This document discusses principles of teaching algebra. It describes a 3-part structure of algebra involving representing elements algebraically, transforming symbolic expressions, and interpreting new forms. Perspectives on algebra's usefulness in fields like science are provided. Principles for school mathematics emphasize patterns, quantitative relationships, and analyzing change. Models for teaching algebra include demonstrating worked examples, conceptual attainment, inductive thinking, advance organizers, and inquiry training. Each model involves multiple phases like confronting problems, gathering data, formulating explanations, and analyzing thinking processes.

1. Carlo Magno, PhD
De La Salle University, Manila

2. Teaching Principles of School
Mathematics

3. Teaching Principles of School
Mathematics

4. Teaching Principles of School
Mathematics

5. Learning Principles in
Mathematics

6. Learning Principles in
Mathematics

7. 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

8. 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.

9. 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

10. 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)

11. Models of teaching
 Concept attainment model
 Inductive thinking model
 Advance organizer
 Inquiry training model

12. 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.

13. 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.

14. Concept Attainment
Model
Phase 1
Presentation of data and
identification of concept
Teacher presents labelled
examples
Students compare attributes in
positive and negative examples
Students generate and test
hypotheses
Students state a definition
according to the essential
attributes

15. 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)

16. Concept Attainment
Model
Phase 2
Testing attainment of the
concept
Students identify unlabeled
examples as yes or no
Teacher confirms
hypothesis, names concept,
and restate definitions
according to essential
attributes
Students generate examples

17. Concept Attainment
Model
 Which of the following requires factoring by
GCF? Grouping? (confirming hypothesis)
 36x2 – 54
 4ax + 16ay + 4x + 4y
 20ac + 14ax – 5c + 20x
 81 + 18y2
 Factor the terms. (confirming the
hypothesis)
 Give your own examples of terms that
requires factoring by GCF? Factoring by
grouping? (generating own examples)

18. Concept Attainment
Model
Phase 3
Analysis of thinking
strategies
Students describe thoughts
Students discuss role of
hypotheses and attributes
Students discuss type and
number of hypotheses

19. 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 + 4y
 20ac + 14ax – 5c + 20x

20. Inductive Thinking Model
 Three postulates:
 Thinking can be taught
 Thinking is an active transaction between
the individual and the data
 Processes of thought evolve by a sequence
that is lawful

21. Inductive Thinking Model
 Strategy 1: Concept formation
 Enumeration and listing
 Grouping
 Labelling and categorizing

22. 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)

23. Inductive Thinking Model
 Strategy 2: Interpretation of data
 identifying critical relationships
 exploring relationships
 making inferences

24. 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)

25. 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

26. 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)

27. Advance Organizer
 Phase 1: Presentation of advance
organizer
 Clarify aims of the lesson
 Present organizer
 Identify defining attributes
 Give examples
 Provide context
 Repeat
 Prompt awareness of learner’s relevant
knowledge and experience

28. Advance Organizer
 Phase 2: Presentation of learning
task and materials
 Present material
 Make logical order of learning material
explicit link material to organizer

29. Advance Organizer
 Phase 3: Strengthening cognitive
organization
 Use principles of integrative reconciliation
 Promote active reception learning
 Elicit critical approach to subject matter
 Clarify

30. Inquiry Training Model
 Suchmans’s theory
 Students inquire naturally when they are
puzzled
 They can become conscious of and learn to
analyze their thinking strategies
 New 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.

31. Inquiry Training Model
 Phase 1: Confrontation with the
problem
 Explain inquiry procedures
 Present 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?

32. Inquiry Training Model
 Phase 2: Data gathering, verification
 Verify the nature of objects and conditions
 Verify 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.

33. Inquiry Training Model
 Phase 3: Data
gathering, experimentation
 Isolate relevant variables
 Hypothesize 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?

34. Inquiry Training Model
 Phase 4: Organizing, formulating an
explanation
 Formulate rules or explanations
 If 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?

35. Watch video

36.  Role playing
 Concept attainment model
 Inductive thinking model
 Advance organizer
 Inquiry training model

37. Insights