Setting the Stage for Presenting and Studying Mathematics

Share Cognitive Science Implications for Teaching Mathematics

Everyone reading these blogs is familiar with or even expert at mathematics. Some are teachers of mathematics. Most have studied mathematics. I am asking you to abandon your favorite ideas of what mathematics is, how it should be presented, and how it should be studied. I do that because I want you to obtain maximum benefit from my essays. I expect to challenge many long-held ideas about teaching, learning, and mathematics. When I and my blog have gone away, evaluate what I have advocated, continue to develop and use these ideas, or discard the entire set of facts.
The activities of an early algebra classroom can be described as follows. We use deductive reasoning to study relations and operations on mathematical objects. The key words in that description are:

  • Relation
  • Operation
  • Object

As a result, we should know the meaning, and examples, of each of relation, operation, and object as we use them in mathematics.
Immediately upon encountering a new “thing” in mathematics we should categorize the “thing” as exactly one of a relation, an operation, or an object. We should similarly categorize all mathematics “things” we already use. Teachers must make this classification clear to their students by pointing it out with each “thing” they discuss. They should discuss the significance and consequences of the classifications.
Here are the base definitions:

  • Def: A binary relation is a comparison of two operands.
  • Def: A unary operation is a calculation involving one operand.
  • Def: A binary operation is a calculation involving two operands.

Examples of mathematical objects, binary relations, unary operations, and binary operations were presented in the blog on 14 JUN 18.
As you proceed beyond the basic algebra courses, other relations and other operations will be introduced. But the three given above will be sufficient in the beginning.
Notice that I have not provided a formal definition of “mathematical object.” If a thing is not some kind of relation or operation, then it is a mathematical object.
It is important to observe that in this beginning into mathematics we encounter three definitions. I will forewarn you that future mathematics discussions will contain many definitions, in fact, most will begin with a definition. If you are an astute student, you should now ask several questions about definitions.

  1. What kind of definitions are used in mathematics?
  2. What is the best way to learn these definitions?
  3. What role do these definitions play in mathematics?

Several other questions are imbedded in these three and will be covered in the discussion of these three.
Definitions used in mathematics are very different than those encountered in normal prose. Question 1 will be discussed in my next blog.
Learning definitions in mathematics MUST begin with memorization. I will explain this statement and attendant controversy in future blogs. As soon as the idea of memorization is suggested, some readers will raise the false dichotomy between memorization and understanding. I will discuss this dichotomy and will emphasize how cognitive psychology currently understands the role of memorization and understanding in the learning process.

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