Module in MC Math 16: Principles and Strategies in Teaching Mathematics

1. Reporter:
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Reolo, Maria Lourdes C.
Lunas, Clarissa Babes B.
Instructor: Mr. Robel Banda
MC MATH 16: Principles and Strategies
of Teaching Mathematics
MODULE 2:UNDERLYING
PRINCIPLES ANDSTRATEGIES

2. INTRODUCTION
Constructivism is an important learning theory that
educators use to help their students learn. Constructivism is
based on the idea that people actively construct or make their
own knowledge, and that reality is determined by your
experiences as a learner.
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3. PIONEERS OFCONTRUCTIVISM
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Piaget (1937)-"thegreatpioneer"
Von Glasersfeld - "the human mind can only know what the human
mindhasmade.
Vygotsky (1962)- His ideas are generally called constructionism,
or social or sociocultural constructivism. He believed in the
primacy of culture in shaping development.

4. Constructivisttheoriesareabout‘knowledgeandhow we
cometoknow'.
The constructivist teacher,by offering appropriate tasks
andopportunitiesfordialogue,guidesthefocus of students'
attention,thus unobtrusively directing their
learning'(Bruner1986).
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5. Process of Mental Construction involves Two Director System (Skemp, 1979):
1.Delta-one-a kind of sensori-motor system which “receives information…
compares this with a goal state, and with the help of a plan which it
constructs from available schemas, takes the operand from its present state to
itsgoalstate.”
2.Delta-two - a goal directed mental activity, whose operands are in delta-
one,anditsjobistooptimizethefunctioningofdeltaone.
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6. 2 Modes of Mental Activity
1.Intuitive-consciousness is centered in delta-one. It
refers to spontaneous processes, those within delta-one,
in which delta-two takes part either not at all, or not
consciously.
2.Reflective- consciousness is centered in delta-two. refers
to conscious activity by delta-two on delta-one.
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7. Basic tenets regarding knowledge
construction.
I.Knowledge is actively created or invented by the child, not passively received
from the environment. This idea can be illustrated by the Piagetian position that
mathematical ideas are made by children, not found like a pebble or accepted
from otherslikeagift(SteffeandCobb1988).
II.Children create new mathematical knowledge by reflecting on their physical
andmentalactions.
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8. III.Learning is a social process in which children grow into the intellectual life of those
around them (Burner 1986), in a social discourse involving explanation, negotiation,
sharing, andevaluation.
IV. When a teacher demands that students use set of mathematical methods, the sense- making
activityofstudentsis seriously curtailed.Studentstendtomimicthemethodsby rotesothatthey
can appear to achieve the teacher's goals. Their beliefs about the nature of mathematics change
from viewing mathematics as sense making to viewing it as learning set procedures that make
littlesense
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9. 2. Students should become
autonomous and self- motivated
in their mathematical activity.
Students’ should believe that
mathematics is a way of
thinking and thinking is the
only way to solve mathematical
problems..
3.Assessment, measurement
and
evaluation what should be a
natural part of the
learning process rather
than an activity completed at
the end of the learning
process.
1. Students should develop
their mathematical
structures that are more
complex, abstract, and
powerful than the ones
they currently possess so that
gradually they will be able to
solve a wide variety of
meaningfulproblems.
Goal of Constructivism
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10. IMPLICATIONS FOR LEARNING
Five steps for constructing new knowledge. They are..
I. Every learnerhas ideas prior tolearning and theseaffecttheway thatthey
make sense ofwhat they are going tolearn (previous knowledge),
II. Learning is not transmitted by linguistic communication but language
is a tool to help students construct knowledge (communicating
language),
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11. III. Learning is a participatory process
(active participation).
IV. Individual constructions should fit with the
accepted views of communities of practice (accepted
views),
V. Knowledge is personally constructed from
new experiences (knowledgeconstruction),
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12. According to Zhao (2003), the “characteristics of
constructivist teaching models include: prompting students
to observe and formulate their own questions; allowing
multiple interpretations and expressions of learning;
encouraging students to work in groups; and in the use of
their peers as resources to learning”.
Teaching Approaches and Role of Teacher
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13. 1.Interactive Teaching Approach
-verymuchusefultoconstructknowledgeofthestudentswith theirown
pace.In thisapproachteacherendeavors:
-To become more sensitive to learner's ideas and questions and provide exploratory experiences
from which thelearners will raise useful questions and suggest sensibleexplanations;
-To carry out with the whole class or with groups of learners, activities to focus on the questions
andideasthatmanyofthelearnershad;
-To act as a team research leader with the class, to help them plan and carry out their own
investigations into their questions, and to help them draw sensible and useful conclusions from
theirfindings.
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14. Teaching Approaches and role of Teacher
Thelearnersworkonthese
tasks in small groups or
individual.
Theclassisreconvenedasa
whole forsharing.
2. Problem - centered Teaching 3.Numerous teaching approaches
• can use in
constructivist ideas
like group work
approach, discovery
or investigative
approach, individual
teaching approach
etc.
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15. Whatever be the constructivist approach be by naming the
main five components are needed exclusively or inclusively in
all the approaches.
1.Orientation
2.Elicitation of ideas
3.Restructuring of ideas
4.Application of ideas
5.Review change in ideas
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16. In these and other 'constructivist' approaches there is a role shift for t
teacher which moves them from 'sage on the stage' to 'guide on the side'. T
shift is likely to involve:
I. Teacher serve in the role of friends, mentors, coaches and facilitators;
II. Negotiating the details of what is to be taught;
III. Valuing the learners' ideas and their autonomy;
IV. Finding appropriate challenging problems and learning activities;
V. Emphasizing cooperation in learning;
VI. Encouraging communication as a form of social interaction;
Role of the Teachers
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17. VII.Trying to find what is going on inside the heads of learners rather than
relying on their overt and often superficial responses;
VIII. Taking an interest in the errors (alternative conceptions) which may
throw light on how the learner is deviating from the teacher's intended path;
IX.Helping the students make connections by linking what is being taught with
prior knowledge and experiences, with other parts of the subject, with other
subjects, and with life outside school;
X. Accepting the notion that learning is skill developmental, and that
children's learning may differ with age as their thinking is constrained because
certain higher intellectual functions including awareness of mental operations
are not available until adolescence;
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18. TEACHING FOR UNDERSTANDING
Defining understanding...
Hiebert and Carpenter (1992: 67) defined mathematical
understanding as involving the building up of a conceptual
‘network’:The mathematics is understood if its mental
representation is part of a network of representations. The
degree of understanding is determined by the number and
strength of its connections. A mathematical idea, procedure, or
fact is understood thoroughly if it is linked to existing networks
with stronger or more numerous connections.
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19. Understanding is not a
dichotomous state, but a
continuum... Everyone
understands to some degree
anything that they know about. It
also follows that understanding
is never complete; for we can
always add more knowledge,
another episode, say, or refine an
image, or see new links between
things we know already. (White
and Gunston 1992:6)
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20. Mathematics Classrooms That Promote Understanding
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1.Direct instruction, the teacher usually demonstrates or models, lectures,
and asks questions that are convergent orclosed‐ended innature.
2.Facilitative methods, the teacher might use investigations and inquiry,
cooperative learning, discussion, and questions that are more
open‐ended.
3.Coaching, the teacher provides students with guided practice and
feedback that highlights ways toimprove their performances.

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DALE'S CONE OF LEARNING
Edgar Dale’s Cone of
Learning is a visual
metaphor for learning
modalities. The
objective of this visual is
not intended to place
value on one learning
modality over another,
rather to show the
difference between
Active Learning and
Passive Learning.

22. THANK
YOU!