give a value for x and c, what will the graph look like? (predicting consequences)

2. Students will plot in a coordinate plane equation for set A and equation for set B. (supporting predictions)

3. Did the slopes turn out the way you predicted? Why or why not? (identifying the prediction)

Advance Organizer

Phase 1: Presentation of advance organizerClarify aims of the lessonPresent organizerIdentify defining attributesGive examplesProvide contextRepeatPrompt awareness of learner’s relevant

knowledge and experience

Advance Organizer

Phase 2: Presentation of learning task and materialsPresent materialMake logical order of learning material

explicit link material to organizer

Advance Organizer

Phase 3: Strengthening cognitive organizationUse principles of integrative reconciliationPromote active reception learningElicit critical approach to subject matterClarify

Inquiry Training Model

Suchmans’s theory Students inquire naturally when they are

puzzledThey can become conscious of and learn to

analyze their thinking strategiesNew strategies can be taught directly and

added to the students existing ones.Cooperative inquiry enriches thinking and

helps students to learn about the tentative, emergent nature of knowledge and to appreciate alternative explanations.

Inquiry Training Model Phase 1: Confrontation with the problem

Explain inquiry proceduresPresent discrepant event

The period T (time in seconds for one complete cycle) of a simple pendulum is related to the length L (in feet) of the pendulum by the formulas 8T2=2L. If one child is on a swing with a 10 – foot chain, then how long does it take to compete one cycle of the swing?

What do you need to do with the pendulum to make it swing faster?

Inquiry Training Model Phase 2: Data gathering, verification

Verify the nature of objects and conditionsVerify the occurrence of the problem

situation

Go to the playground and try elongating the swing. Take the time in seconds. Record it.

Try shortening the swing. Take the time in seconds. Record it.

Inquiry Training Model Phase 3: Data gathering, experimentation

Isolate relevant variablesHypothesize causal relationship

Set up your own simulated swing getting pieces of string and a yoyo.

Record the time of swing for each length: 15 in, 12 in, 10 in, 8 in, 6 in, 4 in

What do you think is the relationship between time and length?

Inquiry Training Model Phase 4: Organizing, formulating an

explanationFormulate rules or explanationsIf time and length are related, what explains

this? Phase 5: Analysis of the inquiry process

Analyze inquiry strategy and develop more effective ones

What other situations can the relationship between time and length be applied?

Watch video

Role playingConcept attainment modelInductive thinking modelAdvance organizerInquiry training model

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teaching algebra revised

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