Learning Mathematics: Implications for Teaching

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Learning Mathematics Implications for Teaching

• Anthony Orton
• Issues in Teaching Mathematics
• (Ed. Orton, A, Wain, G.)

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The Construction of Meaning

• Unless pupils have an adequate understanding of a
  procedure, unless they have in some way made it
  their own, there is often resistance to trying to
  use it.
• Pupils often benefit if they understand why a
  procedure works as well as simply being able to
  carry out the procedure (instrumental
  understanding).

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Constructivism????

• We all have to make sense of the world ourselves,
  and we continue to develop our understanding
  throughout life.
• Hypotheses
• Weak Knowledge is actively constructed by the
  learner, not passively received from the
  environment
• Radical coming to know is an adaptive process
  that organizes ones experiential world it does
  not discover an independent, pre-existing world
  outside the mind of the knower.

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Social Constructivism (Socio-constructivism)

• meaning, or understanding, is being to some
  extent actively negotiated through such
  discussion.
• Teachers role to help students develop skills
  of constructing, evaluating and modifying
  concepts The teachers role therefore is to work
  to improve the quality of the discussions rather
  than to focus from the beginning on the
  correct answer.

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Contructivist Teaching Experiment (Kamii, 1989)

• Children invent arithmetic themselves
• belief that children acquire mathematical
  knowledge not by internalizing rules imposed from
  outside but by construction from the inside
  through their own natural thinking abilities
• errors arise because they are thinking,
• task of teacher is not to correct from outside,
  but to create a situation in which the children
  will inevitably correct themselves.

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The twelve strategies for teachers to become
constructivists(Brooks Brooks)

• (1) Constructivist teachers encourage and accept
  student autonomy and initiative.
• (2) Constructivist teachers use raw data and
  primary sources, along with manipulative,
  interactive, and physical materials.
• (3) When framing tasks, constructivist teachers
  use cognitive terminology such as "classify,"
  "analyze," "predict," and "create."
• (4) Constructivist teachers allow student
  responses to drive lessons, shift instructional
  strategies, and alter content.
• (5) Constructivist teachers inquire about
  students' understanding of concepts before
  sharing their own understandings of those
  concepts.
• (6) Constructivist teachers encourage students to
  engage in dialogue, both with the teacher and
  with one another.

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• (7) Constructivist teachers encourage student
  inquiry by asking thoughtful, open-ended
  questions and encouraging students to ask
  questions of each other.
• (8) Constructivist teachers seek elaboration of
  students' initial responses.
• (9) Constructivist teachers engage students in
  experiences that might engender contradictions to
  their initial hypotheses and then encourage
  discussion.
• (10) Constructivist teachers allow wait time
  after posing questions.
• (11) Constructivist teachers provide time for
  students to construct relationships and create
  metaphors.
• (12) Constructivist teachers nurture students'
  natural curiosity through frequent use of the
  learning cycle model. (The learning cycle model
  consists of discovery, concept introduction, and
  concept application.)

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Actions and changes need to occur for creating
constructivist teachers.

• (1) Structure preservice and inservice teacher
  education around constructivist principles and
  practices.
• (2) Jettison most standardize testing and make
  assessment meaningful for students.
• (3) Focus resources more on teachers'
  professional development than on textbooks and
  workbooks.
• (4) Eliminate letter and number grades.
• (5) Form school-based study groups focused on
  human developmental principles.
• (6) Require annual seminars on teaching and
  learning for administrators and school board
  members.

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Interpretation Construction (ICON) Design Model
John B. Black and Robert O. McClintock
http//www.ilt.columbia.edu/ilt/papers/ICON.html

• Observation Students make observations of
  authentic artifacts anchored in authentic
  situations
• Interpretation Construction Students construct
  interpretations of observations and construct
  arguments for the validity of their
  interpretations
• Contextualization Students access background and
  contextual materials of various sorts to aid
  interpretation and argumentation
• Cognitive Apprenticeship Students serve as
  apprentices to teachers to master observation,
  interpretation and contextualization

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• Collaboration Students collaborate in
  observation, interpretation and contextualization
  
• Multiple Interpretations Students gain cognitive
  flexibility by being exposed to multiple
  interpretations
• Multiple Manifestations Students gain
  transferability by seeing multiple manifestations
  of the same interpretations

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