Reading Mathematics — Foundational Cognitive Science

Cognitive Science as a Foundation for Tips 1 – 5

The Learning Process

There are two classifications of conditions for learning.  There are external conditions of learning which are controlled by the instructional developer or teacher.  There are internal conditions which derive from the stored memories of the learner.

The diagram below is a widely accepted model of the processes involved in the act of learning.


The following is a simplified glimpse of some aspects of the internal conditions of learning.

Information stays in short-term memory for a very short time (measured in seconds) except during an activity called rehearsal.  Information which is to be remembered for use at a later time must be semantically encoded and stored in long-term memory.

During the learning process, information is retrieved from long-term memory into short-term memory where it combines with other items in short-term memory to bring about new kinds of learning.

Suppose the learner has previously learned the meaning of the two terms “mathematical expression” and “equal sign”.  To say he has learned this information means it has been semantically encoded and stored in long-term memory and is ready for retrieval into short-term memory.

When this student is presented with the definition “An equation is a mathematical statement which contains an equal sign”, the two previously learned bits of information are retrieved from long-term memory into short-term memory where they are combined with the new definition to be encoded and stored in long-term memory.  At this point the student has learned the meaning of  the word equation as it is used in mathematics.  I should point out that the process is actually a bit more complex but this example illustrates what must happen during the learning process.

What can go wrong in the above process to prevent learning from happening?  There are many potential problems.   We have absolutely no control over some but the learner, the instructional developer,  and instructor can prevent some problems.

In the above example, it is clear that the learner must retrieve two items from long-term memory.  If that retrieval does not take place, learning does not happen.  Retrieval might fail because the two necessary items are not in long-term memory or they are in long-term memory but not available for retrieval.

Careful sequencing of courses and topics in mathematics education is an attempt to insure that the learner has previously learned requisite material and has it stored in long-term memory ready for retrieval.

Availability for Retrieval

“Semantic encoding is the process involved in moving information from short-term to long-term memory.  This process involves making the information meaningful by tying it to previously learned information structures (schemas) or establishing new structures.  Linkages of this sort would seem to be facilitated through the use of concept maps whereby the learner is enabled to see the structure of the material to be learned.” [Gagne p.68]

“As in the case of individual facts, the learning and storage of larger units of organized verbal information occurs within the context of a network of interconnected and organized propositions previously stored in the learner’s memory.”[Gagne p.84]

Without the use of concept maps some information in long-term memory is simply not available for retrieval.  It can’t be remembered.

There are four common kinds of concept maps with a few other specialized maps that help in certain situations.  The concept map which is most commonly associated with mathematics is called the Hierarchy Concept Map.

Here is a simplified  picture of a Hierarchy Concept Map which might be used for mathematics.


The entries in this concept map are the course names, textbook titles, chapter titles, section titles, concept names, etc. The student who pays attention to these various titles (and the associated  hierarchy) is constructing a concept map which is essential for efficient retrieval of mathematics information from long-term memory.

That is why the previous five tips for reading mathematics encourage you to pay attention to the organizational titles.   They help build the very essential concept map.

Another View

Another way of visualizing how your long-term memory organizes information might be the following.

There is a “room” reserved in your LTM (long-term memory) for mathematics information.  The room is divided into “sections” labeled “Algebra”, “Geometry”, “Analysis”, and so on for each of the major segments of mathematics.

Each of these sections of the room contain numerous “file cabinets”, each reserved for a subset of the room section.  So in the “Algebra” section of the room, among others there will be a file cabinet for Functions.  Inside this file cabinet are folders for topics such as Zeros of functions, Linear functions, Quadratic functions, etc.

This organization permits efficient recovery of information from any one of the folders.

When the learner uses no concept map, it is comparable to having all the millions of mathematics facts strewn about on the floor of the mathematics room.  Obviously in such disarray, virtually nothing is available for retrieval and therefore very little learning can  take place.


One Response to “Reading Mathematics — Foundational Cognitive Science”

  1. Help with Math Says:

    Help with Math…

    […]Reading Mathematics — Foundational Cognitive Science « DrDel on Teaching Math[…]…

Leave a Reply

Fill in your details below or click an icon to log in: Logo

You are commenting using your account. Log Out /  Change )

Google+ photo

You are commenting using your Google+ account. Log Out /  Change )

Twitter picture

You are commenting using your Twitter account. Log Out /  Change )

Facebook photo

You are commenting using your Facebook account. Log Out /  Change )


Connecting to %s

%d bloggers like this: