Understanding “Understanding” Part III

A post where we explore how to define understanding in a cognitive load theory framework.

In my last two blog posts, I discussed the concepts of element interactivity, as well as intrinsic and extraneous cognitive load. We say information has high element interactivity if there are many elements of the information that must be processed together at the same time. High element interactivity generally implies high intrinsic cognitive load. Here, intrinsic cognitive load refers to a working memory load caused by the intrinsic nature of information that we are trying to process. Finally, extraneous cognitive load refers to a working memory load imposed by the pedagogical nature of the information being taught.

Defining Understanding

Now that we know about element interactivity, we can use this concept to define understanding. In a cognitive load setting, understanding is the ability to process all interacting elements in working memory at one time. Since the focus is on interacting elements, it does not make sense to define understanding to individual elements, such as learning one French vocabulary word (cat = chat).

Let’s analyze our previous examples. Consider the math fact 3 + 5 = 8. According to our definition, if a learner is able to answer 3 + 5 = ? correctly, without having process all of the interacting elements, we would say that she has demonstrated understanding of the question at hand. I would argue then, that using a strategy such as tallying up three and five on her fingers would display a lack of understanding. Even beginning with three fingers and counting up to eight, whilst being a more effective strategy, still displays a lack of understanding as she is processing some or all of the elements individually. Of course, I am not arguing that students shouldn’t be permitted to use these counting strategies. It is likely that these are crucial stepping stones in the learning trajectory, and the instructor needs to be mindful of when the student seems ready to move beyond these strategies.

Understanding and Incorrectness

One aspect of the definition that I am curious about is when the learner makes a mistake in the process. Consider solving for x in 3x – 10 = 5. Is it possible for the student to understand, yet be incorrect? Are these mutually exclusive events? Let’s say the student solution is

3x – 10 = 5
3x = -15
x = -5

This is incorrect, but it still shows us that they understand the process of solving for x, and that they can process all of this information in working memory at once. Does understanding come down to a judgement call on the side of the instructor in these cases?

Instructional Implications – A Case for Quick Math Fact Recall

Let’s try to deconstruct our current pedagogy in light of this definition of understanding. Consider all of the multiplication facts that our students must recall. There is element interactivity amongst one individual fact (3 x 4 = 12), as well interaction amongst all of the multiplication facts for three, as well as interaction amongst all facts up to 9 x 9 = 81! Working memory might get overwhelmed, as intrinsic load is high due to the number of facts that must be remembered.

Think also about what our current curriculum states: students should be comfortable with knowing other concepts, such as knowing 3 x 4 = 4 + 4 + 4 = 12, building array models, or knowing about the commutative property.  All of this increases extraneous cognitive load; thus requiring more time and effort for the students to move the facts to long-term memory. I would argue that this is why we have seen a shift to moving recall of the multiplication facts to later grade levels. In British Columbia, students aren’t expected to recall facts for 3s or 4s until Grade 5; and there is no mention of the harder facts like 7s, 8s or 9s.

To compare, I had my multiplication facts memorized by the end of Grade 3 in the 80s in Ontario. Some might argue that we were taught without understanding (this alternate definition is a bit fuzzy, but typically is interpreted as knowing how to complete a question utilizing a model). This is false, as I have many documents showing that we indeed used models. But the key difference here is that the focus of instruction was on automatization of facts, and that models were used to introduce concepts and as help when students weren’t understanding. Models were used to decrease intrinsic load, not to increase extraneous load!

Addition and subtraction models were used to introduce the concepts in Grade 1.

For such a large task, such as learning the multiplication facts, why not have students learn the individual facts first? Using techniques such as interleaved and spaced practice, and introducing new fact families after long exposure to previous ones, would be beneficial for learning. After students are comfortable with recall of the facts, then we can focus our teaching on developing understanding (the fuzzier definition) of how multiplication is connected to other concepts. Of course, once students can recall the multiplication facts, they have displayed understanding in the cognitive load sense, as they can process all of the elements together at once. So why would we want to learn our facts first, before connecting to other concepts? Once the facts are remembered well, then the can be retrieved quickly and efficiently, leading to lowered intrinsic load, and more working memory capacity to work on the current problem of connecting the fact to another concept.

In conclusion, I am not saying that we shouldn’t explain why certain facts are the way they are! This can certainly be done as motivation to the problem, and mixed throughout as needed; however, this should not be the focus of the learning because this increases extraneous load and not all students will successfully move the facts into long-term memory store this way.

Advertisement