Archive for March, 2013

Homework in College Math

March 21, 2013

The concept of homework is understood differently in high school than in college. I do not intend to compare or contrast the two nor do I intend to discuss their relative merits. In this document I will simply point out what homework means and what the related expectations are in many college mathematics classes.

Homework means studying outside of the classroom.

Neither instructors nor students should speak of collecting (turning in) homework. It is impossible to collect studying. There are a few products of homework activity that might be collected by an instructor. If any of the student’s work is collected, the instructor should be obliged to provide meaningful feedback about all aspects of the student’s work.

Studying mathematics is much like studying any other subject. You must learn the vocabulary in order to communicate in the discipline. You must read and listen to what the experts have to say about the subject. In a school environment that usually means reading and studying a designated textbook and any other material (handouts, lectures, websites) supplied by the teacher or school.

Studying mathematics is also much different than studying any other subject. The differences are apparent in two major ways.

  • All definitions in mathematics are stipulative definitions as compared to lexical definitions found in virtually every other subject. Because lexical definitions report common use of the term, one can learn those meanings through repetitive usage. Stipulative definitions stipulate what the term will mean and therefore it is virtually impossible to learn its meaning through repetitive use. Learning a stipulative definition must begin by memorizing the definitions. Other activities then help the learner to deepen his understanding of the term.
  • Mathematics is completely dependent on deductive reasoning. Only deductive arguments are permitted in the justification of a mathematics statement. The student must learn to produce arguments which are based strictly on deductive reasoning and which are completely devoid of inductive reasoning or emotional reasoning. Mathematics requires arguments that flawlessly adhere to the laws of Aristotelian logic.

Part of homework (study) should include answering as many different questions as are available about the topic. If feedback is provided for available questions, study that feedback. Remember the goal is to learn concepts.

Lists of computational exercises are generally very easy to find in textbooks. It is important to use these exercises as a means of evaluating your understanding of the narrative of the text and lecture. Keep in mind that working numerical problems is only one small part of homework. In general mathematics is not learned by working problems, it is learned in order to answer questions.

Homework involves each of the following activities at different times during the process of studying for a typical college mathematics course. Some of these activities are performed daily and some are performed less frequently.

  • Reading
    • Presentations
    • Examples
    • Illustrations
  • Writing
    • Definitions>
    • Major properties
    • Certain formulas
    • Certain procedures
    • Lecture Notes
    • Examples
      • From Textbook
      • Learner Created
      • Learner Created Non-examples
    • Exercises
  • Memorizing
    • Definitions
    • Major properties
    • Certain formulas
    • Certain procedures
  • Identification
    • Generalizations
    • Abstractions
    • Deductive Reasoning
    • Patterns
    • Relationships
    • Mathematical Objects
    • Binary Relations
    • Unary Operations
    • Binary Operations
    • Major concepts
  • Contemplation
    • Implications
    • Relation to Examples
    • Applications
    • Relation to Previous Concepts
  • Ask questions
    • Ask yourself questions
    • Ask your friends questions
    • Ask your instructor questions
  • Answer questions
    • Answer your own questions
    • Answer questions from you friends
    • Answer questions in the textbook
    • Answer questions in class
  • Analysis
    • Individual Concepts
    • Combinations of Concepts
    • Complete Course Content
  • Review
    • Definitions
    • Concepts
    • Structures
    • Algorithms
  • Working problems
    • Analysis
    • Process
    • Reasoning
    • Proper presentation
      • Correct mathematics
      • Correct grammar
      • Complete sentences
      • Proper use of words

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