Activity 1: A Mini-Lesson
In groups of 3 to 4 people, work together to
reach conclusions about the pieces in the
bag.
What do you notice about the shapes?
How do they relate?
Make as many valid and profound
conjectures about them as possible.
What knowledge did you
construct?
 In any right triangle, the sum of the squares
on the legs is equal to the square on the
hypotenuse.
 In any obtuse triangle, the sum of the squares
on the two shorter sides is less than the square
on the longest side.
 In any acute triangle, the sum of the squares
on the two shorter sides is greater than the
square on the longest side.
Activity 2: A Quiz
Read the questions.
In the left column, mark whether you agree
or disagree with each statement.
As we present, write evidence in the
statement column to support or refute your
agreement/disagreement.
You will have a chance for revising your
answers at the end of the presentation.
Earliest Proponents of
Constructivism
Lao Tzu (6th century BC)
Siddhartha Gautama (c 563 to 483 BC)
Heraclitus (540-475 BC)
Constructivists – A Sampler













Giambattista Vico (1668 – 1744)
Immanuel Kant (1724 – 1804)
Arthur Schopenhauer (1788 – 1860)
William James (1842 – 1912)
Hans Vaihinger (1852 – 1933)
John Dewey (1859 – 1952)
Alfred Adler (1870 – 1937)
Johann Herbart (1886 – 1841)
George Kelly (1905 – 1967)
Maria Montessori (1870 – 1952)
Friedrich Hayek (1899 – present)
David Ausubel (1918 – present)
Seymour Papert (1928 – present)
Forms of Constructivism
Cognitive constructivism
– Piaget’s work lead up to this
Social constructivism
– Vygotsky
Radical constructivism
– Ernst von Glasersfeld
Jean Piaget
Aug. 9 1896 – Sept. 16, 1980
Philosopher
Natural scientist
Developmental psychologist
– Known for his extensive study of children and
theory of cognitive development.
“The great pioneer of the constructivist
theory of knowing.” ~ Ernst von Glasersfeld
Stages of Cognitive
Development
Sensorimotor Stage (birth to age 2)
– Children experience the world through
movement and senses and learn object
permanence.
Preoperational Stage (ages 2 to 7)
– Children acquire motor skills and mentally act
on objects with illogical operations.
Stages of Cognitive
Development
Concrete Operational Stage (ages 7 to 11)
– Children begin to think logically about
concrete events.
Formal Operational Stage (age 11 to adult)
– Children begin to develop abstract reasoning
and draw conclusions from the information
available.
Lev Vygotsky (60’s & 70’s)
“All the higher functions
originate as actual
relationships between
individuals.”
The level of learning
developed with adult guidance
or peer collaboration exceeds
that which can be attained
alone.
Jerome Bruner (60’s to present)
 Instruction must be concerned with the
experiences and contexts that make the
student willing and able to learn
(readiness).
 Instruction must be structured so that it
can be easily grasped by the student
(spiral organization).
 Instruction should be designed to facilitate
extrapolation and or fill in the gaps (going
beyond the information given).
Ernst von Glasersfeld
1917 to Present
Emeritus Professor of psychology at the
University of Georgia
Cybernetician
– Study of feedback and desired
concepts in living organisms,
machines and organizations.
Proponent of radical constructivism
– Knowledge is the self-organized cognitive
process of the human brain.
Ernst von Glasersfeld
“If the self, as I suggest, is a relational entity, it
cannot have a locus in the world of experiential
objects. It does not reside in the heart, as Aristotle
thought, or in the brain, as we tend to think today.
It resides in no place at all, but merely manifests
itself in the continuity of our acts of
differentiating and relating and in the intuitive
certainty we have that our experience is truly
ours.”
Glasersfeld [1970]
Constructivism Claims
Knowledge cannot be instructed
(transmitted) by a teacher; it can only be
constructed by the learner.
– Learning-teaching process is interactive in
nature.
Knowledge cannot be represented
symbolically.
– Claim that knowledge, by its very nature, can
not be represented symbolically.
Constructivism Claims
Knowledge can only be communicated in
complex learning situations.
– Children learn all or nearly all of their
mathematics in the context of complex
problems.
It is not possible to apply standard
evaluations to assess learning.
– Objective reality is not uniformly interpretable
by all learners.
Paul Cobb
Professor at Vanderbilt Univ.
2005 Hans Freudenthal Medal from the
International Commission on Mathematical
Instruction
Elected to the National Academy of
Education of the US
Regarded today as one of the leading
sociocultural theorists in math education
Leslie Steffe
Senior Scholar Award from the Special
Interest Group for Research in
Mathematics Education of the American
Educational Research Association - Sp. 07
University of Georgia Distinguished
Research Professor of Math Ed. - 1985
Albert Christ-Janer Award - 1984
Creative Research Medal - 1983
Dina van Hiele-Geldof
& Pierre van Hiele
Level 0: Recognition or Visualization
Level 1: Analysis
Level 2: Ordering or Informal Deductive
Level 3: Deduction or Formal Deductive
Level 4: Rigor
Level 0: Recognition
or Visualization
Children at the visualization level
think about shapes in terms of what
they resemble.
At this level, children are able to sort
shapes into groups that look alike to
them in some way.
Level 1: Analysis
Children at the analysis level think in
terms of properties.
They can list all of the properties of a
figure but don’t see any relationships
between the properties.
They don’t realize that some properties
imply others.
Level 2: Ordering or
Informal Deductive
Children not only think about
properties but also are able to notice
relationships within and between
figures.
Children are able to formulate
meaningful definitions.
Children are also able to make and
follow informal deductive arguments.
Level 3: Deduction or
Formal Deductive
Children think about relationships
between properties of shapes and also
understand relationships between axioms,
definitions, theorems, corollaries, and
postulates.
They understand how to do a formal proof
and understand why it is needed.
Level 4: Rigor
Children can think in terms of
abstract mathematical systems.
College mathematics majors and
mathematicians are at this level.
Implications for Math Teaching
 The levels are not age dependent, but rather, are related more to the
experiences students have had.
 The levels are sequential; children must pass through the levels in order as
their understanding increases (except for gifted children).
 To move from one level to the next, children need to have many experiences
in which they are actively involved in exploring and communicating about
their observations of shapes, properties, and relationships.
 For learning to take place, language must match the child’s level of
understanding. If the language used is above the child’s level of thinking, the
child may only be able to learn procedures and memorize without
understanding.
 It is difficult for two people who are at different levels to communicate
effectively.
 A teacher must realize that the meaning of many terms is different to the
child than it is to the teacher and adjust his or her communication
accordingly.
Implications for Math Teaching
An effective teacher will use the Van
Hiele levels to develop five skill areas
for geometry.
Visual Skills
Verbal Skills
Drawing Skills
Logical Skills
Applied Skills
van Hiele According to Pusey
"Geometry is a course that leaves many children
behind because they have not had much exposure to
it prior to high school or the few experiences they
have had did not require thinking above the visual
level. Thus, students encounter the secondary course
unprepared for the stated goals and objectives for
high school geometry. I implore us as a profession to
not ignore the evidence and research that has sought
to explain why these difficulties arise (like the van
Hiele model). Instead, I propose we use this data to
direct our pedagogical decisions and thereby give
support to children in the learning of geometry."
~ Eleanor Pusey
Examples of the Constructivist
Classroom
 Fourth-grade heat experiment
 Calculus Coffee Cooling Problem
Implications on Education
Constructivist teachers do not
take the role of the "sage on the
stage." Rather, teachers act as
"guides on the side" who
provide students with
opportunities to test the
adequacy of their current
understandings.
Implications on Education
 If learning is based on prior knowledge:
– Teachers must note that knowledge and
provide learning environments that exploit
inconsistencies between learners' current
understandings and the new experiences
before them.
– Teachers cannot assume that all children
understand something in the same way.
Further, children may need different
experiences to advance to different levels of
understanding.
Implications on Education
 If students must apply their current
understandings in new situations in order to build
new knowledge:
– Teachers must engage students with use of
prior knowledge.
– Teachers must have problems that are student
driven not teacher driven (not those that are
primarily important to teachers and the
educational system).
– Teachers can also encourage group interaction.
 If new knowledge is actively built:
– Time is needed to build it.
Lesson Structure
Not intended to be a rigid set of rules
The first objective in a constructivist
lesson is to engage student interest on a
topic that has a broad concept.
– This may be accomplished by doing:
• a demonstration,
• presenting data or
• showing a short film.
Lesson Structure
Not intended to be a rigid set of rules
Ask open-ended questions that probe the
students preconceptions on the topic.
Present some information or data that
does not fit with their existing
understanding.
Lesson Structure
Not intended to be a rigid set of rules
 Have students break into small groups to
formulate their own hypotheses and experiments
that will reconcile their previous understanding
with the discrepant information.
 The role of the teacher during the small group
interaction time is to circulate around the
classroom to be a resource or to ask probing
questions that aid the students in coming to an
understanding of the principle being studied.
Lesson Structure
Not intended to be a rigid set of rules
After sufficient time for
experimentation, the small groups share
their ideas and conclusions with the rest
of the class, which will try to come to a
consensus about what they learned.
Lesson Structure
Not intended to be a rigid set of rules
Higher-level thinking is encouraged.
The students should be challenged
beyond the simple factual response. The
students should be encouraged to
connect and summarize concepts by
analyzing, predicting, justifying, and
defending their ideas
Strategies for Implementing a
Constructivist Lesson
 Starting the lesson
– Consider previous knowledge to frame investigations
– Identify situations where students perceptions may
vary
– Ask Questions
– Consider possible responses to questions
– Note unexpected phenomena
Strategies for Implementing a
Constructivist Lesson
 Continuing the Lesson
– Encourage Cooperative Learning
– Brainstorm Possible Alternatives
– Experiment with Manipulatives
– Design a Model
– Collect and Organize Data
– Students Conduct and Design Experiments
Strategies Continued
 Proposing explanations & solutions
– Communicate information and ideas
– Construct and explain a model
– Construct a new explanation
– Review and critique solutions
– Utilize peer evaluation
– Assemble appropriate closure
– Integrate a solution with existing
knowledge and experiences
– Make Connections
Strategies Continued
 Taking Action
– Make decisions
– Apply knowledge and skills
–
–
–
–
–
Transfer knowledge and skills
Share information and ideas
Ask new questions
Develop products and promote ideas
Use models and ideas to illicit discussions and
acceptance by others
Assessment
• Assessment can be done traditionally using a standard
paper and pencil test.
• Each small group can study/review together for an
evaluation but one person is chosen at random from a
group to take the quiz for the entire group. The idea is
that peer interaction is paramount when learners are
constructing meaning for themselves, hence what one
individual in the group has learned should be the same as
that learned by another individual (Lord, 1994).
• The teacher could also evaluate each small group as a unit
to assess what they have learned.
Issues & Controversies
 Is Constructivism a learning theory or a
pedagogy?
 Behaviorism v. Constructivism
 What does a Constructivist approach in a
mathematics lesson look like?
Effectiveness of Explicit and Constructivist Mathematics Instruction
for Low-Achieving Students in the Netherlands
 Practical Considerations
 The Role of the Computer in Education
Implications in Rural Contexts
Prior knowledge
It must be relevant to them
Pose questions that they would want to
answer
Determine resources – Internet
Culturally appropriate teaching methods
Great Explorations In Math
and Science (GEMS)
 Science and mathematics teachers who need new ideas might look to GEMS
for inspiration. These publications (teacher guides, handbooks, assembly
presentations, and exhibit guides) include many of the essentials of hands-on
science and mathematics instruction. GEMS workbooks (most of which
range from $10 to $15) engage students in direct experience and
experimentation before introducing explanations of principles and concepts.
GEMS integrates mathematics with life, earth, and physical science.
 GEMS offers titles for students from preschool to high school. Many of the
guides offer suggestions for linking activities across the curriculum into
language arts, social studies, and art.
 A product of the Lawrence Hall of Science at the University of California,
Berkeley, the activities and lessons were designed and refined in classrooms
across the country. The growing list of titles now includes 37 teacher's guides
and 5 GEMS handbooks. For more information:
 LHS GEMS
Lawrence Hall of Science
University of California
Berkeley, CA 94720
Telephone: (510) 642-7771.
Before
Agree?
Statement (Give evidence to support your agreement/disagreement.)
1.
The earth is round.
2.
People construct new knowledge and understandings based on what they already know and believe.
3.
Teachers should not tell students anything directly but, instead, should allow them to construct knowledge for themselves.
4.
An ACCLAIM scholar in cohort 3 wrote a thesis on an issue related to constructivism.
5.
There are two types of constructivism: cognitive and social.
6.
The historical roots of constructivism as a psychological theory are most commonly traced to the work of Socrates.
7.
When a teacher presents a student with a novel situation, the student will construct a new cognitive structure on his own.
Assimilating and accommodating this new material is reward enough; the student requires no external reward or motivation.
8.
The book Fish is Fish illustrates the difficulties a teacher might face when presenting a novel situation to students.
9.
A teacher can assess a cognitive structure that a student has created internally.
10. Buddha was a constructivist.
After
Agree?
Writings of Jerome Bruner

Bruner, J. (1960). The Process of Education. Cambridge, MA: Harvard University Press.

Bruner, J. (1966). Toward a Theory of Instruction. Cambridge, MA: Harvard University Press.

Bruner, J. (1973). Going Beyond the Information Given. New York: Norton.

Bruner, J. (1983). Child's Talk: Learning to Use Language. New York: Norton.Bruner, J. (1986).
Actual Minds, Possible Worlds. Cambridge, MA: Harvard University Press.

Bruner, J. (1990). Acts of Meaning. Cambridge, MA: Harvard University Press.

Bruner, J. (1996). The Culture of Education, Cambridge, MA: Harvard University Press.

Bruner, J., Goodnow, J., & Austin, A. (1956). A Study of Thinking. New York: Wiley.

Bruner, Jerome S. In Search of Mind: Essays in Autobiography. New York: Harper & Row, 1983.
Writings of Jean Piaget
 Piaget, J., The Origins of Intelligence in Children. (1952) M. Cook,
trans. New York: International Universities Press.
 Piaget, J., The Child and Reality: Problems of Genetic Psychology.
(1973a) New York: Grossman.
 Piaget, J., The Language and Thought of the Child. (1973b) London:
Routledge and Kegan Paul.
 Piaget, J., The Grasp of Conciousness. (1977) London: Routledge
and Kegan Paul.
 Piaget, J., Success and Understanding. (1978) Cambridge, MA:
Harvard University Press.
Writings of Lev Vygotsky
 Vygotsky, L.S., Thought and Language. (1962)
Cambridge, MA: MIT Press
 Vygotsky, L.S., Mind in Society: The Development of the
Higher Psychological Processes. (1978) Cambridge,
MA: The Harvard University Press
References
Brooks, J.G. and Brooks, M.G. (1993). Alexandria, VA: Association for Supervision and Curriculum
Development
Faulkenberry, E., & Faulkenberry, T. (2006, April). Constructivism in Mathematics Education: A
Historical and Personal Perspective. Texas Science Teacher, 35(1), 17-21. Retrieved June 30, 2007,
from Education Research Complete database.
Kroesbergen, E., Van Luit, J., & Maas, C. (2004, January 1). Effectiveness of Explicit and
Constructivist Mathematics Instruction for Low-Achieving Students in the Netherlands.
Elementary School Journal, 104(3),
. (ERIC Document Reproduction Service No. EJ695971)
Retrieved July 4, 2007, from ERIC database.
Lord, T.R. (1994). Using constructivism to enhance student learning in college biology. Journal of
College Science Teaching, 23 (6), 346-348.
Cobb, Paul. (1994, October). An Exchange: Constructivism in Mathematics and Science Education.
Educational Researcher, Vol. 23, No. 7, 4. Retrieved July 9, 2007, from ERIC database.
Lionni, L. (1970) Fish is Fish. New York Scholastic Press. Marshall, Cl and G.B. Rossman.
Websites
carbon.cudenver.edu/~mryder/itc_data/constructivism.html
inform.umd.edu/UMS+State/UMD-Projects/MCTP/Essays/Constructivism.txt
www.math.uiuc.edu/~castelln/VanHiele.pdf
www.lib.ncsu.edu/theses/available/etd-04012003-202147/unrestricted/etd.pdf
www.sedl.org/pubs/sedletter/v09n03/practice.html
www.mathunion.org/ICMI/Awards/2005/CobbCitation.html
www.projectconstruct.org
www.mathunion.org/ICMI/Awards/2005/CobbCitation.html
Websites
mathforum.org/mathed/constructivism.html
www.math.upatras.gr/~mboudour/articles/constr.html
www.er.ugam.ca/nobel/r21270/cv/Constructivism.html
www.stemnet.nf.ca/%7Eelmurphy/emurphy/cle.html
act-r.psy.Cmu.edu/papers/misapplied.html
wolfweb.unr.edu/homepage/jcannon/ejse/ejsev2n2ed.html
www.uib.no/People/sinia/CSCL/web_struktur-836.htm
www.teach-nology.com/currenttrends/constructivism/
pegasus.cc.edu/~kthompso/projects/lit_constructivist.pdf