STOP DOING MATH LONG ENOUGH TO LEARN IT

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STOP DOING MATH LONG ENOUGH TO LEARN IT

• Principles of Learning
• Delano P. Wegener, Ph.D.
• Spring 2005

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Instruction

• Instruction is much more than presentation of
  information.
• Instruction may include events that are generated
  by a page of print, a picture, a television
  program, a combination of physical objects,
  potentially many other stimuli, as well as
  activities of a teacher.

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Instruction

• Teaching refers to the activities of the teacher.
  Therefore teaching is only one part (I think it
  is an important part) of instruction.
• Instruction is a deliberately arranged set of
  external events designed to support internal
  learning processes.

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Internal Conditions of Learning

• Because the internal conditions of learning are
  beyond our control we will not elaborate beyond
  listing them.

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Internal Conditions of Learning

• Reception of stimuli by receptors
• Registration of information by sensory registers
• Selective perception for storage in short-term
  memory (STM)
• Rehearsal to maintain information in STM
• Semantic encoding for storage in long-term memory
  (LTM)
• Retrieval from LTM to working memory (STM)
• Response generation to effectors
• Performance in the learners environment
• Control of processes through executive strategies

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Internal Conditions of Learning

• Cognitive scientists use that information, but we
  make no explicit use of the internal conditions
  of learning when designing instruction.

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Learning

• Some of the very basic facts and theories about
  learning will help to understand the guidelines,
  presented later, for studying mathematics.

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Learning

• In particular, it is helpful to be aware of
• Conditions of Learning,
• Learning Outcomes,
• Domains of Learning Objectives, and
• Classes of Learning Objectives in the Cognitive
  Domain.

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Learning

• In this seminar we are especially interested in
  how these facts and theories about learning
  pertain to learning mathematics.

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External Conditions of Learning

1. Gaining attention
2. Informing the learner of the objective
3. Stimulating recall of prerequisite learning
4. Presenting the stimulus material
5. Providing learning guidance
6. Eliciting the performance
7. Providing feedback about performance correctness
8. Assessing the performance
9. Enhancing retention and transfer

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External Conditions of Learning

• Because the External Conditions of Learning
  directly affect instruction and what you must do
  to learn, each of these conditions will be
  explained.

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1. Gaining attention

• Stimulation to gain attention to ensure the
  reception of stimuli.
• Various kinds of events can be used to gain the
  students attention. These events might be as
  simple as calling the class to order or as
  complex as the mix of sound, pictures, movement,
  and light as found in the most sophisticated TV
  commercials.

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1. Gaining attention

• An appeal to the students interest is frequently
  employed as a means of gaining attention.
• For adult students we frequently assume they will
  themselves provide the stimulation to gain their
  attention.

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Informing the Learnerof the Objective

• Informing learners of the learning objective
  establishes appropriate performance.
• The student must know the kind of performance
  that is expected as a demonstration that learning
  has taken place.
• In general it is a mistake to assume the student
  will know the objective of the lesson.

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3. Stimulating Recall of Prerequisite Learning

• Reminding learners of previously learned content
  for retrieval from LTM
• Much of learning is the combination of ideas. If
  any of the ideas involved have been learned
  previously, the student should be reminded of
  them so they are retrieved from LTM into STM
  where they are available for immediate recall.

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3. Stimulating Recall of Prerequisite Learning

• At the time of learning, previously learned ideas
  must be readily available. They must therefore be
  recalled from LTM (into STM) prior to the time of
  learning.

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Stimulating Recall of Prerequisite
LearningMathematics

• Suppose we expect the student to learn the
• Fundamental Theorem of Arithmetic
• Any natural number can be expressed as a
  product of prime numbers.
• If the definitions of natural number, product,
  and prime number are not readily available, then
  the Fundamental Theorem of Arithmetic will not be
  learned.

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Stimulating Recall of Prerequisite
LearningMathematics

• It is essential that these definitions be
  recalled from LTM to STM.
• Assuming that these definitions have previously
  been learned, the teacher can insure they are
  recalled into STM by simply reminding the student
  of those definitions.

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Stimulating Recall of Prerequisite
LearningMathematics

• The adult student might be expected to recall
  those definitions simply because the appearance
  of the words is enough of a reminder.

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Presenting the Stimulus MaterialMathematics

• It is important that the proper stimuli be
  presented as a part of the instructional events.
  
• If a mathematical rule is to be learned, then
  that rule must be communicated. Such
  communication, if printed, may use italics, bold
  print, underlining, colors, etc. to emphasize
  particular features.

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Presenting the Stimulus Material

• When young children are learning concepts or
  rules the stimulus material should present
  examples prior to presenting a statement of the
  concept or rule.
• When adults are learning concepts or rules the
  stimulus material should present a statement of
  the concept or rule followed by examples.

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Presenting the Stimulus MaterialMathematics

• Stimulation presentation for the learning of
  concepts and rules requires the use of a variety
  of examples.
• Thus the stimulus presentation for learning the
  mathematical concept of linear function will
  involve examples of functions like
• f(x) 3x, or f(x) 7
• as well as examples like f(x) 3x 7

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5. Providing Learning Guidance

• Communications which have the function of
  providing learning guidance do not provide the
  answer.
• They suggest a line of thought which will lead to
  appropriate combining of previously learned
  concepts allowing the student to learn the
  answer.

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5. Providing Learning Guidance

• Communications designed to provide learning
  guidance should stimulate a direction of thought
  which keeps the student on the right track.

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Providing Learning GuidanceMathematics

• When presenting an example of solving a linear
  equation the instructor does not encourage a
  memorized set of steps to arrive at the answer.
  
• Rather the instructor constantly reminds the
  student of the previously learned two operations
  which generate an equation equivalent to the
  original equation.

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Providing Learning GuidanceMathematics

• All communications in this context are designed
  to keep the student on track to generate a
  sequence of equations, all equivalent to the
  original, terminating in a simplest equation.

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6. Eliciting the Performance

• Suppose the previous five events have taken
  place, enough material has been presented,
  sufficient learning guidance has taken place, and
  the student indicates/believes he has learned the
  concepts.
• It is then time for the student to demonstrate
  both to himself and the instructor that he has
  learned the concept.

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6. Eliciting the PerformanceMathematics

• When studying mathematics, the first five
  External Conditions of Learning have occurred
  only after the student has studied all materials
  (text, lecture, etc.) related to a section of the
  textbook.

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6. Eliciting the PerformanceMathematics

• The student should then turn to the exercises and
  demonstrate to himself that the concept has been
  learned.
• The purpose of homework and or quizzes is to
  demonstrate to both the student and the
  instructor that the student has indeed learned
  the desired concept.

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6. Eliciting the PerformanceMathematics

• Notice that working exercises (at this stage) is
  to demonstrate that the concept has been learned.
  
• It (working exercises) is not a device for
  learning the concept.
• Therefore it is not necessary to work huge
  numbers of exercises.

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7. Providing Feedback About Performance
Correctness

• The important characteristic of feedback
  communication is not its form but its function
• Providing information to the student about
  the correctness of his/her performance relative
  to the Learning Objective.

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Providing Feedback About Performance
CorrectnessMathematics

• Mathematics textbooks generally provide very
  minimal feedback in the form of answers to the
  odd numbered problems.

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Providing Feedback About Performance Correctness
Mathematics

• The Learning Objective is hardly ever find the
  answer to a problem.
• The Learning Objective is to learn to use a
  combination of concepts, rules, processes, etc.
  to solve particular types of problems.
• The feedback in most textbooks is not very useful
  and indeed fosters a misunderstanding of the
  Learning Objective.

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Providing Feedback About Performance Correctness
Mathematics

• Therefore, the mathematics instructor should
  provide feedback which addresses the steps and
  the reasons for the steps used by the student
  when solving a problem.
• That is, the mathematics instructor should
  provide feedback which is directly related to the
  Learning Objective.

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8. Assessing the Performance

• In mathematics classes, assessing student
  performance usually takes the form of quizzes and
  tests.
• With every such assessment tool the instructor
  must be concerned with reliability and validity.

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8. Assessing the Performance

• Is the observation reliable or was the correct
  response obtained by chance?
• Notice that reliability is low for true/false and
  multiple choice questions unless special care is
  taken in the construction of such questions.

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8. Assessing the Performance

• Is the observation valid ?
• That is does correct performance accurately
  reflect the objective?

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8. Assessing the Performance

• Is the observation of correct performance free of
  distortion?
• Distortion might be memorization of an answer or
  recall of an answer from some previous example.
• Notice that when a test contains exercises which
  the student has previously seen, the chance for
  distortion is great.

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9. Enhancing Retention and Transfer

• Arranging a variety of practice to aid future
  retrieval and transfer.
• If concepts, rules, procedures, etc. are to be
  well retained, provision must be made for
  systematic reviews spaced at intervals of weeks
  and months. This is more effective than repeated
  examples immediately following the initial
  learning.

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Enhancing Retention and TransferMathematics

• What this means in mathematics is that it is
  probably ineffective to do large numbers of
  exercises during or immediately after the initial
  learning.

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Enhancing Retention and TransferMathematics

• It is more effective to
• Regularly review those items (definitions, rules,
  concepts, etc.) which are to be memorized and
• Regularly go back to previous sections and work a
  few exercises