CLT Part I – Problems with Problem Solving

“Primary knowledge is learned easily, automatically and unconsciously and cannot be taught. Secondary knowledge is culturally acquired, … learned consciously with effort and should be explicitly taught.” -Sweller, Ayres & Kalyuga

I have started to read “Cognitive Load Theory” by Sweller, Ayres and Kalyuga. I intend to try to reflect on each section as it arises to help solidify my own knowledge and to invite those of you willing to read into my thoughts.

Just as we often do with a mathematical lesson, we first begin with some definitions.

Biologically Primary Knowledge
Geary describes biologically primary knowledge as skills that are readily learned because we have evolved to acquire them for biological survival. Primary knowledge is learnable, but not teachable. For instance, we come to acquire our first language, not by being taught, but by determining ourselves the varying motions of the breath, lips and tongue required to produce certain sounds. We readily acquire this skill because we have evolved to do so, irrespective of our culture. Individuals simply need to be members of a functioning society to attain this biologically primary knowledge. For now, we will assume that any skill area in which we can acquire information without first being explicitly taught is a biologically primary skill.

Biologically Secondary Knowledge
Biologically secondary knowledge, on the other hand, is knowledge that we have not encountered in our evolutionary journey as a species. We can assimilate this knowledge; however, it needs to be explicitly taught within a culture. Secondary knowledge is both learnable and teachable. However, acquiring secondary knowledge takes much effort and is done consciously. For example, while learning to speak is biologically primary, learning proper etiquette when speaking, such as saying “I am a boy.” rather than “I is a boy.” must be explicitly taught.

Problem Solving as Biologically Primary
Reflecting a bit on the phrase “problem solving” we might believe that this is a secondary skill. However, deeper reflection allows us to see that “problem solving” in its simplest sense (i.e. solving a problem) is biologically primary. Certainly our ancestors developed general strategies to solve basic problems such as how to find an efficient way to cross a river. A general problem-solving strategy called means-end analysis is well-known, and is argued to be biologically primary due to the fact that it is utilized in many problem domains, yet is not an explicitly taught skill.

Means-end analysis uses forward and backward reasoning over a problem space. There are two states: the current state and the goal state. Differences between these two states are used to determine actions that can lead to smaller differences between the two states. When an action is performed, we check to see if the current state is equal to the goal state; if not, we recursively continue until the goal state is reached.

For example, we might pose the problem “Who was the seventh president of the United States of America?”. Our current state (unless we know this fact) is that we don’t know his/her name, and our goal state is the name of the seventh president of the USA. We might reason that a Google search would be helpful in determining this information, so we type “seventh president of USA” into Google’s search bar (an action leading to a smaller difference in our two states). Now our current state has a page of relevant websites. Since the current and goal states are still different, we continue by clicking the first link, which is a Wikipedia article on Andrew Jackson (another action in hopes to simplify the difference of the two states). Finally, we reason that if we read the first sentence of this article, we might have our answer. Indeed, Wikipedia reveals that Andrew Jackson was the seventh president, so our current and goal states are equivalent. We are done our means-end analysis.

Instructional Consequences
As you can see from our previous example, means-end analysis, while biologically primary, is actually more complex cognitively than simply knowing the fact that Andrew Jackson was the seventh president. If we know this fact, we can by-pass the strain of using means-end analysis to determine the goal state. Of course, since you are likely an adult, you know how to use a search page relatively easily (you have already assimilated this secondary knowledge) so the strain on your cognitive processes might be minimal. For a more novice learner, the action of searching for this information using Google may be quite a strain on their cognitive processes. In either case, it is important to note that having domain-specific knowledge (i.e. knowing that Andrew Jackson was the seventh president of the United States) helps reduce the cognitive processes associated with general problem solving strategies such as means-end analysis.

To continue this discussion, let’s think about a mathematics problem such as simplifying 51 + 52 + 53 + 54 + … + 100. We could attempt to use means-end analysis by adding 51+52=103, then 103+53=156, then …, but this would be terribly taxing on our working memory (and we might be more likely to make an error due to this strain). If we notice, however, that 51+100=151, and that 52+99=151 and that 53+98=151, …, we may note that 51+52+…+100 = 25 x 151 = 3775. But the crux here is that we may not have thought of the second strategy unless we have either been explicitly shown this, or we have encountered a similar problem (and we reason through this problem thinking of the previous one). Again, we see that having domain-specific knowledge (i.e. having seen a similar problem) reduces the cognitive demand of this problem.

So should we teach problem solving to our students?

Well this depends on what your end-goal is as you teach problem solving. General problem solving skills are biologically primary, so we do not need to teach these skills. For instance, if your end-goal is to teach your students a general skill like “think of a similar problem”, it is probably not a beneficial problem solving session. Teaching domain specific-knowledge, such as giving worked examples, is likely to be very beneficial for students, as they can then use these worked examples to help them reason through future problems. In addition to this, the more domain-specific knowledge a student has, the easier problem solving becomes since domain-specific knowledge often lessens the cognitive processes of a more general problem solving strategy like means-end analysis (see the two examples given above).

1) There are two types of knowledge to think about: biologically primary knowledge, which is learned, but cannot be taught; and biologically secondary knowledge, which can be both learned and taught (but may take more effort than primary knowledge).

2) General problem solving strategies are a part of our biologically primary knowledge. We do not need to explicitly teach these skills. In fact, teaching general problem solving skills such as “think backwards” or “eliminate possibilities” is probably no more useful than a means-end analysis.

3) Having domain-specific knowledge is helpful when faced with a problem solving situation. In fact, the more domain-specific knowledge we have, the more likely we are to be able to find a strategy that will be useful to solve a particular problem.

Of course, I appreciate any thoughts.

6 Comments on “CLT Part I – Problems with Problem Solving

  1. Pingback: Understanding “Understanding” Part I | For the Love of Maths

  2. Pingback: Cognitive Load Theory is Wrong?? – For the Love of Maths

  3. Pingback: What Neuroscience can tell us about Arithmetic Success | For the Love of Maths

  4. Thinking of “simpler problems” or any of the other techniques that were first popularized by Polya in his classic “How to Solve It”, is not likely to be of any use with students having small domain knowledge. In fact, Polya had intended his book for upper level high school, and college math students. Giving students such instruction in problem solving is no more helpful than telling a child to “Be careful” when riding their bicycles. What does “be careful” mean?

    Young students need to be scaffolded through problems, despite the advice to the contrary by the late Grant Wiggins and others with similar philosophies. When confronted with the problem of “How many 3/8 inch intervals there are in 15/16 inches” a “simpler problem” can be modeled by the teacher. Such as “How many 2 ft lengths can an 8 ft board be sawed into?” The student sees that the problem is solved by 8/2. The teacher can then ask how many 1/2 ft lengths the 8 ft board can be sawed into, and so forth until the student can make the connection that the original problem is solved in the same way: by division.

    The notion of teaching problem solving and “understanding” is currently at play in the US with the Common Core math standards, and the thinking is that “explaining one’s answer” is both the evidence and pathway to understanding. It is neither. See

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