# Understanding “Understanding” Part I

A post where we explore some concepts required to define understanding in a cognitive load theory framework.

Elements & Schemata

In one of my earlier posts, I discussed biologically primary and secondary knowledge. In short, primary knowledge is knowledge in which we are biologically programmed to learn, such as how to communicate to others within our culture. Secondary knowledge, however, we are not biologically programmed to learn.

To keep things simple within the framework I want to discuss understanding in, let’s assume that facts and procedures can be divided into two classes: elements and schemata. Elements are single pieces of information that can be processed within our working memory, such as knowing that the number 3 corresponds to the numerical amount three. Once known, elements can be placed together to begin forming schemata. For instance, a schemata for “3” may include knowing that 3 can be mapped to the word “three” or to three objects (cardinal), is the whole number after 2 and before 4 (ordinal), or that the number 3 may be used on your football jersey (nominal).

Schemata, once well-known, can be linked. For instance, a schemata about prime numbers may include knowing that 2, 3 and 5 are the first three prime numbers. In addition to this, elements can form sub-schemata. Our reference to the ordinal, cardinal and nominal interpretations for the number three might all be considered sub-schemata of the overall schemata we have for three. As we know, the beauty of schemata is that, once well-formed, they can enter working memory as a single element, freeing up working memory space for other information.

Element Interactivity

Element interactivity occurs when two or more elements must be processed simultaneously in working memory because they are logically related. Think about the multiplication fact 3 x 4  = 12. There are actually five symbols that must all be interpreted at once due to them being logically connected. There are three numerals: 3, 4 and 12. There is the multiplication operation, which could be interpreted in a couple different ways (as an array, as repeated addition, as a multivariable function that returns the product). Finally, there is the equal sign, which is a symbol referring to the idea of 12 being equivalent in some way to the product of 3 and 4. As a novice learner, all five symbols must be processed individually in the working memory; whereas an expert learner has a well-built schemata that allows them to by-pass having to process all of the symbols every time they see a multiplication fact. In essence, an expert processes one element; whereas a novice may have to process all five elements.

Instructional Implications

As mathematics instructors, we need to be mindful of how the elements of our problems are interacting within the context of teaching our students. High element interactivity necessarily causes more working memory capacity to be used, increasing cognitive load. One potential way to combat curricular competencies involving high element interactivity is to re-visit pre-existing topics and ensuring our students have the well-formed schemata required to ease some of this cognitive load. Think about how challenging linear equations are for our students: they involve complex understanding of integers and fractions, as well as comprehension of how to manipulate all four of the main numerical operations. Before introducing equations, it would seem logical to review operations with integers and fractions so that students can consolidate their knowledge in these areas. By helping to create well-formed schemata in these topics, students can apply more working memory capacity to the new procedures that are intrinsic to linear equations, without applying too much working memory capacity to previous curricular topics. If consolidation does not happen, it is no surprise that the student struggles with linear equations, as the element interactivity is high and too much working memory is being allocated to topics that are not the focus of the lesson.

In my next blog post, I will explore two more interesting topics: intrinsic and extraneous cognitive load. We will see the interplay of element interactivity with these two topics and discuss instructional implications.