Activity 1: A Mini-Lesson

1. 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.

2. 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.

3. 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.

4. Earliest Proponents of Constructivism • Lao Tzu (6th century BC) • Siddhartha Gautama (c 563 to 483 BC) • Heraclitus (540-475 BC)

5. 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)

6. Forms of Constructivism • Cognitive constructivism • Piaget’s work lead up to this • Social constructivism • Vygotsky • Radical constructivism • Ernst von Glasersfeld

7. Jean PiagetAug. 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

8. 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.

9. 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.

10. 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.

11. 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).

12. Ernst 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.

13. 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]

14. ConstructivismClaims • 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.

15. 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.

16. 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

17. 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

18. 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

19. 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.

20. 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.

21. 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.

22. 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.

23. Level 4: Rigor • Children can think in terms of abstract mathematical systems. • College mathematics majors and mathematicians are at this level.

24. 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.

25. 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

26. 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

27. Examples of the Constructivist Classroom • Fourth-grade heat experiment • Calculus Coffee Cooling Problem

28. 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.

29. 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.

30. 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.

31. Lesson 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.

32. Lesson 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.

33. Lesson 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.

34. Lesson 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.

35. Lesson 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

36. 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

37. 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

38. 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

39. 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

40. 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.