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.

Activity 1: A Mini-LessonActivity 1: A Mini-Lesson

What knowledge did you What knowledge did you construct?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 QuizActivity 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 Earliest Proponents of ConstructivismConstructivism

Lao Tzu (6th century BC) Siddhartha Gautama (c 563 to 483 BC)Heraclitus (540-475 BC)

Constructivists – A SamplerConstructivists – 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 ConstructivismForms of Constructivism

Cognitive constructivism– Piaget’s work lead up to this

Social constructivism– Vygotsky

Radical constructivism– Ernst von Glasersfeld

Jean PiagetJean PiagetAug. 9 1896 – Sept. 16, 1980

Philosopher Natural scientistDevelopmental 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 Stages of Cognitive DevelopmentDevelopment

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 Stages of Cognitive DevelopmentDevelopment

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)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)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 GlasersfeldErnst von Glasersfeld1917 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 GlasersfeldErnst 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]

ConstructivismConstructivism ClaimsClaims

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 ClaimsConstructivism 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 CobbPaul 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 SteffeLeslie 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 - 1984Creative Research Medal - 1983

Dina van Hiele-GeldofDina van Hiele-Geldof& Pierre van Hiele& Pierre van Hiele

Level 0: Recognition or VisualizationLevel 1: AnalysisLevel 2: Ordering or Informal DeductiveLevel 3: Deduction or Formal DeductiveLevel 4: Rigor

Level 0: RecognitionLevel 0: Recognitionor Visualizationor 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: AnalysisLevel 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 orLevel 2: Ordering orInformal DeductiveInformal 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 orLevel 3: Deduction orFormal DeductiveFormal 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: RigorLevel 4: Rigor

Children can think in terms of abstract mathematical systems.

College mathematics majors and mathematicians are at this level.

Implications for Math TeachingImplications 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 TeachingImplications for Math Teaching

Visual Skills Verbal Skills Drawing Skills Logical Skills Applied Skills

An effective teacher will use the Van Hiele levels to develop five skill areas for geometry.

van Hiele According to Puseyvan 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 Examples of the Constructivist ClassroomClassroom

Fourth-grade heat experiment

Calculus Coffee Cooling Problem

Implications on EducationImplications 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.

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 EducationImplications 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.

Implications on EducationImplications on Education

Lesson StructureLesson StructureNot 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 StructureLesson StructureNot 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 StructureLesson StructureNot 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 StructureLesson StructureNot 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 StructureLesson StructureNot 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 Strategies for Implementing a Constructivist LessonConstructivist 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 Strategies for Implementing a Constructivist LessonConstructivist 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 ContinuedStrategies 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 ContinuedStrategies 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

AssessmentAssessment

• 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 & ControversiesIssues & Controversies

Is Constructivism a learning theory or a pedagogy?

Behaviorism v. ConstructivismWhat 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 ConsiderationsThe Role of the Computer in Education

Implications in Rural ContextsImplications in Rural Contexts

Prior knowledge It must be relevant to themPose questions that they would want to

answerDetermine resources – InternetCulturally appropriate teaching methods

Great Explorations In MathGreat Explorations In Math and Science (GEMS) 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 GEMSLawrence Hall of ScienceUniversity of CaliforniaBerkeley, CA 94720Telephone: (510) 642-7771.

Before Agree?

Statement (Give evidence to support your agreement/disagreement.) After Agree?

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.

Writings of Jerome BrunerWritings 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 PiagetWritings 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 VygotskyWritings 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

ReferencesReferences

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.

WebsitesWebsites

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

WebsitesWebsites

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