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.  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 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 learner’s environment  Control of processes through executive strategies

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

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

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.

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

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

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. 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 attention  Stimulation to gain attention to ensure the reception of stimuli.  Various kinds of events can be used to gain the student’s 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 attention  An appeal to the student’s 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.

2.Informing the Learner of 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.

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.

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.

3.Stimulating Recall of Prerequisite Learning Mathematics  Suppose we expect the student to learn the Fundamental Theorem of Arithmetic: Fundamental Theorem of Arithmetic: Any natural number can be expressed as a product of prime numbers. 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.

3.Stimulating Recall of Prerequisite Learning Mathematics  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.

3.Stimulating Recall of Prerequisite Learning Mathematics  The adult student might be expected to recall those definitions simply because the appearance of the words is enough of a reminder.

4.Presenting the Stimulus Material Mathematics  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.

4.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.

4.Presenting the Stimulus Material Mathematics  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 f(x) = 3x, or f(x) = 7 as well as examples like f(x) = 3x + 7 as well as examples like f(x) = 3x + 7

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”.

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

5.Providing Learning Guidance Mathematics  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.

5.Providing Learning Guidance Mathematics  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.

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.

6. Eliciting the Performance Mathematics  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.

6. Eliciting the Performance Mathematics  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.

6. Eliciting the Performance Mathematics  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.

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. Providing information to the student about the correctness of his/her performance relative to the Learning Objective.

7.Providing Feedback About Performance Correctness Mathematics  Mathematics textbooks generally provide very minimal feedback in the form of answers to the odd numbered problems.

7.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.

7.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.

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.

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.

8. Assessing the Performance  Is the observation valid ?  That is does correct performance accurately reflect the objective?

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.

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.

9.Enhancing Retention and Transfer Mathematics  What this means in mathematics is that it is probably ineffective to do large numbers of exercises during or immediately after the initial learning.

9.Enhancing Retention and Transfer Mathematics  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