12. Memory

James acknowledges memory as a result of associations between old and new information.  Retentiveness is a fixed property of an individual given that the prior knowledge of students varies.  This blog will ponder effective instructional design strategies that support student memory and retention of math concepts.  

When designing instruction that incorporates worked examples, there are two important considerations a teacher must bear in mind with regards to memory.  First, there are two types of knowledge that a student must retain.  The student must understand the problem type or schema, and the student must know the calculations and procedures associated with the problem type.  Connecting the two types of knowledge is essential for effective worked examples.  James alluded to this when he said, “The connecting is the thinking; and,  if we attend clearly to the connection, the connected thing will certainly be likely to remain within recall”  (p. 70). 

Secondly, the selection and order of examples presented in the lessons need to be chosen with the problem type and procedural knowledge in mind.  An effective presentation of examples will help improve memory of schema and problem solving processes.  The Einstellung effect is the creation of a mechanized state of mind.  It refers to a person’s predisposition to solve a given problem using a specific procedure even though there may be a better or other way of solving the problem. 

Here is an example of the Einstellung effect in an Algebra class. A teacher gives the following examples: 

1.     2x + 5 = 12

2.    3x + 2 = 9

3.    -2x + 1 = 15

4.    -6x + 6 = 24

In the previous examples, the first step is to use the additive inverse to eliminate the integer on the left side of the equal side.  For each one of those problems, students would perform subtraction as their first step.

Now the teacher gives the final example:  2x – 12 = -16.  In this example, students will perform addition as their first step.  However, they have just completed four problems using subtraction as the first step, and may be inclined to continue because that is what all the previous examples had them do.  The students who chose to use addition did not understand the overall concept of inverse relationships when solving equations and therefore, performed the operation they were trained to do given the previous worked examples.  A teacher may want to consider varying problem type in order to avoid problems such as the Einstellung effect, and use techniques and strategies that focus on the problem type and schema, rather than just the procedural calculations. A teacher could use verbal and visual cues to help students form associations between problem types and procedures, therby reducing the chance of errors such as the Einstellung effect. I often use graphic organizers in my Algebra class for this same purpose.    

James defines memory as the result of the association of ideas, and a teacher can elicit the associations through directed cues.  He states, “…the cue is something contiguously associated with the thing recalled” (p. 59).   While worked examples are math specific, other types of examples are used in different subject areas.  What are some effective cues that a teacher can employ to spawn the association of ideas and improve memory of concepts?

William completing a
graphing calculator exercise.