STOP DOING MATH STOP DOING MATH LONG ENOUGH TO LONG ENOUGH TO
LEARN ITLEARN IT
Principles of LearningPrinciples of Learning
Delano P. Wegener, Ph.D.Delano P. Wegener, Ph.D.
Spring 2005Spring 2005
Instruction Instruction Instruction is much more than presentation of Instruction is much more than presentation of
information. information.
Instruction may include events that are Instruction may include events that are generated by a page of print, a picture, a generated by a page of print, a picture, a television program, a combination of physical television program, a combination of physical objects, potentially many other stimuli, as well objects, potentially many other stimuli, as well as activities of a teacher. as activities of a teacher.
Instruction Instruction Teaching refers to the activities of the teacher. Teaching refers to the activities of the teacher.
Therefore teaching is only one part (I think it is Therefore teaching is only one part (I think it is an important part) of instruction.an important part) of instruction.
Instruction is a deliberately arranged Instruction is a deliberately arranged set of external events designed to set of external events designed to support internal learning processes.support internal learning processes.
Internal Conditions of LearningInternal Conditions of Learning
Because the internal conditions of learning are Because the internal conditions of learning are beyond our control we will not elaborate beyond beyond our control we will not elaborate beyond listing them.listing them.
Internal Conditions of Learning Internal Conditions of Learning Reception of stimuli by receptorsReception of stimuli by receptors Registration of information by sensory registersRegistration of information by sensory registers Selective perception for storage in short-term Selective perception for storage in short-term
memory (STM)memory (STM) Rehearsal to maintain information in STMRehearsal to maintain information in STM Semantic encoding for storage in long-term memory Semantic encoding for storage in long-term memory
(LTM)(LTM) Retrieval from LTM to working memory (STM)Retrieval from LTM to working memory (STM) Response generation to effectors Response generation to effectors Performance in the learner’s environmentPerformance in the learner’s environment Control of processes through executive strategiesControl of processes through executive strategies
Internal Conditions of LearningInternal Conditions of Learning Cognitive scientists use that information, but we Cognitive scientists use that information, but we
make no explicit use of the internal conditions make no explicit use of the internal conditions of learning when designing instruction.of learning when designing instruction.
LearningLearning Some of the very basic facts and theories Some of the very basic facts and theories
about learning will help to understand the about learning will help to understand the guidelines, presented later, for studying guidelines, presented later, for studying mathematics. mathematics.
LearningLearning In particular, it is helpful to be aware of:In particular, it is helpful to be aware of:
Conditions of Learning, Conditions of Learning, Learning Outcomes, Learning Outcomes, Domains of Learning Objectives, and Domains of Learning Objectives, and Classes of Learning Objectives in the Cognitive Classes of Learning Objectives in the Cognitive
Domain. Domain.
LearningLearning In this seminar we are especially interested in In this seminar we are especially interested in
how these facts and theories about learning how these facts and theories about learning pertain to learning mathematics.pertain to learning mathematics.
External Conditions of LearningExternal Conditions of Learning1.1. Gaining attentionGaining attention2.2. Informing the learner of the objectiveInforming the learner of the objective3.3. Stimulating recall of prerequisite learningStimulating recall of prerequisite learning4.4. Presenting the stimulus materialPresenting the stimulus material5.5. Providing learning guidanceProviding learning guidance6.6. Eliciting the performanceEliciting the performance7.7. Providing feedback about performance Providing feedback about performance
correctnesscorrectness8.8. Assessing the performanceAssessing the performance9.9. Enhancing retention and transferEnhancing retention and transfer
External Conditions of LearningExternal Conditions of Learning Because the External Conditions of Learning Because the External Conditions of Learning
directly affect instruction and what you must directly affect instruction and what you must do to learn, each of these conditions will be do to learn, each of these conditions will be explained.explained.
1. 1. Gaining attentionGaining attention Stimulation to gain attention to ensure the Stimulation to gain attention to ensure the
reception of stimuli.reception of stimuli.
Various kinds of events can be used to gain Various kinds of events can be used to gain the student’s attention. These events might the student’s attention. These events might be as simple as calling the class to order or be as simple as calling the class to order or as complex as the mix of sound, pictures, as complex as the mix of sound, pictures, movement, and light as found in the most movement, and light as found in the most sophisticated TV commercials. sophisticated TV commercials.
1. 1. Gaining attentionGaining attention An appeal to the student’s interest is An appeal to the student’s interest is
frequently employed as a means of gaining frequently employed as a means of gaining attention. attention.
For adult students we frequently assume they For adult students we frequently assume they will themselves provide the stimulation to gain will themselves provide the stimulation to gain their attention.their attention.
2.2. Informing the LearnerInforming the Learnerof the Objectiveof the Objective
Informing learners of the learning objective Informing learners of the learning objective establishes appropriate performance.establishes appropriate performance.
The student must know the kind of The student must know the kind of performance that is expected as a performance that is expected as a demonstration that learning has taken place.demonstration that learning has taken place.
In general it is a mistake to assume the In general it is a mistake to assume the student will know the objective of the lesson. student will know the objective of the lesson.
3. Stimulating Recall of 3. Stimulating Recall of Prerequisite LearningPrerequisite Learning
Reminding learners of previously learned Reminding learners of previously learned content for retrieval from LTMcontent for retrieval from LTM
Much of learning is the combination of ideas. Much of learning is the combination of ideas. If any of the ideas involved have been If any of the ideas involved have been learned previously, the student should be learned previously, the student should be reminded of them so they are retrieved from reminded of them so they are retrieved from LTM into STM where they are available for LTM into STM where they are available for immediate recall. immediate recall.
3. Stimulating Recall of 3. Stimulating Recall of Prerequisite LearningPrerequisite Learning
At the time of learning, previously learned At the time of learning, previously learned ideas must be readily available. They must ideas must be readily available. They must therefore be recalled from LTM (into STM) therefore be recalled from LTM (into STM) prior to the time of learning.prior to the time of learning.
3.3. Stimulating Recall of Stimulating Recall of Prerequisite LearningPrerequisite Learning
MathematicsMathematics Suppose we expect the student to learn the Suppose we expect the student to learn the Fundamental Theorem of Arithmetic:Fundamental Theorem of Arithmetic:
Any natural number can be expressed as a Any natural number can be expressed as a product of prime numbers.product of prime numbers.
If the definitions of If the definitions of natural numbernatural number, , productproduct, , and and prime numberprime number are not readily available, are not readily available, then the Fundamental Theorem of Arithmetic then the Fundamental Theorem of Arithmetic will not be learned.will not be learned.
3.3. Stimulating Recall of Stimulating Recall of Prerequisite LearningPrerequisite Learning
MathematicsMathematics It is essential that these definitions be It is essential that these definitions be
recalled from LTM to STM. recalled from LTM to STM.
Assuming that these definitions have Assuming that these definitions have previously been learned, the teacher can previously been learned, the teacher can insure they are recalled into STM by simply insure they are recalled into STM by simply reminding the student of those definitions. reminding the student of those definitions.
3.3. Stimulating Recall of Stimulating Recall of Prerequisite LearningPrerequisite Learning
MathematicsMathematics The adult student might be expected to recall The adult student might be expected to recall
those definitions simply because the those definitions simply because the appearance of the words is enough of a appearance of the words is enough of a reminderreminder..
4.4. Presenting the Presenting the Stimulus MaterialStimulus Material
MathematicsMathematics It is important that the proper stimuli be It is important that the proper stimuli be
presented as a part of the instructional presented as a part of the instructional events. events.
If a mathematical rule is to be learned, then If a mathematical rule is to be learned, then that rule must be communicated. Such that rule must be communicated. Such communication, if printed, may use italics, communication, if printed, may use italics, bold print, underlining, colors, etc. to bold print, underlining, colors, etc. to emphasize particular features.emphasize particular features.
4.4. Presenting the Presenting the Stimulus MaterialStimulus Material
When young children are learning concepts When young children are learning concepts or rules the stimulus material should present or rules the stimulus material should present examples prior to presenting a statement of examples prior to presenting a statement of the concept or rule. the concept or rule.
When adults are learning concepts or rules When adults are learning concepts or rules the stimulus material should present a the stimulus material should present a statement of the concept or rule followed by statement of the concept or rule followed by examples.examples.
4.4. Presenting the Presenting the Stimulus MaterialStimulus Material
MathematicsMathematics Stimulation presentation for the learning of Stimulation presentation for the learning of
concepts and rules requires the use of a concepts and rules requires the use of a variety of examples. variety of examples.
Thus the stimulus presentation for learning Thus the stimulus presentation for learning the mathematical concept of “linear function” the mathematical concept of “linear function” will involve examples of functions like will involve examples of functions like
f(x) = 3x, or f(x) = 7 f(x) = 3x, or f(x) = 7
as well as examples like f(x) = 3x + 7as well as examples like f(x) = 3x + 7
5. Providing Learning Guidance5. Providing Learning Guidance
Communications which have the function of Communications which have the function of providing learning guidance do not provide providing learning guidance do not provide “the answer”. “the answer”.
They suggest a line of thought which will lead They suggest a line of thought which will lead to appropriate combining of previously to appropriate combining of previously learned concepts allowing the student to learn learned concepts allowing the student to learn “the answer”.“the answer”.
5. Providing Learning Guidance5. Providing Learning Guidance
Communications designed to provide learning Communications designed to provide learning guidance should stimulate a direction of guidance should stimulate a direction of thought which keeps the student on the right thought which keeps the student on the right track.track.
5.5. Providing Learning GuidanceProviding Learning GuidanceMathematicsMathematics
When presenting an example of solving a When presenting an example of solving a linear equation the instructor does not linear equation the instructor does not encourage a memorized set of steps to arrive encourage a memorized set of steps to arrive at “the answer”. at “the answer”.
Rather the instructor constantly reminds the Rather the instructor constantly reminds the student of the previously learned two student of the previously learned two operations which generate an equation operations which generate an equation equivalent to the original equation. equivalent to the original equation.
5.5. Providing Learning GuidanceProviding Learning GuidanceMathematicsMathematics
All communications in this context are All communications in this context are designed to keep the student “on track” to designed to keep the student “on track” to generate a sequence of equations, all generate a sequence of equations, all equivalent to the original, terminating in a equivalent to the original, terminating in a simplest equation.simplest equation.
6. Eliciting the Performance6. Eliciting the Performance
Suppose the previous five events have taken Suppose the previous five events have taken place, enough material has been presented, place, enough material has been presented, sufficient learning guidance has taken place, sufficient learning guidance has taken place, and the student indicates/believes he has and the student indicates/believes he has learned the concepts. learned the concepts.
It is then time for the student to demonstrate It is then time for the student to demonstrate both to himself and the instructor that he has both to himself and the instructor that he has learned the concept.learned the concept.
6. Eliciting the Performance6. Eliciting the PerformanceMathematicsMathematics
When studying mathematics, the first five When studying mathematics, the first five External Conditions of Learning have External Conditions of Learning have occurred only after the student has studied all occurred only after the student has studied all materials (text, lecture, etc.) related to a materials (text, lecture, etc.) related to a section of the textbook. section of the textbook.
6. Eliciting the Performance6. Eliciting the PerformanceMathematicsMathematics
The student should then turn to the exercises The student should then turn to the exercises and demonstrate to himself that the concept and demonstrate to himself that the concept has been learned. has been learned.
The purpose of homework and or quizzes is The purpose of homework and or quizzes is to demonstrate to both the student and the to demonstrate to both the student and the instructor that the student has indeed learned instructor that the student has indeed learned the desired concept.the desired concept.
6. Eliciting the Performance6. Eliciting the PerformanceMathematicsMathematics
Notice that working exercises (at this stage) is Notice that working exercises (at this stage) is to demonstrate that the concept has been to demonstrate that the concept has been learned. learned.
It (working exercises) is not a device for It (working exercises) is not a device for learning the concept. learning the concept.
Therefore it is not necessary to work huge Therefore it is not necessary to work huge numbers of exercises.numbers of exercises.
7. Providing Feedback About 7. Providing Feedback About Performance CorrectnessPerformance Correctness
The important characteristic of feedback The important characteristic of feedback communication is not its form but its function:communication is not its form but its function:
Providing information to the student about Providing information to the student about the correctness of his/her performance the correctness of his/her performance relative to the Learning Objective.relative to the Learning Objective.
7.7. Providing Feedback About Providing Feedback About Performance CorrectnessPerformance Correctness
MathematicsMathematics
Mathematics textbooks generally provide very Mathematics textbooks generally provide very minimal feedback in the form of answers to minimal feedback in the form of answers to the odd numbered problems. the odd numbered problems.
7.7. Providing Feedback About Providing Feedback About Performance CorrectnessPerformance Correctness
MathematicsMathematics The Learning Objective is hardly ever “find The Learning Objective is hardly ever “find
the answer” to a problem.the answer” to a problem.
The Learning Objective is to learn to use a The Learning Objective is to learn to use a combination of concepts, rules, processes, combination of concepts, rules, processes, etc. to solve particular types of problems. etc. to solve particular types of problems.
The feedback in most textbooks is not very The feedback in most textbooks is not very useful and indeed fosters a misunderstanding useful and indeed fosters a misunderstanding of the Learning Objective.of the Learning Objective.
7.7. Providing Feedback About Providing Feedback About Performance CorrectnessPerformance Correctness
MathematicsMathematics Therefore, the mathematics instructor should Therefore, the mathematics instructor should
provide feedback which addresses the steps provide feedback which addresses the steps and the reasons for the steps used by the and the reasons for the steps used by the student when solving a problem. student when solving a problem.
That is, the mathematics instructor should That is, the mathematics instructor should provide feedback which is directly related to provide feedback which is directly related to the Learning Objective.the Learning Objective.
8. Assessing the Performance8. Assessing the Performance
In mathematics classes, assessing student In mathematics classes, assessing student performance usually takes the form of performance usually takes the form of quizzes and tests. quizzes and tests.
With every such assessment tool the With every such assessment tool the instructor must be concerned with reliability instructor must be concerned with reliability and validity.and validity.
8. Assessing the Performance8. Assessing the Performance
Is the observation reliable or was the correct Is the observation reliable or was the correct response obtained by chance?response obtained by chance?
Notice that reliability is low for true/false and Notice that reliability is low for true/false and multiple choice questions unless special care multiple choice questions unless special care is taken in the construction of such questions.is taken in the construction of such questions.
8. Assessing the Performance8. Assessing the Performance
Is the observation valid ? Is the observation valid ?
That is does correct performance accurately That is does correct performance accurately reflect the objective? reflect the objective?
8. Assessing the Performance8. Assessing the Performance
Is the observation of correct performance free Is the observation of correct performance free of distortion? of distortion?
Distortion might be memorization of an Distortion might be memorization of an answer or recall of an answer from some answer or recall of an answer from some previous example.previous example.
Notice that when a test contains exercises Notice that when a test contains exercises which the student has previously seen, the which the student has previously seen, the chance for distortion is great.chance for distortion is great.
9. Enhancing 9. Enhancing Retention and TransferRetention and Transfer
Arranging a variety of practice to aid future Arranging a variety of practice to aid future retrieval and transfer.retrieval and transfer.
If concepts, rules, procedures, etc. are to be If concepts, rules, procedures, etc. are to be well retained, provision must be made for well retained, provision must be made for systematic reviews spaced at intervals of systematic reviews spaced at intervals of weeks and months. This is more effective weeks and months. This is more effective than repeated examples immediately than repeated examples immediately following the initial learning.following the initial learning.
9.9. Enhancing Enhancing Retention and TransferRetention and Transfer
MathematicsMathematics What this means in mathematics is that it is What this means in mathematics is that it is
probably ineffective to do large numbers of probably ineffective to do large numbers of exercises during or immediately after the exercises during or immediately after the initial learning. initial learning.
9.9. Enhancing Enhancing Retention and TransferRetention and Transfer
MathematicsMathematics It is more effective to:It is more effective to:
Regularly review those items (definitions, Regularly review those items (definitions, rules, concepts, etc.) which are to be rules, concepts, etc.) which are to be memorized and memorized and
Regularly go back to previous sections and Regularly go back to previous sections and work a few exerciseswork a few exercises