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. Instruction. Instruction is much more than presentation of information. - PowerPoint PPT Presentation



    Delano P. Wegener, Ph.D.Spring 2005

  • 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.

  • 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.

  • Internal Conditions of Learning

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

  • Internal Conditions of Learning Reception of stimuli by receptorsRegistration of information by sensory registersSelective perception for storage in short-term memory (STM)Rehearsal to maintain information in STMSemantic encoding for storage in long-term memory (LTM)Retrieval from LTM to working memory (STM)Response generation to effectors Performance in the learners environmentControl of processes through executive strategies

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

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

  • LearningIn 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.

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

  • External Conditions of LearningGaining attentionInforming the learner of the objectiveStimulating recall of prerequisite learningPresenting the stimulus materialProviding learning guidanceEliciting the performanceProviding feedback about performance correctnessAssessing the performanceEnhancing retention and transfer

  • 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.

  • 1. Gaining attentionStimulation 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.

  • 1. Gaining attentionAn 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.

  • Informing the Learnerof the ObjectiveInforming 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.

  • 3. Stimulating Recall of Prerequisite LearningReminding 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.

  • 3. Stimulating Recall of Prerequisite LearningAt 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.

  • Stimulating Recall of Prerequisite LearningMathematicsSuppose 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.

  • Stimulating Recall of Prerequisite LearningMathematicsIt 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.

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

  • Presenting the Stimulus MaterialMathematicsIt 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.

  • Presenting the Stimulus MaterialWhen 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.

  • Presenting the Stimulus MaterialMathematicsStimulation 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

  • 5. Providing Learning GuidanceCommunications 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.

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

  • Providing Learning GuidanceMathematicsWhen 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.

  • Providing Learning GuidanceMathematicsAll 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.

  • 6. Eliciting the PerformanceSuppose 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.

  • 6. Eliciting the PerformanceMathematicsWhen 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.

  • 6. Eliciting the PerformanceMathematicsThe 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.

  • 6. Eliciting the PerformanceMathematicsNotice 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.

  • 7. Providing Feedback About Performance CorrectnessThe 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.

  • Providing Feedback About Performance CorrectnessMathematicsMathematics textbooks generally provide very minimal feedback in the form of answers to the odd numbered problems.

  • Providing Feedback About Performance Correctness MathematicsThe 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.

  • Providing Feedback About Performance Correctness MathematicsTherefore, 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.

  • 8. Assessing the PerformanceIn 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.

  • 8. Assessing the PerformanceIs the observation reliable or was the correct response obtained by chance?