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?
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